diff --git a/docs/pyerrors.html b/docs/pyerrors.html index 6d73dfc3..1e183e0c 100644 --- a/docs/pyerrors.html +++ b/docs/pyerrors.html @@ -3,7 +3,7 @@ - + pyerrors API documentation diff --git a/docs/pyerrors/correlators.html b/docs/pyerrors/correlators.html index 9e26c4c3..40e24e3b 100644 --- a/docs/pyerrors/correlators.html +++ b/docs/pyerrors/correlators.html @@ -3,7 +3,7 @@ - + pyerrors.correlators API documentation diff --git a/docs/pyerrors/covobs.html b/docs/pyerrors/covobs.html index 45c4edb7..dbf6673d 100644 --- a/docs/pyerrors/covobs.html +++ b/docs/pyerrors/covobs.html @@ -3,7 +3,7 @@ - + pyerrors.covobs API documentation diff --git a/docs/pyerrors/dirac.html b/docs/pyerrors/dirac.html index 5339888b..b144dcf0 100644 --- a/docs/pyerrors/dirac.html +++ b/docs/pyerrors/dirac.html @@ -3,7 +3,7 @@ - + pyerrors.dirac API documentation diff --git a/docs/pyerrors/fits.html b/docs/pyerrors/fits.html index f97553bc..98dd2d7b 100644 --- a/docs/pyerrors/fits.html +++ b/docs/pyerrors/fits.html @@ -3,7 +3,7 @@ - + pyerrors.fits API documentation @@ -125,7 +125,7 @@ 14from autograd import elementwise_grad as egrad 15from numdifftools import Jacobian as num_jacobian 16from numdifftools import Hessian as num_hessian - 17from .obs import Obs, derived_observable, covariance, cov_Obs + 17from .obs import Obs, derived_observable, covariance, cov_Obs, invert_corr_cov_cholesky 18 19 20class Fit_result(Sequence): @@ -262,683 +262,745 @@ 151 For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`. 152 In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix). 153 This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning). -154 expected_chisquare : bool -155 If True estimates the expected chisquare which is -156 corrected by effects caused by correlated input data (default False). -157 resplot : bool -158 If True, a plot which displays fit, data and residuals is generated (default False). -159 qqplot : bool -160 If True, a quantile-quantile plot of the fit result is generated (default False). -161 num_grad : bool -162 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). -163 -164 Returns -165 ------- -166 output : Fit_result -167 Parameters and information on the fitted result. -168 ''' -169 output = Fit_result() -170 -171 if (isinstance(x, dict) and isinstance(y, dict) and isinstance(func, dict)): -172 xd = {key: anp.asarray(x[key]) for key in x} -173 yd = y -174 funcd = func -175 output.fit_function = func -176 elif (isinstance(x, dict) or isinstance(y, dict) or isinstance(func, dict)): -177 raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.") -178 else: -179 x = np.asarray(x) -180 xd = {"": x} -181 yd = {"": y} -182 funcd = {"": func} -183 output.fit_function = func -184 -185 if kwargs.get('num_grad') is True: -186 jacobian = num_jacobian -187 hessian = num_hessian -188 else: -189 jacobian = auto_jacobian -190 hessian = auto_hessian -191 -192 key_ls = sorted(list(xd.keys())) -193 -194 if sorted(list(yd.keys())) != key_ls: -195 raise ValueError('x and y dictionaries do not contain the same keys.') -196 -197 if sorted(list(funcd.keys())) != key_ls: -198 raise ValueError('x and func dictionaries do not contain the same keys.') -199 -200 x_all = np.concatenate([np.array(xd[key]) for key in key_ls]) -201 y_all = np.concatenate([np.array(yd[key]) for key in key_ls]) -202 -203 y_f = [o.value for o in y_all] -204 dy_f = [o.dvalue for o in y_all] -205 -206 if len(x_all.shape) > 2: -207 raise ValueError("Unknown format for x values") -208 -209 if np.any(np.asarray(dy_f) <= 0.0): -210 raise Exception("No y errors available, run the gamma method first.") -211 -212 # number of fit parameters -213 n_parms_ls = [] -214 for key in key_ls: -215 if not callable(funcd[key]): -216 raise TypeError('func (key=' + key + ') is not a function.') -217 if np.asarray(xd[key]).shape[-1] != len(yd[key]): -218 raise ValueError('x and y input (key=' + key + ') do not have the same length') -219 for n_loc in range(100): -220 try: -221 funcd[key](np.arange(n_loc), x_all.T[0]) -222 except TypeError: -223 continue -224 except IndexError: -225 continue -226 else: -227 break -228 else: -229 raise RuntimeError("Fit function (key=" + key + ") is not valid.") -230 n_parms_ls.append(n_loc) -231 -232 n_parms = max(n_parms_ls) -233 -234 if len(key_ls) > 1: -235 for key in key_ls: -236 if np.asarray(yd[key]).shape != funcd[key](np.arange(n_parms), xd[key]).shape: -237 raise ValueError(f"Fit function {key} returns the wrong shape ({funcd[key](np.arange(n_parms), xd[key]).shape} instead of {xd[key].shape})\nIf the fit function is just a constant you could try adding x*0 to get the correct shape.") -238 -239 if not silent: -240 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) -241 -242 if priors is not None: -243 if isinstance(priors, (list, np.ndarray)): -244 if n_parms != len(priors): -245 raise ValueError("'priors' does not have the correct length.") -246 -247 loc_priors = [] -248 for i_n, i_prior in enumerate(priors): -249 loc_priors.append(_construct_prior_obs(i_prior, i_n)) +154 inv_chol_cov_matrix [array,list], optional +155 array: shape = (no of y values) X (no of y values) +156 list: for an uncombined fit: [""] +157 for a combined fit: list of keys belonging to the corr_matrix saved in the array, must be the same as the keys of the y dict in alphabetical order +158 If correlated_fit=True is set as well, can provide an inverse covariance matrix (y errors, dy_f included!) of your own choosing for a correlated fit. +159 The matrix must be a lower triangular matrix constructed from a Cholesky decomposition: The function invert_corr_cov_cholesky(corr, inverrdiag) can be +160 used to construct it from a correlation matrix (corr) and the errors dy_f of the data points (inverrdiag = np.diag(1 / np.asarray(dy_f))). For the correct +161 ordering the correlation matrix (corr) can be sorted via the function sort_corr(corr, kl, yd) where kl is the list of keys and yd the y dict. +162 expected_chisquare : bool +163 If True estimates the expected chisquare which is +164 corrected by effects caused by correlated input data (default False). +165 resplot : bool +166 If True, a plot which displays fit, data and residuals is generated (default False). +167 qqplot : bool +168 If True, a quantile-quantile plot of the fit result is generated (default False). +169 num_grad : bool +170 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). +171 +172 Returns +173 ------- +174 output : Fit_result +175 Parameters and information on the fitted result. +176 Examples +177 ------ +178 >>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set +179 >>> import numpy as np +180 >>> from scipy.stats import norm +181 >>> from scipy.linalg import cholesky +182 >>> import pyerrors as pe +183 >>> # generating the random data set +184 >>> num_samples = 400 +185 >>> N = 3 +186 >>> x = np.arange(N) +187 >>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers +188 >>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers +189 >>> r = r1 = r2 = np.zeros((N, N)) +190 >>> y = {} +191 >>> for i in range(N): +192 >>> for j in range(N): +193 >>> r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix +194 >>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors +195 >>> for i in range(N): +196 >>> for j in range(N): +197 >>> r[i, j] *= errl[i] * errl[j] # element in covariance matrix +198 >>> c = cholesky(r, lower=True) +199 >>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined +200 >>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built +201 >>> x_dict = {} +202 >>> y_dict = {} +203 >>> chol_inv_dict = {} +204 >>> data = [] +205 >>> for key in y.keys(): +206 >>> x_dict[key] = x +207 >>> for i in range(N): +208 >>> data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data +209 >>> [o.gamma_method() for o in data] +210 >>> corr = pe.covariance(data, correlation=True) +211 >>> inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data])) +212 >>> chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below +213 >>> y_dict = {'a': data[:3], 'b': data[3:]} +214 >>> # common fit parameter p[0] in combined fit +215 >>> def fit1(p, x): +216 >>> return p[0] + p[1] * x +217 >>> def fit2(p, x): +218 >>> return p[0] + p[2] * x +219 >>> fitf_dict = {'a': fit1, 'b':fit2} +220 >>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']]) +221 Fit with 3 parameters +222 Method: Levenberg-Marquardt +223 `ftol` termination condition is satisfied. +224 chisquare/d.o.f.: 0.5388013574561786 # random +225 fit parameters [1.11897846 0.96361162 0.92325319] # random +226 +227 ''' +228 output = Fit_result() +229 +230 if (isinstance(x, dict) and isinstance(y, dict) and isinstance(func, dict)): +231 xd = {key: anp.asarray(x[key]) for key in x} +232 yd = y +233 funcd = func +234 output.fit_function = func +235 elif (isinstance(x, dict) or isinstance(y, dict) or isinstance(func, dict)): +236 raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.") +237 else: +238 x = np.asarray(x) +239 xd = {"": x} +240 yd = {"": y} +241 funcd = {"": func} +242 output.fit_function = func +243 +244 if kwargs.get('num_grad') is True: +245 jacobian = num_jacobian +246 hessian = num_hessian +247 else: +248 jacobian = auto_jacobian +249 hessian = auto_hessian 250 -251 prior_mask = np.arange(len(priors)) -252 output.priors = loc_priors -253 -254 elif isinstance(priors, dict): -255 loc_priors = [] -256 prior_mask = [] -257 output.priors = {} -258 for pos, prior in priors.items(): -259 if isinstance(pos, int): -260 prior_mask.append(pos) -261 else: -262 raise TypeError("Prior position needs to be an integer.") -263 loc_priors.append(_construct_prior_obs(prior, pos)) +251 key_ls = sorted(list(xd.keys())) +252 +253 if sorted(list(yd.keys())) != key_ls: +254 raise ValueError('x and y dictionaries do not contain the same keys.') +255 +256 if sorted(list(funcd.keys())) != key_ls: +257 raise ValueError('x and func dictionaries do not contain the same keys.') +258 +259 x_all = np.concatenate([np.array(xd[key]) for key in key_ls]) +260 y_all = np.concatenate([np.array(yd[key]) for key in key_ls]) +261 +262 y_f = [o.value for o in y_all] +263 dy_f = [o.dvalue for o in y_all] 264 -265 output.priors[pos] = loc_priors[-1] -266 if max(prior_mask) >= n_parms: -267 raise ValueError("Prior position out of range.") -268 else: -269 raise TypeError("Unkown type for `priors`.") +265 if len(x_all.shape) > 2: +266 raise ValueError("Unknown format for x values") +267 +268 if np.any(np.asarray(dy_f) <= 0.0): +269 raise Exception("No y errors available, run the gamma method first.") 270 -271 p_f = [o.value for o in loc_priors] -272 dp_f = [o.dvalue for o in loc_priors] -273 if np.any(np.asarray(dp_f) <= 0.0): -274 raise Exception("No prior errors available, run the gamma method first.") -275 else: -276 p_f = dp_f = np.array([]) -277 prior_mask = [] -278 loc_priors = [] -279 -280 if 'initial_guess' in kwargs: -281 x0 = kwargs.get('initial_guess') -282 if len(x0) != n_parms: -283 raise ValueError('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) -284 else: -285 x0 = [0.1] * n_parms -286 -287 if priors is None: -288 def general_chisqfunc_uncorr(p, ivars, pr): -289 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -290 return (ivars - model) / dy_f -291 else: -292 def general_chisqfunc_uncorr(p, ivars, pr): -293 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -294 return anp.concatenate(((ivars - model) / dy_f, (p[prior_mask] - pr) / dp_f)) -295 -296 def chisqfunc_uncorr(p): -297 return anp.sum(general_chisqfunc_uncorr(p, y_f, p_f) ** 2) -298 -299 if kwargs.get('correlated_fit') is True: -300 corr = covariance(y_all, correlation=True, **kwargs) -301 covdiag = np.diag(1 / np.asarray(dy_f)) -302 condn = np.linalg.cond(corr) -303 if condn > 0.1 / np.finfo(float).eps: -304 raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})") -305 if condn > 1e13: -306 warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning) -307 chol = np.linalg.cholesky(corr) -308 chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True) +271 # number of fit parameters +272 n_parms_ls = [] +273 for key in key_ls: +274 if not callable(funcd[key]): +275 raise TypeError('func (key=' + key + ') is not a function.') +276 if np.asarray(xd[key]).shape[-1] != len(yd[key]): +277 raise ValueError('x and y input (key=' + key + ') do not have the same length') +278 for n_loc in range(100): +279 try: +280 funcd[key](np.arange(n_loc), x_all.T[0]) +281 except TypeError: +282 continue +283 except IndexError: +284 continue +285 else: +286 break +287 else: +288 raise RuntimeError("Fit function (key=" + key + ") is not valid.") +289 n_parms_ls.append(n_loc) +290 +291 n_parms = max(n_parms_ls) +292 +293 if len(key_ls) > 1: +294 for key in key_ls: +295 if np.asarray(yd[key]).shape != funcd[key](np.arange(n_parms), xd[key]).shape: +296 raise ValueError(f"Fit function {key} returns the wrong shape ({funcd[key](np.arange(n_parms), xd[key]).shape} instead of {xd[key].shape})\nIf the fit function is just a constant you could try adding x*0 to get the correct shape.") +297 +298 if not silent: +299 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) +300 +301 if priors is not None: +302 if isinstance(priors, (list, np.ndarray)): +303 if n_parms != len(priors): +304 raise ValueError("'priors' does not have the correct length.") +305 +306 loc_priors = [] +307 for i_n, i_prior in enumerate(priors): +308 loc_priors.append(_construct_prior_obs(i_prior, i_n)) 309 -310 def general_chisqfunc(p, ivars, pr): -311 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -312 return anp.concatenate((anp.dot(chol_inv, (ivars - model)), (p[prior_mask] - pr) / dp_f)) -313 -314 def chisqfunc(p): -315 return anp.sum(general_chisqfunc(p, y_f, p_f) ** 2) -316 else: -317 general_chisqfunc = general_chisqfunc_uncorr -318 chisqfunc = chisqfunc_uncorr -319 -320 output.method = kwargs.get('method', 'Levenberg-Marquardt') -321 if not silent: -322 print('Method:', output.method) +310 prior_mask = np.arange(len(priors)) +311 output.priors = loc_priors +312 +313 elif isinstance(priors, dict): +314 loc_priors = [] +315 prior_mask = [] +316 output.priors = {} +317 for pos, prior in priors.items(): +318 if isinstance(pos, int): +319 prior_mask.append(pos) +320 else: +321 raise TypeError("Prior position needs to be an integer.") +322 loc_priors.append(_construct_prior_obs(prior, pos)) 323 -324 if output.method != 'Levenberg-Marquardt': -325 if output.method == 'migrad': -326 tolerance = 1e-4 # default value of 1e-1 set by iminuit can be problematic -327 if 'tol' in kwargs: -328 tolerance = kwargs.get('tol') -329 fit_result = iminuit.minimize(chisqfunc_uncorr, x0, tol=tolerance) # Stopping criterion 0.002 * tol * errordef -330 if kwargs.get('correlated_fit') is True: -331 fit_result = iminuit.minimize(chisqfunc, fit_result.x, tol=tolerance) -332 output.iterations = fit_result.nfev -333 else: -334 tolerance = 1e-12 -335 if 'tol' in kwargs: -336 tolerance = kwargs.get('tol') -337 fit_result = scipy.optimize.minimize(chisqfunc_uncorr, x0, method=kwargs.get('method'), tol=tolerance) -338 if kwargs.get('correlated_fit') is True: -339 fit_result = scipy.optimize.minimize(chisqfunc, fit_result.x, method=kwargs.get('method'), tol=tolerance) -340 output.iterations = fit_result.nit -341 -342 chisquare = fit_result.fun -343 -344 else: -345 if 'tol' in kwargs: -346 print('tol cannot be set for Levenberg-Marquardt') -347 -348 def chisqfunc_residuals_uncorr(p): -349 return general_chisqfunc_uncorr(p, y_f, p_f) -350 -351 fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) -352 if kwargs.get('correlated_fit') is True: -353 -354 def chisqfunc_residuals(p): -355 return general_chisqfunc(p, y_f, p_f) -356 -357 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) -358 -359 chisquare = np.sum(fit_result.fun ** 2) -360 assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14) -361 -362 output.iterations = fit_result.nfev -363 -364 if not fit_result.success: -365 raise Exception('The minimization procedure did not converge.') -366 -367 output.chisquare = chisquare -368 output.dof = y_all.shape[-1] - n_parms + len(loc_priors) -369 output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof) -370 if output.dof > 0: -371 output.chisquare_by_dof = output.chisquare / output.dof -372 else: -373 output.chisquare_by_dof = float('nan') -374 -375 output.message = fit_result.message -376 if not silent: -377 print(fit_result.message) -378 print('chisquare/d.o.f.:', output.chisquare_by_dof) -379 print('fit parameters', fit_result.x) -380 -381 def prepare_hat_matrix(): -382 hat_vector = [] -383 for key in key_ls: -384 if (len(xd[key]) != 0): -385 hat_vector.append(jacobian(funcd[key])(fit_result.x, xd[key])) -386 hat_vector = [item for sublist in hat_vector for item in sublist] -387 return hat_vector -388 -389 if kwargs.get('expected_chisquare') is True: -390 if kwargs.get('correlated_fit') is not True: -391 W = np.diag(1 / np.asarray(dy_f)) -392 cov = covariance(y_all) -393 hat_vector = prepare_hat_matrix() -394 A = W @ hat_vector -395 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T -396 expected_chisquare = np.trace((np.identity(y_all.shape[-1]) - P_phi) @ W @ cov @ W) -397 output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare -398 if not silent: -399 print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare) -400 -401 fitp = fit_result.x -402 -403 try: -404 hess = hessian(chisqfunc)(fitp) -405 except TypeError: -406 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None -407 -408 len_y = len(y_f) -409 -410 def chisqfunc_compact(d): -411 return anp.sum(general_chisqfunc(d[:n_parms], d[n_parms: n_parms + len_y], d[n_parms + len_y:]) ** 2) -412 -413 jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f, p_f))) -414 -415 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv -416 try: -417 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:]) -418 except np.linalg.LinAlgError: -419 raise Exception("Cannot invert hessian matrix.") +324 output.priors[pos] = loc_priors[-1] +325 if max(prior_mask) >= n_parms: +326 raise ValueError("Prior position out of range.") +327 else: +328 raise TypeError("Unkown type for `priors`.") +329 +330 p_f = [o.value for o in loc_priors] +331 dp_f = [o.dvalue for o in loc_priors] +332 if np.any(np.asarray(dp_f) <= 0.0): +333 raise Exception("No prior errors available, run the gamma method first.") +334 else: +335 p_f = dp_f = np.array([]) +336 prior_mask = [] +337 loc_priors = [] +338 +339 if 'initial_guess' in kwargs: +340 x0 = kwargs.get('initial_guess') +341 if len(x0) != n_parms: +342 raise ValueError('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) +343 else: +344 x0 = [0.1] * n_parms +345 +346 if priors is None: +347 def general_chisqfunc_uncorr(p, ivars, pr): +348 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) +349 return (ivars - model) / dy_f +350 else: +351 def general_chisqfunc_uncorr(p, ivars, pr): +352 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) +353 return anp.concatenate(((ivars - model) / dy_f, (p[prior_mask] - pr) / dp_f)) +354 +355 def chisqfunc_uncorr(p): +356 return anp.sum(general_chisqfunc_uncorr(p, y_f, p_f) ** 2) +357 +358 if kwargs.get('correlated_fit') is True: +359 if 'inv_chol_cov_matrix' in kwargs: +360 chol_inv = kwargs.get('inv_chol_cov_matrix') +361 if (chol_inv[0].shape[0] != len(dy_f)): +362 raise TypeError('The number of columns of the inverse covariance matrix handed over needs to be equal to the number of y errors.') +363 if (chol_inv[0].shape[0] != chol_inv[0].shape[1]): +364 raise TypeError('The inverse covariance matrix handed over needs to have the same number of rows as columns.') +365 if (chol_inv[1] != key_ls): +366 raise ValueError('The keys of inverse covariance matrix are not the same or do not appear in the same order as the x and y values.') +367 chol_inv = chol_inv[0] +368 else: +369 corr = covariance(y_all, correlation=True, **kwargs) +370 inverrdiag = np.diag(1 / np.asarray(dy_f)) +371 chol_inv = invert_corr_cov_cholesky(corr, inverrdiag) +372 +373 def general_chisqfunc(p, ivars, pr): +374 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) +375 return anp.concatenate((anp.dot(chol_inv, (ivars - model)), (p[prior_mask] - pr) / dp_f)) +376 +377 def chisqfunc(p): +378 return anp.sum(general_chisqfunc(p, y_f, p_f) ** 2) +379 else: +380 general_chisqfunc = general_chisqfunc_uncorr +381 chisqfunc = chisqfunc_uncorr +382 +383 output.method = kwargs.get('method', 'Levenberg-Marquardt') +384 if not silent: +385 print('Method:', output.method) +386 +387 if output.method != 'Levenberg-Marquardt': +388 if output.method == 'migrad': +389 tolerance = 1e-4 # default value of 1e-1 set by iminuit can be problematic +390 if 'tol' in kwargs: +391 tolerance = kwargs.get('tol') +392 fit_result = iminuit.minimize(chisqfunc_uncorr, x0, tol=tolerance) # Stopping criterion 0.002 * tol * errordef +393 if kwargs.get('correlated_fit') is True: +394 fit_result = iminuit.minimize(chisqfunc, fit_result.x, tol=tolerance) +395 output.iterations = fit_result.nfev +396 else: +397 tolerance = 1e-12 +398 if 'tol' in kwargs: +399 tolerance = kwargs.get('tol') +400 fit_result = scipy.optimize.minimize(chisqfunc_uncorr, x0, method=kwargs.get('method'), tol=tolerance) +401 if kwargs.get('correlated_fit') is True: +402 fit_result = scipy.optimize.minimize(chisqfunc, fit_result.x, method=kwargs.get('method'), tol=tolerance) +403 output.iterations = fit_result.nit +404 +405 chisquare = fit_result.fun +406 +407 else: +408 if 'tol' in kwargs: +409 print('tol cannot be set for Levenberg-Marquardt') +410 +411 def chisqfunc_residuals_uncorr(p): +412 return general_chisqfunc_uncorr(p, y_f, p_f) +413 +414 fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) +415 if kwargs.get('correlated_fit') is True: +416 def chisqfunc_residuals(p): +417 return general_chisqfunc(p, y_f, p_f) +418 +419 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) 420 -421 result = [] -422 for i in range(n_parms): -423 result.append(derived_observable(lambda x_all, **kwargs: (x_all[0] + np.finfo(np.float64).eps) / (y_all[0].value + np.finfo(np.float64).eps) * fitp[i], list(y_all) + loc_priors, man_grad=list(deriv_y[i]))) -424 -425 output.fit_parameters = result -426 -427 # Hotelling t-squared p-value for correlated fits. -428 if kwargs.get('correlated_fit') is True: -429 n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all)) -430 output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare, -431 output.dof, n_cov - output.dof) -432 -433 if kwargs.get('resplot') is True: -434 for key in key_ls: -435 residual_plot(xd[key], yd[key], funcd[key], result, title=key) +421 chisquare = np.sum(fit_result.fun ** 2) +422 assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14) +423 +424 output.iterations = fit_result.nfev +425 +426 if not fit_result.success: +427 raise Exception('The minimization procedure did not converge.') +428 +429 output.chisquare = chisquare +430 output.dof = y_all.shape[-1] - n_parms + len(loc_priors) +431 output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof) +432 if output.dof > 0: +433 output.chisquare_by_dof = output.chisquare / output.dof +434 else: +435 output.chisquare_by_dof = float('nan') 436 -437 if kwargs.get('qqplot') is True: -438 for key in key_ls: -439 qqplot(xd[key], yd[key], funcd[key], result, title=key) -440 -441 return output +437 output.message = fit_result.message +438 if not silent: +439 print(fit_result.message) +440 print('chisquare/d.o.f.:', output.chisquare_by_dof) +441 print('fit parameters', fit_result.x) 442 -443 -444def total_least_squares(x, y, func, silent=False, **kwargs): -445 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. -446 -447 Parameters -448 ---------- -449 x : list -450 list of Obs, or a tuple of lists of Obs -451 y : list -452 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. -453 func : object -454 func has to be of the form -455 -456 ```python -457 import autograd.numpy as anp -458 -459 def func(a, x): -460 return a[0] + a[1] * x + a[2] * anp.sinh(x) -461 ``` +443 def prepare_hat_matrix(): +444 hat_vector = [] +445 for key in key_ls: +446 if (len(xd[key]) != 0): +447 hat_vector.append(jacobian(funcd[key])(fit_result.x, xd[key])) +448 hat_vector = [item for sublist in hat_vector for item in sublist] +449 return hat_vector +450 +451 if kwargs.get('expected_chisquare') is True: +452 if kwargs.get('correlated_fit') is not True: +453 W = np.diag(1 / np.asarray(dy_f)) +454 cov = covariance(y_all) +455 hat_vector = prepare_hat_matrix() +456 A = W @ hat_vector +457 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T +458 expected_chisquare = np.trace((np.identity(y_all.shape[-1]) - P_phi) @ W @ cov @ W) +459 output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare +460 if not silent: +461 print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare) 462 -463 For multiple x values func can be of the form +463 fitp = fit_result.x 464 -465 ```python -466 def func(a, x): -467 (x1, x2) = x -468 return a[0] * x1 ** 2 + a[1] * x2 -469 ``` -470 -471 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation -472 will not work. -473 silent : bool, optional -474 If true all output to the console is omitted (default False). -475 initial_guess : list -476 can provide an initial guess for the input parameters. Relevant for non-linear -477 fits with many parameters. -478 expected_chisquare : bool -479 If true prints the expected chisquare which is -480 corrected by effects caused by correlated input data. -481 This can take a while as the full correlation matrix -482 has to be calculated (default False). -483 num_grad : bool -484 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). -485 -486 Notes -487 ----- -488 Based on the orthogonal distance regression module of scipy. -489 -490 Returns -491 ------- -492 output : Fit_result -493 Parameters and information on the fitted result. -494 ''' -495 -496 output = Fit_result() -497 -498 output.fit_function = func -499 -500 x = np.array(x) -501 -502 x_shape = x.shape -503 -504 if kwargs.get('num_grad') is True: -505 jacobian = num_jacobian -506 hessian = num_hessian -507 else: -508 jacobian = auto_jacobian -509 hessian = auto_hessian -510 -511 if not callable(func): -512 raise TypeError('func has to be a function.') -513 -514 for i in range(42): -515 try: -516 func(np.arange(i), x.T[0]) -517 except TypeError: -518 continue -519 except IndexError: -520 continue -521 else: -522 break -523 else: -524 raise RuntimeError("Fit function is not valid.") -525 -526 n_parms = i -527 if not silent: -528 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) -529 -530 x_f = np.vectorize(lambda o: o.value)(x) -531 dx_f = np.vectorize(lambda o: o.dvalue)(x) -532 y_f = np.array([o.value for o in y]) -533 dy_f = np.array([o.dvalue for o in y]) -534 -535 if np.any(np.asarray(dx_f) <= 0.0): -536 raise Exception('No x errors available, run the gamma method first.') -537 -538 if np.any(np.asarray(dy_f) <= 0.0): -539 raise Exception('No y errors available, run the gamma method first.') -540 -541 if 'initial_guess' in kwargs: -542 x0 = kwargs.get('initial_guess') -543 if len(x0) != n_parms: -544 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) -545 else: -546 x0 = [1] * n_parms +465 try: +466 hess = hessian(chisqfunc)(fitp) +467 except TypeError: +468 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None +469 +470 len_y = len(y_f) +471 +472 def chisqfunc_compact(d): +473 return anp.sum(general_chisqfunc(d[:n_parms], d[n_parms: n_parms + len_y], d[n_parms + len_y:]) ** 2) +474 +475 jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f, p_f))) +476 +477 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv +478 try: +479 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:]) +480 except np.linalg.LinAlgError: +481 raise Exception("Cannot invert hessian matrix.") +482 +483 result = [] +484 for i in range(n_parms): +485 result.append(derived_observable(lambda x_all, **kwargs: (x_all[0] + np.finfo(np.float64).eps) / (y_all[0].value + np.finfo(np.float64).eps) * fitp[i], list(y_all) + loc_priors, man_grad=list(deriv_y[i]))) +486 +487 output.fit_parameters = result +488 +489 # Hotelling t-squared p-value for correlated fits. +490 if kwargs.get('correlated_fit') is True: +491 n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all)) +492 output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare, +493 output.dof, n_cov - output.dof) +494 +495 if kwargs.get('resplot') is True: +496 for key in key_ls: +497 residual_plot(xd[key], yd[key], funcd[key], result, title=key) +498 +499 if kwargs.get('qqplot') is True: +500 for key in key_ls: +501 qqplot(xd[key], yd[key], funcd[key], result, title=key) +502 +503 return output +504 +505 +506def total_least_squares(x, y, func, silent=False, **kwargs): +507 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. +508 +509 Parameters +510 ---------- +511 x : list +512 list of Obs, or a tuple of lists of Obs +513 y : list +514 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. +515 func : object +516 func has to be of the form +517 +518 ```python +519 import autograd.numpy as anp +520 +521 def func(a, x): +522 return a[0] + a[1] * x + a[2] * anp.sinh(x) +523 ``` +524 +525 For multiple x values func can be of the form +526 +527 ```python +528 def func(a, x): +529 (x1, x2) = x +530 return a[0] * x1 ** 2 + a[1] * x2 +531 ``` +532 +533 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation +534 will not work. +535 silent : bool, optional +536 If true all output to the console is omitted (default False). +537 initial_guess : list +538 can provide an initial guess for the input parameters. Relevant for non-linear +539 fits with many parameters. +540 expected_chisquare : bool +541 If true prints the expected chisquare which is +542 corrected by effects caused by correlated input data. +543 This can take a while as the full correlation matrix +544 has to be calculated (default False). +545 num_grad : bool +546 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). 547 -548 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) -549 model = Model(func) -550 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) -551 odr.set_job(fit_type=0, deriv=1) -552 out = odr.run() -553 -554 output.residual_variance = out.res_var -555 -556 output.method = 'ODR' +548 Notes +549 ----- +550 Based on the orthogonal distance regression module of scipy. +551 +552 Returns +553 ------- +554 output : Fit_result +555 Parameters and information on the fitted result. +556 ''' 557 -558 output.message = out.stopreason +558 output = Fit_result() 559 -560 output.xplus = out.xplus +560 output.fit_function = func 561 -562 if not silent: -563 print('Method: ODR') -564 print(*out.stopreason) -565 print('Residual variance:', output.residual_variance) -566 -567 if out.info > 3: -568 raise Exception('The minimization procedure did not converge.') -569 -570 m = x_f.size -571 -572 def odr_chisquare(p): -573 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) -574 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) -575 return chisq -576 -577 if kwargs.get('expected_chisquare') is True: -578 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) -579 -580 if kwargs.get('covariance') is not None: -581 cov = kwargs.get('covariance') -582 else: -583 cov = covariance(np.concatenate((y, x.ravel()))) -584 -585 number_of_x_parameters = int(m / x_f.shape[-1]) -586 -587 old_jac = jacobian(func)(out.beta, out.xplus) -588 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) -589 fused_row2 = np.concatenate((jacobian(lambda x, y: func(y, x))(out.xplus, out.beta).reshape(x_f.shape[-1], x_f.shape[-1] * number_of_x_parameters), np.identity(number_of_x_parameters * old_jac.shape[0]))) -590 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) +562 x = np.array(x) +563 +564 x_shape = x.shape +565 +566 if kwargs.get('num_grad') is True: +567 jacobian = num_jacobian +568 hessian = num_hessian +569 else: +570 jacobian = auto_jacobian +571 hessian = auto_hessian +572 +573 if not callable(func): +574 raise TypeError('func has to be a function.') +575 +576 for i in range(42): +577 try: +578 func(np.arange(i), x.T[0]) +579 except TypeError: +580 continue +581 except IndexError: +582 continue +583 else: +584 break +585 else: +586 raise RuntimeError("Fit function is not valid.") +587 +588 n_parms = i +589 if not silent: +590 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) 591 -592 A = W @ new_jac -593 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T -594 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) -595 if expected_chisquare <= 0.0: -596 warnings.warn("Negative expected_chisquare.", RuntimeWarning) -597 expected_chisquare = np.abs(expected_chisquare) -598 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare -599 if not silent: -600 print('chisquare/expected_chisquare:', -601 output.chisquare_by_expected_chisquare) +592 x_f = np.vectorize(lambda o: o.value)(x) +593 dx_f = np.vectorize(lambda o: o.dvalue)(x) +594 y_f = np.array([o.value for o in y]) +595 dy_f = np.array([o.dvalue for o in y]) +596 +597 if np.any(np.asarray(dx_f) <= 0.0): +598 raise Exception('No x errors available, run the gamma method first.') +599 +600 if np.any(np.asarray(dy_f) <= 0.0): +601 raise Exception('No y errors available, run the gamma method first.') 602 -603 fitp = out.beta -604 try: -605 hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel()))) -606 except TypeError: -607 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None -608 -609 def odr_chisquare_compact_x(d): -610 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) -611 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((d[n_parms + m:].reshape(x_shape) - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) -612 return chisq -613 -614 jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) +603 if 'initial_guess' in kwargs: +604 x0 = kwargs.get('initial_guess') +605 if len(x0) != n_parms: +606 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) +607 else: +608 x0 = [1] * n_parms +609 +610 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) +611 model = Model(func) +612 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) +613 odr.set_job(fit_type=0, deriv=1) +614 out = odr.run() 615 -616 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv -617 try: -618 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) -619 except np.linalg.LinAlgError: -620 raise Exception("Cannot invert hessian matrix.") +616 output.residual_variance = out.res_var +617 +618 output.method = 'ODR' +619 +620 output.message = out.stopreason 621 -622 def odr_chisquare_compact_y(d): -623 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) -624 chisq = anp.sum(((d[n_parms + m:] - model) / dy_f) ** 2) + anp.sum(((x_f - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) -625 return chisq -626 -627 jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f))) +622 output.xplus = out.xplus +623 +624 if not silent: +625 print('Method: ODR') +626 print(*out.stopreason) +627 print('Residual variance:', output.residual_variance) 628 -629 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv -630 try: -631 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) -632 except np.linalg.LinAlgError: -633 raise Exception("Cannot invert hessian matrix.") -634 -635 result = [] -636 for i in range(n_parms): -637 result.append(derived_observable(lambda my_var, **kwargs: (my_var[0] + np.finfo(np.float64).eps) / (x.ravel()[0].value + np.finfo(np.float64).eps) * out.beta[i], list(x.ravel()) + list(y), man_grad=list(deriv_x[i]) + list(deriv_y[i]))) +629 if out.info > 3: +630 raise Exception('The minimization procedure did not converge.') +631 +632 m = x_f.size +633 +634 def odr_chisquare(p): +635 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) +636 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) +637 return chisq 638 -639 output.fit_parameters = result -640 -641 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) -642 output.dof = x.shape[-1] - n_parms -643 output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof) -644 -645 return output +639 if kwargs.get('expected_chisquare') is True: +640 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) +641 +642 if kwargs.get('covariance') is not None: +643 cov = kwargs.get('covariance') +644 else: +645 cov = covariance(np.concatenate((y, x.ravel()))) 646 -647 -648def fit_lin(x, y, **kwargs): -649 """Performs a linear fit to y = n + m * x and returns two Obs n, m. -650 -651 Parameters -652 ---------- -653 x : list -654 Can either be a list of floats in which case no xerror is assumed, or -655 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. -656 y : list -657 List of Obs, the dvalues of the Obs are used as yerror for the fit. -658 -659 Returns -660 ------- -661 fit_parameters : list[Obs] -662 LIist of fitted observables. -663 """ +647 number_of_x_parameters = int(m / x_f.shape[-1]) +648 +649 old_jac = jacobian(func)(out.beta, out.xplus) +650 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) +651 fused_row2 = np.concatenate((jacobian(lambda x, y: func(y, x))(out.xplus, out.beta).reshape(x_f.shape[-1], x_f.shape[-1] * number_of_x_parameters), np.identity(number_of_x_parameters * old_jac.shape[0]))) +652 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) +653 +654 A = W @ new_jac +655 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T +656 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) +657 if expected_chisquare <= 0.0: +658 warnings.warn("Negative expected_chisquare.", RuntimeWarning) +659 expected_chisquare = np.abs(expected_chisquare) +660 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare +661 if not silent: +662 print('chisquare/expected_chisquare:', +663 output.chisquare_by_expected_chisquare) 664 -665 def f(a, x): -666 y = a[0] + a[1] * x -667 return y -668 -669 if all(isinstance(n, Obs) for n in x): -670 out = total_least_squares(x, y, f, **kwargs) -671 return out.fit_parameters -672 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): -673 out = least_squares(x, y, f, **kwargs) -674 return out.fit_parameters -675 else: -676 raise TypeError('Unsupported types for x') +665 fitp = out.beta +666 try: +667 hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel()))) +668 except TypeError: +669 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None +670 +671 def odr_chisquare_compact_x(d): +672 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) +673 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((d[n_parms + m:].reshape(x_shape) - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) +674 return chisq +675 +676 jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) 677 -678 -679def qqplot(x, o_y, func, p, title=""): -680 """Generates a quantile-quantile plot of the fit result which can be used to -681 check if the residuals of the fit are gaussian distributed. -682 -683 Returns -684 ------- -685 None -686 """ -687 -688 residuals = [] -689 for i_x, i_y in zip(x, o_y): -690 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) -691 residuals = sorted(residuals) -692 my_y = [o.value for o in residuals] -693 probplot = scipy.stats.probplot(my_y) -694 my_x = probplot[0][0] -695 plt.figure(figsize=(8, 8 / 1.618)) -696 plt.errorbar(my_x, my_y, fmt='o') -697 fit_start = my_x[0] -698 fit_stop = my_x[-1] -699 samples = np.arange(fit_start, fit_stop, 0.01) -700 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') -701 plt.plot(samples, probplot[1][0] * samples + probplot[1][1], zorder=10, label='Least squares fit, r=' + str(np.around(probplot[1][2], 3)), marker='', ls='-') +678 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv +679 try: +680 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) +681 except np.linalg.LinAlgError: +682 raise Exception("Cannot invert hessian matrix.") +683 +684 def odr_chisquare_compact_y(d): +685 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) +686 chisq = anp.sum(((d[n_parms + m:] - model) / dy_f) ** 2) + anp.sum(((x_f - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) +687 return chisq +688 +689 jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f))) +690 +691 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv +692 try: +693 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) +694 except np.linalg.LinAlgError: +695 raise Exception("Cannot invert hessian matrix.") +696 +697 result = [] +698 for i in range(n_parms): +699 result.append(derived_observable(lambda my_var, **kwargs: (my_var[0] + np.finfo(np.float64).eps) / (x.ravel()[0].value + np.finfo(np.float64).eps) * out.beta[i], list(x.ravel()) + list(y), man_grad=list(deriv_x[i]) + list(deriv_y[i]))) +700 +701 output.fit_parameters = result 702 -703 plt.xlabel('Theoretical quantiles') -704 plt.ylabel('Ordered Values') -705 plt.legend(title=title) -706 plt.draw() -707 +703 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) +704 output.dof = x.shape[-1] - n_parms +705 output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof) +706 +707 return output 708 -709def residual_plot(x, y, func, fit_res, title=""): -710 """Generates a plot which compares the fit to the data and displays the corresponding residuals -711 -712 For uncorrelated data the residuals are expected to be distributed ~N(0,1). -713 -714 Returns -715 ------- -716 None -717 """ -718 sorted_x = sorted(x) -719 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) -720 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) -721 x_samples = np.arange(xstart, xstop + 0.01, 0.01) -722 -723 plt.figure(figsize=(8, 8 / 1.618)) -724 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) -725 ax0 = plt.subplot(gs[0]) -726 ax0.errorbar(x, [o.value for o in y], yerr=[o.dvalue for o in y], ls='none', fmt='o', capsize=3, markersize=5, label='Data') -727 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) -728 ax0.set_xticklabels([]) -729 ax0.set_xlim([xstart, xstop]) -730 ax0.set_xticklabels([]) -731 ax0.legend(title=title) -732 -733 residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], np.asarray(x))) / np.asarray([o.dvalue for o in y]) -734 ax1 = plt.subplot(gs[1]) -735 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) -736 ax1.tick_params(direction='out') -737 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) -738 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") -739 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') -740 ax1.set_xlim([xstart, xstop]) -741 ax1.set_ylabel('Residuals') -742 plt.subplots_adjust(wspace=None, hspace=None) -743 plt.draw() +709 +710def fit_lin(x, y, **kwargs): +711 """Performs a linear fit to y = n + m * x and returns two Obs n, m. +712 +713 Parameters +714 ---------- +715 x : list +716 Can either be a list of floats in which case no xerror is assumed, or +717 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. +718 y : list +719 List of Obs, the dvalues of the Obs are used as yerror for the fit. +720 +721 Returns +722 ------- +723 fit_parameters : list[Obs] +724 LIist of fitted observables. +725 """ +726 +727 def f(a, x): +728 y = a[0] + a[1] * x +729 return y +730 +731 if all(isinstance(n, Obs) for n in x): +732 out = total_least_squares(x, y, f, **kwargs) +733 return out.fit_parameters +734 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): +735 out = least_squares(x, y, f, **kwargs) +736 return out.fit_parameters +737 else: +738 raise TypeError('Unsupported types for x') +739 +740 +741def qqplot(x, o_y, func, p, title=""): +742 """Generates a quantile-quantile plot of the fit result which can be used to +743 check if the residuals of the fit are gaussian distributed. 744 -745 -746def error_band(x, func, beta): -747 """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta. -748 -749 Returns -750 ------- -751 err : np.array(Obs) -752 Error band for an array of sample values x -753 """ -754 cov = covariance(beta) -755 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): -756 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) -757 -758 deriv = [] -759 for i, item in enumerate(x): -760 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) -761 -762 err = [] -763 for i, item in enumerate(x): -764 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) -765 err = np.array(err) -766 -767 return err -768 +745 Returns +746 ------- +747 None +748 """ +749 +750 residuals = [] +751 for i_x, i_y in zip(x, o_y): +752 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) +753 residuals = sorted(residuals) +754 my_y = [o.value for o in residuals] +755 probplot = scipy.stats.probplot(my_y) +756 my_x = probplot[0][0] +757 plt.figure(figsize=(8, 8 / 1.618)) +758 plt.errorbar(my_x, my_y, fmt='o') +759 fit_start = my_x[0] +760 fit_stop = my_x[-1] +761 samples = np.arange(fit_start, fit_stop, 0.01) +762 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') +763 plt.plot(samples, probplot[1][0] * samples + probplot[1][1], zorder=10, label='Least squares fit, r=' + str(np.around(probplot[1][2], 3)), marker='', ls='-') +764 +765 plt.xlabel('Theoretical quantiles') +766 plt.ylabel('Ordered Values') +767 plt.legend(title=title) +768 plt.draw() 769 -770def ks_test(objects=None): -771 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. -772 -773 Parameters -774 ---------- -775 objects : list -776 List of fit results to include in the analysis (optional). -777 -778 Returns -779 ------- -780 None -781 """ -782 -783 if objects is None: -784 obs_list = [] -785 for obj in gc.get_objects(): -786 if isinstance(obj, Fit_result): -787 obs_list.append(obj) -788 else: -789 obs_list = objects -790 -791 p_values = [o.p_value for o in obs_list] -792 -793 bins = len(p_values) -794 x = np.arange(0, 1.001, 0.001) -795 plt.plot(x, x, 'k', zorder=1) -796 plt.xlim(0, 1) -797 plt.ylim(0, 1) -798 plt.xlabel('p-value') -799 plt.ylabel('Cumulative probability') -800 plt.title(str(bins) + ' p-values') -801 -802 n = np.arange(1, bins + 1) / np.float64(bins) -803 Xs = np.sort(p_values) -804 plt.step(Xs, n) -805 diffs = n - Xs -806 loc_max_diff = np.argmax(np.abs(diffs)) -807 loc = Xs[loc_max_diff] -808 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) -809 plt.draw() +770 +771def residual_plot(x, y, func, fit_res, title=""): +772 """Generates a plot which compares the fit to the data and displays the corresponding residuals +773 +774 For uncorrelated data the residuals are expected to be distributed ~N(0,1). +775 +776 Returns +777 ------- +778 None +779 """ +780 sorted_x = sorted(x) +781 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) +782 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) +783 x_samples = np.arange(xstart, xstop + 0.01, 0.01) +784 +785 plt.figure(figsize=(8, 8 / 1.618)) +786 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) +787 ax0 = plt.subplot(gs[0]) +788 ax0.errorbar(x, [o.value for o in y], yerr=[o.dvalue for o in y], ls='none', fmt='o', capsize=3, markersize=5, label='Data') +789 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) +790 ax0.set_xticklabels([]) +791 ax0.set_xlim([xstart, xstop]) +792 ax0.set_xticklabels([]) +793 ax0.legend(title=title) +794 +795 residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], np.asarray(x))) / np.asarray([o.dvalue for o in y]) +796 ax1 = plt.subplot(gs[1]) +797 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) +798 ax1.tick_params(direction='out') +799 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) +800 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") +801 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') +802 ax1.set_xlim([xstart, xstop]) +803 ax1.set_ylabel('Residuals') +804 plt.subplots_adjust(wspace=None, hspace=None) +805 plt.draw() +806 +807 +808def error_band(x, func, beta): +809 """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta. 810 -811 print(scipy.stats.kstest(p_values, 'uniform')) -812 -813 -814def _extract_val_and_dval(string): -815 split_string = string.split('(') -816 if '.' in split_string[0] and '.' not in split_string[1][:-1]: -817 factor = 10 ** -len(split_string[0].partition('.')[2]) -818 else: -819 factor = 1 -820 return float(split_string[0]), float(split_string[1][:-1]) * factor -821 -822 -823def _construct_prior_obs(i_prior, i_n): -824 if isinstance(i_prior, Obs): -825 return i_prior -826 elif isinstance(i_prior, str): -827 loc_val, loc_dval = _extract_val_and_dval(i_prior) -828 return cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}") -829 else: -830 raise TypeError("Prior entries need to be 'Obs' or 'str'.") +811 Returns +812 ------- +813 err : np.array(Obs) +814 Error band for an array of sample values x +815 """ +816 cov = covariance(beta) +817 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): +818 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) +819 +820 deriv = [] +821 for i, item in enumerate(x): +822 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) +823 +824 err = [] +825 for i, item in enumerate(x): +826 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) +827 err = np.array(err) +828 +829 return err +830 +831 +832def ks_test(objects=None): +833 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. +834 +835 Parameters +836 ---------- +837 objects : list +838 List of fit results to include in the analysis (optional). +839 +840 Returns +841 ------- +842 None +843 """ +844 +845 if objects is None: +846 obs_list = [] +847 for obj in gc.get_objects(): +848 if isinstance(obj, Fit_result): +849 obs_list.append(obj) +850 else: +851 obs_list = objects +852 +853 p_values = [o.p_value for o in obs_list] +854 +855 bins = len(p_values) +856 x = np.arange(0, 1.001, 0.001) +857 plt.plot(x, x, 'k', zorder=1) +858 plt.xlim(0, 1) +859 plt.ylim(0, 1) +860 plt.xlabel('p-value') +861 plt.ylabel('Cumulative probability') +862 plt.title(str(bins) + ' p-values') +863 +864 n = np.arange(1, bins + 1) / np.float64(bins) +865 Xs = np.sort(p_values) +866 plt.step(Xs, n) +867 diffs = n - Xs +868 loc_max_diff = np.argmax(np.abs(diffs)) +869 loc = Xs[loc_max_diff] +870 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) +871 plt.draw() +872 +873 print(scipy.stats.kstest(p_values, 'uniform')) +874 +875 +876def _extract_val_and_dval(string): +877 split_string = string.split('(') +878 if '.' in split_string[0] and '.' not in split_string[1][:-1]: +879 factor = 10 ** -len(split_string[0].partition('.')[2]) +880 else: +881 factor = 1 +882 return float(split_string[0]), float(split_string[1][:-1]) * factor +883 +884 +885def _construct_prior_obs(i_prior, i_n): +886 if isinstance(i_prior, Obs): +887 return i_prior +888 elif isinstance(i_prior, str): +889 loc_val, loc_dval = _extract_val_and_dval(i_prior) +890 return cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}") +891 else: +892 raise TypeError("Prior entries need to be 'Obs' or 'str'.") @@ -1184,294 +1246,356 @@ Hotelling t-squared p-value for correlated fits. 152 For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`. 153 In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix). 154 This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning). -155 expected_chisquare : bool -156 If True estimates the expected chisquare which is -157 corrected by effects caused by correlated input data (default False). -158 resplot : bool -159 If True, a plot which displays fit, data and residuals is generated (default False). -160 qqplot : bool -161 If True, a quantile-quantile plot of the fit result is generated (default False). -162 num_grad : bool -163 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). -164 -165 Returns -166 ------- -167 output : Fit_result -168 Parameters and information on the fitted result. -169 ''' -170 output = Fit_result() -171 -172 if (isinstance(x, dict) and isinstance(y, dict) and isinstance(func, dict)): -173 xd = {key: anp.asarray(x[key]) for key in x} -174 yd = y -175 funcd = func -176 output.fit_function = func -177 elif (isinstance(x, dict) or isinstance(y, dict) or isinstance(func, dict)): -178 raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.") -179 else: -180 x = np.asarray(x) -181 xd = {"": x} -182 yd = {"": y} -183 funcd = {"": func} -184 output.fit_function = func -185 -186 if kwargs.get('num_grad') is True: -187 jacobian = num_jacobian -188 hessian = num_hessian -189 else: -190 jacobian = auto_jacobian -191 hessian = auto_hessian -192 -193 key_ls = sorted(list(xd.keys())) -194 -195 if sorted(list(yd.keys())) != key_ls: -196 raise ValueError('x and y dictionaries do not contain the same keys.') -197 -198 if sorted(list(funcd.keys())) != key_ls: -199 raise ValueError('x and func dictionaries do not contain the same keys.') -200 -201 x_all = np.concatenate([np.array(xd[key]) for key in key_ls]) -202 y_all = np.concatenate([np.array(yd[key]) for key in key_ls]) -203 -204 y_f = [o.value for o in y_all] -205 dy_f = [o.dvalue for o in y_all] -206 -207 if len(x_all.shape) > 2: -208 raise ValueError("Unknown format for x values") -209 -210 if np.any(np.asarray(dy_f) <= 0.0): -211 raise Exception("No y errors available, run the gamma method first.") -212 -213 # number of fit parameters -214 n_parms_ls = [] -215 for key in key_ls: -216 if not callable(funcd[key]): -217 raise TypeError('func (key=' + key + ') is not a function.') -218 if np.asarray(xd[key]).shape[-1] != len(yd[key]): -219 raise ValueError('x and y input (key=' + key + ') do not have the same length') -220 for n_loc in range(100): -221 try: -222 funcd[key](np.arange(n_loc), x_all.T[0]) -223 except TypeError: -224 continue -225 except IndexError: -226 continue -227 else: -228 break -229 else: -230 raise RuntimeError("Fit function (key=" + key + ") is not valid.") -231 n_parms_ls.append(n_loc) -232 -233 n_parms = max(n_parms_ls) -234 -235 if len(key_ls) > 1: -236 for key in key_ls: -237 if np.asarray(yd[key]).shape != funcd[key](np.arange(n_parms), xd[key]).shape: -238 raise ValueError(f"Fit function {key} returns the wrong shape ({funcd[key](np.arange(n_parms), xd[key]).shape} instead of {xd[key].shape})\nIf the fit function is just a constant you could try adding x*0 to get the correct shape.") -239 -240 if not silent: -241 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) -242 -243 if priors is not None: -244 if isinstance(priors, (list, np.ndarray)): -245 if n_parms != len(priors): -246 raise ValueError("'priors' does not have the correct length.") -247 -248 loc_priors = [] -249 for i_n, i_prior in enumerate(priors): -250 loc_priors.append(_construct_prior_obs(i_prior, i_n)) +155 inv_chol_cov_matrix [array,list], optional +156 array: shape = (no of y values) X (no of y values) +157 list: for an uncombined fit: [""] +158 for a combined fit: list of keys belonging to the corr_matrix saved in the array, must be the same as the keys of the y dict in alphabetical order +159 If correlated_fit=True is set as well, can provide an inverse covariance matrix (y errors, dy_f included!) of your own choosing for a correlated fit. +160 The matrix must be a lower triangular matrix constructed from a Cholesky decomposition: The function invert_corr_cov_cholesky(corr, inverrdiag) can be +161 used to construct it from a correlation matrix (corr) and the errors dy_f of the data points (inverrdiag = np.diag(1 / np.asarray(dy_f))). For the correct +162 ordering the correlation matrix (corr) can be sorted via the function sort_corr(corr, kl, yd) where kl is the list of keys and yd the y dict. +163 expected_chisquare : bool +164 If True estimates the expected chisquare which is +165 corrected by effects caused by correlated input data (default False). +166 resplot : bool +167 If True, a plot which displays fit, data and residuals is generated (default False). +168 qqplot : bool +169 If True, a quantile-quantile plot of the fit result is generated (default False). +170 num_grad : bool +171 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). +172 +173 Returns +174 ------- +175 output : Fit_result +176 Parameters and information on the fitted result. +177 Examples +178 ------ +179 >>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set +180 >>> import numpy as np +181 >>> from scipy.stats import norm +182 >>> from scipy.linalg import cholesky +183 >>> import pyerrors as pe +184 >>> # generating the random data set +185 >>> num_samples = 400 +186 >>> N = 3 +187 >>> x = np.arange(N) +188 >>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers +189 >>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers +190 >>> r = r1 = r2 = np.zeros((N, N)) +191 >>> y = {} +192 >>> for i in range(N): +193 >>> for j in range(N): +194 >>> r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix +195 >>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors +196 >>> for i in range(N): +197 >>> for j in range(N): +198 >>> r[i, j] *= errl[i] * errl[j] # element in covariance matrix +199 >>> c = cholesky(r, lower=True) +200 >>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined +201 >>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built +202 >>> x_dict = {} +203 >>> y_dict = {} +204 >>> chol_inv_dict = {} +205 >>> data = [] +206 >>> for key in y.keys(): +207 >>> x_dict[key] = x +208 >>> for i in range(N): +209 >>> data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data +210 >>> [o.gamma_method() for o in data] +211 >>> corr = pe.covariance(data, correlation=True) +212 >>> inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data])) +213 >>> chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below +214 >>> y_dict = {'a': data[:3], 'b': data[3:]} +215 >>> # common fit parameter p[0] in combined fit +216 >>> def fit1(p, x): +217 >>> return p[0] + p[1] * x +218 >>> def fit2(p, x): +219 >>> return p[0] + p[2] * x +220 >>> fitf_dict = {'a': fit1, 'b':fit2} +221 >>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']]) +222 Fit with 3 parameters +223 Method: Levenberg-Marquardt +224 `ftol` termination condition is satisfied. +225 chisquare/d.o.f.: 0.5388013574561786 # random +226 fit parameters [1.11897846 0.96361162 0.92325319] # random +227 +228 ''' +229 output = Fit_result() +230 +231 if (isinstance(x, dict) and isinstance(y, dict) and isinstance(func, dict)): +232 xd = {key: anp.asarray(x[key]) for key in x} +233 yd = y +234 funcd = func +235 output.fit_function = func +236 elif (isinstance(x, dict) or isinstance(y, dict) or isinstance(func, dict)): +237 raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.") +238 else: +239 x = np.asarray(x) +240 xd = {"": x} +241 yd = {"": y} +242 funcd = {"": func} +243 output.fit_function = func +244 +245 if kwargs.get('num_grad') is True: +246 jacobian = num_jacobian +247 hessian = num_hessian +248 else: +249 jacobian = auto_jacobian +250 hessian = auto_hessian 251 -252 prior_mask = np.arange(len(priors)) -253 output.priors = loc_priors -254 -255 elif isinstance(priors, dict): -256 loc_priors = [] -257 prior_mask = [] -258 output.priors = {} -259 for pos, prior in priors.items(): -260 if isinstance(pos, int): -261 prior_mask.append(pos) -262 else: -263 raise TypeError("Prior position needs to be an integer.") -264 loc_priors.append(_construct_prior_obs(prior, pos)) +252 key_ls = sorted(list(xd.keys())) +253 +254 if sorted(list(yd.keys())) != key_ls: +255 raise ValueError('x and y dictionaries do not contain the same keys.') +256 +257 if sorted(list(funcd.keys())) != key_ls: +258 raise ValueError('x and func dictionaries do not contain the same keys.') +259 +260 x_all = np.concatenate([np.array(xd[key]) for key in key_ls]) +261 y_all = np.concatenate([np.array(yd[key]) for key in key_ls]) +262 +263 y_f = [o.value for o in y_all] +264 dy_f = [o.dvalue for o in y_all] 265 -266 output.priors[pos] = loc_priors[-1] -267 if max(prior_mask) >= n_parms: -268 raise ValueError("Prior position out of range.") -269 else: -270 raise TypeError("Unkown type for `priors`.") +266 if len(x_all.shape) > 2: +267 raise ValueError("Unknown format for x values") +268 +269 if np.any(np.asarray(dy_f) <= 0.0): +270 raise Exception("No y errors available, run the gamma method first.") 271 -272 p_f = [o.value for o in loc_priors] -273 dp_f = [o.dvalue for o in loc_priors] -274 if np.any(np.asarray(dp_f) <= 0.0): -275 raise Exception("No prior errors available, run the gamma method first.") -276 else: -277 p_f = dp_f = np.array([]) -278 prior_mask = [] -279 loc_priors = [] -280 -281 if 'initial_guess' in kwargs: -282 x0 = kwargs.get('initial_guess') -283 if len(x0) != n_parms: -284 raise ValueError('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) -285 else: -286 x0 = [0.1] * n_parms -287 -288 if priors is None: -289 def general_chisqfunc_uncorr(p, ivars, pr): -290 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -291 return (ivars - model) / dy_f -292 else: -293 def general_chisqfunc_uncorr(p, ivars, pr): -294 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -295 return anp.concatenate(((ivars - model) / dy_f, (p[prior_mask] - pr) / dp_f)) -296 -297 def chisqfunc_uncorr(p): -298 return anp.sum(general_chisqfunc_uncorr(p, y_f, p_f) ** 2) -299 -300 if kwargs.get('correlated_fit') is True: -301 corr = covariance(y_all, correlation=True, **kwargs) -302 covdiag = np.diag(1 / np.asarray(dy_f)) -303 condn = np.linalg.cond(corr) -304 if condn > 0.1 / np.finfo(float).eps: -305 raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})") -306 if condn > 1e13: -307 warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning) -308 chol = np.linalg.cholesky(corr) -309 chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True) +272 # number of fit parameters +273 n_parms_ls = [] +274 for key in key_ls: +275 if not callable(funcd[key]): +276 raise TypeError('func (key=' + key + ') is not a function.') +277 if np.asarray(xd[key]).shape[-1] != len(yd[key]): +278 raise ValueError('x and y input (key=' + key + ') do not have the same length') +279 for n_loc in range(100): +280 try: +281 funcd[key](np.arange(n_loc), x_all.T[0]) +282 except TypeError: +283 continue +284 except IndexError: +285 continue +286 else: +287 break +288 else: +289 raise RuntimeError("Fit function (key=" + key + ") is not valid.") +290 n_parms_ls.append(n_loc) +291 +292 n_parms = max(n_parms_ls) +293 +294 if len(key_ls) > 1: +295 for key in key_ls: +296 if np.asarray(yd[key]).shape != funcd[key](np.arange(n_parms), xd[key]).shape: +297 raise ValueError(f"Fit function {key} returns the wrong shape ({funcd[key](np.arange(n_parms), xd[key]).shape} instead of {xd[key].shape})\nIf the fit function is just a constant you could try adding x*0 to get the correct shape.") +298 +299 if not silent: +300 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) +301 +302 if priors is not None: +303 if isinstance(priors, (list, np.ndarray)): +304 if n_parms != len(priors): +305 raise ValueError("'priors' does not have the correct length.") +306 +307 loc_priors = [] +308 for i_n, i_prior in enumerate(priors): +309 loc_priors.append(_construct_prior_obs(i_prior, i_n)) 310 -311 def general_chisqfunc(p, ivars, pr): -312 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -313 return anp.concatenate((anp.dot(chol_inv, (ivars - model)), (p[prior_mask] - pr) / dp_f)) -314 -315 def chisqfunc(p): -316 return anp.sum(general_chisqfunc(p, y_f, p_f) ** 2) -317 else: -318 general_chisqfunc = general_chisqfunc_uncorr -319 chisqfunc = chisqfunc_uncorr -320 -321 output.method = kwargs.get('method', 'Levenberg-Marquardt') -322 if not silent: -323 print('Method:', output.method) +311 prior_mask = np.arange(len(priors)) +312 output.priors = loc_priors +313 +314 elif isinstance(priors, dict): +315 loc_priors = [] +316 prior_mask = [] +317 output.priors = {} +318 for pos, prior in priors.items(): +319 if isinstance(pos, int): +320 prior_mask.append(pos) +321 else: +322 raise TypeError("Prior position needs to be an integer.") +323 loc_priors.append(_construct_prior_obs(prior, pos)) 324 -325 if output.method != 'Levenberg-Marquardt': -326 if output.method == 'migrad': -327 tolerance = 1e-4 # default value of 1e-1 set by iminuit can be problematic -328 if 'tol' in kwargs: -329 tolerance = kwargs.get('tol') -330 fit_result = iminuit.minimize(chisqfunc_uncorr, x0, tol=tolerance) # Stopping criterion 0.002 * tol * errordef -331 if kwargs.get('correlated_fit') is True: -332 fit_result = iminuit.minimize(chisqfunc, fit_result.x, tol=tolerance) -333 output.iterations = fit_result.nfev -334 else: -335 tolerance = 1e-12 -336 if 'tol' in kwargs: -337 tolerance = kwargs.get('tol') -338 fit_result = scipy.optimize.minimize(chisqfunc_uncorr, x0, method=kwargs.get('method'), tol=tolerance) -339 if kwargs.get('correlated_fit') is True: -340 fit_result = scipy.optimize.minimize(chisqfunc, fit_result.x, method=kwargs.get('method'), tol=tolerance) -341 output.iterations = fit_result.nit -342 -343 chisquare = fit_result.fun -344 -345 else: -346 if 'tol' in kwargs: -347 print('tol cannot be set for Levenberg-Marquardt') -348 -349 def chisqfunc_residuals_uncorr(p): -350 return general_chisqfunc_uncorr(p, y_f, p_f) -351 -352 fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) -353 if kwargs.get('correlated_fit') is True: -354 -355 def chisqfunc_residuals(p): -356 return general_chisqfunc(p, y_f, p_f) -357 -358 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) -359 -360 chisquare = np.sum(fit_result.fun ** 2) -361 assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14) -362 -363 output.iterations = fit_result.nfev -364 -365 if not fit_result.success: -366 raise Exception('The minimization procedure did not converge.') -367 -368 output.chisquare = chisquare -369 output.dof = y_all.shape[-1] - n_parms + len(loc_priors) -370 output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof) -371 if output.dof > 0: -372 output.chisquare_by_dof = output.chisquare / output.dof -373 else: -374 output.chisquare_by_dof = float('nan') -375 -376 output.message = fit_result.message -377 if not silent: -378 print(fit_result.message) -379 print('chisquare/d.o.f.:', output.chisquare_by_dof) -380 print('fit parameters', fit_result.x) -381 -382 def prepare_hat_matrix(): -383 hat_vector = [] -384 for key in key_ls: -385 if (len(xd[key]) != 0): -386 hat_vector.append(jacobian(funcd[key])(fit_result.x, xd[key])) -387 hat_vector = [item for sublist in hat_vector for item in sublist] -388 return hat_vector -389 -390 if kwargs.get('expected_chisquare') is True: -391 if kwargs.get('correlated_fit') is not True: -392 W = np.diag(1 / np.asarray(dy_f)) -393 cov = covariance(y_all) -394 hat_vector = prepare_hat_matrix() -395 A = W @ hat_vector -396 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T -397 expected_chisquare = np.trace((np.identity(y_all.shape[-1]) - P_phi) @ W @ cov @ W) -398 output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare -399 if not silent: -400 print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare) -401 -402 fitp = fit_result.x -403 -404 try: -405 hess = hessian(chisqfunc)(fitp) -406 except TypeError: -407 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None -408 -409 len_y = len(y_f) -410 -411 def chisqfunc_compact(d): -412 return anp.sum(general_chisqfunc(d[:n_parms], d[n_parms: n_parms + len_y], d[n_parms + len_y:]) ** 2) -413 -414 jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f, p_f))) -415 -416 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv -417 try: -418 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:]) -419 except np.linalg.LinAlgError: -420 raise Exception("Cannot invert hessian matrix.") +325 output.priors[pos] = loc_priors[-1] +326 if max(prior_mask) >= n_parms: +327 raise ValueError("Prior position out of range.") +328 else: +329 raise TypeError("Unkown type for `priors`.") +330 +331 p_f = [o.value for o in loc_priors] +332 dp_f = [o.dvalue for o in loc_priors] +333 if np.any(np.asarray(dp_f) <= 0.0): +334 raise Exception("No prior errors available, run the gamma method first.") +335 else: +336 p_f = dp_f = np.array([]) +337 prior_mask = [] +338 loc_priors = [] +339 +340 if 'initial_guess' in kwargs: +341 x0 = kwargs.get('initial_guess') +342 if len(x0) != n_parms: +343 raise ValueError('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) +344 else: +345 x0 = [0.1] * n_parms +346 +347 if priors is None: +348 def general_chisqfunc_uncorr(p, ivars, pr): +349 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) +350 return (ivars - model) / dy_f +351 else: +352 def general_chisqfunc_uncorr(p, ivars, pr): +353 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) +354 return anp.concatenate(((ivars - model) / dy_f, (p[prior_mask] - pr) / dp_f)) +355 +356 def chisqfunc_uncorr(p): +357 return anp.sum(general_chisqfunc_uncorr(p, y_f, p_f) ** 2) +358 +359 if kwargs.get('correlated_fit') is True: +360 if 'inv_chol_cov_matrix' in kwargs: +361 chol_inv = kwargs.get('inv_chol_cov_matrix') +362 if (chol_inv[0].shape[0] != len(dy_f)): +363 raise TypeError('The number of columns of the inverse covariance matrix handed over needs to be equal to the number of y errors.') +364 if (chol_inv[0].shape[0] != chol_inv[0].shape[1]): +365 raise TypeError('The inverse covariance matrix handed over needs to have the same number of rows as columns.') +366 if (chol_inv[1] != key_ls): +367 raise ValueError('The keys of inverse covariance matrix are not the same or do not appear in the same order as the x and y values.') +368 chol_inv = chol_inv[0] +369 else: +370 corr = covariance(y_all, correlation=True, **kwargs) +371 inverrdiag = np.diag(1 / np.asarray(dy_f)) +372 chol_inv = invert_corr_cov_cholesky(corr, inverrdiag) +373 +374 def general_chisqfunc(p, ivars, pr): +375 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) +376 return anp.concatenate((anp.dot(chol_inv, (ivars - model)), (p[prior_mask] - pr) / dp_f)) +377 +378 def chisqfunc(p): +379 return anp.sum(general_chisqfunc(p, y_f, p_f) ** 2) +380 else: +381 general_chisqfunc = general_chisqfunc_uncorr +382 chisqfunc = chisqfunc_uncorr +383 +384 output.method = kwargs.get('method', 'Levenberg-Marquardt') +385 if not silent: +386 print('Method:', output.method) +387 +388 if output.method != 'Levenberg-Marquardt': +389 if output.method == 'migrad': +390 tolerance = 1e-4 # default value of 1e-1 set by iminuit can be problematic +391 if 'tol' in kwargs: +392 tolerance = kwargs.get('tol') +393 fit_result = iminuit.minimize(chisqfunc_uncorr, x0, tol=tolerance) # Stopping criterion 0.002 * tol * errordef +394 if kwargs.get('correlated_fit') is True: +395 fit_result = iminuit.minimize(chisqfunc, fit_result.x, tol=tolerance) +396 output.iterations = fit_result.nfev +397 else: +398 tolerance = 1e-12 +399 if 'tol' in kwargs: +400 tolerance = kwargs.get('tol') +401 fit_result = scipy.optimize.minimize(chisqfunc_uncorr, x0, method=kwargs.get('method'), tol=tolerance) +402 if kwargs.get('correlated_fit') is True: +403 fit_result = scipy.optimize.minimize(chisqfunc, fit_result.x, method=kwargs.get('method'), tol=tolerance) +404 output.iterations = fit_result.nit +405 +406 chisquare = fit_result.fun +407 +408 else: +409 if 'tol' in kwargs: +410 print('tol cannot be set for Levenberg-Marquardt') +411 +412 def chisqfunc_residuals_uncorr(p): +413 return general_chisqfunc_uncorr(p, y_f, p_f) +414 +415 fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) +416 if kwargs.get('correlated_fit') is True: +417 def chisqfunc_residuals(p): +418 return general_chisqfunc(p, y_f, p_f) +419 +420 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) 421 -422 result = [] -423 for i in range(n_parms): -424 result.append(derived_observable(lambda x_all, **kwargs: (x_all[0] + np.finfo(np.float64).eps) / (y_all[0].value + np.finfo(np.float64).eps) * fitp[i], list(y_all) + loc_priors, man_grad=list(deriv_y[i]))) -425 -426 output.fit_parameters = result -427 -428 # Hotelling t-squared p-value for correlated fits. -429 if kwargs.get('correlated_fit') is True: -430 n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all)) -431 output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare, -432 output.dof, n_cov - output.dof) -433 -434 if kwargs.get('resplot') is True: -435 for key in key_ls: -436 residual_plot(xd[key], yd[key], funcd[key], result, title=key) +422 chisquare = np.sum(fit_result.fun ** 2) +423 assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14) +424 +425 output.iterations = fit_result.nfev +426 +427 if not fit_result.success: +428 raise Exception('The minimization procedure did not converge.') +429 +430 output.chisquare = chisquare +431 output.dof = y_all.shape[-1] - n_parms + len(loc_priors) +432 output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof) +433 if output.dof > 0: +434 output.chisquare_by_dof = output.chisquare / output.dof +435 else: +436 output.chisquare_by_dof = float('nan') 437 -438 if kwargs.get('qqplot') is True: -439 for key in key_ls: -440 qqplot(xd[key], yd[key], funcd[key], result, title=key) -441 -442 return output +438 output.message = fit_result.message +439 if not silent: +440 print(fit_result.message) +441 print('chisquare/d.o.f.:', output.chisquare_by_dof) +442 print('fit parameters', fit_result.x) +443 +444 def prepare_hat_matrix(): +445 hat_vector = [] +446 for key in key_ls: +447 if (len(xd[key]) != 0): +448 hat_vector.append(jacobian(funcd[key])(fit_result.x, xd[key])) +449 hat_vector = [item for sublist in hat_vector for item in sublist] +450 return hat_vector +451 +452 if kwargs.get('expected_chisquare') is True: +453 if kwargs.get('correlated_fit') is not True: +454 W = np.diag(1 / np.asarray(dy_f)) +455 cov = covariance(y_all) +456 hat_vector = prepare_hat_matrix() +457 A = W @ hat_vector +458 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T +459 expected_chisquare = np.trace((np.identity(y_all.shape[-1]) - P_phi) @ W @ cov @ W) +460 output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare +461 if not silent: +462 print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare) +463 +464 fitp = fit_result.x +465 +466 try: +467 hess = hessian(chisqfunc)(fitp) +468 except TypeError: +469 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None +470 +471 len_y = len(y_f) +472 +473 def chisqfunc_compact(d): +474 return anp.sum(general_chisqfunc(d[:n_parms], d[n_parms: n_parms + len_y], d[n_parms + len_y:]) ** 2) +475 +476 jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f, p_f))) +477 +478 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv +479 try: +480 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:]) +481 except np.linalg.LinAlgError: +482 raise Exception("Cannot invert hessian matrix.") +483 +484 result = [] +485 for i in range(n_parms): +486 result.append(derived_observable(lambda x_all, **kwargs: (x_all[0] + np.finfo(np.float64).eps) / (y_all[0].value + np.finfo(np.float64).eps) * fitp[i], list(y_all) + loc_priors, man_grad=list(deriv_y[i]))) +487 +488 output.fit_parameters = result +489 +490 # Hotelling t-squared p-value for correlated fits. +491 if kwargs.get('correlated_fit') is True: +492 n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all)) +493 output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare, +494 output.dof, n_cov - output.dof) +495 +496 if kwargs.get('resplot') is True: +497 for key in key_ls: +498 residual_plot(xd[key], yd[key], funcd[key], result, title=key) +499 +500 if kwargs.get('qqplot') is True: +501 for key in key_ls: +502 qqplot(xd[key], yd[key], funcd[key], result, title=key) +503 +504 return output @@ -1555,6 +1679,13 @@ If True, use the full inverse covariance matrix in the definition of the chisqua For details about how the covariance matrix is estimated see pyerrors.obs.covariance. In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix). This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning). +
  • inv_chol_cov_matrix [array,list], optional: array: shape = (no of y values) X (no of y values) +list: for an uncombined fit: [""] +for a combined fit: list of keys belonging to the corr_matrix saved in the array, must be the same as the keys of the y dict in alphabetical order +If correlated_fit=True is set as well, can provide an inverse covariance matrix (y errors, dy_f included!) of your own choosing for a correlated fit. +The matrix must be a lower triangular matrix constructed from a Cholesky decomposition: The function invert_corr_cov_cholesky(corr, inverrdiag) can be +used to construct it from a correlation matrix (corr) and the errors dy_f of the data points (inverrdiag = np.diag(1 / np.asarray(dy_f))). For the correct +ordering the correlation matrix (corr) can be sorted via the function sort_corr(corr, kl, yd) where kl is the list of keys and yd the y dict.
  • expected_chisquare (bool): If True estimates the expected chisquare which is corrected by effects caused by correlated input data (default False).
    • @@ -1572,6 +1703,60 @@ Use numerical differentation instead of automatic differentiation to perform the
  • output (Fit_result): Parameters and information on the fitted result.
    • + +
    Examples
    + +
    +
    >>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set
+>>> import numpy as np
+>>> from scipy.stats import norm
+>>> from scipy.linalg import cholesky
+>>> import pyerrors as pe
+>>> # generating the random data set
+>>> num_samples = 400
+>>> N = 3
+>>> x = np.arange(N)
+>>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers
+>>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers
+>>> r = r1 = r2 = np.zeros((N, N))
+>>> y = {}
+>>> for i in range(N):
+>>>    for j in range(N):
+>>>        r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix
+>>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors
+>>> for i in range(N):
+>>>    for j in range(N):
+>>>        r[i, j] *= errl[i] * errl[j] # element in covariance matrix
+>>> c = cholesky(r, lower=True)
+>>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined
+>>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built
+>>> x_dict = {}
+>>> y_dict = {}
+>>> chol_inv_dict = {}
+>>> data = []
+>>> for key in y.keys():
+>>>    x_dict[key] = x
+>>>    for i in range(N):
+>>>        data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data
+>>>    [o.gamma_method() for o in data]
+>>>    corr = pe.covariance(data, correlation=True)
+>>>    inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data]))
+>>>    chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below
+>>> y_dict = {'a': data[:3], 'b': data[3:]}
+>>> # common fit parameter p[0] in combined fit
+>>> def fit1(p, x):
+>>>    return p[0] + p[1] * x
+>>> def fit2(p, x):
+>>>    return p[0] + p[2] * x
+>>> fitf_dict = {'a': fit1, 'b':fit2}
+>>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']])
+Fit with 3 parameters
+Method: Levenberg-Marquardt
+`ftol` termination condition is satisfied.
+chisquare/d.o.f.: 0.5388013574561786 # random
+fit parameters [1.11897846 0.96361162 0.92325319] # random
+
    +
    @@ -1587,208 +1772,208 @@ Parameters and information on the fitted result. - 
    445def total_least_squares(x, y, func, silent=False, **kwargs):
-446    r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
-447
-448    Parameters
-449    ----------
-450    x : list
-451        list of Obs, or a tuple of lists of Obs
-452    y : list
-453        list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
-454    func : object
-455        func has to be of the form
-456
-457        ```python
-458        import autograd.numpy as anp
-459
-460        def func(a, x):
-461            return a[0] + a[1] * x + a[2] * anp.sinh(x)
-462        ```
-463
-464        For multiple x values func can be of the form
-465
-466        ```python
-467        def func(a, x):
-468            (x1, x2) = x
-469            return a[0] * x1 ** 2 + a[1] * x2
-470        ```
-471
-472        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
-473        will not work.
-474    silent : bool, optional
-475        If true all output to the console is omitted (default False).
-476    initial_guess : list
-477        can provide an initial guess for the input parameters. Relevant for non-linear
-478        fits with many parameters.
-479    expected_chisquare : bool
-480        If true prints the expected chisquare which is
-481        corrected by effects caused by correlated input data.
-482        This can take a while as the full correlation matrix
-483        has to be calculated (default False).
-484    num_grad : bool
-485        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
-486
-487    Notes
-488    -----
-489    Based on the orthogonal distance regression module of scipy.
-490
-491    Returns
-492    -------
-493    output : Fit_result
-494        Parameters and information on the fitted result.
-495    '''
-496
-497    output = Fit_result()
-498
-499    output.fit_function = func
-500
-501    x = np.array(x)
-502
-503    x_shape = x.shape
-504
-505    if kwargs.get('num_grad') is True:
-506        jacobian = num_jacobian
-507        hessian = num_hessian
-508    else:
-509        jacobian = auto_jacobian
-510        hessian = auto_hessian
-511
-512    if not callable(func):
-513        raise TypeError('func has to be a function.')
-514
-515    for i in range(42):
-516        try:
-517            func(np.arange(i), x.T[0])
-518        except TypeError:
-519            continue
-520        except IndexError:
-521            continue
-522        else:
-523            break
-524    else:
-525        raise RuntimeError("Fit function is not valid.")
-526
-527    n_parms = i
-528    if not silent:
-529        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
-530
-531    x_f = np.vectorize(lambda o: o.value)(x)
-532    dx_f = np.vectorize(lambda o: o.dvalue)(x)
-533    y_f = np.array([o.value for o in y])
-534    dy_f = np.array([o.dvalue for o in y])
-535
-536    if np.any(np.asarray(dx_f) <= 0.0):
-537        raise Exception('No x errors available, run the gamma method first.')
-538
-539    if np.any(np.asarray(dy_f) <= 0.0):
-540        raise Exception('No y errors available, run the gamma method first.')
-541
-542    if 'initial_guess' in kwargs:
-543        x0 = kwargs.get('initial_guess')
-544        if len(x0) != n_parms:
-545            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
-546    else:
-547        x0 = [1] * n_parms
+            
    507def total_least_squares(x, y, func, silent=False, **kwargs):
+508    r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
+509
+510    Parameters
+511    ----------
+512    x : list
+513        list of Obs, or a tuple of lists of Obs
+514    y : list
+515        list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
+516    func : object
+517        func has to be of the form
+518
+519        ```python
+520        import autograd.numpy as anp
+521
+522        def func(a, x):
+523            return a[0] + a[1] * x + a[2] * anp.sinh(x)
+524        ```
+525
+526        For multiple x values func can be of the form
+527
+528        ```python
+529        def func(a, x):
+530            (x1, x2) = x
+531            return a[0] * x1 ** 2 + a[1] * x2
+532        ```
+533
+534        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
+535        will not work.
+536    silent : bool, optional
+537        If true all output to the console is omitted (default False).
+538    initial_guess : list
+539        can provide an initial guess for the input parameters. Relevant for non-linear
+540        fits with many parameters.
+541    expected_chisquare : bool
+542        If true prints the expected chisquare which is
+543        corrected by effects caused by correlated input data.
+544        This can take a while as the full correlation matrix
+545        has to be calculated (default False).
+546    num_grad : bool
+547        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
 548
-549    data = RealData(x_f, y_f, sx=dx_f, sy=dy_f)
-550    model = Model(func)
-551    odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps)
-552    odr.set_job(fit_type=0, deriv=1)
-553    out = odr.run()
-554
-555    output.residual_variance = out.res_var
-556
-557    output.method = 'ODR'
+549    Notes
+550    -----
+551    Based on the orthogonal distance regression module of scipy.
+552
+553    Returns
+554    -------
+555    output : Fit_result
+556        Parameters and information on the fitted result.
+557    '''
 558
-559    output.message = out.stopreason
+559    output = Fit_result()
 560
-561    output.xplus = out.xplus
+561    output.fit_function = func
 562
-563    if not silent:
-564        print('Method: ODR')
-565        print(*out.stopreason)
-566        print('Residual variance:', output.residual_variance)
-567
-568    if out.info > 3:
-569        raise Exception('The minimization procedure did not converge.')
-570
-571    m = x_f.size
-572
-573    def odr_chisquare(p):
-574        model = func(p[:n_parms], p[n_parms:].reshape(x_shape))
-575        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2)
-576        return chisq
-577
-578    if kwargs.get('expected_chisquare') is True:
-579        W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel()))))
-580
-581        if kwargs.get('covariance') is not None:
-582            cov = kwargs.get('covariance')
-583        else:
-584            cov = covariance(np.concatenate((y, x.ravel())))
-585
-586        number_of_x_parameters = int(m / x_f.shape[-1])
-587
-588        old_jac = jacobian(func)(out.beta, out.xplus)
-589        fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0)))
-590        fused_row2 = np.concatenate((jacobian(lambda x, y: func(y, x))(out.xplus, out.beta).reshape(x_f.shape[-1], x_f.shape[-1] * number_of_x_parameters), np.identity(number_of_x_parameters * old_jac.shape[0])))
-591        new_jac = np.concatenate((fused_row1, fused_row2), axis=1)
+563    x = np.array(x)
+564
+565    x_shape = x.shape
+566
+567    if kwargs.get('num_grad') is True:
+568        jacobian = num_jacobian
+569        hessian = num_hessian
+570    else:
+571        jacobian = auto_jacobian
+572        hessian = auto_hessian
+573
+574    if not callable(func):
+575        raise TypeError('func has to be a function.')
+576
+577    for i in range(42):
+578        try:
+579            func(np.arange(i), x.T[0])
+580        except TypeError:
+581            continue
+582        except IndexError:
+583            continue
+584        else:
+585            break
+586    else:
+587        raise RuntimeError("Fit function is not valid.")
+588
+589    n_parms = i
+590    if not silent:
+591        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
 592
-593        A = W @ new_jac
-594        P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
-595        expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W)
-596        if expected_chisquare <= 0.0:
-597            warnings.warn("Negative expected_chisquare.", RuntimeWarning)
-598            expected_chisquare = np.abs(expected_chisquare)
-599        output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare
-600        if not silent:
-601            print('chisquare/expected_chisquare:',
-602                  output.chisquare_by_expected_chisquare)
+593    x_f = np.vectorize(lambda o: o.value)(x)
+594    dx_f = np.vectorize(lambda o: o.dvalue)(x)
+595    y_f = np.array([o.value for o in y])
+596    dy_f = np.array([o.dvalue for o in y])
+597
+598    if np.any(np.asarray(dx_f) <= 0.0):
+599        raise Exception('No x errors available, run the gamma method first.')
+600
+601    if np.any(np.asarray(dy_f) <= 0.0):
+602        raise Exception('No y errors available, run the gamma method first.')
 603
-604    fitp = out.beta
-605    try:
-606        hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel())))
-607    except TypeError:
-608        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
-609
-610    def odr_chisquare_compact_x(d):
-611        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
-612        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((d[n_parms + m:].reshape(x_shape) - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2)
-613        return chisq
-614
-615    jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel())))
+604    if 'initial_guess' in kwargs:
+605        x0 = kwargs.get('initial_guess')
+606        if len(x0) != n_parms:
+607            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
+608    else:
+609        x0 = [1] * n_parms
+610
+611    data = RealData(x_f, y_f, sx=dx_f, sy=dy_f)
+612    model = Model(func)
+613    odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps)
+614    odr.set_job(fit_type=0, deriv=1)
+615    out = odr.run()
 616
-617    # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv
-618    try:
-619        deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:])
-620    except np.linalg.LinAlgError:
-621        raise Exception("Cannot invert hessian matrix.")
+617    output.residual_variance = out.res_var
+618
+619    output.method = 'ODR'
+620
+621    output.message = out.stopreason
 622
-623    def odr_chisquare_compact_y(d):
-624        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
-625        chisq = anp.sum(((d[n_parms + m:] - model) / dy_f) ** 2) + anp.sum(((x_f - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2)
-626        return chisq
-627
-628    jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f)))
+623    output.xplus = out.xplus
+624
+625    if not silent:
+626        print('Method: ODR')
+627        print(*out.stopreason)
+628        print('Residual variance:', output.residual_variance)
 629
-630    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
-631    try:
-632        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:])
-633    except np.linalg.LinAlgError:
-634        raise Exception("Cannot invert hessian matrix.")
-635
-636    result = []
-637    for i in range(n_parms):
-638        result.append(derived_observable(lambda my_var, **kwargs: (my_var[0] + np.finfo(np.float64).eps) / (x.ravel()[0].value + np.finfo(np.float64).eps) * out.beta[i], list(x.ravel()) + list(y), man_grad=list(deriv_x[i]) + list(deriv_y[i])))
+630    if out.info > 3:
+631        raise Exception('The minimization procedure did not converge.')
+632
+633    m = x_f.size
+634
+635    def odr_chisquare(p):
+636        model = func(p[:n_parms], p[n_parms:].reshape(x_shape))
+637        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2)
+638        return chisq
 639
-640    output.fit_parameters = result
-641
-642    output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel())))
-643    output.dof = x.shape[-1] - n_parms
-644    output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof)
-645
-646    return output
+640    if kwargs.get('expected_chisquare') is True:
+641        W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel()))))
+642
+643        if kwargs.get('covariance') is not None:
+644            cov = kwargs.get('covariance')
+645        else:
+646            cov = covariance(np.concatenate((y, x.ravel())))
+647
+648        number_of_x_parameters = int(m / x_f.shape[-1])
+649
+650        old_jac = jacobian(func)(out.beta, out.xplus)
+651        fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0)))
+652        fused_row2 = np.concatenate((jacobian(lambda x, y: func(y, x))(out.xplus, out.beta).reshape(x_f.shape[-1], x_f.shape[-1] * number_of_x_parameters), np.identity(number_of_x_parameters * old_jac.shape[0])))
+653        new_jac = np.concatenate((fused_row1, fused_row2), axis=1)
+654
+655        A = W @ new_jac
+656        P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
+657        expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W)
+658        if expected_chisquare <= 0.0:
+659            warnings.warn("Negative expected_chisquare.", RuntimeWarning)
+660            expected_chisquare = np.abs(expected_chisquare)
+661        output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare
+662        if not silent:
+663            print('chisquare/expected_chisquare:',
+664                  output.chisquare_by_expected_chisquare)
+665
+666    fitp = out.beta
+667    try:
+668        hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel())))
+669    except TypeError:
+670        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
+671
+672    def odr_chisquare_compact_x(d):
+673        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
+674        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((d[n_parms + m:].reshape(x_shape) - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2)
+675        return chisq
+676
+677    jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel())))
+678
+679    # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv
+680    try:
+681        deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:])
+682    except np.linalg.LinAlgError:
+683        raise Exception("Cannot invert hessian matrix.")
+684
+685    def odr_chisquare_compact_y(d):
+686        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
+687        chisq = anp.sum(((d[n_parms + m:] - model) / dy_f) ** 2) + anp.sum(((x_f - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2)
+688        return chisq
+689
+690    jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f)))
+691
+692    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
+693    try:
+694        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:])
+695    except np.linalg.LinAlgError:
+696        raise Exception("Cannot invert hessian matrix.")
+697
+698    result = []
+699    for i in range(n_parms):
+700        result.append(derived_observable(lambda my_var, **kwargs: (my_var[0] + np.finfo(np.float64).eps) / (x.ravel()[0].value + np.finfo(np.float64).eps) * out.beta[i], list(x.ravel()) + list(y), man_grad=list(deriv_x[i]) + list(deriv_y[i])))
+701
+702    output.fit_parameters = result
+703
+704    output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel())))
+705    output.dof = x.shape[-1] - n_parms
+706    output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof)
+707
+708    return output
 
