diff --git a/docs/pyerrors/fits.html b/docs/pyerrors/fits.html index a8802145..e9ebafa7 100644 --- a/docs/pyerrors/fits.html +++ b/docs/pyerrors/fits.html @@ -109,838 +109,1047 @@ -
1import gc - 2from collections.abc import Sequence - 3import warnings - 4import numpy as np - 5import autograd.numpy as anp - 6import scipy.optimize - 7import scipy.stats - 8import matplotlib.pyplot as plt - 9from matplotlib import gridspec - 10from scipy.odr import ODR, Model, RealData - 11import iminuit - 12from autograd import jacobian as auto_jacobian - 13from autograd import hessian as auto_hessian - 14from autograd import elementwise_grad as egrad - 15from numdifftools import Jacobian as num_jacobian - 16from numdifftools import Hessian as num_hessian - 17from .obs import Obs, derived_observable, covariance, cov_Obs - 18 - 19 - 20class Fit_result(Sequence): - 21 """Represents fit results. - 22 - 23 Attributes - 24 ---------- - 25 fit_parameters : list - 26 results for the individual fit parameters, - 27 also accessible via indices. - 28 chisquare_by_dof : float - 29 reduced chisquare. - 30 p_value : float - 31 p-value of the fit - 32 t2_p_value : float - 33 Hotelling t-squared p-value for correlated fits. - 34 """ - 35 - 36 def __init__(self): - 37 self.fit_parameters = None - 38 - 39 def __getitem__(self, idx): - 40 return self.fit_parameters[idx] - 41 - 42 def __len__(self): - 43 return len(self.fit_parameters) - 44 - 45 def gamma_method(self, **kwargs): - 46 """Apply the gamma method to all fit parameters""" - 47 [o.gamma_method(**kwargs) for o in self.fit_parameters] - 48 - 49 gm = gamma_method - 50 - 51 def __str__(self): - 52 my_str = 'Goodness of fit:\n' - 53 if hasattr(self, 'chisquare_by_dof'): - 54 my_str += '\u03C7\u00b2/d.o.f. = ' + f'{self.chisquare_by_dof:2.6f}' + '\n' - 55 elif hasattr(self, 'residual_variance'): - 56 my_str += 'residual variance = ' + f'{self.residual_variance:2.6f}' + '\n' - 57 if hasattr(self, 'chisquare_by_expected_chisquare'): - 58 my_str += '\u03C7\u00b2/\u03C7\u00b2exp = ' + f'{self.chisquare_by_expected_chisquare:2.6f}' + '\n' - 59 if hasattr(self, 'p_value'): - 60 my_str += 'p-value = ' + f'{self.p_value:2.4f}' + '\n' - 61 if hasattr(self, 't2_p_value'): - 62 my_str += 't\u00B2p-value = ' + f'{self.t2_p_value:2.4f}' + '\n' - 63 my_str += 'Fit parameters:\n' - 64 for i_par, par in enumerate(self.fit_parameters): - 65 my_str += str(i_par) + '\t' + ' ' * int(par >= 0) + str(par).rjust(int(par < 0.0)) + '\n' - 66 return my_str - 67 - 68 def __repr__(self): - 69 m = max(map(len, list(self.__dict__.keys()))) + 1 - 70 return '\n'.join([key.rjust(m) + ': ' + repr(value) for key, value in sorted(self.__dict__.items())]) - 71 - 72 - 73def least_squares(x, y, func, priors=None, silent=False, **kwargs): - 74 r'''Performs a non-linear fit to y = func(x). - 75 - 76 Parameters - 77 ---------- - 78 x : list - 79 list of floats. - 80 y : list - 81 list of Obs. - 82 func : object - 83 fit function, has to be of the form - 84 - 85 ```python - 86 import autograd.numpy as anp - 87 - 88 def func(a, x): - 89 return a[0] + a[1] * x + a[2] * anp.sinh(x) - 90 ``` - 91 - 92 For multiple x values func can be of the form - 93 - 94 ```python - 95 def func(a, x): - 96 (x1, x2) = x - 97 return a[0] * x1 ** 2 + a[1] * x2 - 98 ``` - 99 -100 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation -101 will not work. -102 priors : list, optional -103 priors has to be a list with an entry for every parameter in the fit. The entries can either be -104 Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like -105 0.548(23), 500(40) or 0.5(0.4) -106 silent : bool, optional -107 If true all output to the console is omitted (default False). -108 initial_guess : list -109 can provide an initial guess for the input parameters. Relevant for -110 non-linear fits with many parameters. In case of correlated fits the guess is used to perform -111 an uncorrelated fit which then serves as guess for the correlated fit. -112 method : str, optional -113 can be used to choose an alternative method for the minimization of chisquare. -114 The possible methods are the ones which can be used for scipy.optimize.minimize and -115 migrad of iminuit. If no method is specified, Levenberg-Marquard is used. -116 Reliable alternatives are migrad, Powell and Nelder-Mead. -117 correlated_fit : bool -118 If True, use the full inverse covariance matrix in the definition of the chisquare cost function. -119 For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`. -120 In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix). -121 This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning). -122 At the moment this option only works for `prior==None` and when no `method` is given. -123 expected_chisquare : bool -124 If True estimates the expected chisquare which is -125 corrected by effects caused by correlated input data (default False). -126 resplot : bool -127 If True, a plot which displays fit, data and residuals is generated (default False). -128 qqplot : bool -129 If True, a quantile-quantile plot of the fit result is generated (default False). -130 num_grad : bool -131 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). -132 -133 Returns -134 ------- -135 output : Fit_result -136 Parameters and information on the fitted result. -137 ''' -138 if priors is not None: -139 return _prior_fit(x, y, func, priors, silent=silent, **kwargs) -140 else: -141 return _standard_fit(x, y, func, silent=silent, **kwargs) -142 -143 -144def total_least_squares(x, y, func, silent=False, **kwargs): -145 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. -146 -147 Parameters -148 ---------- -149 x : list -150 list of Obs, or a tuple of lists of Obs -151 y : list -152 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. -153 func : object -154 func has to be of the form -155 -156 ```python -157 import autograd.numpy as anp -158 -159 def func(a, x): -160 return a[0] + a[1] * x + a[2] * anp.sinh(x) -161 ``` -162 -163 For multiple x values func can be of the form -164 -165 ```python -166 def func(a, x): -167 (x1, x2) = x -168 return a[0] * x1 ** 2 + a[1] * x2 -169 ``` -170 -171 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation -172 will not work. -173 silent : bool, optional -174 If true all output to the console is omitted (default False). -175 initial_guess : list -176 can provide an initial guess for the input parameters. Relevant for non-linear -177 fits with many parameters. -178 expected_chisquare : bool -179 If true prints the expected chisquare which is -180 corrected by effects caused by correlated input data. -181 This can take a while as the full correlation matrix -182 has to be calculated (default False). -183 num_grad : bool -184 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). -185 -186 Notes -187 ----- -188 Based on the orthogonal distance regression module of scipy. -189 -190 Returns -191 ------- -192 output : Fit_result -193 Parameters and information on the fitted result. -194 ''' -195 -196 output = Fit_result() -197 -198 output.fit_function = func -199 -200 x = np.array(x) -201 -202 x_shape = x.shape -203 -204 if kwargs.get('num_grad') is True: -205 jacobian = num_jacobian -206 hessian = num_hessian -207 else: -208 jacobian = auto_jacobian -209 hessian = auto_hessian -210 -211 if not callable(func): -212 raise TypeError('func has to be a function.') -213 -214 for i in range(42): -215 try: -216 func(np.arange(i), x.T[0]) -217 except TypeError: -218 continue -219 except IndexError: -220 continue -221 else: -222 break -223 else: -224 raise RuntimeError("Fit function is not valid.") -225 -226 n_parms = i -227 if not silent: -228 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) -229 -230 x_f = np.vectorize(lambda o: o.value)(x) -231 dx_f = np.vectorize(lambda o: o.dvalue)(x) -232 y_f = np.array([o.value for o in y]) -233 dy_f = np.array([o.dvalue for o in y]) -234 -235 if np.any(np.asarray(dx_f) <= 0.0): -236 raise Exception('No x errors available, run the gamma method first.') -237 -238 if np.any(np.asarray(dy_f) <= 0.0): -239 raise Exception('No y errors available, run the gamma method first.') -240 -241 if 'initial_guess' in kwargs: -242 x0 = kwargs.get('initial_guess') -243 if len(x0) != n_parms: -244 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) -245 else: -246 x0 = [1] * n_parms -247 -248 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) -249 model = Model(func) -250 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) -251 odr.set_job(fit_type=0, deriv=1) -252 out = odr.run() -253 -254 output.residual_variance = out.res_var -255 -256 output.method = 'ODR' -257 -258 output.message = out.stopreason -259 -260 output.xplus = out.xplus -261 -262 if not silent: -263 print('Method: ODR') -264 print(*out.stopreason) -265 print('Residual variance:', output.residual_variance) -266 -267 if out.info > 3: -268 raise Exception('The minimization procedure did not converge.') -269 -270 m = x_f.size -271 -272 def odr_chisquare(p): -273 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) -274 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) -275 return chisq -276 -277 if kwargs.get('expected_chisquare') is True: -278 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) -279 -280 if kwargs.get('covariance') is not None: -281 cov = kwargs.get('covariance') -282 else: -283 cov = covariance(np.concatenate((y, x.ravel()))) -284 -285 number_of_x_parameters = int(m / x_f.shape[-1]) -286 -287 old_jac = jacobian(func)(out.beta, out.xplus) -288 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) -289 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]))) -290 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) -291 -292 A = W @ new_jac -293 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T -294 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) -295 if expected_chisquare <= 0.0: -296 warnings.warn("Negative expected_chisquare.", RuntimeWarning) -297 expected_chisquare = np.abs(expected_chisquare) -298 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare -299 if not silent: -300 print('chisquare/expected_chisquare:', -301 output.chisquare_by_expected_chisquare) -302 -303 fitp = out.beta -304 try: -305 hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel()))) -306 except TypeError: -307 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None -308 -309 def odr_chisquare_compact_x(d): -310 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) -311 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) -312 return chisq -313 -314 jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) -315 -316 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv -317 try: -318 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) -319 except np.linalg.LinAlgError: -320 raise Exception("Cannot invert hessian matrix.") -321 -322 def odr_chisquare_compact_y(d): -323 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) -324 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) -325 return chisq -326 -327 jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f))) -328 -329 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv -330 try: -331 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) -332 except np.linalg.LinAlgError: -333 raise Exception("Cannot invert hessian matrix.") -334 -335 result = [] -336 for i in range(n_parms): -337 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]))) -338 -339 output.fit_parameters = result -340 -341 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) -342 output.dof = x.shape[-1] - n_parms -343 output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof) -344 -345 return output -346 -347 -348def _prior_fit(x, y, func, priors, silent=False, **kwargs): -349 output = Fit_result() -350 -351 output.fit_function = func -352 -353 x = np.asarray(x) -354 -355 if kwargs.get('num_grad') is True: -356 hessian = num_hessian -357 else: -358 hessian = auto_hessian -359 -360 if not callable(func): -361 raise TypeError('func has to be a function.') -362 -363 for i in range(100): -364 try: -365 func(np.arange(i), 0) -366 except TypeError: -367 continue -368 except IndexError: -369 continue -370 else: -371 break -372 else: -373 raise RuntimeError("Fit function is not valid.") -374 -375 n_parms = i -376 -377 if n_parms != len(priors): -378 raise Exception('Priors does not have the correct length.') -379 -380 def extract_val_and_dval(string): -381 split_string = string.split('(') -382 if '.' in split_string[0] and '.' not in split_string[1][:-1]: -383 factor = 10 ** -len(split_string[0].partition('.')[2]) -384 else: -385 factor = 1 -386 return float(split_string[0]), float(split_string[1][:-1]) * factor -387 -388 loc_priors = [] -389 for i_n, i_prior in enumerate(priors): -390 if isinstance(i_prior, Obs): -391 loc_priors.append(i_prior) -392 else: -393 loc_val, loc_dval = extract_val_and_dval(i_prior) -394 loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}")) -395 -396 output.priors = loc_priors -397 -398 if not silent: -399 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) -400 -401 y_f = [o.value for o in y] -402 dy_f = [o.dvalue for o in y] -403 -404 if np.any(np.asarray(dy_f) <= 0.0): -405 raise Exception('No y errors available, run the gamma method first.') -406 -407 p_f = [o.value for o in loc_priors] -408 dp_f = [o.dvalue for o in loc_priors] -409 -410 if np.any(np.asarray(dp_f) <= 0.0): -411 raise Exception('No prior errors available, run the gamma method first.') -412 -413 if 'initial_guess' in kwargs: -414 x0 = kwargs.get('initial_guess') -415 if len(x0) != n_parms: -416 raise Exception('Initial guess does not have the correct length.') -417 else: -418 x0 = p_f -419 -420 def chisqfunc(p): -421 model = func(p, x) -422 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((p_f - p) / dp_f) ** 2) -423 return chisq -424 -425 if not silent: -426 print('Method: migrad') -427 -428 m = iminuit.Minuit(chisqfunc, x0) -429 m.errordef = 1 -430 m.print_level = 0 -431 if 'tol' in kwargs: -432 m.tol = kwargs.get('tol') -433 else: -434 m.tol = 1e-4 -435 m.migrad() -436 params = np.asarray(m.values) -437 -438 output.chisquare_by_dof = m.fval / len(x) -439 -440 output.method = 'migrad' -441 -442 if not silent: -443 print('chisquare/d.o.f.:', output.chisquare_by_dof) -444 -445 if not m.fmin.is_valid: -446 raise Exception('The minimization procedure did not converge.') -447 -448 hess = hessian(chisqfunc)(params) -449 hess_inv = np.linalg.pinv(hess) -450 -451 def chisqfunc_compact(d): -452 model = func(d[:n_parms], x) -453 chisq = anp.sum(((d[n_parms: n_parms + len(x)] - model) / dy_f) ** 2) + anp.sum(((d[n_parms + len(x):] - d[:n_parms]) / dp_f) ** 2) -454 return chisq -455 -456 jac_jac = hessian(chisqfunc_compact)(np.concatenate((params, y_f, p_f))) -457 -458 deriv = -hess_inv @ jac_jac[:n_parms, n_parms:] -459 -460 result = [] -461 for i in range(n_parms): -462 result.append(derived_observable(lambda x, **kwargs: (x[0] + np.finfo(np.float64).eps) / (y[0].value + np.finfo(np.float64).eps) * params[i], list(y) + list(loc_priors), man_grad=list(deriv[i]))) -463 -464 output.fit_parameters = result -465 output.chisquare = chisqfunc(np.asarray(params)) -466 -467 if kwargs.get('resplot') is True: -468 residual_plot(x, y, func, result) -469 -470 if kwargs.get('qqplot') is True: -471 qqplot(x, y, func, result) -472 -473 return output -474 -475 -476def _standard_fit(x, y, func, silent=False, **kwargs): -477 -478 output = Fit_result() -479 -480 output.fit_function = func -481 -482 x = np.asarray(x) -483 -484 if kwargs.get('num_grad') is True: -485 jacobian = num_jacobian -486 hessian = num_hessian -487 else: -488 jacobian = auto_jacobian -489 hessian = auto_hessian -490 -491 if x.shape[-1] != len(y): -492 raise Exception('x and y input have to have the same length') -493 -494 if len(x.shape) > 2: -495 raise Exception('Unknown format for x values') -496 -497 if not callable(func): -498 raise TypeError('func has to be a function.') -499 -500 for i in range(42): -501 try: -502 func(np.arange(i), x.T[0]) -503 except TypeError: -504 continue -505 except IndexError: -506 continue -507 else: -508 break -509 else: -510 raise RuntimeError("Fit function is not valid.") -511 -512 n_parms = i -513 -514 if not silent: -515 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) -516 -517 y_f = [o.value for o in y] -518 dy_f = [o.dvalue for o in y] -519 -520 if np.any(np.asarray(dy_f) <= 0.0): -521 raise Exception('No y errors available, run the gamma method first.') -522 -523 if 'initial_guess' in kwargs: -524 x0 = kwargs.get('initial_guess') -525 if len(x0) != n_parms: -526 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) -527 else: -528 x0 = [0.1] * n_parms -529 -530 if kwargs.get('correlated_fit') is True: -531 corr = covariance(y, correlation=True, **kwargs) -532 covdiag = np.diag(1 / np.asarray(dy_f)) -533 condn = np.linalg.cond(corr) -534 if condn > 0.1 / np.finfo(float).eps: -535 raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})") -536 if condn > 1e13: -537 warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning) -538 chol = np.linalg.cholesky(corr) -539 chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True) -540 -541 def chisqfunc_corr(p): -542 model = func(p, x) -543 chisq = anp.sum(anp.dot(chol_inv, (y_f - model)) ** 2) -544 return chisq -545 -546 def chisqfunc(p): -547 model = func(p, x) -548 chisq = anp.sum(((y_f - model) / dy_f) ** 2) -549 return chisq -550 -551 output.method = kwargs.get('method', 'Levenberg-Marquardt') -552 if not silent: -553 print('Method:', output.method) -554 -555 if output.method != 'Levenberg-Marquardt': -556 if output.method == 'migrad': -557 fit_result = iminuit.minimize(chisqfunc, x0, tol=1e-4) # Stopping criterion 0.002 * tol * errordef -558 if kwargs.get('correlated_fit') is True: -559 fit_result = iminuit.minimize(chisqfunc_corr, fit_result.x, tol=1e-4) # Stopping criterion 0.002 * tol * errordef -560 output.iterations = fit_result.nfev -561 else: -562 fit_result = scipy.optimize.minimize(chisqfunc, x0, method=kwargs.get('method'), tol=1e-12) -563 if kwargs.get('correlated_fit') is True: -564 fit_result = scipy.optimize.minimize(chisqfunc_corr, fit_result.x, method=kwargs.get('method'), tol=1e-12) -565 output.iterations = fit_result.nit -566 -567 chisquare = fit_result.fun -568 -569 else: -570 if kwargs.get('correlated_fit') is True: -571 def chisqfunc_residuals_corr(p): -572 model = func(p, x) -573 chisq = anp.dot(chol_inv, (y_f - model)) -574 return chisq -575 -576 def chisqfunc_residuals(p): -577 model = func(p, x) -578 chisq = ((y_f - model) / dy_f) -579 return chisq -580 -581 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) -582 if kwargs.get('correlated_fit') is True: -583 fit_result = scipy.optimize.least_squares(chisqfunc_residuals_corr, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) -584 -585 chisquare = np.sum(fit_result.fun ** 2) -586 if kwargs.get('correlated_fit') is True: -587 assert np.isclose(chisquare, chisqfunc_corr(fit_result.x), atol=1e-14) -588 else: -589 assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14) -590 -591 output.iterations = fit_result.nfev -592 -593 if not fit_result.success: -594 raise Exception('The minimization procedure did not converge.') -595 -596 if x.shape[-1] - n_parms > 0: -597 output.chisquare_by_dof = chisquare / (x.shape[-1] - n_parms) -598 else: -599 output.chisquare_by_dof = float('nan') -600 -601 output.message = fit_result.message -602 if not silent: -603 print(fit_result.message) -604 print('chisquare/d.o.f.:', output.chisquare_by_dof) -605 -606 if kwargs.get('expected_chisquare') is True: -607 if kwargs.get('correlated_fit') is not True: -608 W = np.diag(1 / np.asarray(dy_f)) -609 cov = covariance(y) -610 A = W @ jacobian(func)(fit_result.x, x) -611 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T -612 expected_chisquare = np.trace((np.identity(x.shape[-1]) - P_phi) @ W @ cov @ W) -613 output.chisquare_by_expected_chisquare = chisquare / expected_chisquare -614 if not silent: -615 print('chisquare/expected_chisquare:', -616 