From ee985aca3cf7106f5da9ae5b7c457cf357da385a Mon Sep 17 00:00:00 2001 From: fjosw Date: Tue, 7 Mar 2023 16:40:53 +0000 Subject: [PATCH] Documentation updated --- docs/pyerrors/fits.html | 2645 +++++++++++++++++++-------------------- docs/search.js | 2 +- 2 files changed, 1322 insertions(+), 1325 deletions(-) diff --git a/docs/pyerrors/fits.html b/docs/pyerrors/fits.html index da4ef085..735950cd 100644 --- a/docs/pyerrors/fits.html +++ b/docs/pyerrors/fits.html @@ -259,681 +259,680 @@ 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 At the moment this option only works for `prior==None` and when no `method` is given. -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 (type(x) == dict and type(y) == dict and type(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 (type(x) == dict or type(y) == dict or type(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 Exception('x and y dictionaries do not contain the same keys.') -197 -198 if sorted(list(funcd.keys())) != key_ls: -199 raise Exception('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 Exception('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 Exception('x and y input (key=' + key + ') do not have the same length') -220 for i in range(100): -221 try: -222 funcd[key](np.arange(i), 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 = i -232 n_parms_ls.append(n_parms) -233 n_parms = max(n_parms_ls) -234 if not silent: -235 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) -236 -237 if priors is not None: -238 if isinstance(priors, (list, np.ndarray)): -239 if n_parms != len(priors): -240 raise ValueError("'priors' does not have the correct length.") -241 -242 loc_priors = [] -243 for i_n, i_prior in enumerate(priors): -244 if isinstance(i_prior, Obs): -245 loc_priors.append(i_prior) -246 elif isinstance(i_prior, str): -247 loc_val, loc_dval = _extract_val_and_dval(i_prior) -248 loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}")) -249 else: -250 raise TypeError("Prior entries need to be 'Obs' or 'str'.") -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 if isinstance(prior, Obs): -265 loc_priors.append(prior) -266 elif isinstance(prior, str): -267 loc_val, loc_dval = _extract_val_and_dval(prior) -268 loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(pos) + f"_{np.random.randint(2147483647):010d}")) -269 else: -270 raise TypeError("Prior entries need to be 'Obs' or 'str'.") -271 output.priors[pos] = loc_priors[-1] -272 if max(prior_mask) >= n_parms: -273 raise ValueError("Prior position out of range.") -274 else: -275 raise TypeError("Unkown type for `priors`.") -276 -277 p_f = [o.value for o in loc_priors] -278 dp_f = [o.dvalue for o in loc_priors] -279 if np.any(np.asarray(dp_f) <= 0.0): -280 raise Exception('No prior errors available, run the gamma method first.') -281 else: -282 p_f = dp_f = np.array([]) -283 prior_mask = [] -284 loc_priors = [] -285 -286 if 'initial_guess' in kwargs: -287 x0 = kwargs.get('initial_guess') -288 if len(x0) != n_parms: -289 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) -290 else: -291 x0 = [0.1] * n_parms -292 -293 if priors is None: -294 def general_chisqfunc_uncorr(p, ivars, pr): -295 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -296 return (ivars - model) / dy_f -297 else: -298 def general_chisqfunc_uncorr(p, ivars, pr): -299 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -300 return anp.concatenate(((ivars - model) / dy_f, (p[prior_mask] - pr) / dp_f)) -301 -302 def chisqfunc_uncorr(p): -303 return anp.sum(general_chisqfunc_uncorr(p, y_f, p_f) ** 2) -304 -305 if kwargs.get('correlated_fit') is True: -306 corr = covariance(y_all, correlation=True, **kwargs) -307 covdiag = np.diag(1 / np.asarray(dy_f)) -308 condn = np.linalg.cond(corr) -309 if condn > 0.1 / np.finfo(float).eps: -310 raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})") -311 if condn > 1e13: -312 warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning) -313 chol = np.linalg.cholesky(corr) -314 chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True) -315 -316 def general_chisqfunc(p, ivars, pr): -317 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -318 return anp.concatenate((anp.dot(chol_inv, (ivars - model)), (p[prior_mask] - pr) / dp_f)) -319 -320 def chisqfunc(p): -321 return anp.sum(general_chisqfunc(p, y_f, p_f) ** 2) -322 else: -323 general_chisqfunc = general_chisqfunc_uncorr -324 chisqfunc = chisqfunc_uncorr -325 -326 output.method = kwargs.get('method', 'Levenberg-Marquardt') -327 if not silent: -328 print('Method:', output.method) -329 -330 if output.method != 'Levenberg-Marquardt': -331 if output.method == 'migrad': -332 tolerance = 1e-4 # default value of 1e-1 set by iminuit can be problematic -333 if 'tol' in kwargs: -334 tolerance = kwargs.get('tol') -335 fit_result = iminuit.minimize(chisqfunc_uncorr, x0, tol=tolerance) # Stopping criterion 0.002 * tol * errordef -336 if kwargs.get('correlated_fit') is True: -337 fit_result = iminuit.minimize(chisqfunc, fit_result.x, tol=tolerance) -338 output.iterations = fit_result.nfev -339 else: -340 tolerance = 1e-12 -341 if 'tol' in kwargs: -342 tolerance = kwargs.get('tol') -343 fit_result = scipy.optimize.minimize(chisqfunc_uncorr, x0, method=kwargs.get('method'), tol=tolerance) -344 if kwargs.get('correlated_fit') is True: -345 fit_result = scipy.optimize.minimize(chisqfunc, fit_result.x, method=kwargs.get('method'), tol=tolerance) -346 output.iterations = fit_result.nit -347 -348 chisquare = fit_result.fun -349 -350 else: -351 if 'tol' in kwargs: -352 print('tol cannot be set for Levenberg-Marquardt') -353 -354 def chisqfunc_residuals_uncorr(p): -355 return general_chisqfunc_uncorr(p, y_f, p_f) -356 -357 fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) -358 if kwargs.get('correlated_fit') is True: -359 -360 def chisqfunc_residuals(p): -361 return general_chisqfunc(p, y_f, p_f) -362 -363 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) -364 -365 chisquare = np.sum(fit_result.fun ** 2) -366 assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14) -367 -368 output.iterations = fit_result.nfev -369 -370 if not fit_result.success: -371 raise Exception('The minimization procedure did not converge.') -372 -373 if x_all.shape[-1] - n_parms > 0: -374 output.chisquare = chisquare -375 output.dof = x_all.shape[-1] - n_parms + len(loc_priors) -376 output.chisquare_by_dof = output.chisquare / output.dof -377 output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof) -378 else: -379 output.chisquare_by_dof = float('nan') -380 -381 output.message = fit_result.message -382 if not silent: -383 print(fit_result.message) -384 print('chisquare/d.o.f.:', output.chisquare_by_dof) -385 print('fit parameters', fit_result.x) -386 -387 def prepare_hat_matrix(): -388 hat_vector = [] -389 for key in key_ls: -390 if (len(xd[key]) != 0): -391 hat_vector.append(jacobian(funcd[key])(fit_result.x, xd[key])) -392 hat_vector = [item for sublist in hat_vector for item in sublist] -393 return hat_vector -394 -395 if kwargs.get('expected_chisquare') is True: -396 if kwargs.get('correlated_fit') is not True: -397 W = np.diag(1 / np.asarray(dy_f)) -398 cov = covariance(y_all) -399 hat_vector = prepare_hat_matrix() -400 A = W @ hat_vector -401 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T -402 expected_chisquare = np.trace((np.identity(x_all.shape[-1]) - P_phi) @ W @ cov @ W) -403 output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare -404 if not silent: -405 print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare) -406 -407 fitp = fit_result.x -408 if np.any(np.asarray(dy_f) <= 0.0): -409 raise Exception('No y errors available, run the gamma method first.') -410 -411 try: -412 hess = hessian(chisqfunc)(fitp) -413 except TypeError: -414 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None -415 -416 len_y = len(y_f) -417 -418 def chisqfunc_compact(d): -419 return anp.sum(general_chisqfunc(d[:n_parms], d[n_parms: n_parms + len_y], d[n_parms + len_y:]) ** 2) -420 -421 jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f, p_f))) -422 -423 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv -424 try: -425 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:]) -426 except np.linalg.LinAlgError: -427 raise Exception("Cannot invert hessian matrix.") -428 -429 result = [] -430 for i in range(n_parms): -431 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]))) -432 -433 output.fit_parameters = result -434 -435 # Hotelling t-squared p-value for correlated fits. -436 if kwargs.get('correlated_fit') is True: -437 n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all)) -438 output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare, -439 output.dof, n_cov - output.dof) -440 -441 if kwargs.get('resplot') is True: -442 for key in key_ls: -443 residual_plot(xd[key], yd[key], funcd[key], result, title=key) -444 -445 if kwargs.get('qqplot') is True: -446 for key in key_ls: -447 qqplot(xd[key], yd[key], funcd[key], result, title=key) -448 -449 return output +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 (type(x) == dict and type(y) == dict and type(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 (type(x) == dict or type(y) == dict or type(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 Exception('x and y dictionaries do not contain the same keys.') +196 +197 if sorted(list(funcd.keys())) != key_ls: +198 raise Exception('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 Exception('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 Exception('x and y input (key=' + key + ') do not have the same length') +219 for i in range(100): +220 try: +221 funcd[key](np.arange(i), 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 = i +231 n_parms_ls.append(n_parms) +232 n_parms = max(n_parms_ls) +233 if not silent: +234 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) +235 +236 if priors is not None: +237 if isinstance(priors, (list, np.ndarray)): +238 if n_parms != len(priors): +239 raise ValueError("'priors' does not have the correct length.") +240 +241 loc_priors = [] +242 for i_n, i_prior in enumerate(priors): +243 if isinstance(i_prior, Obs): +244 loc_priors.append(i_prior) +245 elif isinstance(i_prior, str): +246 loc_val, loc_dval = _extract_val_and_dval(i_prior) +247 loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}")) +248 else: +249 raise TypeError("Prior entries need to be 'Obs' or 'str'.") +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 if isinstance(prior, Obs): +264 loc_priors.append(prior) +265 elif isinstance(prior, str): +266 loc_val, loc_dval = _extract_val_and_dval(prior) +267 loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(pos) + f"_{np.random.randint(2147483647):010d}")) +268 else: +269 raise TypeError("Prior entries need to be 'Obs' or 'str'.") +270 output.priors[pos] = loc_priors[-1] +271 if max(prior_mask) >= n_parms: +272 raise ValueError("Prior position out of range.") +273 else: +274 raise TypeError("Unkown type for `priors`.") +275 +276 p_f = [o.value for o in loc_priors] +277 dp_f = [o.dvalue for o in loc_priors] +278 if np.any(np.asarray(dp_f) <= 0.0): +279 raise Exception('No prior errors available, run the gamma method first.') +280 else: +281 p_f = dp_f = np.array([]) +282 prior_mask = [] +283 loc_priors = [] +284 +285 if 'initial_guess' in kwargs: +286 x0 = kwargs.get('initial_guess') +287 if len(x0) != n_parms: +288 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) +289 else: +290 x0 = [0.1] * n_parms +291 +292 if priors is None: +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 (ivars - model) / dy_f +296 else: +297 def general_chisqfunc_uncorr(p, ivars, pr): +298 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) +299 return anp.concatenate(((ivars - model) / dy_f, (p[prior_mask] - pr) / dp_f)) +300 +301 def chisqfunc_uncorr(p): +302 return anp.sum(general_chisqfunc_uncorr(p, y_f, p_f) ** 2) +303 +304 if kwargs.get('correlated_fit') is True: +305 corr = covariance(y_all, correlation=True, **kwargs) +306 covdiag = np.diag(1 / np.asarray(dy_f)) +307 condn = np.linalg.cond(corr) +308 if condn > 0.1 / np.finfo(float).eps: +309 raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})") +310 if condn > 1e13: +311 warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning) +312 chol = np.linalg.cholesky(corr) +313 chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True) +314 +315 def general_chisqfunc(p, ivars, pr): +316 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) +317 return anp.concatenate((anp.dot(chol_inv, (ivars - model)), (p[prior_mask] - pr) / dp_f)) +318 +319 def chisqfunc(p): +320 return anp.sum(general_chisqfunc(p, y_f, p_f) ** 2) +321 else: +322 general_chisqfunc = general_chisqfunc_uncorr +323 chisqfunc = chisqfunc_uncorr +324 +325 output.method = kwargs.get('method', 'Levenberg-Marquardt') +326 if not silent: +327 print('Method:', output.method) +328 +329 if output.method != 'Levenberg-Marquardt': +330 if output.method == 'migrad': +331 tolerance = 1e-4 # default value of 1e-1 set by iminuit can be problematic +332 if 'tol' in kwargs: +333 tolerance = kwargs.get('tol') +334 fit_result = iminuit.minimize(chisqfunc_uncorr, x0, tol=tolerance) # Stopping criterion 0.002 * tol * errordef +335 if kwargs.get('correlated_fit') is True: +336 fit_result = iminuit.minimize(chisqfunc, fit_result.x, tol=tolerance) +337 output.iterations = fit_result.nfev +338 else: +339 tolerance = 1e-12 +340 if 'tol' in kwargs: +341 tolerance = kwargs.get('tol') +342 fit_result = scipy.optimize.minimize(chisqfunc_uncorr, x0, method=kwargs.get('method'), tol=tolerance) +343 if kwargs.get('correlated_fit') is True: +344 fit_result = scipy.optimize.minimize(chisqfunc, fit_result.x, method=kwargs.get('method'), tol=tolerance) +345 output.iterations = fit_result.nit +346 +347 chisquare = fit_result.fun +348 +349 else: +350 if 'tol' in kwargs: +351 print('tol cannot be set for Levenberg-Marquardt') +352 +353 def chisqfunc_residuals_uncorr(p): +354 return general_chisqfunc_uncorr(p, y_f, p_f) +355 +356 fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) +357 if kwargs.get('correlated_fit') is True: +358 +359 def chisqfunc_residuals(p): +360 return general_chisqfunc(p, y_f, p_f) +361 +362 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) +363 +364 chisquare = np.sum(fit_result.fun ** 2) +365 assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14) +366 +367 output.iterations = fit_result.nfev +368 +369 if not fit_result.success: +370 raise Exception('The minimization procedure did not converge.') +371 +372 if x_all.shape[-1] - n_parms > 0: +373 output.chisquare = chisquare +374 output.dof = x_all.shape[-1] - n_parms + len(loc_priors) +375 output.chisquare_by_dof = output.chisquare / output.dof +376 output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof) +377 else: +378 output.chisquare_by_dof = float('nan') +379 +380 output.message = fit_result.message +381 if not silent: +382 print(fit_result.message) +383 print('chisquare/d.o.f.:', output.chisquare_by_dof) +384 print('fit parameters', fit_result.x) +385 +386 def prepare_hat_matrix(): +387 hat_vector = [] +388 for key in key_ls: +389 if (len(xd[key]) != 0): +390 hat_vector.append(jacobian(funcd[key])(fit_result.x, xd[key])) +391 hat_vector = [item for sublist in hat_vector for item in sublist] +392 return hat_vector +393 +394 if kwargs.get('expected_chisquare') is True: +395 if kwargs.get('correlated_fit') is not True: +396 W = np.diag(1 / np.asarray(dy_f)) +397 cov = covariance(y_all) +398 hat_vector = prepare_hat_matrix() +399 A = W @ hat_vector +400 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T +401 expected_chisquare = np.trace((np.identity(x_all.shape[-1]) - P_phi) @ W @ cov @ W) +402 output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare +403 if not silent: +404 print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare) +405 +406 fitp = fit_result.x +407 if np.any(np.asarray(dy_f) <= 0.0): +408 raise Exception('No y errors available, run the gamma method first.') +409 +410 try: +411 hess = hessian(chisqfunc)(fitp) +412 except TypeError: +413 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None +414 +415 len_y = len(y_f) +416 +417 def chisqfunc_compact(d): +418 return anp.sum(general_chisqfunc(d[:n_parms], d[n_parms: n_parms + len_y], d[n_parms + len_y:]) ** 2) +419 +420 jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f, p_f))) +421 +422 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv +423 try: +424 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:]) +425 except np.linalg.LinAlgError: +426 raise Exception("Cannot invert hessian matrix.") +427 +428 result = [] +429 for i in range(n_parms): +430 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]))) +431 +432 output.fit_parameters = result +433 +434 # Hotelling t-squared p-value for correlated fits. +435 if kwargs.get('correlated_fit') is True: +436 n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all)) +437 output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare, +438 output.dof, n_cov - output.dof) +439 +440 if kwargs.get('resplot') is True: +441 for key in key_ls: +442 residual_plot(xd[key], yd[key], funcd[key], result, title=key) +443 +444 if kwargs.get('qqplot') is True: +445 for key in key_ls: +446 qqplot(xd[key], yd[key], funcd[key], result, title=key) +447 +448 return output +449 450 -451 -452def total_least_squares(x, y, func, silent=False, **kwargs): -453 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. -454 -455 Parameters -456 ---------- -457 x : list -458 list of Obs, or a tuple of lists of Obs -459 y : list -460 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. -461 func : object -462 func has to be of the form -463 -464 ```python -465 import autograd.numpy as anp -466 -467 def func(a, x): -468 return a[0] + a[1] * x + a[2] * anp.sinh(x) -469 ``` -470 -471 For multiple x values func can be of the form -472 -473 ```python -474 def func(a, x): -475 (x1, x2) = x -476 return a[0] * x1 ** 2 + a[1] * x2 -477 ``` -478 -479 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation -480 will not work. -481 silent : bool, optional -482 If true all output to the console is omitted (default False). -483 initial_guess : list -484 can provide an initial guess for the input parameters. Relevant for non-linear -485 fits with many parameters. -486 expected_chisquare : bool -487 If true prints the expected chisquare which is -488 corrected by effects caused by correlated input data. -489 This can take a while as the full correlation matrix -490 has to be calculated (default False). -491 num_grad : bool -492 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). -493 -494 Notes -495 ----- -496 Based on the orthogonal distance regression module of scipy. -497 -498 Returns -499 ------- -500 output : Fit_result -501 Parameters and information on the fitted result. -502 ''' -503 -504 output = Fit_result() -505 -506 output.fit_function = func -507 -508 x = np.array(x) -509 -510 x_shape = x.shape -511 -512 if kwargs.get('num_grad') is True: -513 jacobian = num_jacobian -514 hessian = num_hessian -515 else: -516 jacobian = auto_jacobian -517 hessian = auto_hessian -518 -519 if not callable(func): -520 raise TypeError('func has to be a function.') -521 -522 for i in range(42): -523 try: -524 func(np.arange(i), x.T[0]) -525 except TypeError: -526 continue -527 except IndexError: -528 continue -529 else: -530 break -531 else: -532 raise RuntimeError("Fit function is not valid.") -533 -534 n_parms = i -535 if not silent: -536 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) -537 -538 x_f = np.vectorize(lambda o: o.value)(x) -539 dx_f = np.vectorize(lambda o: o.dvalue)(x) -540 y_f = np.array([o.value for o in y]) -541 dy_f = np.array([o.dvalue for o in y]) -542 -543 if np.any(np.asarray(dx_f) <= 0.0): -544 raise Exception('No x errors available, run the gamma method first.') -545 -546 if np.any(np.asarray(dy_f) <= 0.0): -547 raise Exception('No y errors available, run the gamma method first.') -548 -549 if 'initial_guess' in kwargs: -550 x0 = kwargs.get('initial_guess') -551 if len(x0) != n_parms: -552 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) -553 else: -554 x0 = [1] * n_parms -555 -556 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) -557 model = Model(func) -558 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) -559 odr.set_job(fit_type=0, deriv=1) -560 out = odr.run() -561 -562 output.residual_variance = out.res_var -563 -564 output.method = 'ODR' -565 -566 output.message = out.stopreason -567 -568 output.xplus = out.xplus -569 -570 if not silent: -571 print('Method: ODR') -572 print(*out.stopreason) -573 print('Residual variance:', output.residual_variance) -574 -575 if out.info > 3: -576 raise Exception('The minimization procedure did not converge.') -577 -578 m = x_f.size -579 -580 def odr_chisquare(p): -581 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) -582 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) -583 return chisq -584 -585 if kwargs.get('expected_chisquare') is True: -586 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) -587 -588 if kwargs.get('covariance') is not None: -589 cov = kwargs.get('covariance') -590 else: -591 cov = covariance(np.concatenate((y, x.ravel()))) -592 -593 number_of_x_parameters = int(m / x_f.shape[-1]) -594 -595 old_jac = jacobian(func)(out.beta, out.xplus) -596 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) -597 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]))) -598 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) -599 -600 A = W @ new_jac -601 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T -602 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) -603 if expected_chisquare <= 0.0: -604 warnings.warn("Negative expected_chisquare.", RuntimeWarning) -605 expected_chisquare = np.abs(expected_chisquare) -606 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare -607 if not silent: -608 print('chisquare/expected_chisquare:', -609 output.chisquare_by_expected_chisquare) -610 -611 fitp = out.beta -612 try: -613 hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel()))) -614 except TypeError: -615 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None -616 -617 def odr_chisquare_compact_x(d): -618 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) -619 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) -620 return chisq -621 -622 jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) -623 -624 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv -625 try: -626 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) -627 except np.linalg.LinAlgError: -628 raise Exception("Cannot invert hessian matrix.") -629 -630 def odr_chisquare_compact_y(d): -631 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) -632 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) -633 return chisq -634 -635 jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f))) -636 -637 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv -638 try: -639 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) -640 except np.linalg.LinAlgError: -641 raise Exception("Cannot invert hessian matrix.") -642 -643 result = [] -644 for i in range(n_parms): -645 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]))) -646 -647 output.fit_parameters = result -648 -649 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) -650 output.dof = x.shape[-1] - n_parms -651 output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof) -652 -653 return output +451def total_least_squares(x, y, func, silent=False, **kwargs): +452 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. +453 +454 Parameters +455 ---------- +456 x : list +457 list of Obs, or a tuple of lists of Obs +458 y : list +459 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. +460 func : object +461 func has to be of the form +462 +463 ```python +464 import autograd.numpy as anp +465 +466 def func(a, x): +467 return a[0] + a[1] * x + a[2] * anp.sinh(x) +468 ``` +469 +470 For multiple x values func can be of the form +471 +472 ```python +473 def func(a, x): +474 (x1, x2) = x +475 return a[0] * x1 ** 2 + a[1] * x2 +476 ``` +477 +478 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation +479 will not work. +480 silent : bool, optional +481 If true all output to the console is omitted (default False). +482 initial_guess : list +483 can provide an initial guess for the input parameters. Relevant for non-linear +484 fits with many parameters. +485 expected_chisquare : bool +486 If true prints the expected chisquare which is +487 corrected by effects caused by correlated input data. +488 This can take a while as the full correlation matrix +489 has to be calculated (default False). +490 num_grad : bool +491 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). +492 +493 Notes +494 ----- +495 Based on the orthogonal distance regression module of scipy. +496 +497 Returns +498 ------- +499 output : Fit_result +500 Parameters and information on the fitted result. +501 ''' +502 +503 output = Fit_result() +504 +505 output.fit_function = func +506 +507 x = np.array(x) +508 +509 x_shape = x.shape +510 +511 if kwargs.get('num_grad') is True: +512 jacobian = num_jacobian +513 hessian = num_hessian +514 else: +515 jacobian = auto_jacobian +516 hessian = auto_hessian +517 +518 if not callable(func): +519 raise TypeError('func has to be a function.') +520 +521 for i in range(42): +522 try: +523 func(np.arange(i), x.T[0]) +524 except TypeError: +525 continue +526 except IndexError: +527 continue +528 else: +529 break +530 else: +531 raise RuntimeError("Fit function is not valid.") +532 +533 n_parms = i +534 if not silent: +535 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) +536 +537 x_f = np.vectorize(lambda o: o.value)(x) +538 dx_f = np.vectorize(lambda o: o.dvalue)(x) +539 y_f = np.array([o.value for o in y]) +540 dy_f = np.array([o.dvalue for o in y]) +541 +542 if np.any(np.asarray(dx_f) <= 0.0): +543 raise Exception('No x errors available, run the gamma method first.') +544 +545 if np.any(np.asarray(dy_f) <= 0.0): +546 raise Exception('No y errors available, run the gamma method first.') +547 +548 if 'initial_guess' in kwargs: +549 x0 = kwargs.get('initial_guess') +550 if len(x0) != n_parms: +551 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) +552 else: +553 x0 = [1] * n_parms +554 +555 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) +556 model = Model(func) +557 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) +558 odr.set_job(fit_type=0, deriv=1) +559 out = odr.run() +560 +561 output.residual_variance = out.res_var +562 +563 output.method = 'ODR' +564 +565 output.message = out.stopreason +566 +567 output.xplus = out.xplus +568 +569 if not silent: +570 print('Method: ODR') +571 print(*out.stopreason) +572 print('Residual variance:', output.residual_variance) +573 +574 if out.info > 3: +575 raise Exception('The minimization procedure did not converge.') +576 +577 m = x_f.size +578 +579 def odr_chisquare(p): +580 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) +581 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) +582 return chisq +583 +584 if kwargs.get('expected_chisquare') is True: +585 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) +586 +587 if kwargs.get('covariance') is not None: +588 cov = kwargs.get('covariance') +589 else: +590 cov = covariance(np.concatenate((y, x.ravel()))) +591 +592 number_of_x_parameters = int(m / x_f.shape[-1]) +593 +594 old_jac = jacobian(func)(out.beta, out.xplus) +595 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) +596 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]))) +597 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) +598 +599 A = W @ new_jac +600 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T +601 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) +602 if expected_chisquare <= 0.0: +603 warnings.warn("Negative expected_chisquare.", RuntimeWarning) +604 expected_chisquare = np.abs(expected_chisquare) +605 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare +606 if not silent: +607 print('chisquare/expected_chisquare:', +608 output.chisquare_by_expected_chisquare) +609 +610 fitp = out.beta +611 try: +612 hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel()))) +613 except TypeError: +614 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None +615 +616 def odr_chisquare_compact_x(d): +617 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) +618 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) +619 return chisq +620 +621 jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) +622 +623 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv +624 try: +625 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) +626 except np.linalg.LinAlgError: +627 raise Exception("Cannot invert hessian matrix.") +628 +629 def odr_chisquare_compact_y(d): +630 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) +631 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) +632 return chisq +633 +634 jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f))) +635 +636 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv +637 try: +638 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) +639 except np.linalg.LinAlgError: +640 raise Exception("Cannot invert hessian matrix.") +641 +642 result = [] +643 for i in range(n_parms): +644 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]))) +645 +646 output.fit_parameters = result +647 +648 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) +649 output.dof = x.shape[-1] - n_parms +650 output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof) +651 +652 return output +653 654 -655 -656def fit_lin(x, y, **kwargs): -657 """Performs a linear fit to y = n + m * x and returns two Obs n, m. -658 -659 Parameters -660 ---------- -661 x : list -662 Can either be a list of floats in which case no xerror is assumed, or -663 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. -664 y : list -665 List of Obs, the dvalues of the Obs are used as yerror for the fit. -666 -667 Returns -668 ------- -669 fit_parameters : list[Obs] -670 LIist of fitted observables. -671 """ -672 -673 def f(a, x): -674 y = a[0] + a[1] * x -675 return y -676 -677 if all(isinstance(n, Obs) for n in x): -678 out = total_least_squares(x, y, f, **kwargs) -679 return out.fit_parameters -680 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): -681 out = least_squares(x, y, f, **kwargs) -682 return out.fit_parameters -683 else: -684 raise Exception('Unsupported types for x') +655def fit_lin(x, y, **kwargs): +656 """Performs a linear fit to y = n + m * x and returns two Obs n, m. +657 +658 Parameters +659 ---------- +660 x : list +661 Can either be a list of floats in which case no xerror is assumed, or +662 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. +663 y : list +664 List of Obs, the dvalues of the Obs are used as yerror for the fit. +665 +666 Returns +667 ------- +668 fit_parameters : list[Obs] +669 LIist of fitted observables. +670 """ +671 +672 def f(a, x): +673 y = a[0] + a[1] * x +674 return y +675 +676 if all(isinstance(n, Obs) for n in x): +677 out = total_least_squares(x, y, f, **kwargs) +678 return out.fit_parameters +679 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): +680 out = least_squares(x, y, f, **kwargs) +681 return out.fit_parameters +682 else: +683 raise Exception('Unsupported types for x') +684 685 -686 -687def qqplot(x, o_y, func, p, title=""): -688 """Generates a quantile-quantile plot of the fit result which can be used to -689 check if the residuals of the fit are gaussian distributed. -690 -691 Returns -692 ------- -693 None -694 """ -695 -696 residuals = [] -697 for i_x, i_y in zip(x, o_y): -698 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) -699 residuals = sorted(residuals) -700 my_y = [o.value for o in residuals] -701 probplot = scipy.stats.probplot(my_y) -702 