    
 
 
@@ -1862,35 +2047,35 @@ Parameters and information on the fitted result.
    - 
    649def fit_lin(x, y, **kwargs):
-650    """Performs a linear fit to y = n + m * x and returns two Obs n, m.
-651
-652    Parameters
-653    ----------
-654    x : list
-655        Can either be a list of floats in which case no xerror is assumed, or
-656        a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
-657    y : list
-658        List of Obs, the dvalues of the Obs are used as yerror for the fit.
-659
-660    Returns
-661    -------
-662    fit_parameters : list[Obs]
-663        LIist of fitted observables.
-664    """
-665
-666    def f(a, x):
-667        y = a[0] + a[1] * x
-668        return y
-669
-670    if all(isinstance(n, Obs) for n in x):
-671        out = total_least_squares(x, y, f, **kwargs)
-672        return out.fit_parameters
-673    elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
-674        out = least_squares(x, y, f, **kwargs)
-675        return out.fit_parameters
-676    else:
-677        raise TypeError('Unsupported types for x')
+            
    711def fit_lin(x, y, **kwargs):
+712    """Performs a linear fit to y = n + m * x and returns two Obs n, m.
+713
+714    Parameters
+715    ----------
+716    x : list
+717        Can either be a list of floats in which case no xerror is assumed, or
+718        a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
+719    y : list
+720        List of Obs, the dvalues of the Obs are used as yerror for the fit.
+721
+722    Returns
+723    -------
+724    fit_parameters : list[Obs]
+725        LIist of fitted observables.
+726    """
+727
+728    def f(a, x):
+729        y = a[0] + a[1] * x
+730        return y
+731
+732    if all(isinstance(n, Obs) for n in x):
+733        out = total_least_squares(x, y, f, **kwargs)
+734        return out.fit_parameters
+735    elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
+736        out = least_squares(x, y, f, **kwargs)
+737        return out.fit_parameters
+738    else:
+739        raise TypeError('Unsupported types for x')
 
    
 
 
@@ -1927,34 +2112,34 @@ LIist of fitted observables.
    - 
    680def qqplot(x, o_y, func, p, title=""):
-681    """Generates a quantile-quantile plot of the fit result which can be used to
-682       check if the residuals of the fit are gaussian distributed.
-683
-684    Returns
-685    -------
-686    None
-687    """
-688
-689    residuals = []
-690    for i_x, i_y in zip(x, o_y):
-691        residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
-692    residuals = sorted(residuals)
-693    my_y = [o.value for o in residuals]
-694    probplot = scipy.stats.probplot(my_y)
-695    my_x = probplot[0][0]
-696    plt.figure(figsize=(8, 8 / 1.618))
-697    plt.errorbar(my_x, my_y, fmt='o')
-698    fit_start = my_x[0]
-699    fit_stop = my_x[-1]
-700    samples = np.arange(fit_start, fit_stop, 0.01)
-701    plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
-702    plt.plot(samples, probplot[1][0] * samples + probplot[1][1], zorder=10, label='Least squares fit, r=' + str(np.around(probplot[1][2], 3)), marker='', ls='-')
-703
-704    plt.xlabel('Theoretical quantiles')
-705    plt.ylabel('Ordered Values')
-706    plt.legend(title=title)
-707    plt.draw()
+            
    742def qqplot(x, o_y, func, p, title=""):
+743    """Generates a quantile-quantile plot of the fit result which can be used to
+744       check if the residuals of the fit are gaussian distributed.
+745
+746    Returns
+747    -------
+748    None
+749    """
+750
+751    residuals = []
+752    for i_x, i_y in zip(x, o_y):
+753        residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
+754    residuals = sorted(residuals)
+755    my_y = [o.value for o in residuals]
+756    probplot = scipy.stats.probplot(my_y)
+757    my_x = probplot[0][0]
+758    plt.figure(figsize=(8, 8 / 1.618))
+759    plt.errorbar(my_x, my_y, fmt='o')
+760    fit_start = my_x[0]
+761    fit_stop = my_x[-1]
+762    samples = np.arange(fit_start, fit_stop, 0.01)
+763    plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
+764    plt.plot(samples, probplot[1][0] * samples + probplot[1][1], zorder=10, label='Least squares fit, r=' + str(np.around(probplot[1][2], 3)), marker='', ls='-')
+765
+766    plt.xlabel('Theoretical quantiles')
+767    plt.ylabel('Ordered Values')
+768    plt.legend(title=title)
+769    plt.draw()
 
    
 
 
@@ -1981,41 +2166,41 @@ LIist of fitted observables.
    - 
    710def residual_plot(x, y, func, fit_res, title=""):
-711    """Generates a plot which compares the fit to the data and displays the corresponding residuals
-712
-713    For uncorrelated data the residuals are expected to be distributed ~N(0,1).
-714
-715    Returns
-716    -------
-717    None
-718    """
-719    sorted_x = sorted(x)
-720    xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
-721    xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
-722    x_samples = np.arange(xstart, xstop + 0.01, 0.01)
-723
-724    plt.figure(figsize=(8, 8 / 1.618))
-725    gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
-726    ax0 = plt.subplot(gs[0])
-727    ax0.errorbar(x, [o.value for o in y], yerr=[o.dvalue for o in y], ls='none', fmt='o', capsize=3, markersize=5, label='Data')
-728    ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
-729    ax0.set_xticklabels([])
-730    ax0.set_xlim([xstart, xstop])
-731    ax0.set_xticklabels([])
-732    ax0.legend(title=title)
-733
-734    residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], np.asarray(x))) / np.asarray([o.dvalue for o in y])
-735    ax1 = plt.subplot(gs[1])
-736    ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
-737    ax1.tick_params(direction='out')
-738    ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
-739    ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
-740    ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
-741    ax1.set_xlim([xstart, xstop])
-742    ax1.set_ylabel('Residuals')
-743    plt.subplots_adjust(wspace=None, hspace=None)
-744    plt.draw()
+            
    772def residual_plot(x, y, func, fit_res, title=""):
+773    """Generates a plot which compares the fit to the data and displays the corresponding residuals
+774
+775    For uncorrelated data the residuals are expected to be distributed ~N(0,1).
+776
+777    Returns
+778    -------
+779    None
+780    """
+781    sorted_x = sorted(x)
+782    xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
+783    xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
+784    x_samples = np.arange(xstart, xstop + 0.01, 0.01)
+785
+786    plt.figure(figsize=(8, 8 / 1.618))
+787    gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
+788    ax0 = plt.subplot(gs[0])
+789    ax0.errorbar(x, [o.value for o in y], yerr=[o.dvalue for o in y], ls='none', fmt='o', capsize=3, markersize=5, label='Data')
+790    ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
+791    ax0.set_xticklabels([])
+792    ax0.set_xlim([xstart, xstop])
+793    ax0.set_xticklabels([])
+794    ax0.legend(title=title)
+795
+796    residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], np.asarray(x))) / np.asarray([o.dvalue for o in y])
+797    ax1 = plt.subplot(gs[1])
+798    ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
+799    ax1.tick_params(direction='out')
+800    ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
+801    ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
+802    ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
+803    ax1.set_xlim([xstart, xstop])
+804    ax1.set_ylabel('Residuals')
+805    plt.subplots_adjust(wspace=None, hspace=None)
+806    plt.draw()
 
    
 
 
@@ -2043,28 +2228,28 @@ LIist of fitted observables.
    - 
    747def error_band(x, func, beta):
-748    """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
-749
-750    Returns
-751    -------
-752    err : np.array(Obs)
-753        Error band for an array of sample values x
-754    """
-755    cov = covariance(beta)
-756    if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
-757        warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
-758
-759    deriv = []
-760    for i, item in enumerate(x):
-761        deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
-762
-763    err = []
-764    for i, item in enumerate(x):
-765        err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
-766    err = np.array(err)
-767
-768    return err
+            
    809def error_band(x, func, beta):
+810    """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
+811
+812    Returns
+813    -------
+814    err : np.array(Obs)
+815        Error band for an array of sample values x
+816    """
+817    cov = covariance(beta)
+818    if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
+819        warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
+820
+821    deriv = []
+822    for i, item in enumerate(x):
+823        deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
+824
+825    err = []
+826    for i, item in enumerate(x):
+827        err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
+828    err = np.array(err)
+829
+830    return err
 
    
 
 
@@ -2091,48 +2276,48 @@ Error band for an array of sample values x
    - 
    771def ks_test(objects=None):
-772    """Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
-773
-774    Parameters
-775    ----------
-776    objects : list
-777        List of fit results to include in the analysis (optional).
-778
-779    Returns
-780    -------
-781    None
-782    """
-783
-784    if objects is None:
-785        obs_list = []
-786        for obj in gc.get_objects():
-787            if isinstance(obj, Fit_result):
-788                obs_list.append(obj)
-789    else:
-790        obs_list = objects
-791
-792    p_values = [o.p_value for o in obs_list]
-793
-794    bins = len(p_values)
-795    x = np.arange(0, 1.001, 0.001)
-796    plt.plot(x, x, 'k', zorder=1)
-797    plt.xlim(0, 1)
-798    plt.ylim(0, 1)
-799    plt.xlabel('p-value')
-800    plt.ylabel('Cumulative probability')
-801    plt.title(str(bins) + ' p-values')
-802
-803    n = np.arange(1, bins + 1) / np.float64(bins)
-804    Xs = np.sort(p_values)
-805    plt.step(Xs, n)
-806    diffs = n - Xs
-807    loc_max_diff = np.argmax(np.abs(diffs))
-808    loc = Xs[loc_max_diff]
-809    plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
-810    plt.draw()
-811
-812    print(scipy.stats.kstest(p_values, 'uniform'))
+            
    833def ks_test(objects=None):
+834    """Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
+835
+836    Parameters
+837    ----------
+838    objects : list
+839        List of fit results to include in the analysis (optional).
+840
+841    Returns
+842    -------
+843    None
+844    """
+845
+846    if objects is None:
+847        obs_list = []
+848        for obj in gc.get_objects():
+849            if isinstance(obj, Fit_result):
+850                obs_list.append(obj)
+851    else:
+852        obs_list = objects
+853
+854    p_values = [o.p_value for o in obs_list]
+855
+856    bins = len(p_values)
+857    x = np.arange(0, 1.001, 0.001)
+858    plt.plot(x, x, 'k', zorder=1)
+859    plt.xlim(0, 1)
+860    plt.ylim(0, 1)
+861    plt.xlabel('p-value')
+862    plt.ylabel('Cumulative probability')
+863    plt.title(str(bins) + ' p-values')
+864
+865    n = np.arange(1, bins + 1) / np.float64(bins)
+866    Xs = np.sort(p_values)
+867    plt.step(Xs, n)
+868    diffs = n - Xs
+869    loc_max_diff = np.argmax(np.abs(diffs))
+870    loc = Xs[loc_max_diff]
+871    plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
+872    plt.draw()
+873
+874    print(scipy.stats.kstest(p_values, 'uniform'))
 
    
 
 
diff --git a/docs/pyerrors/input.html b/docs/pyerrors/input.html
index c187babd..85bd0cb7 100644
--- a/docs/pyerrors/input.html
+++ b/docs/pyerrors/input.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.input API documentation
 
     
diff --git a/docs/pyerrors/input/bdio.html b/docs/pyerrors/input/bdio.html
index e5bbe3c4..d178f040 100644
--- a/docs/pyerrors/input/bdio.html
+++ b/docs/pyerrors/input/bdio.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.input.bdio API documentation
 
     
diff --git a/docs/pyerrors/input/dobs.html b/docs/pyerrors/input/dobs.html
index b64d10ca..b43a5005 100644
--- a/docs/pyerrors/input/dobs.html
+++ b/docs/pyerrors/input/dobs.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.input.dobs API documentation
 
     
diff --git a/docs/pyerrors/input/hadrons.html b/docs/pyerrors/input/hadrons.html
index ca67a795..7572b4ac 100644
--- a/docs/pyerrors/input/hadrons.html
+++ b/docs/pyerrors/input/hadrons.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.input.hadrons API documentation
 
     
diff --git a/docs/pyerrors/input/json.html b/docs/pyerrors/input/json.html
index 215fd600..b05af178 100644
--- a/docs/pyerrors/input/json.html
+++ b/docs/pyerrors/input/json.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.input.json API documentation
 
     
diff --git a/docs/pyerrors/input/misc.html b/docs/pyerrors/input/misc.html
index e2f3c3d2..7f57d557 100644
--- a/docs/pyerrors/input/misc.html
+++ b/docs/pyerrors/input/misc.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.input.misc API documentation
 
     
diff --git a/docs/pyerrors/input/openQCD.html b/docs/pyerrors/input/openQCD.html
index 4bb0d188..2503c7d4 100644
--- a/docs/pyerrors/input/openQCD.html
+++ b/docs/pyerrors/input/openQCD.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.input.openQCD API documentation
 
     
diff --git a/docs/pyerrors/input/pandas.html b/docs/pyerrors/input/pandas.html
index 09de2b72..51da9832 100644
--- a/docs/pyerrors/input/pandas.html
+++ b/docs/pyerrors/input/pandas.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.input.pandas API documentation
 
     
diff --git a/docs/pyerrors/input/sfcf.html b/docs/pyerrors/input/sfcf.html
index fd1d55f2..afacfd76 100644
--- a/docs/pyerrors/input/sfcf.html
+++ b/docs/pyerrors/input/sfcf.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.input.sfcf API documentation
 
     
diff --git a/docs/pyerrors/input/utils.html b/docs/pyerrors/input/utils.html
index 701ca225..ce6134ae 100644
--- a/docs/pyerrors/input/utils.html
+++ b/docs/pyerrors/input/utils.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.input.utils API documentation
 
     
diff --git a/docs/pyerrors/integrate.html b/docs/pyerrors/integrate.html
index 3b51cd43..0f0e2041 100644
--- a/docs/pyerrors/integrate.html
+++ b/docs/pyerrors/integrate.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.integrate API documentation
 
     
diff --git a/docs/pyerrors/linalg.html b/docs/pyerrors/linalg.html
index 4f6cef85..f1cdf836 100644
--- a/docs/pyerrors/linalg.html
+++ b/docs/pyerrors/linalg.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.linalg API documentation
 
     
diff --git a/docs/pyerrors/misc.html b/docs/pyerrors/misc.html
index 6a9b0b8d..39ab9ed2 100644
--- a/docs/pyerrors/misc.html
+++ b/docs/pyerrors/misc.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.misc API documentation
 
     
diff --git a/docs/pyerrors/mpm.html b/docs/pyerrors/mpm.html
index c2ab9597..19a2e912 100644
--- a/docs/pyerrors/mpm.html
+++ b/docs/pyerrors/mpm.html
@@ -3,7 +3,7 @@
 
     
     
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+    
     pyerrors.mpm API documentation
 
     
diff --git a/docs/pyerrors/obs.html b/docs/pyerrors/obs.html
index 43183151..d48504ba 100644
--- a/docs/pyerrors/obs.html
+++ b/docs/pyerrors/obs.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.obs API documentation
 
     
@@ -295,6 +295,12 @@
             
  • 
                     covariance
             
    • 
+            
  • 
+                    invert_corr_cov_cholesky
+            
    • 
+            
  • 
+                    sort_corr
+            
    • 
             
  • 
                     import_jackknife
             
    • 
@@ -1874,234 +1880,320 @@
     1544    return cov
 1545
 1546
-1547def _smooth_eigenvalues(corr, E):
-1548    """Eigenvalue smoothing as described in hep-lat/9412087
-1549
-1550    corr : np.ndarray
-1551        correlation matrix
-1552    E : integer
-1553        Number of eigenvalues to be left substantially unchanged
-1554    """
-1555    if not (2 < E < corr.shape[0] - 1):
-1556        raise Exception(f"'E' has to be between 2 and the dimension of the correlation matrix minus 1 ({corr.shape[0] - 1}).")
-1557    vals, vec = np.linalg.eigh(corr)
-1558    lambda_min = np.mean(vals[:-E])
-1559    vals[vals < lambda_min] = lambda_min
-1560    vals /= np.mean(vals)
-1561    return vec @ np.diag(vals) @ vec.T
-1562
-1563
-1564def _covariance_element(obs1, obs2):
-1565    """Estimates the covariance of two Obs objects, neglecting autocorrelations."""
+1547def invert_corr_cov_cholesky(corr, inverrdiag):
+1548    """Constructs a lower triangular matrix `chol` via the Cholesky decomposition of the correlation matrix `corr`
+1549       and then returns the inverse covariance matrix `chol_inv` as a lower triangular matrix by solving `chol * x = inverrdiag`.
+1550
+1551    Parameters
+1552    ----------
+1553    corr : np.ndarray
+1554           correlation matrix
+1555    inverrdiag : np.ndarray
+1556              diagonal matrix, the entries are the inverse errors of the data points considered
+1557    """
+1558
+1559    condn = np.linalg.cond(corr)
+1560    if condn > 0.1 / np.finfo(float).eps:
+1561        raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})")
+1562    if condn > 1e13:
+1563        warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning)
+1564    chol = np.linalg.cholesky(corr)
+1565    chol_inv = scipy.linalg.solve_triangular(chol, inverrdiag, lower=True)
 1566
-1567    def calc_gamma(deltas1, deltas2, idx1, idx2, new_idx):
-1568        deltas1 = _reduce_deltas(deltas1, idx1, new_idx)
-1569        deltas2 = _reduce_deltas(deltas2, idx2, new_idx)
-1570        return np.sum(deltas1 * deltas2)
-1571
-1572    if set(obs1.names).isdisjoint(set(obs2.names)):
-1573        return 0.0
-1574
-1575    if not hasattr(obs1, 'e_dvalue') or not hasattr(obs2, 'e_dvalue'):
-1576        raise Exception('The gamma method has to be applied to both Obs first.')
-1577
-1578    dvalue = 0.0
+1567    return chol_inv
+1568
+1569
+1570def sort_corr(corr, kl, yd):
+1571    """ Reorders a correlation matrix to match the alphabetical order of its underlying y data.
+1572
+1573    The ordering of the input correlation matrix `corr` is given by the list of keys `kl`.
+1574    The input dictionary `yd` (with the same keys `kl`) must contain the corresponding y data
+1575    that the correlation matrix is based on.
+1576    This function sorts the list of keys `kl` alphabetically and sorts the matrix `corr`
+1577    according to this alphabetical order such that the sorted matrix `corr_sorted` corresponds
+1578    to the y data `yd` when arranged in an alphabetical order by its keys.
 1579
-1580    for e_name in obs1.mc_names:
-1581
-1582        if e_name not in obs2.mc_names:
-1583            continue
-1584
-1585        idl_d = {}
-1586        for r_name in obs1.e_content[e_name]:
-1587            if r_name not in obs2.e_content[e_name]:
-1588                continue
-1589            idl_d[r_name] = _intersection_idx([obs1.idl[r_name], obs2.idl[r_name]])
-1590
-1591        gamma = 0.0
+1580    Parameters
+1581    ----------
+1582    corr : np.ndarray
+1583        A square correlation matrix constructed using the order of the y data specified by `kl`.
+1584        The dimensions of `corr` should match the total number of y data points in `yd` combined.
+1585    kl : list of str
+1586        A list of keys that denotes the order in which the y data from `yd` was used to build the
+1587        input correlation matrix `corr`.
+1588    yd : dict of list
+1589        A dictionary where each key corresponds to a unique identifier, and its value is a list of
+1590        y data points. The total number of y data points across all keys must match the dimensions
+1591        of `corr`. The lists in the dictionary can be lists of Obs.
 1592
-1593        for r_name in obs1.e_content[e_name]:
-1594            if r_name not in obs2.e_content[e_name]:
-1595                continue
-1596            if len(idl_d[r_name]) == 0:
-1597                continue
-1598            gamma += calc_gamma(obs1.deltas[r_name], obs2.deltas[r_name], obs1.idl[r_name], obs2.idl[r_name], idl_d[r_name])
-1599
-1600        if gamma == 0.0:
-1601            continue
-1602
-1603        gamma_div = 0.0
-1604        for r_name in obs1.e_content[e_name]:
-1605            if r_name not in obs2.e_content[e_name]:
-1606                continue
-1607            if len(idl_d[r_name]) == 0:
-1608                continue
-1609            gamma_div += np.sqrt(calc_gamma(obs1.deltas[r_name], obs1.deltas[r_name], obs1.idl[r_name], obs1.idl[r_name], idl_d[r_name]) * calc_gamma(obs2.deltas[r_name], obs2.deltas[r_name], obs2.idl[r_name], obs2.idl[r_name], idl_d[r_name]))
-1610        gamma /= gamma_div
-1611
-1612        dvalue += gamma
+1593    Returns
+1594    -------
+1595    np.ndarray
+1596        A new, sorted correlation matrix that corresponds to the y data from `yd` when arranged alphabetically by its keys.
+1597
+1598    Example
+1599    -------
+1600    >>> import numpy as np
+1601    >>> import pyerrors as pe
+1602    >>> corr = np.array([[1, 0.2, 0.3], [0.2, 1, 0.4], [0.3, 0.4, 1]])
+1603    >>> kl = ['b', 'a']
+1604    >>> yd = {'a': [1, 2], 'b': [3]}
+1605    >>> sorted_corr = pe.obs.sort_corr(corr, kl, yd)
+1606    >>> print(sorted_corr)
+1607    array([[1. , 0.3, 0.4],
+1608           [0.3, 1. , 0.2],
+1609           [0.4, 0.2, 1. ]])
+1610
+1611    """
+1612    kl_sorted = sorted(kl)
 1613
-1614    for e_name in obs1.cov_names:
-1615
-1616        if e_name not in obs2.cov_names:
-1617            continue
-1618
-1619        dvalue += np.dot(np.transpose(obs1.covobs[e_name].grad), np.dot(obs1.covobs[e_name].cov, obs2.covobs[e_name].grad)).item()
-1620
-1621    return dvalue
-1622
-1623
-1624def import_jackknife(jacks, name, idl=None):
-1625    """Imports jackknife samples and returns an Obs
-1626
-1627    Parameters
-1628    ----------
-1629    jacks : numpy.ndarray
-1630        numpy array containing the mean value as zeroth entry and
-1631        the N jackknife samples as first to Nth entry.
-1632    name : str
-1633        name of the ensemble the samples are defined on.
-1634    """
-1635    length = len(jacks) - 1
-1636    prj = (np.ones((length, length)) - (length - 1) * np.identity(length))
-1637    samples = jacks[1:] @ prj
-1638    mean = np.mean(samples)
-1639    new_obs = Obs([samples - mean], [name], idl=idl, means=[mean])
-1640    new_obs._value = jacks[0]
-1641    return new_obs
-1642
-1643
-1644def import_bootstrap(boots, name, random_numbers):
-1645    """Imports bootstrap samples and returns an Obs
-1646
-1647    Parameters
-1648    ----------
-1649    boots : numpy.ndarray
-1650        numpy array containing the mean value as zeroth entry and
-1651        the N bootstrap samples as first to Nth entry.
-1652    name : str
-1653        name of the ensemble the samples are defined on.
-1654    random_numbers : np.ndarray
-1655        Array of shape (samples, length) containing the random numbers to generate the bootstrap samples,
-1656        where samples is the number of bootstrap samples and length is the length of the original Monte Carlo
-1657        chain to be reconstructed.
-1658    """
-1659    samples, length = random_numbers.shape
-1660    if samples != len(boots) - 1:
-1661        raise ValueError("Random numbers do not have the correct shape.")
-1662
-1663    if samples < length:
-1664        raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.")
+1614    posd = {}
+1615    ofs = 0
+1616    for ki, k in enumerate(kl):
+1617        posd[k] = [i + ofs for i in range(len(yd[k]))]
+1618        ofs += len(posd[k])
+1619
+1620    mapping = []
+1621    for k in kl_sorted:
+1622        for i in range(len(yd[k])):
+1623            mapping.append(posd[k][i])
+1624
+1625    corr_sorted = np.zeros_like(corr)
+1626    for i in range(corr.shape[0]):
+1627        for j in range(corr.shape[0]):
+1628            corr_sorted[i][j] = corr[mapping[i]][mapping[j]]
+1629
+1630    return corr_sorted
+1631
+1632
+1633def _smooth_eigenvalues(corr, E):
+1634    """Eigenvalue smoothing as described in hep-lat/9412087
+1635
+1636    corr : np.ndarray
+1637        correlation matrix
+1638    E : integer
+1639        Number of eigenvalues to be left substantially unchanged
+1640    """
+1641    if not (2 < E < corr.shape[0] - 1):
+1642        raise Exception(f"'E' has to be between 2 and the dimension of the correlation matrix minus 1 ({corr.shape[0] - 1}).")
+1643    vals, vec = np.linalg.eigh(corr)
+1644    lambda_min = np.mean(vals[:-E])
+1645    vals[vals < lambda_min] = lambda_min
+1646    vals /= np.mean(vals)
+1647    return vec @ np.diag(vals) @ vec.T
+1648
+1649
+1650def _covariance_element(obs1, obs2):
+1651    """Estimates the covariance of two Obs objects, neglecting autocorrelations."""
+1652
+1653    def calc_gamma(deltas1, deltas2, idx1, idx2, new_idx):
+1654        deltas1 = _reduce_deltas(deltas1, idx1, new_idx)
+1655        deltas2 = _reduce_deltas(deltas2, idx2, new_idx)
+1656        return np.sum(deltas1 * deltas2)
+1657
+1658    if set(obs1.names).isdisjoint(set(obs2.names)):
+1659        return 0.0
+1660
+1661    if not hasattr(obs1, 'e_dvalue') or not hasattr(obs2, 'e_dvalue'):
+1662        raise Exception('The gamma method has to be applied to both Obs first.')
+1663
+1664    dvalue = 0.0
 1665
-1666    proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length
+1666    for e_name in obs1.mc_names:
 1667
-1668    samples = scipy.linalg.lstsq(proj, boots[1:])[0]
-1669    ret = Obs([samples], [name])
-1670    ret._value = boots[0]
-1671    return ret
-1672
-1673
-1674def merge_obs(list_of_obs):
-1675    """Combine all observables in list_of_obs into one new observable
+1668        if e_name not in obs2.mc_names:
+1669            continue
+1670
+1671        idl_d = {}
+1672        for r_name in obs1.e_content[e_name]:
+1673            if r_name not in obs2.e_content[e_name]:
+1674                continue
+1675            idl_d[r_name] = _intersection_idx([obs1.idl[r_name], obs2.idl[r_name]])
 1676
-1677    Parameters
-1678    ----------
-1679    list_of_obs : list
-1680        list of the Obs object to be combined
-1681
-1682    Notes
-1683    -----
-1684    It is not possible to combine obs which are based on the same replicum
-1685    """
-1686    replist = [item for obs in list_of_obs for item in obs.names]
-1687    if (len(replist) == len(set(replist))) is False:
-1688        raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist)))
-1689    if any([len(o.cov_names) for o in list_of_obs]):
-1690        raise Exception('Not possible to merge data that contains covobs!')
-1691    new_dict = {}
-1692    idl_dict = {}
-1693    for o in list_of_obs:
-1694        new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0)
-1695                        for key in set(o.deltas) | set(o.r_values)})
-1696        idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)})
+1677        gamma = 0.0
+1678
+1679        for r_name in obs1.e_content[e_name]:
+1680            if r_name not in obs2.e_content[e_name]:
+1681                continue
+1682            if len(idl_d[r_name]) == 0:
+1683                continue
+1684            gamma += calc_gamma(obs1.deltas[r_name], obs2.deltas[r_name], obs1.idl[r_name], obs2.idl[r_name], idl_d[r_name])
+1685
+1686        if gamma == 0.0:
+1687            continue
+1688
+1689        gamma_div = 0.0
+1690        for r_name in obs1.e_content[e_name]:
+1691            if r_name not in obs2.e_content[e_name]:
+1692                continue
+1693            if len(idl_d[r_name]) == 0:
+1694                continue
+1695            gamma_div += np.sqrt(calc_gamma(obs1.deltas[r_name], obs1.deltas[r_name], obs1.idl[r_name], obs1.idl[r_name], idl_d[r_name]) * calc_gamma(obs2.deltas[r_name], obs2.deltas[r_name], obs2.idl[r_name], obs2.idl[r_name], idl_d[r_name]))
+1696        gamma /= gamma_div
 1697
-1698    names = sorted(new_dict.keys())
-1699    o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names])
-1700    o.reweighted = np.max([oi.reweighted for oi in list_of_obs])
-1701    return o
-1702
-1703
-1704def cov_Obs(means, cov, name, grad=None):
-1705    """Create an Obs based on mean(s) and a covariance matrix
+1698        dvalue += gamma
+1699
+1700    for e_name in obs1.cov_names:
+1701
+1702        if e_name not in obs2.cov_names:
+1703            continue
+1704
+1705        dvalue += np.dot(np.transpose(obs1.covobs[e_name].grad), np.dot(obs1.covobs[e_name].cov, obs2.covobs[e_name].grad)).item()
 1706
-1707    Parameters
-1708    ----------
-1709    mean : list of floats or float
-1710        N mean value(s) of the new Obs
-1711    cov : list or array
-1712        2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
-1713    name : str
-1714        identifier for the covariance matrix
-1715    grad : list or array
-1716        Gradient of the Covobs wrt. the means belonging to cov.
-1717    """
-1718
-1719    def covobs_to_obs(co):
-1720        """Make an Obs out of a Covobs
-1721
-1722        Parameters
-1723        ----------
-1724        co : Covobs
-1725            Covobs to be embedded into the Obs
-1726        """
-1727        o = Obs([], [], means=[])
-1728        o._value = co.value
-1729        o.names.append(co.name)
-1730        o._covobs[co.name] = co
-1731        o._dvalue = np.sqrt(co.errsq())
-1732        return o
-1733
-1734    ol = []
-1735    if isinstance(means, (float, int)):
-1736        means = [means]
-1737
-1738    for i in range(len(means)):
-1739        ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad)))
-1740    if ol[0].covobs[name].N != len(means):
-1741        raise Exception('You have to provide %d mean values!' % (ol[0].N))
-1742    if len(ol) == 1:
-1743        return ol[0]
-1744    return ol
-1745
-1746
-1747def _determine_gap(o, e_content, e_name):
-1748    gaps = []
-1749    for r_name in e_content[e_name]:
-1750        if isinstance(o.idl[r_name], range):
-1751            gaps.append(o.idl[r_name].step)
-1752        else:
-1753            gaps.append(np.min(np.diff(o.idl[r_name])))
-1754
-1755    gap = min(gaps)
-1756    if not np.all([gi % gap == 0 for gi in gaps]):
-1757        raise Exception(f"Replica for ensemble {e_name} do not have a common spacing.", gaps)
+1707    return dvalue
+1708
+1709
+1710def import_jackknife(jacks, name, idl=None):
+1711    """Imports jackknife samples and returns an Obs
+1712
+1713    Parameters
+1714    ----------
+1715    jacks : numpy.ndarray
+1716        numpy array containing the mean value as zeroth entry and
+1717        the N jackknife samples as first to Nth entry.
+1718    name : str
+1719        name of the ensemble the samples are defined on.
+1720    """
+1721    length = len(jacks) - 1
+1722    prj = (np.ones((length, length)) - (length - 1) * np.identity(length))
+1723    samples = jacks[1:] @ prj
+1724    mean = np.mean(samples)
+1725    new_obs = Obs([samples - mean], [name], idl=idl, means=[mean])
+1726    new_obs._value = jacks[0]
+1727    return new_obs
+1728
+1729
+1730def import_bootstrap(boots, name, random_numbers):
+1731    """Imports bootstrap samples and returns an Obs
+1732
+1733    Parameters
+1734    ----------
+1735    boots : numpy.ndarray
+1736        numpy array containing the mean value as zeroth entry and
+1737        the N bootstrap samples as first to Nth entry.
+1738    name : str
+1739        name of the ensemble the samples are defined on.
+1740    random_numbers : np.ndarray
+1741        Array of shape (samples, length) containing the random numbers to generate the bootstrap samples,
+1742        where samples is the number of bootstrap samples and length is the length of the original Monte Carlo
+1743        chain to be reconstructed.
+1744    """
+1745    samples, length = random_numbers.shape
+1746    if samples != len(boots) - 1:
+1747        raise ValueError("Random numbers do not have the correct shape.")
+1748
+1749    if samples < length:
+1750        raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.")
+1751
+1752    proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length
+1753
+1754    samples = scipy.linalg.lstsq(proj, boots[1:])[0]
+1755    ret = Obs([samples], [name])
+1756    ret._value = boots[0]
+1757    return ret
 1758
-1759    return gap
-1760
-1761
-1762def _check_lists_equal(idl):
-1763    '''
-1764    Use groupby to efficiently check whether all elements of idl are identical.
-1765    Returns True if all elements are equal, otherwise False.
-1766
-1767    Parameters
-1768    ----------
-1769    idl : list of lists, ranges or np.ndarrays
-1770    '''
-1771    g = groupby([np.nditer(el) if isinstance(el, np.ndarray) else el for el in idl])
-1772    if next(g, True) and not next(g, False):
-1773        return True
-1774    return False
+1759
+1760def merge_obs(list_of_obs):
+1761    """Combine all observables in list_of_obs into one new observable
+1762
+1763    Parameters
+1764    ----------
+1765    list_of_obs : list
+1766        list of the Obs object to be combined
+1767
+1768    Notes
+1769    -----
+1770    It is not possible to combine obs which are based on the same replicum
+1771    """
+1772    replist = [item for obs in list_of_obs for item in obs.names]
+1773    if (len(replist) == len(set(replist))) is False:
+1774        raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist)))
+1775    if any([len(o.cov_names) for o in list_of_obs]):
+1776        raise Exception('Not possible to merge data that contains covobs!')
+1777    new_dict = {}
+1778    idl_dict = {}
+1779    for o in list_of_obs:
+1780        new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0)
+1781                        for key in set(o.deltas) | set(o.r_values)})
+1782        idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)})
+1783
+1784    names = sorted(new_dict.keys())
+1785    o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names])
+1786    o.reweighted = np.max([oi.reweighted for oi in list_of_obs])
+1787    return o
+1788
+1789
+1790def cov_Obs(means, cov, name, grad=None):
+1791    """Create an Obs based on mean(s) and a covariance matrix
+1792
+1793    Parameters
+1794    ----------
+1795    mean : list of floats or float
+1796        N mean value(s) of the new Obs
+1797    cov : list or array
+1798        2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
+1799    name : str
+1800        identifier for the covariance matrix
+1801    grad : list or array
+1802        Gradient of the Covobs wrt. the means belonging to cov.
+1803    """
+1804
+1805    def covobs_to_obs(co):
+1806        """Make an Obs out of a Covobs
+1807
+1808        Parameters
+1809        ----------
+1810        co : Covobs
+1811            Covobs to be embedded into the Obs
+1812        """
+1813        o = Obs([], [], means=[])
+1814        o._value = co.value
+1815        o.names.append(co.name)
+1816        o._covobs[co.name] = co
+1817        o._dvalue = np.sqrt(co.errsq())
+1818        return o
+1819
+1820    ol = []
+1821    if isinstance(means, (float, int)):
+1822        means = [means]
+1823
+1824    for i in range(len(means)):
+1825        ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad)))
+1826    if ol[0].covobs[name].N != len(means):
+1827        raise Exception('You have to provide %d mean values!' % (ol[0].N))
+1828    if len(ol) == 1:
+1829        return ol[0]
+1830    return ol
+1831
+1832
+1833def _determine_gap(o, e_content, e_name):
+1834    gaps = []
+1835    for r_name in e_content[e_name]:
+1836        if isinstance(o.idl[r_name], range):
+1837            gaps.append(o.idl[r_name].step)
+1838        else:
+1839            gaps.append(np.min(np.diff(o.idl[r_name])))
+1840
+1841    gap = min(gaps)
+1842    if not np.all([gi % gap == 0 for gi in gaps]):
+1843        raise Exception(f"Replica for ensemble {e_name} do not have a common spacing.", gaps)
+1844
+1845    return gap
+1846
+1847
+1848def _check_lists_equal(idl):
+1849    '''
+1850    Use groupby to efficiently check whether all elements of idl are identical.
+1851    Returns True if all elements are equal, otherwise False.
+1852
+1853    Parameters
+1854    ----------
+1855    idl : list of lists, ranges or np.ndarrays
+1856    '''
+1857    g = groupby([np.nditer(el) if isinstance(el, np.ndarray) else el for el in idl])
+1858    if next(g, True) and not next(g, False):
+1859        return True
+1860    return False
    @@ -5791,6 +5883,180 @@ This construction ensures that the estimated covariance matrix is positive semi- + +
    + +
    + + def + invert_corr_cov_cholesky(corr, inverrdiag): + + + +
    + + 
    1548def invert_corr_cov_cholesky(corr, inverrdiag):
+1549    """Constructs a lower triangular matrix `chol` via the Cholesky decomposition of the correlation matrix `corr`
+1550       and then returns the inverse covariance matrix `chol_inv` as a lower triangular matrix by solving `chol * x = inverrdiag`.
+1551
+1552    Parameters
+1553    ----------
+1554    corr : np.ndarray
+1555           correlation matrix
+1556    inverrdiag : np.ndarray
+1557              diagonal matrix, the entries are the inverse errors of the data points considered
+1558    """
+1559
+1560    condn = np.linalg.cond(corr)
+1561    if condn > 0.1 / np.finfo(float).eps:
+1562        raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})")
+1563    if condn > 1e13:
+1564        warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning)
+1565    chol = np.linalg.cholesky(corr)
+1566    chol_inv = scipy.linalg.solve_triangular(chol, inverrdiag, lower=True)
+1567
+1568    return chol_inv
+
    + + +

    Constructs a lower triangular matrix chol via the Cholesky decomposition of the correlation matrix corr + and then returns the inverse covariance matrix chol_inv as a lower triangular matrix by solving chol * x = inverrdiag.

