diff --git a/docs/pyerrors.html b/docs/pyerrors.html index 6d73dfc3..1e183e0c 100644 --- a/docs/pyerrors.html +++ b/docs/pyerrors.html @@ -3,7 +3,7 @@
- +pyerrors.obs.covariance
.
In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).
This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
+>>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set
+>>> import numpy as np
+>>> from scipy.stats import norm
+>>> from scipy.linalg import cholesky
+>>> import pyerrors as pe
+>>> # generating the random data set
+>>> num_samples = 400
+>>> N = 3
+>>> x = np.arange(N)
+>>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers
+>>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers
+>>> r = r1 = r2 = np.zeros((N, N))
+>>> y = {}
+>>> for i in range(N):
+>>> for j in range(N):
+>>> r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix
+>>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors
+>>> for i in range(N):
+>>> for j in range(N):
+>>> r[i, j] *= errl[i] * errl[j] # element in covariance matrix
+>>> c = cholesky(r, lower=True)
+>>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined
+>>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built
+>>> x_dict = {}
+>>> y_dict = {}
+>>> chol_inv_dict = {}
+>>> data = []
+>>> for key in y.keys():
+>>> x_dict[key] = x
+>>> for i in range(N):
+>>> data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data
+>>> [o.gamma_method() for o in data]
+>>> corr = pe.covariance(data, correlation=True)
+>>> inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data]))
+>>> chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below
+>>> y_dict = {'a': data[:3], 'b': data[3:]}
+>>> # common fit parameter p[0] in combined fit
+>>> def fit1(p, x):
+>>> return p[0] + p[1] * x
+>>> def fit2(p, x):
+>>> return p[0] + p[2] * x
+>>> fitf_dict = {'a': fit1, 'b':fit2}
+>>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']])
+Fit with 3 parameters
+Method: Levenberg-Marquardt
+`ftol` termination condition is satisfied.
+chisquare/d.o.f.: 0.5388013574561786 # random
+fit parameters [1.11897846 0.96361162 0.92325319] # random
+
+445def total_least_squares(x, y, func, silent=False, **kwargs): -446 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. -447 -448 Parameters -449 ---------- -450 x : list -451 list of Obs, or a tuple of lists of Obs -452 y : list -453 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. -454 func : object -455 func has to be of the form -456 -457 ```python -458 import autograd.numpy as anp -459 -460 def func(a, x): -461 return a[0] + a[1] * x + a[2] * anp.sinh(x) -462 ``` -463 -464 For multiple x values func can be of the form -465 -466 ```python -467 def func(a, x): -468 (x1, x2) = x -469 return a[0] * x1 ** 2 + a[1] * x2 -470 ``` -471 -472 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation -473 will not work. -474 silent : bool, optional -475 If true all output to the console is omitted (default False). -476 initial_guess : list -477 can provide an initial guess for the input parameters. Relevant for non-linear -478 fits with many parameters. -479 expected_chisquare : bool -480 If true prints the expected chisquare which is -481 corrected by effects caused by correlated input data. -482 This can take a while as the full correlation matrix -483 has to be calculated (default False). -484 num_grad : bool -485 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). -486 -487 Notes -488 ----- -489 Based on the orthogonal distance regression module of scipy. -490 -491 Returns -492 ------- -493 output : Fit_result -494 Parameters and information on the fitted result. -495 ''' -496 -497 output = Fit_result() -498 -499 output.fit_function = func -500 -501 x = np.array(x) -502 -503 x_shape = x.shape -504 -505 if kwargs.get('num_grad') is True: -506 jacobian = num_jacobian -507 hessian = num_hessian -508 else: -509 jacobian = auto_jacobian -510 hessian = auto_hessian -511 -512 if not callable(func): -513 raise TypeError('func has to be a function.') -514 -515 for i in range(42): -516 try: -517 func(np.arange(i), x.T[0]) -518 except TypeError: -519 continue -520 except IndexError: -521 continue -522 else: -523 break -524 else: -525 raise RuntimeError("Fit function is not valid.") -526 -527 n_parms = i -528 if not silent: -529 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) -530 -531 x_f = np.vectorize(lambda o: o.value)(x) -532 dx_f = np.vectorize(lambda o: o.dvalue)(x) -533 y_f = np.array([o.value for o in y]) -534 dy_f = np.array([o.dvalue for o in y]) -535 -536 if np.any(np.asarray(dx_f) <= 0.0): -537 raise Exception('No x errors available, run the gamma method first.') -538 -539 if np.any(np.asarray(dy_f) <= 0.0): -540 raise Exception('No y errors available, run the gamma method first.') -541 -542 if 'initial_guess' in kwargs: -543 x0 = kwargs.get('initial_guess') -544 if len(x0) != n_parms: -545 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) -546 else: -547 x0 = [1] * n_parms +@@ -1862,35 +2047,35 @@ Parameters and information on the fitted result.507def total_least_squares(x, y, func, silent=False, **kwargs): +508 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. +509 +510 Parameters +511 ---------- +512 x : list +513 list of Obs, or a tuple of lists of Obs +514 y : list +515 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. +516 func : object +517 func has to be of the form +518 +519 ```python +520 import autograd.numpy as anp +521 +522 def func(a, x): +523 return a[0] + a[1] * x + a[2] * anp.sinh(x) +524 ``` +525 +526 For multiple x values func can be of the form +527 +528 ```python +529 def func(a, x): +530 (x1, x2) = x +531 return a[0] * x1 ** 2 + a[1] * x2 +532 ``` +533 +534 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation +535 will not work. +536 silent : bool, optional +537 If true all output to the console is omitted (default False). +538 initial_guess : list +539 can provide an initial guess for the input parameters. Relevant for non-linear +540 fits with many parameters. +541 expected_chisquare : bool +542 If true prints the expected chisquare which is +543 corrected by effects caused by correlated input data. +544 This can take a while as the full correlation matrix +545 has to be calculated (default False). +546 num_grad : bool +547 Use numerical differentation instead of automatic differentiation to perform the error propagation (default False). 548 -549 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) -550 model = Model(func) -551 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) -552 odr.set_job(fit_type=0, deriv=1) -553 out = odr.run() -554 -555 output.residual_variance = out.res_var -556 -557 output.method = 'ODR' +549 Notes +550 ----- +551 Based on the orthogonal distance regression module of scipy. +552 +553 Returns +554 ------- +555 output : Fit_result +556 Parameters and information on the fitted result. +557 ''' 558 -559 output.message = out.stopreason +559 output = Fit_result() 560 -561 output.xplus = out.xplus +561 output.fit_function = func 562 -563 if not silent: -564 print('Method: ODR') -565 print(*out.stopreason) -566 print('Residual variance:', output.residual_variance) -567 -568 if out.info > 3: -569 raise Exception('The minimization procedure did not converge.') -570 -571 m = x_f.size -572 -573 def odr_chisquare(p): -574 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) -575 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) -576 return chisq -577 -578 if kwargs.get('expected_chisquare') is True: -579 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) -580 -581 if kwargs.get('covariance') is not None: -582 cov = kwargs.get('covariance') -583 else: -584 cov = covariance(np.concatenate((y, x.ravel()))) -585 -586 number_of_x_parameters = int(m / x_f.shape[-1]) -587 -588 old_jac = jacobian(func)(out.beta, out.xplus) -589 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) -590 fused_row2 = np.concatenate((jacobian(lambda x, y: func(y, x))(out.xplus, out.beta).reshape(x_f.shape[-1], x_f.shape[-1] * number_of_x_parameters), np.identity(number_of_x_parameters * old_jac.shape[0]))) -591 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) +563 x = np.array(x) +564 +565 x_shape = x.shape +566 +567 if kwargs.get('num_grad') is True: +568 jacobian = num_jacobian +569 hessian = num_hessian +570 else: +571 jacobian = auto_jacobian +572 hessian = auto_hessian +573 +574 if not callable(func): +575 raise TypeError('func has to be a function.') +576 +577 for i in range(42): +578 try: +579 func(np.arange(i), x.T[0]) +580 except TypeError: +581 continue +582 except IndexError: +583 continue +584 else: +585 break +586 else: +587 raise RuntimeError("Fit function is not valid.") +588 +589 n_parms = i +590 if not silent: +591 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) 592 -593 A = W @ new_jac -594 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T -595 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) -596 if expected_chisquare <= 0.0: -597 warnings.warn("Negative expected_chisquare.", RuntimeWarning) -598 expected_chisquare = np.abs(expected_chisquare) -599 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare -600 if not silent: -601 print('chisquare/expected_chisquare:', -602 output.chisquare_by_expected_chisquare) +593 x_f = np.vectorize(lambda o: o.value)(x) +594 dx_f = np.vectorize(lambda o: o.dvalue)(x) +595 y_f = np.array([o.value for o in y]) +596 dy_f = np.array([o.dvalue for o in y]) +597 +598 if np.any(np.asarray(dx_f) <= 0.0): +599 raise Exception('No x errors available, run the gamma method first.') +600 +601 if np.any(np.asarray(dy_f) <= 0.0): +602 raise Exception('No y errors available, run the gamma method first.') 603 -604 fitp = out.beta -605 try: -606 hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel()))) -607 except TypeError: -608 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None -609 -610 def odr_chisquare_compact_x(d): -611 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) -612 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((d[n_parms + m:].reshape(x_shape) - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) -613 return chisq -614 -615 jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) +604 if 'initial_guess' in kwargs: +605 x0 = kwargs.get('initial_guess') +606 if len(x0) != n_parms: +607 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) +608 else: +609 x0 = [1] * n_parms +610 +611 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) +612 model = Model(func) +613 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) +614 odr.set_job(fit_type=0, deriv=1) +615 out = odr.run() 616 -617 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv -618 try: -619 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) -620 except np.linalg.LinAlgError: -621 raise Exception("Cannot invert hessian matrix.") +617 output.residual_variance = out.res_var +618 +619 output.method = 'ODR' +620 +621 output.message = out.stopreason 622 -623 def odr_chisquare_compact_y(d): -624 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) -625 chisq = anp.sum(((d[n_parms + m:] - model) / dy_f) ** 2) + anp.sum(((x_f - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) -626 return chisq -627 -628 jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f))) +623 output.xplus = out.xplus +624 +625 if not silent: +626 print('Method: ODR') +627 print(*out.stopreason) +628 print('Residual variance:', output.residual_variance) 629 -630 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv -631 try: -632 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) -633 except np.linalg.LinAlgError: -634 raise Exception("Cannot invert hessian matrix.") -635 -636 result = [] -637 for i in range(n_parms): -638 result.append(derived_observable(lambda my_var, **kwargs: (my_var[0] + np.finfo(np.float64).eps) / (x.ravel()[0].value + np.finfo(np.float64).eps) * out.beta[i], list(x.ravel()) + list(y), man_grad=list(deriv_x[i]) + list(deriv_y[i]))) +630 if out.info > 3: +631 raise Exception('The minimization procedure did not converge.') +632 +633 m = x_f.size +634 +635 def odr_chisquare(p): +636 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) +637 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) +638 return chisq 639 -640 output.fit_parameters = result -641 -642 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) -643 output.dof = x.shape[-1] - n_parms -644 output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof) -645 -646 return output +640 if kwargs.get('expected_chisquare') is True: +641 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) +642 +643 if kwargs.get('covariance') is not None: +644 cov = kwargs.get('covariance') +645 else: +646 cov = covariance(np.concatenate((y, x.ravel()))) +647 +648 number_of_x_parameters = int(m / x_f.shape[-1]) +649 +650 old_jac = jacobian(func)(out.beta, out.xplus) +651 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) +652 fused_row2 = np.concatenate((jacobian(lambda x, y: func(y, x))(out.xplus, out.beta).reshape(x_f.shape[-1], x_f.shape[-1] * number_of_x_parameters), np.identity(number_of_x_parameters * old_jac.shape[0]))) +653 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) +654 +655 A = W @ new_jac +656 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T +657 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) +658 if expected_chisquare <= 0.0: +659 warnings.warn("Negative expected_chisquare.", RuntimeWarning) +660 expected_chisquare = np.abs(expected_chisquare) +661 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare +662 if not silent: +663 print('chisquare/expected_chisquare:', +664 output.chisquare_by_expected_chisquare) +665 +666 fitp = out.beta +667 try: +668 hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel()))) +669 except TypeError: +670 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None +671 +672 def odr_chisquare_compact_x(d): +673 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) +674 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((d[n_parms + m:].reshape(x_shape) - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) +675 return chisq +676 +677 jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) +678 +679 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv +680 try: +681 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) +682 except np.linalg.LinAlgError: +683 raise Exception("Cannot invert hessian matrix.") +684 +685 def odr_chisquare_compact_y(d): +686 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) +687 chisq = anp.sum(((d[n_parms + m:] - model) / dy_f) ** 2) + anp.sum(((x_f - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) +688 return chisq +689 +690 jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f))) +691 +692 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv +693 try: +694 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) +695 except np.linalg.LinAlgError: +696 raise Exception("Cannot invert hessian matrix.") +697 +698 result = [] +699 for i in range(n_parms): +700 result.append(derived_observable(lambda my_var, **kwargs: (my_var[0] + np.finfo(np.float64).eps) / (x.ravel()[0].value + np.finfo(np.float64).eps) * out.beta[i], list(x.ravel()) + list(y), man_grad=list(deriv_x[i]) + list(deriv_y[i]))) +701 +702 output.fit_parameters = result +703 +704 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) +705 output.dof = x.shape[-1] - n_parms +706 output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof) +707 +708 return output
649def fit_lin(x, y, **kwargs): -650 """Performs a linear fit to y = n + m * x and returns two Obs n, m. -651 -652 Parameters -653 ---------- -654 x : list -655 Can either be a list of floats in which case no xerror is assumed, or -656 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. -657 y : list -658 List of Obs, the dvalues of the Obs are used as yerror for the fit. -659 -660 Returns -661 ------- -662 fit_parameters : list[Obs] -663 LIist of fitted observables. -664 """ -665 -666 def f(a, x): -667 y = a[0] + a[1] * x -668 return y -669 -670 if all(isinstance(n, Obs) for n in x): -671 out = total_least_squares(x, y, f, **kwargs) -672 return out.fit_parameters -673 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): -674 out = least_squares(x, y, f, **kwargs) -675 return out.fit_parameters -676 else: -677 raise TypeError('Unsupported types for x') +@@ -1927,34 +2112,34 @@ LIist of fitted observables.711def fit_lin(x, y, **kwargs): +712 """Performs a linear fit to y = n + m * x and returns two Obs n, m. +713 +714 Parameters +715 ---------- +716 x : list +717 Can either be a list of floats in which case no xerror is assumed, or +718 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. +719 y : list +720 List of Obs, the dvalues of the Obs are used as yerror for the fit. +721 +722 Returns +723 ------- +724 fit_parameters : list[Obs] +725 LIist of fitted observables. +726 """ +727 +728 def f(a, x): +729 y = a[0] + a[1] * x +730 return y +731 +732 if all(isinstance(n, Obs) for n in x): +733 out = total_least_squares(x, y, f, **kwargs) +734 return out.fit_parameters +735 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): +736 out = least_squares(x, y, f, **kwargs) +737 return out.fit_parameters +738 else: +739 raise TypeError('Unsupported types for x')
680def qqplot(x, o_y, func, p, title=""): -681 """Generates a quantile-quantile plot of the fit result which can be used to -682 check if the residuals of the fit are gaussian distributed. -683 -684 Returns -685 ------- -686 None -687 """ -688 -689 residuals = [] -690 for i_x, i_y in zip(x, o_y): -691 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) -692 residuals = sorted(residuals) -693 my_y = [o.value for o in residuals] -694 probplot = scipy.stats.probplot(my_y) -695 my_x = probplot[0][0] -696 plt.figure(figsize=(8, 8 / 1.618)) -697 plt.errorbar(my_x, my_y, fmt='o') -698 fit_start = my_x[0] -699 fit_stop = my_x[-1] -700 samples = np.arange(fit_start, fit_stop, 0.01) -701 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') -702 plt.plot(samples, probplot[1][0] * samples + probplot[1][1], zorder=10, label='Least squares fit, r=' + str(np.around(probplot[1][2], 3)), marker='', ls='-') -703 -704 plt.xlabel('Theoretical quantiles') -705 plt.ylabel('Ordered Values') -706 plt.legend(title=title) -707 plt.draw() +@@ -1981,41 +2166,41 @@ LIist of fitted observables.742def qqplot(x, o_y, func, p, title=""): +743 """Generates a quantile-quantile plot of the fit result which can be used to +744 check if the residuals of the fit are gaussian distributed. +745 +746 Returns +747 ------- +748 None +749 """ +750 +751 residuals = [] +752 for i_x, i_y in zip(x, o_y): +753 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) +754 residuals = sorted(residuals) +755 my_y = [o.value for o in residuals] +756 probplot = scipy.stats.probplot(my_y) +757 my_x = probplot[0][0] +758 plt.figure(figsize=(8, 8 / 1.618)) +759 plt.errorbar(my_x, my_y, fmt='o') +760 fit_start = my_x[0] +761 fit_stop = my_x[-1] +762 samples = np.arange(fit_start, fit_stop, 0.01) +763 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') +764 plt.plot(samples, probplot[1][0] * samples + probplot[1][1], zorder=10, label='Least squares fit, r=' + str(np.around(probplot[1][2], 3)), marker='', ls='-') +765 +766 plt.xlabel('Theoretical quantiles') +767 plt.ylabel('Ordered Values') +768 plt.legend(title=title) +769 plt.draw()
710def residual_plot(x, y, func, fit_res, title=""): -711 """Generates a plot which compares the fit to the data and displays the corresponding residuals -712 -713 For uncorrelated data the residuals are expected to be distributed ~N(0,1). -714 -715 Returns -716 ------- -717 None -718 """ -719 sorted_x = sorted(x) -720 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) -721 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) -722 x_samples = np.arange(xstart, xstop + 0.01, 0.01) -723 -724 plt.figure(figsize=(8, 8 / 1.618)) -725 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) -726 ax0 = plt.subplot(gs[0]) -727 ax0.errorbar(x, [o.value for o in y], yerr=[o.dvalue for o in y], ls='none', fmt='o', capsize=3, markersize=5, label='Data') -728 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) -729 ax0.set_xticklabels([]) -730 ax0.set_xlim([xstart, xstop]) -731 ax0.set_xticklabels([]) -732 ax0.legend(title=title) -733 -734 residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], np.asarray(x))) / np.asarray([o.dvalue for o in y]) -735 ax1 = plt.subplot(gs[1]) -736 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) -737 ax1.tick_params(direction='out') -738 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) -739 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") -740 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') -741 ax1.set_xlim([xstart, xstop]) -742 ax1.set_ylabel('Residuals') -743 plt.subplots_adjust(wspace=None, hspace=None) -744 plt.draw() +@@ -2043,28 +2228,28 @@ LIist of fitted observables.772def residual_plot(x, y, func, fit_res, title=""): +773 """Generates a plot which compares the fit to the data and displays the corresponding residuals +774 +775 For uncorrelated data the residuals are expected to be distributed ~N(0,1). +776 +777 Returns +778 ------- +779 None +780 """ +781 sorted_x = sorted(x) +782 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) +783 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) +784 x_samples = np.arange(xstart, xstop + 0.01, 0.01) +785 +786 plt.figure(figsize=(8, 8 / 1.618)) +787 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) +788 ax0 = plt.subplot(gs[0]) +789 ax0.errorbar(x, [o.value for o in y], yerr=[o.dvalue for o in y], ls='none', fmt='o', capsize=3, markersize=5, label='Data') +790 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) +791 ax0.set_xticklabels([]) +792 ax0.set_xlim([xstart, xstop]) +793 ax0.set_xticklabels([]) +794 ax0.legend(title=title) +795 +796 residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], np.asarray(x))) / np.asarray([o.dvalue for o in y]) +797 ax1 = plt.subplot(gs[1]) +798 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) +799 ax1.tick_params(direction='out') +800 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) +801 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") +802 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') +803 ax1.set_xlim([xstart, xstop]) +804 ax1.set_ylabel('Residuals') +805 plt.subplots_adjust(wspace=None, hspace=None) +806 plt.draw()
747def error_band(x, func, beta): -748 """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta. -749 -750 Returns -751 ------- -752 err : np.array(Obs) -753 Error band for an array of sample values x -754 """ -755 cov = covariance(beta) -756 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): -757 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) -758 -759 deriv = [] -760 for i, item in enumerate(x): -761 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) -762 -763 err = [] -764 for i, item in enumerate(x): -765 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) -766 err = np.array(err) -767 -768 return err +@@ -2091,48 +2276,48 @@ Error band for an array of sample values x809def error_band(x, func, beta): +810 """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta. +811 +812 Returns +813 ------- +814 err : np.array(Obs) +815 Error band for an array of sample values x +816 """ +817 cov = covariance(beta) +818 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): +819 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) +820 +821 deriv = [] +822 for i, item in enumerate(x): +823 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) +824 +825 err = [] +826 for i, item in enumerate(x): +827 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) +828 err = np.array(err) +829 +830 return err
771def ks_test(objects=None): -772 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. -773 -774 Parameters -775 ---------- -776 objects : list -777 List of fit results to include in the analysis (optional). -778 -779 Returns -780 ------- -781 None -782 """ -783 -784 if objects is None: -785 obs_list = [] -786 for obj in gc.get_objects(): -787 if isinstance(obj, Fit_result): -788 obs_list.append(obj) -789 else: -790 obs_list = objects -791 -792 p_values = [o.p_value for o in obs_list] -793 -794 bins = len(p_values) -795 x = np.arange(0, 1.001, 0.001) -796 plt.plot(x, x, 'k', zorder=1) -797 plt.xlim(0, 1) -798 plt.ylim(0, 1) -799 plt.xlabel('p-value') -800 plt.ylabel('Cumulative probability') -801 plt.title(str(bins) + ' p-values') -802 -803 n = np.arange(1, bins + 1) / np.float64(bins) -804 Xs = np.sort(p_values) -805 plt.step(Xs, n) -806 diffs = n - Xs -807 loc_max_diff = np.argmax(np.abs(diffs)) -808 loc = Xs[loc_max_diff] -809 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) -810 plt.draw() -811 -812 print(scipy.stats.kstest(p_values, 'uniform')) +diff --git a/docs/pyerrors/input.html b/docs/pyerrors/input.html index c187babd..85bd0cb7 100644 --- a/docs/pyerrors/input.html +++ b/docs/pyerrors/input.html @@ -3,7 +3,7 @@ - +833def ks_test(objects=None): +834 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. +835 +836 Parameters +837 ---------- +838 objects : list +839 List of fit results to include in the analysis (optional). +840 +841 Returns +842 ------- +843 None +844 """ +845 +846 if objects is None: +847 obs_list = [] +848 for obj in gc.get_objects(): +849 if isinstance(obj, Fit_result): +850 obs_list.append(obj) +851 else: +852 obs_list = objects +853 +854 p_values = [o.p_value for o in obs_list] +855 +856 bins = len(p_values) +857 x = np.arange(0, 1.001, 0.001) +858 plt.plot(x, x, 'k', zorder=1) +859 plt.xlim(0, 1) +860 plt.ylim(0, 1) +861 plt.xlabel('p-value') +862 plt.ylabel('Cumulative probability') +863 plt.title(str(bins) + ' p-values') +864 +865 n = np.arange(1, bins + 1) / np.float64(bins) +866 Xs = np.sort(p_values) +867 plt.step(Xs, n) +868 diffs = n - Xs +869 loc_max_diff = np.argmax(np.abs(diffs)) +870 loc = Xs[loc_max_diff] +871 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) +872 plt.draw() +873 +874 print(scipy.stats.kstest(p_values, 