my_x = probplot[0][0] -703 plt.figure(figsize=(8, 8 / 1.618)) -704 plt.errorbar(my_x, my_y, fmt='o') -705 fit_start = my_x[0] -706 fit_stop = my_x[-1] -707 samples = np.arange(fit_start, fit_stop, 0.01) -708 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') -709 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='-') -710 -711 plt.xlabel('Theoretical quantiles') -712 plt.ylabel('Ordered Values') -713 plt.legend(title=title) -714 plt.draw() +686def qqplot(x, o_y, func, p, title=""): +687 """Generates a quantile-quantile plot of the fit result which can be used to +688 check if the residuals of the fit are gaussian distributed. +689 +690 Returns +691 ------- +692 None +693 """ +694 +695 residuals = [] +696 for i_x, i_y in zip(x, o_y): +697 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) +698 residuals = sorted(residuals) +699 my_y = [o.value for o in residuals] +700 probplot = scipy.stats.probplot(my_y) +701 my_x = probplot[0][0] +702 plt.figure(figsize=(8, 8 / 1.618)) +703 plt.errorbar(my_x, my_y, fmt='o') +704 fit_start = my_x[0] +705 fit_stop = my_x[-1] +706 samples = np.arange(fit_start, fit_stop, 0.01) +707 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') +708 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='-') +709 +710 plt.xlabel('Theoretical quantiles') +711 plt.ylabel('Ordered Values') +712 plt.legend(title=title) +713 plt.draw() +714 715 -716 -717def residual_plot(x, y, func, fit_res, title=""): -718 """Generates a plot which compares the fit to the data and displays the corresponding residuals -719 -720 For uncorrelated data the residuals are expected to be distributed ~N(0,1). -721 -722 Returns -723 ------- -724 None -725 """ -726 sorted_x = sorted(x) -727 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) -728 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) -729 x_samples = np.arange(xstart, xstop + 0.01, 0.01) -730 -731 plt.figure(figsize=(8, 8 / 1.618)) -732 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) -733 ax0 = plt.subplot(gs[0]) -734 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') -735 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) -736 ax0.set_xticklabels([]) -737 ax0.set_xlim([xstart, xstop]) -738 ax0.set_xticklabels([]) -739 ax0.legend(title=title) -740 -741 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]) -742 ax1 = plt.subplot(gs[1]) -743 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) -744 ax1.tick_params(direction='out') -745 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) -746 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") -747 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') -748 ax1.set_xlim([xstart, xstop]) -749 ax1.set_ylabel('Residuals') -750 plt.subplots_adjust(wspace=None, hspace=None) -751 plt.draw() +716def residual_plot(x, y, func, fit_res, title=""): +717 """Generates a plot which compares the fit to the data and displays the corresponding residuals +718 +719 For uncorrelated data the residuals are expected to be distributed ~N(0,1). +720 +721 Returns +722 ------- +723 None +724 """ +725 sorted_x = sorted(x) +726 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) +727 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) +728 x_samples = np.arange(xstart, xstop + 0.01, 0.01) +729 +730 plt.figure(figsize=(8, 8 / 1.618)) +731 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) +732 ax0 = plt.subplot(gs[0]) +733 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') +734 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) +735 ax0.set_xticklabels([]) +736 ax0.set_xlim([xstart, xstop]) +737 ax0.set_xticklabels([]) +738 ax0.legend(title=title) +739 +740 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]) +741 ax1 = plt.subplot(gs[1]) +742 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) +743 ax1.tick_params(direction='out') +744 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) +745 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") +746 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') +747 ax1.set_xlim([xstart, xstop]) +748 ax1.set_ylabel('Residuals') +749 plt.subplots_adjust(wspace=None, hspace=None) +750 plt.draw() +751 752 -753 -754def error_band(x, func, beta): -755 """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta. -756 -757 Returns -758 ------- -759 err : np.array(Obs) -760 Error band for an array of sample values x -761 """ -762 cov = covariance(beta) -763 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): -764 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) -765 -766 deriv = [] -767 for i, item in enumerate(x): -768 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) -769 -770 err = [] -771 for i, item in enumerate(x): -772 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) -773 err = np.array(err) -774 -775 return err +753def error_band(x, func, beta): +754 """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta. +755 +756 Returns +757 ------- +758 err : np.array(Obs) +759 Error band for an array of sample values x +760 """ +761 cov = covariance(beta) +762 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): +763 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) +764 +765 deriv = [] +766 for i, item in enumerate(x): +767 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) +768 +769 err = [] +770 for i, item in enumerate(x): +771 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) +772 err = np.array(err) +773 +774 return err +775 776 -777 -778def ks_test(objects=None): -779 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. -780 -781 Parameters -782 ---------- -783 objects : list -784 List of fit results to include in the analysis (optional). -785 -786 Returns -787 ------- -788 None -789 """ -790 -791 if objects is None: -792 obs_list = [] -793 for obj in gc.get_objects(): -794 if isinstance(obj, Fit_result): -795 obs_list.append(obj) -796 else: -797 obs_list = objects -798 -799 p_values = [o.p_value for o in obs_list] -800 -801 bins = len(p_values) -802 x = np.arange(0, 1.001, 0.001) -803 plt.plot(x, x, 'k', zorder=1) -804 plt.xlim(0, 1) -805 plt.ylim(0, 1) -806 plt.xlabel('p-value') -807 plt.ylabel('Cumulative probability') -808 plt.title(str(bins) + ' p-values') -809 -810 n = np.arange(1, bins + 1) / np.float64(bins) -811 Xs = np.sort(p_values) -812 plt.step(Xs, n) -813 diffs = n - Xs -814 loc_max_diff = np.argmax(np.abs(diffs)) -815 loc = Xs[loc_max_diff] -816 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) -817 plt.draw() -818 -819 print(scipy.stats.kstest(p_values, 'uniform')) +777def ks_test(objects=None): +778 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. +779 +780 Parameters +781 ---------- +782 objects : list +783 List of fit results to include in the analysis (optional). +784 +785 Returns +786 ------- +787 None +788 """ +789 +790 if objects is None: +791 obs_list = [] +792 for obj in gc.get_objects(): +793 if isinstance(obj, Fit_result): +794 obs_list.append(obj) +795 else: +796 obs_list = objects +797 +798 p_values = [o.p_value for o in obs_list] +799 +800 bins = len(p_values) +801 x = np.arange(0, 1.001, 0.001) +802 plt.plot(x, x, 'k', zorder=1) +803 plt.xlim(0, 1) +804 plt.ylim(0, 1) +805 plt.xlabel('p-value') +806 plt.ylabel('Cumulative probability') +807 plt.title(str(bins) + ' p-values') +808 +809 n = np.arange(1, bins + 1) / np.float64(bins) +810 Xs = np.sort(p_values) +811 plt.step(Xs, n) +812 diffs = n - Xs +813 loc_max_diff = np.argmax(np.abs(diffs)) +814 loc = Xs[loc_max_diff] +815 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) +816 plt.draw() +817 +818 print(scipy.stats.kstest(p_values, 'uniform')) +819 820 -821 -822def _extract_val_and_dval(string): -823 split_string = string.split('(') -824 if '.' in split_string[0] and '.' not in split_string[1][:-1]: -825 factor = 10 ** -len(split_string[0].partition('.')[2]) -826 else: -827 factor = 1 -828 return float(split_string[0]), float(split_string[1][:-1]) * factor +821def _extract_val_and_dval(string): +822 split_string = string.split('(') +823 if '.' in split_string[0] and '.' not in split_string[1][:-1]: +824 factor = 10 ** -len(split_string[0].partition('.')[2]) +825 else: +826 factor = 1 +827 return float(split_string[0]), float(split_string[1][:-1]) * factor @@ -1168,302 +1167,301 @@ 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 At the moment this option only works for `prior==None` and when no `method` is given. -156 expected_chisquare : bool -157 If True estimates the expected chisquare which is -158 corrected by effects caused by correlated input data (default False). -159 resplot : bool -160 If True, a plot which displays fit, data and residuals is generated (default False). -161 qqplot : bool -162 If True, a quantile-quantile plot of the fit result is generated (default False). -163 num_grad : bool -164 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). -165 -166 Returns -167 ------- -168 output : Fit_result -169 Parameters and information on the fitted result. -170 ''' -171 output = Fit_result() -172 -173 if (type(x) == dict and type(y) == dict and type(func) == dict): -174 xd = {key: anp.asarray(x[key]) for key in x} -175 yd = y -176 funcd = func -177 output.fit_function = func -178 elif (type(x) == dict or type(y) == dict or type(func) == dict): -179 raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.") -180 else: -181 x = np.asarray(x) -182 xd = {"": x} -183 yd = {"": y} -184 funcd = {"": func} -185 output.fit_function = func -186 -187 if kwargs.get('num_grad') is True: -188 jacobian = num_jacobian -189 hessian = num_hessian -190 else: -191 jacobian = auto_jacobian -192 hessian = auto_hessian -193 -194 key_ls = sorted(list(xd.keys())) -195 -196 if sorted(list(yd.keys())) != key_ls: -197 raise Exception('x and y dictionaries do not contain the same keys.') -198 -199 if sorted(list(funcd.keys())) != key_ls: -200 raise Exception('x and func dictionaries do not contain the same keys.') -201 -202 x_all = np.concatenate([np.array(xd[key]) for key in key_ls]) -203 y_all = np.concatenate([np.array(yd[key]) for key in key_ls]) -204 -205 y_f = [o.value for o in y_all] -206 dy_f = [o.dvalue for o in y_all] -207 -208 if len(x_all.shape) > 2: -209 raise Exception('Unknown format for x values') -210 -211 if np.any(np.asarray(dy_f) <= 0.0): -212 raise Exception('No y errors available, run the gamma method first.') -213 -214 # number of fit parameters -215 n_parms_ls = [] -216 for key in key_ls: -217 if not callable(funcd[key]): -218 raise TypeError('func (key=' + key + ') is not a function.') -219 if np.asarray(xd[key]).shape[-1] != len(yd[key]): -220 raise Exception('x and y input (key=' + key + ') do