    + +
    Parameters
    + + +
    + + +
    +
    + +
    + + def + sort_corr(corr, kl, yd): + + + +
    + + 
    1571def sort_corr(corr, kl, yd):
+1572    """ Reorders a correlation matrix to match the alphabetical order of its underlying y data.
+1573
+1574    The ordering of the input correlation matrix `corr` is given by the list of keys `kl`.
+1575    The input dictionary `yd` (with the same keys `kl`) must contain the corresponding y data
+1576    that the correlation matrix is based on.
+1577    This function sorts the list of keys `kl` alphabetically and sorts the matrix `corr`
+1578    according to this alphabetical order such that the sorted matrix `corr_sorted` corresponds
+1579    to the y data `yd` when arranged in an alphabetical order by its keys.
+1580
+1581    Parameters
+1582    ----------
+1583    corr : np.ndarray
+1584        A square correlation matrix constructed using the order of the y data specified by `kl`.
+1585        The dimensions of `corr` should match the total number of y data points in `yd` combined.
+1586    kl : list of str
+1587        A list of keys that denotes the order in which the y data from `yd` was used to build the
+1588        input correlation matrix `corr`.
+1589    yd : dict of list
+1590        A dictionary where each key corresponds to a unique identifier, and its value is a list of
+1591        y data points. The total number of y data points across all keys must match the dimensions
+1592        of `corr`. The lists in the dictionary can be lists of Obs.
+1593
+1594    Returns
+1595    -------
+1596    np.ndarray
+1597        A new, sorted correlation matrix that corresponds to the y data from `yd` when arranged alphabetically by its keys.
+1598
+1599    Example
+1600    -------
+1601    >>> import numpy as np
+1602    >>> import pyerrors as pe
+1603    >>> corr = np.array([[1, 0.2, 0.3], [0.2, 1, 0.4], [0.3, 0.4, 1]])
+1604    >>> kl = ['b', 'a']
+1605    >>> yd = {'a': [1, 2], 'b': [3]}
+1606    >>> sorted_corr = pe.obs.sort_corr(corr, kl, yd)
+1607    >>> print(sorted_corr)
+1608    array([[1. , 0.3, 0.4],
+1609           [0.3, 1. , 0.2],
+1610           [0.4, 0.2, 1. ]])
+1611
+1612    """
+1613    kl_sorted = sorted(kl)
+1614
+1615    posd = {}
+1616    ofs = 0
+1617    for ki, k in enumerate(kl):
+1618        posd[k] = [i + ofs for i in range(len(yd[k]))]
+1619        ofs += len(posd[k])
+1620
+1621    mapping = []
+1622    for k in kl_sorted:
+1623        for i in range(len(yd[k])):
+1624            mapping.append(posd[k][i])
+1625
+1626    corr_sorted = np.zeros_like(corr)
+1627    for i in range(corr.shape[0]):
+1628        for j in range(corr.shape[0]):
+1629            corr_sorted[i][j] = corr[mapping[i]][mapping[j]]
+1630
+1631    return corr_sorted
+
    + + +

    Reorders a correlation matrix to match the alphabetical order of its underlying y data.

    + +

    The ordering of the input correlation matrix corr is given by the list of keys kl. +The input dictionary yd (with the same keys kl) must contain the corresponding y data +that the correlation matrix is based on. +This function sorts the list of keys kl alphabetically and sorts the matrix corr +according to this alphabetical order such that the sorted matrix corr_sorted corresponds +to the y data yd when arranged in an alphabetical order by its keys.

    + +
    Parameters
    + + + +
    Returns
    + + + +
    Example
    + +
    +
    >>> import numpy as np
+>>> import pyerrors as pe
+>>> corr = np.array([[1, 0.2, 0.3], [0.2, 1, 0.4], [0.3, 0.4, 1]])
+>>> kl = ['b', 'a']
+>>> yd = {'a': [1, 2], 'b': [3]}
+>>> sorted_corr = pe.obs.sort_corr(corr, kl, yd)
+>>> print(sorted_corr)
+array([[1. , 0.3, 0.4],
+       [0.3, 1. , 0.2],
+       [0.4, 0.2, 1. ]])
+
    +
    +
    + +
    @@ -5803,24 +6069,24 @@ This construction ensures that the estimated covariance matrix is positive semi- - 
    1625def import_jackknife(jacks, name, idl=None):
-1626    """Imports jackknife samples and returns an Obs
-1627
-1628    Parameters
-1629    ----------
-1630    jacks : numpy.ndarray
-1631        numpy array containing the mean value as zeroth entry and
-1632        the N jackknife samples as first to Nth entry.
-1633    name : str
-1634        name of the ensemble the samples are defined on.
-1635    """
-1636    length = len(jacks) - 1
-1637    prj = (np.ones((length, length)) - (length - 1) * np.identity(length))
-1638    samples = jacks[1:] @ prj
-1639    mean = np.mean(samples)
-1640    new_obs = Obs([samples - mean], [name], idl=idl, means=[mean])
-1641    new_obs._value = jacks[0]
-1642    return new_obs
+            
    1711def import_jackknife(jacks, name, idl=None):
+1712    """Imports jackknife samples and returns an Obs
+1713
+1714    Parameters
+1715    ----------
+1716    jacks : numpy.ndarray
+1717        numpy array containing the mean value as zeroth entry and
+1718        the N jackknife samples as first to Nth entry.
+1719    name : str
+1720        name of the ensemble the samples are defined on.
+1721    """
+1722    length = len(jacks) - 1
+1723    prj = (np.ones((length, length)) - (length - 1) * np.identity(length))
+1724    samples = jacks[1:] @ prj
+1725    mean = np.mean(samples)
+1726    new_obs = Obs([samples - mean], [name], idl=idl, means=[mean])
+1727    new_obs._value = jacks[0]
+1728    return new_obs
 
    
 
 
@@ -5850,34 +6116,34 @@ name of the ensemble the samples are defined on.
    - 
    1645def import_bootstrap(boots, name, random_numbers):
-1646    """Imports bootstrap samples and returns an Obs
-1647
-1648    Parameters
-1649    ----------
-1650    boots : numpy.ndarray
-1651        numpy array containing the mean value as zeroth entry and
-1652        the N bootstrap samples as first to Nth entry.
-1653    name : str
-1654        name of the ensemble the samples are defined on.
-1655    random_numbers : np.ndarray
-1656        Array of shape (samples, length) containing the random numbers to generate the bootstrap samples,
-1657        where samples is the number of bootstrap samples and length is the length of the original Monte Carlo
-1658        chain to be reconstructed.
-1659    """
-1660    samples, length = random_numbers.shape
-1661    if samples != len(boots) - 1:
-1662        raise ValueError("Random numbers do not have the correct shape.")
-1663
-1664    if samples < length:
-1665        raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.")
-1666
-1667    proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length
-1668
-1669    samples = scipy.linalg.lstsq(proj, boots[1:])[0]
-1670    ret = Obs([samples], [name])
-1671    ret._value = boots[0]
-1672    return ret
+            
    1731def import_bootstrap(boots, name, random_numbers):
+1732    """Imports bootstrap samples and returns an Obs
+1733
+1734    Parameters
+1735    ----------
+1736    boots : numpy.ndarray
+1737        numpy array containing the mean value as zeroth entry and
+1738        the N bootstrap samples as first to Nth entry.
+1739    name : str
+1740        name of the ensemble the samples are defined on.
+1741    random_numbers : np.ndarray
+1742        Array of shape (samples, length) containing the random numbers to generate the bootstrap samples,
+1743        where samples is the number of bootstrap samples and length is the length of the original Monte Carlo
+1744        chain to be reconstructed.
+1745    """
+1746    samples, length = random_numbers.shape
+1747    if samples != len(boots) - 1:
+1748        raise ValueError("Random numbers do not have the correct shape.")
+1749
+1750    if samples < length:
+1751        raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.")
+1752
+1753    proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length
+1754
+1755    samples = scipy.linalg.lstsq(proj, boots[1:])[0]
+1756    ret = Obs([samples], [name])
+1757    ret._value = boots[0]
+1758    return ret
 
    
 
 
@@ -5911,34 +6177,34 @@ chain to be reconstructed.
    - 
    1675def merge_obs(list_of_obs):
-1676    """Combine all observables in list_of_obs into one new observable
-1677
-1678    Parameters
-1679    ----------
-1680    list_of_obs : list
-1681        list of the Obs object to be combined
-1682
-1683    Notes
-1684    -----
-1685    It is not possible to combine obs which are based on the same replicum
-1686    """
-1687    replist = [item for obs in list_of_obs for item in obs.names]
-1688    if (len(replist) == len(set(replist))) is False:
-1689        raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist)))
-1690    if any([len(o.cov_names) for o in list_of_obs]):
-1691        raise Exception('Not possible to merge data that contains covobs!')
-1692    new_dict = {}
-1693    idl_dict = {}
-1694    for o in list_of_obs:
-1695        new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0)
-1696                        for key in set(o.deltas) | set(o.r_values)})
-1697        idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)})
-1698
-1699    names = sorted(new_dict.keys())
-1700    o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names])
-1701    o.reweighted = np.max([oi.reweighted for oi in list_of_obs])
-1702    return o
+            
    1761def merge_obs(list_of_obs):
+1762    """Combine all observables in list_of_obs into one new observable
+1763
+1764    Parameters
+1765    ----------
+1766    list_of_obs : list
+1767        list of the Obs object to be combined
+1768
+1769    Notes
+1770    -----
+1771    It is not possible to combine obs which are based on the same replicum
+1772    """
+1773    replist = [item for obs in list_of_obs for item in obs.names]
+1774    if (len(replist) == len(set(replist))) is False:
+1775        raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist)))
+1776    if any([len(o.cov_names) for o in list_of_obs]):
+1777        raise Exception('Not possible to merge data that contains covobs!')
+1778    new_dict = {}
+1779    idl_dict = {}
+1780    for o in list_of_obs:
+1781        new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0)
+1782                        for key in set(o.deltas) | set(o.r_values)})
+1783        idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)})
+1784
+1785    names = sorted(new_dict.keys())
+1786    o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names])
+1787    o.reweighted = np.max([oi.reweighted for oi in list_of_obs])
+1788    return o
 
    
 
 
@@ -5969,47 +6235,47 @@ list of the Obs object to be combined
    - 
    1705def cov_Obs(means, cov, name, grad=None):
-1706    """Create an Obs based on mean(s) and a covariance matrix
-1707
-1708    Parameters
-1709    ----------
-1710    mean : list of floats or float
-1711        N mean value(s) of the new Obs
-1712    cov : list or array
-1713        2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
-1714    name : str
-1715        identifier for the covariance matrix
-1716    grad : list or array
-1717        Gradient of the Covobs wrt. the means belonging to cov.
-1718    """
-1719
-1720    def covobs_to_obs(co):
-1721        """Make an Obs out of a Covobs
-1722
-1723        Parameters
-1724        ----------
-1725        co : Covobs
-1726            Covobs to be embedded into the Obs
-1727        """
-1728        o = Obs([], [], means=[])
-1729        o._value = co.value
-1730        o.names.append(co.name)
-1731        o._covobs[co.name] = co
-1732        o._dvalue = np.sqrt(co.errsq())
-1733        return o
-1734
-1735    ol = []
-1736    if isinstance(means, (float, int)):
-1737        means = [means]
-1738
-1739    for i in range(len(means)):
-1740        ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad)))
-1741    if ol[0].covobs[name].N != len(means):
-1742        raise Exception('You have to provide %d mean values!' % (ol[0].N))
-1743    if len(ol) == 1:
-1744        return ol[0]
-1745    return ol
+            
    1791def cov_Obs(means, cov, name, grad=None):
+1792    """Create an Obs based on mean(s) and a covariance matrix
+1793
+1794    Parameters
+1795    ----------
+1796    mean : list of floats or float
+1797        N mean value(s) of the new Obs
+1798    cov : list or array
+1799        2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
+1800    name : str
+1801        identifier for the covariance matrix
+1802    grad : list or array
+1803        Gradient of the Covobs wrt. the means belonging to cov.
+1804    """
+1805
+1806    def covobs_to_obs(co):
+1807        """Make an Obs out of a Covobs
+1808
+1809        Parameters
+1810        ----------
+1811        co : Covobs
+1812            Covobs to be embedded into the Obs
+1813        """
+1814        o = Obs([], [], means=[])
+1815        o._value = co.value
+1816        o.names.append(co.name)
+1817        o._covobs[co.name] = co
+1818        o._dvalue = np.sqrt(co.errsq())
+1819        return o
+1820
+1821    ol = []
+1822    if isinstance(means, (float, int)):
+1823        means = [means]
+1824
+1825    for i in range(len(means)):
+1826        ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad)))
+1827    if ol[0].covobs[name].N != len(means):
+1828        raise Exception('You have to provide %d mean values!' % (ol[0].N))
+1829    if len(ol) == 1:
+1830        return ol[0]
+1831    return ol
 
    
 
 
diff --git a/docs/pyerrors/roots.html b/docs/pyerrors/roots.html
index 4a71cb86..b12b5f02 100644
--- a/docs/pyerrors/roots.html
+++ b/docs/pyerrors/roots.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.roots API documentation
 
     
diff --git a/docs/pyerrors/special.html b/docs/pyerrors/special.html
index b1c2aa0c..99bfc775 100644
--- a/docs/pyerrors/special.html
+++ b/docs/pyerrors/special.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.special API documentation
 
     
diff --git a/docs/pyerrors/version.html b/docs/pyerrors/version.html
index 7020c7a1..44587cdb 100644
--- a/docs/pyerrors/version.html
+++ b/docs/pyerrors/version.html
@@ -3,7 +3,7 @@
 
     
     