output.chisquare_by_expected_chisquare) -617 -618 fitp = fit_result.x -619 try: -620 if kwargs.get('correlated_fit') is True: -621 hess = hessian(chisqfunc_corr)(fitp) -622 else: -623 hess = hessian(chisqfunc)(fitp) -624 except TypeError: -625 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None -626 -627 if kwargs.get('correlated_fit') is True: -628 def chisqfunc_compact(d): -629 model = func(d[:n_parms], x) -630 chisq = anp.sum(anp.dot(chol_inv, (d[n_parms:] - model)) ** 2) -631 return chisq -632 -633 else: -634 def chisqfunc_compact(d): -635 model = func(d[:n_parms], x) -636 chisq = anp.sum(((d[n_parms:] - model) / dy_f) ** 2) -637 return chisq -638 -639 jac_jac = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f))) -640 -641 # Compute hess^{-1} @ jac_jac[:n_parms, n_parms:] using LAPACK dgesv -642 try: -643 deriv = -scipy.linalg.solve(hess, jac_jac[:n_parms, n_parms:]) -644 except np.linalg.LinAlgError: -645 raise Exception("Cannot invert hessian matrix.") -646 -647 result = [] -648 for i in range(n_parms): -649 result.append(derived_observable(lambda x, **kwargs: (x[0] + np.finfo(np.float64).eps) / (y[0].value + np.finfo(np.float64).eps) * fit_result.x[i], list(y), man_grad=list(deriv[i]))) -650 -651 output.fit_parameters = result -652 -653 output.chisquare = chisquare -654 output.dof = x.shape[-1] - n_parms -655 output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof) -656 # Hotelling t-squared p-value for correlated fits. -657 if kwargs.get('correlated_fit') is True: -658 n_cov = np.min(np.vectorize(lambda x: x.N)(y)) -659 output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare, -660 output.dof, n_cov - output.dof) -661 -662 if kwargs.get('resplot') is True: -663 residual_plot(x, y, func, result) -664 -665 if kwargs.get('qqplot') is True: -666 qqplot(x, y, func, result) -667 -668 return output -669 -670 -671def fit_lin(x, y, **kwargs): -672 """Performs a linear fit to y = n + m * x and returns two Obs n, m. -673 -674 Parameters -675 ---------- -676 x : list -677 Can either be a list of floats in which case no xerror is assumed, or -678 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. -679 y : list -680 List of Obs, the dvalues of the Obs are used as yerror for the fit. -681 -682 Returns -683 ------- -684 fit_parameters : list[Obs] -685 LIist of fitted observables. -686 """ -687 -688 def f(a, x): -689 y = a[0] + a[1] * x -690 return y -691 -692 if all(isinstance(n, Obs) for n in x): -693 out = total_least_squares(x, y, f, **kwargs) -694 return out.fit_parameters -695 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): -696 out = least_squares(x, y, f, **kwargs) -697 return out.fit_parameters -698 else: -699 raise Exception('Unsupported types for x') -700 -701 -702def qqplot(x, o_y, func, p): -703 """Generates a quantile-quantile plot of the fit result which can be used to -704 check if the residuals of the fit are gaussian distributed. -705 -706 Returns -707 ------- -708 None -709 """ -710 -711 residuals = [] -712 for i_x, i_y in zip(x, o_y): -713 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) -714 residuals = sorted(residuals) -715 my_y = [o.value for o in residuals] -716 probplot = scipy.stats.probplot(my_y) -717 my_x = probplot[0][0] -718 plt.figure(figsize=(8, 8 / 1.618)) -719 plt.errorbar(my_x, my_y, fmt='o') -720 fit_start = my_x[0] -721 fit_stop = my_x[-1] -722 samples = np.arange(fit_start, fit_stop, 0.01) -723 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') -724 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='-') -725 -726 plt.xlabel('Theoretical quantiles') -727 plt.ylabel('Ordered Values') -728 plt.legend() -729 plt.draw() -730 -731 -732def residual_plot(x, y, func, fit_res): -733 """Generates a plot which compares the fit to the data and displays the corresponding residuals -734 -735 Returns -736 ------- -737 None -738 """ -739 sorted_x = sorted(x) -740 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) -741 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) -742 x_samples = np.arange(xstart, xstop + 0.01, 0.01) -743 -744 plt.figure(figsize=(8, 8 / 1.618)) -745 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) -746 ax0 = plt.subplot(gs[0]) -747 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') -748 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) -749 ax0.set_xticklabels([]) -750 ax0.set_xlim([xstart, xstop]) -751 ax0.set_xticklabels([]) -752 ax0.legend() -753 -754 residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], x)) / np.asarray([o.dvalue for o in y]) -755 ax1 = plt.subplot(gs[1]) -756 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) -757 ax1.tick_params(direction='out') -758 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) -759 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") -760 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') -761 ax1.set_xlim([xstart, xstop]) -762 ax1.set_ylabel('Residuals') -763 plt.subplots_adjust(wspace=None, hspace=None) -764 plt.draw() -765 -766 -767def error_band(x, func, beta): -768 """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta. -769 -770 Returns -771 ------- -772 err : np.array(Obs) -773 Error band for an array of sample values x -774 """ -775 cov = covariance(beta) -776 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): -777 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) -778 -779 deriv = [] -780 for i, item in enumerate(x): -781 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) -782 -783 err = [] -784 for i, item in enumerate(x): -785 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) -786 err = np.array(err) -787 -788 return err -789 -790 -791def ks_test(objects=None): -792 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. -793 -794 Parameters -795 ---------- -796 objects : list -797 List of fit results to include in the analysis (optional). -798 -799 Returns -800 ------- -801 None -802 """ -803 -804 if objects is None: -805 obs_list = [] -806 for obj in gc.get_objects(): -807 if isinstance(obj, Fit_result): -808 obs_list.append(obj) -809 else: -810 obs_list = objects -811 -812 p_values = [o.p_value for o in obs_list] -813 -814 bins = len(p_values) -815 x = np.arange(0, 1.001, 0.001) -816 plt.plot(x, x, 'k', zorder=1) -817 plt.xlim(0, 1) -818 plt.ylim(0, 1) -819 plt.xlabel('p-value') -820 plt.ylabel('Cumulative probability') -821 plt.title(str(bins) + ' p-values') -822 -823 n = np.arange(1, bins + 1) / np.float64(bins) -824 Xs = np.sort(p_values) -825 plt.step(Xs, n) -826 diffs = n - Xs -827 loc_max_diff = np.argmax(np.abs(diffs)) -828 loc = Xs[loc_max_diff] -829 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) -830 plt.draw() -831 -832 print(scipy.stats.kstest(p_values, 'uniform')) +@@ -1114,81 +1323,119 @@ Hotelling t-squared p-value for correlated fits.1import gc + 2from collections.abc import Sequence + 3import warnings + 4import numpy as np + 5import autograd.numpy as anp + 6import scipy.optimize + 7import scipy.stats + 8import matplotlib.pyplot as plt + 9from matplotlib import gridspec + 10from scipy.odr import ODR, Model, RealData + 11import iminuit + 12from autograd import jacobian as auto_jacobian + 13from autograd import hessian as auto_hessian + 14from autograd import elementwise_grad as egrad + 15from numdifftools import Jacobian as num_jacobian + 16from numdifftools import Hessian as num_hessian + 17from .obs import Obs, derived_observable, covariance, cov_Obs + 18 + 19 + 20class Fit_result(Sequence): + 21 """Represents fit results. + 22 + 23 Attributes + 24 ---------- + 25 fit_parameters : list + 26 results for the individual fit parameters, + 27 also accessible via indices. + 28 chisquare_by_dof : float + 29 reduced chisquare. + 30 p_value : float + 31 p-value of the fit + 32 t2_p_value : float + 33 Hotelling t-squared p-value for correlated fits. + 34 """ + 35 + 36 def __init__(self): + 37 self.fit_parameters = None + 38 + 39 def __getitem__(self, idx): + 40 return self.fit_parameters[idx] + 41 + 42 def __len__(self): + 43 return len(self.fit_parameters) + 44 + 45 def gamma_method(self, **kwargs): + 46 """Apply the gamma method to all fit parameters""" + 47 [o.gamma_method(**kwargs) for o in self.fit_parameters] + 48 + 49 gm = gamma_method + 50 + 51 def __str__(self): + 52 my_str = 'Goodness of fit:\n' + 53 if hasattr(self, 'chisquare_by_dof'): + 54 my_str += '\u03C7\u00b2/d.o.f. = ' + f'{self.chisquare_by_dof:2.6f}' + '\n' + 55 elif hasattr(self, 'residual_variance'): + 56 my_str += 'residual variance = ' + f'{self.residual_variance:2.6f}' + '\n' + 57 if hasattr(self, 'chisquare_by_expected_chisquare'): + 58 my_str += '\u03C7\u00b2/\u03C7\u00b2exp = ' + f'{self.chisquare_by_expected_chisquare:2.6f}' + '\n' + 59 if hasattr(self, 'p_value'): + 60 my_str += 'p-value = ' + f'{self.p_value:2.4f}' + '\n' + 61 if hasattr(self, 't2_p_value'): + 62 my_str += 't\u00B2p-value = ' + f'{self.t2_p_value:2.4f}' + '\n' + 63 my_str += 'Fit parameters:\n' + 64 for i_par, par in enumerate(self.fit_parameters): + 65 my_str += str(i_par) + '\t' + ' ' * int(par >= 0) + str(par).rjust(int(par < 0.0)) + '\n' + 66 return my_str + 67 + 68 def __repr__(self): + 69 m = max(map(len, list(self.__dict__.keys()))) + 1 + 70 return '\n'.join([key.rjust(m) + ': ' + repr(value) for key, value in sorted(self.__dict__.items())]) + 71 + 72 + 73def least_squares(x, y, func, priors=None, silent=False, **kwargs): + 74 r'''Performs a non-linear fit to y = func(x). + 75 ``` + 76 + 77 Parameters + 78 ---------- + 79 For an uncombined fit: + 80 + 81 x : list + 82 list of floats. + 83 y : list + 84 list of Obs. + 85 func : object + 86 fit function, has to be of the form + 87 + 88 ```python + 89 import autograd.numpy as anp + 90 + 91 def func(a, x): + 92 return a[0] + a[1] * x + a[2] * anp.sinh(x) + 93 ``` + 94 + 95 For multiple x values func can be of the form + 96 + 97 ```python + 98 def func(a, x): + 99 (x1, x2) = x + 100 return a[0] * x1 ** 2 + a[1] * x2 + 101 ``` + 102 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation + 103 will not work. + 104 + 105 OR For a combined fit: + 106 + 107 x : dict + 108 dict of lists. + 109 y : dict + 110 dict of lists of Obs. + 111 funcs : dict + 112 dict of objects + 113 fit functions have to be of the form (here a[0] is the common fit parameter) + 114 ```python + 115 import autograd.numpy as anp + 116 funcs = {"a": func_a, + 117 "b": func_b} + 118 + 119 def func_a(a, x): + 120 return a[1] * anp.exp(-a[0] * x) + 121 + 122 def func_b(a, x): + 123 return a[2] * anp.exp(-a[0] * x) + 124 + 125 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation + 126 will not work. + 127 + 128 priors : list, optional + 129 priors has to be a list with an entry for every parameter in the fit. The entries can either be + 130 Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like + 131 0.548(23), 500(40) or 0.5(0.4) + 132 silent : bool, optional + 133 If true all output to the console is omitted (default False). + 134 initial_guess : list + 135 can provide an initial guess for the input parameters. Relevant for + 136 non-linear fits with many parameters. In case of correlated fits the guess is used to perform + 137 an uncorrelated fit which then serves as guess for the correlated fit. + 138 method : str, optional + 139 can be used to choose an alternative method for the minimization of chisquare. + 140 The possible methods are the ones which can be used for scipy.optimize.minimize and + 141 migrad of iminuit. If no method is specified, Levenberg-Marquard is used. + 142 Reliable alternatives are migrad, Powell and Nelder-Mead. + 143 tol: float, optional + 144 can be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence + 145 to a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly + 146 invalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values + 147 The stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max) + 148 correlated_fit : bool + 149 If True, use the full inverse covariance matrix in the definition of the chisquare cost function. + 150 For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`. + 151 In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix). + 152 This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning). + 153 At the moment this option only works for `prior==None` and when no `method` is given. + 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 if priors is not None: + 170 return _prior_fit(x, y, func, priors, silent=silent, **kwargs) + 171 + 172 elif (type(x) == dict and type(y) == dict and type(func) == dict): + 173 return _combined_fit(x, y, func, silent=silent, **kwargs) + 174 elif (type(x) == dict or type(y) == dict or type(func) == dict): + 175 raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.") + 176 else: + 177 return _standard_fit(x, y, func, silent=silent, **kwargs) + 178 + 179 + 180def total_least_squares(x, y, func, silent=False, **kwargs): + 181 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. + 182 + 183 Parameters + 184 ---------- + 185 x : list + 186 list of Obs, or a tuple of lists of Obs + 187 y : list + 188 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. + 189 func : object + 190 func has to be of the form + 191 + 192 ```python + 193 import autograd.numpy as anp + 194 + 195 def func(a, x): + 196 return a[0] + a[1] * x + a[2] * anp.sinh(x) + 197 ``` + 198 + 199 For multiple x values func can be of the form + 200 + 201 ```python + 202 def func(a, x): + 203 (x1, x2) = x + 204 return a[0] * x1 ** 2 + a[1] * x2 + 205 ``` + 206 + 207 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation + 208 will not work. + 209 silent : bool, optional + 210 If true all output to the console is omitted (default False). + 211 initial_guess : list + 212 can provide an initial guess for the input parameters. Relevant for non-linear + 213 fits with many parameters. + 214 expected_chisquare : bool + 215 If true prints the expected chisquare which is + 216 corrected by effects caused by correlated input data. + 217 This can take a while as the full correlation matrix + 218 has to be calculated (default False). + 219 num_grad : bool + 220 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). + 221 + 222 Notes + 223 ----- + 224 Based on the orthogonal distance regression module of scipy. + 225 + 226 Returns + 227 ------- + 228 output : Fit_result + 229 Parameters and information on the fitted result. + 230 ''' + 231 + 232 output = Fit_result() + 233 + 234 output.fit_function = func + 235 + 236 x = np.array(x) + 237 + 238 x_shape = x.shape + 239 + 240 if kwargs.get('num_grad') is True: + 241 jacobian = num_jacobian + 242 hessian = num_hessian + 243 else: + 244 jacobian = auto_jacobian + 245 hessian = auto_hessian + 246 + 247 if not callable(func): + 248 raise TypeError('func has to be a function.') + 249 + 250 for i in range(42): + 251 try: + 252 func(np.arange(i), x.T[0]) + 253 except TypeError: + 254 continue + 255 except IndexError: + 256 continue + 257 else: + 258 break + 259 else: + 260 raise RuntimeError("Fit function is not valid.") + 261 + 262 n_parms = i + 263 if not silent: + 264 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) + 265 + 266 x_f = np.vectorize(lambda o: o.value)(x) + 267 dx_f = np.vectorize(lambda o: o.dvalue)(x) + 268 y_f = np.array([o.value for o in y]) + 269 dy_f = np.array([o.dvalue for o in y]) + 270 + 271 if np.any(np.asarray(dx_f) <= 0.0): + 272 raise Exception('No x errors available, run the gamma method first.') + 273 + 274 if np.any(np.asarray(dy_f) <= 0.0): + 275 raise Exception('No y errors available, run the gamma method first.') + 276 + 277 if 'initial_guess' in kwargs: + 278 x0 = kwargs.get('initial_guess') + 279 if len(x0) != n_parms: + 280 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) + 281 else: + 282 x0 = [1] * n_parms + 283 + 284 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) + 285 model = Model(func) + 286 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) + 287 odr.set_job(fit_type=0, deriv=1) + 288 out = odr.run() + 289 + 290 output.residual_variance = out.res_var + 291 + 292 output.method = 'ODR' + 293 + 294 output.message = out.stopreason + 295 + 296 output.xplus = out.xplus + 297 + 298 if not silent: + 299 print('Method: ODR') + 300 print(*out.stopreason) + 301 print('Residual variance:', output.residual_variance) + 302 + 303 if out.info > 3: + 304 raise Exception('The minimization procedure did not converge.') + 305 + 306 m = x_f.size + 307 + 308 def odr_chisquare(p): + 309 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) + 310 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) + 311 return chisq + 312 + 313 if kwargs.get('expected_chisquare') is True: + 314 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) + 315 + 316 if kwargs.get('covariance') is not None: + 317 cov = kwargs.get('covariance') + 318 else: + 319 cov = covariance(np.concatenate((y, x.ravel()))) + 320 + 321 number_of_x_parameters = int(m / x_f.shape[-1]) + 322 + 323 old_jac = jacobian(func)(out.beta, out.xplus) + 324 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) + 325 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]))) + 326 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) + 327 + 328 A = W @ new_jac + 329 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T + 330 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) + 331 if expected_chisquare <= 0.0: + 332 warnings.warn("Negative expected_chisquare.", RuntimeWarning) + 333 expected_chisquare = np.abs(expected_chisquare) + 334 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare + 335 if not silent: + 336 print('chisquare/expected_chisquare:', + 337 output.chisquare_by_expected_chisquare) + 338 + 339 fitp = out.beta + 340 try: + 341 hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel()))) + 342 except TypeError: + 343 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None + 344 + 345 def odr_chisquare_compact_x(d): + 346 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) + 347 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) + 348 return chisq + 349 + 350 jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) + 351 + 352 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv + 353 try: + 354 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) + 355 except np.linalg.LinAlgError: + 356 raise Exception("Cannot invert hessian matrix.") + 357 + 358 def odr_chisquare_compact_y(d): + 359 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) + 360 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) + 361 return chisq + 362 + 363 jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f))) + 364 + 365 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv + 366 try: + 367 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) + 368 except np.linalg.LinAlgError: + 369 raise Exception("Cannot invert hessian matrix.") + 370 + 371 result = [] + 372 for i in range(n_parms): + 373 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]))) + 374 + 375 output.fit_parameters = result + 376 + 377 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) + 378 output.dof = x.shape[-1] - n_parms + 379 output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof) + 380 + 381 return output + 382 + 383 + 384def _prior_fit(x, y, func, priors, silent=False, **kwargs): + 385 output = Fit_result() + 386 + 387 output.fit_function = func + 388 + 389 x = np.asarray(x) + 390 + 391 if kwargs.get('num_grad') is True: + 392 hessian = num_hessian + 393 else: + 394 hessian = auto_hessian + 395 + 396 if not callable(func): + 397 raise TypeError('func has to be a function.') + 398 + 399 for i in range(100): + 400 try: + 401 func(np.arange(i), 0) + 402 except TypeError: + 403 continue + 404 except IndexError: + 405 continue + 406 else: + 407 break + 408 else: + 409 raise RuntimeError("Fit function is not valid.") + 410 + 411 n_parms = i + 412 + 413 if n_parms != len(priors): + 414 raise Exception('Priors does not have the correct length.') + 415 + 416 def extract_val_and_dval(string): + 417 split_string = string.split('(') + 418 if '.' in split_string[0] and '.' not in split_string[1][:-1]: + 419 factor = 10 ** -len(split_string[0].partition('.')