not have the same length') -221 for i in range(100): -222 try: -223 funcd[key](np.arange(i), x_all.T[0]) -224 except TypeError: -225 continue -226 except IndexError: -227 continue -228 else: -229 break -230 else: -231 raise RuntimeError("Fit function (key=" + key + ") is not valid.") -232 n_parms = i -233 n_parms_ls.append(n_parms) -234 n_parms = max(n_parms_ls) -235 if not silent: -236 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) -237 -238 if priors is not None: -239 if isinstance(priors, (list, np.ndarray)): -240 if n_parms != len(priors): -241 raise ValueError("'priors' does not have the correct length.") -242 -243 loc_priors = [] -244 for i_n, i_prior in enumerate(priors): -245 if isinstance(i_prior, Obs): -246 loc_priors.append(i_prior) -247 elif isinstance(i_prior, str): -248 loc_val, loc_dval = _extract_val_and_dval(i_prior) -249 loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}")) -250 else: -251 raise TypeError("Prior entries need to be 'Obs' or 'str'.") -252 -253 prior_mask = np.arange(len(priors)) -254 output.priors = loc_priors -255 -256 elif isinstance(priors, dict): -257 loc_priors = [] -258 prior_mask = [] -259 output.priors = {} -260 for pos, prior in priors.items(): -261 if isinstance(pos, int): -262 prior_mask.append(pos) -263 else: -264 raise TypeError("Prior position needs to be an integer.") -265 if isinstance(prior, Obs): -266 loc_priors.append(prior) -267 elif isinstance(prior, str): -268 loc_val, loc_dval = _extract_val_and_dval(prior) -269 loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(pos) + f"_{np.random.randint(2147483647):010d}")) -270 else: -271 raise TypeError("Prior entries need to be 'Obs' or 'str'.") -272 output.priors[pos] = loc_priors[-1] -273 if max(prior_mask) >= n_parms: -274 raise ValueError("Prior position out of range.") -275 else: -276 raise TypeError("Unkown type for `priors`.") -277 -278 p_f = [o.value for o in loc_priors] -279 dp_f = [o.dvalue for o in loc_priors] -280 if np.any(np.asarray(dp_f) <= 0.0): -281 raise Exception('No prior errors available, run the gamma method first.') -282 else: -283 p_f = dp_f = np.array([]) -284 prior_mask = [] -285 loc_priors = [] -286 -287 if 'initial_guess' in kwargs: -288 x0 = kwargs.get('initial_guess') -289 if len(x0) != n_parms: -290 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) -291 else: -292 x0 = [0.1] * n_parms -293 -294 if priors is None: -295 def general_chisqfunc_uncorr(p, ivars, pr): -296 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -297 return (ivars - model) / dy_f -298 else: -299 def general_chisqfunc_uncorr(p, ivars, pr): -300 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -301 return anp.concatenate(((ivars - model) / dy_f, (p[prior_mask] - pr) / dp_f)) -302 -303 def chisqfunc_uncorr(p): -304 return anp.sum(general_chisqfunc_uncorr(p, y_f, p_f) ** 2) -305 -306 if kwargs.get('correlated_fit') is True: -307 corr = covariance(y_all, correlation=True, **kwargs) -308 covdiag = np.diag(1 / np.asarray(dy_f)) -309 condn = np.linalg.cond(corr) -310 if condn > 0.1 / np.finfo(float).eps: -311 raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})") -312 if condn > 1e13: -313 warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning) -314 chol = np.linalg.cholesky(corr) -315 chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True) -316 -317 def general_chisqfunc(p, ivars, pr): -318 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) -319 return anp.concatenate((anp.dot(chol_inv, (ivars - model)), (p[prior_mask] - pr) / dp_f)) -320 -321 def chisqfunc(p): -322 return anp.sum(general_chisqfunc(p, y_f, p_f) ** 2) -323 else: -324 general_chisqfunc = general_chisqfunc_uncorr -325 chisqfunc = chisqfunc_uncorr -326 -327 output.method = kwargs.get('method', 'Levenberg-Marquardt') -328 if not silent: -329 print('Method:', output.method) -330 -331 if output.method != 'Levenberg-Marquardt': -332 if output.method == 'migrad': -333 tolerance = 1e-4 # default value of 1e-1 set by iminuit can be problematic -334 if 'tol' in kwargs: -335 tolerance = kwargs.get('tol') -336 fit_result = iminuit.minimize(chisqfunc_uncorr, x0, tol=tolerance) # Stopping criterion 0.002 * tol * errordef -337 if kwargs.get('correlated_fit') is True: -338 fit_result = iminuit.minimize(chisqfunc, fit_result.x, tol=tolerance) -339 output.iterations = fit_result.nfev -340 else: -341 tolerance = 1e-12 -342 if 'tol' in kwargs: -343 tolerance = kwargs.get('tol') -344 fit_result = scipy.optimize.minimize(chisqfunc_uncorr, x0, method=kwargs.get('method'), tol=tolerance) -345 if kwargs.get('correlated_fit') is True: -346 fit_result = scipy.optimize.minimize(chisqfunc, fit_result.x, method=kwargs.get('method'), tol=tolerance) -347 output.iterations = fit_result.nit -348 -349 chisquare = fit_result.fun -350 -351 else: -352 if 'tol' in kwargs: -353 print('tol cannot be set for Levenberg-Marquardt') -354 -355 def chisqfunc_residuals_uncorr(p): -356 return general_chisqfunc_uncorr(p, y_f, p_f) -357 -358 fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) -359 if kwargs.get('correlated_fit') is True: -360 -361 def chisqfunc_residuals(p): -362 return general_chisqfunc(p, y_f, p_f) -363 -364 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) -365 -366 chisquare = np.sum(fit_result.fun ** 2) -367 assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14) -368 -369 output.iterations = fit_result.nfev -370 -371 if not fit_result.success: -372 raise Exception('The minimization procedure did not converge.') -373 -374 if x_all.shape[-1] - n_parms > 0: -375 output.chisquare = chisquare -376 output.dof = x_all.shape[-1] - n_parms + len(loc_priors) -377 output.chisquare_by_dof = output.chisquare / output.dof -378 output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof) -379 else: -380 output.chisquare_by_dof = float('nan') -381 -382 output.message = fit_result.message -383 if not silent: -384 print(fit_result.message) -385 print('chisquare/d.o.f.:', output.chisquare_by_dof) -386 print('fit parameters', fit_result.x) -387 -388 def prepare_hat_matrix(): -389 hat_vector = [] -390 for key in key_ls: -391 if (len(xd[key]) != 0): -392 hat_vector.append(jacobian(funcd[key])(fit_result.x, xd[key])) -393 hat_vector = [item for sublist in hat_vector for item in sublist] -394 return hat_vector -395 -396 if kwargs.get('expected_chisquare') is True: -397 if kwargs.get('correlated_fit') is not True: -398 W = np.diag(1 / np.asarray(dy_f)) -399 cov = covariance(y_all) -400 hat_vector = prepare_hat_matrix() -401 A = W @ hat_vector -402 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T -403 expected_chisquare = np.trace((np.identity(x_all.shape[-1]) - P_phi) @ W @ cov @ W) -404 output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare -405 if not silent: -406 print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare) -407 -408 fitp = fit_result.x -409 if np.any(np.asarray(dy_f) <= 0.0): -410 raise Exception('No y errors available, run the gamma method first.') -411 -412 try: -413 hess = hessian(chisqfunc)(fitp) -414 except TypeError: -415 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None -416 -417 len_y = len(y_f) -418 -419 def chisqfunc_compact(d): -420 return anp.sum(general_chisqfunc(d[:n_parms], d[n_parms: n_parms + len_y], d[n_parms + len_y:]) ** 2) -421 -422 jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f, p_f))) -423 -424 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv -425 try: -426 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:]) -427 except np.linalg.LinAlgError: -428 raise Exception("Cannot invert hessian matrix.") -429 -430 result = [] -431 for i in range(n_parms): -432 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]))) -433 -434 output.fit_parameters = result -435 -436 # Hotelling t-squared p-value for correlated fits. -437 if kwargs.get('correlated_fit') is True: -438 n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all)) -439 output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare, -440 output.dof, n_cov - output.dof) -441 -442 if kwargs.get('resplot') is True: -443 for key in key_ls: -444 residual_plot(xd[key], yd[key], funcd[key], result, title=key) -445 -446 if kwargs.get('qqplot') is True: -447 for key in key_ls: -448 qqplot(xd[key], yd[key], funcd[key], result, title=key) -449 -450 return output +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 (type(x) == dict and type(y) == dict and type(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 (type(x) == dict or type(y) == dict or type(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 Exception('x and y dictionaries do not contain the same keys.') +197 +198 if sorted(list(funcd.keys())) != key_ls: +199 raise Exception('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 Exception('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 Exception('x and y input (key=' + key + ') do not have the same length') +220 for i in range(100): +221 try: +222 funcd[key](np.arange(i), 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 = i +232 n_parms_ls.append(n_parms) +233 n_parms = max(n_parms_ls) +234 if not silent: +235 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) +236 +237 if priors is not None: +238 if isinstance(priors, (list, np.ndarray)): +239 if n_parms != len(priors): +240 raise ValueError("'priors' does not have the correct length.") +241 +242 loc_priors = [] +243 for i_n, i_prior in enumerate(priors): +244 if isinstance(i_prior, Obs): +245 loc_priors.append(i_prior) +246 elif isinstance(i_prior, str): +247 loc_val, loc_dval = _extract_val_and_dval(i_prior) +248 loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}")) +249 else: +250 raise TypeError("Prior entries need to be 'Obs' or 'str'.") +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 if isinstance(prior, Obs): +265 loc_priors.append(prior) +266 elif isinstance(prior, str): +267 loc_val, loc_dval = _extract_val_and_dval(prior) +268 loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(pos) + f"_{np.random.randint(2147483647):010d}")) +269 else: +270 raise TypeError("Prior entries need to be 'Obs' or 'str'.") +271 output.priors[pos] = loc_priors[-1] +272 if max(prior_mask) >= n_parms: +273 raise ValueError("Prior position out of range.") +274 else: +275 raise TypeError("Unkown type for `priors`.") +276 +277 p_f = [o.value for o in loc_priors] +278 dp_f = [o.dvalue for o in loc_priors] +279 if np.any(np.asarray(dp_f) <= 0.0): +280 raise Exception('No prior errors available, run the gamma method first.') +281 else: +282 p_f = dp_f = np.array([]) +283 prior_mask = [] +284 loc_priors = [] +285 +286 if 'initial_guess' in kwargs: +287 x0 = kwargs.get('initial_guess') +288 