-    
+    
     pyerrors.version API documentation
 
     
diff --git a/docs/search.js b/docs/search.js
index 5a1f6df3..05036446 100644
--- a/docs/search.js
+++ b/docs/search.js
@@ -1,6 +1,6 @@
 window.pdocSearch = (function(){
 /** elasticlunr - http://weixsong.github.io * Copyright (C) 2017 Oliver Nightingale * Copyright (C) 2017 Wei Song * MIT Licensed */!function(){function e(e){if(null===e||"object"!=typeof e)return e;var t=e.constructor();for(var n in e)e.hasOwnProperty(n)&&(t[n]=e[n]);return t}var t=function(e){var n=new t.Index;return n.pipeline.add(t.trimmer,t.stopWordFilter,t.stemmer),e&&e.call(n,n),n};t.version="0.9.5",lunr=t,t.utils={},t.utils.warn=function(e){return function(t){e.console&&console.warn&&console.warn(t)}}(this),t.utils.toString=function(e){return void 0===e||null===e?"":e.toString()},t.EventEmitter=function(){this.events={}},t.EventEmitter.prototype.addListener=function(){var e=Array.prototype.slice.call(arguments),t=e.pop(),n=e;if("function"!=typeof t)throw new TypeError("last argument must be a function");n.forEach(function(e){this.hasHandler(e)||(this.events[e]=[]),this.events[e].push(t)},this)},t.EventEmitter.prototype.removeListener=function(e,t){if(this.hasHandler(e)){var n=this.events[e].indexOf(t);-1!==n&&(this.events[e].splice(n,1),0==this.events[e].length&&delete this.events[e])}},t.EventEmitter.prototype.emit=function(e){if(this.hasHandler(e)){var t=Array.prototype.slice.call(arguments,1);this.events[e].forEach(function(e){e.apply(void 0,t)},this)}},t.EventEmitter.prototype.hasHandler=function(e){return e in this.events},t.tokenizer=function(e){if(!arguments.length||null===e||void 0===e)return[];if(Array.isArray(e)){var n=e.filter(function(e){return null===e||void 0===e?!1:!0});n=n.map(function(e){return t.utils.toString(e).toLowerCase()});var i=[];return n.forEach(function(e){var n=e.split(t.tokenizer.seperator);i=i.concat(n)},this),i}return e.toString().trim().toLowerCase().split(t.tokenizer.seperator)},t.tokenizer.defaultSeperator=/[\s\-]+/,t.tokenizer.seperator=t.tokenizer.defaultSeperator,t.tokenizer.setSeperator=function(e){null!==e&&void 0!==e&&"object"==typeof e&&(t.tokenizer.seperator=e)},t.tokenizer.resetSeperator=function(){t.tokenizer.seperator=t.tokenizer.defaultSeperator},t.tokenizer.getSeperator=function(){return t.tokenizer.seperator},t.Pipeline=function(){this._queue=[]},t.Pipeline.registeredFunctions={},t.Pipeline.registerFunction=function(e,n){n in t.Pipeline.registeredFunctions&&t.utils.warn("Overwriting existing registered function: "+n),e.label=n,t.Pipeline.registeredFunctions[n]=e},t.Pipeline.getRegisteredFunction=function(e){return e in t.Pipeline.registeredFunctions!=!0?null:t.Pipeline.registeredFunctions[e]},t.Pipeline.warnIfFunctionNotRegistered=function(e){var n=e.label&&e.label in this.registeredFunctions;n||t.utils.warn("Function is not registered with pipeline. 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configuration"),this.buildDefaultConfig(n)}},t.Configuration.prototype.buildDefaultConfig=function(e){this.reset(),e.forEach(function(e){this.config[e]={boost:1,bool:"OR",expand:!1}},this)},t.Configuration.prototype.buildUserConfig=function(e,n){var i="OR",o=!1;if(this.reset(),"bool"in e&&(i=e.bool||i),"expand"in e&&(o=e.expand||o),"fields"in e)for(var r in e.fields)if(n.indexOf(r)>-1){var s=e.fields[r],u=o;void 0!=s.expand&&(u=s.expand),this.config[r]={boost:s.boost||0===s.boost?s.boost:1,bool:s.bool||i,expand:u}}else t.utils.warn("field name in user configuration not found in index instance fields");else this.addAllFields2UserConfig(i,o,n)},t.Configuration.prototype.addAllFields2UserConfig=function(e,t,n){n.forEach(function(n){this.config[n]={boost:1,bool:e,expand:t}},this)},t.Configuration.prototype.get=function(){return this.config},t.Configuration.prototype.reset=function(){this.config={}},lunr.SortedSet=function(){this.length=0,this.elements=[]},lunr.SortedSet.load=function(e){var t=new this;return t.elements=e,t.length=e.length,t},lunr.SortedSet.prototype.add=function(){var e,t;for(e=0;e1;){if(r===e)return o;e>r&&(t=o),r>e&&(n=o),i=n-t,o=t+Math.floor(i/2),r=this.elements[o]}return r===e?o:-1},lunr.SortedSet.prototype.locationFor=function(e){for(var t=0,n=this.elements.length,i=n-t,o=t+Math.floor(i/2),r=this.elements[o];i>1;)e>r&&(t=o),r>e&&(n=o),i=n-t,o=t+Math.floor(i/2),r=this.elements[o];return r>e?o:e>r?o+1:void 0},lunr.SortedSet.prototype.intersect=function(e){for(var t=new lunr.SortedSet,n=0,i=0,o=this.length,r=e.length,s=this.elements,u=e.elements;;){if(n>o-1||i>r-1)break;s[n]!==u[i]?s[n]u[i]&&i++:(t.add(s[n]),n++,i++)}return t},lunr.SortedSet.prototype.clone=function(){var e=new lunr.SortedSet;return e.elements=this.toArray(),e.length=e.elements.length,e},lunr.SortedSet.prototype.union=function(e){var t,n,i;this.length>=e.length?(t=this,n=e):(t=e,n=this),i=t.clone();for(var o=0,r=n.toArray();oWhat is pyerrors?\n\n
    pyerrors is a python package for error computation and propagation of Markov chain Monte Carlo data.\nIt is based on the gamma method arXiv:hep-lat/0306017. Some of its features are:
    \n\n
      \n
    • automatic differentiation for exact linear error propagation as suggested in arXiv:1809.01289 (partly based on the autograd package).
      • \n
    • treatment of slow modes in the simulation as suggested in arXiv:1009.5228.
      • \n
    • coherent error propagation for data from different Markov chains.
      • \n
    • non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in arXiv:1809.01289.
      • \n
    • real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...).
      • \n
    \n\n
    More detailed examples can found in the GitHub repository \"badge\".
    \n\n
    If you use pyerrors for research that leads to a publication please consider citing:
    \n\n
      \n
    • Fabian Joswig, Simon Kuberski, Justus T. Kuhlmann, Jan Neuendorf, pyerrors: a python framework for error analysis of Monte Carlo data. Comput.Phys.Commun. 288 (2023) 108750.
      • \n
    • Ulli Wolff, Monte Carlo errors with less errors. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum).
      • \n
    • Alberto Ramos, Automatic differentiation for error analysis of Monte Carlo data. Comput.Phys.Commun. 238 (2019) 19-35.
      • \n
    \n\n
    and
    \n\n
      \n
    • Stefan Schaefer, Rainer Sommer, Francesco Virotta, Critical slowing down and error analysis in lattice QCD simulations. Nucl.Phys.B 845 (2011) 93-119.
      • \n
    \n\n
    where applicable.
    \n\n
    There exist similar publicly available implementations of gamma method error analysis suites in Fortran, Julia and Python.
    \n\n
    Installation
    \n\n
    Install the most recent release using pip and pypi:
    \n\n
    \n
    python -m pip install pyerrors     # Fresh install\npython -m pip install -U pyerrors  # Update\n
    \n
    \n\n
    Install the most recent release using conda and conda-forge:
    \n\n
    \n
    conda install -c conda-forge pyerrors  # Fresh install\nconda update -c conda-forge pyerrors   # Update\n
    \n
    \n\n
    Install the current develop version:
    \n\n
    \n
    python -m pip install -U --no-deps --force-reinstall git+https://github.com/fjosw/pyerrors.git@develop\n
    \n
    \n\n
    (Also works for any feature branch).
    \n\n
    Basic example
    \n\n
    \n
    import numpy as np\nimport pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name']) # Initialize an Obs object\nmy_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object\nmy_new_obs.gamma_method()                     # Estimate the statistical error\nprint(my_new_obs)                             # Print the result to stdout\n> 0.31498(72)\n
    \n
    \n\n
    The Obs class
    \n\n
    pyerrors introduces a new datatype, Obs, which simplifies error propagation and estimation for auto- and cross-correlated data.\nAn Obs object can be initialized with two arguments, the first is a list containing the samples for an observable from a Monte Carlo chain.\nThe samples can either be provided as python list or as numpy array.\nThe second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble. It is crucial for the correct error propagation that observations from the same Monte Carlo history are labeled with the same name. See Multiple ensembles/replica for details.
    \n\n
    \n
    import pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name'])\n
    \n
    \n\n
    Error propagation
    \n\n
    When performing mathematical operations on Obs objects the correct error propagation is intrinsically taken care of using a first order Taylor expansion\n$$\\delta_f^i=\\sum_\\alpha \\bar{f}_\\alpha \\delta_\\alpha^i\\,,\\quad \\delta_\\alpha^i=a_\\alpha^i-\\bar{a}_\\alpha\\,,$$\nas introduced in arXiv:hep-lat/0306017.\nThe required derivatives $\\bar{f}_\\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in arXiv:1809.01289.
    \n\n
    The Obs class is designed such that mathematical numpy functions can be used on Obs just as for regular floats.
    \n\n
    \n
    import numpy as np\nimport pyerrors as pe\n\nmy_obs1 = pe.Obs([samples1], ['ensemble_name'])\nmy_obs2 = pe.Obs([samples2], ['ensemble_name'])\n\nmy_sum = my_obs1 + my_obs2\n\nmy_m_eff = np.log(my_obs1 / my_obs2)\n\niamzero = my_m_eff - my_m_eff\n# Check that value and fluctuations are zero within machine precision\nprint(iamzero == 0.0)\n> True\n
    \n
    \n\n
    Error estimation
    \n\n
    The error estimation within pyerrors is based on the gamma method introduced in arXiv:hep-lat/0306017.\nAfter having arrived at the derived quantity of interest the gamma_method can be called as detailed in the following example.
    \n\n
    \n
    my_sum.gamma_method()\nprint(my_sum)\n> 1.70(57)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 5.72046658e-01 +/- 7.56746598e-02 (33.650%)\n>  t_int         2.71422900e+00 +/- 6.40320983e-01 S = 2.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n
    The gamma_method is not automatically called after every intermediate step in order to prevent computational overhead.
    \n\n
    We use the following definition of the integrated autocorrelation time established in Madras & Sokal 1988\n$$\\tau_\\mathrm{int}=\\frac{1}{2}+\\sum_{t=1}^{W}\\rho(t)\\geq \\frac{1}{2}\\,.$$\nThe window $W$ is determined via the automatic windowing procedure described in arXiv:hep-lat/0306017.\nThe standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the gamma_method as parameter.
    \n\n
    \n
    my_sum.gamma_method(S=3.0)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.30675201e-01 +/- 1.04585650e-01 (37.099%)\n>  t_int         3.29909703e+00 +/- 9.77310102e-01 S = 3.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n
    The integrated autocorrelation time $\\tau_\\mathrm{int}$ and the autocorrelation function $\\rho(W)$ can be monitored via the methods pyerrors.obs.Obs.plot_tauint and pyerrors.obs.Obs.plot_rho.
    \n\n
    If the parameter $S$ is set to zero it is assumed that the dataset does not exhibit any autocorrelation and the window size is chosen to be zero.\nIn this case the error estimate is identical to the sample standard error.
    \n\n
    Exponential tails
    \n\n
    Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\\rho$ as suggested in arXiv:1009.5228. The longest autocorrelation time in the history, $\\tau_\\mathrm{exp}$, can be passed to the gamma_method as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.
    \n\n
    \n
    my_sum.gamma_method(tau_exp=7.2)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.28097762e-01 +/- 5.79077524e-02 (36.947%)\n>  t_int         3.27218667e+00 +/- 7.99583654e-01 tau_exp = 7.20,  N_sigma = 1\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n
    For the full API see pyerrors.obs.Obs.gamma_method.
    \n\n
    Multiple ensembles/replica
    \n\n
    Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their name.
    \n\n
    \n
    obs1 = pe.Obs([samples1], ['ensemble1'])\nobs2 = pe.Obs([samples2], ['ensemble2'])\n\nmy_sum = obs1 + obs2\nmy_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 2 ensembles:\n>   \u00b7 Ensemble 'ensemble1' : 1000 configurations (from 1 to 1000)\n>   \u00b7 Ensemble 'ensemble2' : 500 configurations (from 1 to 500)\n
    \n
    \n\n
    Observables from the same Monte Carlo chain have to be initialized with the same name for correct error propagation. If different names were used in this case the data would be treated as statistically independent resulting in loss of relevant information and a potential over or under estimate of the statistical error.
    \n\n
    pyerrors identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar | in the name of the data set.
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    \n
    obs1 = pe.Obs([samples1], ['ensemble1|r01'])\nobs2 = pe.Obs([samples2], ['ensemble1|r02'])\n\n> my_sum = obs1 + obs2\n> my_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1'\n>     \u00b7 Replicum 'r01' : 1000 configurations (from 1 to 1000)\n>     \u00b7 Replicum 'r02' : 500 configurations (from 1 to 500)\n
    \n
    \n\n
    Error estimation for multiple ensembles
    \n\n
    In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example.
    \n\n
    \n
    pe.Obs.S_dict['ensemble1'] = 2.5\npe.Obs.tau_exp_dict['ensemble2'] = 8.0\npe.Obs.tau_exp_dict['ensemble3'] = 2.0\n
    \n
    \n\n
    In case the gamma_method is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.\nPassing arguments to the gamma_method still dominates over the dictionaries.
    \n\n
    Irregular Monte Carlo chains
    \n\n
    Obs objects defined on irregular Monte Carlo chains can be initialized with the parameter idl.
    \n\n
    \n
    # Observable defined on configurations 20 to 519\nobs1 = pe.Obs([samples1], ['ensemble1'], idl=[range(20, 520)])\nobs1.details()\n> Result         9.98319881e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 20 to 519)\n\n# Observable defined on every second configuration between 5 and 1003\nobs2 = pe.Obs([samples2], ['ensemble1'], idl=[range(5, 1005, 2)])\nobs2.details()\n> Result         9.99100712e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 5 to 1003 in steps of 2)\n\n# Observable defined on configurations 2, 9, 28, 29 and 501\nobs3 = pe.Obs([samples3], ['ensemble1'], idl=[[2, 9, 28, 29, 501]])\nobs3.details()\n> Result         1.01718064e+00\n> 5 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 5 configurations (irregular range)\n
    \n
    \n\n
    Obs objects defined on regular and irregular histories of the same ensemble can be combined with each other and the correct error propagation and estimation is automatically taken care of.
    \n\n
    Warning: Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.\nMake sure to check the autocorrelation time with e.g. pyerrors.obs.Obs.plot_rho or pyerrors.obs.Obs.plot_tauint.
    \n\n
    For the full API see pyerrors.obs.Obs.
    \n\n
    Correlators
    \n\n
    When one is not interested in single observables but correlation functions, pyerrors offers the Corr class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize a Corr objects one needs to arrange the data as a list of Obs
    \n\n
    \n
    my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0      0.7957(80)\n> 1      0.5156(51)\n> 2      0.3227(33)\n> 3      0.2041(21)\n
    \n
    \n\n
    In case the correlation functions are not defined on the outermost timeslices, for example because of fixed boundary conditions, a padding can be introduced.
    \n\n
    \n
    my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3], padding=[1, 1])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0\n> 1      0.7957(80)\n> 2      0.5156(51)\n> 3      0.3227(33)\n> 4      0.2041(21)\n> 5\n
    \n
    \n\n
    The individual entries of a correlator can be accessed via slicing
    \n\n
    \n
    print(my_corr[3])\n> 0.3227(33)\n
    \n
    \n\n
    Error propagation with the Corr class works very similar to Obs objects. Mathematical operations are overloaded and Corr objects can be computed together with other Corr objects, Obs objects or real numbers and integers.
    \n\n
    \n
    my_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr\n
    \n
    \n\n
    pyerrors provides the user with a set of regularly used methods for the manipulation of correlator objects:
    \n\n
      \n
    • Corr.gamma_method applies the gamma method to all entries of the correlator.
      • \n
    • Corr.m_eff to construct effective masses. Various variants for periodic and fixed temporal boundary conditions are available.
      • \n
    • Corr.deriv returns the first derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
      • \n
    • Corr.second_deriv returns the second derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
      • \n
    • Corr.symmetric symmetrizes parity even correlations functions, assuming periodic boundary conditions.
      • \n
    • Corr.anti_symmetric anti-symmetrizes parity odd correlations functions, assuming periodic boundary conditions.
      • \n
    • Corr.T_symmetry averages a correlator with its time symmetry partner, assuming fixed boundary conditions.
      • \n
    • Corr.plateau extracts a plateau value from the correlator in a given range.
      • \n
    • Corr.roll periodically shifts the correlator.
      • \n
    • Corr.reverse reverses the time ordering of the correlator.
      • \n
    • Corr.correlate constructs a disconnected correlation function from the correlator and another Corr or Obs object.
      • \n
    • Corr.reweight reweights the correlator.
      • \n
    \n\n
    pyerrors can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (see pyerrors.correlators.Corr.GEVP).
    \n\n
    For the full API see pyerrors.correlators.Corr.
    \n\n
    Complex valued observables
    \n\n
    pyerrors can handle complex valued observables via the class pyerrors.obs.CObs.\nCObs are initialized with a real and an imaginary part which both can be Obs valued.
    \n\n
    \n
    my_real_part = pe.Obs([samples1], ['ensemble1'])\nmy_imag_part = pe.Obs([samples2], ['ensemble1'])\n\nmy_cobs = pe.CObs(my_real_part, my_imag_part)\nmy_cobs.gamma_method()\nprint(my_cobs)\n> (0.9959(91)+0.659(28)j)\n
    \n
    \n\n
    Elementary mathematical operations are overloaded and samples are properly propagated as for the Obs class.
    \n\n
    \n
    my_derived_cobs = (my_cobs + my_cobs.conjugate()) / np.abs(my_cobs)\nmy_derived_cobs.gamma_method()\nprint(my_derived_cobs)\n> (1.668(23)+0.0j)\n
    \n
    \n\n
    The Covobs class
    \n\n
    In many projects, auxiliary data that is not based on Monte Carlo chains enters. Examples are experimentally determined mesons masses which are used to set the scale or renormalization constants. These numbers come with an error that has to be propagated through the analysis. The Covobs class allows to define such quantities in pyerrors. Furthermore, external input might consist of correlated quantities. An example are the parameters of an interpolation formula, which are defined via mean values and a covariance matrix between all parameters. The contribution of the interpolation formula to the error of a derived quantity therefore might depend on the complete covariance matrix.
    \n\n
    This concept is built into the definition of Covobs. In pyerrors, external input is defined by $M$ mean values, a $M\\times M$ covariance matrix, where $M=1$ is permissible, and a name that uniquely identifies the covariance matrix. Below, we define the pion mass, based on its mean value and error, 134.9768(5). Note, that the square of the error enters cov_Obs, since the second argument of this function is the covariance matrix of the Covobs.
    \n\n
    \n
    import pyerrors.obs as pe\n\nmpi = pe.cov_Obs(134.9768, 0.0005**2, 'pi^0 mass')\nmpi.gamma_method()\nmpi.details()\n> Result         1.34976800e+02 +/- 5.00000000e-04 +/- 0.00000000e+00 (0.000%)\n>  pi^0 mass     5.00000000e-04\n> 0 samples in 1 ensemble:\n>   \u00b7 Covobs   'pi^0 mass'\n
    \n
    \n\n
    The resulting object mpi is an Obs that contains a Covobs. In the following, it may be handled as any other Obs. The contribution of the covariance matrix to the error of an Obs is determined from the $M \\times M$ covariance matrix $\\Sigma$ and the gradient of the Obs with respect to the external quantities, which is the $1\\times M$ Jacobian matrix $J$, via\n$$s = \\sqrt{J^T \\Sigma J}\\,,$$\nwhere the Jacobian is computed for each derived quantity via automatic differentiation.
    \n\n
    Correlated auxiliary data is defined similarly to above, e.g., via
    \n\n
    \n
    RAP = pe.cov_Obs([16.7457, -19.0475], [[3.49591, -6.07560], [-6.07560, 10.5834]], 'R_AP, 1906.03445, (5.3a)')\nprint(RAP)\n> [Obs[16.7(1.9)], Obs[-19.0(3.3)]]\n
    \n
    \n\n
    where RAP now is a list of two Obs that contains the two correlated parameters.
    \n\n
    Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the Covobs class allows to quote the derivative of a result with respect to the external quantities. If these derivatives are published together with the result, small shifts in the definition of external quantities, e.g., the definition of the physical point, can be performed a posteriori based on the published information. This may help to compare results of different groups. The gradient of an Obs o with respect to a covariance matrix with the identifying string k may be accessed via
    \n\n
    \n
    o.covobs[k].grad\n
    \n
    \n\n
    Error propagation in iterative algorithms
    \n\n
    pyerrors supports exact linear error propagation for iterative algorithms like various variants of non-linear least squares fits or root finding. The derivatives required for the error propagation are calculated as described in arXiv:1809.01289.
    \n\n
    Least squares fits
    \n\n
    Standard non-linear least square fits with errors on the dependent but not the independent variables can be performed with pyerrors.fits.least_squares. As default solver the Levenberg-Marquardt algorithm implemented in scipy is used.
    \n\n
    Fit functions have to be of the following form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n    return a[1] * anp.exp(-a[0] * x)\n
    \n
    \n\n
    It is important that numerical functions refer to autograd.numpy instead of numpy for the automatic differentiation in iterative algorithms to work properly.
    \n\n
    Fits can then be performed via
    \n\n
    \n
    fit_result = pe.fits.least_squares(x, y, func)\nprint("\\n", fit_result)\n> Fit with 2 parameters\n> Method: Levenberg-Marquardt\n> `ftol` termination condition is satisfied.\n> chisquare/d.o.f.: 0.9593035785160936\n\n>  Goodness of fit:\n> \u03c7\u00b2/d.o.f. = 0.959304\n> p-value   = 0.5673\n> Fit parameters:\n> 0      0.0548(28)\n> 1      1.933(64)\n
    \n
    \n\n
    where x is a list or numpy.array of floats and y is a list or numpy.array of Obs.
    \n\n
    Data stored in Corr objects can be fitted directly using the Corr.fit method.
    \n\n
    \n
    my_corr = pe.Corr(y)\nfit_result = my_corr.fit(func, fitrange=[12, 25])\n
    \n
    \n\n
    this can simplify working with absolute fit ranges and takes care of gaps in the data automatically.
    \n\n
    For fit functions with multiple independent variables the fit function can be of the form
    \n\n
    \n
    def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n
    pyerrors also supports correlated fits which can be triggered via the parameter correlated_fit=True.\nDetails about how the required covariance matrix is estimated can be found in pyerrors.obs.covariance.\nDirect visualizations of the performed fits can be triggered via resplot=True or qqplot=True.
    \n\n
    For all available options including combined fits to multiple datasets see pyerrors.fits.least_squares.
    \n\n
    Total least squares fits
    \n\n
    pyerrors can also fit data with errors on both the dependent and independent variables using the total least squares method also referred to as orthogonal distance regression as implemented in scipy, see pyerrors.fits.least_squares. The syntax is identical to the standard least squares case, the only difference being that x also has to be a list or numpy.array of Obs.
    \n\n
    For the full API see pyerrors.fits for fits and pyerrors.roots for finding roots of functions.
    \n\n
    Matrix operations
    \n\n
    pyerrors provides wrappers for Obs- and CObs-valued matrix operations based on numpy.linalg. The supported functions include:
    \n\n
      \n
    • inv for the matrix inverse.
      • \n
    • cholseky for the Cholesky decomposition.
      • \n
    • det for the matrix determinant.
      • \n
    • eigh for eigenvalues and eigenvectors of hermitean matrices.
      • \n
    • eig for eigenvalues of general matrices.
      • \n
    • pinv for the Moore-Penrose pseudoinverse.
      • \n
    • svd for the singular-value-decomposition.
      • \n
    \n\n
    For the full API see pyerrors.linalg.
    \n\n
    Export data
    \n\n
    \n\n
    The preferred exported file format within pyerrors is json.gz. Files written to this format are valid JSON files that have been compressed using gzip. The structure of the content is inspired by the dobs format of the ALPHA collaboration. The aim of the format is to facilitate the storage of data in a self-contained way such that, even years after the creation of the file, it is possible to extract all necessary information:
    \n\n
      \n
    • What observables are stored? Possibly: How exactly are they defined.
      • \n
    • How does each single ensemble or external quantity contribute to the error of the observable?
      • \n
    • Who did write the file when and on which machine?
      • \n
    \n\n
    This can be achieved by storing all information in one single file. The export routines of pyerrors are written such that as much information as possible is written automatically as described in the following example
    \n\n
    \n
    my_obs = pe.Obs([samples], ["test_ensemble"])\nmy_obs.tag = "My observable"\n\npe.input.json.dump_to_json(my_obs, "test_output_file", description="This file contains a test observable")\n# For a single observable one can equivalently use the class method dump\nmy_obs.dump("test_output_file", description="This file contains a test observable")\n\ncheck = pe.input.json.load_json("test_output_file")\n\nprint(my_obs == check)\n> True\n
    \n
    \n\n
    The format also allows to directly write out the content of Corr objects or lists and arrays of Obs objects by passing the desired data to pyerrors.input.json.dump_to_json.
    \n\n
    json.gz format specification
    \n\n
    The first entries of the file provide optional auxiliary information:
    \n\n
      \n
    • program is a string that indicates which program was used to write the file.
      • \n
    • version is a string that specifies the version of the format.
      • \n
    • who is a string that specifies the user name of the creator of the file.
      • \n
    • date is a string and contains the creation date of the file.
      • \n
    • host is a string and contains the hostname of the machine where the file has been written.
      • \n
    • description contains information on the content of the file. This field is not filled automatically in pyerrors. The user is advised to provide as detailed information as possible in this field. Examples are: Input files of measurements or simulations, LaTeX formulae or references to publications to specify how the observables have been computed, details on the analysis strategy, ... This field may be any valid JSON type. Strings, arrays or objects (equivalent to dicts in python) are well suited to provide information.
      • \n
    \n\n
    The only necessary entry of the file is the field\n-obsdata, an array that contains the actual data.
    \n\n
    Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of Obs, list, numpy.ndarray, Corr. All Obs inside a structure (with dimension > 0) have to be defined on the same set of configurations. Different structures, that are represented by entries of the array obsdata, are treated independently. Each entry of the array obsdata has the following required entries:
    \n\n
      \n
    • type is a string that specifies the type of the structure. This allows to parse the content to the correct form after reading the file. It is always possible to interpret the content as list of Obs.
      • \n
    • value is an array that contains the mean values of the Obs inside the structure.\nThe following entries are optional:
      • \n
    • layout is a string that specifies the layout of multi-dimensional structures. Examples are \"2, 2\" for a 2x2 dimensional matrix or  \"64, 4, 4\" for a Corr with $T=64$ and 4x4 matrices on each time slices. \"1\" denotes a single Obs. Multi-dimensional structures are stored in row-major format (see below).
      • \n
    • tag is any JSON type. It contains additional information concerning the structure. The tag of an Obs in pyerrors is written here.
      • \n
    • reweighted is a Bool that may be used to specify, whether the Obs in the structure have been reweighted.
      • \n
    • data is an array that contains the data from MC chains. We will define it below.
      • \n
    • cdata is an array that contains the data from external quantities with an error (Covobs in pyerrors). We will define it below.
      • \n
    \n\n
    The array data contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains:
    \n\n
      \n
    • id, a string that contains the name of the ensemble
      • \n
    • replica, an array that contains an entry per replica of the ensemble.
      • \n
    \n\n
    Each entry of replica contains\nname, a string that contains the name of the replica\ndeltas, an array that contains the actual data.
    \n\n
    Each entry in deltas corresponds to one configuration of the replica and has $1+N$ many entries. The first entry is an integer that specifies the configuration number that, together with ensemble and replica name, may be used to uniquely identify the configuration on which the data has been obtained. The following N entries specify the deltas, i.e., the deviation of the observable from the mean value on this configuration, of each Obs inside the structure. Multi-dimensional structures are stored in a row-major format. For primary observables, such as correlation functions, $value + delta_i$ matches the primary data obtained on the configuration.
    \n\n
    The array cdata contains information about the contribution of auxiliary observables, represented by Covobs in pyerrors, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains:
    \n\n
      \n
    • id, a string that identifies the covariance matrix
      • \n
    • layout, a string that defines the dimensions of the $M\\times M$ covariance matrix (has to be \"M, M\" or \"1\").
      • \n
    • cov, an array that contains the $M\\times M$ many entries of the covariance matrix, stored in row-major format.
      • \n
    • grad, an array that contains N entries, one for each Obs inside the structure. Each entry itself is an array, that contains the M gradients of the Nth observable  with respect to the quantity that corresponds to the Mth diagonal entry of the covariance matrix.
      • \n
    \n\n
    A JSON schema that may be used to verify the correctness of a file with respect to the format definition is stored in ./examples/json_schema.json. The schema is a self-descriptive format definition and contains an exemplary file.
    \n\n
    Julia I/O routines for the json.gz format, compatible with ADerrors.jl, can be found here.
    \n"}, "pyerrors.correlators": {"fullname": "pyerrors.correlators", "modulename": "pyerrors.correlators", "kind": "module", "doc": "
    \n"}, "pyerrors.correlators.Corr": {"fullname": "pyerrors.correlators.Corr", "modulename": "pyerrors.correlators", "qualname": "Corr", "kind": "class", "doc": "
    The class for a correlator (time dependent sequence of pe.Obs).
    \n\n
    Everything, this class does, can be achieved using lists or arrays of Obs.\nBut it is simply more convenient to have a dedicated object for correlators.\nOne often wants to add or multiply correlators of the same length at every timeslice and it is inconvenient\nto iterate over all timeslices for every operation. This is especially true, when dealing with matrices.
    \n\n
    The correlator can have two types of content: An Obs at every timeslice OR a matrix at every timeslice.\nOther dependency (eg. spatial) are not supported.
    \n\n
    The Corr class can also deal with missing measurements or paddings for fixed boundary conditions.\nThe missing entries are represented via the None object.
    \n\n
    Initialization
    \n\n
    A simple correlator can be initialized with a list or a one-dimensional array of Obs or Cobs
    \n\n
    \n
    corr11 = pe.Corr([obs1, obs2])\ncorr11 = pe.Corr(np.array([obs1, obs2]))\n
    \n
    \n\n
    A matrix-valued correlator can either be initialized via a two-dimensional array of Corr objects
    \n\n
    \n
    matrix_corr = pe.Corr(np.array([[corr11, corr12], [corr21, corr22]]))\n
    \n
    \n\n
    or alternatively via a three-dimensional array of Obs or CObs of shape (T, N, N) where T is\nthe temporal extent of the correlator and N is the dimension of the matrix.
    \n"}, "pyerrors.correlators.Corr.__init__": {"fullname": "pyerrors.correlators.Corr.__init__", "modulename": "pyerrors.correlators", "qualname": "Corr.__init__", "kind": "function", "doc": "
    Initialize a Corr object.
    \n\n
    Parameters
    \n\n
      \n
    • data_input (list or array):\nlist of Obs or list of arrays of Obs or array of Corrs (see class docstring for details).
      • \n
    • padding (list, optional):\nList with two entries where the first labels the padding\nat the front of the correlator and the second the padding\nat the back.
      • \n
    • prange (list, optional):\nList containing the first and last timeslice of the plateau\nregion identified for this correlator.
      • \n
    \n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "pyerrors.correlators.Corr.tag": {"fullname": "pyerrors.correlators.Corr.tag", "modulename": "pyerrors.correlators", "qualname": "Corr.tag", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.content": {"fullname": "pyerrors.correlators.Corr.content", "modulename": "pyerrors.correlators", "qualname": "Corr.content", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.T": {"fullname": "pyerrors.correlators.Corr.T", "modulename": "pyerrors.correlators", "qualname": "Corr.T", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.prange": {"fullname": "pyerrors.correlators.Corr.prange", "modulename": "pyerrors.correlators", "qualname": "Corr.prange", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.reweighted": {"fullname": "pyerrors.correlators.Corr.reweighted", "modulename": "pyerrors.correlators", "qualname": "Corr.reweighted", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.gamma_method": {"fullname": "pyerrors.correlators.Corr.gamma_method", "modulename": "pyerrors.correlators", "qualname": "Corr.gamma_method", "kind": "function", "doc": "
    Apply the gamma method to the content of the Corr.
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.gm": {"fullname": "pyerrors.correlators.Corr.gm", "modulename": "pyerrors.correlators", "qualname": "Corr.gm", "kind": "function", "doc": "
    Apply the gamma method to the content of the Corr.
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.projected": {"fullname": "pyerrors.correlators.Corr.projected", "modulename": "pyerrors.correlators", "qualname": "Corr.projected", "kind": "function", "doc": "
    We need to project the Correlator with a Vector to get a single value at each timeslice.
    \n\n
    The method can use one or two vectors.\nIf two are specified it returns v1@G@v2 (the order might be very important.)\nBy default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to
    \n", "signature": "(self, vector_l=None, vector_r=None, normalize=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.item": {"fullname": "pyerrors.correlators.Corr.item", "modulename": "pyerrors.correlators", "qualname": "Corr.item", "kind": "function", "doc": "
    Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.
    \n\n
    Parameters
    \n\n
      \n
    • i (int):\nFirst index to be picked.
      • \n
    • j (int):\nSecond index to be picked.
      • \n
    \n", "signature": "(self, i, j):", "funcdef": "def"}, "pyerrors.correlators.Corr.plottable": {"fullname": "pyerrors.correlators.Corr.plottable", "modulename": "pyerrors.correlators", "qualname": "Corr.plottable", "kind": "function", "doc": "
    Outputs the correlator in a plotable format.
    \n\n
    Outputs three lists containing the timeslice index, the value on each\ntimeslice and the error on each timeslice.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.symmetric": {"fullname": "pyerrors.correlators.Corr.symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.symmetric", "kind": "function", "doc": "
    Symmetrize the correlator around x0=0.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.anti_symmetric": {"fullname": "pyerrors.correlators.Corr.anti_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.anti_symmetric", "kind": "function", "doc": "
    Anti-symmetrize the correlator around x0=0.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.is_matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.is_matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.is_matrix_symmetric", "kind": "function", "doc": "
    Checks whether a correlator matrices is symmetric on every timeslice.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.trace": {"fullname": "pyerrors.correlators.Corr.trace", "modulename": "pyerrors.correlators", "qualname": "Corr.trace", "kind": "function", "doc": "
    Calculates the per-timeslice trace of a correlator matrix.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.matrix_symmetric", "kind": "function", "doc": "
    Symmetrizes the correlator matrices on every timeslice.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.GEVP": {"fullname": "pyerrors.correlators.Corr.GEVP", "modulename": "pyerrors.correlators", "qualname": "Corr.GEVP", "kind": "function", "doc": "
    Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.
    \n\n
    The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the\nlargest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing
    \n\n
    \n
    C.GEVP(t0=2)[0]  # Ground state vector(s)\nC.GEVP(t0=2)[:3]  # Vectors for the lowest three states\n
    \n
    \n\n
    Parameters
    \n\n
      \n
    • t0 (int):\nThe time t0 for the right hand side of the GEVP according to $G(t)v_i=\\lambda_i G(t_0)v_i$
      • \n
    • ts (int):\nfixed time $G(t_s)v_i=\\lambda_i G(t_0)v_i$ if sort=None.\nIf sort=\"Eigenvector\" it gives a reference point for the sorting method.
      • \n
    • sort (string):\nIf this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.\n
        \n
      • \"Eigenvalue\": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice. (default)
        • \n
      • \"Eigenvector\": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.\nThe reference state is identified by its eigenvalue at $t=t_s$.
        • \n
      • None: The GEVP is solved only at ts, no sorting is necessary
        • \n
      • \n
    • vector_obs (bool):\nIf True, uncertainties are propagated in the eigenvector computation (default False).
      • \n
    \n\n
    Other Parameters
    \n\n
      \n
    • state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
      • \n
    • method (str):\nMethod used to solve the GEVP.\n
        \n
      • \"eigh\": Use scipy.linalg.eigh to solve the GEVP. (default for vector_obs=False)
        • \n
      • \"cholesky\": Use manually implemented solution via the Cholesky decomposition. Automatically chosen if vector_obs==True.
        • \n
      • \n
    \n", "signature": "(self, t0, ts=None, sort='Eigenvalue', vector_obs=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Eigenvalue": {"fullname": "pyerrors.correlators.Corr.Eigenvalue", "modulename": "pyerrors.correlators", "qualname": "Corr.Eigenvalue", "kind": "function", "doc": "
    Determines the eigenvalue of the GEVP by solving and projecting the correlator
    \n\n
    Parameters
    \n\n
      \n
    • state (int):\nThe state one is interested in ordered by energy. The lowest state is zero.
      • \n
    • All other parameters are identical to the ones of Corr.GEVP.
      • \n
    \n", "signature": "(self, t0, ts=None, state=0, sort='Eigenvalue', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Hankel": {"fullname": "pyerrors.correlators.Corr.Hankel", "modulename": "pyerrors.correlators", "qualname": "Corr.Hankel", "kind": "function", "doc": "
    Constructs an NxN Hankel matrix
    \n\n
    C(t) c(t+1) ... c(t+n-1)\nC(t+1) c(t+2) ... c(t+n)\n.................\nC(t+(n-1)) c(t+n) ... c(t+2(n-1))
    \n\n
    Parameters
    \n\n
      \n
    • N (int):\nDimension of the Hankel matrix
      • \n
    • periodic (bool, optional):\ndetermines whether the matrix is extended periodically
      • \n
    \n", "signature": "(self, N, periodic=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.roll": {"fullname": "pyerrors.correlators.Corr.roll", "modulename": "pyerrors.correlators", "qualname": "Corr.roll", "kind": "function", "doc": "
    Periodically shift the correlator by dt timeslices
    \n\n
    Parameters
    \n\n
      \n
    • dt (int):\nnumber of timeslices
      • \n
    \n", "signature": "(self, dt):", "funcdef": "def"}, "pyerrors.correlators.Corr.reverse": {"fullname": "pyerrors.correlators.Corr.reverse", "modulename": "pyerrors.correlators", "qualname": "Corr.reverse", "kind": "function", "doc": "
    Reverse the time ordering of the Corr
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.thin": {"fullname": "pyerrors.correlators.Corr.thin", "modulename": "pyerrors.correlators", "qualname": "Corr.thin", "kind": "function", "doc": "
    Thin out a correlator to suppress correlations
    \n\n
    Parameters
    \n\n
      \n
    • spacing (int):\nKeep only every 'spacing'th entry of the correlator
      • \n
    • offset (int):\nOffset the equal spacing
      • \n
    \n", "signature": "(self, spacing=2, offset=0):", "funcdef": "def"}, "pyerrors.correlators.Corr.correlate": {"fullname": "pyerrors.correlators.Corr.correlate", "modulename": "pyerrors.correlators", "qualname": "Corr.correlate", "kind": "function", "doc": "
    Correlate the correlator with another correlator or Obs
    \n\n
    Parameters
    \n\n
      \n
    • partner (Obs or Corr):\npartner to correlate the correlator with.\nCan either be an Obs which is correlated with all entries of the\ncorrelator or a Corr of same length.
      • \n
    \n", "signature": "(self, partner):", "funcdef": "def"}, "pyerrors.correlators.Corr.reweight": {"fullname": "pyerrors.correlators.Corr.reweight", "modulename": "pyerrors.correlators", "qualname": "Corr.reweight", "kind": "function", "doc": "
    Reweight the correlator.
    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl.
      • \n
    \n", "signature": "(self, weight, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.T_symmetry": {"fullname": "pyerrors.correlators.Corr.T_symmetry", "modulename": "pyerrors.correlators", "qualname": "Corr.T_symmetry", "kind": "function", "doc": "
    Return the time symmetry average of the correlator and its partner
    \n\n
    Parameters
    \n\n
      \n
    • partner (Corr):\nTime symmetry partner of the Corr
      • \n
    • parity (int):\nParity quantum number of the correlator, can be +1 or -1
      • \n
    \n", "signature": "(self, partner, parity=1):", "funcdef": "def"}, "pyerrors.correlators.Corr.deriv": {"fullname": "pyerrors.correlators.Corr.deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.deriv", "kind": "function", "doc": "
    Return the first derivative of the correlator with respect to x0.
    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, forward, backward, improved, log, default: symmetric
      • \n
    \n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.second_deriv": {"fullname": "pyerrors.correlators.Corr.second_deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.second_deriv", "kind": "function", "doc": "
    Return the second derivative of the correlator with respect to x0.
    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice:\n    - symmetric (default)\n        $$\\tilde{\\partial}^2_0 f(x_0) = f(x_0+1)-2f(x_0)+f(x_0-1)$$\n    - big_symmetric\n        $$\\partial^2_0 f(x_0) = \\frac{f(x_0+2)-2f(x_0)+f(x_0-2)}{4}$$\n    - improved\n        $$\\partial^2_0 f(x_0) = \\frac{-f(x_0+2) + 16 * f(x_0+1) - 30 * f(x_0) + 16 * f(x_0-1) - f(x_0-2)}{12}$$\n    - log\n        $$f(x) = \\tilde{\\partial}^2_0 log(f(x_0))+(\\tilde{\\partial}_0 log(f(x_0)))^2$$
      • \n
    \n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.m_eff": {"fullname": "pyerrors.correlators.Corr.m_eff", "modulename": "pyerrors.correlators", "qualname": "Corr.m_eff", "kind": "function", "doc": "
    Returns the effective mass of the correlator as correlator object
    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\nlog : uses the standard effective mass log(C(t) / C(t+1))\ncosh, periodic : Use periodicity of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.\nsinh : Use anti-periodicity of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.\nSee, e.g., arXiv:1205.5380\narccosh : Uses the explicit form of the symmetrized correlator (not recommended)\nlogsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
      • \n
    • guess (float):\nguess for the root finder, only relevant for the root variant
      • \n
    \n", "signature": "(self, variant='log', guess=1.0):", "funcdef": "def"}, "pyerrors.correlators.Corr.fit": {"fullname": "pyerrors.correlators.Corr.fit", "modulename": "pyerrors.correlators", "qualname": "Corr.fit", "kind": "function", "doc": "
    Fits function to the data
    \n\n
    Parameters
    \n\n
      \n
    • function (obj):\nfunction to fit to the data. See fits.least_squares for details.
      • \n
    • fitrange (list):\nTwo element list containing the timeslices on which the fit is supposed to start and stop.\nCaution: This range is inclusive as opposed to standard python indexing.\nfitrange=[4, 6] corresponds to the three entries 4, 5 and 6.\nIf not specified, self.prange or all timeslices are used.
      • \n
    • silent (bool):\nDecides whether output is printed to the standard output.
      • \n
    \n", "signature": "(self, function, fitrange=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.plateau": {"fullname": "pyerrors.correlators.Corr.plateau", "modulename": "pyerrors.correlators", "qualname": "Corr.plateau", "kind": "function", "doc": "
    Extract a plateau value from a Corr object
    \n\n
    Parameters
    \n\n
      \n
    • plateau_range (list):\nlist with two entries, indicating the first and the last timeslice\nof the plateau region.
      • \n
    • method (str):\nmethod to extract the plateau.\n    'fit' fits a constant to the plateau region\n    'avg', 'average' or 'mean' just average over the given timeslices.
      • \n
    • auto_gamma (bool):\napply gamma_method with default parameters to the Corr. Defaults to None
      • \n
    \n", "signature": "(self, plateau_range=None, method='fit', auto_gamma=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.set_prange": {"fullname": "pyerrors.correlators.Corr.set_prange", "modulename": "pyerrors.correlators", "qualname": "Corr.set_prange", "kind": "function", "doc": "
    Sets the attribute prange of the Corr object.
    \n", "signature": "(self, prange):", "funcdef": "def"}, "pyerrors.correlators.Corr.show": {"fullname": "pyerrors.correlators.Corr.show", "modulename": "pyerrors.correlators", "qualname": "Corr.show", "kind": "function", "doc": "
    Plots the correlator using the tag of the correlator as label if available.
    \n\n
    Parameters
    \n\n
      \n
    • x_range (list):\nlist of two values, determining the range of the x-axis e.g. [4, 8].
      • \n
    • comp (Corr or list of Corr):\nCorrelator or list of correlators which are plotted for comparison.\nThe tags of these correlators are used as labels if available.
      • \n
    • logscale (bool):\nSets y-axis to logscale.
      • \n
    • plateau (Obs):\nPlateau value to be visualized in the figure.
      • \n
    • fit_res (Fit_result):\nFit_result object to be visualized.
      • \n
    • fit_key (str):\nKey for the fit function in Fit_result.fit_function (for combined fits).
      • \n
    • ylabel (str):\nLabel for the y-axis.
      • \n
    • save (str):\npath to file in which the figure should be saved.
      • \n
    • auto_gamma (bool):\nApply the gamma method with standard parameters to all correlators and plateau values before plotting.
      • \n
    • hide_sigma (float):\nHides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
      • \n
    • references (list):\nList of floating point values that are displayed as horizontal lines for reference.
      • \n
    • title (string):\nOptional title of the figure.
      • \n
    \n", "signature": "(\tself,\tx_range=None,\tcomp=None,\ty_range=None,\tlogscale=False,\tplateau=None,\tfit_res=None,\tfit_key=None,\tylabel=None,\tsave=None,\tauto_gamma=False,\thide_sigma=None,\treferences=None,\ttitle=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.spaghetti_plot": {"fullname": "pyerrors.correlators.Corr.spaghetti_plot", "modulename": "pyerrors.correlators", "qualname": "Corr.spaghetti_plot", "kind": "function", "doc": "
    Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.
    \n\n
    Parameters
    \n\n
      \n
    • logscale (bool):\nDetermines whether the scale of the y-axis is logarithmic or standard.
      • \n
    \n", "signature": "(self, logscale=True):", "funcdef": "def"}, "pyerrors.correlators.Corr.dump": {"fullname": "pyerrors.correlators.Corr.dump", "modulename": "pyerrors.correlators", "qualname": "Corr.dump", "kind": "function", "doc": "
    Dumps the Corr into a file of chosen type
    \n\n
    Parameters
    \n\n
      \n
    • filename (str):\nName of the file to be saved.
      • \n
    • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n", "signature": "(self, filename, datatype='json.gz', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.print": {"fullname": "pyerrors.correlators.Corr.print", "modulename": "pyerrors.correlators", "qualname": "Corr.print", "kind": "function", "doc": "
    \n", "signature": "(self, print_range=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.sqrt": {"fullname": "pyerrors.correlators.Corr.sqrt", "modulename": "pyerrors.correlators", "qualname": "Corr.sqrt", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.log": {"fullname": "pyerrors.correlators.Corr.log", "modulename": "pyerrors.correlators", "qualname": "Corr.log", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.exp": {"fullname": "pyerrors.correlators.Corr.exp", "modulename": "pyerrors.correlators", "qualname": "Corr.exp", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sin": {"fullname": "pyerrors.correlators.Corr.sin", "modulename": "pyerrors.correlators", "qualname": "Corr.sin", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cos": {"fullname": "pyerrors.correlators.Corr.cos", "modulename": "pyerrors.correlators", "qualname": "Corr.cos", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tan": {"fullname": "pyerrors.correlators.Corr.tan", "modulename": "pyerrors.correlators", "qualname": "Corr.tan", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sinh": {"fullname": "pyerrors.correlators.Corr.sinh", "modulename": "pyerrors.correlators", "qualname": "Corr.sinh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cosh": {"fullname": "pyerrors.correlators.Corr.cosh", "modulename": "pyerrors.correlators", "qualname": "Corr.cosh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tanh": {"fullname": "pyerrors.correlators.Corr.tanh", "modulename": "pyerrors.correlators", "qualname": "Corr.tanh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsin": {"fullname": "pyerrors.correlators.Corr.arcsin", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsin", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccos": {"fullname": "pyerrors.correlators.Corr.arccos", "modulename": "pyerrors.correlators", "qualname": "Corr.arccos", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctan": {"fullname": "pyerrors.correlators.Corr.arctan", "modulename": "pyerrors.correlators", "qualname": "Corr.arctan", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsinh": {"fullname": "pyerrors.correlators.Corr.arcsinh", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsinh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccosh": {"fullname": "pyerrors.correlators.Corr.arccosh", "modulename": "pyerrors.correlators", "qualname": "Corr.arccosh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctanh": {"fullname": "pyerrors.correlators.Corr.arctanh", "modulename": "pyerrors.correlators", "qualname": "Corr.arctanh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.real": {"fullname": "pyerrors.correlators.Corr.real", "modulename": "pyerrors.correlators", "qualname": "Corr.real", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.imag": {"fullname": "pyerrors.correlators.Corr.imag", "modulename": "pyerrors.correlators", "qualname": "Corr.imag", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.prune": {"fullname": "pyerrors.correlators.Corr.prune", "modulename": "pyerrors.correlators", "qualname": "Corr.prune", "kind": "function", "doc": "
    Project large correlation matrix to lowest states
    \n\n
    This method can be used to reduce the size of an (N x N) correlation matrix\nto (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise\nis still small.
    \n\n
    Parameters
    \n\n
      \n
    • Ntrunc (int):\nRank of the target matrix.
      • \n
    • tproj (int):\nTime where the eigenvectors are evaluated, corresponds to ts in the GEVP method.\nThe default value is 3.
      • \n
    • t0proj (int):\nTime where the correlation matrix is inverted. Choosing t0proj=1 is strongly\ndiscouraged for O(a) improved theories, since the correctness of the procedure\ncannot be granted in this case. The default value is 2.
      • \n
    • basematrix (Corr):\nCorrelation matrix that is used to determine the eigenvectors of the\nlowest states based on a GEVP. basematrix is taken to be the Corr itself if\nis is not specified.
      • \n
    \n\n
    Notes
    \n\n
    We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving\nthe GEVP $$C(t) v_n(t, t_0) = \\lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \\equiv t_\\mathrm{proj}$\nand $t_0 \\equiv t_{0, \\mathrm{proj}}$. The target matrix is projected onto the subspace of the\nresulting eigenvectors $v_n, n=1,\\dots,N_\\mathrm{trunc}$ via\n$$G^\\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large\ncorrelation matrix and to remove some noise that is added by irrelevant operators.\nThis may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated\nbound $t_0 \\leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.
    \n", "signature": "(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.N": {"fullname": "pyerrors.correlators.Corr.N", "modulename": "pyerrors.correlators", "qualname": "Corr.N", "kind": "variable", "doc": "
    \n"}, "pyerrors.covobs": {"fullname": "pyerrors.covobs", "modulename": "pyerrors.covobs", "kind": "module", "doc": "
    \n"}, "pyerrors.covobs.Covobs": {"fullname": "pyerrors.covobs.Covobs", "modulename": "pyerrors.covobs", "qualname": "Covobs", "kind": "class", "doc": "
    \n"}, "pyerrors.covobs.Covobs.__init__": {"fullname": "pyerrors.covobs.Covobs.__init__", "modulename": "pyerrors.covobs", "qualname": "Covobs.__init__", "kind": "function", "doc": "
    Initialize Covobs object.
    \n\n
    Parameters
    \n\n
      \n
    • mean (float):\nMean value of the new Obs
      • \n
    • cov (list or array):\n2d Covariance matrix or 1d diagonal entries
      • \n
    • name (str):\nidentifier for the covariance matrix
      • \n
    • pos (int):\nPosition of the variance belonging to mean in cov.\nIs taken to be 1 if cov is 0-dimensional
      • \n
    • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
      • \n
    \n", "signature": "(mean, cov, name, pos=None, grad=None)"}, "pyerrors.covobs.Covobs.name": {"fullname": "pyerrors.covobs.Covobs.name", "modulename": "pyerrors.covobs", "qualname": "Covobs.name", "kind": "variable", "doc": "
    \n"}, "pyerrors.covobs.Covobs.value": {"fullname": "pyerrors.covobs.Covobs.value", "modulename": "pyerrors.covobs", "qualname": "Covobs.value", "kind": "variable", "doc": "
    \n"}, "pyerrors.covobs.Covobs.errsq": {"fullname": "pyerrors.covobs.Covobs.errsq", "modulename": "pyerrors.covobs", "qualname": "Covobs.errsq", "kind": "function", "doc": "
    Return the variance (= square of the error) of the Covobs
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.covobs.Covobs.cov": {"fullname": "pyerrors.covobs.Covobs.cov", "modulename": "pyerrors.covobs", "qualname": "Covobs.cov", "kind": "variable", "doc": "
    \n"}, "pyerrors.covobs.Covobs.grad": {"fullname": "pyerrors.covobs.Covobs.grad", "modulename": "pyerrors.covobs", "qualname": "Covobs.grad", "kind": "variable", "doc": "
    \n"}, "pyerrors.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "
    \n"}, "pyerrors.dirac.gammaX": {"fullname": "pyerrors.dirac.gammaX", "modulename": "pyerrors.dirac", "qualname": "gammaX", "kind": "variable", "doc": "
    \n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+0.j,  0.+1.j],\n       [ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n       [ 0.+0.j, -0.-1.j,  0.+0.j,  0.+0.j],\n       [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaY": {"fullname": "pyerrors.dirac.gammaY", "modulename": "pyerrors.dirac", "qualname": "gammaY", "kind": "variable", "doc": "
    \n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j],\n       [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n       [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n       [-1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaZ": {"fullname": "pyerrors.dirac.gammaZ", "modulename": "pyerrors.dirac", "qualname": "gammaZ", "kind": "variable", "doc": "
    \n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j,  0.+0.j, -0.-1.j],\n       [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+1.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaT": {"fullname": "pyerrors.dirac.gammaT", "modulename": "pyerrors.dirac", "qualname": "gammaT", "kind": "variable", "doc": "
    \n", "default_value": "array([[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n       [1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gamma": {"fullname": "pyerrors.dirac.gamma", "modulename": "pyerrors.dirac", "qualname": "gamma", "kind": "variable", "doc": "
    \n", "default_value": "array([[[ 0.+0.j,  0.+0.j,  0.+0.j,  0.+1.j],\n        [ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n        [ 0.+0.j, -0.-1.j,  0.+0.j,  0.+0.j],\n        [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j],\n        [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n        [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n        [-1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n        [ 0.+0.j,  0.+0.j,  0.+0.j, -0.-1.j],\n        [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j],\n        [ 0.+0.j,  0.+1.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n        [ 0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j],\n        [ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],\n        [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j]]])"}, "pyerrors.dirac.gamma5": {"fullname": "pyerrors.dirac.gamma5", "modulename": "pyerrors.dirac", "qualname": "gamma5", "kind": "variable", "doc": "
    \n", "default_value": "array([[ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j, -1.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j]])"}, "pyerrors.dirac.identity": {"fullname": "pyerrors.dirac.identity", "modulename": "pyerrors.dirac", "qualname": "identity", "kind": "variable", "doc": "
    \n", "default_value": "array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]])"}, "pyerrors.dirac.epsilon_tensor": {"fullname": "pyerrors.dirac.epsilon_tensor", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor", "kind": "function", "doc": "
    Rank-3 epsilon tensor
    \n\n
    Based on https://codegolf.stackexchange.com/a/160375
    \n\n
    Returns
    \n\n
      \n
    • elem (int):\nElement (i,j,k) of the epsilon tensor of rank 3
      • \n
    \n", "signature": "(i, j, k):", "funcdef": "def"}, "pyerrors.dirac.epsilon_tensor_rank4": {"fullname": "pyerrors.dirac.epsilon_tensor_rank4", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor_rank4", "kind": "function", "doc": "
    Rank-4 epsilon tensor
    \n\n
    Extension of https://codegolf.stackexchange.com/a/160375
    \n\n
    Returns
    \n\n
      \n
    • elem (int):\nElement (i,j,k,o) of the epsilon tensor of rank 4
      • \n
    \n", "signature": "(i, j, k, o):", "funcdef": "def"}, "pyerrors.dirac.Grid_gamma": {"fullname": "pyerrors.dirac.Grid_gamma", "modulename": "pyerrors.dirac", "qualname": "Grid_gamma", "kind": "function", "doc": "
    Returns gamma matrix in Grid labeling.
    \n", "signature": "(gamma_tag):", "funcdef": "def"}, "pyerrors.fits": {"fullname": "pyerrors.fits", "modulename": "pyerrors.fits", "kind": "module", "doc": "
    \n"}, "pyerrors.fits.Fit_result": {"fullname": "pyerrors.fits.Fit_result", "modulename": "pyerrors.fits", "qualname": "Fit_result", "kind": "class", "doc": "
    Represents fit results.
    \n\n
    Attributes
    \n\n
      \n
    • fit_parameters (list):\nresults for the individual fit parameters,\nalso accessible via indices.
      • \n
    • chisquare_by_dof (float):\nreduced chisquare.
      • \n
    • p_value (float):\np-value of the fit
      • \n
    • t2_p_value (float):\nHotelling t-squared p-value for correlated fits.
      • \n
    \n", "bases": "collections.abc.Sequence"}, "pyerrors.fits.Fit_result.fit_parameters": {"fullname": "pyerrors.fits.Fit_result.fit_parameters", "modulename": "pyerrors.fits", "qualname": "Fit_result.fit_parameters", "kind": "variable", "doc": "
    \n"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "
    Apply the gamma method to all fit parameters
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.Fit_result.gm": {"fullname": "pyerrors.fits.Fit_result.gm", "modulename": "pyerrors.fits", "qualname": "Fit_result.gm", "kind": "function", "doc": "
    Apply the gamma method to all fit parameters
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.least_squares": {"fullname": "pyerrors.fits.least_squares", "modulename": "pyerrors.fits", "qualname": "least_squares", "kind": "function", "doc": "
    Performs a non-linear fit to y = func(x).\n    ```
    \n\n
    Parameters
    \n\n
      \n
    • For an uncombined fit:
      • \n
    • x (list):\nlist of floats.
      • \n
    • y (list):\nlist of Obs.
      • \n
    • func (object):\nfit function, has to be of the form
      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(a, x):\n    return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
      \n
      \n\n
      For multiple x values func can be of the form
      \n\n
      \n
      def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
      \n
      \n\n
      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
      • \n
    • OR For a combined fit:
      • \n
    • x (dict):\ndict of lists.
      • \n
    • y (dict):\ndict of lists of Obs.
      • \n
    • funcs (dict):\ndict of objects\nfit functions have to be of the form (here a[0] is the common fit parameter)\n```python\nimport autograd.numpy as anp\nfuncs = {\"a\": func_a,\n        \"b\": func_b}
      \n\n
      def func_a(a, x):\n    return a[1] * anp.exp(-a[0] * x)
      \n\n
      def func_b(a, x):\n    return a[2] * anp.exp(-a[0] * x)
      \n\n
      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
      • \n
    • priors (dict or list, optional):\npriors can either be a dictionary with integer keys and the corresponding priors as values or\na list with an entry for every parameter in the fit. The entries can either be\nObs (e.g. results from a previous fit) or strings containing a value and an error formatted like\n0.548(23), 500(40) or 0.5(0.4)
      • \n
    • silent (bool, optional):\nIf true all output to the console is omitted (default False).
      • \n
    • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for\nnon-linear fits with many parameters. In case of correlated fits the guess is used to perform\nan uncorrelated fit which then serves as guess for the correlated fit.
      • \n
    • method (str, optional):\ncan be used to choose an alternative method for the minimization of chisquare.\nThe possible methods are the ones which can be used for scipy.optimize.minimize and\nmigrad of iminuit. If no method is specified, Levenberg-Marquard is used.\nReliable alternatives are migrad, Powell and Nelder-Mead.
      • \n
    • tol (float, optional):\ncan be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence\nto a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly\ninvalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values\nThe stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
      • \n
    • correlated_fit (bool):\nIf True, use the full inverse covariance matrix in the definition of the chisquare cost function.\nFor details about how the covariance matrix is estimated see pyerrors.obs.covariance.\nIn practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).\nThis procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
      • \n
    • expected_chisquare (bool):\nIf True estimates the expected chisquare which is\ncorrected by effects caused by correlated input data (default False).
      • \n
    • resplot (bool):\nIf True, a plot which displays fit, data and residuals is generated (default False).
      • \n
    • qqplot (bool):\nIf True, a quantile-quantile plot of the fit result is generated (default False).
      • \n
    • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • output (Fit_result):\nParameters and information on the fitted result.
      • \n
    \n", "signature": "(x, y, func, priors=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.total_least_squares": {"fullname": "pyerrors.fits.total_least_squares", "modulename": "pyerrors.fits", "qualname": "total_least_squares", "kind": "function", "doc": "
    Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nlist of Obs, or a tuple of lists of Obs
      • \n
    • y (list):\nlist of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
      • \n
    • func (object):\nfunc has to be of the form
      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(a, x):\n    return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
      \n
      \n\n
      For multiple x values func can be of the form
      \n\n
      \n
      def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
      \n
      \n\n
      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
      • \n
    • silent (bool, optional):\nIf true all output to the console is omitted (default False).
      • \n
    • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for non-linear\nfits with many parameters.
      • \n
    • expected_chisquare (bool):\nIf true prints the expected chisquare which is\ncorrected by effects caused by correlated input data.\nThis can take a while as the full correlation matrix\nhas to be calculated (default False).
      • \n
    • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
      • \n
    \n\n
    Notes
    \n\n
    Based on the orthogonal distance regression module of scipy.
    \n\n
    Returns
    \n\n
      \n
    • output (Fit_result):\nParameters and information on the fitted result.
      • \n
    \n", "signature": "(x, y, func, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.fit_lin": {"fullname": "pyerrors.fits.fit_lin", "modulename": "pyerrors.fits", "qualname": "fit_lin", "kind": "function", "doc": "
    Performs a linear fit to y = n + m * x and returns two Obs n, m.
    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nCan either be a list of floats in which case no xerror is assumed, or\na list of Obs, where the dvalues of the Obs are used as xerror for the fit.
      • \n
    • y (list):\nList of Obs, the dvalues of the Obs are used as yerror for the fit.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • fit_parameters (list[Obs]):\nLIist of fitted observables.
      • \n
    \n", "signature": "(x, y, **kwargs):", "funcdef": "def"}, "pyerrors.fits.qqplot": {"fullname": "pyerrors.fits.qqplot", "modulename": "pyerrors.fits", "qualname": "qqplot", "kind": "function", "doc": "
    Generates a quantile-quantile plot of the fit result which can be used to\n   check if the residuals of the fit are gaussian distributed.
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(x, o_y, func, p, title=''):", "funcdef": "def"}, "pyerrors.fits.residual_plot": {"fullname": "pyerrors.fits.residual_plot", "modulename": "pyerrors.fits", "qualname": "residual_plot", "kind": "function", "doc": "
    Generates a plot which compares the fit to the data and displays the corresponding residuals
    \n\n
    For uncorrelated data the residuals are expected to be distributed ~N(0,1).
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(x, y, func, fit_res, title=''):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "
    Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
    \n\n
    Returns
    \n\n
      \n
    • err (np.array(Obs)):\nError band for an array of sample values x
      • \n
    \n", "signature": "(x, func, beta):", "funcdef": "def"}, "pyerrors.fits.ks_test": {"fullname": "pyerrors.fits.ks_test", "modulename": "pyerrors.fits", "qualname": "ks_test", "kind": "function", "doc": "
    Performs a Kolmogorov\u2013Smirnov test for the p-values of all fit object.
    \n\n
    Parameters
    \n\n
      \n
    • objects (list):\nList of fit results to include in the analysis (optional).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(objects=None):", "funcdef": "def"}, "pyerrors.input": {"fullname": "pyerrors.input", "modulename": "pyerrors.input", "kind": "module", "doc": "
    pyerrors includes an input submodule in which input routines and parsers for the output of various numerical programs are contained.
    \n\n
    Jackknife samples
    \n\n
    For comparison with other analysis workflows pyerrors can also generate jackknife samples from an Obs object or import jackknife samples into an Obs object.\nSee pyerrors.obs.Obs.export_jackknife and pyerrors.obs.import_jackknife for details.
    \n"}, "pyerrors.input.bdio": {"fullname": "pyerrors.input.bdio", "modulename": "pyerrors.input.bdio", "kind": "module", "doc": "
    \n"}, "pyerrors.input.bdio.read_ADerrors": {"fullname": "pyerrors.input.bdio.read_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "read_ADerrors", "kind": "function", "doc": "
    Extract generic MCMC data from a bdio file
    \n\n
    read_ADerrors requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
    \n\n
    all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
    \n\n
    Parameters
    \n\n
      \n
    • file_path -- path to the bdio file
      • \n
    • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (List[Obs]):\nExtracted data
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.write_ADerrors": {"fullname": "pyerrors.input.bdio.write_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "write_ADerrors", "kind": "function", "doc": "
    Write Obs to a bdio file according to ADerrors conventions
    \n\n
    read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
    \n\n
    all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
    \n\n
    Parameters
    \n\n
      \n
    • file_path -- path to the bdio file
      • \n
    • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • success (int):\nreturns 0 is successful
      • \n
    \n", "signature": "(obs_list, file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_mesons": {"fullname": "pyerrors.input.bdio.read_mesons", "modulename": "pyerrors.input.bdio", "qualname": "read_mesons", "kind": "function", "doc": "
    Extract mesons data from a bdio file and return it as a dictionary
    \n\n
    The dictionary can be accessed with a tuple consisting of (type, source_position, kappa1, kappa2)
    \n\n
    read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
    \n\n
    all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
    \n\n
    Parameters
    \n\n
      \n
    • file_path (str):\npath to the bdio file
      • \n
    • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    • start (int):\nThe first configuration to be read (default 1)
      • \n
    • stop (int):\nThe last configuration to be read (default None)
      • \n
    • step (int):\nFixed step size between two measurements (default 1)
      • \n
    • alternative_ensemble_name (str):\nManually overwrite ensemble name
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (dict):\nExtracted meson data
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_dSdm": {"fullname": "pyerrors.input.bdio.read_dSdm", "modulename": "pyerrors.input.bdio", "qualname": "read_dSdm", "kind": "function", "doc": "
    Extract dSdm data from a bdio file and return it as a dictionary
    \n\n
    The dictionary can be accessed with a tuple consisting of (type, kappa)
    \n\n
    read_dSdm requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
    \n\n
    all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
    \n\n
    Parameters
    \n\n
      \n
    • file_path (str):\npath to the bdio file
      • \n
    • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    • start (int):\nThe first configuration to be read (default 1)
      • \n
    • stop (int):\nThe last configuration to be read (default None)
      • \n
    • step (int):\nFixed step size between two measurements (default 1)
      • \n
    • alternative_ensemble_name (str):\nManually overwrite ensemble name
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.dobs": {"fullname": "pyerrors.input.dobs", "modulename": "pyerrors.input.dobs", "kind": "module", "doc": "
    \n"}, "pyerrors.input.dobs.create_pobs_string": {"fullname": "pyerrors.input.dobs.create_pobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_pobs_string", "kind": "function", "doc": "
    Export a list of Obs or structures containing Obs to an xml string\naccording to the Zeuthen pobs format.
    \n\n
    Tags are not written or recovered automatically. The separator | is removed from the replica names.
    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • xml_str (str):\nXML formatted string of the input data
      • \n
    \n", "signature": "(obsl, name, spec='', origin='', symbol=[], enstag=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_pobs": {"fullname": "pyerrors.input.dobs.write_pobs", "modulename": "pyerrors.input.dobs", "qualname": "write_pobs", "kind": "function", "doc": "
    Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen pobs format.
    \n\n
    Tags are not written or recovered automatically. The separator | is removed from the replica names.
    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
      • \n
    • gz (bool):\nIf True, the output is a gzipped xml. If False, the output is an xml file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(\tobsl,\tfname,\tname,\tspec='',\torigin='',\tsymbol=[],\tenstag=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_pobs": {"fullname": "pyerrors.input.dobs.read_pobs", "modulename": "pyerrors.input.dobs", "qualname": "read_pobs", "kind": "function", "doc": "
    Import a list of Obs from an xml.gz file in the Zeuthen pobs format.
    \n\n
    Tags are not written or recovered automatically.
    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • separatior_insertion (str or int):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nNone (default): Replica names remain unchanged.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(fname, full_output=False, gz=True, separator_insertion=None):", "funcdef": "def"}, "pyerrors.input.dobs.import_dobs_string": {"fullname": "pyerrors.input.dobs.import_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "import_dobs_string", "kind": "function", "doc": "
    Import a list of Obs from a string in the Zeuthen dobs format.
    \n\n
    Tags are not written or recovered automatically.
    \n\n
    Parameters
    \n\n
      \n
    • content (str):\nXML string containing the data
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(content, full_output=False, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_dobs": {"fullname": "pyerrors.input.dobs.read_dobs", "modulename": "pyerrors.input.dobs", "qualname": "read_dobs", "kind": "function", "doc": "
    Import a list of Obs from an xml.gz file in the Zeuthen dobs format.
    \n\n
    Tags are not written or recovered automatically.
    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes XML file.
      • \n
    • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(fname, full_output=False, gz=True, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.create_dobs_string": {"fullname": "pyerrors.input.dobs.create_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_dobs_string", "kind": "function", "doc": "
    Generate the string for the export of a list of Obs or structures containing Obs\nto a .xml.gz file according to the Zeuthen dobs format.
    \n\n
    Tags are not written or recovered automatically. The separator |is removed from the replica names.
    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • who (str):\nProvide the name of the person that exports the data.
      • \n
    • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • xml_str (str):\nXML string generated from the data
      • \n
    \n", "signature": "(\tobsl,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_dobs": {"fullname": "pyerrors.input.dobs.write_dobs", "modulename": "pyerrors.input.dobs", "qualname": "write_dobs", "kind": "function", "doc": "
    Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen dobs format.
    \n\n
    Tags are not written or recovered automatically. The separator | is removed from the replica names.
    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • who (str):\nProvide the name of the person that exports the data.
      • \n
    • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
      • \n
    • gz (bool):\nIf True, the output is a gzipped XML. If False, the output is a XML file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(\tobsl,\tfname,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.hadrons": {"fullname": "pyerrors.input.hadrons", "modulename": "pyerrors.input.hadrons", "kind": "module", "doc": "
    \n"}, "pyerrors.input.hadrons.read_hd5": {"fullname": "pyerrors.input.hadrons.read_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_hd5", "kind": "function", "doc": "
    Read hadrons hdf5 file and extract entry based on attributes.
    \n\n
    Parameters
    \n\n
      \n
    • filestem (str):\nFull namestem of the files to read, including the full path.
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • group (str):\nlabel of the group to be extracted.
      • \n
    • attrs (dict or int):\nDictionary containing the attributes. For example
      \n\n
      \n
      attrs = {"gamma_snk": "Gamma5",\n         "gamma_src": "Gamma5"}\n
      \n
      \n\n
      Alternatively an integer can be specified to identify the sub group.\nThis is discouraged as the order in the file is not guaranteed.
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • part (str):\nstring specifying whether to extract the real part ('real'),\nthe imaginary part ('imag') or a complex correlator ('complex').\nDefault 'real'.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr (Corr):\nCorrelator of the source sink combination in question.
      • \n
    \n", "signature": "(filestem, ens_id, group, attrs=None, idl=None, part='real'):", "funcdef": "def"}, "pyerrors.input.hadrons.read_meson_hd5": {"fullname": "pyerrors.input.hadrons.read_meson_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_meson_hd5", "kind": "function", "doc": "
    Read hadrons meson hdf5 file and extract the meson labeled 'meson'
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • meson (str):\nlabel of the meson to be extracted, standard value meson_0 which\ncorresponds to the pseudoscalar pseudoscalar two-point function.
      • \n