'uniform'))pyerrors.input API documentation diff --git a/docs/pyerrors/input/bdio.html b/docs/pyerrors/input/bdio.html index e5bbe3c4..d178f040 100644 --- a/docs/pyerrors/input/bdio.html +++ b/docs/pyerrors/input/bdio.html @@ -3,7 +3,7 @@ - +pyerrors.input.bdio API documentation diff --git a/docs/pyerrors/input/dobs.html b/docs/pyerrors/input/dobs.html index b64d10ca..b43a5005 100644 --- a/docs/pyerrors/input/dobs.html +++ b/docs/pyerrors/input/dobs.html @@ -3,7 +3,7 @@ - +pyerrors.input.dobs API documentation diff --git a/docs/pyerrors/input/hadrons.html b/docs/pyerrors/input/hadrons.html index ca67a795..7572b4ac 100644 --- a/docs/pyerrors/input/hadrons.html +++ b/docs/pyerrors/input/hadrons.html @@ -3,7 +3,7 @@ - +pyerrors.input.hadrons API documentation diff --git a/docs/pyerrors/input/json.html b/docs/pyerrors/input/json.html index 215fd600..b05af178 100644 --- a/docs/pyerrors/input/json.html +++ b/docs/pyerrors/input/json.html @@ -3,7 +3,7 @@ - +pyerrors.input.json API documentation diff --git a/docs/pyerrors/input/misc.html b/docs/pyerrors/input/misc.html index e2f3c3d2..7f57d557 100644 --- a/docs/pyerrors/input/misc.html +++ b/docs/pyerrors/input/misc.html @@ -3,7 +3,7 @@ - +pyerrors.input.misc API documentation diff --git a/docs/pyerrors/input/openQCD.html b/docs/pyerrors/input/openQCD.html index 4bb0d188..2503c7d4 100644 --- a/docs/pyerrors/input/openQCD.html +++ b/docs/pyerrors/input/openQCD.html @@ -3,7 +3,7 @@ - +pyerrors.input.openQCD API documentation diff --git a/docs/pyerrors/input/pandas.html b/docs/pyerrors/input/pandas.html index 09de2b72..51da9832 100644 --- a/docs/pyerrors/input/pandas.html +++ b/docs/pyerrors/input/pandas.html @@ -3,7 +3,7 @@ - +pyerrors.input.pandas API documentation diff --git a/docs/pyerrors/input/sfcf.html b/docs/pyerrors/input/sfcf.html index fd1d55f2..afacfd76 100644 --- a/docs/pyerrors/input/sfcf.html +++ b/docs/pyerrors/input/sfcf.html @@ -3,7 +3,7 @@ - +pyerrors.input.sfcf API documentation diff --git a/docs/pyerrors/input/utils.html b/docs/pyerrors/input/utils.html index 701ca225..ce6134ae 100644 --- a/docs/pyerrors/input/utils.html +++ b/docs/pyerrors/input/utils.html @@ -3,7 +3,7 @@ - +pyerrors.input.utils API documentation diff --git a/docs/pyerrors/integrate.html b/docs/pyerrors/integrate.html index 3b51cd43..0f0e2041 100644 --- a/docs/pyerrors/integrate.html +++ b/docs/pyerrors/integrate.html @@ -3,7 +3,7 @@ - +pyerrors.integrate API documentation diff --git a/docs/pyerrors/linalg.html b/docs/pyerrors/linalg.html index 4f6cef85..f1cdf836 100644 --- a/docs/pyerrors/linalg.html +++ b/docs/pyerrors/linalg.html @@ -3,7 +3,7 @@ - +pyerrors.linalg API documentation diff --git a/docs/pyerrors/misc.html b/docs/pyerrors/misc.html index 6a9b0b8d..39ab9ed2 100644 --- a/docs/pyerrors/misc.html +++ b/docs/pyerrors/misc.html @@ -3,7 +3,7 @@ - +pyerrors.misc API documentation diff --git a/docs/pyerrors/mpm.html b/docs/pyerrors/mpm.html index c2ab9597..19a2e912 100644 --- a/docs/pyerrors/mpm.html +++ b/docs/pyerrors/mpm.html @@ -3,7 +3,7 @@ - +pyerrors.mpm API documentation diff --git a/docs/pyerrors/obs.html b/docs/pyerrors/obs.html index 43183151..d48504ba 100644 --- a/docs/pyerrors/obs.html +++ b/docs/pyerrors/obs.html @@ -3,7 +3,7 @@ - +pyerrors.obs API documentation @@ -295,6 +295,12 @@covariance ++ invert_corr_cov_cholesky + ++ sort_corr + import_jackknife @@ -1874,234 +1880,320 @@ 1544 return cov 1545 1546 -1547def _smooth_eigenvalues(corr, E): -1548 """Eigenvalue smoothing as described in hep-lat/9412087 -1549 -1550 corr : np.ndarray -1551 correlation matrix -1552 E : integer -1553 Number of eigenvalues to be left substantially unchanged -1554 """ -1555 if not (2 < E < corr.shape[0] - 1): -1556 raise Exception(f"'E' has to be between 2 and the dimension of the correlation matrix minus 1 ({corr.shape[0] - 1}).") -1557 vals, vec = np.linalg.eigh(corr) -1558 lambda_min = np.mean(vals[:-E]) -1559 vals[vals < lambda_min] = lambda_min -1560 vals /= np.mean(vals) -1561 return vec @ np.diag(vals) @ vec.T -1562 -1563 -1564def _covariance_element(obs1, obs2): -1565 """Estimates the covariance of two Obs objects, neglecting autocorrelations.""" +1547def invert_corr_cov_cholesky(corr, inverrdiag): +1548 """Constructs a lower triangular matrix `chol` via the Cholesky decomposition of the correlation matrix `corr` +1549 and then returns the inverse covariance matrix `chol_inv` as a lower triangular matrix by solving `chol * x = inverrdiag`. +1550 +1551 Parameters +1552 ---------- +1553 corr : np.ndarray +1554 correlation matrix +1555 inverrdiag : np.ndarray +1556 diagonal matrix, the entries are the inverse errors of the data points considered +1557 """ +1558 +1559 condn = np.linalg.cond(corr) +1560 if condn > 0.1 / np.finfo(float).eps: +1561 raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})") +1562 if condn > 1e13: +1563 warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning) +1564 chol = np.linalg.cholesky(corr) +1565 chol_inv = scipy.linalg.solve_triangular(chol, inverrdiag, lower=True) 1566 -1567 def calc_gamma(deltas1, deltas2, idx1, idx2, new_idx): -1568 deltas1 = _reduce_deltas(deltas1, idx1, new_idx) -1569 deltas2 = _reduce_deltas(deltas2, idx2, new_idx) -1570 return np.sum(deltas1 * deltas2) -1571 -1572 if set(obs1.names).isdisjoint(set(obs2.names)): -1573 return 0.0 -1574 -1575 if not hasattr(obs1, 'e_dvalue') or not hasattr(obs2, 'e_dvalue'): -1576 raise Exception('The gamma method has to be applied to both Obs first.') -1577 -1578 dvalue = 0.0 +1567 return chol_inv +1568 +1569 +1570def sort_corr(corr, kl, yd): +1571 """ Reorders a correlation matrix to match the alphabetical order of its underlying y data. +1572 +1573 The ordering of the input correlation matrix `corr` is given by the list of keys `kl`. +1574 The input dictionary `yd` (with the same keys `kl`) must contain the corresponding y data +1575 that the correlation matrix is based on. +1576 This function sorts the list of keys `kl` alphabetically and sorts the matrix `corr` +1577 according to this alphabetical order such that the sorted matrix `corr_sorted` corresponds +1578 to the y data `yd` when arranged in an alphabetical order by its keys. 1579 -1580 for e_name in obs1.mc_names: -1581 -1582 if e_name not in obs2.mc_names: -1583 continue -1584 -1585 idl_d = {} -1586 for r_name in obs1.e_content[e_name]: -1587 if r_name not in obs2.e_content[e_name]: -1588 continue -1589 idl_d[r_name] = _intersection_idx([obs1.idl[r_name], obs2.idl[r_name]]) -1590 -1591 gamma = 0.0 +1580 Parameters +1581 ---------- +1582 corr : np.ndarray +1583 A square correlation matrix constructed using the order of the y data specified by `kl`. +1584 The dimensions of `corr` should match the total number of y data points in `yd` combined. +1585 kl : list of str +1586 A list of keys that denotes the order in which the y data from `yd` was used to build the +1587 input correlation matrix `corr`. +1588 yd : dict of list +1589 A dictionary where each key corresponds to a unique identifier, and its value is a list of +1590 y data points. The total number of y data points across all keys must match the dimensions +1591 of `corr`. The lists in the dictionary can be lists of Obs. 1592 -1593 for r_name in obs1.e_content[e_name]: -1594 if r_name not in obs2.e_content[e_name]: -1595 continue -1596 if len(idl_d[r_name]) == 0: -1597 continue -1598 gamma += calc_gamma(obs1.deltas[r_name], obs2.deltas[r_name], obs1.idl[r_name], obs2.idl[r_name], idl_d[r_name]) -1599 -1600 if gamma == 0.0: -1601 continue -1602 -1603 gamma_div = 0.0 -1604 for r_name in obs1.e_content[e_name]: -1605 if r_name not in obs2.e_content[e_name]: -1606 continue -1607 if len(idl_d[r_name]) == 0: -1608 continue -1609 gamma_div += np.sqrt(calc_gamma(obs1.deltas[r_name], obs1.deltas[r_name], obs1.idl[r_name], obs1.idl[r_name], idl_d[r_name]) * calc_gamma(obs2.deltas[r_name], obs2.deltas[r_name], obs2.idl[r_name], obs2.idl[r_name], idl_d[r_name])) -1610 gamma /= gamma_div -1611 -1612 dvalue += gamma +1593 Returns +1594 ------- +1595 np.ndarray +1596 A new, sorted correlation matrix that corresponds to the y data from `yd` when arranged alphabetically by its keys. +1597 +1598 Example +1599 ------- +1600 >>> import numpy as np +1601 >>> import pyerrors as pe +1602 >>> corr = np.array([[1, 0.2, 0.3], [0.2, 1, 0.4], [0.3, 0.4, 1]]) +1603 >>> kl = ['b', 'a'] +1604 >>> yd = {'a': [1, 2], 'b': [3]} +1605 >>> sorted_corr = pe.obs.sort_corr(corr, kl, yd) +1606 >>> print(sorted_corr) +1607 array([[1. , 0.3, 0.4], +1608 [0.3, 1. , 0.2], +1609 [0.4, 0.2, 1. ]]) +1610 +1611 """ +1612 kl_sorted = sorted(kl) 1613 -1614 for e_name in obs1.cov_names: -1615 -1616 if e_name not in obs2.cov_names: -1617 continue -1618 -1619 dvalue += np.dot(np.transpose(obs1.covobs[e_name].grad), np.dot(obs1.covobs[e_name].cov, obs2.covobs[e_name].grad)).item() -1620 -1621 return dvalue -1622 -1623 -1624def import_jackknife(jacks, name, idl=None): -1625 """Imports jackknife samples and returns an Obs -1626 -1627 Parameters -1628 ---------- -1629 jacks : numpy.ndarray -1630 numpy array containing the mean value as zeroth entry and -1631 the N jackknife samples as first to Nth entry. -1632 name : str -1633 name of the ensemble the samples are defined on. -1634 """ -1635 length = len(jacks) - 1 -1636 prj = (np.ones((length, length)) - (length - 1) * np.identity(length)) -1637 samples = jacks[1:] @ prj -1638 mean = np.mean(samples) -1639 new_obs = Obs([samples - mean], [name], idl=idl, means=[mean]) -1640 new_obs._value = jacks[0] -1641 return new_obs -1642 -1643 -1644def import_bootstrap(boots, name, random_numbers): -1645 """Imports bootstrap samples and returns an Obs -1646 -1647 Parameters -1648 ---------- -1649 boots : numpy.ndarray -1650 numpy array containing the mean value as zeroth entry and -1651 the N bootstrap samples as first to Nth entry. -1652 name : str -1653 name of the ensemble the samples are defined on. -1654 random_numbers : np.ndarray -1655 Array of shape (samples, length) containing the random numbers to generate the bootstrap samples, -1656 where samples is the number of bootstrap samples and length is the length of the original Monte Carlo -1657 chain to be reconstructed. -1658 """ -1659 samples, length = random_numbers.shape -1660 if samples != len(boots) - 1: -1661 raise ValueError("Random numbers do not have the correct shape.") -1662 -1663 if samples < length: -1664 raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.") +1614 posd = {} +1615 ofs = 0 +1616 for ki, k in enumerate(kl): +1617 posd[k] = [i + ofs for i in range(len(yd[k]))] +1618 ofs += len(posd[k]) +1619 +1620 mapping = [] +1621 for k in kl_sorted: +1622 for i in range(len(yd[k])): +1623 mapping.append(posd[k][i]) +1624 +1625 corr_sorted = np.zeros_like(corr) +1626 for i in range(corr.shape[0]): +1627 for j in range(corr.shape[0]): +1628 corr_sorted[i][j] = corr[mapping[i]][mapping[j]] +1629 +1630 return corr_sorted +1631 +1632 +1633def _smooth_eigenvalues(corr, E): +1634 """Eigenvalue smoothing as described in hep-lat/9412087 +1635 +1636 corr : np.ndarray +1637 correlation matrix +1638 E : integer +1639 Number of eigenvalues to be left substantially unchanged +1640 """ +1641 if not (2 < E < corr.shape[0] - 1): +1642 raise Exception(f"'E' has to be between 2 and the dimension of the correlation matrix minus 1 ({corr.shape[0] - 1}).") +1643 vals, vec = np.linalg.eigh(corr) +1644 lambda_min = np.mean(vals[:-E]) +1645 vals[vals < lambda_min] = lambda_min +1646 vals /= np.mean(vals) +1647 return vec @ np.diag(vals) @ vec.T +1648 +1649 +1650def _covariance_element(obs1, obs2): +1651 """Estimates the covariance of two Obs objects, neglecting autocorrelations.""" +1652 +1653 def calc_gamma(deltas1, deltas2, idx1, idx2, new_idx): +1654 deltas1 = _reduce_deltas(deltas1, idx1, new_idx) +1655 deltas2 = _reduce_deltas(deltas2, idx2, new_idx) +1656 return np.sum(deltas1 * deltas2) +1657 +1658 if set(obs1.names).isdisjoint(set(obs2.names)): +1659 return 0.0 +1660 +1661 if not hasattr(obs1, 'e_dvalue') or not hasattr(obs2, 'e_dvalue'): +1662 raise Exception('The gamma method has to be applied to both Obs first.') +1663 +1664 dvalue = 0.0 1665 -1666 proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length +1666 for e_name in obs1.mc_names: 1667 -1668 samples = scipy.linalg.lstsq(proj, boots[1:])[0] -1669 ret = Obs([samples], [name]) -1670 ret._value = boots[0] -1671 return ret -1672 -1673 -1674def merge_obs(list_of_obs): -1675 """Combine all observables in list_of_obs into one new observable +1668 if e_name not in obs2.mc_names: +1669 continue +1670 +1671 idl_d = {} +1672 for r_name in obs1.e_content[e_name]: +1673 if r_name not in obs2.e_content[e_name]: +1674 continue +1675 idl_d[r_name] = _intersection_idx([obs1.idl[r_name], obs2.idl[r_name]]) 1676 -1677 Parameters -1678 ---------- -1679 list_of_obs : list -1680 list of the Obs object to be combined -1681 -1682 Notes -1683 ----- -1684 It is not possible to combine obs which are based on the same replicum -1685 """ -1686 replist = [item for obs in