if len(x0) != n_parms: +289 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) +290 else: +291 x0 = [0.1] * n_parms +292 +293 if priors is None: +294 def general_chisqfunc_uncorr(p, ivars, pr): +295 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) +296 return (ivars - model) / dy_f +297 else: +298 def general_chisqfunc_uncorr(p, ivars, pr): +299 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) +300 return anp.concatenate(((ivars - model) / dy_f, (p[prior_mask] - pr) / dp_f)) +301 +302 def chisqfunc_uncorr(p): +303 return anp.sum(general_chisqfunc_uncorr(p, y_f, p_f) ** 2) +304 +305 if kwargs.get('correlated_fit') is True: +306 corr = covariance(y_all, correlation=True, **kwargs) +307 covdiag = np.diag(1 / np.asarray(dy_f)) +308 condn = np.linalg.cond(corr) +309 if condn > 0.1 / np.finfo(float).eps: +310 raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})") +311 if condn > 1e13: +312 warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning) +313 chol = np.linalg.cholesky(corr) +314 chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True) +315 +316 def general_chisqfunc(p, ivars, pr): +317 model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls]) +318 return anp.concatenate((anp.dot(chol_inv, (ivars - model)), (p[prior_mask] - pr) / dp_f)) +319 +320 def chisqfunc(p): +321 return anp.sum(general_chisqfunc(p, y_f, p_f) ** 2) +322 else: +323 general_chisqfunc = general_chisqfunc_uncorr +324 chisqfunc = chisqfunc_uncorr +325 +326 output.method = kwargs.get('method', 'Levenberg-Marquardt') +327 if not silent: +328 print('Method:', output.method) +329 +330 if output.method != 'Levenberg-Marquardt': +331 if output.method == 'migrad': +332 tolerance = 1e-4 # default value of 1e-1 set by iminuit can be problematic +333 if 'tol' in kwargs: +334 tolerance = kwargs.get('tol') +335 fit_result = iminuit.minimize(chisqfunc_uncorr, x0, tol=tolerance) # Stopping criterion 0.002 * tol * errordef +336 if kwargs.get('correlated_fit') is True: +337 fit_result = iminuit.minimize(chisqfunc, fit_result.x, tol=tolerance) +338 output.iterations = fit_result.nfev +339 else: +340 tolerance = 1e-12 +341 if 'tol' in kwargs: +342 tolerance = kwargs.get('tol') +343 fit_result = scipy.optimize.minimize(chisqfunc_uncorr, x0, method=kwargs.get('method'), tol=tolerance) +344 if kwargs.get('correlated_fit') is True: +345 fit_result = scipy.optimize.minimize(chisqfunc, fit_result.x, method=kwargs.get('method'), tol=tolerance) +346 output.iterations = fit_result.nit +347 +348 chisquare = fit_result.fun +349 +350 else: +351 if 'tol' in kwargs: +352 print('tol cannot be set for Levenberg-Marquardt') +353 +354 def chisqfunc_residuals_uncorr(p): +355 return general_chisqfunc_uncorr(p, y_f, p_f) +356 +357 fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) +358 if kwargs.get('correlated_fit') is True: +359 +360 def chisqfunc_residuals(p): +361 return general_chisqfunc(p, y_f, p_f) +362 +363 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) +364 +365 chisquare = np.sum(fit_result.fun ** 2) +366 assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14) +367 +368 output.iterations = fit_result.nfev +369 +370 if not fit_result.success: +371 raise Exception('The minimization procedure did not converge.') +372 +373 if x_all.shape[-1] - n_parms > 0: +374 output.chisquare = chisquare +375 output.dof = x_all.shape[-1] - n_parms + len(loc_priors) +376 output.chisquare_by_dof = output.chisquare / output.dof +377 output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof) +378 else: +379 output.chisquare_by_dof = float('nan') +380 +381 output.message = fit_result.message +382 if not silent: +383 print(fit_result.message) +384 print('chisquare/d.o.f.:', output.chisquare_by_dof) +385 print('fit parameters', fit_result.x) +386 +387 def prepare_hat_matrix(): +388 hat_vector = [] +389 for key in key_ls: +390 if (len(xd[key]) != 0): +391 hat_vector.append(jacobian(funcd[key])(fit_result.x, xd[key])) +392 hat_vector = [item for sublist in hat_vector for item in sublist] +393 return hat_vector +394 +395 if kwargs.get('expected_chisquare') is True: +396 if kwargs.get('correlated_fit') is not True: +397 W = np.diag(1 / np.asarray(dy_f)) +398 cov = covariance(y_all) +399 hat_vector = prepare_hat_matrix() +400 A = W @ hat_vector +401 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T +402 expected_chisquare = np.trace((np.identity(x_all.shape[-1]) - P_phi) @ W @ cov @ W) +403 output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare +404 if not silent: +405 print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare) +406 +407 fitp = fit_result.x +408 if np.any(np.asarray(dy_f) <= 0.0): +409 raise Exception('No y errors available, run the gamma method first.') +410 +411 try: +412 hess = hessian(chisqfunc)(fitp) +413 except TypeError: +414 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None +415 +416 len_y = len(y_f) +417 +418 def chisqfunc_compact(d): +419 return anp.sum(general_chisqfunc(d[:n_parms], d[n_parms: n_parms + len_y], d[n_parms + len_y:]) ** 2) +420 +421 jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f, p_f))) +422 +423 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv +424 try: +425 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:]) +426 except np.linalg.LinAlgError: +427 raise Exception("Cannot invert hessian matrix.") +428 +429 result = [] +430 for i in range(n_parms): +431 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]))) +432 +433 output.fit_parameters = result +434 +435 # Hotelling t-squared p-value for correlated fits. +436 if kwargs.get('correlated_fit') is True: +437 n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all)) +438 output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare, +439 output.dof, n_cov - output.dof) +440 +441 if kwargs.get('resplot') is True: +442 for key in key_ls: +443 residual_plot(xd[key], yd[key], funcd[key], result, title=key) +444 +445 if kwargs.get('qqplot') is True: +446 for key in key_ls: +447 qqplot(xd[key], yd[key], funcd[key], result, title=key) +448 +449 return output @@ -1546,8 +1544,7 @@ The stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol If True, use the full inverse covariance matrix in the definition of the chisquare cost function. 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). -At the moment this option only works for prior==None and when no method is given. +This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
  • expected_chisquare (bool): If True estimates the expected chisquare which is corrected by effects caused by correlated input data (default False).
    • @@ -1580,208 +1577,208 @@ Parameters and information on the fitted result. - 
    453def total_least_squares(x, y, func, silent=False, **kwargs):
-454    r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
-455
-456    Parameters
-457    ----------
-458    x : list
-459        list of Obs, or a tuple of lists of Obs
-460    y : list
-461        list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
-462    func : object
-463        func has to be of the form
-464
-465        ```python
-466        import autograd.numpy as anp
-467
-468        def func(a, x):
-469            return a[0] + a[1] * x + a[2] * anp.sinh(x)
-470        ```
-471
-472        For multiple x values func can be of the form
-473
-474        ```python
-475        def func(a, x):
-476            (x1, x2) = x
-477            return a[0] * x1 ** 2 + a[1] * x2
-478        ```
-479
-480        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
-481        will not work.
-482    silent : bool, optional
-483        If true all output to the console is omitted (default False).
-484    initial_guess : list
-485        can provide an initial guess for the input parameters. Relevant for non-linear
-486        fits with many parameters.
-487    expected_chisquare : bool
-488        If true prints the expected chisquare which is
-489        corrected by effects caused by correlated input data.
-490        This can take a while as the full correlation matrix
-491        has to be calculated (default False).
-492    num_grad : bool
-493        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
-494
-495    Notes
-496    -----
-497    Based on the orthogonal distance regression module of scipy.
-498
-499    Returns
-500    -------
-501    output : Fit_result
-502        Parameters and information on the fitted result.
-503    '''
-504
-505    output = Fit_result()
-506
-507    output.fit_function = func
-508
-509    x = np.array(x)
-510
-511    x_shape = x.shape
-512
-513    if kwargs.get('num_grad') is True:
-514        jacobian = num_jacobian
-515        hessian = num_hessian
-516    else:
-517        jacobian = auto_jacobian
-518        hessian = auto_hessian
-519
-520    if not callable(func):
-521        raise TypeError('func has to be a function.')
-522
-523    for i in range(42):
-524        try:
-525            func(np.arange(i), x.T[0])
-526        except TypeError:
-527            continue
-528        except IndexError:
-529            continue
-530        else:
-531            break
-532    else:
-533        raise RuntimeError("Fit function is not valid.")
-534
-535    n_parms = i
-536    if not silent:
-537        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
-538
-539    x_f = np.vectorize(lambda o: o.value)(x)
-540    dx_f = np.vectorize(lambda o: o.dvalue)(x)
-541    y_f = np.array([o.value for o in y])
-542    dy_f = np.array([o.dvalue for o in y])
-543
-544    if np.any(np.asarray(dx_f) <= 0.0):
-545        raise Exception('No x errors available, run the gamma method first.')
-546
-547    if np.any(np.asarray(dy_f) <= 0.0):
-548        raise Exception('No y errors available, run the gamma method first.')
-549
-550    if 'initial_guess' in kwargs:
-551        x0 = kwargs.get('initial_guess')
-552        if len(x0) != n_parms:
-553            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
-554    else:
-555        x0 = [1] * n_parms
-556
-557    data = RealData(x_f, y_f, sx=dx_f, sy=dy_f)
-558    model = Model(func)
-559    odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps)
-560    odr.set_job(fit_type=0, deriv=1)
-561    out = odr.run()
-562
-563    output.residual_variance = out.res_var
-564
-565    output.method = 'ODR'
-566
-567    output.message = out.stopreason
-568
-569    output.xplus = out.xplus
-570
-571    if not silent:
-572        print('Method: ODR')
-573        print(*out.stopreason)
-574        print('Residual variance:', output.residual_variance)
-575
-576    if out.info > 3:
-577        raise Exception('The minimization procedure did not converge.')