    • gammas (tuple of strings):\nInstrad of a meson label one can also provide a tuple of two strings\nindicating the gamma matrices at sink and source (gamma_snk, gamma_src).\n(\"Gamma5\", \"Gamma5\") corresponds to the pseudoscalar pseudoscalar\ntwo-point function. The gammas argument dominateds over meson.
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr (Corr):\nCorrelator of the source sink combination in question.
      • \n
    \n", "signature": "(path, filestem, ens_id, meson='meson_0', idl=None, gammas=None):", "funcdef": "def"}, "pyerrors.input.hadrons.extract_t0_hd5": {"fullname": "pyerrors.input.hadrons.extract_t0_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "extract_t0_hd5", "kind": "function", "doc": "
    Read hadrons FlowObservables hdf5 file and extract t0
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • obs (str):\nlabel of the observable from which t0 should be extracted.\nOptions: 'Clover energy density' and 'Plaquette energy density'
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
      • \n
    \n", "signature": "(\tpath,\tfilestem,\tens_id,\tobs='Clover energy density',\tfit_range=5,\tidl=None,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.hadrons.read_DistillationContraction_hd5": {"fullname": "pyerrors.input.hadrons.read_DistillationContraction_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_DistillationContraction_hd5", "kind": "function", "doc": "
    Read hadrons DistillationContraction hdf5 files in given directory structure
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the directories to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • diagrams (list):\nList of strings of the diagrams to extract, e.g. [\"direct\", \"box\", \"cross\"].
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (dict):\nextracted DistillationContration data
      • \n
    \n", "signature": "(path, ens_id, diagrams=['direct'], idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.Npr_matrix": {"fullname": "pyerrors.input.hadrons.Npr_matrix", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix", "kind": "class", "doc": "
    ndarray(shape, dtype=float, buffer=None, offset=0,\n        strides=None, order=None)
    \n\n
    An array object represents a multidimensional, homogeneous array\nof fixed-size items.  An associated data-type object describes the\nformat of each element in the array (its byte-order, how many bytes it\noccupies in memory, whether it is an integer, a floating point number,\nor something else, etc.)
    \n\n
    Arrays should be constructed using array, zeros or empty (refer\nto the See Also section below).  The parameters given here refer to\na low-level method (ndarray(...)) for instantiating an array.
    \n\n
    For more information, refer to the numpy module and examine the\nmethods and attributes of an array.
    \n\n
    Parameters
    \n\n
      \n
    • (for the __new__ method; see Notes below)
      • \n
    • shape (tuple of ints):\nShape of created array.
      • \n
    • dtype (data-type, optional):\nAny object that can be interpreted as a numpy data type.
      • \n
    • buffer (object exposing buffer interface, optional):\nUsed to fill the array with data.
      • \n
    • offset (int, optional):\nOffset of array data in buffer.
      • \n
    • strides (tuple of ints, optional):\nStrides of data in memory.
      • \n
    • order ({'C', 'F'}, optional):\nRow-major (C-style) or column-major (Fortran-style) order.
      • \n
    \n\n
    Attributes
    \n\n
      \n
    • T (ndarray):\nTranspose of the array.
      • \n
    • data (buffer):\nThe array's elements, in memory.
      • \n
    • dtype (dtype object):\nDescribes the format of the elements in the array.
      • \n
    • flags (dict):\nDictionary containing information related to memory use, e.g.,\n'C_CONTIGUOUS', 'OWNDATA', 'WRITEABLE', etc.
      • \n
    • flat (numpy.flatiter object):\nFlattened version of the array as an iterator.  The iterator\nallows assignments, e.g., x.flat = 3 (See ndarray.flat for\nassignment examples; TODO).
      • \n
    • imag (ndarray):\nImaginary part of the array.
      • \n
    • real (ndarray):\nReal part of the array.
      • \n
    • size (int):\nNumber of elements in the array.
      • \n
    • itemsize (int):\nThe memory use of each array element in bytes.
      • \n
    • nbytes (int):\nThe total number of bytes required to store the array data,\ni.e., itemsize * size.
      • \n
    • ndim (int):\nThe array's number of dimensions.
      • \n
    • shape (tuple of ints):\nShape of the array.
      • \n
    • strides (tuple of ints):\nThe step-size required to move from one element to the next in\nmemory. For example, a contiguous (3, 4) array of type\nint16 in C-order has strides (8, 2).  This implies that\nto move from element to element in memory requires jumps of 2 bytes.\nTo move from row-to-row, one needs to jump 8 bytes at a time\n(2 * 4).
      • \n
    • ctypes (ctypes object):\nClass containing properties of the array needed for interaction\nwith ctypes.
      • \n
    • base (ndarray):\nIf the array is a view into another array, that array is its base\n(unless that array is also a view).  The base array is where the\narray data is actually stored.
      • \n
    \n\n
    See Also
    \n\n
    array:  Construct an array.
    \nzeros:  Create an array, each element of which is zero.
    \nempty:  Create an array, but leave its allocated memory unchanged (i.e.,\nit contains \"garbage\").
    \ndtype:  Create a data-type.
    \nnumpy.typing.NDArray:  An ndarray alias :term:generic <generic type>\nw.r.t. its dtype.type <numpy.dtype.type>.  
    \n\n
    Notes
    \n\n
    There are two modes of creating an array using __new__:
    \n\n
      \n
    1. If buffer is None, then only shape, dtype, and order\nare used.
      2. \n
    2. If buffer is an object exposing the buffer interface, then\nall keywords are interpreted.
      3. \n
    \n\n
    No __init__ method is needed because the array is fully initialized\nafter the __new__ method.
    \n\n
    Examples
    \n\n
    These examples illustrate the low-level ndarray constructor.  Refer\nto the See Also section above for easier ways of constructing an\nndarray.
    \n\n
    First mode, buffer is None:
    \n\n
    \n
    >>> import numpy as np\n>>> np.ndarray(shape=(2,2), dtype=float, order='F')\narray([[0.0e+000, 0.0e+000], # random\n       [     nan, 2.5e-323]])\n
    \n
    \n\n
    Second mode:
    \n\n
    \n
    >>> np.ndarray((2,), buffer=np.array([1,2,3]),\n...            offset=np.int_().itemsize,\n...            dtype=int) # offset = 1*itemsize, i.e. skip first element\narray([2, 3])\n
    \n
    \n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "
    Gamma_5 hermitean conjugate
    \n\n
    Uses the fact that the propagator is gamma5 hermitean, so just the\nin and out momenta of the propagator are exchanged.
    \n"}, "pyerrors.input.hadrons.read_ExternalLeg_hd5": {"fullname": "pyerrors.input.hadrons.read_ExternalLeg_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_ExternalLeg_hd5", "kind": "function", "doc": "
    Read hadrons ExternalLeg hdf5 file and output an array of CObs
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (Npr_matrix):\nread Cobs-matrix
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Bilinear_hd5": {"fullname": "pyerrors.input.hadrons.read_Bilinear_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Bilinear_hd5", "kind": "function", "doc": "
    Read hadrons Bilinear hdf5 file and output an array of CObs
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result_dict (dict[Npr_matrix]):\nextracted Bilinears
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Fourquark_hd5": {"fullname": "pyerrors.input.hadrons.read_Fourquark_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Fourquark_hd5", "kind": "function", "doc": "
    Read hadrons FourquarkFullyConnected hdf5 file and output an array of CObs
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • vertices (list):\nVertex functions to be extracted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result_dict (dict):\nextracted fourquark matrizes
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None, vertices=['VA', 'AV']):", "funcdef": "def"}, "pyerrors.input.json": {"fullname": "pyerrors.input.json", "modulename": "pyerrors.input.json", "kind": "module", "doc": "
    \n"}, "pyerrors.input.json.create_json_string": {"fullname": "pyerrors.input.json.create_json_string", "modulename": "pyerrors.input.json", "qualname": "create_json_string", "kind": "function", "doc": "
    Generate the string for the export of a list of Obs or structures containing Obs\nto a .json(.gz) file
    \n\n
    Parameters
    \n\n
      \n
    • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • json_string (str):\nString for export to .json(.gz) file
      • \n
    \n", "signature": "(ol, description='', indent=1):", "funcdef": "def"}, "pyerrors.input.json.dump_to_json": {"fullname": "pyerrors.input.json.dump_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_to_json", "kind": "function", "doc": "
    Export a list of Obs or structures containing Obs to a .json(.gz) file.\nDict keys that are not JSON-serializable such as floats are converted to strings.
    \n\n
    Parameters
    \n\n
      \n
    • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Null
      • \n
    \n", "signature": "(ol, fname, description='', indent=1, gz=True):", "funcdef": "def"}, "pyerrors.input.json.import_json_string": {"fullname": "pyerrors.input.json.import_json_string", "modulename": "pyerrors.input.json", "qualname": "import_json_string", "kind": "function", "doc": "
    Reconstruct a list of Obs or structures containing Obs from a json string.
    \n\n
    The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
    \n\n
    Parameters
    \n\n
      \n
    • json_string (str):\njson string containing the data.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nreconstructed list of observables from the json string
      • \n
    • or
      • \n
    • result (Obs):\nonly one observable if the list only has one entry
      • \n
    • or
      • \n
    • result (dict):\nif full_output=True
      • \n
    \n", "signature": "(json_string, verbose=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.load_json": {"fullname": "pyerrors.input.json.load_json", "modulename": "pyerrors.input.json", "qualname": "load_json", "kind": "function", "doc": "
    Import a list of Obs or structures containing Obs from a .json(.gz) file.
    \n\n
    The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nreconstructed list of observables from the json string
      • \n
    • or
      • \n
    • result (Obs):\nonly one observable if the list only has one entry
      • \n
    • or
      • \n
    • result (dict):\nif full_output=True
      • \n
    \n", "signature": "(fname, verbose=True, gz=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.dump_dict_to_json": {"fullname": "pyerrors.input.json.dump_dict_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_dict_to_json", "kind": "function", "doc": "
    Export a dict of Obs or structures containing Obs to a .json(.gz) file
    \n\n
    Parameters
    \n\n
      \n
    • od (dict):\nDict of JSON valid structures and objects that will be exported.\nAt the moment, these objects can be either of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    • reps (str):\nSpecify the structure of the placeholder in exported dict to be reps[0-9]+.
      • \n
    • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(od, fname, description='', indent=1, reps='DICTOBS', gz=True):", "funcdef": "def"}, "pyerrors.input.json.load_json_dict": {"fullname": "pyerrors.input.json.load_json_dict", "modulename": "pyerrors.input.json", "qualname": "load_json_dict", "kind": "function", "doc": "
    Import a dict of Obs or structures containing Obs from a .json(.gz) file.
    \n\n
    The following structures are supported: Obs, list, numpy.ndarray, Corr
    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    • reps (str):\nSpecify the structure of the placeholder in imported dict to be reps[0-9]+.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (Obs / list / Corr):\nRead data
      • \n
    • or
      • \n
    • data (dict):\nRead data and meta-data
      • \n
    \n", "signature": "(fname, verbose=True, gz=True, full_output=False, reps='DICTOBS'):", "funcdef": "def"}, "pyerrors.input.misc": {"fullname": "pyerrors.input.misc", "modulename": "pyerrors.input.misc", "kind": "module", "doc": "
    \n"}, "pyerrors.input.misc.fit_t0": {"fullname": "pyerrors.input.misc.fit_t0", "modulename": "pyerrors.input.misc", "qualname": "fit_t0", "kind": "function", "doc": "
    Compute the root of (flow-based) data based on a dictionary that contains\nthe necessary information in key-value pairs a la (flow time: observable at flow time).
    \n\n
    It is assumed that the data is monotonically increasing and passes zero from below.\nNo exception is thrown if this is not the case (several roots, no monotonic increase).\nAn exception is thrown if no root can be found in the data.
    \n\n
    A linear fit in the vicinity of the root is performed to exctract the root from the\ntwo fit parameters.
    \n\n
    Parameters
    \n\n
      \n
    • t2E_dict (dict):\nDictionary with pairs of (flow time: observable at flow time) where the flow times\nare of type float and the observables of type Obs.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data. (Default: False)
      • \n
    • observable (str):\nKeyword to identify the observable to print the correct ylabel (if plot_fit is True)\nfor the observables 't0' and 'w0'. No y label is printed otherwise. (Default: 't0')
      • \n
    \n\n
    Returns
    \n\n
      \n
    • root (Obs):\nThe root of the data series.
      • \n
    \n", "signature": "(t2E_dict, fit_range, plot_fit=False, observable='t0'):", "funcdef": "def"}, "pyerrors.input.misc.read_pbp": {"fullname": "pyerrors.input.misc.read_pbp", "modulename": "pyerrors.input.misc", "qualname": "read_pbp", "kind": "function", "doc": "
    Read pbp format from given folder structure.
    \n\n
    Parameters
    \n\n
      \n
    • r_start (list):\nlist which contains the first config to be read for each replicum
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nlist of observables read
      • \n
    \n", "signature": "(path, prefix, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD": {"fullname": "pyerrors.input.openQCD", "modulename": "pyerrors.input.openQCD", "kind": "module", "doc": "
    \n"}, "pyerrors.input.openQCD.read_rwms": {"fullname": "pyerrors.input.openQCD.read_rwms", "modulename": "pyerrors.input.openQCD", "qualname": "read_rwms", "kind": "function", "doc": "
    Read rwms format from given folder structure. Returns a list of length nrw
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath that contains the data files
      • \n
    • prefix (str):\nall files in path that start with prefix are considered as input files.\nMay be used together postfix to consider only special file endings.\nPrefix is ignored, if the keyword 'files' is used.
      • \n
    • version (str):\nversion of openQCD, default 2.0
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.ms1' for openQCD-files
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • print_err (bool):\nPrint additional information that is useful for debugging.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • rwms (Obs):\nReweighting factors read
      • \n
    \n", "signature": "(path, prefix, version='2.0', names=None, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_t0": {"fullname": "pyerrors.input.openQCD.extract_t0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_t0", "kind": "function", "doc": "
    Extract t0/a^2 from given .ms.dat files. Returns t0 as Obs.
    \n\n
    It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t^2 - c (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.
    \n\n
    It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to .ms.dat files
      • \n
    • prefix (str):\nEnsemble prefix
      • \n
    • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
      • \n
    • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
      • \n
    • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • postfix (str):\nPostfix of measurement file (Default: ms)
      • \n
    • c (float):\nConstant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1.
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • plaquette (bool):\nIf true extract the plaquette estimate of t0 instead.
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
      • \n
    • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
      • \n
    \n\n
    Returns
    \n\n
      \n
    • t0 (Obs):\nExtracted t0
      • \n
    \n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_w0": {"fullname": "pyerrors.input.openQCD.extract_w0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_w0", "kind": "function", "doc": "
    Extract w0/a from given .ms.dat files. Returns w0 as Obs.
    \n\n
    It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t d(t^2)/dt -  (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.
    \n\n
    It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to .ms.dat files
      • \n
    • prefix (str):\nEnsemble prefix
      • \n
    • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
      • \n
    • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
      • \n
    • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • postfix (str):\nPostfix of measurement file (Default: ms)
      • \n
    • c (float):\nConstant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1.
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • plaquette (bool):\nIf true extract the plaquette estimate of w0 instead.
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of w0 is shown together with the data.
      • \n
    • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
      • \n
    \n\n
    Returns
    \n\n
      \n
    • w0 (Obs):\nExtracted w0
      • \n
    \n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop": {"fullname": "pyerrors.input.openQCD.read_qtop", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop", "kind": "function", "doc": "
    Read the topologial charge based on openQCD gradient flow measurements.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • version (str):\nEither openQCD or sfqcd, depending on the data.
      • \n
    • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
      • \n
    • integer_charge (bool):\nIf True, the charge is rounded towards the nearest integer on each config.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (Obs):\nRead topological charge
      • \n
    \n", "signature": "(path, prefix, c, dtr_cnfg=1, version='openQCD', **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_gf_coupling": {"fullname": "pyerrors.input.openQCD.read_gf_coupling", "modulename": "pyerrors.input.openQCD", "qualname": "read_gf_coupling", "kind": "function", "doc": "
    Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.
    \n\n
    Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used.
      • \n
    \n", "signature": "(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.qtop_projection": {"fullname": "pyerrors.input.openQCD.qtop_projection", "modulename": "pyerrors.input.openQCD", "qualname": "qtop_projection", "kind": "function", "doc": "
    Returns the projection to the topological charge sector defined by target.
    \n\n
    Parameters
    \n\n
      \n
    • path (Obs):\nTopological charge.
      • \n
    • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • reto (Obs):\nprojection to the topological charge sector defined by target
      • \n
    \n", "signature": "(qtop, target=0):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop_sector": {"fullname": "pyerrors.input.openQCD.read_qtop_sector", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop_sector", "kind": "function", "doc": "
    Constructs reweighting factors to a specified topological sector.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L
      • \n
    • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of trajectories\nbetween two configs.\nif it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • version (str):\nversion string of the openQCD (sfqcd) version used to create\nthe ensemble. Default is 2.0. May also be set to sfqcd.
      • \n
    • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
      • \n
    • r_start (list):\noffset of the first ensemble, making it easier to match\nlater on with other Obs
      • \n
    • r_stop (list):\nlast configurations that need to be read (per replicum)
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • reto (Obs):\nprojection to the topological charge sector defined by target
      • \n
    \n", "signature": "(path, prefix, c, target=0, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_ms5_xsf": {"fullname": "pyerrors.input.openQCD.read_ms5_xsf", "modulename": "pyerrors.input.openQCD", "qualname": "read_ms5_xsf", "kind": "function", "doc": "
    Read data from files in the specified directory with the specified prefix and quark combination extension, and return a Corr object containing the data.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nThe directory to search for the files in.
      • \n
    • prefix (str):\nThe prefix to match the files against.
      • \n
    • qc (str):\nThe quark combination extension to match the files against.
      • \n
    • corr (str):\nThe correlator to extract data for.
      • \n
    • sep (str, optional):\nThe separator to use when parsing the replika names.
      • \n
    • **kwargs: Additional keyword arguments. The following keyword arguments are recognized:
      \n\n
        \n
      • names (List[str]): A list of names to use for the replicas.
        • \n
      • files (List[str]): A list of files to read data from.
        • \n
      • idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given.
        • \n
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Corr: A complex valued Corr object containing the data read from the files. In case of boudary to bulk correlators.
      • \n
    • or
      • \n
    • CObs: A complex valued CObs object containing the data read from the files. In case of boudary to boundary correlators.
      • \n
    \n\n
    Raises
    \n\n
      \n
    • FileNotFoundError: If no files matching the specified prefix and quark combination extension are found in the specified directory.
      • \n
    • IOError: If there is an error reading a file.
      • \n
    • struct.error: If there is an error unpacking binary data.
      • \n
    \n", "signature": "(path, prefix, qc, corr, sep='r', **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas": {"fullname": "pyerrors.input.pandas", "modulename": "pyerrors.input.pandas", "kind": "module", "doc": "
    \n"}, "pyerrors.input.pandas.to_sql": {"fullname": "pyerrors.input.pandas.to_sql", "modulename": "pyerrors.input.pandas", "qualname": "to_sql", "kind": "function", "doc": "
    Write DataFrame including Obs or Corr valued columns to sqlite database.
    \n\n
    Parameters
    \n\n
      \n
    • df (pandas.DataFrame):\nDataframe to be written to the database.
      • \n
    • table_name (str):\nName of the table in the database.
      • \n
    • db (str):\nPath to the sqlite database.
      • \n
    • if exists (str):\nHow to behave if table already exists. Options 'fail', 'replace', 'append'.
      • \n
    • gz (bool):\nIf True the json strings are gzipped.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(df, table_name, db, if_exists='fail', gz=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.read_sql": {"fullname": "pyerrors.input.pandas.read_sql", "modulename": "pyerrors.input.pandas", "qualname": "read_sql", "kind": "function", "doc": "
    Execute SQL query on sqlite database and obtain DataFrame including Obs or Corr valued columns.
    \n\n
    Parameters
    \n\n
      \n
    • sql (str):\nSQL query to be executed.
      • \n
    • db (str):\nPath to the sqlite database.
      • \n
    • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
      • \n
    \n", "signature": "(sql, db, auto_gamma=False, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.dump_df": {"fullname": "pyerrors.input.pandas.dump_df", "modulename": "pyerrors.input.pandas", "qualname": "dump_df", "kind": "function", "doc": "
    Exports a pandas DataFrame containing Obs valued columns to a (gzipped) csv file.
    \n\n
    Before making use of pandas to_csv functionality Obs objects are serialized via the standardized\njson format of pyerrors.
    \n\n
    Parameters
    \n\n
      \n
    • df (pandas.DataFrame):\nDataframe to be dumped to a file.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • gz (bool):\nIf True, the output is a gzipped csv file. If False, the output is a csv file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(df, fname, gz=True):", "funcdef": "def"}, "pyerrors.input.pandas.load_df": {"fullname": "pyerrors.input.pandas.load_df", "modulename": "pyerrors.input.pandas", "qualname": "load_df", "kind": "function", "doc": "
    Imports a pandas DataFrame from a csv.(gz) file in which Obs objects are serialized as json strings.
    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
      • \n
    \n", "signature": "(fname, auto_gamma=False, gz=True):", "funcdef": "def"}, "pyerrors.input.sfcf": {"fullname": "pyerrors.input.sfcf", "modulename": "pyerrors.input.sfcf", "kind": "module", "doc": "
    \n"}, "pyerrors.input.sfcf.sep": {"fullname": "pyerrors.input.sfcf.sep", "modulename": "pyerrors.input.sfcf", "qualname": "sep", "kind": "variable", "doc": "
    \n", "default_value": "'/'"}, "pyerrors.input.sfcf.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "
    Read sfcf files from given folder structure.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to the sfcf files.
      • \n
    • prefix (str):\nPrefix of the sfcf files.
      • \n
    • name (str):\nName of the correlation function to read.
      • \n
    • quarks (str):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
      • \n
    • corr_type (str):\nType of correlation function to read. Can be\n
        \n
      • 'bi' for boundary-inner
        • \n
      • 'bb' for boundary-boundary
        • \n
      • 'bib' for boundary-inner-boundary
        • \n
      • \n
    • noffset (int):\nOffset of the source (only relevant when wavefunctions are used)
      • \n
    • wf (int):\nID of wave function
      • \n
    • wf2 (int):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
      • \n
    • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • ens_name (str):\nreplaces the name of the ensemble
      • \n
    • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
      • \n
    • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
      • \n
    • replica (list):\nlist of replica to be read, default is all
      • \n
    • files (list):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
      • \n
    • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
      • \n
    \n", "signature": "(\tpath,\tprefix,\tname,\tquarks='.*',\tcorr_type='bi',\tnoffset=0,\twf=0,\twf2=0,\tversion='1.0c',\tcfg_separator='n',\tsilent=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.sfcf.read_sfcf_multi": {"fullname": "pyerrors.input.sfcf.read_sfcf_multi", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf_multi", "kind": "function", "doc": "
    Read sfcf files from given folder structure.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to the sfcf files.
      • \n
    • prefix (str):\nPrefix of the sfcf files.
      • \n
    • name (str):\nName of the correlation function to read.
      • \n
    • quarks_list (list[str]):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
      • \n
    • corr_type_list (list[str]):\nType of correlation function to read. Can be\n
        \n
      • 'bi' for boundary-inner
        • \n
      • 'bb' for boundary-boundary
        • \n
      • 'bib' for boundary-inner-boundary
        • \n
      • \n
    • noffset_list (list[int]):\nOffset of the source (only relevant when wavefunctions are used)
      • \n
    • wf_list (int):\nID of wave function
      • \n
    • wf2_list (list[int]):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
      • \n
    • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • ens_name (str):\nreplaces the name of the ensemble
      • \n
    • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
      • \n
    • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
      • \n
    • replica (list):\nlist of replica to be read, default is all
      • \n
    • files (list[list[int]]):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
      • \n
    • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (dict[list[Obs]]):\ndict with one of the following properties:\nif keyed_out:\n    dict[key] = list[Obs]\n    where key has the form name/quarks/offset/wf/wf2\nif not keyed_out:\n    dict[name][quarks][offset][wf][wf2] = list[Obs]
      • \n
    \n", "signature": "(\tpath,\tprefix,\tname_list,\tquarks_list=['.*'],\tcorr_type_list=['bi'],\tnoffset_list=[0],\twf_list=[0],\twf2_list=[0],\tversion='1.0c',\tcfg_separator='n',\tsilent=False,\tkeyed_out=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.utils": {"fullname": "pyerrors.input.utils", "modulename": "pyerrors.input.utils", "kind": "module", "doc": "
    Utilities for the input
    \n"}, "pyerrors.input.utils.sort_names": {"fullname": "pyerrors.input.utils.sort_names", "modulename": "pyerrors.input.utils", "qualname": "sort_names", "kind": "function", "doc": "
    Sorts a list of names of replika with searches for r and id in the replikum string.\nIf this search fails, a fallback method is used,\nwhere the strings are simply compared and the first diffeing numeral is used for differentiation.
    \n\n
    Parameters
    \n\n
      \n
    • ll (list):\nlist to sort
      • \n
    \n\n
    Returns
    \n\n
      \n
    • ll (list):\nsorted list
      • \n
    \n", "signature": "(ll):", "funcdef": "def"}, "pyerrors.input.utils.check_idl": {"fullname": "pyerrors.input.utils.check_idl", "modulename": "pyerrors.input.utils", "qualname": "check_idl", "kind": "function", "doc": "
    Checks if list of configurations is contained in an idl
    \n\n
    Parameters
    \n\n
      \n
    • idl (range or list):\nidl of the current replicum
      • \n
    • che (list):\nlist of configurations to be checked against
      • \n
    \n\n
    Returns
    \n\n
      \n
    • miss_str (str):\nstring with integers of which idls are missing
      • \n
    \n", "signature": "(idl, che):", "funcdef": "def"}, "pyerrors.input.utils.check_params": {"fullname": "pyerrors.input.utils.check_params", "modulename": "pyerrors.input.utils", "qualname": "check_params", "kind": "function", "doc": "
    Check if, for sfcf, the parameter hashes at the end of the parameter files are in fact the expected one.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nmeasurement path, same as for sfcf read method
      • \n
    • param_hash (str):\nexpected parameter hash
      • \n
    • prefix (str):\ndata prefix to find the appropriate replicum folders in path
      • \n
    • param_prefix (str):\nprefix of the parameter file. Defaults to 'parameters_'
      • \n
    \n\n
    Returns
    \n\n
      \n
    • nums (dict):\ndictionary of faulty parameter files sorted by the replica paths
      • \n
    \n", "signature": "(path, param_hash, prefix, param_prefix='parameters_'):", "funcdef": "def"}, "pyerrors.integrate": {"fullname": "pyerrors.integrate", "modulename": "pyerrors.integrate", "kind": "module", "doc": "
    \n"}, "pyerrors.integrate.quad": {"fullname": "pyerrors.integrate.quad", "modulename": "pyerrors.integrate", "qualname": "quad", "kind": "function", "doc": "
    Performs a (one-dimensional) numeric integration of f(p, x) from a to b.
    \n\n
    The integration is performed using scipy.integrate.quad().\nAll parameters that can be passed to scipy.integrate.quad may also be passed to this function.\nThe output is the same as for scipy.integrate.quad, the first element being an Obs.
    \n\n
    Parameters
    \n\n
      \n
    • func (object):\nfunction to integrate, has to be of the form
      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(p, x):\n    return p[0] + p[1] * x + p[2] * anp.sinh(x)\n
      \n
      \n\n
      where x is the integration variable.
      • \n
    • p (list of floats or Obs):\nparameters of the function func.
      • \n
    • a (float or Obs):\nLower limit of integration (use -numpy.inf for -infinity).
      • \n
    • b (float or Obs):\nUpper limit of integration (use -numpy.inf for -infinity).
      • \n
    • All parameters of scipy.integrate.quad
      • \n
    \n\n
    Returns
    \n\n
      \n
    • y (Obs):\nThe integral of func from a to b.
      • \n
    • abserr (float):\nAn estimate of the absolute error in the result.
      • \n
    • infodict (dict):\nA dictionary containing additional information.\nRun scipy.integrate.quad_explain() for more information.
      • \n
    • message: A convergence message.
      • \n
    • explain: Appended only with 'cos' or 'sin' weighting and infinite\nintegration limits, it contains an explanation of the codes in\ninfodict['ierlst']
      • \n
    \n", "signature": "(func, p, a, b, **kwargs):", "funcdef": "def"}, "pyerrors.linalg": {"fullname": "pyerrors.linalg", "modulename": "pyerrors.linalg", "kind": "module", "doc": "
    \n"}, "pyerrors.linalg.matmul": {"fullname": "pyerrors.linalg.matmul", "modulename": "pyerrors.linalg", "qualname": "matmul", "kind": "function", "doc": "
    Matrix multiply all operands.
    \n\n
    Parameters
    \n\n
      \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    • This implementation is faster compared to standard multiplication via the @ operator.
      • \n
    \n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.jack_matmul": {"fullname": "pyerrors.linalg.jack_matmul", "modulename": "pyerrors.linalg", "qualname": "jack_matmul", "kind": "function", "doc": "
    Matrix multiply both operands making use of the jackknife approximation.
    \n\n
    Parameters
    \n\n
      \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    • For large matrices this is considerably faster compared to matmul.
      • \n
    \n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.einsum": {"fullname": "pyerrors.linalg.einsum", "modulename": "pyerrors.linalg", "qualname": "einsum", "kind": "function", "doc": "
    Wrapper for numpy.einsum
    \n\n
    Parameters
    \n\n
      \n
    • subscripts (str):\nSubscripts for summation (see numpy documentation for details)
      • \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    \n", "signature": "(subscripts, *operands):", "funcdef": "def"}, "pyerrors.linalg.inv": {"fullname": "pyerrors.linalg.inv", "modulename": "pyerrors.linalg", "qualname": "inv", "kind": "function", "doc": "
    Inverse of Obs or CObs valued matrices.
    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.cholesky": {"fullname": "pyerrors.linalg.cholesky", "modulename": "pyerrors.linalg", "qualname": "cholesky", "kind": "function", "doc": "
    Cholesky decomposition of Obs valued matrices.
    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.det": {"fullname": "pyerrors.linalg.det", "modulename": "pyerrors.linalg", "qualname": "det", "kind": "function", "doc": "
    Determinant of Obs valued matrices.
    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.eigh": {"fullname": "pyerrors.linalg.eigh", "modulename": "pyerrors.linalg", "qualname": "eigh", "kind": "function", "doc": "
    Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eig": {"fullname": "pyerrors.linalg.eig", "modulename": "pyerrors.linalg", "qualname": "eig", "kind": "function", "doc": "
    Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig.
    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eigv": {"fullname": "pyerrors.linalg.eigv", "modulename": "pyerrors.linalg", "qualname": "eigv", "kind": "function", "doc": "
    Computes the eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.pinv": {"fullname": "pyerrors.linalg.pinv", "modulename": "pyerrors.linalg", "qualname": "pinv", "kind": "function", "doc": "
    Computes the Moore-Penrose pseudoinverse of a matrix of Obs.
    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.svd": {"fullname": "pyerrors.linalg.svd", "modulename": "pyerrors.linalg", "qualname": "svd", "kind": "function", "doc": "
    Computes the singular value decomposition of a matrix of Obs.
    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.misc": {"fullname": "pyerrors.misc", "modulename": "pyerrors.misc", "kind": "module", "doc": "
    \n"}, "pyerrors.misc.print_config": {"fullname": "pyerrors.misc.print_config", "modulename": "pyerrors.misc", "qualname": "print_config", "kind": "function", "doc": "
    Print information about version of python, pyerrors and dependencies.
    \n", "signature": "():", "funcdef": "def"}, "pyerrors.misc.errorbar": {"fullname": "pyerrors.misc.errorbar", "modulename": "pyerrors.misc", "qualname": "errorbar", "kind": "function", "doc": "
    pyerrors wrapper for the errorbars method of matplotlib
    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nA list of x-values which can be Obs.
      • \n
    • y (list):\nA list of y-values which can be Obs.
      • \n
    • axes ((matplotlib.pyplot.axes)):\nThe axes to plot on. default is plt.
      • \n
    \n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.10.14/x64/lib/python3.10/site-packages/matplotlib/pyplot.py'>,\t**kwargs):", "funcdef": "def"}, "pyerrors.misc.dump_object": {"fullname": "pyerrors.misc.dump_object", "modulename": "pyerrors.misc", "qualname": "dump_object", "kind": "function", "doc": "
    Dump object into pickle file.
    \n\n
    Parameters
    \n\n
      \n
    • obj (object):\nobject to be saved in the pickle file
      • \n
    • name (str):\nname of the file
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(obj, name, **kwargs):", "funcdef": "def"}, "pyerrors.misc.load_object": {"fullname": "pyerrors.misc.load_object", "modulename": "pyerrors.misc", "qualname": "load_object", "kind": "function", "doc": "
    Load object from pickle file.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the file
      • \n
    \n\n
    Returns
    \n\n
      \n
    • object (Obs):\nLoaded Object
      • \n
    \n", "signature": "(path):", "funcdef": "def"}, "pyerrors.misc.pseudo_Obs": {"fullname": "pyerrors.misc.pseudo_Obs", "modulename": "pyerrors.misc", "qualname": "pseudo_Obs", "kind": "function", "doc": "
    Generate an Obs object with given value, dvalue and name for test purposes
    \n\n
    Parameters
    \n\n
      \n
    • value (float):\ncentral value of the Obs to be generated.
      • \n
    • dvalue (float):\nerror of the Obs to be generated.
      • \n
    • name (str):\nname of the ensemble for which the Obs is to be generated.
      • \n
    • samples (int):\nnumber of samples for the Obs (default 1000).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (Obs):\nGenerated Observable
      • \n
    \n", "signature": "(value, dvalue, name, samples=1000):", "funcdef": "def"}, "pyerrors.misc.gen_correlated_data": {"fullname": "pyerrors.misc.gen_correlated_data", "modulename": "pyerrors.misc", "qualname": "gen_correlated_data", "kind": "function", "doc": "
    Generate observables with given covariance and autocorrelation times.
    \n\n
    Parameters
    \n\n
      \n
    • means (list):\nlist containing the mean value of each observable.
      • \n
    • cov (numpy.ndarray):\ncovariance matrix for the data to be generated.
      • \n
    • name (str):\nensemble name for the data to be geneated.
      • \n
    • tau (float or list):\ncan either be a real number or a list with an entry for\nevery dataset.
      • \n
    • samples (int):\nnumber of samples to be generated for each observable.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr_obs (list[Obs]):\nGenerated observable list
      • \n
    \n", "signature": "(means, cov, name, tau=0.5, samples=1000):", "funcdef": "def"}, "pyerrors.mpm": {"fullname": "pyerrors.mpm", "modulename": "pyerrors.mpm", "kind": "module", "doc": "
    \n"}, "pyerrors.mpm.matrix_pencil_method": {"fullname": "pyerrors.mpm.matrix_pencil_method", "modulename": "pyerrors.mpm", "qualname": "matrix_pencil_method", "kind": "function", "doc": "
    Matrix pencil method to extract k energy levels from data
    \n\n
    Implementation of the matrix pencil method based on\neq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)
    \n\n
    Parameters
    \n\n
      \n
    • data (list):\ncan be a list of Obs for the analysis of a single correlator, or a list of lists\nof Obs if several correlators are to analyzed at once.
      • \n
    • k (int):\nNumber of states to extract (default 1).
      • \n
    • p (int):\nmatrix pencil parameter which filters noise. The optimal value is expected between\nlen(data)/3 and 2*len(data)/3. The computation is more expensive the closer p is\nto len(data)/2 but could possibly suppress more noise (default len(data)//2).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • energy_levels (list[Obs]):\nExtracted energy levels
      • \n
    \n", "signature": "(corrs, k=1, p=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs": {"fullname": "pyerrors.obs", "modulename": "pyerrors.obs", "kind": "module", "doc": "
    \n"}, "pyerrors.obs.Obs": {"fullname": "pyerrors.obs.Obs", "modulename": "pyerrors.obs", "qualname": "Obs", "kind": "class", "doc": "
    Class for a general observable.
    \n\n
    Instances of Obs are the basic objects of a pyerrors error analysis.\nThey are initialized with a list which contains arrays of samples for\ndifferent ensembles/replica and another list of same length which contains\nthe names of the ensembles/replica. Mathematical operations can be\nperformed on instances. The result is another instance of Obs. The error of\nan instance can be computed with the gamma_method. Also contains additional\nmethods for output and visualization of the error calculation.
    \n\n
    Attributes
    \n\n
      \n
    • S_global (float):\nStandard value for S (default 2.0)
      • \n
    • S_dict (dict):\nDictionary for S values. If an entry for a given ensemble\nexists this overwrites the standard value for that ensemble.
      • \n
    • tau_exp_global (float):\nStandard value for tau_exp (default 0.0)
      • \n
    • tau_exp_dict (dict):\nDictionary for tau_exp values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
      • \n
    • N_sigma_global (float):\nStandard value for N_sigma (default 1.0)
      • \n
    • N_sigma_dict (dict):\nDictionary for N_sigma values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
      • \n
    \n"}, "pyerrors.obs.Obs.__init__": {"fullname": "pyerrors.obs.Obs.__init__", "modulename": "pyerrors.obs", "qualname": "Obs.__init__", "kind": "function", "doc": "
    Initialize Obs object.
    \n\n
    Parameters
    \n\n
      \n
    • samples (list):\nlist of numpy arrays containing the Monte Carlo samples
      • \n
    • names (list):\nlist of strings labeling the individual samples
      • \n
    • idl (list, optional):\nlist of ranges or lists on which the samples are defined
      • \n
    \n", "signature": "(samples, names, idl=None, **kwargs)"}, "pyerrors.obs.Obs.S_global": {"fullname": "pyerrors.obs.Obs.S_global", "modulename": "pyerrors.obs", "qualname": "Obs.S_global", "kind": "variable", "doc": "
    \n", "default_value": "2.0"}, "pyerrors.obs.Obs.S_dict": {"fullname": "pyerrors.obs.Obs.S_dict", "modulename": "pyerrors.obs", "qualname": "Obs.S_dict", "kind": "variable", "doc": "
    \n", "default_value": "{}"}, "pyerrors.obs.Obs.tau_exp_global": {"fullname": "pyerrors.obs.Obs.tau_exp_global", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_global", "kind": "variable", "doc": "
    \n", "default_value": "0.0"}, "pyerrors.obs.Obs.tau_exp_dict": {"fullname": "pyerrors.obs.Obs.tau_exp_dict", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_dict", "kind": "variable", "doc": "
    \n", "default_value": "{}"}, "pyerrors.obs.Obs.N_sigma_global": {"fullname": "pyerrors.obs.Obs.N_sigma_global", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_global", "kind": "variable", "doc": "
    \n", "default_value": "1.0"}, "pyerrors.obs.Obs.N_sigma_dict": {"fullname": "pyerrors.obs.Obs.N_sigma_dict", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_dict", "kind": "variable", "doc": "
    \n", "default_value": "{}"}, "pyerrors.obs.Obs.names": {"fullname": "pyerrors.obs.Obs.names", "modulename": "pyerrors.obs", "qualname": "Obs.names", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.shape": {"fullname": "pyerrors.obs.Obs.shape", "modulename": "pyerrors.obs", "qualname": "Obs.shape", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.r_values": {"fullname": "pyerrors.obs.Obs.r_values", "modulename": "pyerrors.obs", "qualname": "Obs.r_values", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.deltas": {"fullname": "pyerrors.obs.Obs.deltas", "modulename": "pyerrors.obs", "qualname": "Obs.deltas", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.N": {"fullname": "pyerrors.obs.Obs.N", "modulename": "pyerrors.obs", "qualname": "Obs.N", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.idl": {"fullname": "pyerrors.obs.Obs.idl", "modulename": "pyerrors.obs", "qualname": "Obs.idl", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.ddvalue": {"fullname": "pyerrors.obs.Obs.ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.ddvalue", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.reweighted": {"fullname": "pyerrors.obs.Obs.reweighted", "modulename": "pyerrors.obs", "qualname": "Obs.reweighted", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.tag": {"fullname": "pyerrors.obs.Obs.tag", "modulename": "pyerrors.obs", "qualname": "Obs.tag", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.value": {"fullname": "pyerrors.obs.Obs.value", "modulename": "pyerrors.obs", "qualname": "Obs.value", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.dvalue": {"fullname": "pyerrors.obs.Obs.dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.dvalue", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_names": {"fullname": "pyerrors.obs.Obs.e_names", "modulename": "pyerrors.obs", "qualname": "Obs.e_names", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.cov_names": {"fullname": "pyerrors.obs.Obs.cov_names", "modulename": "pyerrors.obs", "qualname": "Obs.cov_names", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.mc_names": {"fullname": "pyerrors.obs.Obs.mc_names", "modulename": "pyerrors.obs", "qualname": "Obs.mc_names", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_content": {"fullname": "pyerrors.obs.Obs.e_content", "modulename": "pyerrors.obs", "qualname": "Obs.e_content", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.covobs": {"fullname": "pyerrors.obs.Obs.covobs", "modulename": "pyerrors.obs", "qualname": "Obs.covobs", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.gamma_method": {"fullname": "pyerrors.obs.Obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "Obs.gamma_method", "kind": "function", "doc": "
    Estimate the error and related properties of the Obs.
    \n\n
    Parameters
    \n\n
      \n
    • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
      • \n
    • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
      • \n
    • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
      • \n
    • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
      • \n
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.gm": {"fullname": "pyerrors.obs.Obs.gm", "modulename": "pyerrors.obs", "qualname": "Obs.gm", "kind": "function", "doc": "
    Estimate the error and related properties of the Obs.
    \n\n
    Parameters
    \n\n
      \n
    • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
      • \n
    • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
      • \n
    • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
      • \n
    • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
      • \n
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.details": {"fullname": "pyerrors.obs.Obs.details", "modulename": "pyerrors.obs", "qualname": "Obs.details", "kind": "function", "doc": "
    Output detailed properties of the Obs.
    \n\n
    Parameters
    \n\n
      \n
    • ens_content (bool):\nprint details about the ensembles and replica if true.
      • \n
    \n", "signature": "(self, ens_content=True):", "funcdef": "def"}, "pyerrors.obs.Obs.reweight": {"fullname": "pyerrors.obs.Obs.reweight", "modulename": "pyerrors.obs", "qualname": "Obs.reweight", "kind": "function", "doc": "
    Reweight the obs with given rewighting factors.
    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
      • \n
    \n", "signature": "(self, weight):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero_within_error": {"fullname": "pyerrors.obs.Obs.is_zero_within_error", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero_within_error", "kind": "function", "doc": "
    Checks whether the observable is zero within 'sigma' standard errors.
    \n\n
    Parameters
    \n\n
      \n
    • sigma (int):\nNumber of standard errors used for the check.
      • \n
    • Works only properly when the gamma method was run.
      • \n
    \n", "signature": "(self, sigma=1):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero": {"fullname": "pyerrors.obs.Obs.is_zero", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero", "kind": "function", "doc": "
    Checks whether the observable is zero within a given tolerance.
    \n\n
    Parameters
    \n\n
      \n
    • atol (float):\nAbsolute tolerance (for details see numpy documentation).
      • \n
    \n", "signature": "(self, atol=1e-10):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_tauint": {"fullname": "pyerrors.obs.Obs.plot_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.plot_tauint", "kind": "function", "doc": "
    Plot integrated autocorrelation time for each ensemble.
    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rho": {"fullname": "pyerrors.obs.Obs.plot_rho", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rho", "kind": "function", "doc": "
    Plot normalized autocorrelation function time for each ensemble.
    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rep_dist": {"fullname": "pyerrors.obs.Obs.plot_rep_dist", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rep_dist", "kind": "function", "doc": "
    Plot replica distribution for each ensemble with more than one replicum.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_history": {"fullname": "pyerrors.obs.Obs.plot_history", "modulename": "pyerrors.obs", "qualname": "Obs.plot_history", "kind": "function", "doc": "
    Plot derived Monte Carlo history for each ensemble
    \n\n
    Parameters
    \n\n
      \n
    • expand (bool):\nshow expanded history for irregular Monte Carlo chains (default: True).
      • \n
    \n", "signature": "(self, expand=True):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_piechart": {"fullname": "pyerrors.obs.Obs.plot_piechart", "modulename": "pyerrors.obs", "qualname": "Obs.plot_piechart", "kind": "function", "doc": "
    Plot piechart which shows the fractional contribution of each\nensemble to the error and returns a dictionary containing the fractions.
    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.dump": {"fullname": "pyerrors.obs.Obs.dump", "modulename": "pyerrors.obs", "qualname": "Obs.dump", "kind": "function", "doc": "
    Dump the Obs to a file 'name' of chosen format.
    \n\n
    Parameters
    \n\n
      \n
    • filename (str):\nname of the file to be saved.
      • \n
    • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
      • \n
    • description (str):\nDescription for output file, only relevant for json.gz format.
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n", "signature": "(self, filename, datatype='json.gz', description='', **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.export_jackknife": {"fullname": "pyerrors.obs.Obs.export_jackknife", "modulename": "pyerrors.obs", "qualname": "Obs.export_jackknife", "kind": "function", "doc": "
    Export jackknife samples from the Obs
    \n\n
    Returns
    \n\n
      \n
    • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N jackknife samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived jackknife samples\nshould agree with samples from a full jackknife analysis up to O(1/N).
      • \n
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.export_bootstrap": {"fullname": "pyerrors.obs.Obs.export_bootstrap", "modulename": "pyerrors.obs", "qualname": "Obs.export_bootstrap", "kind": "function", "doc": "
    Export bootstrap samples from the Obs
    \n\n
    Parameters
    \n\n
      \n
    • samples (int):\nNumber of bootstrap samples to generate.
      • \n
    • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples.\nIf not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
      • \n
    • save_rng (str):\nSave the random numbers to a file if a path is specified.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N import_bootstrap samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived bootstrap samples\nshould agree with samples from a full bootstrap analysis up to O(1/N).
      • \n
    \n", "signature": "(self, samples=500, random_numbers=None, save_rng=None):", "funcdef": "def"}, "pyerrors.obs.Obs.sqrt": {"fullname": "pyerrors.obs.Obs.sqrt", "modulename": "pyerrors.obs", "qualname": "Obs.sqrt", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.log": {"fullname": "pyerrors.obs.Obs.log", "modulename": "pyerrors.obs", "qualname": "Obs.log", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.exp": {"fullname": "pyerrors.obs.Obs.exp", "modulename": "pyerrors.obs", "qualname": "Obs.exp", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sin": {"fullname": "pyerrors.obs.Obs.sin", "modulename": "pyerrors.obs", "qualname": "Obs.sin", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cos": {"fullname": "pyerrors.obs.Obs.cos", "modulename": "pyerrors.obs", "qualname": "Obs.cos", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tan": {"fullname": "pyerrors.obs.Obs.tan", "modulename": "pyerrors.obs", "qualname": "Obs.tan", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsin": {"fullname": "pyerrors.obs.Obs.arcsin", "modulename": "pyerrors.obs", "qualname": "Obs.arcsin", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccos": {"fullname": "pyerrors.obs.Obs.arccos", "modulename": "pyerrors.obs", "qualname": "Obs.arccos", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctan": {"fullname": "pyerrors.obs.Obs.arctan", "modulename": "pyerrors.obs", "qualname": "Obs.arctan", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sinh": {"fullname": "pyerrors.obs.Obs.sinh", "modulename": "pyerrors.obs", "qualname": "Obs.sinh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cosh": {"fullname": "pyerrors.obs.Obs.cosh", "modulename": "pyerrors.obs", "qualname": "Obs.cosh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tanh": {"fullname": "pyerrors.obs.Obs.tanh", "modulename": "pyerrors.obs", "qualname": "Obs.tanh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsinh": {"fullname": "pyerrors.obs.Obs.arcsinh", "modulename": "pyerrors.obs", "qualname": "Obs.arcsinh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccosh": {"fullname": "pyerrors.obs.Obs.arccosh", "modulename": "pyerrors.obs", "qualname": "Obs.arccosh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctanh": {"fullname": "pyerrors.obs.Obs.arctanh", "modulename": "pyerrors.obs", "qualname": "Obs.arctanh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.N_sigma": {"fullname": "pyerrors.obs.Obs.N_sigma", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.S": {"fullname": "pyerrors.obs.Obs.S", "modulename": "pyerrors.obs", "qualname": "Obs.S", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_ddvalue": {"fullname": "pyerrors.obs.Obs.e_ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_ddvalue", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_drho": {"fullname": "pyerrors.obs.Obs.e_drho", "modulename": "pyerrors.obs", "qualname": "Obs.e_drho", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_dtauint": {"fullname": "pyerrors.obs.Obs.e_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_dtauint", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_dvalue": {"fullname": "pyerrors.obs.Obs.e_dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_dvalue", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_n_dtauint": {"fullname": "pyerrors.obs.Obs.e_n_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_dtauint", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_n_tauint": {"fullname": "pyerrors.obs.Obs.e_n_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_tauint", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_rho": {"fullname": "pyerrors.obs.Obs.e_rho", "modulename": "pyerrors.obs", "qualname": "Obs.e_rho", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_tauint": {"fullname": "pyerrors.obs.Obs.e_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_tauint", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_windowsize": {"fullname": "pyerrors.obs.Obs.e_windowsize", "modulename": "pyerrors.obs", "qualname": "Obs.e_windowsize", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.tau_exp": {"fullname": "pyerrors.obs.Obs.tau_exp", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.CObs": {"fullname": "pyerrors.obs.CObs", "modulename": "pyerrors.obs", "qualname": "CObs", "kind": "class", "doc": "
    Class for a complex valued observable.
    \n"}, "pyerrors.obs.CObs.__init__": {"fullname": "pyerrors.obs.CObs.__init__", "modulename": "pyerrors.obs", "qualname": "CObs.__init__", "kind": "function", "doc": "
    \n", "signature": "(real, imag=0.0)"}, "pyerrors.obs.CObs.tag": {"fullname": "pyerrors.obs.CObs.tag", "modulename": "pyerrors.obs", "qualname": "CObs.tag", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.CObs.real": {"fullname": "pyerrors.obs.CObs.real", "modulename": "pyerrors.obs", "qualname": "CObs.real", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.CObs.imag": {"fullname": "pyerrors.obs.CObs.imag", "modulename": "pyerrors.obs", "qualname": "CObs.imag", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.CObs.gamma_method": {"fullname": "pyerrors.obs.CObs.gamma_method", "modulename": "pyerrors.obs", "qualname": "CObs.gamma_method", "kind": "function", "doc": "
    Executes the gamma_method for the real and the imaginary part.
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.CObs.is_zero": {"fullname": "pyerrors.obs.CObs.is_zero", "modulename": "pyerrors.obs", "qualname": "CObs.is_zero", "kind": "function", "doc": "
    Checks whether both real and imaginary part are zero within machine precision.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs.conjugate": {"fullname": "pyerrors.obs.CObs.conjugate", "modulename": "pyerrors.obs", "qualname": "CObs.conjugate", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.gamma_method": {"fullname": "pyerrors.obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "gamma_method", "kind": "function", "doc": "
    Vectorized version of the gamma_method applicable to lists or arrays of Obs.
    \n\n
    See docstring of pe.Obs.gamma_method for details.
    \n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.gm": {"fullname": "pyerrors.obs.gm", "modulename": "pyerrors.obs", "qualname": "gm", "kind": "function", "doc": "
    Vectorized version of the gamma_method applicable to lists or arrays of Obs.
    \n\n
    See docstring of pe.Obs.gamma_method for details.
    \n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.derived_observable": {"fullname": "pyerrors.obs.derived_observable", "modulename": "pyerrors.obs", "qualname": "derived_observable", "kind": "function", "doc": "
    Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.
    \n\n
    Parameters
    \n\n
      \n
    • func (object):\narbitrary function of the form func(data, **kwargs). For the\nautomatic differentiation to work, all numpy functions have to have\nthe autograd wrapper (use 'import autograd.numpy as anp').
      • \n
    • data (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
      • \n
    • num_grad (bool):\nif True, numerical derivatives are used instead of autograd\n(default False). To control the numerical differentiation the\nkwargs of numdifftools.step_generators.MaxStepGenerator\ncan be used.
      • \n
    • man_grad (list):\nmanually supply a list or an array which contains the jacobian\nof func. Use cautiously, supplying the wrong derivative will\nnot be intercepted.
      • \n
    \n\n
    Notes
    \n\n
    For simple mathematical operations it can be practical to use anonymous\nfunctions. For the ratio of two observables one can e.g. use
    \n\n
    new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])
    \n", "signature": "(func, data, array_mode=False, **kwargs):", "funcdef": "def"}, "pyerrors.obs.reweight": {"fullname": "pyerrors.obs.reweight", "modulename": "pyerrors.obs", "qualname": "reweight", "kind": "function", "doc": "
    Reweight a list of observables.
    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • obs (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
      • \n
    \n", "signature": "(weight, obs, **kwargs):", "funcdef": "def"}, "pyerrors.obs.correlate": {"fullname": "pyerrors.obs.correlate", "modulename": "pyerrors.obs", "qualname": "correlate", "kind": "function", "doc": "
    Correlate two observables.
    \n\n
    Parameters
    \n\n
      \n
    • obs_a (Obs):\nFirst observable
      • \n
    • obs_b (Obs):\nSecond observable
      • \n
    \n\n
    Notes
    \n\n
    Keep in mind to only correlate primary observables which have not been reweighted\nyet. The reweighting has to be applied after correlating the observables.\nCurrently only works if ensembles are identical (this is not strictly necessary).
    \n", "signature": "(obs_a, obs_b):", "funcdef": "def"}, "pyerrors.obs.covariance": {"fullname": "pyerrors.obs.covariance", "modulename": "pyerrors.obs", "qualname": "covariance", "kind": "function", "doc": "
    Calculates the error covariance matrix of a set of observables.
    \n\n
    WARNING: This function should be used with care, especially for observables with support on multiple\n         ensembles with differing autocorrelations. See the notes below for details.
    \n\n
    The gamma method has to be applied first to all observables.
    \n\n
    Parameters
    \n\n
      \n
    • obs (list or numpy.ndarray):\nList or one dimensional array of Obs
      • \n
    • visualize (bool):\nIf True plots the corresponding normalized correlation matrix (default False).
      • \n
    • correlation (bool):\nIf True the correlation matrix instead of the error covariance matrix is returned (default False).
      • \n
    • smooth (None or int):\nIf smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue\nsmoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the\nlargest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely\nsmall ones.
      • \n
    \n\n
    Notes
    \n\n
    The error covariance is defined such that it agrees with the squared standard error for two identical observables\n$$\\operatorname{cov}(a,a)=\\sum_{s=1}^N\\delta_a^s\\delta_a^s/N^2=\\Gamma_{aa}(0)/N=\\operatorname{var}(a)/N=\\sigma_a^2$$\nin the absence of autocorrelation.\nThe error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite\n$$\\sum_{i,j}v_i\\Gamma_{ij}(0)v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i,j}v_i\\delta_i^s\\delta_j^s v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i}|v_i\\delta_i^s|^2\\geq 0\\,,$$ for every $v\\in\\mathbb{R}^M$, while such an identity does not hold for larger windows/lags.\nFor observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.\n$$\\tau_{\\mathrm{int}, ij}=\\sqrt{\\tau_{\\mathrm{int}, i}\\times \\tau_{\\mathrm{int}, j}}$$\nThis construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).
    \n", "signature": "(obs, visualize=False, correlation=False, smooth=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs.import_jackknife": {"fullname": "pyerrors.obs.import_jackknife", "modulename": "pyerrors.obs", "qualname": "import_jackknife", "kind": "function", "doc": "
    Imports jackknife samples and returns an Obs
    \n\n
    Parameters
    \n\n
      \n
    • jacks (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N jackknife samples as first to Nth entry.
      • \n
    • name (str):\nname of the ensemble the samples are defined on.
      • \n
    \n", "signature": "(jacks, name, idl=None):", "funcdef": "def"}, "pyerrors.obs.import_bootstrap": {"fullname": "pyerrors.obs.import_bootstrap", "modulename": "pyerrors.obs", "qualname": "import_bootstrap", "kind": "function", "doc": "
    Imports bootstrap samples and returns an Obs
    \n\n
    Parameters
    \n\n
      \n
    • boots (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N bootstrap samples as first to Nth entry.
      • \n
    • name (str):\nname of the ensemble the samples are defined on.
      • \n
    • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples,\nwhere samples is the number of bootstrap samples and length is the length of the original Monte Carlo\nchain to be reconstructed.
      • \n
    \n", "signature": "(boots, name, random_numbers):", "funcdef": "def"}, "pyerrors.obs.merge_obs": {"fullname": "pyerrors.obs.merge_obs", "modulename": "pyerrors.obs", "qualname": "merge_obs", "kind": "function", "doc": "
    Combine all observables in list_of_obs into one new observable
    \n\n
    Parameters
    \n\n
      \n
    • list_of_obs (list):\nlist of the Obs object to be combined
      • \n
    \n\n
    Notes
    \n\n
    It is not possible to combine obs which are based on the same replicum
    \n", "signature": "(list_of_obs):", "funcdef": "def"}, "pyerrors.obs.cov_Obs": {"fullname": "pyerrors.obs.cov_Obs", "modulename": "pyerrors.obs", "qualname": "cov_Obs", "kind": "function", "doc": "
    Create an Obs based on mean(s) and a covariance matrix
    \n\n
    Parameters
    \n\n
      \n
    • mean (list of floats or float):\nN mean value(s) of the new Obs
      • \n
    • cov (list or array):\n2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
      • \n
    • name (str):\nidentifier for the covariance matrix
      • \n
    • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
      • \n
    \n", "signature": "(means, cov, name, grad=None):", "funcdef": "def"}, "pyerrors.roots": {"fullname": "pyerrors.roots", "modulename": "pyerrors.roots", "kind": "module", "doc": "
    \n"}, "pyerrors.roots.find_root": {"fullname": "pyerrors.roots.find_root", "modulename": "pyerrors.roots", "qualname": "find_root", "kind": "function", "doc": "
    Finds the root of the function func(x, d) where d is an Obs.
    \n\n
    Parameters
    \n\n
      \n
    • d (Obs):\nObs passed to the function.
      • \n
    • func (object):\nFunction to be minimized. Any numpy functions have to use the autograd.numpy wrapper.\nExample:
      \n\n
      \n
      import autograd.numpy as anp\ndef root_func(x, d):\n    return anp.exp(-x ** 2) - d\n
      \n
      • \n
    • guess (float):\nInitial guess for the minimization.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (Obs):\nObs valued root of the function.
      • \n
    \n", "signature": "(d, func, guess=1.0, **kwargs):", "funcdef": "def"}, "pyerrors.special": {"fullname": "pyerrors.special", "modulename": "pyerrors.special", "kind": "module", "doc": "
    \n"}, "pyerrors.special.beta": {"fullname": "pyerrors.special.beta", "modulename": "pyerrors.special", "qualname": "beta", "kind": "function", "doc": "
    beta(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    beta(a, b, out=None)
    \n\n
    Beta function.
    \n\n
    This function is defined in 1 as
    \n\n
    $$B(a, b) = \\int_0^1 t^{a-1}(1-t)^{b-1}dt\n        = \\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)},$$
    \n\n
    where \\( \\Gamma \\) is the gamma function.
    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nReal-valued arguments
      • \n
    • out (ndarray, optional):\nOptional output array for the function result
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the beta function
      • \n
    \n\n
    See Also
    \n\n
    gamma:  the gamma function
    \nbetainc:   the regularized incomplete beta function
    \nbetaln:  the natural logarithm of the absolute\nvalue of the beta function  
    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n
    The beta function relates to the gamma function by the\ndefinition given above:
    \n\n
    \n
    >>> sc.beta(2, 3)\n0.08333333333333333\n>>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3)\n0.08333333333333333\n
    \n
    \n\n
    As this relationship demonstrates, the beta function\nis symmetric:
    \n\n
    \n
    >>> sc.beta(1.7, 2.4)\n0.16567527689031739\n>>> sc.beta(2.4, 1.7)\n0.16567527689031739\n
    \n
    \n\n
    This function satisfies \\( B(1, b) = 1/b \\):
    \n\n
    \n
    >>> sc.beta(1, 4)\n0.25\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      NIST Digital Library of Mathematical Functions,\nEq. 5.12.1. https://dlmf.nist.gov/5.12 ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betainc": {"fullname": "pyerrors.special.betainc", "modulename": "pyerrors.special", "qualname": "betainc", "kind": "function", "doc": "
    betainc(x1, x2, x3, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    betainc(a, b, x, out=None)
    \n\n
    Regularized incomplete beta function.
    \n\n
    Computes the regularized incomplete beta function, defined as 1:
    \n\n
    $$I_x(a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^x\nt^{a-1}(1-t)^{b-1}dt,$$
    \n\n
    for \\( 0 \\leq x \\leq 1 \\).
    \n\n
    This function is the cumulative distribution function for the beta\ndistribution; its range is [0, 1].
    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nPositive, real-valued parameters
      • \n
    • x (array_like):\nReal-valued such that \\( 0 \\leq x \\leq 1 \\),\nthe upper limit of integration
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the regularized incomplete beta function
      • \n
    \n\n
    See Also
    \n\n
    beta:  beta function
    \nbetaincinv:  inverse of the regularized incomplete beta function
    \nbetaincc:  complement of the regularized incomplete beta function
    \nscipy.stats.beta:  beta distribution  
    \n\n
    Notes
    \n\n
    The term regularized in the name of this function refers to the\nscaling of the function by the gamma function terms shown in the\nformula.  When not qualified as regularized, the name incomplete\nbeta function often refers to just the integral expression,\nwithout the gamma terms.  One can use the function beta from\nscipy.special to get this \"nonregularized\" incomplete beta\nfunction by multiplying the result of betainc(a, b, x) by\nbeta(a, b).
    \n\n
    References
    \n\n
    Examples
    \n\n
    Let \\( B(a, b) \\) be the beta function.
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n
    The coefficient in terms of gamma is equal to\n\\( 1/B(a, b) \\). Also, when \\( x=1 \\)\nthe integral is equal to \\( B(a, b) \\).\nTherefore, \\( I_{x=1}(a, b) = 1 \\) for any \\( a, b \\).
    \n\n
    \n
    >>> sc.betainc(0.2, 3.5, 1.0)\n1.0\n
    \n
    \n\n
    It satisfies\n\\( I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b)) \\),\nwhere \\( F \\) is the hypergeometric function hyp2f1:
    \n\n
    \n
    >>> a, b, x = 1.4, 3.1, 0.5\n>>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))\n0.8148904036225295\n>>> sc.betainc(a, b, x)\n0.8148904036225296\n
    \n
    \n\n
    This functions satisfies the relationship\n\\( I_x(a, b) = 1 - I_{1-x}(b, a) \\):
    \n\n
    \n
    >>> sc.betainc(2.2, 3.1, 0.4)\n0.49339638807619446\n>>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)\n0.49339638807619446\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/8.17 ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betaln": {"fullname": "pyerrors.special.betaln", "modulename": "pyerrors.special", "qualname": "betaln", "kind": "function", "doc": "
    betaln(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    betaln(a, b, out=None)
    \n\n
    Natural logarithm of absolute value of beta function.
    \n\n
    Computes ln(abs(beta(a, b))).
    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nPositive, real-valued parameters
      • \n
    • out (ndarray, optional):\nOptional output array for function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the betaln function
      • \n
    \n\n
    See Also
    \n\n
    gamma:  the gamma function
    \nbetainc:   the regularized incomplete beta function
    \nbeta:  the beta function  
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import betaln, beta\n
    \n
    \n\n
    Verify that, for moderate values of a and b, betaln(a, b)\nis the same as log(beta(a, b)):
    \n\n
    \n
    >>> betaln(3, 4)\n-4.0943445622221\n
    \n
    \n\n
    \n
    >>> np.log(beta(3, 4))\n-4.0943445622221\n
    \n
    \n\n
    In the following beta(a, b) underflows to 0, so we can't compute\nthe logarithm of the actual value.
    \n\n
    \n
    >>> a = 400\n>>> b = 900\n>>> beta(a, b)\n0.0\n
    \n
    \n\n
    We can compute the logarithm of beta(a, b) by using betaln:
    \n\n
    \n
    >>> betaln(a, b)\n-804.3069951764146\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.polygamma": {"fullname": "pyerrors.special.polygamma", "modulename": "pyerrors.special", "qualname": "polygamma", "kind": "function", "doc": "
    Polygamma functions.
    \n\n
    Defined as \\( \\psi^{(n)}(x) \\) where \\( \\psi \\) is the\ndigamma function. See [dlmf]_ for details.
    \n\n
    Parameters
    \n\n
      \n
    • n (array_like):\nThe order of the derivative of the digamma function; must be\nintegral
      • \n
    • x (array_like):\nReal valued input
      • \n
    \n\n
    Returns
    \n\n
      \n
    • ndarray: Function results
      • \n
    \n\n
    See Also
    \n\n
    digamma  
    \n\n
    References
    \n\n
    .. [dlmf] NIST, Digital Library of Mathematical Functions,\n    https://dlmf.nist.gov/5.15
    \n\n
    Examples
    \n\n
    \n
    >>> from scipy import special\n>>> x = [2, 3, 25.5]\n>>> special.polygamma(1, x)\narray([ 0.64493407,  0.39493407,  0.03999467])\n>>> special.polygamma(0, x) == special.psi(x)\narray([ True,  True,  True], dtype=bool)\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.psi": {"fullname": "pyerrors.special.psi", "modulename": "pyerrors.special", "qualname": "psi", "kind": "function", "doc": "
    psi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    psi(z, out=None)
    \n\n
    The digamma function.
    \n\n
    The logarithmic derivative of the gamma function evaluated at z.
    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex argument.
      • \n
    • out (ndarray, optional):\nArray for the computed values of psi.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • digamma (scalar or ndarray):\nComputed values of psi.
      • \n
    \n\n
    Notes
    \n\n
    For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.
    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    Verify psi(z) = psi(z + 1) - 1/z:
    \n\n
    \n
    >>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      2. \n\n
    2. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      3. \n\n
    3. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      4. \n\n
    4. \n
      Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
      \n
      5. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.digamma": {"fullname": "pyerrors.special.digamma", "modulename": "pyerrors.special", "qualname": "digamma", "kind": "function", "doc": "
    psi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    psi(z, out=None)
    \n\n
    The digamma function.
    \n\n
    The logarithmic derivative of the gamma function evaluated at z.
    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex argument.
      • \n
    • out (ndarray, optional):\nArray for the computed values of psi.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • digamma (scalar or ndarray):\nComputed values of psi.
      • \n
    \n\n
    Notes
    \n\n
    For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.
    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    Verify psi(z) = psi(z + 1) - 1/z:
    \n\n
    \n
    >>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      2. \n\n
    2. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      3. \n\n
    3. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      4. \n\n
    4. \n
      Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
      \n
      5. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gamma": {"fullname": "pyerrors.special.gamma", "modulename": "pyerrors.special", "qualname": "gamma", "kind": "function", "doc": "
    gamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    gamma(z, out=None)
    \n\n
    gamma function.
    \n\n
    The gamma function is defined as
    \n\n
    $$\\Gamma(z) = \\int_0^\\infty t^{z-1} e^{-t} dt$$
    \n\n
    for \\( \\Re(z) > 0 \\) and is extended to the rest of the complex\nplane by analytic continuation. See [dlmf]_ for more details.
    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex valued argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the gamma function
      • \n
    \n\n
    Notes
    \n\n
    The gamma function is often referred to as the generalized\nfactorial since \\( \\Gamma(n + 1) = n! \\) for natural numbers\n\\( n \\). More generally it satisfies the recurrence relation\n\\( \\Gamma(z + 1) = z \\cdot \\Gamma(z) \\) for complex \\( z \\),\nwhich, combined with the fact that \\( \\Gamma(1) = 1 \\), implies\nthe above identity for \\( z = n \\).
    \n\n
    References
    \n\n
    .. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5.2#E1
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import gamma, factorial\n
    \n
    \n\n
    \n
    >>> gamma([0, 0.5, 1, 5])\narray([         inf,   1.77245385,   1.        ,  24.        ])\n
    \n
    \n\n
    \n
    >>> z = 2.5 + 1j\n>>> gamma(z)\n(0.77476210455108352+0.70763120437959293j)\n>>> gamma(z+1), z*gamma(z)  # Recurrence property\n((1.2292740569981171+2.5438401155000685j),\n (1.2292740569981158+2.5438401155000658j))\n
    \n
    \n\n
    \n
    >>> gamma(0.5)**2  # gamma(0.5) = sqrt(pi)\n3.1415926535897927\n
    \n
    \n\n
    Plot gamma(x) for real x
    \n\n
    \n
    >>> x = np.linspace(-3.5, 5.5, 2251)\n>>> y = gamma(x)\n
    \n
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')\n>>> k = np.arange(1, 7)\n>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,\n...          label='(x-1)!, x = 1, 2, ...')\n>>> plt.xlim(-3.5, 5.5)\n>>> plt.ylim(-10, 25)\n>>> plt.grid()\n>>> plt.xlabel('x')\n>>> plt.legend(loc='lower right')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaln": {"fullname": "pyerrors.special.gammaln", "modulename": "pyerrors.special", "qualname": "gammaln", "kind": "function", "doc": "
    gammaln(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    gammaln(x, out=None)
    \n\n
    Logarithm of the absolute value of the gamma function.
    \n\n
    Defined as
    \n\n
    $$\\ln(\\lvert\\Gamma(x)\\rvert)$$
    \n\n
    where \\( \\Gamma \\) is the gamma function. For more details on\nthe gamma function, see [dlmf]_.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the log of the absolute value of gamma
      • \n
    \n\n
    See Also
    \n\n
    gammasgn:  sign of the gamma function
    \nloggamma:  principal branch of the logarithm of the gamma function  
    \n\n
    Notes
    \n\n
    It is the same function as the Python standard library function\nmath.lgamma().
    \n\n
    When used in conjunction with gammasgn, this function is useful\nfor working in logspace on the real axis without having to deal\nwith complex numbers via the relation exp(gammaln(x)) =\ngammasgn(x) * gamma(x).
    \n\n
    For complex-valued log-gamma, use loggamma instead of gammaln.
    \n\n
    References
    \n\n
    .. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import scipy.special as sc\n
    \n
    \n\n
    It has two positive zeros.
    \n\n
    \n
    >>> sc.gammaln([1, 2])\narray([0., 0.])\n
    \n
    \n\n
    It has poles at nonpositive integers.
    \n\n
    \n
    >>> sc.gammaln([0, -1, -2, -3, -4])\narray([inf, inf, inf, inf, inf])\n
    \n
    \n\n
    It asymptotically approaches x * log(x) (Stirling's formula).
    \n\n
    \n