list_of_obs for item in obs.names] -1687 if (len(replist) == len(set(replist))) is False: -1688 raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist))) -1689 if any([len(o.cov_names) for o in list_of_obs]): -1690 raise Exception('Not possible to merge data that contains covobs!') -1691 new_dict = {} -1692 idl_dict = {} -1693 for o in list_of_obs: -1694 new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0) -1695 for key in set(o.deltas) | set(o.r_values)}) -1696 idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)}) +1677 gamma = 0.0 +1678 +1679 for r_name in obs1.e_content[e_name]: +1680 if r_name not in obs2.e_content[e_name]: +1681 continue +1682 if len(idl_d[r_name]) == 0: +1683 continue +1684 gamma += calc_gamma(obs1.deltas[r_name], obs2.deltas[r_name], obs1.idl[r_name], obs2.idl[r_name], idl_d[r_name]) +1685 +1686 if gamma == 0.0: +1687 continue +1688 +1689 gamma_div = 0.0 +1690 for r_name in obs1.e_content[e_name]: +1691 if r_name not in obs2.e_content[e_name]: +1692 continue +1693 if len(idl_d[r_name]) == 0: +1694 continue +1695 gamma_div += np.sqrt(calc_gamma(obs1.deltas[r_name], obs1.deltas[r_name], obs1.idl[r_name], obs1.idl[r_name], idl_d[r_name]) * calc_gamma(obs2.deltas[r_name], obs2.deltas[r_name], obs2.idl[r_name], obs2.idl[r_name], idl_d[r_name])) +1696 gamma /= gamma_div 1697 -1698 names = sorted(new_dict.keys()) -1699 o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names]) -1700 o.reweighted = np.max([oi.reweighted for oi in list_of_obs]) -1701 return o -1702 -1703 -1704def cov_Obs(means, cov, name, grad=None): -1705 """Create an Obs based on mean(s) and a covariance matrix +1698 dvalue += gamma +1699 +1700 for e_name in obs1.cov_names: +1701 +1702 if e_name not in obs2.cov_names: +1703 continue +1704 +1705 dvalue += np.dot(np.transpose(obs1.covobs[e_name].grad), np.dot(obs1.covobs[e_name].cov, obs2.covobs[e_name].grad)).item() 1706 -1707 Parameters -1708 ---------- -1709 mean : list of floats or float -1710 N mean value(s) of the new Obs -1711 cov : list or array -1712 2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance -1713 name : str -1714 identifier for the covariance matrix -1715 grad : list or array -1716 Gradient of the Covobs wrt. the means belonging to cov. -1717 """ -1718 -1719 def covobs_to_obs(co): -1720 """Make an Obs out of a Covobs -1721 -1722 Parameters -1723 ---------- -1724 co : Covobs -1725 Covobs to be embedded into the Obs -1726 """ -1727 o = Obs([], [], means=[]) -1728 o._value = co.value -1729 o.names.append(co.name) -1730 o._covobs[co.name] = co -1731 o._dvalue = np.sqrt(co.errsq()) -1732 return o -1733 -1734 ol = [] -1735 if isinstance(means, (float, int)): -1736 means = [means] -1737 -1738 for i in range(len(means)): -1739 ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad))) -1740 if ol[0].covobs[name].N != len(means): -1741 raise Exception('You have to provide %d mean values!' % (ol[0].N)) -1742 if len(ol) == 1: -1743 return ol[0] -1744 return ol -1745 -1746 -1747def _determine_gap(o, e_content, e_name): -1748 gaps = [] -1749 for r_name in e_content[e_name]: -1750 if isinstance(o.idl[r_name], range): -1751 gaps.append(o.idl[r_name].step) -1752 else: -1753 gaps.append(np.min(np.diff(o.idl[r_name]))) -1754 -1755 gap = min(gaps) -1756 if not np.all([gi % gap == 0 for gi in gaps]): -1757 raise Exception(f"Replica for ensemble {e_name} do not have a common spacing.", gaps) +1707 return dvalue +1708 +1709 +1710def import_jackknife(jacks, name, idl=None): +1711 """Imports jackknife samples and returns an Obs +1712 +1713 Parameters +1714 ---------- +1715 jacks : numpy.ndarray +1716 numpy array containing the mean value as zeroth entry and +1717 the N jackknife samples as first to Nth entry. +1718 name : str +1719 name of the ensemble the samples are defined on. +1720 """ +1721 length = len(jacks) - 1 +1722 prj = (np.ones((length, length)) - (length - 1) * np.identity(length)) +1723 samples = jacks[1:] @ prj +1724 mean = np.mean(samples) +1725 new_obs = Obs([samples - mean], [name], idl=idl, means=[mean]) +1726 new_obs._value = jacks[0] +1727 return new_obs +1728 +1729 +1730def import_bootstrap(boots, name, random_numbers): +1731 """Imports bootstrap samples and returns an Obs +1732 +1733 Parameters +1734 ---------- +1735 boots : numpy.ndarray +1736 numpy array containing the mean value as zeroth entry and +1737 the N bootstrap samples as first to Nth entry. +1738 name : str +1739 name of the ensemble the samples are defined on. +1740 random_numbers : np.ndarray +1741 Array of shape (samples, length) containing the random numbers to generate the bootstrap samples, +1742 where samples is the number of bootstrap samples and length is the length of the original Monte Carlo +1743 chain to be reconstructed. +1744 """ +1745 samples, length = random_numbers.shape +1746 if samples != len(boots) - 1: +1747 raise ValueError("Random numbers do not have the correct shape.") +1748 +1749 if samples < length: +1750 raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.") +1751 +1752 proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length +1753 +1754 samples = scipy.linalg.lstsq(proj, boots[1:])[0] +1755 ret = Obs([samples], [name]) +1756 ret._value = boots[0] +1757 return ret 1758 -1759 return gap -1760 -1761 -1762def _check_lists_equal(idl): -1763 ''' -1764 Use groupby to efficiently check whether all elements of idl are identical. -1765 Returns True if all elements are equal, otherwise False. -1766 -1767 Parameters -1768 ---------- -1769 idl : list of lists, ranges or np.ndarrays -1770 ''' -1771 g = groupby([np.nditer(el) if isinstance(el, np.ndarray) else el for el in idl]) -1772 if next(g, True) and not next(g, False): -1773 return True -1774 return False +1759 +1760def merge_obs(list_of_obs): +1761 """Combine all observables in list_of_obs into one new observable +1762 +1763 Parameters +1764 ---------- +1765 list_of_obs : list +1766 list of the Obs object to be combined +1767 +1768 Notes +1769 ----- +1770 It is not possible to combine obs which are based on the same replicum +1771 """ +1772 replist = [item for obs in list_of_obs for item in obs.names] +1773 if (len(replist) == len(set(replist))) is False: +1774 raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist))) +1775 if any([len(o.cov_names) for o in list_of_obs]): +1776 raise Exception('Not possible to merge data that contains covobs!') +1777 new_dict = {} +1778 idl_dict = {} +1779 for o in list_of_obs: +1780 new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0) +1781 for key in set(o.deltas) | set(o.r_values)}) +1782 idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)}) +1783 +1784 names = sorted(new_dict.keys()) +1785 o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names]) +1786 o.reweighted = np.max([oi.reweighted for oi in list_of_obs]) +1787 return o +1788 +1789 +1790def cov_Obs(means, cov, name, grad=None): +1791 """Create an Obs based on mean(s) and a covariance matrix +1792 +1793 Parameters +1794 ---------- +1795 mean : list of floats or float +1796 N mean value(s) of the new Obs +1797 cov : list or array +1798 2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance +1799 name : str +1800 identifier for the covariance matrix +1801 grad : list or array +1802 Gradient of the Covobs wrt. the means belonging to cov. +1803 """ +1804 +1805 def covobs_to_obs(co): +1806 """Make an Obs out of a Covobs +1807 +1808 Parameters +1809 ---------- +1810 co : Covobs +1811 Covobs to be embedded into the Obs +1812 """ +1813 o = Obs([], [], means=[]) +1814 o._value = co.value +1815 o.names.append(co.name) +1816 o._covobs[co.name] = co +1817 o._dvalue = np.sqrt(co.errsq()) +1818 return o +1819 +1820 ol = [] +1821 if isinstance(means, (float, int)): +1822 means = [means] +1823 +1824 for i in range(len(means)): +1825 ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad))) +1826 if ol[0].covobs[name].N != len(means): +1827 raise Exception('You have to provide %d mean values!' % (ol[0].N)) +1828 if len(ol) == 1: +1829 return ol[0] +1830 return ol +1831 +1832 +1833def _determine_gap(o, e_content, e_name): +1834 gaps = [] +1835 for r_name in e_content[e_name]: +1836 if isinstance(o.idl[r_name], range): +1837 gaps.append(o.idl[r_name].step) +1838 else: +1839 gaps.append(np.min(np.diff(o.idl[r_name]))) +1840 +1841 gap = min(gaps) +1842 if not np.all([gi % gap == 0 for gi in gaps]): +1843 raise Exception(f"Replica for ensemble {e_name} do not have a common spacing.", gaps) +1844 +1845 return gap +1846 +1847 +1848def _check_lists_equal(idl): +1849 ''' +1850 Use groupby to efficiently check whether all elements of idl are identical. +1851 Returns True if all elements are equal, otherwise False. +1852 +1853 Parameters +1854 ---------- +1855 idl : list of lists, ranges or np.ndarrays +1856 ''' +1857 g = groupby([np.nditer(el) if isinstance(el, np.ndarray) else el for el in idl]) +1858 if next(g, True) and not next(g, False): +1859 return True +1860 return False
1548def invert_corr_cov_cholesky(corr, inverrdiag): +1549 """Constructs a lower triangular matrix `chol` via the Cholesky decomposition of the correlation matrix `corr` +1550 and then returns the inverse covariance matrix `chol_inv` as a lower triangular matrix by solving `chol * x = inverrdiag`. +1551 +1552 Parameters +1553 ---------- +1554 corr : np.ndarray +1555 correlation matrix +1556 inverrdiag : np.ndarray +1557 diagonal matrix, the entries are the inverse errors of the data points considered +1558 """ +1559 +1560 condn = np.linalg.cond(corr) +1561 if condn > 0.1 / np.finfo(float).eps: +1562 raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})") +1563 if condn > 1e13: +1564 warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning) +1565 chol = np.linalg.cholesky(corr) +1566 chol_inv = scipy.linalg.solve_triangular(chol, inverrdiag, lower=True) +1567 +1568 return chol_inv +
Constructs a lower triangular matrix chol
via the Cholesky decomposition of the correlation matrix corr
+ and then returns the inverse covariance matrix chol_inv
as a lower triangular matrix by solving chol * x = inverrdiag
.
1571def sort_corr(corr, kl, yd): +1572 """ Reorders a correlation matrix to match the alphabetical order of its underlying y data. +1573 +1574 The ordering of the input correlation matrix `corr` is given by the list of keys `kl`. +1575 The input dictionary `yd` (with the same keys `kl`) must contain the corresponding y data +1576 that the correlation matrix is based on. +1577 This function sorts the list of keys `kl` alphabetically and sorts the matrix `corr` +1578 according to this alphabetical order such that the sorted matrix `corr_sorted` corresponds +1579 to the y data `yd` when arranged in an alphabetical order by its keys. +1580 +1581 Parameters +1582 ---------- +1583 corr : np.ndarray +1584 A square correlation matrix constructed using the order of the y data specified by `kl`. +1585 The dimensions of `corr` should match the total number of y data points in `yd` combined. +1586 kl : list of str +1587 A list of keys that denotes the order in which the y data from `yd` was used to build the +1588 input correlation matrix `corr`. +1589 yd : dict of list +1590 A dictionary where each key corresponds to a unique identifier, and its value is a list of +1591 y data points. The total number of y data points across all keys must match the dimensions +1592 of `corr`. The lists in the dictionary can be lists of Obs. +1593 +1594 Returns +1595 ------- +1596 np.ndarray +1597 A new, sorted correlation matrix that corresponds to the y data from `yd` when arranged alphabetically by its keys. +1598 +1599 Example +1600 ------- +1601 >>> import numpy as np +1602 >>> import pyerrors as pe +1603 >>> corr = np.array([[1, 0.2, 0.3], [0.2, 1, 0.4], [0.3, 0.4, 1]]) +1604 >>> kl = ['b', 'a'] +1605 >>> yd = {'a': [1, 2], 'b': [3]} +1606 >>> sorted_corr = pe.obs.sort_corr(corr, kl, yd) +1607 >>> print(sorted_corr) +1608 array([[1. , 0.3, 0.4], +1609 [0.3, 1. , 0.2], +1610 [0.4, 0.2, 1. ]]) +1611 +1612 """ +1613 kl_sorted = sorted(kl) +1614 +1615 posd = {} +1616 ofs = 0 +1617 for ki, k in enumerate(kl): +1618 posd[k] = [i + ofs for i in range(len(yd[k]))] +1619 ofs += len(posd[k]) +1620 +1621 mapping = [] +1622 for k in kl_sorted: +1623 for i in range(len(yd[k])): +1624 mapping.append(posd[k][i]) +1625 +1626 corr_sorted = np.zeros_like(corr) +1627 for i in range(corr.shape[0]): +1628 for j in range(corr.shape[0]): +1629 corr_sorted[i][j] = corr[mapping[i]][mapping[j]] +1630 +1631 return corr_sorted +
Reorders a correlation matrix to match the alphabetical order of its underlying y data.
+ +The ordering of the input correlation matrix corr
is given by the list of keys kl
.
+The input dictionary yd
(with the same keys kl
) must contain the corresponding y data
+that the correlation matrix is based on.
+This function sorts the list of keys kl
alphabetically and sorts the matrix corr
+according to this alphabetical order such that the sorted matrix corr_sorted
corresponds
+to the y data yd
when arranged in an alphabetical order by its keys.
kl
.