[2]) + 420 else: + 421 factor = 1 + 422 return float(split_string[0]), float(split_string[1][:-1]) * factor + 423 + 424 loc_priors = [] + 425 for i_n, i_prior in enumerate(priors): + 426 if isinstance(i_prior, Obs): + 427 loc_priors.append(i_prior) + 428 else: + 429 loc_val, loc_dval = extract_val_and_dval(i_prior) + 430 loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}")) + 431 + 432 output.priors = loc_priors + 433 + 434 if not silent: + 435 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) + 436 + 437 y_f = [o.value for o in y] + 438 dy_f = [o.dvalue for o in y] + 439 + 440 if np.any(np.asarray(dy_f) <= 0.0): + 441 raise Exception('No y errors available, run the gamma method first.') + 442 + 443 p_f = [o.value for o in loc_priors] + 444 dp_f = [o.dvalue for o in loc_priors] + 445 + 446 if np.any(np.asarray(dp_f) <= 0.0): + 447 raise Exception('No prior errors available, run the gamma method first.') + 448 + 449 if 'initial_guess' in kwargs: + 450 x0 = kwargs.get('initial_guess') + 451 if len(x0) != n_parms: + 452 raise Exception('Initial guess does not have the correct length.') + 453 else: + 454 x0 = p_f + 455 + 456 def chisqfunc(p): + 457 model = func(p, x) + 458 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((p_f - p) / dp_f) ** 2) + 459 return chisq + 460 + 461 if not silent: + 462 print('Method: migrad') + 463 + 464 m = iminuit.Minuit(chisqfunc, x0) + 465 m.errordef = 1 + 466 m.print_level = 0 + 467 if 'tol' in kwargs: + 468 m.tol = kwargs.get('tol') + 469 else: + 470 m.tol = 1e-4 + 471 m.migrad() + 472 params = np.asarray(m.values) + 473 + 474 output.chisquare_by_dof = m.fval / len(x) + 475 + 476 output.method = 'migrad' + 477 + 478 if not silent: + 479 print('chisquare/d.o.f.:', output.chisquare_by_dof) + 480 + 481 if not m.fmin.is_valid: + 482 raise Exception('The minimization procedure did not converge.') + 483 + 484 hess = hessian(chisqfunc)(params) + 485 hess_inv = np.linalg.pinv(hess) + 486 + 487 def chisqfunc_compact(d): + 488 model = func(d[:n_parms], x) + 489 chisq = anp.sum(((d[n_parms: n_parms + len(x)] - model) / dy_f) ** 2) + anp.sum(((d[n_parms + len(x):] - d[:n_parms]) / dp_f) ** 2) + 490 return chisq + 491 + 492 jac_jac = hessian(chisqfunc_compact)(np.concatenate((params, y_f, p_f))) + 493 + 494 deriv = -hess_inv @ jac_jac[:n_parms, n_parms:] + 495 + 496 result = [] + 497 for i in range(n_parms): + 498 result.append(derived_observable(lambda x, **kwargs: (x[0] + np.finfo(np.float64).eps) / (y[0].value + np.finfo(np.float64).eps) * params[i], list(y) + list(loc_priors), man_grad=list(deriv[i]))) + 499 + 500 output.fit_parameters = result + 501 output.chisquare = chisqfunc(np.asarray(params)) + 502 + 503 if kwargs.get('resplot') is True: + 504 residual_plot(x, y, func, result) + 505 + 506 if kwargs.get('qqplot') is True: + 507 qqplot(x, y, func, result) + 508 + 509 return output + 510 + 511 + 512def _standard_fit(x, y, func, silent=False, **kwargs): + 513 output = Fit_result() + 514 + 515 output.fit_function = func + 516 + 517 x = np.asarray(x) + 518 + 519 if kwargs.get('num_grad') is True: + 520 jacobian = num_jacobian + 521 hessian = num_hessian + 522 else: + 523 jacobian = auto_jacobian + 524 hessian = auto_hessian + 525 + 526 if x.shape[-1] != len(y): + 527 raise Exception('x and y input have to have the same length') + 528 + 529 if len(x.shape) > 2: + 530 raise Exception('Unknown format for x values') + 531 + 532 if not callable(func): + 533 raise TypeError('func has to be a function.') + 534 + 535 for i in range(42): + 536 try: + 537 func(np.arange(i), x.T[0]) + 538 except TypeError: + 539 continue + 540 except IndexError: + 541 continue + 542 else: + 543 break + 544 else: + 545 raise RuntimeError("Fit function is not valid.") + 546 + 547 n_parms = i + 548 + 549 if not silent: + 550 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) + 551 + 552 y_f = [o.value for o in y] + 553 dy_f = [o.dvalue for o in y] + 554 + 555 if np.any(np.asarray(dy_f) <= 0.0): + 556 raise Exception('No y errors available, run the gamma method first.') + 557 + 558 if 'initial_guess' in kwargs: + 559 x0 = kwargs.get('initial_guess') + 560 if len(x0) != n_parms: + 561 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) + 562 else: + 563 x0 = [0.1] * n_parms + 564 + 565 if kwargs.get('correlated_fit') is True: + 566 corr = covariance(y, correlation=True, **kwargs) + 567 covdiag = np.diag(1 / np.asarray(dy_f)) + 568 condn = np.linalg.cond(corr) + 569 if condn > 0.1 / np.finfo(float).eps: + 570 raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})") + 571 if condn > 1e13: + 572 warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning) + 573 chol = np.linalg.cholesky(corr) + 574 chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True) + 575 + 576 def chisqfunc_corr(p): + 577 model = func(p, x) + 578 chisq = anp.sum(anp.dot(chol_inv, (y_f - model)) ** 2) + 579 return chisq + 580 + 581 def chisqfunc(p): + 582 model = func(p, x) + 583 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + 584 return chisq + 585 + 586 output.method = kwargs.get('method', 'Levenberg-Marquardt') + 587 if not silent: + 588 print('Method:', output.method) + 589 + 590 if output.method != 'Levenberg-Marquardt': + 591 if output.method == 'migrad': + 592 fit_result = iminuit.minimize(chisqfunc, x0, tol=1e-4) # Stopping criterion 0.002 * tol * errordef + 593 if kwargs.get('correlated_fit') is True: + 594 fit_result = iminuit.minimize(chisqfunc_corr, fit_result.x, tol=1e-4) # Stopping criterion 0.002 * tol * errordef + 595 output.iterations = fit_result.nfev + 596 else: + 597 fit_result = scipy.optimize.minimize(chisqfunc, x0, method=kwargs.get('method'), tol=1e-12) + 598 if kwargs.get('correlated_fit') is True: + 599 fit_result = scipy.optimize.minimize(chisqfunc_corr, fit_result.x, method=kwargs.get('method'), tol=1e-12) + 600 output.iterations = fit_result.nit + 601 + 602 chisquare = fit_result.fun + 603 + 604 else: + 605 if kwargs.get('correlated_fit') is True: + 606 def chisqfunc_residuals_corr(p): + 607 model = func(p, x) + 608 chisq = anp.dot(chol_inv, (y_f - model)) + 609 return chisq + 610 + 611 def chisqfunc_residuals(p): + 612 model = func(p, x) + 613 chisq = ((y_f - model) / dy_f) + 614 return chisq + 615 + 616 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) + 617 if kwargs.get('correlated_fit') is True: + 618 fit_result = scipy.optimize.least_squares(chisqfunc_residuals_corr, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) + 619 + 620 chisquare = np.sum(fit_result.fun ** 2) + 621 if kwargs.get('correlated_fit') is True: + 622 assert np.isclose(chisquare, chisqfunc_corr(fit_result.x), atol=1e-14) + 623 else: + 624 assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14) + 625 + 626 output.iterations = fit_result.nfev + 627 + 628 if not fit_result.success: + 629 raise Exception('The minimization procedure did not converge.') + 630 + 631 if x.shape[-1] - n_parms > 0: + 632 output.chisquare_by_dof = chisquare / (x.shape[-1] - n_parms) + 633 else: + 634 output.chisquare_by_dof = float('nan') + 635 + 636 output.message = fit_result.message + 637 if not silent: + 638 print(fit_result.message) + 639 print('chisquare/d.o.f.:', output.chisquare_by_dof) + 640 + 641 if kwargs.get('expected_chisquare') is True: + 642 if kwargs.get('correlated_fit') is not True: + 643 W = np.diag(1 / np.asarray(dy_f)) + 644 cov = covariance(y) + 645 A = W @ jacobian(func)(fit_result.x, x) + 646 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T + 647 expected_chisquare = np.trace((np.identity(x.shape[-1]) - P_phi) @ W @ cov @ W) + 648 output.chisquare_by_expected_chisquare = chisquare / expected_chisquare + 649 if not silent: + 650 print('chisquare/expected_chisquare:', + 651 output.chisquare_by_expected_chisquare) + 652 + 653 fitp = fit_result.x + 654 try: + 655 if kwargs.get('correlated_fit') is True: + 656 hess = hessian(chisqfunc_corr)(fitp) + 657 else: + 658 hess = hessian(chisqfunc)(fitp) + 659 except TypeError: + 660 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None + 661 + 662 if kwargs.get('correlated_fit') is True: + 663 def chisqfunc_compact(d): + 664 model = func(d[:n_parms], x) + 665 chisq = anp.sum(anp.dot(chol_inv, (d[n_parms:] - model)) ** 2) + 666 return chisq + 667 + 668 else: + 669 def chisqfunc_compact(d): + 670 model = func(d[:n_parms], x) + 671 chisq = anp.sum(((d[n_parms:] - model) / dy_f) ** 2) + 672 return chisq + 673 + 674 jac_jac = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f))) + 675 + 676 # Compute hess^{-1} @ jac_jac[:n_parms, n_parms:] using LAPACK dgesv + 677 try: + 678 deriv = -scipy.linalg.solve(hess, jac_jac[:n_parms, n_parms:]) + 679 except np.linalg.LinAlgError: + 680 raise Exception("Cannot invert hessian matrix.") + 681 + 682 result = [] + 683 for i in range(n_parms): + 684 result.append(derived_observable(lambda x, **kwargs: (x[0] + np.finfo(np.float64).eps) / (y[0].value + np.finfo(np.float64).eps) * fit_result.x[i], list(y), man_grad=list(deriv[i]))) + 685 + 686 output.fit_parameters = result + 687 + 688 output.chisquare = chisquare + 689 output.dof = x.shape[-1] - n_parms + 690 output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof) + 691 # Hotelling t-squared p-value for correlated fits. + 692 if kwargs.get('correlated_fit') is True: + 693 n_cov = np.min(np.vectorize(lambda x: x.N)(y)) + 694 output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare, + 695 output.dof, n_cov - output.dof) + 696 + 697 if kwargs.get('resplot') is True: + 698 residual_plot(x, y, func, result) + 699 + 700 if kwargs.get('qqplot') is True: + 701 qqplot(x, y, func, result) + 702 + 703 return output + 704 + 705 + 706def _combined_fit(x, y, func, silent=False, **kwargs): + 707 + 708 if kwargs.get('correlated_fit') is True: + 709 raise Exception("Correlated fit has not been implemented yet") + 710 + 711 output = Fit_result() + 712 output.fit_function = func + 713 + 714 if kwargs.get('num_grad') is True: + 715 jacobian = num_jacobian + 716 hessian = num_hessian + 717 else: + 718 jacobian = auto_jacobian + 719 hessian = auto_hessian + 720 + 721 key_ls = sorted(list(x.keys())) + 722 + 723 if sorted(list(y.keys())) != key_ls: + 724 raise Exception('x and y dictionaries do not contain the same keys.') + 725 + 726 if sorted(list(func.keys())) != key_ls: + 727 raise Exception('x and func dictionaries do not contain the same keys.') + 728 + 729 x_all = np.concatenate([np.array(x[key]) for key in key_ls]) + 730 y_all = np.concatenate([np.array(y[key]) for key in key_ls]) + 731 + 732 y_f = [o.value for o in y_all] + 733 dy_f = [o.dvalue for o in y_all] + 734 + 735 if len(x_all.shape) > 2: + 736 raise Exception('Unknown format for x values') + 737 + 738 # number of fit parameters + 739 n_parms_ls = [] + 740 for key in key_ls: + 741 if not callable(func[key]): + 742 raise TypeError('func (key=' + key + ') is not a function.') + 743 if len(x[key]) != len(y[key]): + 744 raise Exception('x and y input (key=' + key + ') do not have the same length') + 745 for i in range(100): + 746 try: + 747 func[key](np.arange(i), x_all.T[0]) + 748 except TypeError: + 749 continue + 750 except IndexError: + 751 continue + 752 else: + 753 break + 754 else: + 755 raise RuntimeError("Fit function (key=" + key + ") is not valid.") + 756 n_parms = i + 757 n_parms_ls.append(n_parms) + 758 n_parms = max(n_parms_ls) + 759 if not silent: + 760 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) + 761 + 762 if 'initial_guess' in kwargs: + 763 x0 = kwargs.get('initial_guess') + 764 if len(x0) != n_parms: + 765 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) + 766 else: + 767 x0 = [0.1] * n_parms + 768 + 769 def chisqfunc(p): + 770 func_list = np.concatenate([[func[k]] * len(x[k]) for k in key_ls]) + 771 model = anp.array([func_list[i](p, x_all[i]) for i in range(len(x_all))]) + 772 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + 773 return chisq + 774 + 775 output.method = kwargs.get('method', 'Levenberg-Marquardt') + 776 if not silent: + 777 print('Method:', output.method) + 778 + 779 if output.method != 'Levenberg-Marquardt': + 780 if output.method == 'migrad': + 781 tolerance = 1e-4 # default value of 1e-1 set by iminuit can be problematic + 782 if 'tol' in kwargs: + 783 tolerance = kwargs.get('tol') + 784 fit_result = iminuit.minimize(chisqfunc, x0, tol=tolerance) # Stopping criterion 0.002 * tol * errordef + 785 output.iterations = fit_result.nfev + 786 else: + 787 tolerance = 1e-12 + 788 if 'tol' in kwargs: + 789 tolerance = kwargs.get('tol') + 790 fit_result = scipy.optimize.minimize(chisqfunc, x0, method=kwargs.get('method'), tol=tolerance) + 791 output.iterations = fit_result.nit + 792 + 793 chisquare = fit_result.fun + 794 + 795 else: + 796 def chisqfunc_residuals(p): + 797 model = np.concatenate([np.array(func[key](p, np.asarray(x[key]))) for key in key_ls]) + 798 chisq = ((y_f - model) / dy_f) + 799 return chisq + 800 if 'tol' in kwargs: + 801 print('tol cannot be set for Levenberg-Marquardt') + 802 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) + 803 + 804 chisquare = np.sum(fit_result.fun ** 2) + 805 assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14) + 806 output.iterations = fit_result.nfev + 807 + 808 output.message = fit_result.message + 809 + 810 if not fit_result.success: + 811 raise Exception('The minimization procedure did not converge.') + 812 + 813 if x_all.shape[-1] - n_parms > 0: + 814 output.chisquare = chisquare + 815 output.dof = x_all.shape[-1] - n_parms + 816 output.chisquare_by_dof = output.chisquare / output.dof + 817 output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof) + 818 else: + 819 output.chisquare_by_dof = float('nan') + 820 + 821 if not silent: + 822 print(fit_result.message) + 823 print('chisquare/d.o.f.:', output.chisquare_by_dof) + 824 print('fit parameters', fit_result.x) + 825 + 826 def chisqfunc_compact(d): + 827 func_list = np.concatenate([[func[k]] * len(x[k]) for k in key_ls]) + 828 model = anp.array([func_list[i](d[:n_parms], x_all[i]) for i in range(len(x_all))]) + 829 chisq = anp.sum(((d[n_parms:] - model) / dy_f) ** 2) + 830 return chisq + 831 + 832 def prepare_hat_matrix(): + 833 hat_vector = [] + 834 for key in key_ls: + 835 x_array = np.asarray(x[key]) + 836 if (len(x_array) != 0): + 837 hat_vector.append(jacobian(func[key])(fit_result.x, x_array)) + 838 hat_vector = [item for sublist in hat_vector for item in sublist] + 839 return hat_vector + 840 + 841 fitp = fit_result.x + 842 + 843 if np.any(np.asarray(dy_f) <= 0.0): + 844 raise Exception('No y errors available, run the gamma method first.') + 845 + 846 try: + 847 hess = hessian(chisqfunc)(fitp) + 848 except TypeError: + 849 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None + 850 + 851 jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f))) + 852 + 853 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv + 854 try: + 855 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:]) + 856 except np.linalg.LinAlgError: + 857 raise Exception("Cannot invert hessian matrix.") + 858 + 859 if kwargs.get('expected_chisquare') is True: + 860 if kwargs.get('correlated_fit') is not True: + 861 W = np.diag(1 / np.asarray(dy_f)) + 862 cov = covariance(y_all) + 863 hat_vector = prepare_hat_matrix() + 864 A = W @ hat_vector # hat_vector = 'jacobian(func)(fit_result.x, x)' + 865 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T + 866 expected_chisquare = np.trace((np.identity(x_all.shape[-1]) - P_phi) @ W @ cov @ W) + 867 output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare + 868 if not silent: + 869 print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare) + 870 + 871 result = [] + 872 for i in range(n_parms): + 873 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), man_grad=list(deriv_y[i]))) + 874 + 875 output.fit_parameters = result + 876 + 877 return output + 878 + 879 + 880def fit_lin(x, y, **kwargs): + 881 """Performs a linear fit to y = n + m * x and returns two Obs n, m. + 882 + 883 Parameters + 884 ---------- + 885 x : list + 886 Can either be a list of floats in which case no xerror is assumed, or + 887 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. + 888 y : list + 889 List of Obs, the dvalues of the Obs are used as yerror for the fit. + 890 + 891 Returns + 892 ------- + 893 fit_parameters : list[Obs] + 894 LIist of fitted observables. + 895 """ + 896 + 897 def f(a, x): + 898 y = a[0] + a[1] * x + 899 return y + 900 + 901 if all(isinstance(n, Obs) for n in x): + 902 out = total_least_squares(x, y, f, **kwargs) + 903 return out.fit_parameters + 904 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): + 905 out = least_squares(x, y, f, **kwargs) + 906 return out.fit_parameters + 907 else: + 908 raise Exception('Unsupported types for x') + 909 + 910 + 911def qqplot(x, o_y, func, p): + 912 """Generates a quantile-quantile plot of the fit result which can be used to + 913 check if the residuals of the fit are gaussian distributed. + 914 + 915 Returns + 916 ------- + 917 None + 918 """ + 919 + 920 residuals = [] + 921 for i_x, i_y in zip(x, o_y): + 922 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) + 923 residuals = sorted(residuals) + 924 my_y = [o.value for o in residuals] + 925 probplot = scipy.stats.probplot(my_y) + 926 my_x = probplot[0][0] + 927 plt.figure(figsize=(8, 8 / 1.618)) + 928 plt.errorbar(my_x, my_y, fmt='o') + 929 fit_start = my_x[0] + 930 fit_stop = my_x[-1] + 931 samples = np.arange(fit_start, fit_stop, 0.01) + 932 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') + 933 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='-') + 934 + 935 plt.xlabel('Theoretical quantiles') + 936 plt.ylabel('Ordered Values') + 937 plt.legend() + 938 plt.draw() + 939 + 940 + 941def residual_plot(x, y, func, fit_res): + 942 """Generates a plot which compares the fit to the data and displays the corresponding residuals + 943 + 944 Returns + 945 ------- + 946 None + 947 """ + 948 sorted_x = sorted(x) + 949 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) + 950 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) + 951 x_samples = np.arange(xstart, xstop + 0.01, 0.01) + 952 + 953 plt.figure(figsize=(8, 8 / 1.618)) + 954 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) + 955 ax0 = plt.subplot(gs[0]) + 956 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') + 957 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) + 958 ax0.set_xticklabels([]) + 959 ax0.set_xlim([xstart, xstop]) + 960 ax0.set_xticklabels([]) + 961 ax0.legend() + 962 + 963 residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], x)) / np.asarray([o.dvalue for o in y]) + 964 ax1 = plt.subplot(gs[1]) + 965 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) + 966 ax1.tick_params(direction='out') + 967 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) + 968 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") + 969 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') + 970 ax1.set_xlim([xstart, xstop]) + 971 ax1.set_ylabel('Residuals') + 972 plt.subplots_adjust(wspace=None, hspace=None) + 973 plt.draw() + 974 + 975 + 976def error_band(x, func, beta): + 977 """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta. + 978 + 979 Returns + 980 ------- + 981 err : np.array(Obs) + 982 Error band for an array of sample values x + 983 """ + 984 cov = covariance(beta) + 985 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): + 986 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) + 987 + 988 deriv = [] + 989 for i, item in enumerate(x): + 990 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) + 991 + 992 err = [] + 993 for i, item in enumerate(x): + 994 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) + 995 err = np.array(err) + 996 + 997 return err + 998 + 999 +1000def ks_test(objects=None): +1001 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. +1002 +1003 Parameters +1004 ---------- +1005 objects : list +1006 List of fit results to include in the