    >>> x = np.array([1e10, 1e20, 1e40, 1e80])\n>>> sc.gammaln(x)\narray([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82])\n>>> x * np.log(x)\narray([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammainc": {"fullname": "pyerrors.special.gammainc", "modulename": "pyerrors.special", "qualname": "gammainc", "kind": "function", "doc": "
    gammainc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    gammainc(a, x, out=None)
    \n\n
    Regularized lower incomplete gamma function.
    \n\n
    It is defined as
    \n\n
    $$P(a, x) = \\frac{1}{\\Gamma(a)} \\int_0^x t^{a - 1}e^{-t} dt$$
    \n\n
    for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nPositive parameter
      • \n
    • x (array_like):\nNonnegative argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the lower incomplete gamma function
      • \n
    \n\n
    See Also
    \n\n
    gammaincc:  regularized upper incomplete gamma function
    \ngammaincinv:  inverse of the regularized lower incomplete gamma function
    \ngammainccinv:  inverse of the regularized upper incomplete gamma function  
    \n\n
    Notes
    \n\n
    The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammaincc is the regularized upper\nincomplete gamma function.
    \n\n
    The implementation largely follows that of [boost]_.
    \n\n
    References
    \n\n
    .. [dlmf] NIST Digital Library of Mathematical functions\n          https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n   https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n
    It is the CDF of the gamma distribution, so it starts at 0 and\nmonotonically increases to 1.
    \n\n
    \n
    >>> sc.gammainc(0.5, [0, 1, 10, 100])\narray([0.        , 0.84270079, 0.99999226, 1.        ])\n
    \n
    \n\n
    It is equal to one minus the upper incomplete gamma function.
    \n\n
    \n
    >>> a, x = 0.5, 0.4\n>>> sc.gammainc(a, x)\n0.6289066304773024\n>>> 1 - sc.gammaincc(a, x)\n0.6289066304773024\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaincc": {"fullname": "pyerrors.special.gammaincc", "modulename": "pyerrors.special", "qualname": "gammaincc", "kind": "function", "doc": "
    gammaincc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    gammaincc(a, x, out=None)
    \n\n
    Regularized upper incomplete gamma function.
    \n\n
    It is defined as
    \n\n
    $$Q(a, x) = \\frac{1}{\\Gamma(a)} \\int_x^\\infty t^{a - 1}e^{-t} dt$$
    \n\n
    for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nPositive parameter
      • \n
    • x (array_like):\nNonnegative argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the upper incomplete gamma function
      • \n
    \n\n
    See Also
    \n\n
    gammainc:  regularized lower incomplete gamma function
    \ngammaincinv:  inverse of the regularized lower incomplete gamma function
    \ngammainccinv:  inverse of the regularized upper incomplete gamma function  
    \n\n
    Notes
    \n\n
    The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammainc is the regularized lower\nincomplete gamma function.
    \n\n
    The implementation largely follows that of [boost]_.
    \n\n
    References
    \n\n
    .. [dlmf] NIST Digital Library of Mathematical functions\n          https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n   https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n
    It is the survival function of the gamma distribution, so it\nstarts at 1 and monotonically decreases to 0.
    \n\n
    \n
    >>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000])\narray([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45,\n       0.00000000e+00])\n
    \n
    \n\n
    It is equal to one minus the lower incomplete gamma function.
    \n\n
    \n
    >>> a, x = 0.5, 0.4\n>>> sc.gammaincc(a, x)\n0.37109336952269756\n>>> 1 - sc.gammainc(a, x)\n0.37109336952269756\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammasgn": {"fullname": "pyerrors.special.gammasgn", "modulename": "pyerrors.special", "qualname": "gammasgn", "kind": "function", "doc": "
    gammasgn(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    gammasgn(x, out=None)
    \n\n
    Sign of the gamma function.
    \n\n
    It is defined as
    \n\n
    $$\\text{gammasgn}(x) =\n\\begin{cases}\n  +1 & \\Gamma(x) > 0 \\\n  -1 & \\Gamma(x) < 0\n\\end{cases}$$
    \n\n
    where \\( \\Gamma \\) is the gamma function; see gamma. This\ndefinition is complete since the gamma function is never zero;\nsee the discussion after [dlmf]_.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Sign of the gamma function
      • \n
    \n\n
    See Also
    \n\n
    gamma:  the gamma function
    \ngammaln:  log of the absolute value of the gamma function
    \nloggamma:  analytic continuation of the log of the gamma function  
    \n\n
    Notes
    \n\n
    The gamma function can be computed as gammasgn(x) *\nnp.exp(gammaln(x)).
    \n\n
    References
    \n\n
    .. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5.2#E1
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import scipy.special as sc\n
    \n
    \n\n
    It is 1 for x > 0.
    \n\n
    \n
    >>> sc.gammasgn([1, 2, 3, 4])\narray([1., 1., 1., 1.])\n
    \n
    \n\n
    It alternates between -1 and 1 for negative integers.
    \n\n
    \n
    >>> sc.gammasgn([-0.5, -1.5, -2.5, -3.5])\narray([-1.,  1., -1.,  1.])\n
    \n
    \n\n
    It can be used to compute the gamma function.
    \n\n
    \n
    >>> x = [1.5, 0.5, -0.5, -1.5]\n>>> sc.gammasgn(x) * np.exp(sc.gammaln(x))\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n>>> sc.gamma(x)\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.rgamma": {"fullname": "pyerrors.special.rgamma", "modulename": "pyerrors.special", "qualname": "rgamma", "kind": "function", "doc": "
    rgamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    rgamma(z, out=None)
    \n\n
    Reciprocal of the gamma function.
    \n\n
    Defined as \\( 1 / \\Gamma(z) \\), where \\( \\Gamma \\) is the\ngamma function. For more on the gamma function see gamma.
    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex valued input
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Function results
      • \n
    \n\n
    See Also
    \n\n
    gamma,, gammaln,, loggamma  
    \n\n
    Notes
    \n\n
    The gamma function has no zeros and has simple poles at\nnonpositive integers, so rgamma is an entire function with zeros\nat the nonpositive integers. See the discussion in [dlmf]_ for\nmore details.
    \n\n
    References
    \n\n
    .. [dlmf] Nist, Digital Library of Mathematical functions,\n    https://dlmf.nist.gov/5.2#i
    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n
    It is the reciprocal of the gamma function.
    \n\n
    \n
    >>> sc.rgamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n>>> 1 / sc.gamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n
    \n
    \n\n
    It is zero at nonpositive integers.
    \n\n
    \n
    >>> sc.rgamma([0, -1, -2, -3])\narray([0., 0., 0., 0.])\n
    \n
    \n\n
    It rapidly underflows to zero along the positive real axis.
    \n\n
    \n
    >>> sc.rgamma([10, 100, 179])\narray([2.75573192e-006, 1.07151029e-156, 0.00000000e+000])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.multigammaln": {"fullname": "pyerrors.special.multigammaln", "modulename": "pyerrors.special", "qualname": "multigammaln", "kind": "function", "doc": "
    Returns the log of multivariate gamma, also sometimes called the\ngeneralized gamma.
    \n\n
    Parameters
    \n\n
      \n
    • a (ndarray):\nThe multivariate gamma is computed for each item of a.
      • \n
    • d (int):\nThe dimension of the space of integration.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (ndarray):\nThe values of the log multivariate gamma at the given points a.
      • \n
    \n\n
    Notes
    \n\n
    The formal definition of the multivariate gamma of dimension d for a real\na is
    \n\n
    $$\\Gamma_d(a) = \\int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA$$
    \n\n
    with the condition \\( a > (d-1)/2 \\), and \\( A > 0 \\) being the set of\nall the positive definite matrices of dimension d.  Note that a is a\nscalar: the integrand only is multivariate, the argument is not (the\nfunction is defined over a subset of the real set).
    \n\n
    This can be proven to be equal to the much friendlier equation
    \n\n
    $$\\Gamma_d(a) = \\pi^{d(d-1)/4} \\prod_{i=1}^{d} \\Gamma(a - (i-1)/2).$$
    \n\n
    References
    \n\n
    R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in\nprobability and mathematical statistics).
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import multigammaln, gammaln\n>>> a = 23.5\n>>> d = 10\n>>> multigammaln(a, d)\n454.1488605074416\n
    \n
    \n\n
    Verify that the result agrees with the logarithm of the equation\nshown above:
    \n\n
    \n
    >>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()\n454.1488605074416\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.kn": {"fullname": "pyerrors.special.kn", "modulename": "pyerrors.special", "qualname": "kn", "kind": "function", "doc": "
    Modified Bessel function of the second kind of integer order n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j0": {"fullname": "pyerrors.special.j0", "modulename": "pyerrors.special", "qualname": "j0", "kind": "function", "doc": "
    j0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    j0(x, out=None)
    \n\n
    Bessel function of the first kind of order 0.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 0 at x.
      • \n
    \n\n
    See Also
    \n\n
    jv:  Bessel function of real order and complex argument.
    \nspherical_jn:  spherical Bessel functions.  
    \n\n
    Notes
    \n\n
    The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval the following rational approximation is used:
    \n\n
    $$J_0(x) \\approx (w - r_1^2)(w - r_2^2) \\frac{P_3(w)}{Q_8(w)},$$
    \n\n
    where \\( w = x^2 \\) and \\( r_1 \\), \\( r_2 \\) are the zeros of\n\\( J_0 \\), and \\( P_3 \\) and \\( Q_8 \\) are polynomials of degrees 3\nand 8, respectively.
    \n\n
    In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
    \n\n
    This function is a wrapper for the Cephes 1 routine j0.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import j0\n>>> j0(1.)\n0.7651976865579665\n
    \n
    \n\n
    Calculate the function at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> j0(np.array([-2., 0., 4.]))\narray([ 0.22389078,  1.        , -0.39714981])\n
    \n
    \n\n
    Plot the function from -20 to 20.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y0": {"fullname": "pyerrors.special.y0", "modulename": "pyerrors.special", "qualname": "y0", "kind": "function", "doc": "
    y0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    y0(x, out=None)
    \n\n
    Bessel function of the second kind of order 0.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 0 at x.
      • \n
    \n\n
    See Also
    \n\n
    j0:  Bessel function of the first kind of order 0
    \nyv:  Bessel function of the first kind  
    \n\n
    Notes
    \n\n
    The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval a rational approximation \\( R(x) \\) is employed to\ncompute,
    \n\n
    $$Y_0(x) = R(x) + \\frac{2 \\log(x) J_0(x)}{\\pi},$$
    \n\n
    where \\( J_0 \\) is the Bessel function of the first kind of order 0.
    \n\n
    In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
    \n\n
    This function is a wrapper for the Cephes 1 routine y0.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import y0\n>>> y0(1.)\n0.08825696421567697\n
    \n
    \n\n
    Calculate at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> y0(np.array([0.5, 2., 3.]))\narray([-0.44451873,  0.51037567,  0.37685001])\n
    \n
    \n\n
    Plot the function from 0 to 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j1": {"fullname": "pyerrors.special.j1", "modulename": "pyerrors.special", "qualname": "j1", "kind": "function", "doc": "
    j1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    j1(x, out=None)
    \n\n
    Bessel function of the first kind of order 1.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 1 at x.
      • \n
    \n\n
    See Also
    \n\n
    jv:  Bessel function of the first kind
    \nspherical_jn:  spherical Bessel functions.  
    \n\n
    Notes
    \n\n
    The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 24 term Chebyshev expansion is used. In the second, the\nasymptotic trigonometric representation is employed using two rational\nfunctions of degree 5/5.
    \n\n
    This function is a wrapper for the Cephes 1 routine j1.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import j1\n>>> j1(1.)\n0.44005058574493355\n
    \n
    \n\n
    Calculate the function at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> j1(np.array([-2., 0., 4.]))\narray([-0.57672481,  0.        , -0.06604333])\n
    \n
    \n\n
    Plot the function from -20 to 20.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y1": {"fullname": "pyerrors.special.y1", "modulename": "pyerrors.special", "qualname": "y1", "kind": "function", "doc": "
    y1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    y1(x, out=None)
    \n\n
    Bessel function of the second kind of order 1.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 1 at x.
      • \n
    \n\n
    See Also
    \n\n
    j1:  Bessel function of the first kind of order 1
    \nyn:  Bessel function of the second kind
    \nyv:  Bessel function of the second kind  
    \n\n
    Notes
    \n\n
    The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 25 term Chebyshev expansion is used, and computing\n\\( J_1 \\) (the Bessel function of the first kind) is required. In the\nsecond, the asymptotic trigonometric representation is employed using two\nrational functions of degree 5/5.
    \n\n
    This function is a wrapper for the Cephes 1 routine y1.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import y1\n>>> y1(1.)\n-0.7812128213002888\n
    \n
    \n\n
    Calculate at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> y1(np.array([0.5, 2., 3.]))\narray([-1.47147239, -0.10703243,  0.32467442])\n
    \n
    \n\n
    Plot the function from 0 to 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.jn": {"fullname": "pyerrors.special.jn", "modulename": "pyerrors.special", "qualname": "jn", "kind": "function", "doc": "
    jv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    jv(v, z, out=None)
    \n\n
    Bessel function of the first kind of real order and complex argument.
    \n\n
    Parameters
    \n\n
      \n
    • v (array_like):\nOrder (float).
      • \n
    • z (array_like):\nArgument (float or complex).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function, \\( J_v(z) \\).
      • \n
    \n\n
    See Also
    \n\n
    jve:  \\( J_v \\) with leading exponential behavior stripped off.
    \nspherical_jn:  spherical Bessel functions.
    \nj0:  faster version of this function for order 0.
    \nj1:  faster version of this function for order 1.  
    \n\n
    Notes
    \n\n
    For positive v values, the computation is carried out using the AMOS\n1 zbesj routine, which exploits the connection to the modified\nBessel function \\( I_v \\),
    \n\n
    $$J_v(z) = \\exp(v\\pi\\imath/2) I_v(-\\imath z)\\qquad (\\Im z > 0)
    \n\n
    J_v(z) = \\exp(-v\\pi\\imath/2) I_v(\\imath z)\\qquad (\\Im z < 0)$$
    \n\n
    For negative v values the formula,
    \n\n
    $$J_{-v}(z) = J_v(z) \\cos(\\pi v) - Y_v(z) \\sin(\\pi v)$$
    \n\n
    is used, where \\( Y_v(z) \\) is the Bessel function of the second\nkind, computed using the AMOS routine zbesy.  Note that the second\nterm is exactly zero for integer v; to improve accuracy the second\nterm is explicitly omitted for v values such that v = floor(v).
    \n\n
    Not to be confused with the spherical Bessel functions (see spherical_jn).
    \n\n
    References
    \n\n
    Examples
    \n\n
    Evaluate the function of order 0 at one point.
    \n\n
    \n
    >>> from scipy.special import jv\n>>> jv(0, 1.)\n0.7651976865579666\n
    \n
    \n\n
    Evaluate the function at one point for different orders.
    \n\n
    \n
    >>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)\n(0.7651976865579666, 0.44005058574493355, 0.24029783912342725)\n
    \n
    \n\n
    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
    \n\n
    \n
    >>> jv([0, 1, 1.5], 1.)\narray([0.76519769, 0.44005059, 0.24029784])\n
    \n
    \n\n
    Evaluate the function at several points for order 0 by providing an\narray for z.
    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> jv(0, points)\narray([ 0.22389078,  1.        , -0.26005195])\n
    \n
    \n\n
    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:
    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> jv(orders, points)\narray([[ 0.22389078,  1.        , -0.26005195],\n       [-0.57672481,  0.        ,  0.33905896]])\n
    \n
    \n\n
    Plot the functions of order 0 to 3 from -10 to 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, jv(i, x), label=f'$J_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.yn": {"fullname": "pyerrors.special.yn", "modulename": "pyerrors.special", "qualname": "yn", "kind": "function", "doc": "
    yn(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    yn(n, x, out=None)
    \n\n
    Bessel function of the second kind of integer order and real argument.
    \n\n
    Parameters
    \n\n
      \n
    • n (array_like):\nOrder (integer).
      • \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function, \\( Y_n(x) \\).
      • \n
    \n\n
    See Also
    \n\n
    yv:  For real order and real or complex argument.
    \ny0:  faster implementation of this function for order 0
    \ny1:  faster implementation of this function for order 1  
    \n\n
    Notes
    \n\n
    Wrapper for the Cephes 1 routine yn.
    \n\n
    The function is evaluated by forward recurrence on n, starting with\nvalues computed by the Cephes routines y0 and y1. If n = 0 or 1,\nthe routine for y0 or y1 is called directly.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Evaluate the function of order 0 at one point.
    \n\n
    \n
    >>> from scipy.special import yn\n>>> yn(0, 1.)\n0.08825696421567697\n
    \n
    \n\n
    Evaluate the function at one point for different orders.
    \n\n
    \n
    >>> yn(0, 1.), yn(1, 1.), yn(2, 1.)\n(0.08825696421567697, -0.7812128213002888, -1.6506826068162546)\n
    \n
    \n\n
    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
    \n\n
    \n
    >>> yn([0, 1, 2], 1.)\narray([ 0.08825696, -0.78121282, -1.65068261])\n
    \n
    \n\n
    Evaluate the function at several points for order 0 by providing an\narray for z.
    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([0.5, 3., 8.])\n>>> yn(0, points)\narray([-0.44451873,  0.37685001,  0.22352149])\n
    \n
    \n\n
    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:
    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> yn(orders, points)\narray([[-0.44451873,  0.37685001,  0.22352149],\n       [-1.47147239,  0.32467442, -0.15806046]])\n
    \n
    \n\n
    Plot the functions of order 0 to 3 from 0 to 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, yn(i, x), label=f'$Y_{i!r}$')\n>>> ax.set_ylim(-3, 1)\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i0": {"fullname": "pyerrors.special.i0", "modulename": "pyerrors.special", "qualname": "i0", "kind": "function", "doc": "
    i0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    i0(x, out=None)
    \n\n
    Modified Bessel function of order 0.
    \n\n
    Defined as,
    \n\n
    $$I_0(x) = \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{(k!)^2} = J_0(\\imath x),$$
    \n\n
    where \\( J_0 \\) is the Bessel function of the first kind of order 0.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float)
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • I (scalar or ndarray):\nValue of the modified Bessel function of order 0 at x.
      • \n
    \n\n
    See Also
    \n\n
    iv:  Modified Bessel function of any order
    \ni0e:  Exponentially scaled modified Bessel function of order 0  
    \n\n
    Notes
    \n\n
    The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
    \n\n
    This function is a wrapper for the Cephes 1 routine i0.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import i0\n>>> i0(1.)\n1.2660658777520082\n
    \n
    \n\n
    Calculate at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> i0(np.array([-2., 0., 3.5]))\narray([2.2795853 , 1.        , 7.37820343])\n
    \n
    \n\n
    Plot the function from -10 to 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i1": {"fullname": "pyerrors.special.i1", "modulename": "pyerrors.special", "qualname": "i1", "kind": "function", "doc": "
    i1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    i1(x, out=None)
    \n\n
    Modified Bessel function of order 1.
    \n\n
    Defined as,
    \n\n
    $$I_1(x) = \\frac{1}{2}x \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{k! (k + 1)!}\n       = -\\imath J_1(\\imath x),$$
    \n\n
    where \\( J_1 \\) is the Bessel function of the first kind of order 1.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float)
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • I (scalar or ndarray):\nValue of the modified Bessel function of order 1 at x.
      • \n
    \n\n
    See Also
    \n\n
    iv:  Modified Bessel function of the first kind
    \ni1e:  Exponentially scaled modified Bessel function of order 1  
    \n\n
    Notes
    \n\n
    The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
    \n\n
    This function is a wrapper for the Cephes 1 routine i1.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import i1\n>>> i1(1.)\n0.5651591039924851\n
    \n
    \n\n
    Calculate the function at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> i1(np.array([-2., 0., 6.]))\narray([-1.59063685,  0.        , 61.34193678])\n
    \n
    \n\n
    Plot the function between -10 and 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.iv": {"fullname": "pyerrors.special.iv", "modulename": "pyerrors.special", "qualname": "iv", "kind": "function", "doc": "
    iv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    iv(v, z, out=None)
    \n\n
    Modified Bessel function of the first kind of real order.
    \n\n
    Parameters
    \n\n
      \n
    • v (array_like):\nOrder. If z is of real type and negative, v must be integer\nvalued.
      • \n
    • z (array_like of float or complex):\nArgument.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the modified Bessel function.
      • \n
    \n\n
    See Also
    \n\n
    ive:  This function with leading exponential behavior stripped off.
    \ni0:  Faster version of this function for order 0.
    \ni1:  Faster version of this function for order 1.  
    \n\n
    Notes
    \n\n
    For real z and \\( v \\in [-50, 50] \\), the evaluation is carried out\nusing Temme's method 1.  For larger orders, uniform asymptotic\nexpansions are applied.
    \n\n
    For complex z and positive v, the AMOS 2 zbesi routine is\ncalled. It uses a power series for small z, the asymptotic expansion\nfor large abs(z), the Miller algorithm normalized by the Wronskian\nand a Neumann series for intermediate magnitudes, and the uniform\nasymptotic expansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large\norders. Backward recurrence is used to generate sequences or reduce\norders when necessary.
    \n\n
    The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
    \n\n
    $$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
    \n\n
    (valid when the real part of z is positive).  For negative v, the\nformula
    \n\n
    $$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
    \n\n
    is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Evaluate the function of order 0 at one point.
    \n\n
    \n
    >>> from scipy.special import iv\n>>> iv(0, 1.)\n1.2660658777520084\n
    \n
    \n\n
    Evaluate the function at one point for different orders.
    \n\n
    \n
    >>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.)\n(1.2660658777520084, 0.565159103992485, 0.2935253263474798)\n
    \n
    \n\n
    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
    \n\n
    \n
    >>> iv([0, 1, 1.5], 1.)\narray([1.26606588, 0.5651591 , 0.29352533])\n
    \n
    \n\n
    Evaluate the function at several points for order 0 by providing an\narray for z.
    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> iv(0, points)\narray([2.2795853 , 1.        , 4.88079259])\n
    \n
    \n\n
    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:
    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> iv(orders, points)\narray([[ 2.2795853 ,  1.        ,  4.88079259],\n       [-1.59063685,  0.        ,  3.95337022]])\n
    \n
    \n\n
    Plot the functions of order 0 to 3 from -5 to 5.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, iv(i, x), label=f'$I_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Temme, Journal of Computational Physics, vol 21, 343 (1976) ↩
      \n
      2. \n\n
    2. \n
      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
      \n
      3. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.ive": {"fullname": "pyerrors.special.ive", "modulename": "pyerrors.special", "qualname": "ive", "kind": "function", "doc": "
    ive(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    ive(v, z, out=None)
    \n\n
    Exponentially scaled modified Bessel function of the first kind.
    \n\n
    Defined as::
    \n\n
    ive(v, z) = iv(v, z) * exp(-abs(z.real))\n
    \n\n
    For imaginary numbers without a real part, returns the unscaled\nBessel function of the first kind iv.
    \n\n
    Parameters
    \n\n
      \n
    • v (array_like of float):\nOrder.
      • \n
    • z (array_like of float or complex):\nArgument.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the exponentially scaled modified Bessel function.
      • \n
    \n\n
    See Also
    \n\n
    iv:  Modified Bessel function of the first kind
    \ni0e:  Faster implementation of this function for order 0
    \ni1e:  Faster implementation of this function for order 1  
    \n\n
    Notes
    \n\n
    For positive v, the AMOS 1 zbesi routine is called. It uses a\npower series for small z, the asymptotic expansion for large\nabs(z), the Miller algorithm normalized by the Wronskian and a\nNeumann series for intermediate magnitudes, and the uniform asymptotic\nexpansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large orders.\nBackward recurrence is used to generate sequences or reduce orders when\nnecessary.
    \n\n
    The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
    \n\n
    $$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
    \n\n
    (valid when the real part of z is positive).  For negative v, the\nformula
    \n\n
    $$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
    \n\n
    is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.
    \n\n
    ive is useful for large arguments z: for these, iv easily overflows,\nwhile ive does not due to the exponential scaling.
    \n\n
    References
    \n\n
    Examples
    \n\n
    In the following example iv returns infinity whereas ive still returns\na finite number.
    \n\n
    \n
    >>> from scipy.special import iv, ive\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> iv(3, 1000.), ive(3, 1000.)\n(inf, 0.01256056218254712)\n
    \n
    \n\n
    Evaluate the function at one point for different orders by\nproviding a list or NumPy array as argument for the v parameter:
    \n\n
    \n
    >>> ive([0, 1, 1.5], 1.)\narray([0.46575961, 0.20791042, 0.10798193])\n
    \n
    \n\n
    Evaluate the function at several points for order 0 by providing an\narray for z.
    \n\n
    \n
    >>> points = np.array([-2., 0., 3.])\n>>> ive(0, points)\narray([0.30850832, 1.        , 0.24300035])\n
    \n
    \n\n
    Evaluate the function at several points for different orders by\nproviding arrays for both v for z. Both arrays have to be\nbroadcastable to the correct shape. To calculate the orders 0, 1\nand 2 for a 1D array of points:
    \n\n
    \n
    >>> ive([[0], [1], [2]], points)\narray([[ 0.30850832,  1.        ,  0.24300035],\n       [-0.21526929,  0.        ,  0.19682671],\n       [ 0.09323903,  0.        ,  0.11178255]])\n
    \n
    \n\n
    Plot the functions of order 0 to 3 from -5 to 5.
    \n\n
    \n
    >>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, ive(i, x), label=fr'$I_{i!r}(z)\\cdot e^{{-|z|}}$')\n>>> ax.legend()\n>>> ax.set_xlabel(r"$z$")\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erf": {"fullname": "pyerrors.special.erf", "modulename": "pyerrors.special", "qualname": "erf", "kind": "function", "doc": "
    erf(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    erf(z, out=None)
    \n\n
    Returns the error function of complex argument.
    \n\n
    It is defined as 2/sqrt(pi)*integral(exp(-t**2), t=0..z).
    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nInput array.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (scalar or ndarray):\nThe values of the error function at the given points x.
      • \n
    \n\n
    See Also
    \n\n
    erfc,, erfinv,, erfcinv,, wofz,, erfcx,, erfi  
    \n\n
    Notes
    \n\n
    The cumulative of the unit normal distribution is given by\nPhi(z) = 1/2[1 + erf(z/sqrt(2))].
    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erf(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erf(x)$')\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfc": {"fullname": "pyerrors.special.erfc", "modulename": "pyerrors.special", "qualname": "erfc", "kind": "function", "doc": "
    erfc(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    erfc(x, out=None)
    \n\n
    Complementary error function, 1 - erf(x).
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal or complex valued argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the complementary error function
      • \n
    \n\n
    See Also
    \n\n
    erf,, erfi,, erfcx,, dawsn,, wofz  
    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erfc(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erfc(x)$')\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfinv": {"fullname": "pyerrors.special.erfinv", "modulename": "pyerrors.special", "qualname": "erfinv", "kind": "function", "doc": "
    erfinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    erfinv(y, out=None)
    \n\n
    Inverse of the error function.
    \n\n
    Computes the inverse of the error function.
    \n\n
    In the complex domain, there is no unique complex number w satisfying\nerf(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, -1 < x < 1, there is a unique real\nnumber satisfying erf(erfinv(x)) = x.
    \n\n
    Parameters
    \n\n
      \n
    • y (ndarray):\nArgument at which to evaluate. Domain: [-1, 1]
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • erfinv (scalar or ndarray):\nThe inverse of erf of y, element-wise
      • \n
    \n\n
    See Also
    \n\n
    erf:  Error function of a complex argument
    \nerfc:  Complementary error function, 1 - erf(x)
    \nerfcinv:  Inverse of the complementary error function  
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfinv, erf\n
    \n
    \n\n
    \n
    >>> erfinv(0.5)\n0.4769362762044699\n
    \n
    \n\n
    \n
    >>> y = np.linspace(-1.0, 1.0, num=9)\n>>> x = erfinv(y)\n>>> x\narray([       -inf, -0.81341985, -0.47693628, -0.22531206,  0.        ,\n        0.22531206,  0.47693628,  0.81341985,         inf])\n
    \n
    \n\n
    Verify that erf(erfinv(y)) is y.
    \n\n
    \n
    >>> erf(x)\narray([-1.  , -0.75, -0.5 , -0.25,  0.  ,  0.25,  0.5 ,  0.75,  1.  ])\n
    \n
    \n\n
    Plot the function:
    \n\n
    \n
    >>> y = np.linspace(-1, 1, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfinv(y)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfcinv": {"fullname": "pyerrors.special.erfcinv", "modulename": "pyerrors.special", "qualname": "erfcinv", "kind": "function", "doc": "
    erfcinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    erfcinv(y, out=None)
    \n\n
    Inverse of the complementary error function.
    \n\n
    Computes the inverse of the complementary error function.
    \n\n
    In the complex domain, there is no unique complex number w satisfying\nerfc(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, 0 < x < 2, there is a unique real\nnumber satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)).
    \n\n
    It is related to inverse of the error function by erfcinv(1-x) = erfinv(x)
    \n\n
    Parameters
    \n\n
      \n
    • y (ndarray):\nArgument at which to evaluate. Domain: [0, 2]
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • erfcinv (scalar or ndarray):\nThe inverse of erfc of y, element-wise
      • \n
    \n\n
    See Also
    \n\n
    erf:  Error function of a complex argument
    \nerfc:  Complementary error function, 1 - erf(x)
    \nerfinv:  Inverse of the error function  
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfcinv\n
    \n
    \n\n
    \n
    >>> erfcinv(0.5)\n0.4769362762044699\n
    \n
    \n\n
    \n
    >>> y = np.linspace(0.0, 2.0, num=11)\n>>> erfcinv(y)\narray([        inf,  0.9061938 ,  0.59511608,  0.37080716,  0.17914345,\n       -0.        , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,\n              -inf])\n
    \n
    \n\n
    Plot the function:
    \n\n
    \n
    >>> y = np.linspace(0, 2, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfcinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfcinv(y)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logit": {"fullname": "pyerrors.special.logit", "modulename": "pyerrors.special", "qualname": "logit", "kind": "function", "doc": "
    logit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    \"\"\"\nlogit(x, out=None)
    \n\n
    Logit ufunc for ndarrays.
    \n\n
    The logit function is defined as logit(p) = log(p/(1-p)).\nNote that logit(0) = -inf, logit(1) = inf, and logit(p)\nfor p<0 or p>1 yields nan.
    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nThe ndarray to apply logit to element-wise.
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare logit of the corresponding entry of x.
      • \n
    \n\n
    See Also
    \n\n
    expit  
    \n\n
    Notes
    \n\n
    As a ufunc logit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs 
    \n\n
    New in version 0.10.0.
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import logit, expit\n
    \n
    \n\n
    \n
    >>> logit([0, 0.25, 0.5, 0.75, 1])\narray([       -inf, -1.09861229,  0.        ,  1.09861229,         inf])\n
    \n
    \n\n
    expit is the inverse of logit:
    \n\n
    \n
    >>> expit(logit([0.1, 0.75, 0.999]))\narray([ 0.1  ,  0.75 ,  0.999])\n
    \n
    \n\n
    Plot logit(x) for x in [0, 1]:
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> x = np.linspace(0, 1, 501)\n>>> y = logit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.ylim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('logit(x)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.expit": {"fullname": "pyerrors.special.expit", "modulename": "pyerrors.special", "qualname": "expit", "kind": "function", "doc": "
    expit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    expit(x, out=None)
    \n\n
    Expit (a.k.a. logistic sigmoid) ufunc for ndarrays.
    \n\n
    The expit function, also known as the logistic sigmoid function, is\ndefined as expit(x) = 1/(1+exp(-x)).  It is the inverse of the\nlogit function.
    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nThe ndarray to apply expit to element-wise.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare expit of the corresponding entry of x.
      • \n
    \n\n
    See Also
    \n\n
    logit  
    \n\n
    Notes
    \n\n
    As a ufunc expit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs 
    \n\n
    New in version 0.10.0.
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import expit, logit\n
    \n
    \n\n
    \n
    >>> expit([-np.inf, -1.5, 0, 1.5, np.inf])\narray([ 0.        ,  0.18242552,  0.5       ,  0.81757448,  1.        ])\n
    \n
    \n\n
    logit is the inverse of expit:
    \n\n
    \n
    >>> logit(expit([-2.5, 0, 3.1, 5.0]))\narray([-2.5,  0. ,  3.1,  5. ])\n
    \n
    \n\n
    Plot expit(x) for x in [-6, 6]:
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-6, 6, 121)\n>>> y = expit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.xlim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('expit(x)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logsumexp": {"fullname": "pyerrors.special.logsumexp", "modulename": "pyerrors.special", "qualname": "logsumexp", "kind": "function", "doc": "
    Compute the log of the sum of exponentials of input elements.
    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nInput array.
      • \n
    • axis (None or int or tuple of ints, optional):\nAxis or axes over which the sum is taken. By default axis is None,\nand all elements are summed.
      \n\n
      New in version 0.11.0.
      • \n
    • b (array-like, optional):\nScaling factor for exp(a) must be of the same shape as a or\nbroadcastable to a. These values may be negative in order to\nimplement subtraction.
      \n\n
      New in version 0.12.0.
      • \n
    • keepdims (bool, optional):\nIf this is set to True, the axes which are reduced are left in the\nresult as dimensions with size one. With this option, the result\nwill broadcast correctly against the original array.
      \n\n
      New in version 0.15.0.
      • \n
    • return_sign (bool, optional):\nIf this is set to True, the result will be a pair containing sign\ninformation; if False, results that are negative will be returned\nas NaN. Default is False (no sign information).
      \n\n
      New in version 0.16.0.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (ndarray):\nThe result, np.log(np.sum(np.exp(a))) calculated in a numerically\nmore stable way. If b is given then np.log(np.sum(b*np.exp(a)))\nis returned. If return_sign is True, res contains the log of\nthe absolute value of the argument.
      • \n
    • sgn (ndarray):\nIf return_sign is True, this will be an array of floating-point\nnumbers matching res containing +1, 0, -1 (for real-valued inputs)\nor a complex phase (for complex inputs). This gives the sign of the\nargument of the logarithm in res.\nIf return_sign is False, only one result is returned.
      • \n
    \n\n
    See Also
    \n\n
    numpy.logaddexp,, numpy.logaddexp2  
    \n\n
    Notes
    \n\n
    NumPy has a logaddexp function which is very similar to logsumexp, but\nonly handles two arguments. logaddexp.reduce is similar to this\nfunction, but may be less stable.
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import logsumexp\n>>> a = np.arange(10)\n>>> logsumexp(a)\n9.4586297444267107\n>>> np.log(np.sum(np.exp(a)))\n9.4586297444267107\n
    \n
    \n\n
    With weights
    \n\n
    \n
    >>> a = np.arange(10)\n>>> b = np.arange(10, 0, -1)\n>>> logsumexp(a, b=b)\n9.9170178533034665\n>>> np.log(np.sum(b*np.exp(a)))\n9.9170178533034647\n
    \n
    \n\n
    Returning a sign flag
    \n\n
    \n
    >>> logsumexp([1,2],b=[1,-1],return_sign=True)\n(1.5413248546129181, -1.0)\n
    \n
    \n\n
    Notice that logsumexp does not directly support masked arrays. To use it\non a masked array, convert the mask into zero weights:
    \n\n
    \n
    >>> a = np.ma.array([np.log(2), 2, np.log(3)],\n...                  mask=[False, True, False])\n>>> b = (~a.mask).astype(int)\n>>> logsumexp(a.data, b=b), np.log(5)\n1.6094379124341005, 1.6094379124341005\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.version": {"fullname": "pyerrors.version", "modulename": "pyerrors.version", "kind": "module", "doc": "
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    What is pyerrors?
    \n\n
    pyerrors is a python package for error computation and propagation of Markov chain Monte Carlo data.\nIt is based on the gamma method arXiv:hep-lat/0306017. Some of its features are:
    \n\n
      \n
    • automatic differentiation for exact linear error propagation as suggested in arXiv:1809.01289 (partly based on the autograd package).
      • \n
    • treatment of slow modes in the simulation as suggested in arXiv:1009.5228.
      • \n
    • coherent error propagation for data from different Markov chains.
      • \n
    • non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in arXiv:1809.01289.
      • \n
    • real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...).
      • \n
    \n\n
    More detailed examples can found in the GitHub repository \"badge\".
    \n\n
    If you use pyerrors for research that leads to a publication please consider citing:
    \n\n
      \n
    • Fabian Joswig, Simon Kuberski, Justus T. Kuhlmann, Jan Neuendorf, pyerrors: a python framework for error analysis of Monte Carlo data. Comput.Phys.Commun. 288 (2023) 108750.
      • \n
    • Ulli Wolff, Monte Carlo errors with less errors. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum).
      • \n
    • Alberto Ramos, Automatic differentiation for error analysis of Monte Carlo data. Comput.Phys.Commun. 238 (2019) 19-35.
      • \n
    \n\n
    and
    \n\n
      \n
    • Stefan Schaefer, Rainer Sommer, Francesco Virotta, Critical slowing down and error analysis in lattice QCD simulations. Nucl.Phys.B 845 (2011) 93-119.
      • \n
    \n\n
    where applicable.
    \n\n
    There exist similar publicly available implementations of gamma method error analysis suites in Fortran, Julia and Python.
    \n\n
    Installation
    \n\n
    Install the most recent release using pip and pypi:
    \n\n
    \n
    python -m pip install pyerrors     # Fresh install\npython -m pip install -U pyerrors  # Update\n
    \n
    \n\n
    Install the most recent release using conda and conda-forge:
    \n\n
    \n
    conda install -c conda-forge pyerrors  # Fresh install\nconda update -c conda-forge pyerrors   # Update\n
    \n
    \n\n
    Install the current develop version:
    \n\n
    \n
    python -m pip install -U --no-deps --force-reinstall git+https://github.com/fjosw/pyerrors.git@develop\n
    \n
    \n\n
    (Also works for any feature branch).
    \n\n
    Basic example
    \n\n
    \n
    import numpy as np\nimport pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name']) # Initialize an Obs object\nmy_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object\nmy_new_obs.gamma_method()                     # Estimate the statistical error\nprint(my_new_obs)                             # Print the result to stdout\n> 0.31498(72)\n
    \n
    \n\n
    The Obs class
    \n\n
    pyerrors introduces a new datatype, Obs, which simplifies error propagation and estimation for auto- and cross-correlated data.\nAn Obs object can be initialized with two arguments, the first is a list containing the samples for an observable from a Monte Carlo chain.\nThe samples can either be provided as python list or as numpy array.\nThe second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble. It is crucial for the correct error propagation that observations from the same Monte Carlo history are labeled with the same name. See Multiple ensembles/replica for details.
    \n\n
    \n
    import pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name'])\n
    \n
    \n\n
    Error propagation
    \n\n
    When performing mathematical operations on Obs objects the correct error propagation is intrinsically taken care of using a first order Taylor expansion\n$$\\delta_f^i=\\sum_\\alpha \\bar{f}_\\alpha \\delta_\\alpha^i\\,,\\quad \\delta_\\alpha^i=a_\\alpha^i-\\bar{a}_\\alpha\\,,$$\nas introduced in arXiv:hep-lat/0306017.\nThe required derivatives $\\bar{f}_\\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in arXiv:1809.01289.
    \n\n
    The Obs class is designed such that mathematical numpy functions can be used on Obs just as for regular floats.
    \n\n
    \n
    import numpy as np\nimport pyerrors as pe\n\nmy_obs1 = pe.Obs([samples1], ['ensemble_name'])\nmy_obs2 = pe.Obs([samples2], ['ensemble_name'])\n\nmy_sum = my_obs1 + my_obs2\n\nmy_m_eff = np.log(my_obs1 / my_obs2)\n\niamzero = my_m_eff - my_m_eff\n# Check that value and fluctuations are zero within machine precision\nprint(iamzero == 0.0)\n> True\n
    \n
    \n\n
    Error estimation
    \n\n
    The error estimation within pyerrors is based on the gamma method introduced in arXiv:hep-lat/0306017.\nAfter having arrived at the derived quantity of interest the gamma_method can be called as detailed in the following example.
    \n\n
    \n
    my_sum.gamma_method()\nprint(my_sum)\n> 1.70(57)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 5.72046658e-01 +/- 7.56746598e-02 (33.650%)\n>  t_int         2.71422900e+00 +/- 6.40320983e-01 S = 2.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n
    The gamma_method is not automatically called after every intermediate step in order to prevent computational overhead.
    \n\n
    We use the following definition of the integrated autocorrelation time established in Madras & Sokal 1988\n$$\\tau_\\mathrm{int}=\\frac{1}{2}+\\sum_{t=1}^{W}\\rho(t)\\geq \\frac{1}{2}\\,.$$\nThe window $W$ is determined via the automatic windowing procedure described in arXiv:hep-lat/0306017.\nThe standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the gamma_method as parameter.
    \n\n
    \n
    my_sum.gamma_method(S=3.0)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.30675201e-01 +/- 1.04585650e-01 (37.099%)\n>  t_int         3.29909703e+00 +/- 9.77310102e-01 S = 3.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n
    The integrated autocorrelation time $\\tau_\\mathrm{int}$ and the autocorrelation function $\\rho(W)$ can be monitored via the methods pyerrors.obs.Obs.plot_tauint and pyerrors.obs.Obs.plot_rho.
    \n\n
    If the parameter $S$ is set to zero it is assumed that the dataset does not exhibit any autocorrelation and the window size is chosen to be zero.\nIn this case the error estimate is identical to the sample standard error.
    \n\n
    Exponential tails
    \n\n
    Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\\rho$ as suggested in arXiv:1009.5228. The longest autocorrelation time in the history, $\\tau_\\mathrm{exp}$, can be passed to the gamma_method as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.
    \n\n
    \n
    my_sum.gamma_method(tau_exp=7.2)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.28097762e-01 +/- 5.79077524e-02 (36.947%)\n>  t_int         3.27218667e+00 +/- 7.99583654e-01 tau_exp = 7.20,  N_sigma = 1\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n
    For the full API see pyerrors.obs.Obs.gamma_method.
    \n\n
    Multiple ensembles/replica
    \n\n
    Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their name.
    \n\n
    \n
    obs1 = pe.Obs([samples1], ['ensemble1'])\nobs2 = pe.Obs([samples2], ['ensemble2'])\n\nmy_sum = obs1 + obs2\nmy_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 2 ensembles:\n>   \u00b7 Ensemble 'ensemble1' : 1000 configurations (from 1 to 1000)\n>   \u00b7 Ensemble 'ensemble2' : 500 configurations (from 1 to 500)\n
    \n
    \n\n
    Observables from the same Monte Carlo chain have to be initialized with the same name for correct error propagation. If different names were used in this case the data would be treated as statistically independent resulting in loss of relevant information and a potential over or under estimate of the statistical error.
    \n\n
    pyerrors identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar | in the name of the data set.
    \n\n
    \n
    obs1 = pe.Obs([samples1], ['ensemble1|r01'])\nobs2 = pe.Obs([samples2], ['ensemble1|r02'])\n\n> my_sum = obs1 + obs2\n> my_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1'\n>     \u00b7 Replicum 'r01' : 1000 configurations (from 1 to 1000)\n>     \u00b7 Replicum 'r02' : 500 configurations (from 1 to 500)\n
    \n
    \n\n
    Error estimation for multiple ensembles
    \n\n
    In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example.
    \n\n
    \n
    pe.Obs.S_dict['ensemble1'] = 2.5\npe.Obs.tau_exp_dict['ensemble2'] = 8.0\npe.Obs.tau_exp_dict['ensemble3'] = 2.0\n
    \n
    \n\n
    In case the gamma_method is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.\nPassing arguments to the gamma_method still dominates over the dictionaries.
    \n\n
    Irregular Monte Carlo chains
    \n\n
    Obs objects defined on irregular Monte Carlo chains can be initialized with the parameter idl.
    \n\n
    \n
    # Observable defined on configurations 20 to 519\nobs1 = pe.Obs([samples1], ['ensemble1'], idl=[range(20, 520)])\nobs1.details()\n> Result         9.98319881e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 20 to 519)\n\n# Observable defined on every second configuration between 5 and 1003\nobs2 = pe.Obs([samples2], ['ensemble1'], idl=[range(5, 1005, 2)])\nobs2.details()\n> Result         9.99100712e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 5 to 1003 in steps of 2)\n\n# Observable defined on configurations 2, 9, 28, 29 and 501\nobs3 = pe.Obs([samples3], ['ensemble1'], idl=[[2, 9, 28, 29, 501]])\nobs3.details()\n> Result         1.01718064e+00\n> 5 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 5 configurations (irregular range)\n
    \n
    \n\n
    Obs objects defined on regular and irregular histories of the same ensemble can be combined with each other and the correct error propagation and estimation is automatically taken care of.
    \n\n
    Warning: Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.\nMake sure to check the autocorrelation time with e.g. pyerrors.obs.Obs.plot_rho or pyerrors.obs.Obs.plot_tauint.
    \n\n
    For the full API see pyerrors.obs.Obs.
    \n\n
    Correlators
    \n\n
    When one is not interested in single observables but correlation functions, pyerrors offers the Corr class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize a Corr objects one needs to arrange the data as a list of Obs
    \n\n
    \n
    my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0      0.7957(80)\n> 1      0.5156(51)\n> 2      0.3227(33)\n> 3      0.2041(21)\n
    \n
    \n\n
    In case the correlation functions are not defined on the outermost timeslices, for example because of fixed boundary conditions, a padding can be introduced.
    \n\n
    \n
    my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3], padding=[1, 1])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0\n> 1      0.7957(80)\n> 2      0.5156(51)\n> 3      0.3227(33)\n> 4      0.2041(21)\n> 5\n
    \n
    \n\n
    The individual entries of a correlator can be accessed via slicing
    \n\n
    \n
    print(my_corr[3])\n> 0.3227(33)\n
    \n
    \n\n
    Error propagation with the Corr class works very similar to Obs objects. Mathematical operations are overloaded and Corr objects can be computed together with other Corr objects, Obs objects or real numbers and integers.
    \n\n
    \n
    my_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr\n
    \n
    \n\n
    pyerrors provides the user with a set of regularly used methods for the manipulation of correlator objects:
    \n\n
      \n
    • Corr.gamma_method applies the gamma method to all entries of the correlator.
      • \n
    • Corr.m_eff to construct effective masses. Various variants for periodic and fixed temporal boundary conditions are available.
      • \n
    • Corr.deriv returns the first derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
      • \n
    • Corr.second_deriv returns the second derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
      • \n
    • Corr.symmetric symmetrizes parity even correlations functions, assuming periodic boundary conditions.
      • \n
    • Corr.anti_symmetric anti-symmetrizes parity odd correlations functions, assuming periodic boundary conditions.
      • \n
    • Corr.T_symmetry averages a correlator with its time symmetry partner, assuming fixed boundary conditions.
      • \n
    • Corr.plateau extracts a plateau value from the correlator in a given range.
      • \n
    • Corr.roll periodically shifts the correlator.
      • \n
    • Corr.reverse reverses the time ordering of the correlator.
      • \n
    • Corr.correlate constructs a disconnected correlation function from the correlator and another Corr or Obs object.
      • \n
    • Corr.reweight reweights the correlator.
      • \n
    \n\n
    pyerrors can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (see pyerrors.correlators.Corr.GEVP).
    \n\n
    For the full API see pyerrors.correlators.Corr.
    \n\n
    Complex valued observables
    \n\n
    pyerrors can handle complex valued observables via the class pyerrors.obs.CObs.\nCObs are initialized with a real and an imaginary part which both can be Obs valued.
    \n\n
    \n
    my_real_part = pe.Obs([samples1], ['ensemble1'])\nmy_imag_part = pe.Obs([samples2], ['ensemble1'])\n\nmy_cobs = pe.CObs(my_real_part, my_imag_part)\nmy_cobs.gamma_method()\nprint(my_cobs)\n> (0.9959(91)+0.659(28)j)\n
    \n
    \n\n
    Elementary mathematical operations are overloaded and samples are properly propagated as for the Obs class.
    \n\n
    \n
    my_derived_cobs = (my_cobs + my_cobs.conjugate()) / np.abs(my_cobs)\nmy_derived_cobs.gamma_method()\nprint(my_derived_cobs)\n> (1.668(23)+0.0j)\n
    \n
    \n\n
    The Covobs class
    \n\n
    In many projects, auxiliary data that is not based on Monte Carlo chains enters. Examples are experimentally determined mesons masses which are used to set the scale or renormalization constants. These numbers come with an error that has to be propagated through the analysis. The Covobs class allows to define such quantities in pyerrors. Furthermore, external input might consist of correlated quantities. An example are the parameters of an interpolation formula, which are defined via mean values and a covariance matrix between all parameters. The contribution of the interpolation formula to the error of a derived quantity therefore might depend on the complete covariance matrix.
    \n\n
    This concept is built into the definition of Covobs. In pyerrors, external input is defined by $M$ mean values, a $M\\times M$ covariance matrix, where $M=1$ is permissible, and a name that uniquely identifies the covariance matrix. Below, we define the pion mass, based on its mean value and error, 134.9768(5). Note, that the square of the error enters cov_Obs, since the second argument of this function is the covariance matrix of the Covobs.
    \n\n
    \n
    import pyerrors.obs as pe\n\nmpi = pe.cov_Obs(134.9768, 0.0005**2, 'pi^0 mass')\nmpi.gamma_method()\nmpi.details()\n> Result         1.34976800e+02 +/- 5.00000000e-04 +/- 0.00000000e+00 (0.000%)\n>  pi^0 mass     5.00000000e-04\n> 0 samples in 1 ensemble:\n>   \u00b7 Covobs   'pi^0 mass'\n
    \n
    \n\n
    The resulting object mpi is an Obs that contains a Covobs. In the following, it may be handled as any other Obs. The contribution of the covariance matrix to the error of an Obs is determined from the $M \\times M$ covariance matrix $\\Sigma$ and the gradient of the Obs with respect to the external quantities, which is the $1\\times M$ Jacobian matrix $J$, via\n$$s = \\sqrt{J^T \\Sigma J}\\,,$$\nwhere the Jacobian is computed for each derived quantity via automatic differentiation.
    \n\n
    Correlated auxiliary data is defined similarly to above, e.g., via
    \n\n
    \n
    RAP = pe.cov_Obs([16.7457, -19.0475], [[3.49591, -6.07560], [-6.07560, 10.5834]], 'R_AP, 1906.03445, (5.3a)')\nprint(RAP)\n> [Obs[16.7(1.9)], Obs[-19.0(3.3)]]\n
    \n
    \n\n
    where RAP now is a list of two Obs that contains the two correlated parameters.
    \n\n
    Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the Covobs class allows to quote the derivative of a result with respect to the external quantities. If these derivatives are published together with the result, small shifts in the definition of external quantities, e.g., the definition of the physical point, can be performed a posteriori based on the published information. This may help to compare results of different groups. The gradient of an Obs o with respect to a covariance matrix with the identifying string k may be accessed via
    \n\n
    \n
    o.covobs[k].grad\n
    \n
    \n\n
    Error propagation in iterative algorithms
    \n\n
    pyerrors supports exact linear error propagation for iterative algorithms like various variants of non-linear least squares fits or root finding. The derivatives required for the error propagation are calculated as described in arXiv:1809.01289.
    \n\n
    Least squares fits
    \n\n
    Standard non-linear least square fits with errors on the dependent but not the independent variables can be performed with pyerrors.fits.least_squares. As default solver the Levenberg-Marquardt algorithm implemented in scipy is used.
    \n\n
    Fit functions have to be of the following form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n    return a[1] * anp.exp(-a[0] * x)\n
    \n
    \n\n
    It is important that numerical functions refer to autograd.numpy instead of numpy for the automatic differentiation in iterative algorithms to work properly.
    \n\n
    Fits can then be performed via
    \n\n
    \n
    fit_result = pe.fits.least_squares(x, y, func)\nprint("\\n", fit_result)\n> Fit with 2 parameters\n> Method: Levenberg-Marquardt\n> `ftol` termination condition is satisfied.\n> chisquare/d.o.f.: 0.9593035785160936\n\n>  Goodness of fit:\n> \u03c7\u00b2/d.o.f. = 0.959304\n> p-value   = 0.5673\n> Fit parameters:\n> 0      0.0548(28)\n> 1      1.933(64)\n
    \n
    \n\n
    where x is a list or numpy.array of floats and y is a list or numpy.array of Obs.
    \n\n
    Data stored in Corr objects can be fitted directly using the Corr.fit method.
    \n\n
    \n
    my_corr = pe.Corr(y)\nfit_result = my_corr.fit(func, fitrange=[12, 25])\n
    \n
    \n\n
    this can simplify working with absolute fit ranges and takes care of gaps in the data automatically.
    \n\n
    For fit functions with multiple independent variables the fit function can be of the form
    \n\n
    \n
    def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n
    pyerrors also supports correlated fits which can be triggered via the parameter correlated_fit=True.\nDetails about how the required covariance matrix is estimated can be found in pyerrors.obs.covariance.\nDirect visualizations of the performed fits can be triggered via resplot=True or qqplot=True.
    \n\n
    For all available options including combined fits to multiple datasets see pyerrors.fits.least_squares.
    \n\n
    Total least squares fits
    \n\n
    pyerrors can also fit data with errors on both the dependent and independent variables using the total least squares method also referred to as orthogonal distance regression as implemented in scipy, see pyerrors.fits.least_squares. The syntax is identical to the standard least squares case, the only difference being that x also has to be a list or numpy.array of Obs.
    \n\n
    For the full API see pyerrors.fits for fits and pyerrors.roots for finding roots of functions.
    \n\n
    Matrix operations
    \n\n
    pyerrors provides wrappers for Obs- and CObs-valued matrix operations based on numpy.linalg. The supported functions include:
    \n\n
      \n
    • inv for the matrix inverse.
      • \n
    • cholseky for the Cholesky decomposition.
      • \n
    • det for the matrix determinant.
      • \n
    • eigh for eigenvalues and eigenvectors of hermitean matrices.
      • \n
    • eig for eigenvalues of general matrices.
      • \n
    • pinv for the Moore-Penrose pseudoinverse.
      • \n
    • svd for the singular-value-decomposition.
      • \n
    \n\n
    For the full API see pyerrors.linalg.
    \n\n
    Export data
    \n\n
    \n\n
    The preferred exported file format within pyerrors is json.gz. Files written to this format are valid JSON files that have been compressed using gzip. The structure of the content is inspired by the dobs format of the ALPHA collaboration. The aim of the format is to facilitate the storage of data in a self-contained way such that, even years after the creation of the file, it is possible to extract all necessary information:
    \n\n
      \n
    • What observables are stored? Possibly: How exactly are they defined.
      • \n
    • How does each single ensemble or external quantity contribute to the error of the observable?
      • \n
    • Who did write the file when and on which machine?
      • \n
    \n\n
    This can be achieved by storing all information in one single file. The export routines of pyerrors are written such that as much information as possible is written automatically as described in the following example
    \n\n
    \n
    my_obs = pe.Obs([samples], ["test_ensemble"])\nmy_obs.tag = "My observable"\n\npe.input.json.dump_to_json(my_obs, "test_output_file", description="This file contains a test observable")\n# For a single observable one can equivalently use the class method dump\nmy_obs.dump("test_output_file", description="This file contains a test observable")\n\ncheck = pe.input.json.load_json("test_output_file")\n\nprint(my_obs == check)\n> True\n
    \n
    \n\n
    The format also allows to directly write out the content of Corr objects or lists and arrays of Obs objects by passing the desired data to pyerrors.input.json.dump_to_json.
    \n\n
    json.gz format specification
    \n\n
    The first entries of the file provide optional auxiliary information:
    \n\n
      \n
    • program is a string that indicates which program was used to write the file.
      • \n
    • version is a string that specifies the version of the format.
      • \n
    • who is a string that specifies the user name of the creator of the file.
      • \n
    • date is a string and contains the creation date of the file.
      • \n
    • host is a string and contains the hostname of the machine where the file has been written.
      • \n
    • description contains information on the content of the file. This field is not filled automatically in pyerrors. The user is advised to provide as detailed information as possible in this field. Examples are: Input files of measurements or simulations, LaTeX formulae or references to publications to specify how the observables have been computed, details on the analysis strategy, ... This field may be any valid JSON type. Strings, arrays or objects (equivalent to dicts in python) are well suited to provide information.
      • \n
    \n\n
    The only necessary entry of the file is the field\n-obsdata, an array that contains the actual data.
    \n\n
    Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of Obs, list, numpy.ndarray, Corr. All Obs inside a structure (with dimension > 0) have to be defined on the same set of configurations. Different structures, that are represented by entries of the array obsdata, are treated independently. Each entry of the array obsdata has the following required entries:
    \n\n
      \n
    • type is a string that specifies the type of the structure. This allows to parse the content to the correct form after reading the file. It is always possible to interpret the content as list of Obs.
      • \n
    • value is an array that contains the mean values of the Obs inside the structure.\nThe following entries are optional:
      • \n
    • layout is a string that specifies the layout of multi-dimensional structures. Examples are \"2, 2\" for a 2x2 dimensional matrix or  \"64, 4, 4\" for a Corr with $T=64$ and 4x4 matrices on each time slices. \"1\" denotes a single Obs. Multi-dimensional structures are stored in row-major format (see below).
      • \n
    • tag is any JSON type. It contains additional information concerning the structure. The tag of an Obs in pyerrors is written here.
      • \n
    • reweighted is a Bool that may be used to specify, whether the Obs in the structure have been reweighted.
      • \n
    • data is an array that contains the data from MC chains. We will define it below.
      • \n
    • cdata is an array that contains the data from external quantities with an error (Covobs in pyerrors). We will define it below.
      • \n
    \n\n
    The array data contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains:
    \n\n
      \n
    • id, a string that contains the name of the ensemble
      • \n
    • replica, an array that contains an entry per replica of the ensemble.
      • \n
    \n\n
    Each entry of replica contains\nname, a string that contains the name of the replica\ndeltas, an array that contains the actual data.
    \n\n
    Each entry in deltas corresponds to one configuration of the replica and has $1+N$ many entries. The first entry is an integer that specifies the configuration number that, together with ensemble and replica name, may be used to uniquely identify the configuration on which the data has been obtained. The following N entries specify the deltas, i.e., the deviation of the observable from the mean value on this configuration, of each Obs inside the structure. Multi-dimensional structures are stored in a row-major format. For primary observables, such as correlation functions, $value + delta_i$ matches the primary data obtained on the configuration.
    \n\n
    The array cdata contains information about the contribution of auxiliary observables, represented by Covobs in pyerrors, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains:
    \n\n
      \n
    • id, a string that identifies the covariance matrix
      • \n
    • layout, a string that defines the dimensions of the $M\\times M$ covariance matrix (has to be \"M, M\" or \"1\").
      • \n
    • cov, an array that contains the $M\\times M$ many entries of the covariance matrix, stored in row-major format.
      • \n
    • grad, an array that contains N entries, one for each Obs inside the structure. Each entry itself is an array, that contains the M gradients of the Nth observable  with respect to the quantity that corresponds to the Mth diagonal entry of the covariance matrix.
      • \n
    \n\n
    A JSON schema that may be used to verify the correctness of a file with respect to the format definition is stored in ./examples/json_schema.json. The schema is a self-descriptive format definition and contains an exemplary file.
    \n\n
    Julia I/O routines for the json.gz format, compatible with ADerrors.jl, can be found here.
    \n"}, "pyerrors.correlators": {"fullname": "pyerrors.correlators", "modulename": "pyerrors.correlators", "kind": "module", "doc": "
    \n"}, "pyerrors.correlators.Corr": {"fullname": "pyerrors.correlators.Corr", "modulename": "pyerrors.correlators", "qualname": "Corr", "kind": "class", "doc": "
    The class for a correlator (time dependent sequence of pe.Obs).
    \n\n
    Everything, this class does, can be achieved using lists or arrays of Obs.\nBut it is simply more convenient to have a dedicated object for correlators.\nOne often wants to add or multiply correlators of the same length at every timeslice and it is inconvenient\nto iterate over all timeslices for every operation. This is especially true, when dealing with matrices.
    \n\n
    The correlator can have two types of content: An Obs at every timeslice OR a matrix at every timeslice.\nOther dependency (eg. spatial) are not supported.
    \n\n
    The Corr class can also deal with missing measurements or paddings for fixed boundary conditions.\nThe missing entries are represented via the None object.
    \n\n
    Initialization
    \n\n
    A simple correlator can be initialized with a list or a one-dimensional array of Obs or Cobs
    \n\n
    \n
    corr11 = pe.Corr([obs1, obs2])\ncorr11 = pe.Corr(np.array([obs1, obs2]))\n
    \n
    \n\n
    A matrix-valued correlator can either be initialized via a two-dimensional array of Corr objects
    \n\n
    \n
    matrix_corr = pe.Corr(np.array([[corr11, corr12], [corr21, corr22]]))\n
    \n
    \n\n
    or alternatively via a three-dimensional array of Obs or CObs of shape (T, N, N) where T is\nthe temporal extent of the correlator and N is the dimension of the matrix.
    \n"}, "pyerrors.correlators.Corr.__init__": {"fullname": "pyerrors.correlators.Corr.__init__", "modulename": "pyerrors.correlators", "qualname": "Corr.__init__", "kind": "function", "doc": "
    Initialize a Corr object.
    \n\n
    Parameters
    \n\n
      \n
    • data_input (list or array):\nlist of Obs or list of arrays of Obs or array of Corrs (see class docstring for details).
      • \n
    • padding (list, optional):\nList with two entries where the first labels the padding\nat the front of the correlator and the second the padding\nat the back.
      • \n
    • prange (list, optional):\nList containing the first and last timeslice of the plateau\nregion identified for this correlator.
      • \n
    \n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "pyerrors.correlators.Corr.tag": {"fullname": "pyerrors.correlators.Corr.tag", "modulename": "pyerrors.correlators", "qualname": "Corr.tag", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.content": {"fullname": "pyerrors.correlators.Corr.content", "modulename": "pyerrors.correlators", "qualname": "Corr.content", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.T": {"fullname": "pyerrors.correlators.Corr.T", "modulename": "pyerrors.correlators", "qualname": "Corr.T", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.prange": {"fullname": "pyerrors.correlators.Corr.prange", "modulename": "pyerrors.correlators", "qualname": "Corr.prange", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.reweighted": {"fullname": "pyerrors.correlators.Corr.reweighted", "modulename": "pyerrors.correlators", "qualname": "Corr.reweighted", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.gamma_method": {"fullname": "pyerrors.correlators.Corr.gamma_method", "modulename": "pyerrors.correlators", "qualname": "Corr.gamma_method", "kind": "function", "doc": "
    Apply the gamma method to the content of the Corr.
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.gm": {"fullname": "pyerrors.correlators.Corr.gm", "modulename": "pyerrors.correlators", "qualname": "Corr.gm", "kind": "function", "doc": "
    Apply the gamma method to the content of the Corr.
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.projected": {"fullname": "pyerrors.correlators.Corr.projected", "modulename": "pyerrors.correlators", "qualname": "Corr.projected", "kind": "function", "doc": "
    We need to project the Correlator with a Vector to get a single value at each timeslice.
    \n\n
    The method can use one or two vectors.\nIf two are specified it returns v1@G@v2 (the order might be very important.)\nBy default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to
    \n", "signature": "(self, vector_l=None, vector_r=None, normalize=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.item": {"fullname": "pyerrors.correlators.Corr.item", "modulename": "pyerrors.correlators", "qualname": "Corr.item", "kind": "function", "doc": "
    Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.
    \n\n
    Parameters
    \n\n
      \n
    • i (int):\nFirst index to be picked.
      • \n
    • j (int):\nSecond index to be picked.
      • \n
    \n", "signature": "(self, i, j):", "funcdef": "def"}, "pyerrors.correlators.Corr.plottable": {"fullname": "pyerrors.correlators.Corr.plottable", "modulename": "pyerrors.correlators", "qualname": "Corr.plottable", "kind": "function", "doc": "
    Outputs the correlator in a plotable format.
    \n\n
    Outputs three lists containing the timeslice index, the value on each\ntimeslice and the error on each timeslice.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.symmetric": {"fullname": "pyerrors.correlators.Corr.symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.symmetric", "kind": "function", "doc": "
    Symmetrize the correlator around x0=0.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.anti_symmetric": {"fullname": "pyerrors.correlators.Corr.anti_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.anti_symmetric", "kind": "function", "doc": "
    Anti-symmetrize the correlator around x0=0.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.is_matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.is_matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.is_matrix_symmetric", "kind": "function", "doc": "
    Checks whether a correlator matrices is symmetric on every timeslice.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.trace": {"fullname": "pyerrors.correlators.Corr.trace", "modulename": "pyerrors.correlators", "qualname": "Corr.trace", "kind": "function", "doc": "
    Calculates the per-timeslice trace of a correlator matrix.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.matrix_symmetric", "kind": "function", "doc": "
    Symmetrizes the correlator matrices on every timeslice.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.GEVP": {"fullname": "pyerrors.correlators.Corr.GEVP", "modulename": "pyerrors.correlators", "qualname": "Corr.GEVP", "kind": "function", "doc": "
    Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.
    \n\n
    The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the\nlargest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing
    \n\n
    \n
    C.GEVP(t0=2)[0]  # Ground state vector(s)\nC.GEVP(t0=2)[:3]  # Vectors for the lowest three states\n
    \n
    \n\n
    Parameters
    \n\n
      \n
    • t0 (int):\nThe time t0 for the right hand side of the GEVP according to $G(t)v_i=\\lambda_i G(t_0)v_i$
      • \n
    • ts (int):\nfixed time $G(t_s)v_i=\\lambda_i G(t_0)v_i$ if sort=None.\nIf sort=\"Eigenvector\" it gives a reference point for the sorting method.
      • \n
    • sort (string):\nIf this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.\n
        \n
      • \"Eigenvalue\": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice. (default)
        • \n
      • \"Eigenvector\": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.\nThe reference state is identified by its eigenvalue at $t=t_s$.
        • \n
      • None: The GEVP is solved only at ts, no sorting is necessary
        • \n
      • \n
    • vector_obs (bool):\nIf True, uncertainties are propagated in the eigenvector computation (default False).
      • \n
    \n\n
    Other Parameters
    \n\n
      \n
    • state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
      • \n
    • method (str):\nMethod used to solve the GEVP.\n
        \n
      • \"eigh\": Use scipy.linalg.eigh to solve the GEVP. (default for vector_obs=False)
        • \n
      • \"cholesky\": Use manually implemented solution via the Cholesky decomposition. Automatically chosen if vector_obs==True.
        • \n
      • \n
    \n", "signature": "(self, t0, ts=None, sort='Eigenvalue', vector_obs=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Eigenvalue": {"fullname": "pyerrors.correlators.Corr.Eigenvalue", "modulename": "pyerrors.correlators", "qualname": "Corr.Eigenvalue", "kind": "function", "doc": "
    Determines the eigenvalue of the GEVP by solving and projecting the correlator
    \n\n
    Parameters
    \n\n
      \n
    • state (int):\nThe state one is interested in ordered by energy. The lowest state is zero.
      • \n
    • All other parameters are identical to the ones of Corr.GEVP.
      • \n
    \n", "signature": "(self, t0, ts=None, state=0, sort='Eigenvalue', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Hankel": {"fullname": "pyerrors.correlators.Corr.Hankel", "modulename": "pyerrors.correlators", "qualname": "Corr.Hankel", "kind": "function", "doc": "
    Constructs an NxN Hankel matrix
    \n\n
    C(t) c(t+1) ... c(t+n-1)\nC(t+1) c(t+2) ... c(t+n)\n.................\nC(t+(n-1)) c(t+n) ... c(t+2(n-1))
    \n\n
    Parameters
    \n\n
      \n
    • N (int):\nDimension of the Hankel matrix
      • \n
    • periodic (bool, optional):\ndetermines whether the matrix is extended periodically
      • \n
    \n", "signature": "(self, N, periodic=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.roll": {"fullname": "pyerrors.correlators.Corr.roll", "modulename": "pyerrors.correlators", "qualname": "Corr.roll", "kind": "function", "doc": "
    Periodically shift the correlator by dt timeslices
    \n\n
    Parameters
    \n\n
      \n
    • dt (int):\nnumber of timeslices
      • \n
    \n", "signature": "(self, dt):", "funcdef": "def"}, "pyerrors.correlators.Corr.reverse": {"fullname": "pyerrors.correlators.Corr.reverse", "modulename": "pyerrors.correlators", "qualname": "Corr.reverse", "kind": "function", "doc": "
    Reverse the time ordering of the Corr
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.thin": {"fullname": "pyerrors.correlators.Corr.thin", "modulename": "pyerrors.correlators", "qualname": "Corr.thin", "kind": "function", "doc": "
    Thin out a correlator to suppress correlations
    \n\n
    Parameters
    \n\n
      \n
    • spacing (int):\nKeep only every 'spacing'th entry of the correlator
      • \n
    • offset (int):\nOffset the equal spacing
      • \n
    \n", "signature": "(self, spacing=2, offset=0):", "funcdef": "def"}, "pyerrors.correlators.Corr.correlate": {"fullname": "pyerrors.correlators.Corr.correlate", "modulename": "pyerrors.correlators", "qualname": "Corr.correlate", "kind": "function", "doc": "
    Correlate the correlator with another correlator or Obs
    \n\n
    Parameters
    \n\n
      \n
    • partner (Obs or Corr):\npartner to correlate the correlator with.\nCan either be an Obs which is correlated with all entries of the\ncorrelator or a Corr of same length.
      • \n
    \n", "signature": "(self, partner):", "funcdef": "def"}, "pyerrors.correlators.Corr.reweight": {"fullname": "pyerrors.correlators.Corr.reweight", "modulename": "pyerrors.correlators", "qualname": "Corr.reweight", "kind": "function", "doc": "
    Reweight the correlator.
    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl.
      • \n
    \n", "signature": "(self, weight, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.T_symmetry": {"fullname": "pyerrors.correlators.Corr.T_symmetry", "modulename": "pyerrors.correlators", "qualname": "Corr.T_symmetry", "kind": "function", "doc": "
    Return the time symmetry average of the correlator and its partner
    \n\n
    Parameters
    \n\n
      \n
    • partner (Corr):\nTime symmetry partner of the Corr
      • \n
    • parity (int):\nParity quantum number of the correlator, can be +1 or -1
      • \n
    \n", "signature": "(self, partner, parity=1):", "funcdef": "def"}, "pyerrors.correlators.Corr.deriv": {"fullname": "pyerrors.correlators.Corr.deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.deriv", "kind": "function", "doc": "
    Return the first derivative of the correlator with respect to x0.
    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, forward, backward, improved, log, default: symmetric
      • \n
    \n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.second_deriv": {"fullname": "pyerrors.correlators.Corr.second_deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.second_deriv", "kind": "function", "doc": "
    Return the second derivative of the correlator with respect to x0.
    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice:\n    - symmetric (default)\n        $$\\tilde{\\partial}^2_0 f(x_0) = f(x_0+1)-2f(x_0)+f(x_0-1)$$\n    - big_symmetric\n        $$\\partial^2_0 f(x_0) = \\frac{f(x_0+2)-2f(x_0)+f(x_0-2)}{4}$$\n    - improved\n        $$\\partial^2_0 f(x_0) = \\frac{-f(x_0+2) + 16 * f(x_0+1) - 30 * f(x_0) + 16 * f(x_0-1) - f(x_0-2)}{12}$$\n    - log\n        $$f(x) = \\tilde{\\partial}^2_0 log(f(x_0))+(\\tilde{\\partial}_0 log(f(x_0)))^2$$
      • \n
    \n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.m_eff": {"fullname": "pyerrors.correlators.Corr.m_eff", "modulename": "pyerrors.correlators", "qualname": "Corr.m_eff", "kind": "function", "doc": "
    Returns the effective mass of the correlator as correlator object
    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\nlog : uses the standard effective mass log(C(t) / C(t+1))\ncosh, periodic : Use periodicity of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.\nsinh : Use anti-periodicity of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.\nSee, e.g., arXiv:1205.5380\narccosh : Uses the explicit form of the symmetrized correlator (not recommended)\nlogsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
      • \n
    • guess (float):\nguess for the root finder, only relevant for the root variant
      • \n