+The dimensions of corr
should match the total number of y data points in yd
combined.yd
was used to build the
+input correlation matrix corr
.corr
. The lists in the dictionary can be lists of Obs.yd
when arranged alphabetically by its keys.>>> import numpy as np
+>>> import pyerrors as pe
+>>> corr = np.array([[1, 0.2, 0.3], [0.2, 1, 0.4], [0.3, 0.4, 1]])
+>>> kl = ['b', 'a']
+>>> yd = {'a': [1, 2], 'b': [3]}
+>>> sorted_corr = pe.obs.sort_corr(corr, kl, yd)
+>>> print(sorted_corr)
+array([[1. , 0.3, 0.4],
+ [0.3, 1. , 0.2],
+ [0.4, 0.2, 1. ]])
+
+1625def import_jackknife(jacks, name, idl=None): -1626 """Imports jackknife samples and returns an Obs -1627 -1628 Parameters -1629 ---------- -1630 jacks : numpy.ndarray -1631 numpy array containing the mean value as zeroth entry and -1632 the N jackknife samples as first to Nth entry. -1633 name : str -1634 name of the ensemble the samples are defined on. -1635 """ -1636 length = len(jacks) - 1 -1637 prj = (np.ones((length, length)) - (length - 1) * np.identity(length)) -1638 samples = jacks[1:] @ prj -1639 mean = np.mean(samples) -1640 new_obs = Obs([samples - mean], [name], idl=idl, means=[mean]) -1641 new_obs._value = jacks[0] -1642 return new_obs +@@ -5850,34 +6116,34 @@ name of the ensemble the samples are defined on.1711def import_jackknife(jacks, name, idl=None): +1712 """Imports jackknife samples and returns an Obs +1713 +1714 Parameters +1715 ---------- +1716 jacks : numpy.ndarray +1717 numpy array containing the mean value as zeroth entry and +1718 the N jackknife samples as first to Nth entry. +1719 name : str +1720 name of the ensemble the samples are defined on. +1721 """ +1722 length = len(jacks) - 1 +1723 prj = (np.ones((length, length)) - (length - 1) * np.identity(length)) +1724 samples = jacks[1:] @ prj +1725 mean = np.mean(samples) +1726 new_obs = Obs([samples - mean], [name], idl=idl, means=[mean]) +1727 new_obs._value = jacks[0] +1728 return new_obs
1645def import_bootstrap(boots, name, random_numbers): -1646 """Imports bootstrap samples and returns an Obs -1647 -1648 Parameters -1649 ---------- -1650 boots : numpy.ndarray -1651 numpy array containing the mean value as zeroth entry and -1652 the N bootstrap samples as first to Nth entry. -1653 name : str -1654 name of the ensemble the samples are defined on. -1655 random_numbers : np.ndarray -1656 Array of shape (samples, length) containing the random numbers to generate the bootstrap samples, -1657 where samples is the number of bootstrap samples and length is the length of the original Monte Carlo -1658 chain to be reconstructed. -1659 """ -1660 samples, length = random_numbers.shape -1661 if samples != len(boots) - 1: -1662 raise ValueError("Random numbers do not have the correct shape.") -1663 -1664 if samples < length: -1665 raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.") -1666 -1667 proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length -1668 -1669 samples = scipy.linalg.lstsq(proj, boots[1:])[0] -1670 ret = Obs([samples], [name]) -1671 ret._value = boots[0] -1672 return ret +@@ -5911,34 +6177,34 @@ chain to be reconstructed.1731def import_bootstrap(boots, name, random_numbers): +1732 """Imports bootstrap samples and returns an Obs +1733 +1734 Parameters +1735 ---------- +1736 boots : numpy.ndarray +1737 numpy array containing the mean value as zeroth entry and +1738 the N bootstrap samples as first to Nth entry. +1739 name : str +1740 name of the ensemble the samples are defined on. +1741 random_numbers : np.ndarray +1742 Array of shape (samples, length) containing the random numbers to generate the bootstrap samples, +1743 where samples is the number of bootstrap samples and length is the length of the original Monte Carlo +1744 chain to be reconstructed. +1745 """ +1746 samples, length = random_numbers.shape +1747 if samples != len(boots) - 1: +1748 raise ValueError("Random numbers do not have the correct shape.") +1749 +1750 if samples < length: +1751 raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.") +1752 +1753 proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length +1754 +1755 samples = scipy.linalg.lstsq(proj, boots[1:])[0] +1756 ret = Obs([samples], [name]) +1757 ret._value = boots[0] +1758 return ret
1675def merge_obs(list_of_obs): -1676 """Combine all observables in list_of_obs into one new observable -1677 -1678 Parameters -1679 ---------- -1680 list_of_obs : list -1681 list of the Obs object to be combined -1682 -1683 Notes -1684 ----- -1685 It is not possible to combine obs which are based on the same replicum -1686 """ -1687 replist = [item for obs in list_of_obs for item in obs.names] -1688 if (len(replist) == len(set(replist))) is False: -1689 raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist))) -1690 if any([len(o.cov_names) for o in list_of_obs]): -1691 raise Exception('Not possible to merge data that contains covobs!') -1692 new_dict = {} -1693 idl_dict = {} -1694 for o in list_of_obs: -1695 new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0) -1696 for key in set(o.deltas) | set(o.r_values)}) -1697 idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)}) -1698 -1699 names = sorted(new_dict.keys()) -1700 o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names]) -1701 o.reweighted = np.max([oi.reweighted for oi in list_of_obs]) -1702 return o +@@ -5969,47 +6235,47 @@ list of the Obs object to be combined1761def merge_obs(list_of_obs): +1762 """Combine all observables in list_of_obs into one new observable +1763 +1764 Parameters +1765 ---------- +1766 list_of_obs : list +1767 list of the Obs object to be combined +1768 +1769 Notes +1770 ----- +1771 It is not possible to combine obs which are based on the same replicum +1772 """ +1773 replist = [item for obs in list_of_obs for item in obs.names] +1774 if (len(replist) == len(set(replist))) is False: +1775 raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist))) +1776 if any([len(o.cov_names) for o in list_of_obs]): +1777 raise Exception('Not possible to merge data that contains covobs!') +1778 new_dict = {} +1779 idl_dict = {} +1780 for o in list_of_obs: +1781 new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0) +1782 for key in set(o.deltas) | set(o.r_values)}) +1783 idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)}) +1784 +1785 names = sorted(new_dict.keys()) +1786 o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names]) +1787 o.reweighted = np.max([oi.reweighted for oi in list_of_obs]) +1788 return o
1705def cov_Obs(means, cov, name, grad=None): -1706 """Create an Obs based on mean(s) and a covariance matrix -1707 -1708 Parameters -1709 ---------- -1710 mean : list of floats or float -1711 N mean value(s) of the new Obs -1712 cov : list or array -1713 2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance -1714 name : str -1715 identifier for the covariance matrix -1716 grad : list or array -1717 Gradient of the Covobs wrt. the means belonging to cov. -1718 """ -1719 -1720 def covobs_to_obs(co): -1721 """Make an Obs out of a Covobs -1722 -1723 Parameters -1724 ---------- -1725 co : Covobs -1726 Covobs to be embedded into the Obs -1727 """ -1728 o = Obs([], [], means=[]) -1729 o._value = co.value -1730 o.names.append(co.name) -1731 o._covobs[co.name] = co -1732 o._dvalue = np.sqrt(co.errsq()) -1733 return o -1734 -1735 ol = [] -1736 if isinstance(means, (float, int)): -1737 means = [means] -1738 -1739 for i in range(len(means)): -1740 ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad))) -1741 if ol[0].covobs[name].N != len(means): -1742 raise Exception('You have to provide %d mean values!' % (ol[0].N)) -1743 if len(ol) == 1: -1744 return ol[0] -1745 return ol +diff --git a/docs/pyerrors/roots.html b/docs/pyerrors/roots.html index 4a71cb86..b12b5f02 100644 --- a/docs/pyerrors/roots.html +++ b/docs/pyerrors/roots.html @@ -3,7 +3,7 @@ - +1791def cov_Obs(means, cov, name, grad=None): +1792 """Create an Obs based on mean(s) and a covariance matrix +1793 +1794 Parameters +1795 ---------- +1796 mean : list of floats or float +1797 N mean value(s) of the new Obs +1798 cov : list or array +1799 2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance +1800 name : str +1801 identifier for the covariance matrix +1802 grad : list or array +1803 Gradient of the Covobs wrt. the means belonging to cov. +1804 """ +1805 +1806 def covobs_to_obs(co): +1807 """Make an Obs out of a Covobs +1808 +1809 Parameters +1810 ---------- +1811 co : Covobs +1812 Covobs to be embedded into the Obs +1813 """ +1814 o = Obs([], [], means=[]) +1815 o._value = co.value +1816 o.names.append(co.name) +1817 o._covobs[co.name] = co +1818 o._dvalue = np.sqrt(co.errsq()) +1819 return o +1820 +1821 ol = [] +1822 if isinstance(means, (float, int)): +1823 means = [means] +1824 +1825 for i in range(len(means)): +1826 ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad))) +1827 if ol[0].covobs[name].N != len(means): +1828 raise Exception('You have to provide %d mean values!' % (ol[0].N)) +1829 if len(ol) == 1: +1830 return ol[0] +1831 return olpyerrors.roots API documentation diff --git a/docs/pyerrors/special.html b/docs/pyerrors/special.html index b1c2aa0c..99bfc775 100644 --- a/docs/pyerrors/special.html +++ b/docs/pyerrors/special.html @@ -3,7 +3,7 @@ - +pyerrors.special API documentation diff --git a/docs/pyerrors/version.html b/docs/pyerrors/version.html index 7020c7a1..44587cdb 100644 --- a/docs/pyerrors/version.html +++ b/docs/pyerrors/version.html @@ -3,7 +3,7 @@ - +pyerrors.version API documentation diff --git a/docs/search.js b/docs/search.js index 5a1f6df3..05036446 100644 --- a/docs/search.js +++ b/docs/search.js @@ -1,6 +1,6 @@ window.pdocSearch = (function(){ /** elasticlunr - http://weixsong.github.io * Copyright (C) 2017 Oliver Nightingale * Copyright (C) 2017 Wei Song * MIT Licensed */!function(){function e(e){if(null===e||"object"!=typeof e)return e;var t=e.constructor();for(var n in e)e.hasOwnProperty(n)&&(t[n]=e[n]);return t}var t=function(e){var n=new t.Index;return n.pipeline.add(t.trimmer,t.stopWordFilter,t.stemmer),e&&e.call(n,n),n};t.version="0.9.5",lunr=t,t.utils={},t.utils.warn=function(e){return function(t){e.console&&console.warn&&console.warn(t)}}(this),t.utils.toString=function(e){return void 0===e||null===e?"":e.toString()},t.EventEmitter=function(){this.events={}},t.EventEmitter.prototype.addListener=function(){var e=Array.prototype.slice.call(arguments),t=e.pop(),n=e;if("function"!=typeof t)throw new TypeError("last argument must be a function");n.forEach(function(e){this.hasHandler(e)||(this.events[e]=[]),this.events[e].push(t)},this)},t.EventEmitter.prototype.removeListener=function(e,t){if(this.hasHandler(e)){var n=this.events[e].indexOf(t);-1!==n&&(this.events[e].splice(n,1),0==this.events[e].length&&delete this.events[e])}},t.EventEmitter.prototype.emit=function(e){if(this.hasHandler(e)){var t=Array.prototype.slice.call(arguments,1);this.events[e].forEach(function(e){e.apply(void 0,t)},this)}},t.EventEmitter.prototype.hasHandler=function(e){return e in this.events},t.tokenizer=function(e){if(!arguments.length||null===e||void 0===e)return[];if(Array.isArray(e)){var n=e.filter(function(e){return null===e||void 0===e?!1:!0});n=n.map(function(e){return t.utils.toString(e).toLowerCase()});var i=[];return n.forEach(function(e){var n=e.split(t.tokenizer.seperator);i=i.concat(n)},this),i}return e.toString().trim().toLowerCase().split(t.tokenizer.seperator)},t.tokenizer.defaultSeperator=/[\s\-]+/,t.tokenizer.seperator=t.tokenizer.defaultSeperator,t.tokenizer.setSeperator=function(e){null!==e&&void 0!==e&&"object"==typeof e&&(t.tokenizer.seperator=e)},t.tokenizer.resetSeperator=function(){t.tokenizer.seperator=t.tokenizer.defaultSeperator},t.tokenizer.getSeperator=function(){return t.tokenizer.seperator},t.Pipeline=function(){this._queue=[]},t.Pipeline.registeredFunctions={},t.Pipeline.registerFunction=function(e,n){n in t.Pipeline.registeredFunctions&&t.utils.warn("Overwriting existing registered function: "+n),e.label=n,t.Pipeline.registeredFunctions[n]=e},t.Pipeline.getRegisteredFunction=function(e){return e in t.Pipeline.registeredFunctions!=!0?null:t.Pipeline.registeredFunctions[e]},t.Pipeline.warnIfFunctionNotRegistered=function(e){var n=e.label&&e.label in this.registeredFunctions;n||t.utils.warn("Function is not registered with pipeline. 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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. Comput.Phys.Commun. 288 (2023) 108750.
\n- Ulli Wolff, Monte Carlo errors with less errors. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum).
\n- Alberto Ramos, Automatic differentiation for error analysis of Monte Carlo data. Comput.Phys.Commun. 238 (2019) 19-35.