-578
-579    m = x_f.size
-580
-581    def odr_chisquare(p):
-582        model = func(p[:n_parms], p[n_parms:].reshape(x_shape))
-583        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2)
-584        return chisq
-585
-586    if kwargs.get('expected_chisquare') is True:
-587        W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel()))))
-588
-589        if kwargs.get('covariance') is not None:
-590            cov = kwargs.get('covariance')
-591        else:
-592            cov = covariance(np.concatenate((y, x.ravel())))
-593
-594        number_of_x_parameters = int(m / x_f.shape[-1])
-595
-596        old_jac = jacobian(func)(out.beta, out.xplus)
-597        fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0)))
-598        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])))
-599        new_jac = np.concatenate((fused_row1, fused_row2), axis=1)
-600
-601        A = W @ new_jac
-602        P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
-603        expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W)
-604        if expected_chisquare <= 0.0:
-605            warnings.warn("Negative expected_chisquare.", RuntimeWarning)
-606            expected_chisquare = np.abs(expected_chisquare)
-607        output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare
-608        if not silent:
-609            print('chisquare/expected_chisquare:',
-610                  output.chisquare_by_expected_chisquare)
-611
-612    fitp = out.beta
-613    try:
-614        hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel())))
-615    except TypeError:
-616        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
-617
-618    def odr_chisquare_compact_x(d):
-619        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
-620        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)
-621        return chisq
-622
-623    jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel())))
-624
-625    # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv
-626    try:
-627        deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:])
-628    except np.linalg.LinAlgError:
-629        raise Exception("Cannot invert hessian matrix.")
-630
-631    def odr_chisquare_compact_y(d):
-632        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
-633        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)
-634        return chisq
-635
-636    jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f)))
-637
-638    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
-639    try:
-640        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:])
-641    except np.linalg.LinAlgError:
-642        raise Exception("Cannot invert hessian matrix.")
-643
-644    result = []
-645    for i in range(n_parms):
-646        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])))
-647
-648    output.fit_parameters = result
-649
-650    output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel())))
-651    output.dof = x.shape[-1] - n_parms
-652    output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof)
-653
-654    return output
+            
    452def total_least_squares(x, y, func, silent=False, **kwargs):
+453    r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
+454
+455    Parameters
+456    ----------
+457    x : list
+458        list of Obs, or a tuple of lists of Obs
+459    y : list
+460        list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
+461    func : object
+462        func has to be of the form
+463
+464        ```python
+465        import autograd.numpy as anp
+466
+467        def func(a, x):
+468            return a[0] + a[1] * x + a[2] * anp.sinh(x)
+469        ```
+470
+471        For multiple x values func can be of the form
+472
+473        ```python
+474        def func(a, x):
+475            (x1, x2) = x
+476            return a[0] * x1 ** 2 + a[1] * x2
+477        ```
+478
+479        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
+480        will not work.
+481    silent : bool, optional
+482        If true all output to the console is omitted (default False).
+483    initial_guess : list
+484        can provide an initial guess for the input parameters. Relevant for non-linear
+485        fits with many parameters.
+486    expected_chisquare : bool
+487        If true prints the expected chisquare which is
+488        corrected by effects caused by correlated input data.
+489        This can take a while as the full correlation matrix
+490        has to be calculated (default False).
+491    num_grad : bool
+492        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
+493
+494    Notes
+495    -----
+496    Based on the orthogonal distance regression module of scipy.
+497
+498    Returns
+499    -------
+500    output : Fit_result
+501        Parameters and information on the fitted result.
+502    '''
+503
+504    output = Fit_result()
+505
+506    output.fit_function = func
+507
+508    x = np.array(x)
+509
+510    x_shape = x.shape
+511
+512    if kwargs.get('num_grad') is True:
+513        jacobian = num_jacobian
+514        hessian = num_hessian
+515    else:
+516        jacobian = auto_jacobian
+517        hessian = auto_hessian
+518
+519    if not callable(func):
+520        raise TypeError('func has to be a function.')
+521
+522    for i in range(42):
+523        try:
+524            func(np.arange(i), x.T[0])
+525        except TypeError:
+526            continue
+527        except IndexError:
+528            continue
+529        else:
+530            break
+531    else:
+532        raise RuntimeError("Fit function is not valid.")
+533
+534    n_parms = i
+535    if not silent:
+536        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
+537
+538    x_f = np.vectorize(lambda o: o.value)(x)
+539    dx_f = np.vectorize(lambda o: o.dvalue)(x)
+540    y_f = np.array([o.value for o in y])
+541    dy_f = np.array([o.dvalue for o in y])
+542
+543    if np.any(np.asarray(dx_f) <= 0.0):
+544        raise Exception('No x errors available, run the gamma method first.')
+545
+546    if np.any(np.asarray(dy_f) <= 0.0):
+547        raise Exception('No y errors available, run the gamma method first.')
+548
+549    if 'initial_guess' in kwargs:
+550        x0 = kwargs.get('initial_guess')
+551        if len(x0) != n_parms:
+552            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
+553    else:
+554        x0 = [1] * n_parms
+555
+556    data = RealData(x_f, y_f, sx=dx_f, sy=dy_f)
+557    model = Model(func)
+558    odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps)
+559    odr.set_job(fit_type=0, deriv=1)
+560    out = odr.run()
+561
+562    output.residual_variance = out.res_var
+563
+564    output.method = 'ODR'
+565
+566    output.message = out.stopreason
+567
+568    output.xplus = out.xplus
+569
+570    if not silent:
+571        print('Method: ODR')
+572        print(*out.stopreason)
+573        print('Residual variance:', output.residual_variance)
+574
+575    if out.info > 3:
+576        raise Exception('The minimization procedure did not converge.')
+577
+578    m = x_f.size
+579
+580    def odr_chisquare(p):
+581        model = func(p[:n_parms], p[n_parms:].reshape(x_shape))
+582        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2)
+583        return chisq
+584
+585    if kwargs.get('expected_chisquare') is True:
+586        W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel()))))
+587
+588        if kwargs.get('covariance') is not None:
+589            cov = kwargs.get('covariance')
+590        else:
+591            cov = covariance(np.concatenate((y, x.ravel())))
+592
+593        number_of_x_parameters = int(m / x_f.shape[-1])
+594
+595        old_jac = jacobian(func)(out.beta, out.xplus)
+596        fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0)))
+597        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])))
+598        new_jac = np.concatenate((fused_row1, fused_row2), axis=1)
+599
+600        A = W @ new_jac
+601        P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
+602        expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W)
+603        if expected_chisquare <= 0.0:
+604            warnings.warn("Negative expected_chisquare.", RuntimeWarning)
+605            expected_chisquare = np.abs(expected_chisquare)
+606        output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare
+607        if not silent:
+608            print('chisquare/expected_chisquare:',
+609                  output.chisquare_by_expected_chisquare)
+610
+611    fitp = out.beta
+612    try:
+613        hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel())))
+614    except TypeError:
+615        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
+616
+617    def odr_chisquare_compact_x(d):
+618        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
+619        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)
+620        return chisq
+621
+622    jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel())))
+623
+624    # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv
+625    try:
+626        deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:])
+627    except np.linalg.LinAlgError:
+628        raise Exception("Cannot invert hessian matrix.")
+629
+630    def odr_chisquare_compact_y(d):
+631        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
+632        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)
+633        return chisq
+634
+635    jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f)))
+636
+637    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
+638    try:
+639        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:])
+640    except np.linalg.LinAlgError:
+641        raise Exception("Cannot invert hessian matrix.")
+642
+643    result = []
+644    for i in range(n_parms):
+645        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])))
+646
+647    output.fit_parameters = result
+648
+649    output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel())))
+650    output.dof = x.shape[-1] - n_parms
+651    output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof)
+652
+653    return output
 
    
 
 
@@ -1855,35 +1852,35 @@ Parameters and information on the fitted result.
    - 
    657def fit_lin(x, y, **kwargs):
-658    """Performs a linear fit to y = n + m * x and returns two Obs n, m.
-659
-660    Parameters
-661    ----------
-662    x : list
-663        Can either be a list of floats in which case no xerror is assumed, or
-664        a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
-665    y : list
-666        List of Obs, the dvalues of the Obs are used as yerror for the fit.
-667
-668    Returns
-669    -------
-670    fit_parameters : list[Obs]
-671        LIist of fitted observables.
-672    """
-673
-674    def f(a, x):
-675        y = a[0] + a[1] * x
-676        return y
-677
-678    if all(isinstance(n, Obs) for n in x):
-679        out = total_least_squares(x, y, f, **kwargs)
-680        return out.fit_parameters
-681    elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
-682        out = least_squares(x, y, f, **kwargs)
-683        return out.fit_parameters
-684    else:
-685        raise Exception('Unsupported types for x')
+            
    656def fit_lin(x, y, **kwargs):
+657    """Performs a linear fit to y = n + m * x and returns two Obs n, m.
+658
+659    Parameters
+660    ----------
+661    x : list
+662        Can either be a list of floats in which case no xerror is assumed, or
+663        a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
+664    y : list
+665        List of Obs, the dvalues of the Obs are used as yerror for the fit.
+666
+667    Returns
+668    -------
+669    fit_parameters : list[Obs]
+670        LIist of fitted observables.
+671    """
+672
+673    def f(a, x):
+674        y = a[0] + a[1] * x
+675        return y
+676
+677    if all(isinstance(n, Obs) for n in x):
+678        out = total_least_squares(x, y, f, **kwargs)
+679        return out.fit_parameters
+680    elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
+681        out = least_squares(x, y, f, **kwargs)
+682        return out.fit_parameters
+683    else:
+684        raise Exception('Unsupported types for x')
 
    
 
 
@@ -1920,34 +1917,34 @@ LIist of fitted observables.
    - 
    688def qqplot(x, o_y, func, p, title=""):
-689    """Generates a quantile-quantile plot of the fit result which can be used to
-690       check if the residuals of the fit are gaussian distributed.