    \n", "signature": "(self, variant='log', guess=1.0):", "funcdef": "def"}, "pyerrors.correlators.Corr.fit": {"fullname": "pyerrors.correlators.Corr.fit", "modulename": "pyerrors.correlators", "qualname": "Corr.fit", "kind": "function", "doc": "
    Fits function to the data
    \n\n
    Parameters
    \n\n
      \n
    • function (obj):\nfunction to fit to the data. See fits.least_squares for details.
      • \n
    • fitrange (list):\nTwo element list containing the timeslices on which the fit is supposed to start and stop.\nCaution: This range is inclusive as opposed to standard python indexing.\nfitrange=[4, 6] corresponds to the three entries 4, 5 and 6.\nIf not specified, self.prange or all timeslices are used.
      • \n
    • silent (bool):\nDecides whether output is printed to the standard output.
      • \n
    \n", "signature": "(self, function, fitrange=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.plateau": {"fullname": "pyerrors.correlators.Corr.plateau", "modulename": "pyerrors.correlators", "qualname": "Corr.plateau", "kind": "function", "doc": "
    Extract a plateau value from a Corr object
    \n\n
    Parameters
    \n\n
      \n
    • plateau_range (list):\nlist with two entries, indicating the first and the last timeslice\nof the plateau region.
      • \n
    • method (str):\nmethod to extract the plateau.\n    'fit' fits a constant to the plateau region\n    'avg', 'average' or 'mean' just average over the given timeslices.
      • \n
    • auto_gamma (bool):\napply gamma_method with default parameters to the Corr. Defaults to None
      • \n
    \n", "signature": "(self, plateau_range=None, method='fit', auto_gamma=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.set_prange": {"fullname": "pyerrors.correlators.Corr.set_prange", "modulename": "pyerrors.correlators", "qualname": "Corr.set_prange", "kind": "function", "doc": "
    Sets the attribute prange of the Corr object.
    \n", "signature": "(self, prange):", "funcdef": "def"}, "pyerrors.correlators.Corr.show": {"fullname": "pyerrors.correlators.Corr.show", "modulename": "pyerrors.correlators", "qualname": "Corr.show", "kind": "function", "doc": "
    Plots the correlator using the tag of the correlator as label if available.
    \n\n
    Parameters
    \n\n
      \n
    • x_range (list):\nlist of two values, determining the range of the x-axis e.g. [4, 8].
      • \n
    • comp (Corr or list of Corr):\nCorrelator or list of correlators which are plotted for comparison.\nThe tags of these correlators are used as labels if available.
      • \n
    • logscale (bool):\nSets y-axis to logscale.
      • \n
    • plateau (Obs):\nPlateau value to be visualized in the figure.
      • \n
    • fit_res (Fit_result):\nFit_result object to be visualized.
      • \n
    • fit_key (str):\nKey for the fit function in Fit_result.fit_function (for combined fits).
      • \n
    • ylabel (str):\nLabel for the y-axis.
      • \n
    • save (str):\npath to file in which the figure should be saved.
      • \n
    • auto_gamma (bool):\nApply the gamma method with standard parameters to all correlators and plateau values before plotting.
      • \n
    • hide_sigma (float):\nHides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
      • \n
    • references (list):\nList of floating point values that are displayed as horizontal lines for reference.
      • \n
    • title (string):\nOptional title of the figure.
      • \n
    \n", "signature": "(\tself,\tx_range=None,\tcomp=None,\ty_range=None,\tlogscale=False,\tplateau=None,\tfit_res=None,\tfit_key=None,\tylabel=None,\tsave=None,\tauto_gamma=False,\thide_sigma=None,\treferences=None,\ttitle=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.spaghetti_plot": {"fullname": "pyerrors.correlators.Corr.spaghetti_plot", "modulename": "pyerrors.correlators", "qualname": "Corr.spaghetti_plot", "kind": "function", "doc": "
    Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.
    \n\n
    Parameters
    \n\n
      \n
    • logscale (bool):\nDetermines whether the scale of the y-axis is logarithmic or standard.
      • \n
    \n", "signature": "(self, logscale=True):", "funcdef": "def"}, "pyerrors.correlators.Corr.dump": {"fullname": "pyerrors.correlators.Corr.dump", "modulename": "pyerrors.correlators", "qualname": "Corr.dump", "kind": "function", "doc": "
    Dumps the Corr into a file of chosen type
    \n\n
    Parameters
    \n\n
      \n
    • filename (str):\nName of the file to be saved.
      • \n
    • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n", "signature": "(self, filename, datatype='json.gz', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.print": {"fullname": "pyerrors.correlators.Corr.print", "modulename": "pyerrors.correlators", "qualname": "Corr.print", "kind": "function", "doc": "
    \n", "signature": "(self, print_range=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.sqrt": {"fullname": "pyerrors.correlators.Corr.sqrt", "modulename": "pyerrors.correlators", "qualname": "Corr.sqrt", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.log": {"fullname": "pyerrors.correlators.Corr.log", "modulename": "pyerrors.correlators", "qualname": "Corr.log", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.exp": {"fullname": "pyerrors.correlators.Corr.exp", "modulename": "pyerrors.correlators", "qualname": "Corr.exp", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sin": {"fullname": "pyerrors.correlators.Corr.sin", "modulename": "pyerrors.correlators", "qualname": "Corr.sin", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cos": {"fullname": "pyerrors.correlators.Corr.cos", "modulename": "pyerrors.correlators", "qualname": "Corr.cos", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tan": {"fullname": "pyerrors.correlators.Corr.tan", "modulename": "pyerrors.correlators", "qualname": "Corr.tan", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sinh": {"fullname": "pyerrors.correlators.Corr.sinh", "modulename": "pyerrors.correlators", "qualname": "Corr.sinh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cosh": {"fullname": "pyerrors.correlators.Corr.cosh", "modulename": "pyerrors.correlators", "qualname": "Corr.cosh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tanh": {"fullname": "pyerrors.correlators.Corr.tanh", "modulename": "pyerrors.correlators", "qualname": "Corr.tanh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsin": {"fullname": "pyerrors.correlators.Corr.arcsin", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsin", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccos": {"fullname": "pyerrors.correlators.Corr.arccos", "modulename": "pyerrors.correlators", "qualname": "Corr.arccos", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctan": {"fullname": "pyerrors.correlators.Corr.arctan", "modulename": "pyerrors.correlators", "qualname": "Corr.arctan", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsinh": {"fullname": "pyerrors.correlators.Corr.arcsinh", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsinh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccosh": {"fullname": "pyerrors.correlators.Corr.arccosh", "modulename": "pyerrors.correlators", "qualname": "Corr.arccosh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctanh": {"fullname": "pyerrors.correlators.Corr.arctanh", "modulename": "pyerrors.correlators", "qualname": "Corr.arctanh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.real": {"fullname": "pyerrors.correlators.Corr.real", "modulename": "pyerrors.correlators", "qualname": "Corr.real", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.imag": {"fullname": "pyerrors.correlators.Corr.imag", "modulename": "pyerrors.correlators", "qualname": "Corr.imag", "kind": "variable", "doc": "
    \n"}, "pyerrors.correlators.Corr.prune": {"fullname": "pyerrors.correlators.Corr.prune", "modulename": "pyerrors.correlators", "qualname": "Corr.prune", "kind": "function", "doc": "
    Project large correlation matrix to lowest states
    \n\n
    This method can be used to reduce the size of an (N x N) correlation matrix\nto (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise\nis still small.
    \n\n
    Parameters
    \n\n
      \n
    • Ntrunc (int):\nRank of the target matrix.
      • \n
    • tproj (int):\nTime where the eigenvectors are evaluated, corresponds to ts in the GEVP method.\nThe default value is 3.
      • \n
    • t0proj (int):\nTime where the correlation matrix is inverted. Choosing t0proj=1 is strongly\ndiscouraged for O(a) improved theories, since the correctness of the procedure\ncannot be granted in this case. The default value is 2.
      • \n
    • basematrix (Corr):\nCorrelation matrix that is used to determine the eigenvectors of the\nlowest states based on a GEVP. basematrix is taken to be the Corr itself if\nis is not specified.
      • \n
    \n\n
    Notes
    \n\n
    We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving\nthe GEVP $$C(t) v_n(t, t_0) = \\lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \\equiv t_\\mathrm{proj}$\nand $t_0 \\equiv t_{0, \\mathrm{proj}}$. The target matrix is projected onto the subspace of the\nresulting eigenvectors $v_n, n=1,\\dots,N_\\mathrm{trunc}$ via\n$$G^\\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large\ncorrelation matrix and to remove some noise that is added by irrelevant operators.\nThis may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated\nbound $t_0 \\leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.
    \n", "signature": "(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.N": {"fullname": "pyerrors.correlators.Corr.N", "modulename": "pyerrors.correlators", "qualname": "Corr.N", "kind": "variable", "doc": "
    \n"}, "pyerrors.covobs": {"fullname": "pyerrors.covobs", "modulename": "pyerrors.covobs", "kind": "module", "doc": "
    \n"}, "pyerrors.covobs.Covobs": {"fullname": "pyerrors.covobs.Covobs", "modulename": "pyerrors.covobs", "qualname": "Covobs", "kind": "class", "doc": "
    \n"}, "pyerrors.covobs.Covobs.__init__": {"fullname": "pyerrors.covobs.Covobs.__init__", "modulename": "pyerrors.covobs", "qualname": "Covobs.__init__", "kind": "function", "doc": "
    Initialize Covobs object.
    \n\n
    Parameters
    \n\n
      \n
    • mean (float):\nMean value of the new Obs
      • \n
    • cov (list or array):\n2d Covariance matrix or 1d diagonal entries
      • \n
    • name (str):\nidentifier for the covariance matrix
      • \n
    • pos (int):\nPosition of the variance belonging to mean in cov.\nIs taken to be 1 if cov is 0-dimensional
      • \n
    • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
      • \n
    \n", "signature": "(mean, cov, name, pos=None, grad=None)"}, "pyerrors.covobs.Covobs.name": {"fullname": "pyerrors.covobs.Covobs.name", "modulename": "pyerrors.covobs", "qualname": "Covobs.name", "kind": "variable", "doc": "
    \n"}, "pyerrors.covobs.Covobs.value": {"fullname": "pyerrors.covobs.Covobs.value", "modulename": "pyerrors.covobs", "qualname": "Covobs.value", "kind": "variable", "doc": "
    \n"}, "pyerrors.covobs.Covobs.errsq": {"fullname": "pyerrors.covobs.Covobs.errsq", "modulename": "pyerrors.covobs", "qualname": "Covobs.errsq", "kind": "function", "doc": "
    Return the variance (= square of the error) of the Covobs
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.covobs.Covobs.cov": {"fullname": "pyerrors.covobs.Covobs.cov", "modulename": "pyerrors.covobs", "qualname": "Covobs.cov", "kind": "variable", "doc": "
    \n"}, "pyerrors.covobs.Covobs.grad": {"fullname": "pyerrors.covobs.Covobs.grad", "modulename": "pyerrors.covobs", "qualname": "Covobs.grad", "kind": "variable", "doc": "
    \n"}, "pyerrors.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "
    \n"}, "pyerrors.dirac.gammaX": {"fullname": "pyerrors.dirac.gammaX", "modulename": "pyerrors.dirac", "qualname": "gammaX", "kind": "variable", "doc": "
    \n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+0.j,  0.+1.j],\n       [ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n       [ 0.+0.j, -0.-1.j,  0.+0.j,  0.+0.j],\n       [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaY": {"fullname": "pyerrors.dirac.gammaY", "modulename": "pyerrors.dirac", "qualname": "gammaY", "kind": "variable", "doc": "
    \n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j],\n       [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n       [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n       [-1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaZ": {"fullname": "pyerrors.dirac.gammaZ", "modulename": "pyerrors.dirac", "qualname": "gammaZ", "kind": "variable", "doc": "
    \n", "default_value": "array([[ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j,  0.+0.j, -0.-1.j],\n       [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+1.j,  0.+0.j,  0.+0.j]])"}, "pyerrors.dirac.gammaT": {"fullname": "pyerrors.dirac.gammaT", "modulename": "pyerrors.dirac", "qualname": "gammaT", "kind": "variable", "doc": "
    \n", "default_value": "array([[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n       [1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gamma": {"fullname": "pyerrors.dirac.gamma", "modulename": "pyerrors.dirac", "qualname": "gamma", "kind": "variable", "doc": "
    \n", "default_value": "array([[[ 0.+0.j,  0.+0.j,  0.+0.j,  0.+1.j],\n        [ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n        [ 0.+0.j, -0.-1.j,  0.+0.j,  0.+0.j],\n        [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j],\n        [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n        [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n        [-1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  0.+1.j,  0.+0.j],\n        [ 0.+0.j,  0.+0.j,  0.+0.j, -0.-1.j],\n        [-0.-1.j,  0.+0.j,  0.+0.j,  0.+0.j],\n        [ 0.+0.j,  0.+1.j,  0.+0.j,  0.+0.j]],\n\n       [[ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],\n        [ 0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j],\n        [ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],\n        [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j]]])"}, "pyerrors.dirac.gamma5": {"fullname": "pyerrors.dirac.gamma5", "modulename": "pyerrors.dirac", "qualname": "gamma5", "kind": "variable", "doc": "
    \n", "default_value": "array([[ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j, -1.+0.j,  0.+0.j],\n       [ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j]])"}, "pyerrors.dirac.identity": {"fullname": "pyerrors.dirac.identity", "modulename": "pyerrors.dirac", "qualname": "identity", "kind": "variable", "doc": "
    \n", "default_value": "array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]])"}, "pyerrors.dirac.epsilon_tensor": {"fullname": "pyerrors.dirac.epsilon_tensor", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor", "kind": "function", "doc": "
    Rank-3 epsilon tensor
    \n\n
    Based on https://codegolf.stackexchange.com/a/160375
    \n\n
    Returns
    \n\n
      \n
    • elem (int):\nElement (i,j,k) of the epsilon tensor of rank 3
      • \n
    \n", "signature": "(i, j, k):", "funcdef": "def"}, "pyerrors.dirac.epsilon_tensor_rank4": {"fullname": "pyerrors.dirac.epsilon_tensor_rank4", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor_rank4", "kind": "function", "doc": "
    Rank-4 epsilon tensor
    \n\n
    Extension of https://codegolf.stackexchange.com/a/160375
    \n\n
    Returns
    \n\n
      \n
    • elem (int):\nElement (i,j,k,o) of the epsilon tensor of rank 4
      • \n
    \n", "signature": "(i, j, k, o):", "funcdef": "def"}, "pyerrors.dirac.Grid_gamma": {"fullname": "pyerrors.dirac.Grid_gamma", "modulename": "pyerrors.dirac", "qualname": "Grid_gamma", "kind": "function", "doc": "
    Returns gamma matrix in Grid labeling.
    \n", "signature": "(gamma_tag):", "funcdef": "def"}, "pyerrors.fits": {"fullname": "pyerrors.fits", "modulename": "pyerrors.fits", "kind": "module", "doc": "
    \n"}, "pyerrors.fits.Fit_result": {"fullname": "pyerrors.fits.Fit_result", "modulename": "pyerrors.fits", "qualname": "Fit_result", "kind": "class", "doc": "
    Represents fit results.
    \n\n
    Attributes
    \n\n
      \n
    • fit_parameters (list):\nresults for the individual fit parameters,\nalso accessible via indices.
      • \n
    • chisquare_by_dof (float):\nreduced chisquare.
      • \n
    • p_value (float):\np-value of the fit
      • \n
    • t2_p_value (float):\nHotelling t-squared p-value for correlated fits.
      • \n
    \n", "bases": "collections.abc.Sequence"}, "pyerrors.fits.Fit_result.fit_parameters": {"fullname": "pyerrors.fits.Fit_result.fit_parameters", "modulename": "pyerrors.fits", "qualname": "Fit_result.fit_parameters", "kind": "variable", "doc": "
    \n"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "
    Apply the gamma method to all fit parameters
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.Fit_result.gm": {"fullname": "pyerrors.fits.Fit_result.gm", "modulename": "pyerrors.fits", "qualname": "Fit_result.gm", "kind": "function", "doc": "
    Apply the gamma method to all fit parameters
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.least_squares": {"fullname": "pyerrors.fits.least_squares", "modulename": "pyerrors.fits", "qualname": "least_squares", "kind": "function", "doc": "
    Performs a non-linear fit to y = func(x).\n    ```
    \n\n
    Parameters
    \n\n
      \n
    • For an uncombined fit:
      • \n
    • x (list):\nlist of floats.
      • \n
    • y (list):\nlist of Obs.
      • \n
    • func (object):\nfit function, has to be of the form
      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(a, x):\n    return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
      \n
      \n\n
      For multiple x values func can be of the form
      \n\n
      \n
      def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
      \n
      \n\n
      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
      • \n
    • OR For a combined fit:
      • \n
    • x (dict):\ndict of lists.
      • \n
    • y (dict):\ndict of lists of Obs.
      • \n
    • funcs (dict):\ndict of objects\nfit functions have to be of the form (here a[0] is the common fit parameter)\n```python\nimport autograd.numpy as anp\nfuncs = {\"a\": func_a,\n        \"b\": func_b}
      \n\n
      def func_a(a, x):\n    return a[1] * anp.exp(-a[0] * x)
      \n\n
      def func_b(a, x):\n    return a[2] * anp.exp(-a[0] * x)
      \n\n
      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
      • \n
    • priors (dict or list, optional):\npriors can either be a dictionary with integer keys and the corresponding priors as values or\na list with an entry for every parameter in the fit. The entries can either be\nObs (e.g. results from a previous fit) or strings containing a value and an error formatted like\n0.548(23), 500(40) or 0.5(0.4)
      • \n
    • silent (bool, optional):\nIf true all output to the console is omitted (default False).
      • \n
    • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for\nnon-linear fits with many parameters. In case of correlated fits the guess is used to perform\nan uncorrelated fit which then serves as guess for the correlated fit.
      • \n
    • method (str, optional):\ncan be used to choose an alternative method for the minimization of chisquare.\nThe possible methods are the ones which can be used for scipy.optimize.minimize and\nmigrad of iminuit. If no method is specified, Levenberg-Marquard is used.\nReliable alternatives are migrad, Powell and Nelder-Mead.
      • \n
    • tol (float, optional):\ncan be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence\nto a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly\ninvalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values\nThe stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
      • \n
    • correlated_fit (bool):\nIf True, use the full inverse covariance matrix in the definition of the chisquare cost function.\nFor details about how the covariance matrix is estimated see pyerrors.obs.covariance.\nIn practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).\nThis procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
      • \n
    • inv_chol_cov_matrix [array,list], optional: array: shape = (no of y values) X (no of y values)\nlist:   for an uncombined fit: [\"\"]\nfor a combined fit: list of keys belonging to the corr_matrix saved in the array, must be the same as the keys of the y dict in alphabetical order\nIf correlated_fit=True is set as well, can provide an inverse covariance matrix (y errors, dy_f included!) of your own choosing for a correlated fit.\nThe matrix must be a lower triangular matrix constructed from a Cholesky decomposition: The function invert_corr_cov_cholesky(corr, inverrdiag) can be\nused to construct it from a correlation matrix (corr) and the errors dy_f of the data points (inverrdiag = np.diag(1 / np.asarray(dy_f))). For the correct\nordering the correlation matrix (corr) can be sorted via the function sort_corr(corr, kl, yd) where kl is the list of keys and yd the y dict.
      • \n
    • expected_chisquare (bool):\nIf True estimates the expected chisquare which is\ncorrected by effects caused by correlated input data (default False).
      • \n
    • resplot (bool):\nIf True, a plot which displays fit, data and residuals is generated (default False).
      • \n
    • qqplot (bool):\nIf True, a quantile-quantile plot of the fit result is generated (default False).
      • \n
    • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • output (Fit_result):\nParameters and information on the fitted result.
      • \n
    \n\n
    Examples
    \n\n
    \n
    >>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set\n>>> import numpy as np\n>>> from scipy.stats import norm\n>>> from scipy.linalg import cholesky\n>>> import pyerrors as pe\n>>> # generating the random data set\n>>> num_samples = 400\n>>> N = 3\n>>> x = np.arange(N)\n>>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers\n>>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers\n>>> r = r1 = r2 = np.zeros((N, N))\n>>> y = {}\n>>> for i in range(N):\n>>>    for j in range(N):\n>>>        r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix\n>>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors\n>>> for i in range(N):\n>>>    for j in range(N):\n>>>        r[i, j] *= errl[i] * errl[j] # element in covariance matrix\n>>> c = cholesky(r, lower=True)\n>>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined\n>>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built\n>>> x_dict = {}\n>>> y_dict = {}\n>>> chol_inv_dict = {}\n>>> data = []\n>>> for key in y.keys():\n>>>    x_dict[key] = x\n>>>    for i in range(N):\n>>>        data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data\n>>>    [o.gamma_method() for o in data]\n>>>    corr = pe.covariance(data, correlation=True)\n>>>    inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data]))\n>>>    chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below\n>>> y_dict = {'a': data[:3], 'b': data[3:]}\n>>> # common fit parameter p[0] in combined fit\n>>> def fit1(p, x):\n>>>    return p[0] + p[1] * x\n>>> def fit2(p, x):\n>>>    return p[0] + p[2] * x\n>>> fitf_dict = {'a': fit1, 'b':fit2}\n>>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']])\nFit with 3 parameters\nMethod: Levenberg-Marquardt\n`ftol` termination condition is satisfied.\nchisquare/d.o.f.: 0.5388013574561786 # random\nfit parameters [1.11897846 0.96361162 0.92325319] # random\n
    \n
    \n", "signature": "(x, y, func, priors=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.total_least_squares": {"fullname": "pyerrors.fits.total_least_squares", "modulename": "pyerrors.fits", "qualname": "total_least_squares", "kind": "function", "doc": "
    Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nlist of Obs, or a tuple of lists of Obs
      • \n
    • y (list):\nlist of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
      • \n
    • func (object):\nfunc has to be of the form
      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(a, x):\n    return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
      \n
      \n\n
      For multiple x values func can be of the form
      \n\n
      \n
      def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
      \n
      \n\n
      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
      • \n
    • silent (bool, optional):\nIf true all output to the console is omitted (default False).
      • \n
    • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for non-linear\nfits with many parameters.
      • \n
    • expected_chisquare (bool):\nIf true prints the expected chisquare which is\ncorrected by effects caused by correlated input data.\nThis can take a while as the full correlation matrix\nhas to be calculated (default False).
      • \n
    • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
      • \n
    \n\n
    Notes
    \n\n
    Based on the orthogonal distance regression module of scipy.
    \n\n
    Returns
    \n\n
      \n
    • output (Fit_result):\nParameters and information on the fitted result.
      • \n
    \n", "signature": "(x, y, func, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.fit_lin": {"fullname": "pyerrors.fits.fit_lin", "modulename": "pyerrors.fits", "qualname": "fit_lin", "kind": "function", "doc": "
    Performs a linear fit to y = n + m * x and returns two Obs n, m.
    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nCan either be a list of floats in which case no xerror is assumed, or\na list of Obs, where the dvalues of the Obs are used as xerror for the fit.
      • \n
    • y (list):\nList of Obs, the dvalues of the Obs are used as yerror for the fit.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • fit_parameters (list[Obs]):\nLIist of fitted observables.
      • \n
    \n", "signature": "(x, y, **kwargs):", "funcdef": "def"}, "pyerrors.fits.qqplot": {"fullname": "pyerrors.fits.qqplot", "modulename": "pyerrors.fits", "qualname": "qqplot", "kind": "function", "doc": "
    Generates a quantile-quantile plot of the fit result which can be used to\n   check if the residuals of the fit are gaussian distributed.
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(x, o_y, func, p, title=''):", "funcdef": "def"}, "pyerrors.fits.residual_plot": {"fullname": "pyerrors.fits.residual_plot", "modulename": "pyerrors.fits", "qualname": "residual_plot", "kind": "function", "doc": "
    Generates a plot which compares the fit to the data and displays the corresponding residuals
    \n\n
    For uncorrelated data the residuals are expected to be distributed ~N(0,1).
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(x, y, func, fit_res, title=''):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "
    Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
    \n\n
    Returns
    \n\n
      \n
    • err (np.array(Obs)):\nError band for an array of sample values x
      • \n
    \n", "signature": "(x, func, beta):", "funcdef": "def"}, "pyerrors.fits.ks_test": {"fullname": "pyerrors.fits.ks_test", "modulename": "pyerrors.fits", "qualname": "ks_test", "kind": "function", "doc": "
    Performs a Kolmogorov\u2013Smirnov test for the p-values of all fit object.
    \n\n
    Parameters
    \n\n
      \n
    • objects (list):\nList of fit results to include in the analysis (optional).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(objects=None):", "funcdef": "def"}, "pyerrors.input": {"fullname": "pyerrors.input", "modulename": "pyerrors.input", "kind": "module", "doc": "
    pyerrors includes an input submodule in which input routines and parsers for the output of various numerical programs are contained.
    \n\n
    Jackknife samples
    \n\n
    For comparison with other analysis workflows pyerrors can also generate jackknife samples from an Obs object or import jackknife samples into an Obs object.\nSee pyerrors.obs.Obs.export_jackknife and pyerrors.obs.import_jackknife for details.
    \n"}, "pyerrors.input.bdio": {"fullname": "pyerrors.input.bdio", "modulename": "pyerrors.input.bdio", "kind": "module", "doc": "
    \n"}, "pyerrors.input.bdio.read_ADerrors": {"fullname": "pyerrors.input.bdio.read_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "read_ADerrors", "kind": "function", "doc": "
    Extract generic MCMC data from a bdio file
    \n\n
    read_ADerrors requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
    \n\n
    all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
    \n\n
    Parameters
    \n\n
      \n
    • file_path -- path to the bdio file
      • \n
    • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (List[Obs]):\nExtracted data
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.write_ADerrors": {"fullname": "pyerrors.input.bdio.write_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "write_ADerrors", "kind": "function", "doc": "
    Write Obs to a bdio file according to ADerrors conventions
    \n\n
    read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
    \n\n
    all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
    \n\n
    Parameters
    \n\n
      \n
    • file_path -- path to the bdio file
      • \n
    • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • success (int):\nreturns 0 is successful
      • \n
    \n", "signature": "(obs_list, file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_mesons": {"fullname": "pyerrors.input.bdio.read_mesons", "modulename": "pyerrors.input.bdio", "qualname": "read_mesons", "kind": "function", "doc": "
    Extract mesons data from a bdio file and return it as a dictionary
    \n\n
    The dictionary can be accessed with a tuple consisting of (type, source_position, kappa1, kappa2)
    \n\n
    read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
    \n\n
    all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
    \n\n
    Parameters
    \n\n
      \n
    • file_path (str):\npath to the bdio file
      • \n
    • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    • start (int):\nThe first configuration to be read (default 1)
      • \n
    • stop (int):\nThe last configuration to be read (default None)
      • \n
    • step (int):\nFixed step size between two measurements (default 1)
      • \n
    • alternative_ensemble_name (str):\nManually overwrite ensemble name
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (dict):\nExtracted meson data
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_dSdm": {"fullname": "pyerrors.input.bdio.read_dSdm", "modulename": "pyerrors.input.bdio", "qualname": "read_dSdm", "kind": "function", "doc": "
    Extract dSdm data from a bdio file and return it as a dictionary
    \n\n
    The dictionary can be accessed with a tuple consisting of (type, kappa)
    \n\n
    read_dSdm requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
    \n\n
    all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
    \n\n
    Parameters
    \n\n
      \n
    • file_path (str):\npath to the bdio file
      • \n
    • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    • start (int):\nThe first configuration to be read (default 1)
      • \n
    • stop (int):\nThe last configuration to be read (default None)
      • \n
    • step (int):\nFixed step size between two measurements (default 1)
      • \n
    • alternative_ensemble_name (str):\nManually overwrite ensemble name
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.dobs": {"fullname": "pyerrors.input.dobs", "modulename": "pyerrors.input.dobs", "kind": "module", "doc": "
    \n"}, "pyerrors.input.dobs.create_pobs_string": {"fullname": "pyerrors.input.dobs.create_pobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_pobs_string", "kind": "function", "doc": "
    Export a list of Obs or structures containing Obs to an xml string\naccording to the Zeuthen pobs format.
    \n\n
    Tags are not written or recovered automatically. The separator | is removed from the replica names.
    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • xml_str (str):\nXML formatted string of the input data
      • \n
    \n", "signature": "(obsl, name, spec='', origin='', symbol=[], enstag=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_pobs": {"fullname": "pyerrors.input.dobs.write_pobs", "modulename": "pyerrors.input.dobs", "qualname": "write_pobs", "kind": "function", "doc": "
    Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen pobs format.
    \n\n
    Tags are not written or recovered automatically. The separator | is removed from the replica names.
    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
      • \n
    • gz (bool):\nIf True, the output is a gzipped xml. If False, the output is an xml file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(\tobsl,\tfname,\tname,\tspec='',\torigin='',\tsymbol=[],\tenstag=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_pobs": {"fullname": "pyerrors.input.dobs.read_pobs", "modulename": "pyerrors.input.dobs", "qualname": "read_pobs", "kind": "function", "doc": "
    Import a list of Obs from an xml.gz file in the Zeuthen pobs format.
    \n\n
    Tags are not written or recovered automatically.
    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • separatior_insertion (str or int):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nNone (default): Replica names remain unchanged.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(fname, full_output=False, gz=True, separator_insertion=None):", "funcdef": "def"}, "pyerrors.input.dobs.import_dobs_string": {"fullname": "pyerrors.input.dobs.import_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "import_dobs_string", "kind": "function", "doc": "
    Import a list of Obs from a string in the Zeuthen dobs format.
    \n\n
    Tags are not written or recovered automatically.
    \n\n
    Parameters
    \n\n
      \n
    • content (str):\nXML string containing the data
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(content, full_output=False, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_dobs": {"fullname": "pyerrors.input.dobs.read_dobs", "modulename": "pyerrors.input.dobs", "qualname": "read_dobs", "kind": "function", "doc": "
    Import a list of Obs from an xml.gz file in the Zeuthen dobs format.
    \n\n
    Tags are not written or recovered automatically.
    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes XML file.
      • \n
    • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(fname, full_output=False, gz=True, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.create_dobs_string": {"fullname": "pyerrors.input.dobs.create_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_dobs_string", "kind": "function", "doc": "
    Generate the string for the export of a list of Obs or structures containing Obs\nto a .xml.gz file according to the Zeuthen dobs format.
    \n\n
    Tags are not written or recovered automatically. The separator |is removed from the replica names.
    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • who (str):\nProvide the name of the person that exports the data.
      • \n
    • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • xml_str (str):\nXML string generated from the data
      • \n
    \n", "signature": "(\tobsl,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_dobs": {"fullname": "pyerrors.input.dobs.write_dobs", "modulename": "pyerrors.input.dobs", "qualname": "write_dobs", "kind": "function", "doc": "
    Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen dobs format.
    \n\n
    Tags are not written or recovered automatically. The separator | is removed from the replica names.
    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • who (str):\nProvide the name of the person that exports the data.
      • \n
    • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
      • \n
    • gz (bool):\nIf True, the output is a gzipped XML. If False, the output is a XML file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(\tobsl,\tfname,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.hadrons": {"fullname": "pyerrors.input.hadrons", "modulename": "pyerrors.input.hadrons", "kind": "module", "doc": "
    \n"}, "pyerrors.input.hadrons.read_hd5": {"fullname": "pyerrors.input.hadrons.read_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_hd5", "kind": "function", "doc": "
    Read hadrons hdf5 file and extract entry based on attributes.
    \n\n
    Parameters
    \n\n
      \n
    • filestem (str):\nFull namestem of the files to read, including the full path.
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • group (str):\nlabel of the group to be extracted.
      • \n
    • attrs (dict or int):\nDictionary containing the attributes. For example
      \n\n
      \n
      attrs = {"gamma_snk": "Gamma5",\n         "gamma_src": "Gamma5"}\n
      \n
      \n\n
      Alternatively an integer can be specified to identify the sub group.\nThis is discouraged as the order in the file is not guaranteed.
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • part (str):\nstring specifying whether to extract the real part ('real'),\nthe imaginary part ('imag') or a complex correlator ('complex').\nDefault 'real'.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr (Corr):\nCorrelator of the source sink combination in question.
      • \n
    \n", "signature": "(filestem, ens_id, group, attrs=None, idl=None, part='real'):", "funcdef": "def"}, "pyerrors.input.hadrons.read_meson_hd5": {"fullname": "pyerrors.input.hadrons.read_meson_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_meson_hd5", "kind": "function", "doc": "
    Read hadrons meson hdf5 file and extract the meson labeled 'meson'
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • meson (str):\nlabel of the meson to be extracted, standard value meson_0 which\ncorresponds to the pseudoscalar pseudoscalar two-point function.
      • \n
    • gammas (tuple of strings):\nInstrad of a meson label one can also provide a tuple of two strings\nindicating the gamma matrices at sink and source (gamma_snk, gamma_src).\n(\"Gamma5\", \"Gamma5\") corresponds to the pseudoscalar pseudoscalar\ntwo-point function. The gammas argument dominateds over meson.
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr (Corr):\nCorrelator of the source sink combination in question.
      • \n
    \n", "signature": "(path, filestem, ens_id, meson='meson_0', idl=None, gammas=None):", "funcdef": "def"}, "pyerrors.input.hadrons.extract_t0_hd5": {"fullname": "pyerrors.input.hadrons.extract_t0_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "extract_t0_hd5", "kind": "function", "doc": "
    Read hadrons FlowObservables hdf5 file and extract t0
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • obs (str):\nlabel of the observable from which t0 should be extracted.\nOptions: 'Clover energy density' and 'Plaquette energy density'
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
      • \n
    \n", "signature": "(\tpath,\tfilestem,\tens_id,\tobs='Clover energy density',\tfit_range=5,\tidl=None,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.hadrons.read_DistillationContraction_hd5": {"fullname": "pyerrors.input.hadrons.read_DistillationContraction_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_DistillationContraction_hd5", "kind": "function", "doc": "
    Read hadrons DistillationContraction hdf5 files in given directory structure
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the directories to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • diagrams (list):\nList of strings of the diagrams to extract, e.g. [\"direct\", \"box\", \"cross\"].
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (dict):\nextracted DistillationContration data
      • \n
    \n", "signature": "(path, ens_id, diagrams=['direct'], idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.Npr_matrix": {"fullname": "pyerrors.input.hadrons.Npr_matrix", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix", "kind": "class", "doc": "
    ndarray(shape, dtype=float, buffer=None, offset=0,\n        strides=None, order=None)
    \n\n
    An array object represents a multidimensional, homogeneous array\nof fixed-size items.  An associated data-type object describes the\nformat of each element in the array (its byte-order, how many bytes it\noccupies in memory, whether it is an integer, a floating point number,\nor something else, etc.)
    \n\n
    Arrays should be constructed using array, zeros or empty (refer\nto the See Also section below).  The parameters given here refer to\na low-level method (ndarray(...)) for instantiating an array.
    \n\n
    For more information, refer to the numpy module and examine the\nmethods and attributes of an array.
    \n\n
    Parameters
    \n\n
      \n
    • (for the __new__ method; see Notes below)
      • \n
    • shape (tuple of ints):\nShape of created array.
      • \n
    • dtype (data-type, optional):\nAny object that can be interpreted as a numpy data type.
      • \n
    • buffer (object exposing buffer interface, optional):\nUsed to fill the array with data.
      • \n
    • offset (int, optional):\nOffset of array data in buffer.
      • \n
    • strides (tuple of ints, optional):\nStrides of data in memory.
      • \n
    • order ({'C', 'F'}, optional):\nRow-major (C-style) or column-major (Fortran-style) order.
      • \n
    \n\n
    Attributes
    \n\n
      \n
    • T (ndarray):\nTranspose of the array.
      • \n
    • data (buffer):\nThe array's elements, in memory.
      • \n
    • dtype (dtype object):\nDescribes the format of the elements in the array.
      • \n
    • flags (dict):\nDictionary containing information related to memory use, e.g.,\n'C_CONTIGUOUS', 'OWNDATA', 'WRITEABLE', etc.
      • \n
    • flat (numpy.flatiter object):\nFlattened version of the array as an iterator.  The iterator\nallows assignments, e.g., x.flat = 3 (See ndarray.flat for\nassignment examples; TODO).
      • \n
    • imag (ndarray):\nImaginary part of the array.
      • \n
    • real (ndarray):\nReal part of the array.
      • \n
    • size (int):\nNumber of elements in the array.
      • \n
    • itemsize (int):\nThe memory use of each array element in bytes.
      • \n
    • nbytes (int):\nThe total number of bytes required to store the array data,\ni.e., itemsize * size.
      • \n
    • ndim (int):\nThe array's number of dimensions.
      • \n
    • shape (tuple of ints):\nShape of the array.
      • \n
    • strides (tuple of ints):\nThe step-size required to move from one element to the next in\nmemory. For example, a contiguous (3, 4) array of type\nint16 in C-order has strides (8, 2).  This implies that\nto move from element to element in memory requires jumps of 2 bytes.\nTo move from row-to-row, one needs to jump 8 bytes at a time\n(2 * 4).
      • \n
    • ctypes (ctypes object):\nClass containing properties of the array needed for interaction\nwith ctypes.
      • \n
    • base (ndarray):\nIf the array is a view into another array, that array is its base\n(unless that array is also a view).  The base array is where the\narray data is actually stored.
      • \n
    \n\n
    See Also
    \n\n
    array:  Construct an array.
    \nzeros:  Create an array, each element of which is zero.
    \nempty:  Create an array, but leave its allocated memory unchanged (i.e.,\nit contains \"garbage\").
    \ndtype:  Create a data-type.
    \nnumpy.typing.NDArray:  An ndarray alias :term:generic <generic type>\nw.r.t. its dtype.type <numpy.dtype.type>.  
    \n\n
    Notes
    \n\n
    There are two modes of creating an array using __new__:
    \n\n
      \n
    1. If buffer is None, then only shape, dtype, and order\nare used.
      2. \n
    2. If buffer is an object exposing the buffer interface, then\nall keywords are interpreted.
      3. \n
    \n\n
    No __init__ method is needed because the array is fully initialized\nafter the __new__ method.
    \n\n
    Examples
    \n\n
    These examples illustrate the low-level ndarray constructor.  Refer\nto the See Also section above for easier ways of constructing an\nndarray.
    \n\n
    First mode, buffer is None:
    \n\n
    \n
    >>> import numpy as np\n>>> np.ndarray(shape=(2,2), dtype=float, order='F')\narray([[0.0e+000, 0.0e+000], # random\n       [     nan, 2.5e-323]])\n
    \n
    \n\n
    Second mode:
    \n\n
    \n
    >>> np.ndarray((2,), buffer=np.array([1,2,3]),\n...            offset=np.int_().itemsize,\n...            dtype=int) # offset = 1*itemsize, i.e. skip first element\narray([2, 3])\n
    \n
    \n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "
    Gamma_5 hermitean conjugate
    \n\n
    Uses the fact that the propagator is gamma5 hermitean, so just the\nin and out momenta of the propagator are exchanged.
    \n"}, "pyerrors.input.hadrons.read_ExternalLeg_hd5": {"fullname": "pyerrors.input.hadrons.read_ExternalLeg_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_ExternalLeg_hd5", "kind": "function", "doc": "
    Read hadrons ExternalLeg hdf5 file and output an array of CObs
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (Npr_matrix):\nread Cobs-matrix
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Bilinear_hd5": {"fullname": "pyerrors.input.hadrons.read_Bilinear_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Bilinear_hd5", "kind": "function", "doc": "
    Read hadrons Bilinear hdf5 file and output an array of CObs
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result_dict (dict[Npr_matrix]):\nextracted Bilinears
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Fourquark_hd5": {"fullname": "pyerrors.input.hadrons.read_Fourquark_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Fourquark_hd5", "kind": "function", "doc": "
    Read hadrons FourquarkFullyConnected hdf5 file and output an array of CObs
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • vertices (list):\nVertex functions to be extracted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result_dict (dict):\nextracted fourquark matrizes
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None, vertices=['VA', 'AV']):", "funcdef": "def"}, "pyerrors.input.json": {"fullname": "pyerrors.input.json", "modulename": "pyerrors.input.json", "kind": "module", "doc": "
    \n"}, "pyerrors.input.json.create_json_string": {"fullname": "pyerrors.input.json.create_json_string", "modulename": "pyerrors.input.json", "qualname": "create_json_string", "kind": "function", "doc": "
    Generate the string for the export of a list of Obs or structures containing Obs\nto a .json(.gz) file
    \n\n
    Parameters
    \n\n
      \n
    • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • json_string (str):\nString for export to .json(.gz) file
      • \n
    \n", "signature": "(ol, description='', indent=1):", "funcdef": "def"}, "pyerrors.input.json.dump_to_json": {"fullname": "pyerrors.input.json.dump_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_to_json", "kind": "function", "doc": "
    Export a list of Obs or structures containing Obs to a .json(.gz) file.\nDict keys that are not JSON-serializable such as floats are converted to strings.
    \n\n
    Parameters
    \n\n
      \n
    • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Null
      • \n
    \n", "signature": "(ol, fname, description='', indent=1, gz=True):", "funcdef": "def"}, "pyerrors.input.json.import_json_string": {"fullname": "pyerrors.input.json.import_json_string", "modulename": "pyerrors.input.json", "qualname": "import_json_string", "kind": "function", "doc": "
    Reconstruct a list of Obs or structures containing Obs from a json string.
    \n\n
    The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
    \n\n
    Parameters
    \n\n
      \n
    • json_string (str):\njson string containing the data.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nreconstructed list of observables from the json string
      • \n
    • or
      • \n
    • result (Obs):\nonly one observable if the list only has one entry
      • \n
    • or
      • \n
    • result (dict):\nif full_output=True
      • \n
    \n", "signature": "(json_string, verbose=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.load_json": {"fullname": "pyerrors.input.json.load_json", "modulename": "pyerrors.input.json", "qualname": "load_json", "kind": "function", "doc": "
    Import a list of Obs or structures containing Obs from a .json(.gz) file.
    \n\n
    The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nreconstructed list of observables from the json string
      • \n
    • or
      • \n
    • result (Obs):\nonly one observable if the list only has one entry
      • \n
    • or
      • \n
    • result (dict):\nif full_output=True
      • \n
    \n", "signature": "(fname, verbose=True, gz=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.dump_dict_to_json": {"fullname": "pyerrors.input.json.dump_dict_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_dict_to_json", "kind": "function", "doc": "
    Export a dict of Obs or structures containing Obs to a .json(.gz) file
    \n\n
    Parameters
    \n\n
      \n
    • od (dict):\nDict of JSON valid structures and objects that will be exported.\nAt the moment, these objects can be either of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    • reps (str):\nSpecify the structure of the placeholder in exported dict to be reps[0-9]+.
      • \n
    • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(od, fname, description='', indent=1, reps='DICTOBS', gz=True):", "funcdef": "def"}, "pyerrors.input.json.load_json_dict": {"fullname": "pyerrors.input.json.load_json_dict", "modulename": "pyerrors.input.json", "qualname": "load_json_dict", "kind": "function", "doc": "
    Import a dict of Obs or structures containing Obs from a .json(.gz) file.
    \n\n
    The following structures are supported: Obs, list, numpy.ndarray, Corr
    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    • reps (str):\nSpecify the structure of the placeholder in imported dict to be reps[0-9]+.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (Obs / list / Corr):\nRead data
      • \n
    • or
      • \n
    • data (dict):\nRead data and meta-data
      • \n
    \n", "signature": "(fname, verbose=True, gz=True, full_output=False, reps='DICTOBS'):", "funcdef": "def"}, "pyerrors.input.misc": {"fullname": "pyerrors.input.misc", "modulename": "pyerrors.input.misc", "kind": "module", "doc": "
    \n"}, "pyerrors.input.misc.fit_t0": {"fullname": "pyerrors.input.misc.fit_t0", "modulename": "pyerrors.input.misc", "qualname": "fit_t0", "kind": "function", "doc": "
    Compute the root of (flow-based) data based on a dictionary that contains\nthe necessary information in key-value pairs a la (flow time: observable at flow time).
    \n\n
    It is assumed that the data is monotonically increasing and passes zero from below.\nNo exception is thrown if this is not the case (several roots, no monotonic increase).\nAn exception is thrown if no root can be found in the data.
    \n\n
    A linear fit in the vicinity of the root is performed to exctract the root from the\ntwo fit parameters.
    \n\n
    Parameters
    \n\n
      \n
    • t2E_dict (dict):\nDictionary with pairs of (flow time: observable at flow time) where the flow times\nare of type float and the observables of type Obs.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data. (Default: False)
      • \n
    • observable (str):\nKeyword to identify the observable to print the correct ylabel (if plot_fit is True)\nfor the observables 't0' and 'w0'. No y label is printed otherwise. (Default: 't0')
      • \n
    \n\n
    Returns
    \n\n
      \n
    • root (Obs):\nThe root of the data series.
      • \n
    \n", "signature": "(t2E_dict, fit_range, plot_fit=False, observable='t0'):", "funcdef": "def"}, "pyerrors.input.misc.read_pbp": {"fullname": "pyerrors.input.misc.read_pbp", "modulename": "pyerrors.input.misc", "qualname": "read_pbp", "kind": "function", "doc": "
    Read pbp format from given folder structure.
    \n\n
    Parameters
    \n\n
      \n
    • r_start (list):\nlist which contains the first config to be read for each replicum
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nlist of observables read
      • \n
    \n", "signature": "(path, prefix, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD": {"fullname": "pyerrors.input.openQCD", "modulename": "pyerrors.input.openQCD", "kind": "module", "doc": "
    \n"}, "pyerrors.input.openQCD.read_rwms": {"fullname": "pyerrors.input.openQCD.read_rwms", "modulename": "pyerrors.input.openQCD", "qualname": "read_rwms", "kind": "function", "doc": "
    Read rwms format from given folder structure. Returns a list of length nrw
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath that contains the data files
      • \n
    • prefix (str):\nall files in path that start with prefix are considered as input files.\nMay be used together postfix to consider only special file endings.\nPrefix is ignored, if the keyword 'files' is used.
      • \n
    • version (str):\nversion of openQCD, default 2.0
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.ms1' for openQCD-files
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • print_err (bool):\nPrint additional information that is useful for debugging.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • rwms (Obs):\nReweighting factors read
      • \n
    \n", "signature": "(path, prefix, version='2.0', names=None, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_t0": {"fullname": "pyerrors.input.openQCD.extract_t0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_t0", "kind": "function", "doc": "
    Extract t0/a^2 from given .ms.dat files. Returns t0 as Obs.
    \n\n
    It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t^2 - c (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.
    \n\n
    It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to .ms.dat files
      • \n
    • prefix (str):\nEnsemble prefix
      • \n
    • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
      • \n
    • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
      • \n
    • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • postfix (str):\nPostfix of measurement file (Default: ms)
      • \n
    • c (float):\nConstant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1.
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • plaquette (bool):\nIf true extract the plaquette estimate of t0 instead.
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
      • \n
    • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
      • \n
    \n\n
    Returns
    \n\n
      \n
    • t0 (Obs):\nExtracted t0
      • \n
    \n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_w0": {"fullname": "pyerrors.input.openQCD.extract_w0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_w0", "kind": "function", "doc": "
    Extract w0/a from given .ms.dat files. Returns w0 as Obs.
    \n\n
    It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t d(t^2)/dt -  (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.
    \n\n
    It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to .ms.dat files
      • \n
    • prefix (str):\nEnsemble prefix
      • \n
    • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
      • \n
    • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
      • \n
    • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • postfix (str):\nPostfix of measurement file (Default: ms)
      • \n
    • c (float):\nConstant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1.
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • plaquette (bool):\nIf true extract the plaquette estimate of w0 instead.
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of w0 is shown together with the data.
      • \n
    • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
      • \n
    \n\n
    Returns
    \n\n
      \n
    • w0 (Obs):\nExtracted w0
      • \n
    \n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop": {"fullname": "pyerrors.input.openQCD.read_qtop", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop", "kind": "function", "doc": "
    Read the topologial charge based on openQCD gradient flow measurements.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • version (str):\nEither openQCD or sfqcd, depending on the data.
      • \n
    • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
      • \n
    • integer_charge (bool):\nIf True, the charge is rounded towards the nearest integer on each config.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (Obs):\nRead topological charge
      • \n
    \n", "signature": "(path, prefix, c, dtr_cnfg=1, version='openQCD', **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_gf_coupling": {"fullname": "pyerrors.input.openQCD.read_gf_coupling", "modulename": "pyerrors.input.openQCD", "qualname": "read_gf_coupling", "kind": "function", "doc": "
    Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.
    \n\n
    Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used.
      • \n
    \n", "signature": "(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.qtop_projection": {"fullname": "pyerrors.input.openQCD.qtop_projection", "modulename": "pyerrors.input.openQCD", "qualname": "qtop_projection", "kind": "function", "doc": "
    Returns the projection to the topological charge sector defined by target.
    \n\n
    Parameters
    \n\n
      \n
    • path (Obs):\nTopological charge.
      • \n
    • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • reto (Obs):\nprojection to the topological charge sector defined by target
      • \n
    \n", "signature": "(qtop, target=0):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop_sector": {"fullname": "pyerrors.input.openQCD.read_qtop_sector", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop_sector", "kind": "function", "doc": "
    Constructs reweighting factors to a specified topological sector.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L
      • \n
    • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of trajectories\nbetween two configs.\nif it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • version (str):\nversion string of the openQCD (sfqcd) version used to create\nthe ensemble. Default is 2.0. May also be set to sfqcd.
      • \n
    • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
      • \n
    • r_start (list):\noffset of the first ensemble, making it easier to match\nlater on with other Obs
      • \n
    • r_stop (list):\nlast configurations that need to be read (per replicum)
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • reto (Obs):\nprojection to the topological charge sector defined by target
      • \n
    \n", "signature": "(path, prefix, c, target=0, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_ms5_xsf": {"fullname": "pyerrors.input.openQCD.read_ms5_xsf", "modulename": "pyerrors.input.openQCD", "qualname": "read_ms5_xsf", "kind": "function", "doc": "
    Read data from files in the specified directory with the specified prefix and quark combination extension, and return a Corr object containing the data.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nThe directory to search for the files in.
      • \n
    • prefix (str):\nThe prefix to match the files against.
      • \n
    • qc (str):\nThe quark combination extension to match the files against.
      • \n
    • corr (str):\nThe correlator to extract data for.
      • \n
    • sep (str, optional):\nThe separator to use when parsing the replika names.
      • \n
    • **kwargs: Additional keyword arguments. The following keyword arguments are recognized:
      \n\n
        \n
      • names (List[str]): A list of names to use for the replicas.
        • \n
      • files (List[str]): A list of files to read data from.
        • \n
      • idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given.
        • \n
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Corr: A complex valued Corr object containing the data read from the files. In case of boudary to bulk correlators.
      • \n
    • or
      • \n
    • CObs: A complex valued CObs object containing the data read from the files. In case of boudary to boundary correlators.
      • \n
    \n\n
    Raises
    \n\n
      \n
    • FileNotFoundError: If no files matching the specified prefix and quark combination extension are found in the specified directory.
      • \n
    • IOError: If there is an error reading a file.
      • \n
    • struct.error: If there is an error unpacking binary data.
      • \n
    \n", "signature": "(path, prefix, qc, corr, sep='r', **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas": {"fullname": "pyerrors.input.pandas", "modulename": "pyerrors.input.pandas", "kind": "module", "doc": "
    \n"}, "pyerrors.input.pandas.to_sql": {"fullname": "pyerrors.input.pandas.to_sql", "modulename": "pyerrors.input.pandas", "qualname": "to_sql", "kind": "function", "doc": "
    Write DataFrame including Obs or Corr valued columns to sqlite database.
    \n\n
    Parameters
    \n\n
      \n
    • df (pandas.DataFrame):\nDataframe to be written to the database.
      • \n
    • table_name (str):\nName of the table in the database.
      • \n
    • db (str):\nPath to the sqlite database.
      • \n
    • if exists (str):\nHow to behave if table already exists. Options 'fail', 'replace', 'append'.
      • \n
    • gz (bool):\nIf True the json strings are gzipped.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(df, table_name, db, if_exists='fail', gz=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.read_sql": {"fullname": "pyerrors.input.pandas.read_sql", "modulename": "pyerrors.input.pandas", "qualname": "read_sql", "kind": "function", "doc": "
    Execute SQL query on sqlite database and obtain DataFrame including Obs or Corr valued columns.
    \n\n
    Parameters
    \n\n
      \n
    • sql (str):\nSQL query to be executed.
      • \n
    • db (str):\nPath to the sqlite database.
      • \n
    • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
      • \n
    \n", "signature": "(sql, db, auto_gamma=False, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.dump_df": {"fullname": "pyerrors.input.pandas.dump_df", "modulename": "pyerrors.input.pandas", "qualname": "dump_df", "kind": "function", "doc": "
    Exports a pandas DataFrame containing Obs valued columns to a (gzipped) csv file.
    \n\n
    Before making use of pandas to_csv functionality Obs objects are serialized via the standardized\njson format of pyerrors.
    \n\n
    Parameters
    \n\n
      \n
    • df (pandas.DataFrame):\nDataframe to be dumped to a file.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • gz (bool):\nIf True, the output is a gzipped csv file. If False, the output is a csv file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(df, fname, gz=True):", "funcdef": "def"}, "pyerrors.input.pandas.load_df": {"fullname": "pyerrors.input.pandas.load_df", "modulename": "pyerrors.input.pandas", "qualname": "load_df", "kind": "function", "doc": "
    Imports a pandas DataFrame from a csv.(gz) file in which Obs objects are serialized as json strings.
    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
      • \n
    \n", "signature": "(fname, auto_gamma=False, gz=True):", "funcdef": "def"}, "pyerrors.input.sfcf": {"fullname": "pyerrors.input.sfcf", "modulename": "pyerrors.input.sfcf", "kind": "module", "doc": "
    \n"}, "pyerrors.input.sfcf.sep": {"fullname": "pyerrors.input.sfcf.sep", "modulename": "pyerrors.input.sfcf", "qualname": "sep", "kind": "variable", "doc": "
    \n", "default_value": "'/'"}, "pyerrors.input.sfcf.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "
    Read sfcf files from given folder structure.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to the sfcf files.
      • \n
    • prefix (str):\nPrefix of the sfcf files.
      • \n
    • name (str):\nName of the correlation function to read.
      • \n
    • quarks (str):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
      • \n
    • corr_type (str):\nType of correlation function to read. Can be\n
        \n
      • 'bi' for boundary-inner
        • \n
      • 'bb' for boundary-boundary
        • \n
      • 'bib' for boundary-inner-boundary
        • \n
      • \n
    • noffset (int):\nOffset of the source (only relevant when wavefunctions are used)
      • \n
    • wf (int):\nID of wave function
      • \n
    • wf2 (int):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
      • \n
    • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • ens_name (str):\nreplaces the name of the ensemble
      • \n
    • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
      • \n
    • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
      • \n
    • replica (list):\nlist of replica to be read, default is all
      • \n
    • files (list):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
      • \n
    • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
      • \n
    \n", "signature": "(\tpath,\tprefix,\tname,\tquarks='.*',\tcorr_type='bi',\tnoffset=0,\twf=0,\twf2=0,\tversion='1.0c',\tcfg_separator='n',\tsilent=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.sfcf.read_sfcf_multi": {"fullname": "pyerrors.input.sfcf.read_sfcf_multi", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf_multi", "kind": "function", "doc": "
    Read sfcf files from given folder structure.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to the sfcf files.
      • \n
    • prefix (str):\nPrefix of the sfcf files.
      • \n
    • name (str):\nName of the correlation function to read.
      • \n
    • quarks_list (list[str]):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
      • \n
    • corr_type_list (list[str]):\nType of correlation function to read. Can be\n
        \n
      • 'bi' for boundary-inner
        • \n
      • 'bb' for boundary-boundary
        • \n
      • 'bib' for boundary-inner-boundary
        • \n
      • \n
    • noffset_list (list[int]):\nOffset of the source (only relevant when wavefunctions are used)
      • \n
    • wf_list (int):\nID of wave function
      • \n
    • wf2_list (list[int]):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
      • \n
    • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • ens_name (str):\nreplaces the name of the ensemble
      • \n
    • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
      • \n
    • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
      • \n
    • replica (list):\nlist of replica to be read, default is all
      • \n
    • files (list[list[int]]):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
      • \n
    • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (dict[list[Obs]]):\ndict with one of the following properties:\nif keyed_out:\n    dict[key] = list[Obs]\n    where key has the form name/quarks/offset/wf/wf2\nif not keyed_out:\n    dict[name][quarks][offset][wf][wf2] = list[Obs]
      • \n
    \n", "signature": "(\tpath,\tprefix,\tname_list,\tquarks_list=['.*'],\tcorr_type_list=['bi'],\tnoffset_list=[0],\twf_list=[0],\twf2_list=[0],\tversion='1.0c',\tcfg_separator='n',\tsilent=False,\tkeyed_out=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.utils": {"fullname": "pyerrors.input.utils", "modulename": "pyerrors.input.utils", "kind": "module", "doc": "
    Utilities for the input
    \n"}, "pyerrors.input.utils.sort_names": {"fullname": "pyerrors.input.utils.sort_names", "modulename": "pyerrors.input.utils", "qualname": "sort_names", "kind": "function", "doc": "
    Sorts a list of names of replika with searches for r and id in the replikum string.\nIf this search fails, a fallback method is used,\nwhere the strings are simply compared and the first diffeing numeral is used for differentiation.
    \n\n
    Parameters
    \n\n
      \n
    • ll (list):\nlist to sort
      • \n
    \n\n
    Returns
    \n\n
      \n
    • ll (list):\nsorted list
      • \n
    \n", "signature": "(ll):", "funcdef": "def"}, "pyerrors.input.utils.check_idl": {"fullname": "pyerrors.input.utils.check_idl", "modulename": "pyerrors.input.utils", "qualname": "check_idl", "kind": "function", "doc": "
    Checks if list of configurations is contained in an idl
    \n\n
    Parameters
    \n\n
      \n
    • idl (range or list):\nidl of the current replicum
      • \n
    • che (list):\nlist of configurations to be checked against
      • \n
    \n\n
    Returns
    \n\n
      \n
    • miss_str (str):\nstring with integers of which idls are missing
      • \n
    \n", "signature": "(idl, che):", "funcdef": "def"}, "pyerrors.input.utils.check_params": {"fullname": "pyerrors.input.utils.check_params", "modulename": "pyerrors.input.utils", "qualname": "check_params", "kind": "function", "doc": "
    Check if, for sfcf, the parameter hashes at the end of the parameter files are in fact the expected one.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nmeasurement path, same as for sfcf read method
      • \n
    • param_hash (str):\nexpected parameter hash
      • \n
    • prefix (str):\ndata prefix to find the appropriate replicum folders in path
      • \n
    • param_prefix (str):\nprefix of the parameter file. Defaults to 'parameters_'
      • \n
    \n\n
    Returns
    \n\n
      \n
    • nums (dict):\ndictionary of faulty parameter files sorted by the replica paths
      • \n
    \n", "signature": "(path, param_hash, prefix, param_prefix='parameters_'):", "funcdef": "def"}, "pyerrors.integrate": {"fullname": "pyerrors.integrate", "modulename": "pyerrors.integrate", "kind": "module", "doc": "
    \n"}, "pyerrors.integrate.quad": {"fullname": "pyerrors.integrate.quad", "modulename": "pyerrors.integrate", "qualname": "quad", "kind": "function", "doc": "
    Performs a (one-dimensional) numeric integration of f(p, x) from a to b.
    \n\n
    The integration is performed using scipy.integrate.quad().\nAll parameters that can be passed to scipy.integrate.quad may also be passed to this function.\nThe output is the same as for scipy.integrate.quad, the first element being an Obs.
    \n\n
    Parameters
    \n\n
      \n
    • func (object):\nfunction to integrate, has to be of the form
      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(p, x):\n    return p[0] + p[1] * x + p[2] * anp.sinh(x)\n
      \n
      \n\n
      where x is the integration variable.
      • \n
    • p (list of floats or Obs):\nparameters of the function func.
      • \n
    • a (float or Obs):\nLower limit of integration (use -numpy.inf for -infinity).
      • \n
    • b (float or Obs):\nUpper limit of integration (use -numpy.inf for -infinity).
      • \n
    • All parameters of scipy.integrate.quad
      • \n
    \n\n
    Returns
    \n\n
      \n
    • y (Obs):\nThe integral of func from a to b.
      • \n
    • abserr (float):\nAn estimate of the absolute error in the result.
      • \n
    • infodict (dict):\nA dictionary containing additional information.\nRun scipy.integrate.quad_explain() for more information.
      • \n
    • message: A convergence message.
      • \n
    • explain: Appended only with 'cos' or 'sin' weighting and infinite\nintegration limits, it contains an explanation of the codes in\ninfodict['ierlst']
      • \n
    \n", "signature": "(func, p, a, b, **kwargs):", "funcdef": "def"}, "pyerrors.linalg": {"fullname": "pyerrors.linalg", "modulename": "pyerrors.linalg", "kind": "module", "doc": "
    \n"}, "pyerrors.linalg.matmul": {"fullname": "pyerrors.linalg.matmul", "modulename": "pyerrors.linalg", "qualname": "matmul", "kind": "function", "doc": "
    Matrix multiply all operands.
    \n\n
    Parameters
    \n\n
      \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    • This implementation is faster compared to standard multiplication via the @ operator.
      • \n
    \n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.jack_matmul": {"fullname": "pyerrors.linalg.jack_matmul", "modulename": "pyerrors.linalg", "qualname": "jack_matmul", "kind": "function", "doc": "
    Matrix multiply both operands making use of the jackknife approximation.
    \n\n
    Parameters
    \n\n
      \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    • For large matrices this is considerably faster compared to matmul.
      • \n
    \n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.einsum": {"fullname": "pyerrors.linalg.einsum", "modulename": "pyerrors.linalg", "qualname": "einsum", "kind": "function", "doc": "
    Wrapper for numpy.einsum
    \n\n
    Parameters
    \n\n
      \n
    • subscripts (str):\nSubscripts for summation (see numpy documentation for details)
      • \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    \n", "signature": "(subscripts, *operands):", "funcdef": "def"}, "pyerrors.linalg.inv": {"fullname": "pyerrors.linalg.inv", "modulename": "pyerrors.linalg", "qualname": "inv", "kind": "function", "doc": "
    Inverse of Obs or CObs valued matrices.
    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.cholesky": {"fullname": "pyerrors.linalg.cholesky", "modulename": "pyerrors.linalg", "qualname": "cholesky", "kind": "function", "doc": "
    Cholesky decomposition of Obs valued matrices.
    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.det": {"fullname": "pyerrors.linalg.det", "modulename": "pyerrors.linalg", "qualname": "det", "kind": "function", "doc": "
    Determinant of Obs valued matrices.
    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.eigh": {"fullname": "pyerrors.linalg.eigh", "modulename": "pyerrors.linalg", "qualname": "eigh", "kind": "function", "doc": "
    Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eig": {"fullname": "pyerrors.linalg.eig", "modulename": "pyerrors.linalg", "qualname": "eig", "kind": "function", "doc": "
    Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig.
    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eigv": {"fullname": "pyerrors.linalg.eigv", "modulename": "pyerrors.linalg", "qualname": "eigv", "kind": "function", "doc": "
    Computes the eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.pinv": {"fullname": "pyerrors.linalg.pinv", "modulename": "pyerrors.linalg", "qualname": "pinv", "kind": "function", "doc": "
    Computes the Moore-Penrose pseudoinverse of a matrix of Obs.
    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.svd": {"fullname": "pyerrors.linalg.svd", "modulename": "pyerrors.linalg", "qualname": "svd", "kind": "function", "doc": "
    Computes the singular value decomposition of a matrix of Obs.
    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.misc": {"fullname": "pyerrors.misc", "modulename": "pyerrors.misc", "kind": "module", "doc": "
    \n"}, "pyerrors.misc.print_config": {"fullname": "pyerrors.misc.print_config", "modulename": "pyerrors.misc", "qualname": "print_config", "kind": "function", "doc": "
    Print information about version of python, pyerrors and dependencies.
    \n", "signature": "():", "funcdef": "def"}, "pyerrors.misc.errorbar": {"fullname": "pyerrors.misc.errorbar", "modulename": "pyerrors.misc", "qualname": "errorbar", "kind": "function", "doc": "
    pyerrors wrapper for the errorbars method of matplotlib
    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nA list of x-values which can be Obs.
      • \n
    • y (list):\nA list of y-values which can be Obs.
      • \n
    • axes ((matplotlib.pyplot.axes)):\nThe axes to plot on. default is plt.
      • \n
    \n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.10.14/x64/lib/python3.10/site-packages/matplotlib/pyplot.py'>,\t**kwargs):", "funcdef": "def"}, "pyerrors.misc.dump_object": {"fullname": "pyerrors.misc.dump_object", "modulename": "pyerrors.misc", "qualname": "dump_object", "kind": "function", "doc": "
    Dump object into pickle file.
    \n\n
    Parameters
    \n\n
      \n
    • obj (object):\nobject to be saved in the pickle file
      • \n
    • name (str):\nname of the file
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(obj, name, **kwargs):", "funcdef": "def"}, "pyerrors.misc.load_object": {"fullname": "pyerrors.misc.load_object", "modulename": "pyerrors.misc", "qualname": "load_object", "kind": "function", "doc": "
    Load object from pickle file.
    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the file
      • \n
    \n\n
    Returns
    \n\n
      \n
    • object (Obs):\nLoaded Object
      • \n
    \n", "signature": "(path):", "funcdef": "def"}, "pyerrors.misc.pseudo_Obs": {"fullname": "pyerrors.misc.pseudo_Obs", "modulename": "pyerrors.misc", "qualname": "pseudo_Obs", "kind": "function", "doc": "
    Generate an Obs object with given value, dvalue and name for test purposes
    \n\n
    Parameters
    \n\n
      \n
    • value (float):\ncentral value of the Obs to be generated.
      • \n
    • dvalue (float):\nerror of the Obs to be generated.
      • \n
    • name (str):\nname of the ensemble for which the Obs is to be generated.
      • \n
    • samples (int):\nnumber of samples for the Obs (default 1000).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (Obs):\nGenerated Observable
      • \n
    \n", "signature": "(value, dvalue, name, samples=1000):", "funcdef": "def"}, "pyerrors.misc.gen_correlated_data": {"fullname": "pyerrors.misc.gen_correlated_data", "modulename": "pyerrors.misc", "qualname": "gen_correlated_data", "kind": "function", "doc": "
    Generate observables with given covariance and autocorrelation times.
    \n\n
    Parameters
    \n\n
      \n
    • means (list):\nlist containing the mean value of each observable.
      • \n
    • cov (numpy.ndarray):\ncovariance matrix for the data to be generated.
      • \n
    • name (str):\nensemble name for the data to be geneated.
      • \n
    • tau (float or list):\ncan either be a real number or a list with an entry for\nevery dataset.
      • \n
    • samples (int):\nnumber of samples to be generated for each observable.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr_obs (list[Obs]):\nGenerated observable list
      • \n
    \n", "signature": "(means, cov, name, tau=0.5, samples=1000):", "funcdef": "def"}, "pyerrors.mpm": {"fullname": "pyerrors.mpm", "modulename": "pyerrors.mpm", "kind": "module", "doc": "
    \n"}, "pyerrors.mpm.matrix_pencil_method": {"fullname": "pyerrors.mpm.matrix_pencil_method", "modulename": "pyerrors.mpm", "qualname": "matrix_pencil_method", "kind": "function", "doc": "
    Matrix pencil method to extract k energy levels from data
    \n\n
    Implementation of the matrix pencil method based on\neq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)
    \n\n
    Parameters
    \n\n
      \n
    • data (list):\ncan be a list of Obs for the analysis of a single correlator, or a list of lists\nof Obs if several correlators are to analyzed at once.
      • \n
    • k (int):\nNumber of states to extract (default 1).
      • \n
    • p (int):\nmatrix pencil parameter which filters noise. The optimal value is expected between\nlen(data)/3 and 2*len(data)/3. The computation is more expensive the closer p is\nto len(data)/2 but could possibly suppress more noise (default len(data)//2).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • energy_levels (list[Obs]):\nExtracted energy levels
      • \n
    \n", "signature": "(corrs, k=1, p=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs": {"fullname": "pyerrors.obs", "modulename": "pyerrors.obs", "kind": "module", "doc": "
    \n"}, "pyerrors.obs.Obs": {"fullname": "pyerrors.obs.Obs", "modulename": "pyerrors.obs", "qualname": "Obs", "kind": "class", "doc": "
    Class for a general observable.
    \n\n
    Instances of Obs are the basic objects of a pyerrors error analysis.\nThey are initialized with a list which contains arrays of samples for\ndifferent ensembles/replica and another list of same length which contains\nthe names of the ensembles/replica. Mathematical operations can be\nperformed on instances. The result is another instance of Obs. The error of\nan instance can be computed with the gamma_method. Also contains additional\nmethods for output and visualization of the error calculation.
    \n\n
    Attributes
    \n\n
      \n
    • S_global (float):\nStandard value for S (default 2.0)
      • \n
    • S_dict (dict):\nDictionary for S values. If an entry for a given ensemble\nexists this overwrites the standard value for that ensemble.
      • \n
    • tau_exp_global (float):\nStandard value for tau_exp (default 0.0)
      • \n
    • tau_exp_dict (dict):\nDictionary for tau_exp values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
      • \n
    • N_sigma_global (float):\nStandard value for N_sigma (default 1.0)
      • \n
    • N_sigma_dict (dict):\nDictionary for N_sigma values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
      • \n
    \n"}, "pyerrors.obs.Obs.__init__": {"fullname": "pyerrors.obs.Obs.__init__", "modulename": "pyerrors.obs", "qualname": "Obs.__init__", "kind": "function", "doc": "