\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\nInstallation
\n\nInstall the most recent release using pip and pypi:
\n\n\n\n\n\npython -m pip install pyerrors # Fresh install\npython -m pip install -U pyerrors # Update\n
Install the most recent release using conda and conda-forge:
\n\n\n\n\n\nconda install -c conda-forge pyerrors # Fresh install\nconda update -c conda-forge pyerrors # Update\n
Install the current
\n\ndevelop
version:\n\n\n\npython -m pip install -U --no-deps --force-reinstall git+https://github.com/fjosw/pyerrors.git@develop\n
(Also works for any feature branch).
\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
The
\n\ngamma_method
is not automatically called after every intermediate step in order to prevent computational overhead.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
.\nDirect visualizations of the performed fits can be triggered viaresplot=True
orqqplot=True
.For all available options including combined fits to multiple datasets see
\n\npyerrors.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 as 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\n\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 matrix at every timeslice.\nOther dependency (eg. spatial) are not supported.
\n\nThe Corr class can also deal with missing measurements or paddings for fixed boundary conditions.\nThe missing entries are represented via the
\n\nNone
object.Initialization
\n\nA simple correlator can be initialized with a list or a one-dimensional array of
\n\nObs
orCobs
\n\n\n\ncorr11 = pe.Corr([obs1, obs2])\ncorr11 = pe.Corr(np.array([obs1, obs2]))\n
A matrix-valued correlator can either be initialized via a two-dimensional array of
\n\nCorr
objects\n\n\n\nmatrix_corr = pe.Corr(np.array([[corr11, corr12], [corr21, corr22]]))\n
or alternatively via a three-dimensional array of
\n"}, "pyerrors.correlators.Corr.__init__": {"fullname": "pyerrors.correlators.Corr.__init__", "modulename": "pyerrors.correlators", "qualname": "Corr.__init__", "kind": "function", "doc": "Obs
orCObs
of shape (T, N, N) where T is\nthe temporal extent of the correlator and N is the dimension of the matrix.Initialize a Corr object.
\n\nParameters
\n\n\n
\n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "pyerrors.correlators.Corr.tag": {"fullname": "pyerrors.correlators.Corr.tag", "modulename": "pyerrors.correlators", "qualname": "Corr.tag", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.content": {"fullname": "pyerrors.correlators.Corr.content", "modulename": "pyerrors.correlators", "qualname": "Corr.content", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.T": {"fullname": "pyerrors.correlators.Corr.T", "modulename": "pyerrors.correlators", "qualname": "Corr.T", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.prange": {"fullname": "pyerrors.correlators.Corr.prange", "modulename": "pyerrors.correlators", "qualname": "Corr.prange", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.reweighted": {"fullname": "pyerrors.correlators.Corr.reweighted", "modulename": "pyerrors.correlators", "qualname": "Corr.reweighted", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.gamma_method": {"fullname": "pyerrors.correlators.Corr.gamma_method", "modulename": "pyerrors.correlators", "qualname": "Corr.gamma_method", "kind": "function", "doc": "- data_input (list or array):\nlist of Obs or list of arrays of Obs or array of Corrs (see class docstring for details).
\n- padding (list, optional):\nList with two entries where the first labels the padding\nat the front of the correlator and the second the padding\nat the back.
\n- prange (list, optional):\nList containing the first and last timeslice of the plateau\nregion identified for this correlator.
\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.trace": {"fullname": "pyerrors.correlators.Corr.trace", "modulename": "pyerrors.correlators", "qualname": "Corr.trace", "kind": "function", "doc": "Calculates the per-timeslice trace of a correlator matrix.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.matrix_symmetric", "kind": "function", "doc": "Symmetrizes the correlator matrices on every timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.GEVP": {"fullname": "pyerrors.correlators.Corr.GEVP", "modulename": "pyerrors.correlators", "qualname": "Corr.GEVP", "kind": "function", "doc": "Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.
\n\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. (default)
\n- \"Eigenvector\": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.\nThe reference state is identified by its eigenvalue at $t=t_s$.
\n- None: The GEVP is solved only at ts, no sorting is necessary
\n- vector_obs (bool):\nIf True, uncertainties are propagated in the eigenvector computation (default False).
\nOther Parameters
\n\n\n
\n", "signature": "(self, t0, ts=None, sort='Eigenvalue', vector_obs=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Eigenvalue": {"fullname": "pyerrors.correlators.Corr.Eigenvalue", "modulename": "pyerrors.correlators", "qualname": "Corr.Eigenvalue", "kind": "function", "doc": "- state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
\n- method (str):\nMethod used to solve the GEVP.\n
\n\n
- \"eigh\": Use scipy.linalg.eigh to solve the GEVP. (default for vector_obs=False)
\n- \"cholesky\": Use manually implemented solution via the Cholesky decomposition. Automatically chosen if vector_obs==True.
\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', **kwargs):", "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- parity (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:\n - symmetric (default)\n $$\\tilde{\\partial}^2_0 f(x_0) = f(x_0+1)-2f(x_0)+f(x_0-1)$$\n - big_symmetric\n $$\\partial^2_0 f(x_0) = \\frac{f(x_0+2)-2f(x_0)+f(x_0-2)}{4}$$\n - improved\n $$\\partial^2_0 f(x_0) = \\frac{-f(x_0+2) + 16 * f(x_0+1) - 30 * f(x_0) + 16 * f(x_0-1) - f(x_0-2)}{12}$$\n - log\n $$f(x) = \\tilde{\\partial}^2_0 log(f(x_0))+(\\tilde{\\partial}_0 log(f(x_0)))^2$$
\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 periodicity of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.\nsinh : Use anti-periodicity of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.\nSee, e.g., arXiv:1205.5380\narccosh : Uses the explicit form of the symmetrized correlator (not recommended)\nlogsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
\n- guess (float):\nguess for the root finder, only relevant for the root variant
\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,\tfit_key=None,\tylabel=None,\tsave=None,\tauto_gamma=False,\thide_sigma=None,\treferences=None,\ttitle=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.spaghetti_plot": {"fullname": "pyerrors.correlators.Corr.spaghetti_plot", "modulename": "pyerrors.correlators", "qualname": "Corr.spaghetti_plot", "kind": "function", "doc": "- x_range (list):\nlist of two values, determining the range of the x-axis e.g. [4, 8].
\n- comp (Corr or list of Corr):\nCorrelator or list of correlators which are plotted for comparison.\nThe tags of these correlators are used as labels if available.
\n- logscale (bool):\nSets y-axis to logscale.
\n- plateau (Obs):\nPlateau value to be visualized in the figure.
\n- fit_res (Fit_result):\nFit_result object to be visualized.
\n- fit_key (str):\nKey for the fit function in Fit_result.fit_function (for combined fits).
\n- ylabel (str):\nLabel for the y-axis.
\n- save (str):\npath to file in which the figure should be saved.
\n- auto_gamma (bool):\nApply the gamma method with standard parameters to all correlators and plateau values before plotting.
\n- hide_sigma (float):\nHides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
\n- references (list):\nList of floating point values that are displayed as horizontal lines for reference.
\n- title (string):\nOptional title of the figure.
\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.real": {"fullname": "pyerrors.correlators.Corr.real", "modulename": "pyerrors.correlators", "qualname": "Corr.real", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.imag": {"fullname": "pyerrors.correlators.Corr.imag", "modulename": "pyerrors.correlators", "qualname": "Corr.imag", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.prune": {"fullname": "pyerrors.correlators.Corr.prune", "modulename": "pyerrors.correlators", "qualname": "Corr.prune", "kind": "function", "doc": "- 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.correlators.Corr.N": {"fullname": "pyerrors.correlators.Corr.N", "modulename": "pyerrors.correlators", "qualname": "Corr.N", "kind": "variable", "doc": "\n"}, "pyerrors.covobs": {"fullname": "pyerrors.covobs", "modulename": "pyerrors.covobs", "kind": "module", "doc": "\n"}, "pyerrors.covobs.Covobs": {"fullname": "pyerrors.covobs.Covobs", "modulename": "pyerrors.covobs", "qualname": "Covobs", "kind": "class", "doc": "\n"}, "pyerrors.covobs.Covobs.__init__": {"fullname": "pyerrors.covobs.Covobs.__init__", "modulename": "pyerrors.covobs", "qualname": "Covobs.__init__", "kind": "function", "doc": "Initialize Covobs object.
\n\nParameters
\n\n\n
\n", "signature": "(mean, cov, name, pos=None, grad=None)"}, "pyerrors.covobs.Covobs.name": {"fullname": "pyerrors.covobs.Covobs.name", "modulename": "pyerrors.covobs", "qualname": "Covobs.name", "kind": "variable", "doc": "\n"}, "pyerrors.covobs.Covobs.value": {"fullname": "pyerrors.covobs.Covobs.value", "modulename": "pyerrors.covobs", "qualname": "Covobs.value", "kind": "variable", "doc": "\n"}, "pyerrors.covobs.Covobs.errsq": {"fullname": "pyerrors.covobs.Covobs.errsq", "modulename": "pyerrors.covobs", "qualname": "Covobs.errsq", "kind": "function", "doc": "- 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.covobs.Covobs.cov": {"fullname": "pyerrors.covobs.Covobs.cov", "modulename": "pyerrors.covobs", "qualname": "Covobs.cov", "kind": "variable", "doc": "\n"}, "pyerrors.covobs.Covobs.grad": {"fullname": "pyerrors.covobs.Covobs.grad", "modulename": "pyerrors.covobs", "qualname": "Covobs.grad", "kind": "variable", "doc": "\n"}, "pyerrors.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "\n"}, "pyerrors.dirac.gammaX": {"fullname": "pyerrors.dirac.gammaX", "modulename": "pyerrors.dirac", "qualname": "gammaX", "kind": "variable", "doc": "\n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+0.j, 0.+1.j],\n [ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, -0.-1.j, 0.+0.j, 0.+0.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaY": {"fullname": "pyerrors.dirac.gammaY", "modulename": "pyerrors.dirac", "qualname": "gammaY", "kind": "variable", "doc": "\n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j],\n [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [-1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaZ": {"fullname": "pyerrors.dirac.gammaZ", "modulename": "pyerrors.dirac", "qualname": "gammaZ", "kind": "variable", "doc": "\n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -0.-1.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+1.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaT": {"fullname": "pyerrors.dirac.gammaT", "modulename": "pyerrors.dirac", "qualname": "gammaT", "kind": "variable", "doc": "\n", "default_value": "array([[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n [1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gamma": {"fullname": "pyerrors.dirac.gamma", "modulename": "pyerrors.dirac", "qualname": "gamma", "kind": "variable", "doc": "\n", "default_value": "array([[[ 0.+0.j, 0.+0.j, 0.+0.j, 0.+1.j],\n [ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, -0.-1.j, 0.+0.j, 0.+0.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j],\n [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [-1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -0.-1.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+1.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n [ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]]])"}, "pyerrors.dirac.gamma5": {"fullname": "pyerrors.dirac.gamma5", "modulename": "pyerrors.dirac", "qualname": "gamma5", "kind": "variable", "doc": "\n", "default_value": "array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, -1.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j]])"}, "pyerrors.dirac.identity": {"fullname": "pyerrors.dirac.identity", "modulename": "pyerrors.dirac", "qualname": "identity", "kind": "variable", "doc": "\n", "default_value": "array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]])"}, "pyerrors.dirac.epsilon_tensor": {"fullname": "pyerrors.dirac.epsilon_tensor", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor", "kind": "function", "doc": "Rank-3 epsilon tensor
\n\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.fit_parameters": {"fullname": "pyerrors.fits.Fit_result.fit_parameters", "modulename": "pyerrors.fits", "qualname": "Fit_result.fit_parameters", "kind": "variable", "doc": "\n"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "- 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 (dict or list, optional):\npriors can either be a dictionary with integer keys and the corresponding priors as values or\na list with an entry for every parameter in the fit. The entries can either be\nObs (e.g. results from a previous fit) or strings containing a value and an error formatted like\n0.548(23), 500(40) or 0.5(0.4)
\n- silent (bool, optional):\nIf true all output to the console is omitted (default False).
\n- initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for\nnon-linear fits with many parameters. In case of correlated fits the guess is used to perform\nan uncorrelated fit which then serves as guess for the correlated fit.
\n- method (str, optional):\ncan be used to choose an alternative method for the minimization of chisquare.\nThe possible methods are the ones which can be used for scipy.optimize.minimize and\nmigrad of iminuit. If no method is specified, Levenberg-Marquard is used.\nReliable alternatives are migrad, Powell and Nelder-Mead.
\n- tol (float, optional):\ncan be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence\nto a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly\ninvalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values\nThe stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
\n- correlated_fit (bool):\nIf True, use the full inverse covariance matrix in the definition of the chisquare cost function.\nFor details about how the covariance matrix is estimated see
\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).- 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, title=''):", "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\nFor uncorrelated data the residuals are expected to be distributed ~N(0,1).
\n\nReturns
\n\n\n
\n", "signature": "(x, y, func, fit_res, title=''):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "- 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- 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, 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- 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": "(fname, full_output=False, gz=True, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.create_dobs_string": {"fullname": "pyerrors.input.dobs.create_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_dobs_string", "kind": "function", "doc": "- 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_hd5": {"fullname": "pyerrors.input.hadrons.read_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_hd5", "kind": "function", "doc": "- None
\nRead hadrons hdf5 file and extract entry based on attributes.
\n\nParameters
\n\n\n
\n\n- filestem (str):\nFull namestem of the files to read, including the full path.