-691
-692    Returns
-693    -------
-694    None
-695    """
-696
-697    residuals = []
-698    for i_x, i_y in zip(x, o_y):
-699        residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
-700    residuals = sorted(residuals)
-701    my_y = [o.value for o in residuals]
-702    probplot = scipy.stats.probplot(my_y)
-703    my_x = probplot[0][0]
-704    plt.figure(figsize=(8, 8 / 1.618))
-705    plt.errorbar(my_x, my_y, fmt='o')
-706    fit_start = my_x[0]
-707    fit_stop = my_x[-1]
-708    samples = np.arange(fit_start, fit_stop, 0.01)
-709    plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
-710    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='-')
-711
-712    plt.xlabel('Theoretical quantiles')
-713    plt.ylabel('Ordered Values')
-714    plt.legend(title=title)
-715    plt.draw()
+            
    687def qqplot(x, o_y, func, p, title=""):
+688    """Generates a quantile-quantile plot of the fit result which can be used to
+689       check if the residuals of the fit are gaussian distributed.
+690
+691    Returns
+692    -------
+693    None
+694    """
+695
+696    residuals = []
+697    for i_x, i_y in zip(x, o_y):
+698        residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
+699    residuals = sorted(residuals)
+700    my_y = [o.value for o in residuals]
+701    probplot = scipy.stats.probplot(my_y)
+702    my_x = probplot[0][0]
+703    plt.figure(figsize=(8, 8 / 1.618))
+704    plt.errorbar(my_x, my_y, fmt='o')
+705    fit_start = my_x[0]
+706    fit_stop = my_x[-1]
+707    samples = np.arange(fit_start, fit_stop, 0.01)
+708    plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
+709    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='-')
+710
+711    plt.xlabel('Theoretical quantiles')
+712    plt.ylabel('Ordered Values')
+713    plt.legend(title=title)
+714    plt.draw()
 
    
 
 
@@ -1974,41 +1971,41 @@ LIist of fitted observables.
    - 
    718def residual_plot(x, y, func, fit_res, title=""):
-719    """Generates a plot which compares the fit to the data and displays the corresponding residuals
-720
-721    For uncorrelated data the residuals are expected to be distributed ~N(0,1).
-722
-723    Returns
-724    -------
-725    None
-726    """
-727    sorted_x = sorted(x)
-728    xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
-729    xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
-730    x_samples = np.arange(xstart, xstop + 0.01, 0.01)
-731
-732    plt.figure(figsize=(8, 8 / 1.618))
-733    gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
-734    ax0 = plt.subplot(gs[0])
-735    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')
-736    ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
-737    ax0.set_xticklabels([])
-738    ax0.set_xlim([xstart, xstop])
-739    ax0.set_xticklabels([])
-740    ax0.legend(title=title)
-741
-742    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])
-743    ax1 = plt.subplot(gs[1])
-744    ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
-745    ax1.tick_params(direction='out')
-746    ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
-747    ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
-748    ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
-749    ax1.set_xlim([xstart, xstop])
-750    ax1.set_ylabel('Residuals')
-751    plt.subplots_adjust(wspace=None, hspace=None)
-752    plt.draw()
+            
    717def residual_plot(x, y, func, fit_res, title=""):
+718    """Generates a plot which compares the fit to the data and displays the corresponding residuals
+719
+720    For uncorrelated data the residuals are expected to be distributed ~N(0,1).
+721
+722    Returns
+723    -------
+724    None
+725    """
+726    sorted_x = sorted(x)
+727    xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
+728    xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
+729    x_samples = np.arange(xstart, xstop + 0.01, 0.01)
+730
+731    plt.figure(figsize=(8, 8 / 1.618))
+732    gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
+733    ax0 = plt.subplot(gs[0])
+734    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')
+735    ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
+736    ax0.set_xticklabels([])
+737    ax0.set_xlim([xstart, xstop])
+738    ax0.set_xticklabels([])
+739    ax0.legend(title=title)
+740
+741    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])
+742    ax1 = plt.subplot(gs[1])
+743    ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
+744    ax1.tick_params(direction='out')
+745    ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
+746    ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
+747    ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
+748    ax1.set_xlim([xstart, xstop])
+749    ax1.set_ylabel('Residuals')
+750    plt.subplots_adjust(wspace=None, hspace=None)
+751    plt.draw()
 
    
 
 
@@ -2036,28 +2033,28 @@ LIist of fitted observables.
    - 
    755def error_band(x, func, beta):
-756    """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
-757
-758    Returns
-759    -------
-760    err : np.array(Obs)
-761        Error band for an array of sample values x
-762    """
-763    cov = covariance(beta)
-764    if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
-765        warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
-766
-767    deriv = []
-768    for i, item in enumerate(x):
-769        deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
-770
-771    err = []
-772    for i, item in enumerate(x):
-773        err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
-774    err = np.array(err)
-775
-776    return err
+            
    754def error_band(x, func, beta):
+755    """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
+756
+757    Returns
+758    -------
+759    err : np.array(Obs)
+760        Error band for an array of sample values x
+761    """
+762    cov = covariance(beta)
+763    if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
+764        warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
+765
+766    deriv = []
+767    for i, item in enumerate(x):
+768        deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
+769
+770    err = []
+771    for i, item in enumerate(x):
+772        err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
+773    err = np.array(err)
+774
+775    return err
 
    
 
 
@@ -2084,48 +2081,48 @@ Error band for an array of sample values x
    - 
    779def ks_test(objects=None):
-780    """Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
-781
-782    Parameters
-783    ----------
-784    objects : list
-785        List of fit results to include in the analysis (optional).
-786
-787    Returns
-788    -------
-789    None
-790    """
-791
-792    if objects is None:
-793        obs_list = []
-794        for obj in gc.get_objects():
-795            if isinstance(obj, Fit_result):
-796                obs_list.append(obj)
-797    else:
-798        obs_list = objects
-799
-800    p_values = [o.p_value for o in obs_list]
-801
-802    bins = len(p_values)
-803    x = np.arange(0, 1.001, 0.001)
-804    plt.plot(x, x, 'k', zorder=1)
-805    plt.xlim(0, 1)
-806    plt.ylim(0, 1)
-807    plt.xlabel('p-value')
-808    plt.ylabel('Cumulative probability')
-809    plt.title(str(bins) + ' p-values')
-810
-811    n = np.arange(1, bins + 1) / np.float64(bins)
-812    Xs = np.sort(p_values)
-813    plt.step(Xs, n)
-814    diffs = n - Xs
-815    loc_max_diff = np.argmax(np.abs(diffs))
-816    loc = Xs[loc_max_diff]
-817    plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
-818    plt.draw()
-819
-820    print(scipy.stats.kstest(p_values, 'uniform'))
+            
    778def ks_test(objects=None):
+779    """Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
+780
+781    Parameters
+782    ----------
+783    objects : list
+784        List of fit results to include in the analysis (optional).
+785
+786    Returns
+787    -------
+788    None
+789    """
+790
+791    if objects is None:
+792        obs_list = []
+793        for obj in gc.get_objects():
+794            if isinstance(obj, Fit_result):
+795                obs_list.append(obj)
+796    else:
+797        obs_list = objects
+798
+799    p_values = [o.p_value for o in obs_list]
+800
+801    bins = len(p_values)
+802    x = np.arange(0, 1.001, 0.001)
+803    plt.plot(x, x, 'k', zorder=1)
+804    plt.xlim(0, 1)
+805    plt.ylim(0, 1)
+806    plt.xlabel('p-value')
+807    plt.ylabel('Cumulative probability')
+808    plt.title(str(bins) + ' p-values')
+809
+810    n = np.arange(1, bins + 1) / np.float64(bins)
+811    Xs = np.sort(p_values)
+812    plt.step(Xs, n)
+813    diffs = n - Xs
+814    loc_max_diff = np.argmax(np.abs(diffs))
+815    loc = Xs[loc_max_diff]
+816    plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
+817    plt.draw()
+818
+819    print(scipy.stats.kstest(p_values, 'uniform'))
 
    
 
 
diff --git a/docs/search.js b/docs/search.js
index 6757d66d..2a5638c2 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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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. [arXiv:2209.14371 [hep-lat]].
      • \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
    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
    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.
    \n\n
    Direct visualizations of the performed fits can be triggered via resplot=True or qqplot=True. For all available options 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 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
    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 GEVP\nmatrix at every timeslice. Other dependency (eg. spatial) are not supported.
    \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
      • \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 indentified for this correlator.
      • \n
    \n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "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.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.
        • \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
      • \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
    \n", "signature": "(self, t0, ts=None, sort='Eigenvalue', **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'):", "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
    • partity (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: symmetric, improved, log, default: symmetric
      • \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 periodicitiy 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-periodicitiy 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
    • 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,\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.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.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.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.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "
    \n"}, "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.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).\nAt the moment this option only works for prior==None and when no method is given.
      • \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_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 source and sink.\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.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
    >>> 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
    \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.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 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 - 0.3\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
    • 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": "(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, **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
      • \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.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "
    Read sfcf c format 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',\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.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.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.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.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/3.10.10/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.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.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.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.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.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.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.version": {"fullname": "pyerrors.version", "modulename": "pyerrors.version", "kind": "module", "doc": "
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+    /** pdoc search index */const docs = {"version": "0.9.5", "fields": ["qualname", "fullname", "annotation", "default_value", "signature", "bases", "doc"], "ref": "fullname", "documentStore": {"docs": {"pyerrors": {"fullname": "pyerrors", "modulename": "pyerrors", "kind": "module", "doc": "
    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. [arXiv:2209.14371 [hep-lat]].
      • \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
    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
    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.
    \n\n
    Direct visualizations of the performed fits can be triggered via resplot=True or qqplot=True. For all available options 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 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
    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 GEVP\nmatrix at every timeslice. Other dependency (eg. spatial) are not supported.
    \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
      • \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 indentified for this correlator.
      • \n
    \n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "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.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.
        • \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
      • \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
    \n", "signature": "(self, t0, ts=None, sort='Eigenvalue', **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'):", "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
    • partity (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: symmetric, improved, log, default: symmetric
      • \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 periodicitiy 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-periodicitiy 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
    • 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,\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.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.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.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.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "
    \n"}, "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.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_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 source and sink.\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.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
    >>> 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
    \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.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 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 - 0.3\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
    • 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": "(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, **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
      • \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.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "
    Read sfcf c format 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',\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.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.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.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.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/3.10.10/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.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.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.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.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.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.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.version": {"fullname": "pyerrors.version", "modulename": "pyerrors.version", "kind": "module", "doc": "
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