    Initialize Obs object.
    \n\n
    Parameters
    \n\n
      \n
    • samples (list):\nlist of numpy arrays containing the Monte Carlo samples
      • \n
    • names (list):\nlist of strings labeling the individual samples
      • \n
    • idl (list, optional):\nlist of ranges or lists on which the samples are defined
      • \n
    \n", "signature": "(samples, names, idl=None, **kwargs)"}, "pyerrors.obs.Obs.S_global": {"fullname": "pyerrors.obs.Obs.S_global", "modulename": "pyerrors.obs", "qualname": "Obs.S_global", "kind": "variable", "doc": "
    \n", "default_value": "2.0"}, "pyerrors.obs.Obs.S_dict": {"fullname": "pyerrors.obs.Obs.S_dict", "modulename": "pyerrors.obs", "qualname": "Obs.S_dict", "kind": "variable", "doc": "
    \n", "default_value": "{}"}, "pyerrors.obs.Obs.tau_exp_global": {"fullname": "pyerrors.obs.Obs.tau_exp_global", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_global", "kind": "variable", "doc": "
    \n", "default_value": "0.0"}, "pyerrors.obs.Obs.tau_exp_dict": {"fullname": "pyerrors.obs.Obs.tau_exp_dict", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_dict", "kind": "variable", "doc": "
    \n", "default_value": "{}"}, "pyerrors.obs.Obs.N_sigma_global": {"fullname": "pyerrors.obs.Obs.N_sigma_global", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_global", "kind": "variable", "doc": "
    \n", "default_value": "1.0"}, "pyerrors.obs.Obs.N_sigma_dict": {"fullname": "pyerrors.obs.Obs.N_sigma_dict", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_dict", "kind": "variable", "doc": "
    \n", "default_value": "{}"}, "pyerrors.obs.Obs.names": {"fullname": "pyerrors.obs.Obs.names", "modulename": "pyerrors.obs", "qualname": "Obs.names", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.shape": {"fullname": "pyerrors.obs.Obs.shape", "modulename": "pyerrors.obs", "qualname": "Obs.shape", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.r_values": {"fullname": "pyerrors.obs.Obs.r_values", "modulename": "pyerrors.obs", "qualname": "Obs.r_values", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.deltas": {"fullname": "pyerrors.obs.Obs.deltas", "modulename": "pyerrors.obs", "qualname": "Obs.deltas", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.N": {"fullname": "pyerrors.obs.Obs.N", "modulename": "pyerrors.obs", "qualname": "Obs.N", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.idl": {"fullname": "pyerrors.obs.Obs.idl", "modulename": "pyerrors.obs", "qualname": "Obs.idl", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.ddvalue": {"fullname": "pyerrors.obs.Obs.ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.ddvalue", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.reweighted": {"fullname": "pyerrors.obs.Obs.reweighted", "modulename": "pyerrors.obs", "qualname": "Obs.reweighted", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.tag": {"fullname": "pyerrors.obs.Obs.tag", "modulename": "pyerrors.obs", "qualname": "Obs.tag", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.value": {"fullname": "pyerrors.obs.Obs.value", "modulename": "pyerrors.obs", "qualname": "Obs.value", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.dvalue": {"fullname": "pyerrors.obs.Obs.dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.dvalue", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_names": {"fullname": "pyerrors.obs.Obs.e_names", "modulename": "pyerrors.obs", "qualname": "Obs.e_names", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.cov_names": {"fullname": "pyerrors.obs.Obs.cov_names", "modulename": "pyerrors.obs", "qualname": "Obs.cov_names", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.mc_names": {"fullname": "pyerrors.obs.Obs.mc_names", "modulename": "pyerrors.obs", "qualname": "Obs.mc_names", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_content": {"fullname": "pyerrors.obs.Obs.e_content", "modulename": "pyerrors.obs", "qualname": "Obs.e_content", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.covobs": {"fullname": "pyerrors.obs.Obs.covobs", "modulename": "pyerrors.obs", "qualname": "Obs.covobs", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.gamma_method": {"fullname": "pyerrors.obs.Obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "Obs.gamma_method", "kind": "function", "doc": "
    Estimate the error and related properties of the Obs.
    \n\n
    Parameters
    \n\n
      \n
    • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
      • \n
    • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
      • \n
    • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
      • \n
    • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
      • \n
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.gm": {"fullname": "pyerrors.obs.Obs.gm", "modulename": "pyerrors.obs", "qualname": "Obs.gm", "kind": "function", "doc": "
    Estimate the error and related properties of the Obs.
    \n\n
    Parameters
    \n\n
      \n
    • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
      • \n
    • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
      • \n
    • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
      • \n
    • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
      • \n
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.details": {"fullname": "pyerrors.obs.Obs.details", "modulename": "pyerrors.obs", "qualname": "Obs.details", "kind": "function", "doc": "
    Output detailed properties of the Obs.
    \n\n
    Parameters
    \n\n
      \n
    • ens_content (bool):\nprint details about the ensembles and replica if true.
      • \n
    \n", "signature": "(self, ens_content=True):", "funcdef": "def"}, "pyerrors.obs.Obs.reweight": {"fullname": "pyerrors.obs.Obs.reweight", "modulename": "pyerrors.obs", "qualname": "Obs.reweight", "kind": "function", "doc": "
    Reweight the obs with given rewighting factors.
    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
      • \n
    \n", "signature": "(self, weight):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero_within_error": {"fullname": "pyerrors.obs.Obs.is_zero_within_error", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero_within_error", "kind": "function", "doc": "
    Checks whether the observable is zero within 'sigma' standard errors.
    \n\n
    Parameters
    \n\n
      \n
    • sigma (int):\nNumber of standard errors used for the check.
      • \n
    • Works only properly when the gamma method was run.
      • \n
    \n", "signature": "(self, sigma=1):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero": {"fullname": "pyerrors.obs.Obs.is_zero", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero", "kind": "function", "doc": "
    Checks whether the observable is zero within a given tolerance.
    \n\n
    Parameters
    \n\n
      \n
    • atol (float):\nAbsolute tolerance (for details see numpy documentation).
      • \n
    \n", "signature": "(self, atol=1e-10):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_tauint": {"fullname": "pyerrors.obs.Obs.plot_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.plot_tauint", "kind": "function", "doc": "
    Plot integrated autocorrelation time for each ensemble.
    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rho": {"fullname": "pyerrors.obs.Obs.plot_rho", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rho", "kind": "function", "doc": "
    Plot normalized autocorrelation function time for each ensemble.
    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rep_dist": {"fullname": "pyerrors.obs.Obs.plot_rep_dist", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rep_dist", "kind": "function", "doc": "
    Plot replica distribution for each ensemble with more than one replicum.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_history": {"fullname": "pyerrors.obs.Obs.plot_history", "modulename": "pyerrors.obs", "qualname": "Obs.plot_history", "kind": "function", "doc": "
    Plot derived Monte Carlo history for each ensemble
    \n\n
    Parameters
    \n\n
      \n
    • expand (bool):\nshow expanded history for irregular Monte Carlo chains (default: True).
      • \n
    \n", "signature": "(self, expand=True):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_piechart": {"fullname": "pyerrors.obs.Obs.plot_piechart", "modulename": "pyerrors.obs", "qualname": "Obs.plot_piechart", "kind": "function", "doc": "
    Plot piechart which shows the fractional contribution of each\nensemble to the error and returns a dictionary containing the fractions.
    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.dump": {"fullname": "pyerrors.obs.Obs.dump", "modulename": "pyerrors.obs", "qualname": "Obs.dump", "kind": "function", "doc": "
    Dump the Obs to a file 'name' of chosen format.
    \n\n
    Parameters
    \n\n
      \n
    • filename (str):\nname of the file to be saved.
      • \n
    • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
      • \n
    • description (str):\nDescription for output file, only relevant for json.gz format.
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n", "signature": "(self, filename, datatype='json.gz', description='', **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.export_jackknife": {"fullname": "pyerrors.obs.Obs.export_jackknife", "modulename": "pyerrors.obs", "qualname": "Obs.export_jackknife", "kind": "function", "doc": "
    Export jackknife samples from the Obs
    \n\n
    Returns
    \n\n
      \n
    • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N jackknife samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived jackknife samples\nshould agree with samples from a full jackknife analysis up to O(1/N).
      • \n
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.export_bootstrap": {"fullname": "pyerrors.obs.Obs.export_bootstrap", "modulename": "pyerrors.obs", "qualname": "Obs.export_bootstrap", "kind": "function", "doc": "
    Export bootstrap samples from the Obs
    \n\n
    Parameters
    \n\n
      \n
    • samples (int):\nNumber of bootstrap samples to generate.
      • \n
    • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples.\nIf not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
      • \n
    • save_rng (str):\nSave the random numbers to a file if a path is specified.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N import_bootstrap samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived bootstrap samples\nshould agree with samples from a full bootstrap analysis up to O(1/N).
      • \n
    \n", "signature": "(self, samples=500, random_numbers=None, save_rng=None):", "funcdef": "def"}, "pyerrors.obs.Obs.sqrt": {"fullname": "pyerrors.obs.Obs.sqrt", "modulename": "pyerrors.obs", "qualname": "Obs.sqrt", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.log": {"fullname": "pyerrors.obs.Obs.log", "modulename": "pyerrors.obs", "qualname": "Obs.log", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.exp": {"fullname": "pyerrors.obs.Obs.exp", "modulename": "pyerrors.obs", "qualname": "Obs.exp", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sin": {"fullname": "pyerrors.obs.Obs.sin", "modulename": "pyerrors.obs", "qualname": "Obs.sin", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cos": {"fullname": "pyerrors.obs.Obs.cos", "modulename": "pyerrors.obs", "qualname": "Obs.cos", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tan": {"fullname": "pyerrors.obs.Obs.tan", "modulename": "pyerrors.obs", "qualname": "Obs.tan", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsin": {"fullname": "pyerrors.obs.Obs.arcsin", "modulename": "pyerrors.obs", "qualname": "Obs.arcsin", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccos": {"fullname": "pyerrors.obs.Obs.arccos", "modulename": "pyerrors.obs", "qualname": "Obs.arccos", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctan": {"fullname": "pyerrors.obs.Obs.arctan", "modulename": "pyerrors.obs", "qualname": "Obs.arctan", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sinh": {"fullname": "pyerrors.obs.Obs.sinh", "modulename": "pyerrors.obs", "qualname": "Obs.sinh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cosh": {"fullname": "pyerrors.obs.Obs.cosh", "modulename": "pyerrors.obs", "qualname": "Obs.cosh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tanh": {"fullname": "pyerrors.obs.Obs.tanh", "modulename": "pyerrors.obs", "qualname": "Obs.tanh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsinh": {"fullname": "pyerrors.obs.Obs.arcsinh", "modulename": "pyerrors.obs", "qualname": "Obs.arcsinh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccosh": {"fullname": "pyerrors.obs.Obs.arccosh", "modulename": "pyerrors.obs", "qualname": "Obs.arccosh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctanh": {"fullname": "pyerrors.obs.Obs.arctanh", "modulename": "pyerrors.obs", "qualname": "Obs.arctanh", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.N_sigma": {"fullname": "pyerrors.obs.Obs.N_sigma", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.S": {"fullname": "pyerrors.obs.Obs.S", "modulename": "pyerrors.obs", "qualname": "Obs.S", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_ddvalue": {"fullname": "pyerrors.obs.Obs.e_ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_ddvalue", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_drho": {"fullname": "pyerrors.obs.Obs.e_drho", "modulename": "pyerrors.obs", "qualname": "Obs.e_drho", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_dtauint": {"fullname": "pyerrors.obs.Obs.e_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_dtauint", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_dvalue": {"fullname": "pyerrors.obs.Obs.e_dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_dvalue", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_n_dtauint": {"fullname": "pyerrors.obs.Obs.e_n_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_dtauint", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_n_tauint": {"fullname": "pyerrors.obs.Obs.e_n_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_tauint", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_rho": {"fullname": "pyerrors.obs.Obs.e_rho", "modulename": "pyerrors.obs", "qualname": "Obs.e_rho", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_tauint": {"fullname": "pyerrors.obs.Obs.e_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_tauint", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.e_windowsize": {"fullname": "pyerrors.obs.Obs.e_windowsize", "modulename": "pyerrors.obs", "qualname": "Obs.e_windowsize", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.Obs.tau_exp": {"fullname": "pyerrors.obs.Obs.tau_exp", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.CObs": {"fullname": "pyerrors.obs.CObs", "modulename": "pyerrors.obs", "qualname": "CObs", "kind": "class", "doc": "
    Class for a complex valued observable.
    \n"}, "pyerrors.obs.CObs.__init__": {"fullname": "pyerrors.obs.CObs.__init__", "modulename": "pyerrors.obs", "qualname": "CObs.__init__", "kind": "function", "doc": "
    \n", "signature": "(real, imag=0.0)"}, "pyerrors.obs.CObs.tag": {"fullname": "pyerrors.obs.CObs.tag", "modulename": "pyerrors.obs", "qualname": "CObs.tag", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.CObs.real": {"fullname": "pyerrors.obs.CObs.real", "modulename": "pyerrors.obs", "qualname": "CObs.real", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.CObs.imag": {"fullname": "pyerrors.obs.CObs.imag", "modulename": "pyerrors.obs", "qualname": "CObs.imag", "kind": "variable", "doc": "
    \n"}, "pyerrors.obs.CObs.gamma_method": {"fullname": "pyerrors.obs.CObs.gamma_method", "modulename": "pyerrors.obs", "qualname": "CObs.gamma_method", "kind": "function", "doc": "
    Executes the gamma_method for the real and the imaginary part.
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.CObs.is_zero": {"fullname": "pyerrors.obs.CObs.is_zero", "modulename": "pyerrors.obs", "qualname": "CObs.is_zero", "kind": "function", "doc": "
    Checks whether both real and imaginary part are zero within machine precision.
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs.conjugate": {"fullname": "pyerrors.obs.CObs.conjugate", "modulename": "pyerrors.obs", "qualname": "CObs.conjugate", "kind": "function", "doc": "
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.gamma_method": {"fullname": "pyerrors.obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "gamma_method", "kind": "function", "doc": "
    Vectorized version of the gamma_method applicable to lists or arrays of Obs.
    \n\n
    See docstring of pe.Obs.gamma_method for details.
    \n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.gm": {"fullname": "pyerrors.obs.gm", "modulename": "pyerrors.obs", "qualname": "gm", "kind": "function", "doc": "
    Vectorized version of the gamma_method applicable to lists or arrays of Obs.
    \n\n
    See docstring of pe.Obs.gamma_method for details.
    \n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.derived_observable": {"fullname": "pyerrors.obs.derived_observable", "modulename": "pyerrors.obs", "qualname": "derived_observable", "kind": "function", "doc": "
    Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.
    \n\n
    Parameters
    \n\n
      \n
    • func (object):\narbitrary function of the form func(data, **kwargs). For the\nautomatic differentiation to work, all numpy functions have to have\nthe autograd wrapper (use 'import autograd.numpy as anp').
      • \n
    • data (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
      • \n
    • num_grad (bool):\nif True, numerical derivatives are used instead of autograd\n(default False). To control the numerical differentiation the\nkwargs of numdifftools.step_generators.MaxStepGenerator\ncan be used.
      • \n
    • man_grad (list):\nmanually supply a list or an array which contains the jacobian\nof func. Use cautiously, supplying the wrong derivative will\nnot be intercepted.
      • \n
    \n\n
    Notes
    \n\n
    For simple mathematical operations it can be practical to use anonymous\nfunctions. For the ratio of two observables one can e.g. use
    \n\n
    new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])
    \n", "signature": "(func, data, array_mode=False, **kwargs):", "funcdef": "def"}, "pyerrors.obs.reweight": {"fullname": "pyerrors.obs.reweight", "modulename": "pyerrors.obs", "qualname": "reweight", "kind": "function", "doc": "
    Reweight a list of observables.
    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • obs (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
      • \n
    \n", "signature": "(weight, obs, **kwargs):", "funcdef": "def"}, "pyerrors.obs.correlate": {"fullname": "pyerrors.obs.correlate", "modulename": "pyerrors.obs", "qualname": "correlate", "kind": "function", "doc": "
    Correlate two observables.
    \n\n
    Parameters
    \n\n
      \n
    • obs_a (Obs):\nFirst observable
      • \n
    • obs_b (Obs):\nSecond observable
      • \n
    \n\n
    Notes
    \n\n
    Keep in mind to only correlate primary observables which have not been reweighted\nyet. The reweighting has to be applied after correlating the observables.\nCurrently only works if ensembles are identical (this is not strictly necessary).
    \n", "signature": "(obs_a, obs_b):", "funcdef": "def"}, "pyerrors.obs.covariance": {"fullname": "pyerrors.obs.covariance", "modulename": "pyerrors.obs", "qualname": "covariance", "kind": "function", "doc": "
    Calculates the error covariance matrix of a set of observables.
    \n\n
    WARNING: This function should be used with care, especially for observables with support on multiple\n         ensembles with differing autocorrelations. See the notes below for details.
    \n\n
    The gamma method has to be applied first to all observables.
    \n\n
    Parameters
    \n\n
      \n
    • obs (list or numpy.ndarray):\nList or one dimensional array of Obs
      • \n
    • visualize (bool):\nIf True plots the corresponding normalized correlation matrix (default False).
      • \n
    • correlation (bool):\nIf True the correlation matrix instead of the error covariance matrix is returned (default False).
      • \n
    • smooth (None or int):\nIf smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue\nsmoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the\nlargest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely\nsmall ones.
      • \n
    \n\n
    Notes
    \n\n
    The error covariance is defined such that it agrees with the squared standard error for two identical observables\n$$\\operatorname{cov}(a,a)=\\sum_{s=1}^N\\delta_a^s\\delta_a^s/N^2=\\Gamma_{aa}(0)/N=\\operatorname{var}(a)/N=\\sigma_a^2$$\nin the absence of autocorrelation.\nThe error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite\n$$\\sum_{i,j}v_i\\Gamma_{ij}(0)v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i,j}v_i\\delta_i^s\\delta_j^s v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i}|v_i\\delta_i^s|^2\\geq 0\\,,$$ for every $v\\in\\mathbb{R}^M$, while such an identity does not hold for larger windows/lags.\nFor observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.\n$$\\tau_{\\mathrm{int}, ij}=\\sqrt{\\tau_{\\mathrm{int}, i}\\times \\tau_{\\mathrm{int}, j}}$$\nThis construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).
    \n", "signature": "(obs, visualize=False, correlation=False, smooth=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs.invert_corr_cov_cholesky": {"fullname": "pyerrors.obs.invert_corr_cov_cholesky", "modulename": "pyerrors.obs", "qualname": "invert_corr_cov_cholesky", "kind": "function", "doc": "
    Constructs a lower triangular matrix chol via the Cholesky decomposition of the correlation matrix corr\n   and then returns the inverse covariance matrix chol_inv as a lower triangular matrix by solving chol * x = inverrdiag.
    \n\n
    Parameters
    \n\n
      \n
    • corr (np.ndarray):\ncorrelation matrix
      • \n
    • inverrdiag (np.ndarray):\ndiagonal matrix, the entries are the inverse errors of the data points considered
      • \n
    \n", "signature": "(corr, inverrdiag):", "funcdef": "def"}, "pyerrors.obs.sort_corr": {"fullname": "pyerrors.obs.sort_corr", "modulename": "pyerrors.obs", "qualname": "sort_corr", "kind": "function", "doc": "
    Reorders a correlation matrix to match the alphabetical order of its underlying y data.
    \n\n
    The ordering of the input correlation matrix corr is given by the list of keys kl.\nThe input dictionary yd (with the same keys kl) must contain the corresponding y data\nthat the correlation matrix is based on.\nThis function sorts the list of keys kl alphabetically and sorts the matrix corr\naccording to this alphabetical order such that the sorted matrix corr_sorted corresponds\nto the y data yd when arranged in an alphabetical order by its keys.
    \n\n
    Parameters
    \n\n
      \n
    • corr (np.ndarray):\nA square correlation matrix constructed using the order of the y data specified by kl.\nThe dimensions of corr should match the total number of y data points in yd combined.
      • \n
    • kl (list of str):\nA list of keys that denotes the order in which the y data from yd was used to build the\ninput correlation matrix corr.
      • \n
    • yd (dict of list):\nA dictionary where each key corresponds to a unique identifier, and its value is a list of\ny data points. The total number of y data points across all keys must match the dimensions\nof corr. The lists in the dictionary can be lists of Obs.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • np.ndarray: A new, sorted correlation matrix that corresponds to the y data from yd when arranged alphabetically by its keys.
      • \n
    \n\n
    Example
    \n\n
    \n
    >>> import numpy as np\n>>> import pyerrors as pe\n>>> corr = np.array([[1, 0.2, 0.3], [0.2, 1, 0.4], [0.3, 0.4, 1]])\n>>> kl = ['b', 'a']\n>>> yd = {'a': [1, 2], 'b': [3]}\n>>> sorted_corr = pe.obs.sort_corr(corr, kl, yd)\n>>> print(sorted_corr)\narray([[1. , 0.3, 0.4],\n       [0.3, 1. , 0.2],\n       [0.4, 0.2, 1. ]])\n
    \n
    \n", "signature": "(corr, kl, yd):", "funcdef": "def"}, "pyerrors.obs.import_jackknife": {"fullname": "pyerrors.obs.import_jackknife", "modulename": "pyerrors.obs", "qualname": "import_jackknife", "kind": "function", "doc": "
    Imports jackknife samples and returns an Obs
    \n\n
    Parameters
    \n\n
      \n
    • jacks (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N jackknife samples as first to Nth entry.
      • \n
    • name (str):\nname of the ensemble the samples are defined on.
      • \n
    \n", "signature": "(jacks, name, idl=None):", "funcdef": "def"}, "pyerrors.obs.import_bootstrap": {"fullname": "pyerrors.obs.import_bootstrap", "modulename": "pyerrors.obs", "qualname": "import_bootstrap", "kind": "function", "doc": "
    Imports bootstrap samples and returns an Obs
    \n\n
    Parameters
    \n\n
      \n
    • boots (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N bootstrap samples as first to Nth entry.
      • \n
    • name (str):\nname of the ensemble the samples are defined on.
      • \n
    • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples,\nwhere samples is the number of bootstrap samples and length is the length of the original Monte Carlo\nchain to be reconstructed.
      • \n
    \n", "signature": "(boots, name, random_numbers):", "funcdef": "def"}, "pyerrors.obs.merge_obs": {"fullname": "pyerrors.obs.merge_obs", "modulename": "pyerrors.obs", "qualname": "merge_obs", "kind": "function", "doc": "
    Combine all observables in list_of_obs into one new observable
    \n\n
    Parameters
    \n\n
      \n
    • list_of_obs (list):\nlist of the Obs object to be combined
      • \n
    \n\n
    Notes
    \n\n
    It is not possible to combine obs which are based on the same replicum
    \n", "signature": "(list_of_obs):", "funcdef": "def"}, "pyerrors.obs.cov_Obs": {"fullname": "pyerrors.obs.cov_Obs", "modulename": "pyerrors.obs", "qualname": "cov_Obs", "kind": "function", "doc": "
    Create an Obs based on mean(s) and a covariance matrix
    \n\n
    Parameters
    \n\n
      \n
    • mean (list of floats or float):\nN mean value(s) of the new Obs
      • \n
    • cov (list or array):\n2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
      • \n
    • name (str):\nidentifier for the covariance matrix
      • \n
    • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
      • \n
    \n", "signature": "(means, cov, name, grad=None):", "funcdef": "def"}, "pyerrors.roots": {"fullname": "pyerrors.roots", "modulename": "pyerrors.roots", "kind": "module", "doc": "
    \n"}, "pyerrors.roots.find_root": {"fullname": "pyerrors.roots.find_root", "modulename": "pyerrors.roots", "qualname": "find_root", "kind": "function", "doc": "
    Finds the root of the function func(x, d) where d is an Obs.
    \n\n
    Parameters
    \n\n
      \n
    • d (Obs):\nObs passed to the function.
      • \n
    • func (object):\nFunction to be minimized. Any numpy functions have to use the autograd.numpy wrapper.\nExample:
      \n\n
      \n
      import autograd.numpy as anp\ndef root_func(x, d):\n    return anp.exp(-x ** 2) - d\n
      \n
      • \n
    • guess (float):\nInitial guess for the minimization.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (Obs):\nObs valued root of the function.
      • \n
    \n", "signature": "(d, func, guess=1.0, **kwargs):", "funcdef": "def"}, "pyerrors.special": {"fullname": "pyerrors.special", "modulename": "pyerrors.special", "kind": "module", "doc": "
    \n"}, "pyerrors.special.beta": {"fullname": "pyerrors.special.beta", "modulename": "pyerrors.special", "qualname": "beta", "kind": "function", "doc": "
    beta(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    beta(a, b, out=None)
    \n\n
    Beta function.
    \n\n
    This function is defined in 1 as
    \n\n
    $$B(a, b) = \\int_0^1 t^{a-1}(1-t)^{b-1}dt\n        = \\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)},$$
    \n\n
    where \\( \\Gamma \\) is the gamma function.
    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nReal-valued arguments
      • \n
    • out (ndarray, optional):\nOptional output array for the function result
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the beta function
      • \n
    \n\n
    See Also
    \n\n
    gamma:  the gamma function
    \nbetainc:   the regularized incomplete beta function
    \nbetaln:  the natural logarithm of the absolute\nvalue of the beta function  
    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n
    The beta function relates to the gamma function by the\ndefinition given above:
    \n\n
    \n
    >>> sc.beta(2, 3)\n0.08333333333333333\n>>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3)\n0.08333333333333333\n
    \n
    \n\n
    As this relationship demonstrates, the beta function\nis symmetric:
    \n\n
    \n
    >>> sc.beta(1.7, 2.4)\n0.16567527689031739\n>>> sc.beta(2.4, 1.7)\n0.16567527689031739\n
    \n
    \n\n
    This function satisfies \\( B(1, b) = 1/b \\):
    \n\n
    \n
    >>> sc.beta(1, 4)\n0.25\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      NIST Digital Library of Mathematical Functions,\nEq. 5.12.1. https://dlmf.nist.gov/5.12 ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betainc": {"fullname": "pyerrors.special.betainc", "modulename": "pyerrors.special", "qualname": "betainc", "kind": "function", "doc": "
    betainc(x1, x2, x3, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    betainc(a, b, x, out=None)
    \n\n
    Regularized incomplete beta function.
    \n\n
    Computes the regularized incomplete beta function, defined as 1:
    \n\n
    $$I_x(a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^x\nt^{a-1}(1-t)^{b-1}dt,$$
    \n\n
    for \\( 0 \\leq x \\leq 1 \\).
    \n\n
    This function is the cumulative distribution function for the beta\ndistribution; its range is [0, 1].
    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nPositive, real-valued parameters
      • \n
    • x (array_like):\nReal-valued such that \\( 0 \\leq x \\leq 1 \\),\nthe upper limit of integration
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the regularized incomplete beta function
      • \n
    \n\n
    See Also
    \n\n
    beta:  beta function
    \nbetaincinv:  inverse of the regularized incomplete beta function
    \nbetaincc:  complement of the regularized incomplete beta function
    \nscipy.stats.beta:  beta distribution  
    \n\n
    Notes
    \n\n
    The term regularized in the name of this function refers to the\nscaling of the function by the gamma function terms shown in the\nformula.  When not qualified as regularized, the name incomplete\nbeta function often refers to just the integral expression,\nwithout the gamma terms.  One can use the function beta from\nscipy.special to get this \"nonregularized\" incomplete beta\nfunction by multiplying the result of betainc(a, b, x) by\nbeta(a, b).
    \n\n
    References
    \n\n
    Examples
    \n\n
    Let \\( B(a, b) \\) be the beta function.
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n
    The coefficient in terms of gamma is equal to\n\\( 1/B(a, b) \\). Also, when \\( x=1 \\)\nthe integral is equal to \\( B(a, b) \\).\nTherefore, \\( I_{x=1}(a, b) = 1 \\) for any \\( a, b \\).
    \n\n
    \n
    >>> sc.betainc(0.2, 3.5, 1.0)\n1.0\n
    \n
    \n\n
    It satisfies\n\\( I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b)) \\),\nwhere \\( F \\) is the hypergeometric function hyp2f1:
    \n\n
    \n
    >>> a, b, x = 1.4, 3.1, 0.5\n>>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))\n0.8148904036225295\n>>> sc.betainc(a, b, x)\n0.8148904036225296\n
    \n
    \n\n
    This functions satisfies the relationship\n\\( I_x(a, b) = 1 - I_{1-x}(b, a) \\):
    \n\n
    \n
    >>> sc.betainc(2.2, 3.1, 0.4)\n0.49339638807619446\n>>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)\n0.49339638807619446\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/8.17 ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betaln": {"fullname": "pyerrors.special.betaln", "modulename": "pyerrors.special", "qualname": "betaln", "kind": "function", "doc": "
    betaln(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    betaln(a, b, out=None)
    \n\n
    Natural logarithm of absolute value of beta function.
    \n\n
    Computes ln(abs(beta(a, b))).
    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nPositive, real-valued parameters
      • \n
    • out (ndarray, optional):\nOptional output array for function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the betaln function
      • \n
    \n\n
    See Also
    \n\n
    gamma:  the gamma function
    \nbetainc:   the regularized incomplete beta function
    \nbeta:  the beta function  
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import betaln, beta\n
    \n
    \n\n
    Verify that, for moderate values of a and b, betaln(a, b)\nis the same as log(beta(a, b)):
    \n\n
    \n
    >>> betaln(3, 4)\n-4.0943445622221\n
    \n
    \n\n
    \n
    >>> np.log(beta(3, 4))\n-4.0943445622221\n
    \n
    \n\n
    In the following beta(a, b) underflows to 0, so we can't compute\nthe logarithm of the actual value.
    \n\n
    \n
    >>> a = 400\n>>> b = 900\n>>> beta(a, b)\n0.0\n
    \n
    \n\n
    We can compute the logarithm of beta(a, b) by using betaln:
    \n\n
    \n
    >>> betaln(a, b)\n-804.3069951764146\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.polygamma": {"fullname": "pyerrors.special.polygamma", "modulename": "pyerrors.special", "qualname": "polygamma", "kind": "function", "doc": "
    Polygamma functions.
    \n\n
    Defined as \\( \\psi^{(n)}(x) \\) where \\( \\psi \\) is the\ndigamma function. See [dlmf]_ for details.
    \n\n
    Parameters
    \n\n
      \n
    • n (array_like):\nThe order of the derivative of the digamma function; must be\nintegral
      • \n
    • x (array_like):\nReal valued input
      • \n
    \n\n
    Returns
    \n\n
      \n
    • ndarray: Function results
      • \n
    \n\n
    See Also
    \n\n
    digamma  
    \n\n
    References
    \n\n
    .. [dlmf] NIST, Digital Library of Mathematical Functions,\n    https://dlmf.nist.gov/5.15
    \n\n
    Examples
    \n\n
    \n
    >>> from scipy import special\n>>> x = [2, 3, 25.5]\n>>> special.polygamma(1, x)\narray([ 0.64493407,  0.39493407,  0.03999467])\n>>> special.polygamma(0, x) == special.psi(x)\narray([ True,  True,  True], dtype=bool)\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.psi": {"fullname": "pyerrors.special.psi", "modulename": "pyerrors.special", "qualname": "psi", "kind": "function", "doc": "
    psi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    psi(z, out=None)
    \n\n
    The digamma function.
    \n\n
    The logarithmic derivative of the gamma function evaluated at z.
    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex argument.
      • \n
    • out (ndarray, optional):\nArray for the computed values of psi.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • digamma (scalar or ndarray):\nComputed values of psi.
      • \n
    \n\n
    Notes
    \n\n
    For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.
    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    Verify psi(z) = psi(z + 1) - 1/z:
    \n\n
    \n
    >>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      2. \n\n
    2. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      3. \n\n
    3. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      4. \n\n
    4. \n
      Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
      \n
      5. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.digamma": {"fullname": "pyerrors.special.digamma", "modulename": "pyerrors.special", "qualname": "digamma", "kind": "function", "doc": "
    psi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    psi(z, out=None)
    \n\n
    The digamma function.
    \n\n
    The logarithmic derivative of the gamma function evaluated at z.
    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex argument.
      • \n
    • out (ndarray, optional):\nArray for the computed values of psi.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • digamma (scalar or ndarray):\nComputed values of psi.
      • \n
    \n\n
    Notes
    \n\n
    For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.
    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    Verify psi(z) = psi(z + 1) - 1/z:
    \n\n
    \n
    >>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      2. \n\n
    2. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      3. \n\n
    3. \n
      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
      \n
      4. \n\n
    4. \n
      Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
      \n
      5. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gamma": {"fullname": "pyerrors.special.gamma", "modulename": "pyerrors.special", "qualname": "gamma", "kind": "function", "doc": "
    gamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    gamma(z, out=None)
    \n\n
    gamma function.
    \n\n
    The gamma function is defined as
    \n\n
    $$\\Gamma(z) = \\int_0^\\infty t^{z-1} e^{-t} dt$$
    \n\n
    for \\( \\Re(z) > 0 \\) and is extended to the rest of the complex\nplane by analytic continuation. See [dlmf]_ for more details.
    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex valued argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the gamma function
      • \n
    \n\n
    Notes
    \n\n
    The gamma function is often referred to as the generalized\nfactorial since \\( \\Gamma(n + 1) = n! \\) for natural numbers\n\\( n \\). More generally it satisfies the recurrence relation\n\\( \\Gamma(z + 1) = z \\cdot \\Gamma(z) \\) for complex \\( z \\),\nwhich, combined with the fact that \\( \\Gamma(1) = 1 \\), implies\nthe above identity for \\( z = n \\).
    \n\n
    References
    \n\n
    .. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5.2#E1
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import gamma, factorial\n
    \n
    \n\n
    \n
    >>> gamma([0, 0.5, 1, 5])\narray([         inf,   1.77245385,   1.        ,  24.        ])\n
    \n
    \n\n
    \n
    >>> z = 2.5 + 1j\n>>> gamma(z)\n(0.77476210455108352+0.70763120437959293j)\n>>> gamma(z+1), z*gamma(z)  # Recurrence property\n((1.2292740569981171+2.5438401155000685j),\n (1.2292740569981158+2.5438401155000658j))\n
    \n
    \n\n
    \n
    >>> gamma(0.5)**2  # gamma(0.5) = sqrt(pi)\n3.1415926535897927\n
    \n
    \n\n
    Plot gamma(x) for real x
    \n\n
    \n
    >>> x = np.linspace(-3.5, 5.5, 2251)\n>>> y = gamma(x)\n
    \n
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')\n>>> k = np.arange(1, 7)\n>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,\n...          label='(x-1)!, x = 1, 2, ...')\n>>> plt.xlim(-3.5, 5.5)\n>>> plt.ylim(-10, 25)\n>>> plt.grid()\n>>> plt.xlabel('x')\n>>> plt.legend(loc='lower right')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaln": {"fullname": "pyerrors.special.gammaln", "modulename": "pyerrors.special", "qualname": "gammaln", "kind": "function", "doc": "
    gammaln(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    gammaln(x, out=None)
    \n\n
    Logarithm of the absolute value of the gamma function.
    \n\n
    Defined as
    \n\n
    $$\\ln(\\lvert\\Gamma(x)\\rvert)$$
    \n\n
    where \\( \\Gamma \\) is the gamma function. For more details on\nthe gamma function, see [dlmf]_.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the log of the absolute value of gamma
      • \n
    \n\n
    See Also
    \n\n
    gammasgn:  sign of the gamma function
    \nloggamma:  principal branch of the logarithm of the gamma function  
    \n\n
    Notes
    \n\n
    It is the same function as the Python standard library function\nmath.lgamma().
    \n\n
    When used in conjunction with gammasgn, this function is useful\nfor working in logspace on the real axis without having to deal\nwith complex numbers via the relation exp(gammaln(x)) =\ngammasgn(x) * gamma(x).
    \n\n
    For complex-valued log-gamma, use loggamma instead of gammaln.
    \n\n
    References
    \n\n
    .. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import scipy.special as sc\n
    \n
    \n\n
    It has two positive zeros.
    \n\n
    \n
    >>> sc.gammaln([1, 2])\narray([0., 0.])\n
    \n
    \n\n
    It has poles at nonpositive integers.
    \n\n
    \n
    >>> sc.gammaln([0, -1, -2, -3, -4])\narray([inf, inf, inf, inf, inf])\n
    \n
    \n\n
    It asymptotically approaches x * log(x) (Stirling's formula).
    \n\n
    \n
    >>> x = np.array([1e10, 1e20, 1e40, 1e80])\n>>> sc.gammaln(x)\narray([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82])\n>>> x * np.log(x)\narray([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammainc": {"fullname": "pyerrors.special.gammainc", "modulename": "pyerrors.special", "qualname": "gammainc", "kind": "function", "doc": "
    gammainc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    gammainc(a, x, out=None)
    \n\n
    Regularized lower incomplete gamma function.
    \n\n
    It is defined as
    \n\n
    $$P(a, x) = \\frac{1}{\\Gamma(a)} \\int_0^x t^{a - 1}e^{-t} dt$$
    \n\n
    for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nPositive parameter
      • \n
    • x (array_like):\nNonnegative argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the lower incomplete gamma function
      • \n
    \n\n
    See Also
    \n\n
    gammaincc:  regularized upper incomplete gamma function
    \ngammaincinv:  inverse of the regularized lower incomplete gamma function
    \ngammainccinv:  inverse of the regularized upper incomplete gamma function  
    \n\n
    Notes
    \n\n
    The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammaincc is the regularized upper\nincomplete gamma function.
    \n\n
    The implementation largely follows that of [boost]_.
    \n\n
    References
    \n\n
    .. [dlmf] NIST Digital Library of Mathematical functions\n          https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n   https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n
    It is the CDF of the gamma distribution, so it starts at 0 and\nmonotonically increases to 1.
    \n\n
    \n
    >>> sc.gammainc(0.5, [0, 1, 10, 100])\narray([0.        , 0.84270079, 0.99999226, 1.        ])\n
    \n
    \n\n
    It is equal to one minus the upper incomplete gamma function.
    \n\n
    \n
    >>> a, x = 0.5, 0.4\n>>> sc.gammainc(a, x)\n0.6289066304773024\n>>> 1 - sc.gammaincc(a, x)\n0.6289066304773024\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaincc": {"fullname": "pyerrors.special.gammaincc", "modulename": "pyerrors.special", "qualname": "gammaincc", "kind": "function", "doc": "
    gammaincc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    gammaincc(a, x, out=None)
    \n\n
    Regularized upper incomplete gamma function.
    \n\n
    It is defined as
    \n\n
    $$Q(a, x) = \\frac{1}{\\Gamma(a)} \\int_x^\\infty t^{a - 1}e^{-t} dt$$
    \n\n
    for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nPositive parameter
      • \n
    • x (array_like):\nNonnegative argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the upper incomplete gamma function
      • \n
    \n\n
    See Also
    \n\n
    gammainc:  regularized lower incomplete gamma function
    \ngammaincinv:  inverse of the regularized lower incomplete gamma function
    \ngammainccinv:  inverse of the regularized upper incomplete gamma function  
    \n\n
    Notes
    \n\n
    The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammainc is the regularized lower\nincomplete gamma function.
    \n\n
    The implementation largely follows that of [boost]_.
    \n\n
    References
    \n\n
    .. [dlmf] NIST Digital Library of Mathematical functions\n          https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n   https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n
    It is the survival function of the gamma distribution, so it\nstarts at 1 and monotonically decreases to 0.
    \n\n
    \n
    >>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000])\narray([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45,\n       0.00000000e+00])\n
    \n
    \n\n
    It is equal to one minus the lower incomplete gamma function.
    \n\n
    \n
    >>> a, x = 0.5, 0.4\n>>> sc.gammaincc(a, x)\n0.37109336952269756\n>>> 1 - sc.gammainc(a, x)\n0.37109336952269756\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammasgn": {"fullname": "pyerrors.special.gammasgn", "modulename": "pyerrors.special", "qualname": "gammasgn", "kind": "function", "doc": "
    gammasgn(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    gammasgn(x, out=None)
    \n\n
    Sign of the gamma function.
    \n\n
    It is defined as
    \n\n
    $$\\text{gammasgn}(x) =\n\\begin{cases}\n  +1 & \\Gamma(x) > 0 \\\n  -1 & \\Gamma(x) < 0\n\\end{cases}$$
    \n\n
    where \\( \\Gamma \\) is the gamma function; see gamma. This\ndefinition is complete since the gamma function is never zero;\nsee the discussion after [dlmf]_.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Sign of the gamma function
      • \n
    \n\n
    See Also
    \n\n
    gamma:  the gamma function
    \ngammaln:  log of the absolute value of the gamma function
    \nloggamma:  analytic continuation of the log of the gamma function  
    \n\n
    Notes
    \n\n
    The gamma function can be computed as gammasgn(x) *\nnp.exp(gammaln(x)).
    \n\n
    References
    \n\n
    .. [dlmf] NIST Digital Library of Mathematical Functions\n          https://dlmf.nist.gov/5.2#E1
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import scipy.special as sc\n
    \n
    \n\n
    It is 1 for x > 0.
    \n\n
    \n
    >>> sc.gammasgn([1, 2, 3, 4])\narray([1., 1., 1., 1.])\n
    \n
    \n\n
    It alternates between -1 and 1 for negative integers.
    \n\n
    \n
    >>> sc.gammasgn([-0.5, -1.5, -2.5, -3.5])\narray([-1.,  1., -1.,  1.])\n
    \n
    \n\n
    It can be used to compute the gamma function.
    \n\n
    \n
    >>> x = [1.5, 0.5, -0.5, -1.5]\n>>> sc.gammasgn(x) * np.exp(sc.gammaln(x))\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n>>> sc.gamma(x)\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.rgamma": {"fullname": "pyerrors.special.rgamma", "modulename": "pyerrors.special", "qualname": "rgamma", "kind": "function", "doc": "
    rgamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    rgamma(z, out=None)
    \n\n
    Reciprocal of the gamma function.
    \n\n
    Defined as \\( 1 / \\Gamma(z) \\), where \\( \\Gamma \\) is the\ngamma function. For more on the gamma function see gamma.
    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex valued input
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Function results
      • \n
    \n\n
    See Also
    \n\n
    gamma,, gammaln,, loggamma  
    \n\n
    Notes
    \n\n
    The gamma function has no zeros and has simple poles at\nnonpositive integers, so rgamma is an entire function with zeros\nat the nonpositive integers. See the discussion in [dlmf]_ for\nmore details.
    \n\n
    References
    \n\n
    .. [dlmf] Nist, Digital Library of Mathematical functions,\n    https://dlmf.nist.gov/5.2#i
    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n
    It is the reciprocal of the gamma function.
    \n\n
    \n
    >>> sc.rgamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n>>> 1 / sc.gamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n
    \n
    \n\n
    It is zero at nonpositive integers.
    \n\n
    \n
    >>> sc.rgamma([0, -1, -2, -3])\narray([0., 0., 0., 0.])\n
    \n
    \n\n
    It rapidly underflows to zero along the positive real axis.
    \n\n
    \n
    >>> sc.rgamma([10, 100, 179])\narray([2.75573192e-006, 1.07151029e-156, 0.00000000e+000])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.multigammaln": {"fullname": "pyerrors.special.multigammaln", "modulename": "pyerrors.special", "qualname": "multigammaln", "kind": "function", "doc": "
    Returns the log of multivariate gamma, also sometimes called the\ngeneralized gamma.
    \n\n
    Parameters
    \n\n
      \n
    • a (ndarray):\nThe multivariate gamma is computed for each item of a.
      • \n
    • d (int):\nThe dimension of the space of integration.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (ndarray):\nThe values of the log multivariate gamma at the given points a.
      • \n
    \n\n
    Notes
    \n\n
    The formal definition of the multivariate gamma of dimension d for a real\na is
    \n\n
    $$\\Gamma_d(a) = \\int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA$$
    \n\n
    with the condition \\( a > (d-1)/2 \\), and \\( A > 0 \\) being the set of\nall the positive definite matrices of dimension d.  Note that a is a\nscalar: the integrand only is multivariate, the argument is not (the\nfunction is defined over a subset of the real set).
    \n\n
    This can be proven to be equal to the much friendlier equation
    \n\n
    $$\\Gamma_d(a) = \\pi^{d(d-1)/4} \\prod_{i=1}^{d} \\Gamma(a - (i-1)/2).$$
    \n\n
    References
    \n\n
    R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in\nprobability and mathematical statistics).
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import multigammaln, gammaln\n>>> a = 23.5\n>>> d = 10\n>>> multigammaln(a, d)\n454.1488605074416\n
    \n
    \n\n
    Verify that the result agrees with the logarithm of the equation\nshown above:
    \n\n
    \n
    >>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()\n454.1488605074416\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.kn": {"fullname": "pyerrors.special.kn", "modulename": "pyerrors.special", "qualname": "kn", "kind": "function", "doc": "
    Modified Bessel function of the second kind of integer order n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j0": {"fullname": "pyerrors.special.j0", "modulename": "pyerrors.special", "qualname": "j0", "kind": "function", "doc": "
    j0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    j0(x, out=None)
    \n\n
    Bessel function of the first kind of order 0.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 0 at x.
      • \n
    \n\n
    See Also
    \n\n
    jv:  Bessel function of real order and complex argument.
    \nspherical_jn:  spherical Bessel functions.  
    \n\n
    Notes
    \n\n
    The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval the following rational approximation is used:
    \n\n
    $$J_0(x) \\approx (w - r_1^2)(w - r_2^2) \\frac{P_3(w)}{Q_8(w)},$$
    \n\n
    where \\( w = x^2 \\) and \\( r_1 \\), \\( r_2 \\) are the zeros of\n\\( J_0 \\), and \\( P_3 \\) and \\( Q_8 \\) are polynomials of degrees 3\nand 8, respectively.
    \n\n
    In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
    \n\n
    This function is a wrapper for the Cephes 1 routine j0.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import j0\n>>> j0(1.)\n0.7651976865579665\n
    \n
    \n\n
    Calculate the function at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> j0(np.array([-2., 0., 4.]))\narray([ 0.22389078,  1.        , -0.39714981])\n
    \n
    \n\n
    Plot the function from -20 to 20.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y0": {"fullname": "pyerrors.special.y0", "modulename": "pyerrors.special", "qualname": "y0", "kind": "function", "doc": "
    y0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    y0(x, out=None)
    \n\n
    Bessel function of the second kind of order 0.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 0 at x.
      • \n
    \n\n
    See Also
    \n\n
    j0:  Bessel function of the first kind of order 0
    \nyv:  Bessel function of the first kind  
    \n\n
    Notes
    \n\n
    The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval a rational approximation \\( R(x) \\) is employed to\ncompute,
    \n\n
    $$Y_0(x) = R(x) + \\frac{2 \\log(x) J_0(x)}{\\pi},$$
    \n\n
    where \\( J_0 \\) is the Bessel function of the first kind of order 0.
    \n\n
    In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
    \n\n
    This function is a wrapper for the Cephes 1 routine y0.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import y0\n>>> y0(1.)\n0.08825696421567697\n
    \n
    \n\n
    Calculate at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> y0(np.array([0.5, 2., 3.]))\narray([-0.44451873,  0.51037567,  0.37685001])\n
    \n
    \n\n
    Plot the function from 0 to 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j1": {"fullname": "pyerrors.special.j1", "modulename": "pyerrors.special", "qualname": "j1", "kind": "function", "doc": "
    j1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    j1(x, out=None)
    \n\n
    Bessel function of the first kind of order 1.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 1 at x.
      • \n
    \n\n
    See Also
    \n\n
    jv:  Bessel function of the first kind
    \nspherical_jn:  spherical Bessel functions.  
    \n\n
    Notes
    \n\n
    The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 24 term Chebyshev expansion is used. In the second, the\nasymptotic trigonometric representation is employed using two rational\nfunctions of degree 5/5.
    \n\n
    This function is a wrapper for the Cephes 1 routine j1.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import j1\n>>> j1(1.)\n0.44005058574493355\n
    \n
    \n\n
    Calculate the function at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> j1(np.array([-2., 0., 4.]))\narray([-0.57672481,  0.        , -0.06604333])\n
    \n
    \n\n
    Plot the function from -20 to 20.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y1": {"fullname": "pyerrors.special.y1", "modulename": "pyerrors.special", "qualname": "y1", "kind": "function", "doc": "
    y1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    y1(x, out=None)
    \n\n
    Bessel function of the second kind of order 1.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 1 at x.
      • \n
    \n\n
    See Also
    \n\n
    j1:  Bessel function of the first kind of order 1
    \nyn:  Bessel function of the second kind
    \nyv:  Bessel function of the second kind  
    \n\n
    Notes
    \n\n
    The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 25 term Chebyshev expansion is used, and computing\n\\( J_1 \\) (the Bessel function of the first kind) is required. In the\nsecond, the asymptotic trigonometric representation is employed using two\nrational functions of degree 5/5.
    \n\n
    This function is a wrapper for the Cephes 1 routine y1.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import y1\n>>> y1(1.)\n-0.7812128213002888\n
    \n
    \n\n
    Calculate at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> y1(np.array([0.5, 2., 3.]))\narray([-1.47147239, -0.10703243,  0.32467442])\n
    \n
    \n\n
    Plot the function from 0 to 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.jn": {"fullname": "pyerrors.special.jn", "modulename": "pyerrors.special", "qualname": "jn", "kind": "function", "doc": "
    jv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    jv(v, z, out=None)
    \n\n
    Bessel function of the first kind of real order and complex argument.
    \n\n
    Parameters
    \n\n
      \n
    • v (array_like):\nOrder (float).
      • \n
    • z (array_like):\nArgument (float or complex).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function, \\( J_v(z) \\).
      • \n
    \n\n
    See Also
    \n\n
    jve:  \\( J_v \\) with leading exponential behavior stripped off.
    \nspherical_jn:  spherical Bessel functions.
    \nj0:  faster version of this function for order 0.
    \nj1:  faster version of this function for order 1.  
    \n\n
    Notes
    \n\n
    For positive v values, the computation is carried out using the AMOS\n1 zbesj routine, which exploits the connection to the modified\nBessel function \\( I_v \\),
    \n\n
    $$J_v(z) = \\exp(v\\pi\\imath/2) I_v(-\\imath z)\\qquad (\\Im z > 0)
    \n\n
    J_v(z) = \\exp(-v\\pi\\imath/2) I_v(\\imath z)\\qquad (\\Im z < 0)$$
    \n\n
    For negative v values the formula,
    \n\n
    $$J_{-v}(z) = J_v(z) \\cos(\\pi v) - Y_v(z) \\sin(\\pi v)$$
    \n\n
    is used, where \\( Y_v(z) \\) is the Bessel function of the second\nkind, computed using the AMOS routine zbesy.  Note that the second\nterm is exactly zero for integer v; to improve accuracy the second\nterm is explicitly omitted for v values such that v = floor(v).
    \n\n
    Not to be confused with the spherical Bessel functions (see spherical_jn).
    \n\n
    References
    \n\n
    Examples
    \n\n
    Evaluate the function of order 0 at one point.
    \n\n
    \n
    >>> from scipy.special import jv\n>>> jv(0, 1.)\n0.7651976865579666\n
    \n
    \n\n
    Evaluate the function at one point for different orders.
    \n\n
    \n
    >>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)\n(0.7651976865579666, 0.44005058574493355, 0.24029783912342725)\n
    \n
    \n\n
    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
    \n\n
    \n
    >>> jv([0, 1, 1.5], 1.)\narray([0.76519769, 0.44005059, 0.24029784])\n
    \n
    \n\n
    Evaluate the function at several points for order 0 by providing an\narray for z.
    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> jv(0, points)\narray([ 0.22389078,  1.        , -0.26005195])\n
    \n
    \n\n
    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:
    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> jv(orders, points)\narray([[ 0.22389078,  1.        , -0.26005195],\n       [-0.57672481,  0.        ,  0.33905896]])\n
    \n
    \n\n
    Plot the functions of order 0 to 3 from -10 to 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, jv(i, x), label=f'$J_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.yn": {"fullname": "pyerrors.special.yn", "modulename": "pyerrors.special", "qualname": "yn", "kind": "function", "doc": "
    yn(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    yn(n, x, out=None)
    \n\n
    Bessel function of the second kind of integer order and real argument.
    \n\n
    Parameters
    \n\n
      \n
    • n (array_like):\nOrder (integer).
      • \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function, \\( Y_n(x) \\).
      • \n
    \n\n
    See Also
    \n\n
    yv:  For real order and real or complex argument.
    \ny0:  faster implementation of this function for order 0
    \ny1:  faster implementation of this function for order 1  
    \n\n
    Notes
    \n\n
    Wrapper for the Cephes 1 routine yn.
    \n\n
    The function is evaluated by forward recurrence on n, starting with\nvalues computed by the Cephes routines y0 and y1. If n = 0 or 1,\nthe routine for y0 or y1 is called directly.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Evaluate the function of order 0 at one point.
    \n\n
    \n
    >>> from scipy.special import yn\n>>> yn(0, 1.)\n0.08825696421567697\n
    \n
    \n\n
    Evaluate the function at one point for different orders.
    \n\n
    \n
    >>> yn(0, 1.), yn(1, 1.), yn(2, 1.)\n(0.08825696421567697, -0.7812128213002888, -1.6506826068162546)\n
    \n
    \n\n
    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
    \n\n
    \n
    >>> yn([0, 1, 2], 1.)\narray([ 0.08825696, -0.78121282, -1.65068261])\n
    \n
    \n\n
    Evaluate the function at several points for order 0 by providing an\narray for z.
    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([0.5, 3., 8.])\n>>> yn(0, points)\narray([-0.44451873,  0.37685001,  0.22352149])\n
    \n
    \n\n
    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:
    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> yn(orders, points)\narray([[-0.44451873,  0.37685001,  0.22352149],\n       [-1.47147239,  0.32467442, -0.15806046]])\n
    \n
    \n\n
    Plot the functions of order 0 to 3 from 0 to 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, yn(i, x), label=f'$Y_{i!r}$')\n>>> ax.set_ylim(-3, 1)\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i0": {"fullname": "pyerrors.special.i0", "modulename": "pyerrors.special", "qualname": "i0", "kind": "function", "doc": "
    i0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    i0(x, out=None)
    \n\n
    Modified Bessel function of order 0.
    \n\n
    Defined as,
    \n\n
    $$I_0(x) = \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{(k!)^2} = J_0(\\imath x),$$
    \n\n
    where \\( J_0 \\) is the Bessel function of the first kind of order 0.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float)
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • I (scalar or ndarray):\nValue of the modified Bessel function of order 0 at x.
      • \n
    \n\n
    See Also
    \n\n
    iv:  Modified Bessel function of any order
    \ni0e:  Exponentially scaled modified Bessel function of order 0  
    \n\n
    Notes
    \n\n
    The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
    \n\n
    This function is a wrapper for the Cephes 1 routine i0.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import i0\n>>> i0(1.)\n1.2660658777520082\n
    \n
    \n\n
    Calculate at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> i0(np.array([-2., 0., 3.5]))\narray([2.2795853 , 1.        , 7.37820343])\n
    \n
    \n\n
    Plot the function from -10 to 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i1": {"fullname": "pyerrors.special.i1", "modulename": "pyerrors.special", "qualname": "i1", "kind": "function", "doc": "
    i1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    i1(x, out=None)
    \n\n
    Modified Bessel function of order 1.
    \n\n
    Defined as,
    \n\n
    $$I_1(x) = \\frac{1}{2}x \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{k! (k + 1)!}\n       = -\\imath J_1(\\imath x),$$
    \n\n
    where \\( J_1 \\) is the Bessel function of the first kind of order 1.
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float)
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • I (scalar or ndarray):\nValue of the modified Bessel function of order 1 at x.
      • \n
    \n\n
    See Also
    \n\n
    iv:  Modified Bessel function of the first kind
    \ni1e:  Exponentially scaled modified Bessel function of order 1  
    \n\n
    Notes
    \n\n
    The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
    \n\n
    This function is a wrapper for the Cephes 1 routine i1.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Calculate the function at one point:
    \n\n
    \n
    >>> from scipy.special import i1\n>>> i1(1.)\n0.5651591039924851\n
    \n
    \n\n
    Calculate the function at several points:
    \n\n
    \n
    >>> import numpy as np\n>>> i1(np.array([-2., 0., 6.]))\narray([-1.59063685,  0.        , 61.34193678])\n
    \n
    \n\n
    Plot the function between -10 and 10.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.iv": {"fullname": "pyerrors.special.iv", "modulename": "pyerrors.special", "qualname": "iv", "kind": "function", "doc": "
    iv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    iv(v, z, out=None)
    \n\n
    Modified Bessel function of the first kind of real order.
    \n\n
    Parameters
    \n\n
      \n
    • v (array_like):\nOrder. If z is of real type and negative, v must be integer\nvalued.
      • \n
    • z (array_like of float or complex):\nArgument.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the modified Bessel function.
      • \n
    \n\n
    See Also
    \n\n
    ive:  This function with leading exponential behavior stripped off.
    \ni0:  Faster version of this function for order 0.
    \ni1:  Faster version of this function for order 1.  
    \n\n
    Notes
    \n\n
    For real z and \\( v \\in [-50, 50] \\), the evaluation is carried out\nusing Temme's method 1.  For larger orders, uniform asymptotic\nexpansions are applied.
    \n\n
    For complex z and positive v, the AMOS 2 zbesi routine is\ncalled. It uses a power series for small z, the asymptotic expansion\nfor large abs(z), the Miller algorithm normalized by the Wronskian\nand a Neumann series for intermediate magnitudes, and the uniform\nasymptotic expansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large\norders. Backward recurrence is used to generate sequences or reduce\norders when necessary.
    \n\n
    The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
    \n\n
    $$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
    \n\n
    (valid when the real part of z is positive).  For negative v, the\nformula
    \n\n
    $$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
    \n\n
    is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.
    \n\n
    References
    \n\n
    Examples
    \n\n
    Evaluate the function of order 0 at one point.
    \n\n
    \n
    >>> from scipy.special import iv\n>>> iv(0, 1.)\n1.2660658777520084\n
    \n
    \n\n
    Evaluate the function at one point for different orders.
    \n\n
    \n
    >>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.)\n(1.2660658777520084, 0.565159103992485, 0.2935253263474798)\n
    \n
    \n\n
    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:
    \n\n
    \n
    >>> iv([0, 1, 1.5], 1.)\narray([1.26606588, 0.5651591 , 0.29352533])\n
    \n
    \n\n
    Evaluate the function at several points for order 0 by providing an\narray for z.
    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> iv(0, points)\narray([2.2795853 , 1.        , 4.88079259])\n
    \n
    \n\n
    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:
    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> iv(orders, points)\narray([[ 2.2795853 ,  1.        ,  4.88079259],\n       [-1.59063685,  0.        ,  3.95337022]])\n
    \n
    \n\n
    Plot the functions of order 0 to 3 from -5 to 5.
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, iv(i, x), label=f'$I_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Temme, Journal of Computational Physics, vol 21, 343 (1976) ↩
      \n
      2. \n\n
    2. \n
      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
      \n
      3. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.ive": {"fullname": "pyerrors.special.ive", "modulename": "pyerrors.special", "qualname": "ive", "kind": "function", "doc": "
    ive(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    ive(v, z, out=None)
    \n\n
    Exponentially scaled modified Bessel function of the first kind.
    \n\n
    Defined as::
    \n\n
    ive(v, z) = iv(v, z) * exp(-abs(z.real))\n
    \n\n
    For imaginary numbers without a real part, returns the unscaled\nBessel function of the first kind iv.
    \n\n
    Parameters
    \n\n
      \n
    • v (array_like of float):\nOrder.
      • \n
    • z (array_like of float or complex):\nArgument.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the exponentially scaled modified Bessel function.
      • \n
    \n\n
    See Also
    \n\n
    iv:  Modified Bessel function of the first kind
    \ni0e:  Faster implementation of this function for order 0
    \ni1e:  Faster implementation of this function for order 1  
    \n\n
    Notes
    \n\n
    For positive v, the AMOS 1 zbesi routine is called. It uses a\npower series for small z, the asymptotic expansion for large\nabs(z), the Miller algorithm normalized by the Wronskian and a\nNeumann series for intermediate magnitudes, and the uniform asymptotic\nexpansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large orders.\nBackward recurrence is used to generate sequences or reduce orders when\nnecessary.
    \n\n
    The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
    \n\n
    $$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
    \n\n
    (valid when the real part of z is positive).  For negative v, the\nformula
    \n\n
    $$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
    \n\n
    is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.
    \n\n
    ive is useful for large arguments z: for these, iv easily overflows,\nwhile ive does not due to the exponential scaling.
    \n\n
    References
    \n\n
    Examples
    \n\n
    In the following example iv returns infinity whereas ive still returns\na finite number.
    \n\n
    \n
    >>> from scipy.special import iv, ive\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> iv(3, 1000.), ive(3, 1000.)\n(inf, 0.01256056218254712)\n
    \n
    \n\n
    Evaluate the function at one point for different orders by\nproviding a list or NumPy array as argument for the v parameter:
    \n\n
    \n
    >>> ive([0, 1, 1.5], 1.)\narray([0.46575961, 0.20791042, 0.10798193])\n
    \n
    \n\n
    Evaluate the function at several points for order 0 by providing an\narray for z.
    \n\n
    \n
    >>> points = np.array([-2., 0., 3.])\n>>> ive(0, points)\narray([0.30850832, 1.        , 0.24300035])\n
    \n
    \n\n
    Evaluate the function at several points for different orders by\nproviding arrays for both v for z. Both arrays have to be\nbroadcastable to the correct shape. To calculate the orders 0, 1\nand 2 for a 1D array of points:
    \n\n
    \n
    >>> ive([[0], [1], [2]], points)\narray([[ 0.30850832,  1.        ,  0.24300035],\n       [-0.21526929,  0.        ,  0.19682671],\n       [ 0.09323903,  0.        ,  0.11178255]])\n
    \n
    \n\n
    Plot the functions of order 0 to 3 from -5 to 5.
    \n\n
    \n
    >>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, ive(i, x), label=fr'$I_{i!r}(z)\\cdot e^{{-|z|}}$')\n>>> ax.legend()\n>>> ax.set_xlabel(r"$z$")\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n
      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erf": {"fullname": "pyerrors.special.erf", "modulename": "pyerrors.special", "qualname": "erf", "kind": "function", "doc": "
    erf(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    erf(z, out=None)
    \n\n
    Returns the error function of complex argument.
    \n\n
    It is defined as 2/sqrt(pi)*integral(exp(-t**2), t=0..z).
    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nInput array.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (scalar or ndarray):\nThe values of the error function at the given points x.
      • \n
    \n\n
    See Also
    \n\n
    erfc,, erfinv,, erfcinv,, wofz,, erfcx,, erfi  
    \n\n
    Notes
    \n\n
    The cumulative of the unit normal distribution is given by\nPhi(z) = 1/2[1 + erf(z/sqrt(2))].
    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erf(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erf(x)$')\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfc": {"fullname": "pyerrors.special.erfc", "modulename": "pyerrors.special", "qualname": "erfc", "kind": "function", "doc": "
    erfc(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    erfc(x, out=None)
    \n\n
    Complementary error function, 1 - erf(x).
    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal or complex valued argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the complementary error function
      • \n
    \n\n
    See Also
    \n\n
    erf,, erfi,, erfcx,, dawsn,, wofz  
    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erfc(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erfc(x)$')\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfinv": {"fullname": "pyerrors.special.erfinv", "modulename": "pyerrors.special", "qualname": "erfinv", "kind": "function", "doc": "
    erfinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    erfinv(y, out=None)
    \n\n
    Inverse of the error function.
    \n\n
    Computes the inverse of the error function.
    \n\n
    In the complex domain, there is no unique complex number w satisfying\nerf(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, -1 < x < 1, there is a unique real\nnumber satisfying erf(erfinv(x)) = x.
    \n\n
    Parameters
    \n\n
      \n
    • y (ndarray):\nArgument at which to evaluate. Domain: [-1, 1]
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • erfinv (scalar or ndarray):\nThe inverse of erf of y, element-wise
      • \n
    \n\n
    See Also
    \n\n
    erf:  Error function of a complex argument
    \nerfc:  Complementary error function, 1 - erf(x)
    \nerfcinv:  Inverse of the complementary error function  
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfinv, erf\n
    \n
    \n\n
    \n
    >>> erfinv(0.5)\n0.4769362762044699\n
    \n
    \n\n
    \n
    >>> y = np.linspace(-1.0, 1.0, num=9)\n>>> x = erfinv(y)\n>>> x\narray([       -inf, -0.81341985, -0.47693628, -0.22531206,  0.        ,\n        0.22531206,  0.47693628,  0.81341985,         inf])\n
    \n
    \n\n
    Verify that erf(erfinv(y)) is y.
    \n\n
    \n
    >>> erf(x)\narray([-1.  , -0.75, -0.5 , -0.25,  0.  ,  0.25,  0.5 ,  0.75,  1.  ])\n
    \n
    \n\n
    Plot the function:
    \n\n
    \n
    >>> y = np.linspace(-1, 1, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfinv(y)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfcinv": {"fullname": "pyerrors.special.erfcinv", "modulename": "pyerrors.special", "qualname": "erfcinv", "kind": "function", "doc": "
    erfcinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    erfcinv(y, out=None)
    \n\n
    Inverse of the complementary error function.
    \n\n
    Computes the inverse of the complementary error function.
    \n\n
    In the complex domain, there is no unique complex number w satisfying\nerfc(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, 0 < x < 2, there is a unique real\nnumber satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)).
    \n\n
    It is related to inverse of the error function by erfcinv(1-x) = erfinv(x)
    \n\n
    Parameters
    \n\n
      \n
    • y (ndarray):\nArgument at which to evaluate. Domain: [0, 2]
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • erfcinv (scalar or ndarray):\nThe inverse of erfc of y, element-wise
      • \n
    \n\n
    See Also
    \n\n
    erf:  Error function of a complex argument
    \nerfc:  Complementary error function, 1 - erf(x)
    \nerfinv:  Inverse of the error function  
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfcinv\n
    \n
    \n\n
    \n
    >>> erfcinv(0.5)\n0.4769362762044699\n
    \n
    \n\n
    \n
    >>> y = np.linspace(0.0, 2.0, num=11)\n>>> erfcinv(y)\narray([        inf,  0.9061938 ,  0.59511608,  0.37080716,  0.17914345,\n       -0.        , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,\n              -inf])\n
    \n
    \n\n
    Plot the function:
    \n\n
    \n
    >>> y = np.linspace(0, 2, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfcinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfcinv(y)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logit": {"fullname": "pyerrors.special.logit", "modulename": "pyerrors.special", "qualname": "logit", "kind": "function", "doc": "
    logit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    \"\"\"\nlogit(x, out=None)
    \n\n
    Logit ufunc for ndarrays.
    \n\n
    The logit function is defined as logit(p) = log(p/(1-p)).\nNote that logit(0) = -inf, logit(1) = inf, and logit(p)\nfor p<0 or p>1 yields nan.
    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nThe ndarray to apply logit to element-wise.
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare logit of the corresponding entry of x.
      • \n
    \n\n
    See Also
    \n\n
    expit  
    \n\n
    Notes
    \n\n
    As a ufunc logit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs 
    \n\n
    New in version 0.10.0.
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import logit, expit\n
    \n
    \n\n
    \n
    >>> logit([0, 0.25, 0.5, 0.75, 1])\narray([       -inf, -1.09861229,  0.        ,  1.09861229,         inf])\n
    \n
    \n\n
    expit is the inverse of logit:
    \n\n
    \n
    >>> expit(logit([0.1, 0.75, 0.999]))\narray([ 0.1  ,  0.75 ,  0.999])\n
    \n
    \n\n
    Plot logit(x) for x in [0, 1]:
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> x = np.linspace(0, 1, 501)\n>>> y = logit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.ylim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('logit(x)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.expit": {"fullname": "pyerrors.special.expit", "modulename": "pyerrors.special", "qualname": "expit", "kind": "function", "doc": "
    expit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
    \n\n
    expit(x, out=None)
    \n\n
    Expit (a.k.a. logistic sigmoid) ufunc for ndarrays.
    \n\n
    The expit function, also known as the logistic sigmoid function, is\ndefined as expit(x) = 1/(1+exp(-x)).  It is the inverse of the\nlogit function.
    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nThe ndarray to apply expit to element-wise.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare expit of the corresponding entry of x.
      • \n
    \n\n
    See Also
    \n\n
    logit  
    \n\n
    Notes
    \n\n
    As a ufunc expit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs 
    \n\n
    New in version 0.10.0.
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import expit, logit\n
    \n
    \n\n
    \n
    >>> expit([-np.inf, -1.5, 0, 1.5, np.inf])\narray([ 0.        ,  0.18242552,  0.5       ,  0.81757448,  1.        ])\n
    \n
    \n\n
    logit is the inverse of expit:
    \n\n
    \n
    >>> logit(expit([-2.5, 0, 3.1, 5.0]))\narray([-2.5,  0. ,  3.1,  5. ])\n
    \n
    \n\n
    Plot expit(x) for x in [-6, 6]:
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-6, 6, 121)\n>>> y = expit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.xlim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('expit(x)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logsumexp": {"fullname": "pyerrors.special.logsumexp", "modulename": "pyerrors.special", "qualname": "logsumexp", "kind": "function", "doc": "
    Compute the log of the sum of exponentials of input elements.
    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nInput array.
      • \n
    • axis (None or int or tuple of ints, optional):\nAxis or axes over which the sum is taken. By default axis is None,\nand all elements are summed.
      \n\n
      New in version 0.11.0.
      • \n
    • b (array-like, optional):\nScaling factor for exp(a) must be of the same shape as a or\nbroadcastable to a. These values may be negative in order to\nimplement subtraction.
      \n\n
      New in version 0.12.0.
      • \n
    • keepdims (bool, optional):\nIf this is set to True, the axes which are reduced are left in the\nresult as dimensions with size one. With this option, the result\nwill broadcast correctly against the original array.
      \n\n
      New in version 0.15.0.
      • \n
    • return_sign (bool, optional):\nIf this is set to True, the result will be a pair containing sign\ninformation; if False, results that are negative will be returned\nas NaN. Default is False (no sign information).
      \n\n
      New in version 0.16.0.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (ndarray):\nThe result, np.log(np.sum(np.exp(a))) calculated in a numerically\nmore stable way. If b is given then np.log(np.sum(b*np.exp(a)))\nis returned. If return_sign is True, res contains the log of\nthe absolute value of the argument.
      • \n
    • sgn (ndarray):\nIf return_sign is True, this will be an array of floating-point\nnumbers matching res containing +1, 0, -1 (for real-valued inputs)\nor a complex phase (for complex inputs). This gives the sign of the\nargument of the logarithm in res.\nIf return_sign is False, only one result is returned.
      • \n
    \n\n
    See Also
    \n\n
    numpy.logaddexp,, numpy.logaddexp2  
    \n\n
    Notes
    \n\n
    NumPy has a logaddexp function which is very similar to logsumexp, but\nonly handles two arguments. logaddexp.reduce is similar to this\nfunction, but may be less stable.
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import logsumexp\n>>> a = np.arange(10)\n>>> logsumexp(a)\n9.4586297444267107\n>>> np.log(np.sum(np.exp(a)))\n9.4586297444267107\n
    \n
    \n\n
    With weights
    \n\n
    \n
    >>> a = np.arange(10)\n>>> b = np.arange(10, 0, -1)\n>>> logsumexp(a, b=b)\n9.9170178533034665\n>>> np.log(np.sum(b*np.exp(a)))\n9.9170178533034647\n
    \n
    \n\n
    Returning a sign flag
    \n\n
    \n
    >>> logsumexp([1,2],b=[1,-1],return_sign=True)\n(1.5413248546129181, -1.0)\n
    \n
    \n\n
    Notice that logsumexp does not directly support masked arrays. To use it\non a masked array, convert the mask into zero weights:
    \n\n
    \n
    >>> a = np.ma.array([np.log(2), 2, np.log(3)],\n...                  mask=[False, True, False])\n>>> b = (~a.mask).astype(int)\n>>> logsumexp(a.data, b=b), np.log(5)\n1.6094379124341005, 1.6094379124341005\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.version": {"fullname": "pyerrors.version", "modulename": "pyerrors.version", "kind": "module", "doc": "
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