\n- ens_id (str):\nname of the ensemble, required for internal bookkeeping
\n- group (str):\nlabel of the group to be extracted.
\n- \n
attrs (dict or int):\nDictionary containing the attributes. For example
\n\n\n\n\n\nattrs = {"gamma_snk": "Gamma5",\n "gamma_src": "Gamma5"}\n
Alternatively an integer can be specified to identify the sub group.\nThis is discouraged as the order in the file is not guaranteed.
- idl (range):\nIf specified only configurations in the given range are read in.
\n- part (str):\nstring specifying whether to extract the real part ('real'),\nthe imaginary part ('imag') or a complex correlator ('complex').\nDefault 'real'.
\nReturns
\n\n\n
\n", "signature": "(filestem, ens_id, group, attrs=None, idl=None, part='real'):", "funcdef": "def"}, "pyerrors.input.hadrons.read_meson_hd5": {"fullname": "pyerrors.input.hadrons.read_meson_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_meson_hd5", "kind": "function", "doc": "- corr (Corr):\nCorrelator of the source sink combination in question.
\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 sink and source (gamma_snk, gamma_src).\n(\"Gamma5\", \"Gamma5\") corresponds to the pseudoscalar pseudoscalar\ntwo-point function. The gammas argument dominateds over meson.
\n- idl (range):\nIf specified only configurations in the given range are read in.
\nReturns
\n\n\n
\n", "signature": "(path, filestem, ens_id, meson='meson_0', idl=None, gammas=None):", "funcdef": "def"}, "pyerrors.input.hadrons.extract_t0_hd5": {"fullname": "pyerrors.input.hadrons.extract_t0_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "extract_t0_hd5", "kind": "function", "doc": "- corr (Corr):\nCorrelator of the source sink combination in question.
\nRead hadrons FlowObservables hdf5 file and extract t0
\n\nParameters
\n\n\n
\n", "signature": "(\tpath,\tfilestem,\tens_id,\tobs='Clover energy density',\tfit_range=5,\tidl=None,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.hadrons.read_DistillationContraction_hd5": {"fullname": "pyerrors.input.hadrons.read_DistillationContraction_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_DistillationContraction_hd5", "kind": "function", "doc": "- path (str):\npath to the files to read
\n- filestem (str):\nnamestem of the files to read
\n- ens_id (str):\nname of the ensemble, required for internal bookkeeping
\n- obs (str):\nlabel of the observable from which t0 should be extracted.\nOptions: 'Clover energy density' and 'Plaquette energy density'
\n- fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
\n- idl (range):\nIf specified only configurations in the given range are read in.
\n- plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
\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>>> import numpy as np\n>>> np.ndarray(shape=(2,2), dtype=float, order='F')\narray([[0.0e+000, 0.0e+000], # random\n [ nan, 2.5e-323]])\n
Second mode:
\n\n\n\n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "\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.\nDict keys that are not JSON-serializable such as floats are converted to strings.
\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.fit_t0": {"fullname": "pyerrors.input.misc.fit_t0", "modulename": "pyerrors.input.misc", "qualname": "fit_t0", "kind": "function", "doc": "- data (Obs / list / Corr):\nRead data
\n- or
\n- data (dict):\nRead data and meta-data
\nCompute the root of (flow-based) data based on a dictionary that contains\nthe necessary information in key-value pairs a la (flow time: observable at flow time).
\n\nIt is assumed that the data is monotonically increasing and passes zero from below.\nNo exception is thrown if this is not the case (several roots, no monotonic increase).\nAn exception is thrown if no root can be found in the data.
\n\nA linear fit in the vicinity of the root is performed to exctract the root from the\ntwo fit parameters.
\n\nParameters
\n\n\n
\n\n- t2E_dict (dict):\nDictionary with pairs of (flow time: observable at flow time) where the flow times\nare of type float and the observables of type Obs.
\n- fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit.
\n- plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data. (Default: False)
\n- observable (str):\nKeyword to identify the observable to print the correct ylabel (if plot_fit is True)\nfor the observables 't0' and 'w0'. No y label is printed otherwise. (Default: 't0')
\nReturns
\n\n\n
\n", "signature": "(t2E_dict, fit_range, plot_fit=False, observable='t0'):", "funcdef": "def"}, "pyerrors.input.misc.read_pbp": {"fullname": "pyerrors.input.misc.read_pbp", "modulename": "pyerrors.input.misc", "qualname": "read_pbp", "kind": "function", "doc": "- root (Obs):\nThe root of the data series.
\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/a^2 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- c (where c=0.3 by default)\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- postfix (str):\nPostfix of measurement file (Default: ms)
\n- c (float):\nConstant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1.
\n- r_start (list):\nlist which contains the first config to be read for each replicum.
\n- r_stop (list):\nlist which contains the last config to be read for each replicum.
\n- r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
\n- plaquette (bool):\nIf true extract the plaquette estimate of t0 instead.
\n- names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
\n- files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
\n- plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
\n- assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
\nReturns
\n\n\n
\n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_w0": {"fullname": "pyerrors.input.openQCD.extract_w0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_w0", "kind": "function", "doc": "- t0 (Obs):\nExtracted t0
\nExtract w0/a from given .ms.dat files. Returns w0 as Obs.
\n\nIt is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t d(t^2
\n\n)/dt - (where c=0.3 by default)\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- postfix (str):\nPostfix of measurement file (Default: ms)
\n- c (float):\nConstant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1.
\n- r_start (list):\nlist which contains the first config to be read for each replicum.
\n- r_stop (list):\nlist which contains the last config to be read for each replicum.
\n- r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
\n- plaquette (bool):\nIf true extract the plaquette estimate of w0 instead.
\n- names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
\n- files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
\n- plot_fit (bool):\nIf true, the fit for the extraction of w0 is shown together with the data.
\n- assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
\nReturns
\n\n\n
\n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop": {"fullname": "pyerrors.input.openQCD.read_qtop", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop", "kind": "function", "doc": "- w0 (Obs):\nExtracted w0
\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.
\n- files (List[str]): A list of files to read data from.
\n- idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given.
\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.sep": {"fullname": "pyerrors.input.sfcf.sep", "modulename": "pyerrors.input.sfcf", "qualname": "sep", "kind": "variable", "doc": "\n", "default_value": "'/'"}, "pyerrors.input.sfcf.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "- data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
\nRead sfcf files 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',\tsilent=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.sfcf.read_sfcf_multi": {"fullname": "pyerrors.input.sfcf.read_sfcf_multi", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf_multi", "kind": "function", "doc": "- result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
\nRead sfcf files 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_list (list[str]):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
\n- corr_type_list (list[str]):\nType of correlation function to read. Can be\n
\n\n
- 'bi' for boundary-inner
\n- 'bb' for boundary-boundary
\n- 'bib' for boundary-inner-boundary
\n- noffset_list (list[int]):\nOffset of the source (only relevant when wavefunctions are used)
\n- wf_list (int):\nID of wave function
\n- wf2_list (list[int]):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
\n- im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
\n- names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
\n- ens_name (str):\nreplaces the name of the ensemble
\n- version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
\n- cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
\n- replica (list):\nlist of replica to be read, default is all
\n- files (list[list[int]]):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
\n- check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
\nReturns
\n\n\n
\n", "signature": "(\tpath,\tprefix,\tname_list,\tquarks_list=['.*'],\tcorr_type_list=['bi'],\tnoffset_list=[0],\twf_list=[0],\twf2_list=[0],\tversion='1.0c',\tcfg_separator='n',\tsilent=False,\tkeyed_out=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.utils": {"fullname": "pyerrors.input.utils", "modulename": "pyerrors.input.utils", "kind": "module", "doc": "- result (dict[list[Obs]]):\ndict with one of the following properties:\nif keyed_out:\n dict[key] = list[Obs]\n where key has the form name/quarks/offset/wf/wf2\nif not keyed_out:\n dict[name][quarks][offset][wf][wf2] = list[Obs]
\nUtilities for the input
\n"}, "pyerrors.input.utils.sort_names": {"fullname": "pyerrors.input.utils.sort_names", "modulename": "pyerrors.input.utils", "qualname": "sort_names", "kind": "function", "doc": "Sorts a list of names of replika with searches for
\n\nr
andid
in the replikum string.\nIf this search fails, a fallback method is used,\nwhere the strings are simply compared and the first diffeing numeral is used for differentiation.Parameters
\n\n\n
\n\n- ll (list):\nlist to sort
\nReturns
\n\n\n
\n", "signature": "(ll):", "funcdef": "def"}, "pyerrors.input.utils.check_idl": {"fullname": "pyerrors.input.utils.check_idl", "modulename": "pyerrors.input.utils", "qualname": "check_idl", "kind": "function", "doc": "- ll (list):\nsorted list
\nChecks 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.input.utils.check_params": {"fullname": "pyerrors.input.utils.check_params", "modulename": "pyerrors.input.utils", "qualname": "check_params", "kind": "function", "doc": "- miss_str (str):\nstring with integers of which idls are missing
\nCheck if, for sfcf, the parameter hashes at the end of the parameter files are in fact the expected one.
\n\nParameters
\n\n\n
\n\n- path (str):\nmeasurement path, same as for sfcf read method
\n- param_hash (str):\nexpected parameter hash
\n- prefix (str):\ndata prefix to find the appropriate replicum folders in path
\n- param_prefix (str):\nprefix of the parameter file. Defaults to 'parameters_'
\nReturns
\n\n\n
\n", "signature": "(path, param_hash, prefix, param_prefix='parameters_'):", "funcdef": "def"}, "pyerrors.integrate": {"fullname": "pyerrors.integrate", "modulename": "pyerrors.integrate", "kind": "module", "doc": "\n"}, "pyerrors.integrate.quad": {"fullname": "pyerrors.integrate.quad", "modulename": "pyerrors.integrate", "qualname": "quad", "kind": "function", "doc": "- nums (dict):\ndictionary of faulty parameter files sorted by the replica paths
\nPerforms a (one-dimensional) numeric integration of f(p, x) from a to b.
\n\nThe integration is performed using scipy.integrate.quad().\nAll parameters that can be passed to scipy.integrate.quad may also be passed to this function.\nThe output is the same as for scipy.integrate.quad, the first element being an Obs.
\n\nParameters
\n\n\n
\n\n- \n
func (object):\nfunction to integrate, has to be of the form
\n\n\n\n\n\nimport autograd.numpy as anp\n\ndef func(p, x):\n return p[0] + p[1] * x + p[2] * anp.sinh(x)\n
where x is the integration variable.
- p (list of floats or Obs):\nparameters of the function func.
\n- a (float or Obs):\nLower limit of integration (use -numpy.inf for -infinity).
\n- b (float or Obs):\nUpper limit of integration (use -numpy.inf for -infinity).
\n- All parameters of scipy.integrate.quad
\nReturns
\n\n\n
\n", "signature": "(func, p, a, b, **kwargs):", "funcdef": "def"}, "pyerrors.linalg": {"fullname": "pyerrors.linalg", "modulename": "pyerrors.linalg", "kind": "module", "doc": "\n"}, "pyerrors.linalg.matmul": {"fullname": "pyerrors.linalg.matmul", "modulename": "pyerrors.linalg", "qualname": "matmul", "kind": "function", "doc": "- y (Obs):\nThe integral of func from
\na
tob
.- abserr (float):\nAn estimate of the absolute error in the result.
\n- infodict (dict):\nA dictionary containing additional information.\nRun scipy.integrate.quad_explain() for more information.
\n- message: A convergence message.
\n- explain: Appended only with 'cos' or 'sin' weighting and infinite\nintegration limits, it contains an explanation of the codes in\ninfodict['ierlst']
\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.eigv": {"fullname": "pyerrors.linalg.eigv", "modulename": "pyerrors.linalg", "qualname": "eigv", "kind": "function", "doc": "Computes the eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.pinv": {"fullname": "pyerrors.linalg.pinv", "modulename": "pyerrors.linalg", "qualname": "pinv", "kind": "function", "doc": "Computes the Moore-Penrose pseudoinverse of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.svd": {"fullname": "pyerrors.linalg.svd", "modulename": "pyerrors.linalg", "qualname": "svd", "kind": "function", "doc": "Computes the singular value decomposition of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.misc": {"fullname": "pyerrors.misc", "modulename": "pyerrors.misc", "kind": "module", "doc": "\n"}, "pyerrors.misc.print_config": {"fullname": "pyerrors.misc.print_config", "modulename": "pyerrors.misc", "qualname": "print_config", "kind": "function", "doc": "Print information about version of python, pyerrors and dependencies.