analysis (optional). +1007 +1008 Returns +1009 ------- +1010 None +1011 """ +1012 +1013 if objects is None: +1014 obs_list = [] +1015 for obj in gc.get_objects(): +1016 if isinstance(obj, Fit_result): +1017 obs_list.append(obj) +1018 else: +1019 obs_list = objects +1020 +1021 p_values = [o.p_value for o in obs_list] +1022 +1023 bins = len(p_values) +1024 x = np.arange(0, 1.001, 0.001) +1025 plt.plot(x, x, 'k', zorder=1) +1026 plt.xlim(0, 1) +1027 plt.ylim(0, 1) +1028 plt.xlabel('p-value') +1029 plt.ylabel('Cumulative probability') +1030 plt.title(str(bins) + ' p-values') +1031 +1032 n = np.arange(1, bins + 1) / np.float64(bins) +1033 Xs = np.sort(p_values) +1034 plt.step(Xs, n) +1035 diffs = n - Xs +1036 loc_max_diff = np.argmax(np.abs(diffs)) +1037 loc = Xs[loc_max_diff] +1038 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) +1039 plt.draw() +1040 +1041 print(scipy.stats.kstest(p_values, 'uniform'))-74def least_squares(x, y, func, priors=None, silent=False, **kwargs): 75 r'''Performs a non-linear fit to y = func(x). - 76 - 77 Parameters - 78 ---------- - 79 x : list - 80 list of floats. - 81 y : list - 82 list of Obs. - 83 func : object - 84 fit function, has to be of the form - 85 - 86 ```python - 87 import autograd.numpy as anp + 76 ``` + 77 + 78 Parameters + 79 ---------- + 80 For an uncombined fit: + 81 + 82 x : list + 83 list of floats. + 84 y : list + 85 list of Obs. + 86 func : object + 87 fit function, has to be of the form 88 - 89 def func(a, x): - 90 return a[0] + a[1] * x + a[2] * anp.sinh(x) - 91 ``` - 92 - 93 For multiple x values func can be of the form - 94 - 95 ```python - 96 def func(a, x): - 97 (x1, x2) = x - 98 return a[0] * x1 ** 2 + a[1] * x2 - 99 ``` -100 -101 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation -102 will not work. -103 priors : list, optional -104 priors has to be a list with an entry for every parameter in the fit. The entries can either be -105 Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like -106 0.548(23), 500(40) or 0.5(0.4) -107 silent : bool, optional -108 If true all output to the console is omitted (default False). -109 initial_guess : list -110 can provide an initial guess for the input parameters. Relevant for -111 non-linear fits with many parameters. In case of correlated fits the guess is used to perform -112 an uncorrelated fit which then serves as guess for the correlated fit. -113 method : str, optional -114 can be used to choose an alternative method for the minimization of chisquare. -115 The possible methods are the ones which can be used for scipy.optimize.minimize and -116 migrad of iminuit. If no method is specified, Levenberg-Marquard is used. -117 Reliable alternatives are migrad, Powell and Nelder-Mead. -118 correlated_fit : bool -119 If True, use the full inverse covariance matrix in the definition of the chisquare cost function. -120 For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`. -121 In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix). -122 This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning). -123 At the moment this option only works for `prior==None` and when no `method` is given. -124 expected_chisquare : bool -125 If True estimates the expected chisquare which is -126 corrected by effects caused by correlated input data (default False). -127 resplot : bool -128 If True, a plot which displays fit, data and residuals is generated (default False). -129 qqplot : bool -130 If True, a quantile-quantile plot of the fit result is generated (default False). -131 num_grad : bool -132 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). -133 -134 Returns -135 ------- -136 output : Fit_result -137 Parameters and information on the fitted result. -138 ''' -139 if priors is not None: -140 return _prior_fit(x, y, func, priors, silent=silent, **kwargs) -141 else: -142 return _standard_fit(x, y, func, silent=silent, **kwargs) + 89 ```python + 90 import autograd.numpy as anp + 91 + 92 def func(a, x): + 93 return a[0] + a[1] * x + a[2] * anp.sinh(x) + 94 ``` + 95 + 96 For multiple x values func can be of the form + 97 + 98 ```python + 99 def func(a, x): +100 (x1, x2) = x +101 return a[0] * x1 ** 2 + a[1] * x2 +102 ``` +103 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation +104 will not work. +105 +106 OR For a combined fit: +107 +108 x : dict +109 dict of lists. +110 y : dict +111 dict of lists of Obs. +112 funcs : dict +113 dict of objects +114 fit functions have to be of the form (here a[0] is the common fit parameter) +115 ```python +116 import autograd.numpy as anp +117 funcs = {"a": func_a, +118 "b": func_b} +119 +120 def func_a(a, x): +121 return a[1] * anp.exp(-a[0] * x) +122 +123 def func_b(a, x): +124 return a[2] * anp.exp(-a[0] * x) +125 +126 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation +127 will not work. +128 +129 priors : list, optional +130 priors has to be a list with an entry for every parameter in the fit. The entries can either be +131 Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like +132 0.548(23), 500(40) or 0.5(0.4) +133 silent : bool, optional +134 If true all output to the console is omitted (default False). +135 initial_guess : list +136 can provide an initial guess for the input parameters. Relevant for +137 non-linear fits with many parameters. In case of correlated fits the guess is used to perform +138 an uncorrelated fit which then serves as guess for the correlated fit. +139 method : str, optional +140 can be used to choose an alternative method for the minimization of chisquare. +141 The possible methods are the ones which can be used for scipy.optimize.minimize and +142 migrad of iminuit. If no method is specified, Levenberg-Marquard is used. +143 Reliable alternatives are migrad, Powell and Nelder-Mead. +144 tol: float, optional +145 can be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence +146 to a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly +147 invalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values +148 The stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max) +149 correlated_fit : bool +150 If True, use the full inverse covariance matrix in the definition of the chisquare cost function. +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 if priors is not None: +171 return _prior_fit(x, y, func, priors, silent=silent, **kwargs) +172 +173 elif (type(x) == dict and type(y) == dict and type(func) == dict): +174 return _combined_fit(x, y, func, silent=silent, **kwargs) +175 elif (type(x) == dict or type(y) == dict or type(func) == dict): +176 raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.") +177 else: +178 return _standard_fit(x, y, func, silent=silent, **kwargs)-Performs a non-linear fit to y = func(x).
++Performs a non-linear fit to y = func(x). + ```
Parameters
+
- For an uncombined fit:
- x (list): list of floats.
- y (list): @@ -1213,6 +1460,27 @@ fit function, has to be of the form
It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation +will not work.
+OR For a combined fit: +x (dict): +dict of lists. +y (dict): +dict of lists of Obs. +funcs (dict): +dict of objects +fit functions have to be of the form (here a[0] is the common fit parameter) +```python +import autograd.numpy as anp +funcs = {"a": func_a, + "b": func_b}
+ +def func_a(a, x): + return a[1] * anp.exp(-a[0] * x)
+ +def func_b(a, x): + return a[2] * anp.exp(-a[0] * x)
+It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation will not work.
priors (list, optional): @@ -1230,6 +1498,11 @@ can be used to choose an alternative method for the minimization of chisquare. The possible methods are the ones which can be used for scipy.optimize.minimize and migrad of iminuit. If no method is specified, Levenberg-Marquard is used. Reliable alternatives are migrad, Powell and Nelder-Mead. +tol (float, optional): +can be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence +to a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly +invalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values +The stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max) correlated_fit (bool): 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
. @@ -1268,208 +1541,208 @@ Parameters and information on the fitted result.-145def total_least_squares(x, y, func, silent=False, **kwargs): -146 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. -147 -148 Parameters -149 ---------- -150 x : list -151 list of Obs, or a tuple of lists of Obs -152 y : list -153 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. -154 func : object -155 func has to be of the form -156 -157 ```python -158 import autograd.numpy as anp -159 -160 def func(a, x): -161 return a[0] + a[1] * x + a[2] * anp.sinh(x) -162 ``` -163 -164 For multiple x values func can be of the form -165 -166 ```python -167 def func(a, x): -168 (x1, x2) = x -169 return a[0] * x1 ** 2 + a[1] * x2 -170 ``` -171 -172 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation -173 will not work. -174 silent : bool, optional -175 If true all output to the console is omitted (default False). -176 initial_guess : list -177 can provide an initial guess for the input parameters. Relevant for non-linear -178 fits with many parameters. -179 expected_chisquare : bool -180 If true prints the expected chisquare which is -181 corrected by effects caused by correlated input data. -182 This can take a while as the full correlation matrix -183 has to be calculated (default False). -184 num_grad : bool -185 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). -186 -187 Notes -188 ----- -189 Based on the orthogonal distance regression module of scipy. -190 -191 Returns -192 ------- -193 output : Fit_result -194 Parameters and information on the fitted result. -195 ''' -196 -197 output = Fit_result() -198 -199 output.fit_function = func -200 -201 x = np.array(x) -202 -203 x_shape = x.shape -204 -205 if kwargs.get('num_grad') is True: -206 jacobian = num_jacobian -207 hessian = num_hessian -208 else: -209 jacobian = auto_jacobian -210 hessian = auto_hessian -211 -212 if not callable(func): -213 raise TypeError('func has to be a function.') -214 -215 for i in range(42): -216 try: -217 func(np.arange(i), x.T[0]) -218 except TypeError: -219 continue -220 except IndexError: -221 continue -222 else: -223 break -224 else: -225 raise RuntimeError("Fit function is not valid.") +@@ -1543,35 +1816,35 @@ Parameters and information on the fitted result.181def total_least_squares(x, y, func, silent=False, **kwargs): +182 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. +183 +184 Parameters +185 ---------- +186 x : list +187 list of Obs, or a tuple of lists of Obs +188 y : list +189 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. +190 func : object +191 func has to be of the form +192 +193 ```python +194 import autograd.numpy as anp +195 +196 def func(a, x): +197 return a[0] + a[1] * x + a[2] * anp.sinh(x) +198 ``` +199 +200 For multiple x values func can be of the form +201 +202 ```python +203 def func(a, x): +204 (x1, x2) = x +205 return a[0] * x1 ** 2 + a[1] * x2 +206 ``` +207 +208 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation +209 will not work. +210 silent : bool, optional +211 If true all output to the console is omitted (default False). +212 initial_guess : list +213 can provide an initial guess for the input parameters. Relevant for non-linear +214 fits with many parameters. +215 expected_chisquare : bool +216 If true prints the expected chisquare which is +217 corrected by effects caused by correlated input data. +218 This can take a while as the full correlation matrix +219 has to be calculated (default False). +220 num_grad : bool +221 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). +222 +223 Notes +224 ----- +225 Based on the orthogonal distance regression module of scipy. 226 -227 n_parms = i -228 if not silent: -229 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) -230 -231 x_f = np.vectorize(lambda o: o.value)(x) -232 dx_f = np.vectorize(lambda o: o.dvalue)(x) -233 y_f = np.array([o.value for o in y]) -234 dy_f = np.array([o.dvalue for o in y]) -235 -236 if np.any(np.asarray(dx_f) <= 0.0): -237 raise Exception('No x errors available, run the gamma method first.') +227 Returns +228 ------- +229 output : Fit_result +230 Parameters and information on the fitted result. +231 ''' +232 +233 output = Fit_result() +234 +235 output.fit_function = func +236 +237 x = np.array(x) 238 -239 if np.any(np.asarray(dy_f) <= 0.0): -240 raise Exception('No y errors available, run the gamma method first.') -241 -242 if 'initial_guess' in kwargs: -243 x0 = kwargs.get('initial_guess') -244 if len(x0) != n_parms: -245 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) -246 else: -247 x0 = [1] * n_parms -248 -249 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) -250 model = Model(func) -251 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) -252 odr.set_job(fit_type=0, deriv=1) -253 out = odr.run() -254 -255 output.residual_variance = out.res_var -256 -257 output.method = 'ODR' -258 -259 output.message = out.stopreason -260 -261 output.xplus = out.xplus +239 x_shape = x.shape +240 +241 if kwargs.get('num_grad') is True: +242 jacobian = num_jacobian +243 hessian = num_hessian +244 else: +245 jacobian = auto_jacobian +246 hessian = auto_hessian +247 +248 if not callable(func): +249 raise TypeError('func has to be a function.') +250 +251 for i in range(42): +252 try: +253 func(np.arange(i), x.T[0]) +254 except TypeError: +255 continue +256 except IndexError: +257 continue +258 else: +259 break +260 else: +261 raise RuntimeError("Fit function is not valid.") 262 -263 if not silent: -264 print('Method: ODR') -265 print(*out.stopreason) -266 print('Residual variance:', output.residual_variance) -267 -268 if out.info > 3: -269 raise Exception('The minimization procedure did not converge.') -270 -271 m = x_f.size -272 -273 def odr_chisquare(p): -274 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) -275 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) -276 return chisq +263 n_parms = i +264 if not silent: +265 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) +266 +267 x_f = np.vectorize(lambda o: o.value)(x) +268 dx_f = np.vectorize(lambda o: o.dvalue)(x) +269 y_f = np.array([o.value for o in y]) +270 dy_f = np.array([o.dvalue for o in y]) +271 +272 if np.any(np.asarray(dx_f) <= 0.0): +273 raise Exception('No x errors available, run the gamma method first.') +274 +275 if np.any(np.asarray(dy_f) <= 0.0): +276 raise Exception('No y errors available, run the gamma method first.') 277 -278 if kwargs.get('expected_chisquare') is True: -279 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) -280 -281 if kwargs.get('covariance') is not None: -282 cov = kwargs.get('covariance') -283 else: -284 cov = covariance(np.concatenate((y, x.ravel()))) -285 -286 number_of_x_parameters = int(m / x_f.shape[-1]) -287 -288 old_jac = jacobian(func)(out.beta, out.xplus) -289 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) -290 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]))) -291 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) +278 if 'initial_guess' in kwargs: +279 x0 = kwargs.get('initial_guess') +280 if len(x0) != n_parms: +281 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) +282 else: +283 x0 = [1] * n_parms +284 +285 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) +286 model = Model(func) +287 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) +288 odr.set_job(fit_type=0, deriv=1) +289 out = odr.run() +290 +291 output.residual_variance = out.res_var 292 -293 A = W @ new_jac -294 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T -295 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) -296 if expected_chisquare <= 0.0: -297 warnings.warn("Negative expected_chisquare.", RuntimeWarning) -298 expected_chisquare = np.abs(expected_chisquare) -299 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare -300 if not silent: -301 print('chisquare/expected_chisquare:', -302 output.chisquare_by_expected_chisquare) +293 output.method = 'ODR' +294 +295 output.message = out.stopreason +296 +297 output.xplus = out.xplus +298 +299 if not silent: +300 print('Method: ODR') +301 print(*out.stopreason) +302 print('Residual variance:', output.residual_variance) 303 -304 fitp = out.beta -305 try: -306 hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel()))) -307 except TypeError: -308 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None -309 -310 def odr_chisquare_compact_x(d): -311 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) -312 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) -313 return chisq -314 -315 jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) +304 if out.info > 3: +305 raise Exception('The minimization procedure did not converge.') +306 +307 m = x_f.size +308 +309 def odr_chisquare(p): +310 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) +311 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) +312 return chisq +313 +314 if kwargs.get('expected_chisquare') is True: +315 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) 316 -317 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv -318 try: -319 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) -320 except np.linalg.LinAlgError: -321 raise Exception("Cannot invert hessian matrix.") -322 -323 def odr_chisquare_compact_y(d): -324 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) -325 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) -326 return chisq -327 -328 jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f))) -329 -330 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv -331 try: -332 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) -333 except np.linalg.LinAlgError: -334 raise Exception("Cannot invert hessian matrix.") -335 -336 result = [] -337 for i in range(n_parms): -338 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]))) +317 if kwargs.get('covariance') is not None: +318 cov = kwargs.get('covariance') +319 else: +320 cov = covariance(np.concatenate((y, x.ravel()))) +321 +322 number_of_x_parameters = int(m / x_f.shape[-1]) +323 +324 old_jac = jacobian(func)(out.beta, out.xplus) +325 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) +326 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]))) +327 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) +328 +329 A = W @ new_jac +330 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T +331 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) +332 if expected_chisquare <= 0.0: +333 warnings.warn("Negative expected_chisquare.", RuntimeWarning) +334 expected_chisquare = np.abs(expected_chisquare) +335 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare +336 if not silent: +337 print('chisquare/expected_chisquare:', +338 output.chisquare_by_expected_chisquare) 339 -340 output.fit_parameters = result -341 -342 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) -343 output.dof = x.shape[-1] - n_parms -344 output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof) +340 fitp = out.beta +341 try: +342 hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel()))) +343 except TypeError: +344 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None 345 -346 return output +346 def odr_chisquare_compact_x(d): +347 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) +348 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) +349 return chisq +350 +351 jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) +352 +353 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv +354 try: +355 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) +356 except np.linalg.LinAlgError: +357 raise Exception("Cannot invert hessian matrix.") +358 +359 def odr_chisquare_compact_y(d): +360 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) +361 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) +362 return chisq +363 +364 jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f))) +365 +366 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv +367 try: +368 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) +369 except np.linalg.LinAlgError: +370 raise Exception("Cannot invert hessian matrix.") +371 +372 result = [] +373 for i in range(n_parms): +374 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]))) +375 +376 output.fit_parameters = result +377 +378 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) +379 output.dof = x.shape[-1] - n_parms +380 output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof) +381 +382 return output-672def fit_lin(x, y, **kwargs): -673 """Performs a linear fit to y = n + m * x and returns two Obs n, m. -674 -675 Parameters -676 ---------- -677 x : list -678 Can either be a list of floats in which case no xerror is assumed, or -679 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. -680 y : list -681 List of Obs, the dvalues of the Obs are used as yerror for the fit. -682 -683 Returns -684 ------- -685 fit_parameters : list[Obs] -686 LIist of fitted observables. -687 """ -688 -689 def f(a, x): -690 y = a[0] + a[1] * x -691 return y -692 -693 if all(isinstance(n, Obs) for n in x): -694 out = total_least_squares(x, y, f, **kwargs) -695 return out.fit_parameters -696 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): -697 out = least_squares(x, y, f, **kwargs) -698 return out.fit_parameters -699 else: -700 raise Exception('Unsupported types for x') +@@ -1608,34 +1881,34 @@ LIist of fitted observables.881def fit_lin(x, y, **kwargs): +882 """Performs a linear fit to y = n + m * x and returns two Obs n, m. +883 +884 Parameters +885 ---------- +886 x : list +887 Can either be a list of floats in which case no xerror is assumed, or +888 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. +889 y : list +890 List of Obs, the dvalues of the Obs are used as yerror for the fit. +891 +892 Returns +893 ------- +894 fit_parameters : list[Obs] +895 LIist of fitted observables. +896 """ +897 +898 def f(a, x): +899 y = a[0] + a[1] * x +900 return y +901 +902 if all(isinstance(n, Obs) for n in x): +903 out = total_least_squares(x, y, f, **kwargs) +904 return out.fit_parameters +905 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): +906 out = least_squares(x, y, f, **kwargs) +907 return out.fit_parameters +908 else: +909 raise Exception('Unsupported types for x')-703def qqplot(x, o_y, func, p): -704 """Generates a quantile-quantile plot of the fit result which can be used to -705 check if the residuals of the fit are gaussian distributed. -706 -707 Returns -708 ------- -709 None -710 """ -711 -712 residuals = [] -713 for i_x, i_y in zip(x, o_y): -714 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) -715 residuals = sorted(residuals) -716 my_y = [o.value for o in residuals] -717 probplot = scipy.stats.probplot(my_y) -718 my_x = probplot[0][0] -719 plt.figure(figsize=(8, 8 / 1.618)) -720 plt.errorbar(my_x, my_y, fmt='o') -721 fit_start = my_x[0] -722 fit_stop = my_x[-1] -723 samples = np.arange(fit_start, fit_stop, 0.01) -724 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') -725 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='-') -726 -727 plt.xlabel('Theoretical quantiles') -728 plt.ylabel('Ordered Values') -729 plt.legend() -730 plt.draw() +@@ -1662,39 +1935,39 @@ LIist of fitted observables.912def qqplot(x, o_y, func, p): +913 """Generates a quantile-quantile plot of the fit result which can be used to +914 check if the residuals of the fit are gaussian distributed. +915 +916 Returns +917 ------- +918 None +919 """ +920 +921 residuals = [] +922 for i_x, i_y in zip(x, o_y): +923 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) +924 residuals = sorted(residuals) +925 my_y = [o.value for o in residuals] +926 probplot = scipy.stats.probplot(my_y) +927 my_x = probplot[0][0] +928 plt.figure(figsize=(8, 8 / 1.618)) +929 plt.errorbar(my_x, my_y, fmt='o') +930 fit_start = my_x[0] +931 fit_stop = my_x[-1] +932 samples = np.arange(fit_start, fit_stop, 0.01) +933 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') +934 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='-') +935 +936 plt.xlabel('Theoretical quantiles') +937 plt.ylabel('Ordered Values') +938 plt.legend() +939 plt.draw()-733def residual_plot(x, y, func, fit_res): -734 """Generates a plot which compares the fit to the data and displays the corresponding residuals -735 -736 Returns -737 ------- -738 None -739 """ -740 sorted_x = sorted(x) -741 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) -742 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) -743 x_samples = np.arange(xstart, xstop + 0.01, 0.01) -744 -745 plt.figure(figsize=(8, 8 / 1.618)) -746 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) -747 ax0 = plt.subplot(gs[0]) -748 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') -749 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) -750 ax0.set_xticklabels([]) -751 ax0.set_xlim([xstart, xstop]) -752 ax0.set_xticklabels([]) -753 ax0.legend() -754 -755 residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], x)) / np.asarray([o.dvalue for o in y]) -756 ax1 = plt.subplot(gs[1]) -757 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) -758 ax1.tick_params(direction='out') -759 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) -760 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") -761 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') -762 ax1.set_xlim([xstart, xstop]) -763 ax1.set_ylabel('Residuals') -764 plt.subplots_adjust(wspace=None, hspace=None) -765 plt.draw() +@@ -1720,28 +1993,28 @@ LIist of fitted observables.942def residual_plot(x, y, func, fit_res): +943 """Generates a plot which compares the fit to the data and displays the corresponding residuals +944 +945 Returns +946 ------- +947 None +948 """ +949 sorted_x = sorted(x) +950 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) +951 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) +952 x_samples = np.arange(xstart, xstop + 0.01, 0.01) +953 +954 plt.figure(figsize=(8, 8 / 1.618)) +955 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) +956 ax0 = plt.subplot(gs[0]) +957 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') +958 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) +959 ax0.set_xticklabels([]) +960 ax0.set_xlim([xstart, xstop]) +961 ax0.set_xticklabels([]) +962 ax0.legend() +963 +964 residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], x)) / np.asarray([o.dvalue for o in y]) +965 ax1 = plt.subplot(gs[1]) +966 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) +967 ax1.tick_params(direction='out') +968 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) +969 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") +970 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') +971 ax1.set_xlim([xstart, xstop]) +972 ax1.set_ylabel('Residuals') +973 plt.subplots_adjust(wspace=None, hspace=None) +974 plt.draw()-768def error_band(x, func, beta): -769 """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta. -770 -771 Returns -772 ------- -773 err : np.array(Obs) -774 Error band for an array of sample values x -775 """ -776 cov = covariance(beta) -777 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): -778 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) -779 -780 deriv = [] -781 for i, item in enumerate(x): -782 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) -783 -784 err = [] -785 for i, item in enumerate(x): -786 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) -787 err = np.array(err) -788 -789 return err +@@ -1768,48 +2041,48 @@ Error band for an array of sample values x977def error_band(x, func, beta): +978 """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta. +979 +980 Returns +981 ------- +982 err : np.array(Obs) +983 Error band for an array of sample values x +984 """ +985 cov = covariance(beta) +986 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): +987 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) +988 +989 deriv = [] +990 for i, item in enumerate(x): +991 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) +992 +993 err = [] +994 for i, item in enumerate(x): +995 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) +996 err = np.array(err) +997 +998 return err792def ks_test(objects=None): -793 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. -794 -795 Parameters -796 ---------- -797 objects : list -798 List of fit results to include in the analysis (optional). -799 -800 Returns -801 ------- -802 None -803 """ -804 -805 if objects is None: -806 obs_list = [] -807 for obj in gc.get_objects(): -808 if isinstance(obj, Fit_result): -809 obs_list.append(obj) -810 else: -811 obs_list = objects -812 -813 p_values = [o.p_value for o in obs_list] -814 -815 bins = len(p_values) -816 x = np.arange(0, 1.001, 0.001) -817 plt.plot(x, x, 'k', zorder=1) -818 plt.xlim(0, 1) -819 plt.ylim(0, 1) -820 plt.xlabel('p-value') -821 plt.ylabel('Cumulative probability') -822 plt.title(str(bins) + ' p-values') -823 -824 n = np.arange(1, bins + 1) / np.float64(bins) -825 Xs = np.sort(p_values) -826 plt.step(Xs, n) -827 diffs = n - Xs -828 loc_max_diff = np.argmax(np.abs(diffs)) -829 loc = Xs[loc_max_diff] -830 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) -831 plt.draw() -832 -833 print(scipy.stats.kstest(p_values, 'uniform')) +diff --git a/docs/search.js b/docs/search.js index f329e039..36205d10 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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u=r[s];u=Math.sqrt(u),this.index[n].addToken(s,{ref:i,tf:u})}},this),n&&this.eventEmitter.emit("add",e,this)}},t.Index.prototype.removeDocByRef=function(e){if(e&&this.documentStore.isDocStored()!==!1&&this.documentStore.hasDoc(e)){var t=this.documentStore.getDoc(e);this.removeDoc(t,!1)}},t.Index.prototype.removeDoc=function(e,n){if(e){var n=void 0===n?!0:n,i=e[this._ref];this.documentStore.hasDoc(i)&&(this.documentStore.removeDoc(i),this._fields.forEach(function(n){var o=this.pipeline.run(t.tokenizer(e[n]));o.forEach(function(e){this.index[n].removeToken(e,i)},this)},this),n&&this.eventEmitter.emit("remove",e,this))}},t.Index.prototype.updateDoc=function(e,t){var t=void 0===t?!0:t;this.removeDocByRef(e[this._ref],!1),this.addDoc(e,!1),t&&this.eventEmitter.emit("update",e,this)},t.Index.prototype.idf=function(e,t){var n="@"+t+"/"+e;if(Object.prototype.hasOwnProperty.call(this._idfCache,n))return this._idfCache[n];var i=this.index[t].getDocFreq(e),o=1+Math.log(this.documentStore.length/(i+1));return this._idfCache[n]=o,o},t.Index.prototype.getFields=function(){return this._fields.slice()},t.Index.prototype.search=function(e,n){if(!e)return[];e="string"==typeof e?{any:e}:JSON.parse(JSON.stringify(e));var i=null;null!=n&&(i=JSON.stringify(n));for(var o=new t.Configuration(i,this.getFields()).get(),r={},s=Object.keys(e),u=0;u1001def ks_test(objects=None): +1002 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. +1003 +1004 Parameters +1005 ---------- +1006 objects : list +1007 List of fit results to include in the analysis (optional). +1008 +1009 Returns +1010 ------- +1011 None +1012 """ +1013 +1014 if objects is None: +1015 obs_list = [] +1016 for obj in gc.get_objects(): +1017 if isinstance(obj, Fit_result): +1018 obs_list.append(obj) +1019 else: +1020 obs_list = objects +1021 +1022 p_values = [o.p_value for o in obs_list] +1023 +1024 bins = len(p_values) +1025 x = np.arange(0, 1.001, 0.001) +1026 plt.plot(x, x, 'k', zorder=1) +1027 plt.xlim(0, 1) +1028 plt.ylim(0, 1) +1029 plt.xlabel('p-value') +1030 plt.ylabel('Cumulative probability') +1031 plt.title(str(bins) + ' p-values') +1032 +1033 n = np.arange(1, bins + 1) / np.float64(bins) +1034 Xs = np.sort(p_values) +1035 plt.step(Xs, n) +1036 diffs = n - Xs +1037 loc_max_diff = np.argmax(np.abs(diffs)) +1038 loc = Xs[loc_max_diff] +1039 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) +1040 plt.draw() +1041 +1042 print(scipy.stats.kstest(p_values, 'uniform'))0&&t.push(e);for(var i in n)"docs"!==i&&"df"!==i&&this.expandToken(e+i,t,n[i]);return t},t.InvertedIndex.prototype.toJSON=function(){return{root:this.root}},t.Configuration=function(e,n){var e=e||"";if(void 0==n||null==n)throw new Error("fields should not be null");this.config={};var i;try{i=JSON.parse(e),this.buildUserConfig(i,n)}catch(o){t.utils.warn("user 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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;e 1;){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();o What is pyerrors?\n\n \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...).
\nMore detailed examples can found in the GitHub repository
\n\n.
If you use
\n\npyerrors
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.
\nand
\n\n\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.
\nwhere applicable.
\n\nThere exist similar publicly available implementations of gamma method error analysis suites in Fortran, Julia and Python.
\n\nBasic example
\n\n\n\n\n\nimport 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
The
\n\nObs
class\n\n
pyerrors
introduces a new datatype,Obs
, which simplifies error propagation and estimation for auto- and cross-correlated data.\nAnObs
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\nimport pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name'])\n
Error propagation
\n\nWhen performing mathematical operations on
\n\nObs
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.The
\n\nObs
class is designed such that mathematical numpy functions can be used onObs
just as for regular floats.\n\n\n\nimport 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
Error estimation
\n\nThe error estimation within
\n\npyerrors
is based on the gamma method introduced in arXiv:hep-lat/0306017.\nAfter having arrived at the derived quantity of interest thegamma_method
can be called as detailed in the following example.\n\n\n\nmy_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
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
\n\ngamma_method
as parameter.\n\n\n\nmy_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
The integrated autocorrelation time $\\tau_\\mathrm{int}$ and the autocorrelation function $\\rho(W)$ can be monitored via the methods
\n\npyerrors.obs.Obs.plot_tauint
andpyerrors.obs.Obs.plot_rho
.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\nExponential tails
\n\nSlow 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
\n\ngamma_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\nmy_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
For the full API see
\n\npyerrors.obs.Obs.gamma_method
.Multiple ensembles/replica
\n\nError propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their
\n\nname
.\n\n\n\nobs1 = 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
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\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\nobs1 = 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
Error estimation for multiple ensembles
\n\nIn 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\n\n\npe.Obs.S_dict['ensemble1'] = 2.5\npe.Obs.tau_exp_dict['ensemble2'] = 8.0\npe.Obs.tau_exp_dict['ensemble3'] = 2.0\n
In case the
\n\ngamma_method
is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.\nPassing arguments to thegamma_method
still dominates over the dictionaries.Irregular Monte Carlo chains
\n\n\n\n
Obs
objects defined on irregular Monte Carlo chains can be initialized with the parameteridl
.\n\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
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.Warning: Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.\nMake sure to check the autocorrelation time with e.g.
\n\npyerrors.obs.Obs.plot_rho
orpyerrors.obs.Obs.plot_tauint
.For the full API see
\n\npyerrors.obs.Obs
.Correlators
\n\nWhen one is not interested in single observables but correlation functions,
\n\npyerrors
offers theCorr
class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize aCorr
objects one needs to arrange the data as a list ofObs
\n\n\n\nmy_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
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\n\n\nmy_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
The individual entries of a correlator can be accessed via slicing
\n\n\n\n\n\nprint(my_corr[3])\n> 0.3227(33)\n
Error propagation with the
\n\nCorr
class works very similar toObs
objects. Mathematical operations are overloaded andCorr
objects can be computed together with otherCorr
objects,Obs
objects or real numbers and integers.\n\n\n\nmy_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr\n
\n\n
pyerrors
provides the user with a set of regularly used methods for the manipulation of correlator objects:\n
\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 asCorr
. Different discretizations of the numerical derivative are available.- \n
Corr.second_deriv
returns the second derivative of the correlator asCorr
. 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 anotherCorr
orObs
object.- \n
Corr.reweight
reweights the correlator.\n\n
pyerrors
can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (seepyerrors.correlators.Corr.GEVP
).For the full API see
\n\npyerrors.correlators.Corr
.Complex valued observables
\n\n\n\n
pyerrors
can handle complex valued observables via the classpyerrors.obs.CObs
.\nCObs
are initialized with a real and an imaginary part which both can beObs
valued.\n\n\n\nmy_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
Elementary mathematical operations are overloaded and samples are properly propagated as for the
\n\nObs
class.\n\n\n\nmy_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
The
\n\nCovobs
classIn 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
\n\nCovobs
class allows to define such quantities inpyerrors
. 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.This concept is built into the definition of
\n\nCovobs
. Inpyerrors
, 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 enterscov_Obs
, since the second argument of this function is the covariance matrix of theCovobs
.\n\n\n\nimport 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
The resulting object
\n\nmpi
is anObs
that contains aCovobs
. In the following, it may be handled as any otherObs
. The contribution of the covariance matrix to the error of anObs
is determined from the $M \\times M$ covariance matrix $\\Sigma$ and the gradient of theObs
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.Correlated auxiliary data is defined similarly to above, e.g., via
\n\n\n\n\n\nRAP = 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
where
\n\nRAP
now is a list of twoObs
that contains the two correlated parameters.Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the
\n\nCovobs
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 anObs
o
with respect to a covariance matrix with the identifying stringk
may be accessed via\n\n\n\no.covobs[k].grad\n
Error propagation in iterative algorithms
\n\n\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.Least squares fits
\n\nStandard non-linear least square fits with errors on the dependent but not the independent variables can be performed with
\n\npyerrors.fits.least_squares
. As default solver the Levenberg-Marquardt algorithm implemented in scipy is used.Fit functions have to be of the following form
\n\n\n\n\n\nimport autograd.numpy as anp\n\ndef func(a, x):\n return a[1] * anp.exp(-a[0] * x)\n
It is important that numerical functions refer to
\n\nautograd.numpy
instead ofnumpy
for the automatic differentiation in iterative algorithms to work properly.Fits can then be performed via
\n\n\n\n\n\nfit_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
where x is a
\n\nlist
ornumpy.array
offloats
and y is alist
ornumpy.array
ofObs
.Data stored in
\n\nCorr
objects can be fitted directly using theCorr.fit
method.\n\n\n\nmy_corr = pe.Corr(y)\nfit_result = my_corr.fit(func, fitrange=[12, 25])\n
this can simplify working with absolute fit ranges and takes care of gaps in the data automatically.
\n\nFor fit functions with multiple independent variables the fit function can be of the form
\n\n\n\n\n\ndef func(a, x):\n (x1, x2) = x\n return a[0] * x1 ** 2 + a[1] * x2\n
\n\n
pyerrors
also supports correlated fits which can be triggered via the parametercorrelated_fit=True
.\nDetails about how the required covariance matrix is estimated can be found inpyerrors.obs.covariance
.Direct visualizations of the performed fits can be triggered via
\n\nresplot=True
orqqplot=True
. For all available options seepyerrors.fits.least_squares
.Total least squares fits
\n\n\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, seepyerrors.fits.least_squares
. The syntax is identical to the standard least squares case, the only difference being thatx
also has to be alist
ornumpy.array
ofObs
.For the full API see
\n\npyerrors.fits
for fits andpyerrors.roots
for finding roots of functions.Matrix operations
\n\n\n\n
pyerrors
provides wrappers forObs
- andCObs
-valued matrix operations based onnumpy.linalg
. The supported functions include:\n
\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.For the full API see
\n\npyerrors.linalg
.Export data
\n\nThe preferred exported file format within
\n\npyerrors
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?
\nThis can be achieved by storing all information in one single file. The export routines of
\n\npyerrors
are written such that as much information as possible is written automatically as described in the following example\n\n\n\nmy_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
The format also allows to directly write out the content of
\n\nCorr
objects or lists and arrays ofObs
objects by passing the desired data topyerrors.input.json.dump_to_json
.json.gz format specification
\n\nThe first entries of the file provide optional auxiliary information:
\n\n\n
\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 inpyerrors
. 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.The only necessary entry of the file is the field\n-
\n\nobsdata
, an array that contains the actual data.Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of
\n\nObs
,list
,numpy.ndarray
,Corr
. AllObs
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 arrayobsdata
, are treated independently. Each entry of the arrayobsdata
has the following required entries:\n
\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. Thetag
of anObs
inpyerrors
is written here.- \n
reweighted
is a Bool that may be used to specify, whether theObs
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
inpyerrors
). We will define it below.The array
\n\ndata
contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains:\n
\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.Each entry of
\n\nreplica
contains\nname
, a string that contains the name of the replica\ndeltas
, an array that contains the actual data.Each entry in
\n\ndeltas
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 eachObs
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.The array
\n\ncdata
contains information about the contribution of auxiliary observables, represented byCovobs
inpyerrors
, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains:\n
\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 eachObs
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.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\nJulia 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\nEverything, 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\nThe 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\nParameters
\n\n\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": "- 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.
\nApply 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\nThe 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\nParameters
\n\n\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": "- i (int):\nFirst index to be picked.
\n- j (int):\nSecond index to be picked.
\nOutputs the correlator in a plotable format.
\n\nOutputs 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\nThe 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\n\n\nC.GEVP(t0=2)[0] # Ground state vector(s)\nC.GEVP(t0=2)[:3] # Vectors for the lowest three states\n
Parameters
\n\n\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\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 $t=t_s$.
\nOther Parameters
\n\n\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": "- state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
\nDetermines the eigenvalue of the GEVP by solving and projecting the correlator
\n\nParameters
\n\n\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": "- 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.
\nConstructs an NxN Hankel matrix
\n\nC(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\nParameters
\n\n\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": "- N (int):\nDimension of the Hankel matrix
\n- periodic (bool, optional):\ndetermines whether the matrix is extended periodically
\nPeriodically shift the correlator by dt timeslices
\n\nParameters
\n\n\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": "- dt (int):\nnumber of timeslices
\nReverse 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\nParameters
\n\n\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": "- spacing (int):\nKeep only every 'spacing'th entry of the correlator
\n- offset (int):\nOffset the equal spacing
\nCorrelate the correlator with another correlator or Obs
\n\nParameters
\n\n\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": "- 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.
\nReweight the correlator.
\n\nParameters
\n\n\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": "- 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.
\nReturn the time symmetry average of the correlator and its partner
\n\nParameters
\n\n\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": "- partner (Corr):\nTime symmetry partner of the Corr
\n- partity (int):\nParity quantum number of the correlator, can be +1 or -1
\nReturn the first derivative of the correlator with respect to x0.
\n\nParameters
\n\n\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": "- variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, forward, backward, improved, log, default: symmetric
\nReturn the second derivative of the correlator with respect to x0.
\n\nParameters
\n\n\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": "- variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, improved, log, default: symmetric
\nReturns the effective mass of the correlator as correlator object
\n\nParameters
\n\n\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": "- 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
\nFits function to the data
\n\nParameters
\n\n\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": "- 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.\n
\nfitrange=[4, 6]
corresponds to the three entries 4, 5 and 6.\nIf not specified, self.prange or all timeslices are used.- silent (bool):\nDecides whether output is printed to the standard output.
\nExtract a plateau value from a Corr object
\n\nParameters
\n\n\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": "- 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
\nSets 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\nParameters
\n\n\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": "- 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.
\nProduces a spaghetti plot of the correlator suited to monitor exceptional configurations.
\n\nParameters
\n\n\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": "- logscale (bool):\nDetermines whether the scale of the y-axis is logarithmic or standard.
\nDumps the Corr into a file of chosen type
\n\nParameters
\n\n\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": "- 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 '.')
\nProject large correlation matrix to lowest states
\n\nThis 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\nParameters
\n\n\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.
\nNotes
\n\nWe 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\nParameters
\n\n\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": "- 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.
\nReturn 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\nBased on https://codegolf.stackexchange.com/a/160375
\n\nReturns
\n\n\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": "- elem (int):\nElement (i,j,k) of the epsilon tensor of rank 3
\nRank-4 epsilon tensor
\n\nExtension of https://codegolf.stackexchange.com/a/160375
\n\nReturns
\n\n\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": "- elem (int):\nElement (i,j,k,o) of the epsilon tensor of rank 4
\nReturns 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\nAttributes
\n\n\n
\n", "bases": "collections.abc.Sequence"}, "pyerrors.fits.Fit_result.__init__": {"fullname": "pyerrors.fits.Fit_result.__init__", "modulename": "pyerrors.fits", "qualname": "Fit_result.__init__", "kind": "function", "doc": "\n", "signature": "()"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "- 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.
\nApply 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\nParameters
\n\n\n
\n\n- x (list):\nlist of floats.
\n- y (list):\nlist of Obs.
\n- \n
func (object):\nfit function, has to be of the form
\n\n\n\n\n\nimport autograd.numpy as anp\n\ndef func(a, x):\n return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
For multiple x values func can be of the form
\n\n\n\n\n\ndef func(a, x):\n (x1, x2) = x\n return a[0] * x1 ** 2 + a[1] * x2\n
It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
- priors (list, optional):\npriors has to be a 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- 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
\npyerrors.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 forprior==None
and when nomethod
is given.- 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).
\nReturns
\n\n\n
\n", "signature": "(x, y, func, priors=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.total_least_squares": {"fullname": "pyerrors.fits.total_least_squares", "modulename": "pyerrors.fits", "qualname": "total_least_squares", "kind": "function", "doc": "- output (Fit_result):\nParameters and information on the fitted result.
\nPerforms a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
\n\nParameters
\n\n\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- \n
func (object):\nfunc has to be of the form
\n\n\n\n\n\nimport autograd.numpy as anp\n\ndef func(a, x):\n return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
For multiple x values func can be of the form
\n\n\n\n\n\ndef func(a, x):\n (x1, x2) = x\n return a[0] * x1 ** 2 + a[1] * x2\n
It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
- 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).
\nNotes
\n\nBased on the orthogonal distance regression module of scipy.
\n\nReturns
\n\n\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": "- output (Fit_result):\nParameters and information on the fitted result.
\nPerforms a linear fit to y = n + m * x and returns two Obs n, m.
\n\nParameters
\n\n\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.
\nReturns
\n\n\n
\n", "signature": "(x, y, **kwargs):", "funcdef": "def"}, "pyerrors.fits.qqplot": {"fullname": "pyerrors.fits.qqplot", "modulename": "pyerrors.fits", "qualname": "qqplot", "kind": "function", "doc": "- fit_parameters (list[Obs]):\nLIist of fitted observables.
\nGenerates 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\nReturns
\n\n\n
\n", "signature": "(x, o_y, func, p):", "funcdef": "def"}, "pyerrors.fits.residual_plot": {"fullname": "pyerrors.fits.residual_plot", "modulename": "pyerrors.fits", "qualname": "residual_plot", "kind": "function", "doc": "- None
\nGenerates a plot which compares the fit to the data and displays the corresponding residuals
\n\nReturns
\n\n\n
\n", "signature": "(x, y, func, fit_res):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "- None
\nCalculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
\n\nReturns
\n\n\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": "- err (np.array(Obs)):\nError band for an array of sample values x
\nPerforms a Kolmogorov\u2013Smirnov test for the p-values of all fit object.
\n\nParameters
\n\n\n
\n\n- objects (list):\nList of fit results to include in the analysis (optional).
\nReturns
\n\n\n
\n", "signature": "(objects=None):", "funcdef": "def"}, "pyerrors.input": {"fullname": "pyerrors.input", "modulename": "pyerrors.input", "kind": "module", "doc": "- None
\n\n\n
pyerrors
includes aninput
submodule in which input routines and parsers for the output of various numerical programs are contained.Jackknife samples
\n\nFor comparison with other analysis workflows
\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": "pyerrors
can also generate jackknife samples from anObs
object or import jackknife samples into anObs
object.\nSeepyerrors.obs.Obs.export_jackknife
andpyerrors.obs.import_jackknife
for details.Extract generic MCMC data from a bdio file
\n\nread_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\nall: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\nParameters
\n\n\n
\n\n- file_path -- path to the bdio file
\n- bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
\nReturns
\n\n\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": "- data (List[Obs]):\nExtracted data
\nWrite Obs to a bdio file according to ADerrors conventions
\n\nread_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\nall: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\nParameters
\n\n\n
\n\n- file_path -- path to the bdio file
\n- bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
\nReturns
\n\n\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": "- success (int):\nreturns 0 is successful
\nExtract mesons data from a bdio file and return it as a dictionary
\n\nThe dictionary can be accessed with a tuple consisting of (type, source_position, kappa1, kappa2)
\n\nread_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\nall: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\nParameters
\n\n\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
\nReturns
\n\n\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": "- data (dict):\nExtracted meson data
\nExtract dSdm data from a bdio file and return it as a dictionary
\n\nThe dictionary can be accessed with a tuple consisting of (type, kappa)
\n\nread_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\nall: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\nParameters
\n\n\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": "- 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
\nExport a list of Obs or structures containing Obs to an xml string\naccording to the Zeuthen pobs format.
\n\nTags are not written or recovered automatically. The separator | is removed from the replica names.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- xml_str (str):\nXML formatted string of the input data
\nExport a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen pobs format.
\n\nTags are not written or recovered automatically. The separator | is removed from the replica names.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- None
\nImport a list of Obs from an xml.gz file in the Zeuthen pobs format.
\n\nTags are not written or recovered automatically.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- res (list[Obs]):\nImported data
\n- or
\n- res (dict):\nImported data and meta-data
\nImport a list of Obs from a string in the Zeuthen dobs format.
\n\nTags are not written or recovered automatically.
\n\nParameters
\n\n\n
\n\n- content (str):\nXML string containing the data
\n- noemtpy (bool):\nIf True, ensembles with no contribution to the Obs are not included.\nIf False, ensembles are included as written in the file, possibly with vanishing entries.
\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.
\nReturns
\n\n\n
\n", "signature": "(content, noempty=False, 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": "- res (list[Obs]):\nImported data
\n- or
\n- res (dict):\nImported data and meta-data
\nImport a list of Obs from an xml.gz file in the Zeuthen dobs format.
\n\nTags are not written or recovered automatically.
\n\nParameters
\n\n\n
\n\n- fname (str):\nFilename of the input file.
\n- noemtpy (bool):\nIf True, ensembles with no contribution to the Obs are not included.\nIf False, ensembles are included as written in 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 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.
\nReturns
\n\n\n
\n", "signature": "(\tfname,\tnoempty=False,\tfull_output=False,\tgz=True,\tseparator_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": "- res (list[Obs]):\nImported data
\n- or
\n- res (dict):\nImported data and meta-data
\nGenerate 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\nTags are not written or recovered automatically. The separator |is removed from the replica names.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- xml_str (str):\nXML string generated from the data
\nExport a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen dobs format.
\n\nTags are not written or recovered automatically. The separator | is removed from the replica names.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- None
\nRead hadrons meson hdf5 file and extract the meson labeled 'meson'
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- corr (Corr):\nCorrelator of the source sink combination in question.
\nRead hadrons DistillationContraction hdf5 files in given directory structure
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- result (dict):\nextracted DistillationContration data
\nndarray(shape, dtype=float, buffer=None, offset=0,\n strides=None, order=None)
\n\nAn 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\nArrays should be constructed using
\n\narray
,zeros
orempty
(refer\nto the See Also section below). The parameters given here refer to\na low-level method (ndarray(...)
) for instantiating an array.For more information, refer to the
\n\nnumpy
module and examine the\nmethods and attributes of an array.Parameters
\n\n\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.
\nAttributes
\n\n\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.,
\nx.flat = 3
(Seendarray.flat
for\nassignment examples; TODO).- 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.,
\nitemsize * size
.- 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
\n(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
).- 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
\nbase
\n(unless that array is also a view). Thebase
array is where the\narray data is actually stored.See Also
\n\n\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. itsdtype.type <numpy.dtype.type>
.Notes
\n\nThere are two modes of creating an array using
\n\n__new__
:\n
\n\n- If
\nbuffer
is None, then onlyshape
,dtype
, andorder
\nare used.- If
\nbuffer
is an object exposing the buffer interface, then\nall keywords are interpreted.No
\n\n__init__
method is needed because the array is fully initialized\nafter the__new__
method.Examples
\n\nThese examples illustrate the low-level
\n\nndarray
constructor. Refer\nto theSee Also
section above for easier ways of constructing an\nndarray.First mode,
\n\nbuffer
is None:\n\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
Second mode:
\n\n\n\n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.__init__": {"fullname": "pyerrors.input.hadrons.Npr_matrix.__init__", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.__init__", "kind": "function", "doc": "\n", "signature": "()"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "\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
Gamma_5 hermitean conjugate
\n\nUses 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\nParameters
\n\n\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.
\nReturns
\n\n\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": "- result (Npr_matrix):\nread Cobs-matrix
\nRead hadrons Bilinear hdf5 file and output an array of CObs
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- result_dict (dict[Npr_matrix]):\nextracted Bilinears
\nRead hadrons FourquarkFullyConnected hdf5 file and output an array of CObs
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- result_dict (dict):\nextracted fourquark matrizes
\nGenerate the string for the export of a list of Obs or structures containing Obs\nto a .json(.gz) file
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- json_string (str):\nString for export to .json(.gz) file
\nExport a list of Obs or structures containing Obs to a .json(.gz) file
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- Null
\nReconstruct a list of Obs or structures containing Obs from a json string.
\n\nThe following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- 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
\nImport a list of Obs or structures containing Obs from a .json(.gz) file.
\n\nThe following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- 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
\nExport a dict of Obs or structures containing Obs to a .json(.gz) file
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- None
\nImport a dict of Obs or structures containing Obs from a .json(.gz) file.
\n\nThe following structures are supported: Obs, list, numpy.ndarray, Corr
\n\nParameters
\n\n\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]+.
\nReturns
\n\n\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": "- data (Obs / list / Corr):\nRead data
\n- or
\n- data (dict):\nRead data and meta-data
\nRead pbp format from given folder structure.
\n\nParameters
\n\n\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
\nReturns
\n\n\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": "- result (list[Obs]):\nlist of observables read
\nRead rwms format from given folder structure. Returns a list of length nrw
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- rwms (Obs):\nReweighting factors read
\nExtract t0 from given .ms.dat files. Returns t0 as Obs.
\n\nIt is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t^2
\n\n- 0.3\nis fitted with a linear function\nfrom which the exact root is extracted. 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\nParameters
\n\n\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
\nReturns
\n\n\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": "- t0 (Obs):\nExtracted t0
\nRead the topologial charge based on openQCD gradient flow measurements.
\n\nParameters
\n\n\n
\n\n- path (str):\npath of the measurement files
\n- prefix (str):\nprefix of the measurement files, e.g.
\n_id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files. - 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.
\nReturns
\n\n\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": "- result (Obs):\nRead topological charge
\nRead the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.
\n\nNote: 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\nParameters
\n\n\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": "- path (str):\npath of the measurement files
\n- prefix (str):\nprefix of the measurement files, e.g.
\n_id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files. - 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.
\nReturns the projection to the topological charge sector defined by target.
\n\nParameters
\n\n\n
\n\n- path (Obs):\nTopological charge.
\n- target (int):\nSpecifies the topological sector to be reweighted to (default 0)
\nReturns
\n\n\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": "- reto (Obs):\nprojection to the topological charge sector defined by target
\nConstructs reweighting factors to a specified topological sector.
\n\nParameters
\n\n\n
\n\n- path (str):\npath of the measurement files
\n- prefix (str):\nprefix of the measurement files, e.g.
\n_id0_r0.ms.dat - 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.
\nReturns
\n\n\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": "- reto (Obs):\nprojection to the topological charge sector defined by target
\nRead data from files in the specified directory with the specified prefix and quark combination extension, and return a
\n\nCorr
object containing the data.Parameters
\n\n\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- \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.
\nReturns
\n\n\n
\n\n- Corr: A complex valued
\nCorr
object containing the data read from the files. In case of boudary to bulk correlators.- or
\n- CObs: A complex valued
\nCObs
object containing the data read from the files. In case of boudary to boundary correlators.Raises
\n\n\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": "- 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.
\nWrite DataFrame including Obs or Corr valued columns to sqlite database.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- None
\nExecute SQL query on sqlite database and obtain DataFrame including Obs or Corr valued columns.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
\nExports a pandas DataFrame containing Obs valued columns to a (gzipped) csv file.
\n\nBefore making use of pandas to_csv functionality Obs objects are serialized via the standardized\njson format of pyerrors.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- None
\nImports a pandas DataFrame from a csv.(gz) file in which Obs objects are serialized as json strings.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
\nRead sfcf c format from given folder structure.
\n\nParameters
\n\n\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\n
- 'bi' for boundary-inner
\n- 'bb' for boundary-boundary
\n- 'bib' for boundary-inner-boundary
\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
\nReturns
\n\n\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": "- result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
\nUtilities 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\nParameters
\n\n\n
\n\n- idl (range or list):\nidl of the current replicum
\n- che (list):\nlist of configurations to be checked against
\nReturns
\n\n\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": "- miss_str (str):\nstring with integers of which idls are missing
\nMatrix multiply all operands.
\n\nParameters
\n\n\n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.jack_matmul": {"fullname": "pyerrors.linalg.jack_matmul", "modulename": "pyerrors.linalg", "qualname": "jack_matmul", "kind": "function", "doc": "- 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.
\nMatrix multiply both operands making use of the jackknife approximation.
\n\nParameters
\n\n\n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.einsum": {"fullname": "pyerrors.linalg.einsum", "modulename": "pyerrors.linalg", "qualname": "einsum", "kind": "function", "doc": "- 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.
\nWrapper for numpy.einsum
\n\nParameters
\n\n\n
\n", "signature": "(subscripts, *operands):", "funcdef": "def"}, "pyerrors.linalg.inv": {"fullname": "pyerrors.linalg.inv", "modulename": "pyerrors.linalg", "qualname": "inv", "kind": "function", "doc": "- 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.
\nInverse 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\nParameters
\n\n\n
\n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.10.9/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": "- 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.
\nDump object into pickle file.
\n\nParameters
\n\n\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 '.')
\nReturns
\n\n\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": "- None
\nLoad object from pickle file.
\n\nParameters
\n\n\n
\n\n- path (str):\npath to the file
\nReturns
\n\n\n
\n", "signature": "(path):", "funcdef": "def"}, "pyerrors.misc.pseudo_Obs": {"fullname": "pyerrors.misc.pseudo_Obs", "modulename": "pyerrors.misc", "qualname": "pseudo_Obs", "kind": "function", "doc": "- object (Obs):\nLoaded Object
\nGenerate an Obs object with given value, dvalue and name for test purposes
\n\nParameters
\n\n\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).
\nReturns
\n\n\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": "- res (Obs):\nGenerated Observable
\nGenerate observables with given covariance and autocorrelation times.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- corr_obs (list[Obs]):\nGenerated observable list
\nMatrix pencil method to extract k energy levels from data
\n\nImplementation of the matrix pencil method based on\neq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)
\n\nParameters
\n\n\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).
\nReturns
\n\n\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": "- energy_levels (list[Obs]):\nExtracted energy levels
\nClass for a general observable.
\n\nInstances 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\nAttributes
\n\n\n
\n"}, "pyerrors.obs.Obs.__init__": {"fullname": "pyerrors.obs.Obs.__init__", "modulename": "pyerrors.obs", "qualname": "Obs.__init__", "kind": "function", "doc": "- 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.
\nInitialize Obs object.
\n\nParameters
\n\n\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": "- 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
\nEstimate the error and related properties of the Obs.
\n\nParameters
\n\n\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": "- 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)
\nEstimate the error and related properties of the Obs.
\n\nParameters
\n\n\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": "- 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)
\nOutput detailed properties of the Obs.
\n\nParameters
\n\n\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": "- ens_content (bool):\nprint details about the ensembles and replica if true.
\nReweight the obs with given rewighting factors.
\n\nParameters
\n\n\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": "- 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.
\nChecks whether the observable is zero within 'sigma' standard errors.
\n\nParameters
\n\n\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": "- sigma (int):\nNumber of standard errors used for the check.
\n- Works only properly when the gamma method was run.
\nChecks whether the observable is zero within a given tolerance.
\n\nParameters
\n\n\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": "- atol (float):\nAbsolute tolerance (for details see numpy documentation).
\nPlot integrated autocorrelation time for each ensemble.
\n\nParameters
\n\n\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": "- save (str):\nsaves the figure to a file named 'save' if.
\nPlot normalized autocorrelation function time for each ensemble.
\n\nParameters
\n\n\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": "- save (str):\nsaves the figure to a file named 'save' if.
\nPlot 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\nParameters
\n\n\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": "- expand (bool):\nshow expanded history for irregular Monte Carlo chains (default: True).
\nPlot piechart which shows the fractional contribution of each\nensemble to the error and returns a dictionary containing the fractions.
\n\nParameters
\n\n\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": "- save (str):\nsaves the figure to a file named 'save' if.
\nDump the Obs to a file 'name' of chosen format.
\n\nParameters
\n\n\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": "- 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 '.')
\nExport jackknife samples from the Obs
\n\nReturns
\n\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": "- 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).
\nClass 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\nParameters
\n\n\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.
\nNotes
\n\nFor simple mathematical operations it can be practical to use anonymous\nfunctions. For the ratio of two observables one can e.g. use
\n\nnew_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\nParameters
\n\n\n
\n", "signature": "(weight, obs, **kwargs):", "funcdef": "def"}, "pyerrors.obs.correlate": {"fullname": "pyerrors.obs.correlate", "modulename": "pyerrors.obs", "qualname": "correlate", "kind": "function", "doc": "- 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.
\nCorrelate two observables.
\n\nParameters
\n\n\n
\n\n- obs_a (Obs):\nFirst observable
\n- obs_b (Obs):\nSecond observable
\nNotes
\n\nKeep 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\nWARNING: 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\nThe gamma method has to be applied first to all observables.
\n\nParameters
\n\n\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.
\nNotes
\n\nThe 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\nParameters
\n\n\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": "- 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.
\nCombine all observables in list_of_obs into one new observable
\n\nParameters
\n\n\n
\n\n- list_of_obs (list):\nlist of the Obs object to be combined
\nNotes
\n\nIt 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\nParameters
\n\n\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": "- 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.
\nFinds the root of the function func(x, d) where d is an
\n\nObs
.Parameters
\n\n\n
\n\n- d (Obs):\nObs passed to the function.
\n- \n
func (object):\nFunction to be minimized. Any numpy functions have to use the autograd.numpy wrapper.\nExample:
\n\n\n\nimport autograd.numpy as anp\ndef root_func(x, d):\n return anp.exp(-x ** 2) - d\n
- \n
guess (float):\nInitial guess for the minimization.
Returns
\n\n\n
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\nObs
valued root of the function.What is pyerrors?
\n\n\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...).
\nMore detailed examples can found in the GitHub repository
\n\n.
If you use
\n\npyerrors
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.
\nand
\n\n\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.
\nwhere applicable.
\n\nThere exist similar publicly available implementations of gamma method error analysis suites in Fortran, Julia and Python.
\n\nBasic example
\n\n\n\n\n\nimport 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
The
\n\nObs
class\n\n
pyerrors
introduces a new datatype,Obs
, which simplifies error propagation and estimation for auto- and cross-correlated data.\nAnObs
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\nimport pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name'])\n
Error propagation
\n\nWhen performing mathematical operations on
\n\nObs
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.The
\n\nObs
class is designed such that mathematical numpy functions can be used onObs
just as for regular floats.\n\n\n\nimport 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
Error estimation
\n\nThe error estimation within
\n\npyerrors
is based on the gamma method introduced in arXiv:hep-lat/0306017.\nAfter having arrived at the derived quantity of interest thegamma_method
can be called as detailed in the following example.\n\n\n\nmy_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
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
\n\ngamma_method
as parameter.\n\n\n\nmy_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
The integrated autocorrelation time $\\tau_\\mathrm{int}$ and the autocorrelation function $\\rho(W)$ can be monitored via the methods
\n\npyerrors.obs.Obs.plot_tauint
andpyerrors.obs.Obs.plot_rho
.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\nExponential tails
\n\nSlow 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
\n\ngamma_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\nmy_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
For the full API see
\n\npyerrors.obs.Obs.gamma_method
.Multiple ensembles/replica
\n\nError propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their
\n\nname
.\n\n\n\nobs1 = 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
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\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\nobs1 = 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
Error estimation for multiple ensembles
\n\nIn 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\n\n\npe.Obs.S_dict['ensemble1'] = 2.5\npe.Obs.tau_exp_dict['ensemble2'] = 8.0\npe.Obs.tau_exp_dict['ensemble3'] = 2.0\n
In case the
\n\ngamma_method
is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.\nPassing arguments to thegamma_method
still dominates over the dictionaries.Irregular Monte Carlo chains
\n\n\n\n
Obs
objects defined on irregular Monte Carlo chains can be initialized with the parameteridl
.\n\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
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.Warning: Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.\nMake sure to check the autocorrelation time with e.g.
\n\npyerrors.obs.Obs.plot_rho
orpyerrors.obs.Obs.plot_tauint
.For the full API see
\n\npyerrors.obs.Obs
.Correlators
\n\nWhen one is not interested in single observables but correlation functions,
\n\npyerrors
offers theCorr
class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize aCorr
objects one needs to arrange the data as a list ofObs
\n\n\n\nmy_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
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\n\n\nmy_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
The individual entries of a correlator can be accessed via slicing
\n\n\n\n\n\nprint(my_corr[3])\n> 0.3227(33)\n
Error propagation with the
\n\nCorr
class works very similar toObs
objects. Mathematical operations are overloaded andCorr
objects can be computed together with otherCorr
objects,Obs
objects or real numbers and integers.\n\n\n\nmy_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr\n
\n\n
pyerrors
provides the user with a set of regularly used methods for the manipulation of correlator objects:\n
\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 asCorr
. Different discretizations of the numerical derivative are available.- \n
Corr.second_deriv
returns the second derivative of the correlator asCorr
. 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 anotherCorr
orObs
object.- \n
Corr.reweight
reweights the correlator.\n\n
pyerrors
can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (seepyerrors.correlators.Corr.GEVP
).For the full API see
\n\npyerrors.correlators.Corr
.Complex valued observables
\n\n\n\n
pyerrors
can handle complex valued observables via the classpyerrors.obs.CObs
.\nCObs
are initialized with a real and an imaginary part which both can beObs
valued.\n\n\n\nmy_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
Elementary mathematical operations are overloaded and samples are properly propagated as for the
\n\nObs
class.\n\n\n\nmy_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
The
\n\nCovobs
classIn 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
\n\nCovobs
class allows to define such quantities inpyerrors
. 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.This concept is built into the definition of
\n\nCovobs
. Inpyerrors
, 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 enterscov_Obs
, since the second argument of this function is the covariance matrix of theCovobs
.\n\n\n\nimport 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
The resulting object
\n\nmpi
is anObs
that contains aCovobs
. In the following, it may be handled as any otherObs
. The contribution of the covariance matrix to the error of anObs
is determined from the $M \\times M$ covariance matrix $\\Sigma$ and the gradient of theObs
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.Correlated auxiliary data is defined similarly to above, e.g., via
\n\n\n\n\n\nRAP = 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
where
\n\nRAP
now is a list of twoObs
that contains the two correlated parameters.Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the
\n\nCovobs
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 anObs
o
with respect to a covariance matrix with the identifying stringk
may be accessed via\n\n\n\no.covobs[k].grad\n
Error propagation in iterative algorithms
\n\n\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.Least squares fits
\n\nStandard non-linear least square fits with errors on the dependent but not the independent variables can be performed with
\n\npyerrors.fits.least_squares
. As default solver the Levenberg-Marquardt algorithm implemented in scipy is used.Fit functions have to be of the following form
\n\n\n\n\n\nimport autograd.numpy as anp\n\ndef func(a, x):\n return a[1] * anp.exp(-a[0] * x)\n
It is important that numerical functions refer to
\n\nautograd.numpy
instead ofnumpy
for the automatic differentiation in iterative algorithms to work properly.Fits can then be performed via
\n\n\n\n\n\nfit_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
where x is a
\n\nlist
ornumpy.array
offloats
and y is alist
ornumpy.array
ofObs
.Data stored in
\n\nCorr
objects can be fitted directly using theCorr.fit
method.\n\n\n\nmy_corr = pe.Corr(y)\nfit_result = my_corr.fit(func, fitrange=[12, 25])\n
this can simplify working with absolute fit ranges and takes care of gaps in the data automatically.
\n\nFor fit functions with multiple independent variables the fit function can be of the form
\n\n\n\n\n\ndef func(a, x):\n (x1, x2) = x\n return a[0] * x1 ** 2 + a[1] * x2\n
\n\n
pyerrors
also supports correlated fits which can be triggered via the parametercorrelated_fit=True
.\nDetails about how the required covariance matrix is estimated can be found inpyerrors.obs.covariance
.Direct visualizations of the performed fits can be triggered via
\n\nresplot=True
orqqplot=True
. For all available options seepyerrors.fits.least_squares
.Total least squares fits
\n\n\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, seepyerrors.fits.least_squares
. The syntax is identical to the standard least squares case, the only difference being thatx
also has to be alist
ornumpy.array
ofObs
.For the full API see
\n\npyerrors.fits
for fits andpyerrors.roots
for finding roots of functions.Matrix operations
\n\n\n\n
pyerrors
provides wrappers forObs
- andCObs
-valued matrix operations based onnumpy.linalg
. The supported functions include:\n
\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.For the full API see
\n\npyerrors.linalg
.Export data
\n\nThe preferred exported file format within
\n\npyerrors
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?
\nThis can be achieved by storing all information in one single file. The export routines of
\n\npyerrors
are written such that as much information as possible is written automatically as described in the following example\n\n\n\nmy_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
The format also allows to directly write out the content of
\n\nCorr
objects or lists and arrays ofObs
objects by passing the desired data topyerrors.input.json.dump_to_json
.json.gz format specification
\n\nThe first entries of the file provide optional auxiliary information:
\n\n\n
\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 inpyerrors
. 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.The only necessary entry of the file is the field\n-
\n\nobsdata
, an array that contains the actual data.Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of
\n\nObs
,list
,numpy.ndarray
,Corr
. AllObs
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 arrayobsdata
, are treated independently. Each entry of the arrayobsdata
has the following required entries:\n
\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. Thetag
of anObs
inpyerrors
is written here.- \n
reweighted
is a Bool that may be used to specify, whether theObs
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
inpyerrors
). We will define it below.The array
\n\ndata
contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains:\n
\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.Each entry of
\n\nreplica
contains\nname
, a string that contains the name of the replica\ndeltas
, an array that contains the actual data.Each entry in
\n\ndeltas
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 eachObs
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.The array
\n\ncdata
contains information about the contribution of auxiliary observables, represented byCovobs
inpyerrors
, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains:\n
\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 eachObs
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.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\nJulia 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\nEverything, 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\nThe 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\nParameters
\n\n\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": "- 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.
\nApply 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\nThe 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\nParameters
\n\n\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": "- i (int):\nFirst index to be picked.
\n- j (int):\nSecond index to be picked.
\nOutputs the correlator in a plotable format.
\n\nOutputs 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\nThe 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\n\n\nC.GEVP(t0=2)[0] # Ground state vector(s)\nC.GEVP(t0=2)[:3] # Vectors for the lowest three states\n
Parameters
\n\n\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\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 $t=t_s$.
\nOther Parameters
\n\n\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": "- state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
\nDetermines the eigenvalue of the GEVP by solving and projecting the correlator
\n\nParameters
\n\n\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": "- 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.
\nConstructs an NxN Hankel matrix
\n\nC(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\nParameters
\n\n\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": "- N (int):\nDimension of the Hankel matrix
\n- periodic (bool, optional):\ndetermines whether the matrix is extended periodically
\nPeriodically shift the correlator by dt timeslices
\n\nParameters
\n\n\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": "- dt (int):\nnumber of timeslices
\nReverse 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\nParameters
\n\n\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": "- spacing (int):\nKeep only every 'spacing'th entry of the correlator
\n- offset (int):\nOffset the equal spacing
\nCorrelate the correlator with another correlator or Obs
\n\nParameters
\n\n\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": "- 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.
\nReweight the correlator.
\n\nParameters
\n\n\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": "- 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.
\nReturn the time symmetry average of the correlator and its partner
\n\nParameters
\n\n\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": "- partner (Corr):\nTime symmetry partner of the Corr
\n- partity (int):\nParity quantum number of the correlator, can be +1 or -1
\nReturn the first derivative of the correlator with respect to x0.
\n\nParameters
\n\n\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": "- variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, forward, backward, improved, log, default: symmetric
\nReturn the second derivative of the correlator with respect to x0.
\n\nParameters
\n\n\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": "- variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, improved, log, default: symmetric
\nReturns the effective mass of the correlator as correlator object
\n\nParameters
\n\n\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": "- 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
\nFits function to the data
\n\nParameters
\n\n\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": "- 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.\n
\nfitrange=[4, 6]
corresponds to the three entries 4, 5 and 6.\nIf not specified, self.prange or all timeslices are used.- silent (bool):\nDecides whether output is printed to the standard output.
\nExtract a plateau value from a Corr object
\n\nParameters
\n\n\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": "- 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
\nSets 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\nParameters
\n\n\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": "- 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.
\nProduces a spaghetti plot of the correlator suited to monitor exceptional configurations.
\n\nParameters
\n\n\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": "- logscale (bool):\nDetermines whether the scale of the y-axis is logarithmic or standard.
\nDumps the Corr into a file of chosen type
\n\nParameters
\n\n\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": "- 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 '.')
\nProject large correlation matrix to lowest states
\n\nThis 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\nParameters
\n\n\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.
\nNotes
\n\nWe 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\nParameters
\n\n\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": "- 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.
\nReturn 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\nBased on https://codegolf.stackexchange.com/a/160375
\n\nReturns
\n\n\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": "- elem (int):\nElement (i,j,k) of the epsilon tensor of rank 3
\nRank-4 epsilon tensor
\n\nExtension of https://codegolf.stackexchange.com/a/160375
\n\nReturns
\n\n\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": "- elem (int):\nElement (i,j,k,o) of the epsilon tensor of rank 4
\nReturns 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\nAttributes
\n\n\n
\n", "bases": "collections.abc.Sequence"}, "pyerrors.fits.Fit_result.__init__": {"fullname": "pyerrors.fits.Fit_result.__init__", "modulename": "pyerrors.fits", "qualname": "Fit_result.__init__", "kind": "function", "doc": "\n", "signature": "()"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "- 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.
\nApply 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\nParameters
\n\n\n
\n\n- For an uncombined fit:
\n- x (list):\nlist of floats.
\n- y (list):\nlist of Obs.
\n- \n
func (object):\nfit function, has to be of the form
\n\n\n\n\n\nimport autograd.numpy as anp\n\ndef func(a, x):\n return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
For multiple x values func can be of the form
\n\n\n\n\n\ndef func(a, x):\n (x1, x2) = x\n return a[0] * x1 ** 2 + a[1] * x2\n
It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
- OR For a combined fit:
\n- x (dict):\ndict of lists.
\n- y (dict):\ndict of lists of Obs.
\n- \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\ndef func_a(a, x):\n return a[1] * anp.exp(-a[0] * x)
\n\ndef func_b(a, x):\n return a[2] * anp.exp(-a[0] * x)
\n\nIt is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
- priors (list, optional):\npriors has to be a 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
\npyerrors.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 forprior==None
and when nomethod
is given.- 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).
\nReturns
\n\n\n
\n", "signature": "(x, y, func, priors=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.total_least_squares": {"fullname": "pyerrors.fits.total_least_squares", "modulename": "pyerrors.fits", "qualname": "total_least_squares", "kind": "function", "doc": "- output (Fit_result):\nParameters and information on the fitted result.
\nPerforms a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
\n\nParameters
\n\n\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- \n
func (object):\nfunc has to be of the form
\n\n\n\n\n\nimport autograd.numpy as anp\n\ndef func(a, x):\n return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
For multiple x values func can be of the form
\n\n\n\n\n\ndef func(a, x):\n (x1, x2) = x\n return a[0] * x1 ** 2 + a[1] * x2\n
It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
- 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).
\nNotes
\n\nBased on the orthogonal distance regression module of scipy.
\n\nReturns
\n\n\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": "- output (Fit_result):\nParameters and information on the fitted result.
\nPerforms a linear fit to y = n + m * x and returns two Obs n, m.
\n\nParameters
\n\n\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.
\nReturns
\n\n\n
\n", "signature": "(x, y, **kwargs):", "funcdef": "def"}, "pyerrors.fits.qqplot": {"fullname": "pyerrors.fits.qqplot", "modulename": "pyerrors.fits", "qualname": "qqplot", "kind": "function", "doc": "- fit_parameters (list[Obs]):\nLIist of fitted observables.
\nGenerates 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\nReturns
\n\n\n
\n", "signature": "(x, o_y, func, p):", "funcdef": "def"}, "pyerrors.fits.residual_plot": {"fullname": "pyerrors.fits.residual_plot", "modulename": "pyerrors.fits", "qualname": "residual_plot", "kind": "function", "doc": "- None
\nGenerates a plot which compares the fit to the data and displays the corresponding residuals
\n\nReturns
\n\n\n
\n", "signature": "(x, y, func, fit_res):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "- None
\nCalculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
\n\nReturns
\n\n\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": "- err (np.array(Obs)):\nError band for an array of sample values x
\nPerforms a Kolmogorov\u2013Smirnov test for the p-values of all fit object.
\n\nParameters
\n\n\n
\n\n- objects (list):\nList of fit results to include in the analysis (optional).
\nReturns
\n\n\n
\n", "signature": "(objects=None):", "funcdef": "def"}, "pyerrors.input": {"fullname": "pyerrors.input", "modulename": "pyerrors.input", "kind": "module", "doc": "- None
\n\n\n
pyerrors
includes aninput
submodule in which input routines and parsers for the output of various numerical programs are contained.Jackknife samples
\n\nFor comparison with other analysis workflows
\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": "pyerrors
can also generate jackknife samples from anObs
object or import jackknife samples into anObs
object.\nSeepyerrors.obs.Obs.export_jackknife
andpyerrors.obs.import_jackknife
for details.Extract generic MCMC data from a bdio file
\n\nread_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\nall: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\nParameters
\n\n\n
\n\n- file_path -- path to the bdio file
\n- bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
\nReturns
\n\n\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": "- data (List[Obs]):\nExtracted data
\nWrite Obs to a bdio file according to ADerrors conventions
\n\nread_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\nall: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\nParameters
\n\n\n
\n\n- file_path -- path to the bdio file
\n- bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
\nReturns
\n\n\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": "- success (int):\nreturns 0 is successful
\nExtract mesons data from a bdio file and return it as a dictionary
\n\nThe dictionary can be accessed with a tuple consisting of (type, source_position, kappa1, kappa2)
\n\nread_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\nall: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\nParameters
\n\n\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
\nReturns
\n\n\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": "- data (dict):\nExtracted meson data
\nExtract dSdm data from a bdio file and return it as a dictionary
\n\nThe dictionary can be accessed with a tuple consisting of (type, kappa)
\n\nread_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\nall: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\nParameters
\n\n\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": "- 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
\nExport a list of Obs or structures containing Obs to an xml string\naccording to the Zeuthen pobs format.
\n\nTags are not written or recovered automatically. The separator | is removed from the replica names.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- xml_str (str):\nXML formatted string of the input data
\nExport a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen pobs format.
\n\nTags are not written or recovered automatically. The separator | is removed from the replica names.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- None
\nImport a list of Obs from an xml.gz file in the Zeuthen pobs format.
\n\nTags are not written or recovered automatically.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- res (list[Obs]):\nImported data
\n- or
\n- res (dict):\nImported data and meta-data
\nImport a list of Obs from a string in the Zeuthen dobs format.
\n\nTags are not written or recovered automatically.
\n\nParameters
\n\n\n
\n\n- content (str):\nXML string containing the data
\n- noemtpy (bool):\nIf True, ensembles with no contribution to the Obs are not included.\nIf False, ensembles are included as written in the file, possibly with vanishing entries.
\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.
\nReturns
\n\n\n
\n", "signature": "(content, noempty=False, 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": "- res (list[Obs]):\nImported data
\n- or
\n- res (dict):\nImported data and meta-data
\nImport a list of Obs from an xml.gz file in the Zeuthen dobs format.
\n\nTags are not written or recovered automatically.
\n\nParameters
\n\n\n
\n\n- fname (str):\nFilename of the input file.
\n- noemtpy (bool):\nIf True, ensembles with no contribution to the Obs are not included.\nIf False, ensembles are included as written in 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 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.
\nReturns
\n\n\n
\n", "signature": "(\tfname,\tnoempty=False,\tfull_output=False,\tgz=True,\tseparator_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": "- res (list[Obs]):\nImported data
\n- or
\n- res (dict):\nImported data and meta-data
\nGenerate 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\nTags are not written or recovered automatically. The separator |is removed from the replica names.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- xml_str (str):\nXML string generated from the data
\nExport a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen dobs format.
\n\nTags are not written or recovered automatically. The separator | is removed from the replica names.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- None
\nRead hadrons meson hdf5 file and extract the meson labeled 'meson'
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- corr (Corr):\nCorrelator of the source sink combination in question.
\nRead hadrons DistillationContraction hdf5 files in given directory structure
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- result (dict):\nextracted DistillationContration data
\nndarray(shape, dtype=float, buffer=None, offset=0,\n strides=None, order=None)
\n\nAn 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\nArrays should be constructed using
\n\narray
,zeros
orempty
(refer\nto the See Also section below). The parameters given here refer to\na low-level method (ndarray(...)
) for instantiating an array.For more information, refer to the
\n\nnumpy
module and examine the\nmethods and attributes of an array.Parameters
\n\n\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.
\nAttributes
\n\n\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.,
\nx.flat = 3
(Seendarray.flat
for\nassignment examples; TODO).- 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.,
\nitemsize * size
.- 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
\n(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
).- 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
\nbase
\n(unless that array is also a view). Thebase
array is where the\narray data is actually stored.See Also
\n\n\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. itsdtype.type <numpy.dtype.type>
.Notes
\n\nThere are two modes of creating an array using
\n\n__new__
:\n
\n\n- If
\nbuffer
is None, then onlyshape
,dtype
, andorder
\nare used.- If
\nbuffer
is an object exposing the buffer interface, then\nall keywords are interpreted.No
\n\n__init__
method is needed because the array is fully initialized\nafter the__new__
method.Examples
\n\nThese examples illustrate the low-level
\n\nndarray
constructor. Refer\nto theSee Also
section above for easier ways of constructing an\nndarray.First mode,
\n\nbuffer
is None:\n\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
Second mode:
\n\n\n\n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.__init__": {"fullname": "pyerrors.input.hadrons.Npr_matrix.__init__", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.__init__", "kind": "function", "doc": "\n", "signature": "()"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "\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
Gamma_5 hermitean conjugate
\n\nUses 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\nParameters
\n\n\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.
\nReturns
\n\n\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": "- result (Npr_matrix):\nread Cobs-matrix
\nRead hadrons Bilinear hdf5 file and output an array of CObs
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- result_dict (dict[Npr_matrix]):\nextracted Bilinears
\nRead hadrons FourquarkFullyConnected hdf5 file and output an array of CObs
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- result_dict (dict):\nextracted fourquark matrizes
\nGenerate the string for the export of a list of Obs or structures containing Obs\nto a .json(.gz) file
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- json_string (str):\nString for export to .json(.gz) file
\nExport a list of Obs or structures containing Obs to a .json(.gz) file
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- Null
\nReconstruct a list of Obs or structures containing Obs from a json string.
\n\nThe following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- 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
\nImport a list of Obs or structures containing Obs from a .json(.gz) file.
\n\nThe following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- 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
\nExport a dict of Obs or structures containing Obs to a .json(.gz) file
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- None
\nImport a dict of Obs or structures containing Obs from a .json(.gz) file.
\n\nThe following structures are supported: Obs, list, numpy.ndarray, Corr
\n\nParameters
\n\n\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]+.
\nReturns
\n\n\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": "- data (Obs / list / Corr):\nRead data
\n- or
\n- data (dict):\nRead data and meta-data
\nRead pbp format from given folder structure.
\n\nParameters
\n\n\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
\nReturns
\n\n\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": "- result (list[Obs]):\nlist of observables read
\nRead rwms format from given folder structure. Returns a list of length nrw
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- rwms (Obs):\nReweighting factors read
\nExtract t0 from given .ms.dat files. Returns t0 as Obs.
\n\nIt is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t^2
\n\n- 0.3\nis fitted with a linear function\nfrom which the exact root is extracted. 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\nParameters
\n\n\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
\nReturns
\n\n\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": "- t0 (Obs):\nExtracted t0
\nRead the topologial charge based on openQCD gradient flow measurements.
\n\nParameters
\n\n\n
\n\n- path (str):\npath of the measurement files
\n- prefix (str):\nprefix of the measurement files, e.g.
\n_id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files. - 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.
\nReturns
\n\n\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": "- result (Obs):\nRead topological charge
\nRead the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.
\n\nNote: 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\nParameters
\n\n\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": "- path (str):\npath of the measurement files
\n- prefix (str):\nprefix of the measurement files, e.g.
\n_id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files. - 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.
\nReturns the projection to the topological charge sector defined by target.
\n\nParameters
\n\n\n
\n\n- path (Obs):\nTopological charge.
\n- target (int):\nSpecifies the topological sector to be reweighted to (default 0)
\nReturns
\n\n\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": "- reto (Obs):\nprojection to the topological charge sector defined by target
\nConstructs reweighting factors to a specified topological sector.
\n\nParameters
\n\n\n
\n\n- path (str):\npath of the measurement files
\n- prefix (str):\nprefix of the measurement files, e.g.
\n_id0_r0.ms.dat - 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.
\nReturns
\n\n\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": "- reto (Obs):\nprojection to the topological charge sector defined by target
\nRead data from files in the specified directory with the specified prefix and quark combination extension, and return a
\n\nCorr
object containing the data.Parameters
\n\n\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- \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.
\nReturns
\n\n\n
\n\n- Corr: A complex valued
\nCorr
object containing the data read from the files. In case of boudary to bulk correlators.- or
\n- CObs: A complex valued
\nCObs
object containing the data read from the files. In case of boudary to boundary correlators.Raises
\n\n\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": "- 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.
\nWrite DataFrame including Obs or Corr valued columns to sqlite database.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- None
\nExecute SQL query on sqlite database and obtain DataFrame including Obs or Corr valued columns.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
\nExports a pandas DataFrame containing Obs valued columns to a (gzipped) csv file.
\n\nBefore making use of pandas to_csv functionality Obs objects are serialized via the standardized\njson format of pyerrors.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- None
\nImports a pandas DataFrame from a csv.(gz) file in which Obs objects are serialized as json strings.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
\nRead sfcf c format from given folder structure.
\n\nParameters
\n\n\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\n
- 'bi' for boundary-inner
\n- 'bb' for boundary-boundary
\n- 'bib' for boundary-inner-boundary
\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
\nReturns
\n\n\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": "- result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
\nUtilities 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\nParameters
\n\n\n
\n\n- idl (range or list):\nidl of the current replicum
\n- che (list):\nlist of configurations to be checked against
\nReturns
\n\n\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": "- miss_str (str):\nstring with integers of which idls are missing
\nMatrix multiply all operands.
\n\nParameters
\n\n\n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.jack_matmul": {"fullname": "pyerrors.linalg.jack_matmul", "modulename": "pyerrors.linalg", "qualname": "jack_matmul", "kind": "function", "doc": "- 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.
\nMatrix multiply both operands making use of the jackknife approximation.
\n\nParameters
\n\n\n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.einsum": {"fullname": "pyerrors.linalg.einsum", "modulename": "pyerrors.linalg", "qualname": "einsum", "kind": "function", "doc": "- 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.
\nWrapper for numpy.einsum
\n\nParameters
\n\n\n
\n", "signature": "(subscripts, *operands):", "funcdef": "def"}, "pyerrors.linalg.inv": {"fullname": "pyerrors.linalg.inv", "modulename": "pyerrors.linalg", "qualname": "inv", "kind": "function", "doc": "- 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.
\nInverse 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\nParameters
\n\n\n
\n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.10.9/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": "- 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.
\nDump object into pickle file.
\n\nParameters
\n\n\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 '.')
\nReturns
\n\n\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": "- None
\nLoad object from pickle file.
\n\nParameters
\n\n\n
\n\n- path (str):\npath to the file
\nReturns
\n\n\n
\n", "signature": "(path):", "funcdef": "def"}, "pyerrors.misc.pseudo_Obs": {"fullname": "pyerrors.misc.pseudo_Obs", "modulename": "pyerrors.misc", "qualname": "pseudo_Obs", "kind": "function", "doc": "- object (Obs):\nLoaded Object
\nGenerate an Obs object with given value, dvalue and name for test purposes
\n\nParameters
\n\n\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).
\nReturns
\n\n\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": "- res (Obs):\nGenerated Observable
\nGenerate observables with given covariance and autocorrelation times.
\n\nParameters
\n\n\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.
\nReturns
\n\n\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": "- corr_obs (list[Obs]):\nGenerated observable list
\nMatrix pencil method to extract k energy levels from data
\n\nImplementation of the matrix pencil method based on\neq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)
\n\nParameters
\n\n\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).
\nReturns
\n\n\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": "- energy_levels (list[Obs]):\nExtracted energy levels
\nClass for a general observable.
\n\nInstances 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\nAttributes
\n\n\n
\n"}, "pyerrors.obs.Obs.__init__": {"fullname": "pyerrors.obs.Obs.__init__", "modulename": "pyerrors.obs", "qualname": "Obs.__init__", "kind": "function", "doc": "- 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.
\nInitialize Obs object.
\n\nParameters
\n\n\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": "- 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
\nEstimate the error and related properties of the Obs.
\n\nParameters
\n\n\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": "- 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)
\nEstimate the error and related properties of the Obs.
\n\nParameters
\n\n\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": "- 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)
\nOutput detailed properties of the Obs.
\n\nParameters
\n\n\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": "- ens_content (bool):\nprint details about the ensembles and replica if true.
\nReweight the obs with given rewighting factors.
\n\nParameters
\n\n\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": "- 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.
\nChecks whether the observable is zero within 'sigma' standard errors.
\n\nParameters
\n\n\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": "- sigma (int):\nNumber of standard errors used for the check.
\n- Works only properly when the gamma method was run.
\nChecks whether the observable is zero within a given tolerance.
\n\nParameters
\n\n\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": "- atol (float):\nAbsolute tolerance (for details see numpy documentation).
\nPlot integrated autocorrelation time for each ensemble.
\n\nParameters
\n\n\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": "- save (str):\nsaves the figure to a file named 'save' if.
\nPlot normalized autocorrelation function time for each ensemble.
\n\nParameters
\n\n\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": "- save (str):\nsaves the figure to a file named 'save' if.
\nPlot 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\nParameters
\n\n\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": "- expand (bool):\nshow expanded history for irregular Monte Carlo chains (default: True).
\nPlot piechart which shows the fractional contribution of each\nensemble to the error and returns a dictionary containing the fractions.
\n\nParameters
\n\n\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": "- save (str):\nsaves the figure to a file named 'save' if.
\nDump the Obs to a file 'name' of chosen format.
\n\nParameters
\n\n\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": "- 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 '.')
\nExport jackknife samples from the Obs
\n\nReturns
\n\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": "- 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).
\nClass 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\nParameters
\n\n\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.
\nNotes
\n\nFor simple mathematical operations it can be practical to use anonymous\nfunctions. For the ratio of two observables one can e.g. use
\n\nnew_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\nParameters
\n\n\n
\n", "signature": "(weight, obs, **kwargs):", "funcdef": "def"}, "pyerrors.obs.correlate": {"fullname": "pyerrors.obs.correlate", "modulename": "pyerrors.obs", "qualname": "correlate", "kind": "function", "doc": "- 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.
\nCorrelate two observables.
\n\nParameters
\n\n\n
\n\n- obs_a (Obs):\nFirst observable
\n- obs_b (Obs):\nSecond observable
\nNotes
\n\nKeep 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\nWARNING: 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\nThe gamma method has to be applied first to all observables.
\n\nParameters
\n\n\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.
\nNotes
\n\nThe 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\nParameters
\n\n\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": "- 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.
\nCombine all observables in list_of_obs into one new observable
\n\nParameters
\n\n\n
\n\n- list_of_obs (list):\nlist of the Obs object to be combined
\nNotes
\n\nIt 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\nParameters
\n\n\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": "- 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.
\nFinds the root of the function func(x, d) where d is an
\n\nObs
.Parameters
\n\n\n
\n\n- d (Obs):\nObs passed to the function.
\n- \n
func (object):\nFunction to be minimized. Any numpy functions have to use the autograd.numpy wrapper.\nExample:
\n\n\n\nimport autograd.numpy as anp\ndef root_func(x, d):\n return anp.exp(-x ** 2) - d\n
- \n
guess (float):\nInitial guess for the minimization.
Returns
\n\n\n
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"pyerrors.input.dobs.write_dobs": {"tf": 1}, "pyerrors.input.openQCD.read_qtop": {"tf": 1.4142135623730951}, "pyerrors.input.openQCD.read_gf_coupling": {"tf": 1.4142135623730951}, "pyerrors.input.openQCD.read_qtop_sector": {"tf": 1.4142135623730951}}, "df": 10}}}}}}}}}}, "pipeline": ["trimmer"], "_isPrebuiltIndex": true}; // mirrored in build-search-index.js (part 1) // Also split on html tags. this is a cheap heuristic, but good enough.- res (Obs):\n
\nObs
valued root of the function.