\n", "signature": "():", "funcdef": "def"}, "pyerrors.misc.errorbar": {"fullname": "pyerrors.misc.errorbar", "modulename": "pyerrors.misc", "qualname": "errorbar", "kind": "function", "doc": "pyerrors wrapper for the errorbars method of matplotlib
\n\nParameters
\n\n\n
\n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.10.14/x64/lib/python3.10/site-packages/matplotlib/pyplot.py'>,\t**kwargs):", "funcdef": "def"}, "pyerrors.misc.dump_object": {"fullname": "pyerrors.misc.dump_object", "modulename": "pyerrors.misc", "qualname": "dump_object", "kind": "function", "doc": "- 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.S_global": {"fullname": "pyerrors.obs.Obs.S_global", "modulename": "pyerrors.obs", "qualname": "Obs.S_global", "kind": "variable", "doc": "\n", "default_value": "2.0"}, "pyerrors.obs.Obs.S_dict": {"fullname": "pyerrors.obs.Obs.S_dict", "modulename": "pyerrors.obs", "qualname": "Obs.S_dict", "kind": "variable", "doc": "\n", "default_value": "{}"}, "pyerrors.obs.Obs.tau_exp_global": {"fullname": "pyerrors.obs.Obs.tau_exp_global", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_global", "kind": "variable", "doc": "\n", "default_value": "0.0"}, "pyerrors.obs.Obs.tau_exp_dict": {"fullname": "pyerrors.obs.Obs.tau_exp_dict", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_dict", "kind": "variable", "doc": "\n", "default_value": "{}"}, "pyerrors.obs.Obs.N_sigma_global": {"fullname": "pyerrors.obs.Obs.N_sigma_global", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_global", "kind": "variable", "doc": "\n", "default_value": "1.0"}, "pyerrors.obs.Obs.N_sigma_dict": {"fullname": "pyerrors.obs.Obs.N_sigma_dict", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_dict", "kind": "variable", "doc": "\n", "default_value": "{}"}, "pyerrors.obs.Obs.names": {"fullname": "pyerrors.obs.Obs.names", "modulename": "pyerrors.obs", "qualname": "Obs.names", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.shape": {"fullname": "pyerrors.obs.Obs.shape", "modulename": "pyerrors.obs", "qualname": "Obs.shape", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.r_values": {"fullname": "pyerrors.obs.Obs.r_values", "modulename": "pyerrors.obs", "qualname": "Obs.r_values", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.deltas": {"fullname": "pyerrors.obs.Obs.deltas", "modulename": "pyerrors.obs", "qualname": "Obs.deltas", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.N": {"fullname": "pyerrors.obs.Obs.N", "modulename": "pyerrors.obs", "qualname": "Obs.N", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.idl": {"fullname": "pyerrors.obs.Obs.idl", "modulename": "pyerrors.obs", "qualname": "Obs.idl", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.ddvalue": {"fullname": "pyerrors.obs.Obs.ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.ddvalue", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.reweighted": {"fullname": "pyerrors.obs.Obs.reweighted", "modulename": "pyerrors.obs", "qualname": "Obs.reweighted", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.tag": {"fullname": "pyerrors.obs.Obs.tag", "modulename": "pyerrors.obs", "qualname": "Obs.tag", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.value": {"fullname": "pyerrors.obs.Obs.value", "modulename": "pyerrors.obs", "qualname": "Obs.value", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.dvalue": {"fullname": "pyerrors.obs.Obs.dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.dvalue", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_names": {"fullname": "pyerrors.obs.Obs.e_names", "modulename": "pyerrors.obs", "qualname": "Obs.e_names", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.cov_names": {"fullname": "pyerrors.obs.Obs.cov_names", "modulename": "pyerrors.obs", "qualname": "Obs.cov_names", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.mc_names": {"fullname": "pyerrors.obs.Obs.mc_names", "modulename": "pyerrors.obs", "qualname": "Obs.mc_names", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_content": {"fullname": "pyerrors.obs.Obs.e_content", "modulename": "pyerrors.obs", "qualname": "Obs.e_content", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.covobs": {"fullname": "pyerrors.obs.Obs.covobs", "modulename": "pyerrors.obs", "qualname": "Obs.covobs", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.gamma_method": {"fullname": "pyerrors.obs.Obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "Obs.gamma_method", "kind": "function", "doc": "- 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.export_bootstrap": {"fullname": "pyerrors.obs.Obs.export_bootstrap", "modulename": "pyerrors.obs", "qualname": "Obs.export_bootstrap", "kind": "function", "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).
\nExport bootstrap samples from the Obs
\n\nParameters
\n\n\n
\n\n- samples (int):\nNumber of bootstrap samples to generate.
\n- random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples.\nIf not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
\n- save_rng (str):\nSave the random numbers to a file if a path is specified.
\nReturns
\n\n\n
\n", "signature": "(self, samples=500, random_numbers=None, save_rng=None):", "funcdef": "def"}, "pyerrors.obs.Obs.sqrt": {"fullname": "pyerrors.obs.Obs.sqrt", "modulename": "pyerrors.obs", "qualname": "Obs.sqrt", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.log": {"fullname": "pyerrors.obs.Obs.log", "modulename": "pyerrors.obs", "qualname": "Obs.log", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.exp": {"fullname": "pyerrors.obs.Obs.exp", "modulename": "pyerrors.obs", "qualname": "Obs.exp", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sin": {"fullname": "pyerrors.obs.Obs.sin", "modulename": "pyerrors.obs", "qualname": "Obs.sin", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cos": {"fullname": "pyerrors.obs.Obs.cos", "modulename": "pyerrors.obs", "qualname": "Obs.cos", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tan": {"fullname": "pyerrors.obs.Obs.tan", "modulename": "pyerrors.obs", "qualname": "Obs.tan", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsin": {"fullname": "pyerrors.obs.Obs.arcsin", "modulename": "pyerrors.obs", "qualname": "Obs.arcsin", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccos": {"fullname": "pyerrors.obs.Obs.arccos", "modulename": "pyerrors.obs", "qualname": "Obs.arccos", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctan": {"fullname": "pyerrors.obs.Obs.arctan", "modulename": "pyerrors.obs", "qualname": "Obs.arctan", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sinh": {"fullname": "pyerrors.obs.Obs.sinh", "modulename": "pyerrors.obs", "qualname": "Obs.sinh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cosh": {"fullname": "pyerrors.obs.Obs.cosh", "modulename": "pyerrors.obs", "qualname": "Obs.cosh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tanh": {"fullname": "pyerrors.obs.Obs.tanh", "modulename": "pyerrors.obs", "qualname": "Obs.tanh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsinh": {"fullname": "pyerrors.obs.Obs.arcsinh", "modulename": "pyerrors.obs", "qualname": "Obs.arcsinh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccosh": {"fullname": "pyerrors.obs.Obs.arccosh", "modulename": "pyerrors.obs", "qualname": "Obs.arccosh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctanh": {"fullname": "pyerrors.obs.Obs.arctanh", "modulename": "pyerrors.obs", "qualname": "Obs.arctanh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.N_sigma": {"fullname": "pyerrors.obs.Obs.N_sigma", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.S": {"fullname": "pyerrors.obs.Obs.S", "modulename": "pyerrors.obs", "qualname": "Obs.S", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_ddvalue": {"fullname": "pyerrors.obs.Obs.e_ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_ddvalue", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_drho": {"fullname": "pyerrors.obs.Obs.e_drho", "modulename": "pyerrors.obs", "qualname": "Obs.e_drho", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_dtauint": {"fullname": "pyerrors.obs.Obs.e_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_dtauint", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_dvalue": {"fullname": "pyerrors.obs.Obs.e_dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_dvalue", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_n_dtauint": {"fullname": "pyerrors.obs.Obs.e_n_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_dtauint", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_n_tauint": {"fullname": "pyerrors.obs.Obs.e_n_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_tauint", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_rho": {"fullname": "pyerrors.obs.Obs.e_rho", "modulename": "pyerrors.obs", "qualname": "Obs.e_rho", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_tauint": {"fullname": "pyerrors.obs.Obs.e_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_tauint", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_windowsize": {"fullname": "pyerrors.obs.Obs.e_windowsize", "modulename": "pyerrors.obs", "qualname": "Obs.e_windowsize", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.tau_exp": {"fullname": "pyerrors.obs.Obs.tau_exp", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp", "kind": "variable", "doc": "\n"}, "pyerrors.obs.CObs": {"fullname": "pyerrors.obs.CObs", "modulename": "pyerrors.obs", "qualname": "CObs", "kind": "class", "doc": "- numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N import_bootstrap samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived bootstrap samples\nshould agree with samples from a full bootstrap analysis up to O(1/N).
\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.tag": {"fullname": "pyerrors.obs.CObs.tag", "modulename": "pyerrors.obs", "qualname": "CObs.tag", "kind": "variable", "doc": "\n"}, "pyerrors.obs.CObs.real": {"fullname": "pyerrors.obs.CObs.real", "modulename": "pyerrors.obs", "qualname": "CObs.real", "kind": "variable", "doc": "\n"}, "pyerrors.obs.CObs.imag": {"fullname": "pyerrors.obs.CObs.imag", "modulename": "pyerrors.obs", "qualname": "CObs.imag", "kind": "variable", "doc": "\n"}, "pyerrors.obs.CObs.gamma_method": {"fullname": "pyerrors.obs.CObs.gamma_method", "modulename": "pyerrors.obs", "qualname": "CObs.gamma_method", "kind": "function", "doc": "Executes the gamma_method for the real and the imaginary part.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.CObs.is_zero": {"fullname": "pyerrors.obs.CObs.is_zero", "modulename": "pyerrors.obs", "qualname": "CObs.is_zero", "kind": "function", "doc": "Checks whether both real and imaginary part are zero within machine precision.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs.conjugate": {"fullname": "pyerrors.obs.CObs.conjugate", "modulename": "pyerrors.obs", "qualname": "CObs.conjugate", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.gamma_method": {"fullname": "pyerrors.obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "gamma_method", "kind": "function", "doc": "Vectorized version of the gamma_method applicable to lists or arrays of Obs.
\n\nSee docstring of pe.Obs.gamma_method for details.
\n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.gm": {"fullname": "pyerrors.obs.gm", "modulename": "pyerrors.obs", "qualname": "gm", "kind": "function", "doc": "Vectorized version of the gamma_method applicable to lists or arrays of Obs.
\n\nSee docstring of pe.Obs.gamma_method for details.
\n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.derived_observable": {"fullname": "pyerrors.obs.derived_observable", "modulename": "pyerrors.obs", "qualname": "derived_observable", "kind": "function", "doc": "Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.
\n\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.import_bootstrap": {"fullname": "pyerrors.obs.import_bootstrap", "modulename": "pyerrors.obs", "qualname": "import_bootstrap", "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.
\nImports bootstrap samples and returns an Obs
\n\nParameters
\n\n\n
\n", "signature": "(boots, name, random_numbers):", "funcdef": "def"}, "pyerrors.obs.merge_obs": {"fullname": "pyerrors.obs.merge_obs", "modulename": "pyerrors.obs", "qualname": "merge_obs", "kind": "function", "doc": "- boots (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N bootstrap samples as first to Nth entry.
\n- name (str):\nname of the ensemble the samples are defined on.
\n- random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples,\nwhere samples is the number of bootstrap samples and length is the length of the original Monte Carlo\nchain to be reconstructed.
\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
\n", "signature": "(d, func, guess=1.0, **kwargs):", "funcdef": "def"}, "pyerrors.special": {"fullname": "pyerrors.special", "modulename": "pyerrors.special", "kind": "module", "doc": "\n"}, "pyerrors.special.beta": {"fullname": "pyerrors.special.beta", "modulename": "pyerrors.special", "qualname": "beta", "kind": "function", "doc": "- res (Obs):\n
\nObs
valued root of the function.beta(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nbeta(a, b, out=None)
\n\nBeta function.
\n\nThis function is defined in 1 as
\n\n$$B(a, b) = \\int_0^1 t^{a-1}(1-t)^{b-1}dt\n = \\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)},$$
\n\nwhere \\( \\Gamma \\) is the gamma function.
\n\nParameters
\n\n\n
\n\n- a, b (array_like):\nReal-valued arguments
\n- out (ndarray, optional):\nOptional output array for the function result
\nReturns
\n\n\n
\n\n- scalar or ndarray: Value of the beta function
\nSee Also
\n\n\n\n
gamma
: the gamma function
\nbetainc
: the regularized incomplete beta function
\nbetaln
: the natural logarithm of the absolute\nvalue of the beta functionReferences
\n\nExamples
\n\n\n\n\n\n>>> import scipy.special as sc\n
The beta function relates to the gamma function by the\ndefinition given above:
\n\n\n\n\n\n>>> sc.beta(2, 3)\n0.08333333333333333\n>>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3)\n0.08333333333333333\n
As this relationship demonstrates, the beta function\nis symmetric:
\n\n\n\n\n\n>>> sc.beta(1.7, 2.4)\n0.16567527689031739\n>>> sc.beta(2.4, 1.7)\n0.16567527689031739\n
This function satisfies \\( B(1, b) = 1/b \\):
\n\n\n\n\n\n>>> sc.beta(1, 4)\n0.25\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betainc": {"fullname": "pyerrors.special.betainc", "modulename": "pyerrors.special", "qualname": "betainc", "kind": "function", "doc": "
\n\n
\n- \n
\nNIST Digital Library of Mathematical Functions,\nEq. 5.12.1. https://dlmf.nist.gov/5.12 ↩
\nbetainc(x1, x2, x3, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nbetainc(a, b, x, out=None)
\n\nRegularized incomplete beta function.
\n\nComputes the regularized incomplete beta function, defined as 1:
\n\n$$I_x(a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^x\nt^{a-1}(1-t)^{b-1}dt,$$
\n\nfor \\( 0 \\leq x \\leq 1 \\).
\n\nThis function is the cumulative distribution function for the beta\ndistribution; its range is [0, 1].
\n\nParameters
\n\n\n
\n\n- a, b (array_like):\nPositive, real-valued parameters
\n- x (array_like):\nReal-valued such that \\( 0 \\leq x \\leq 1 \\),\nthe upper limit of integration
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Value of the regularized incomplete beta function
\nSee Also
\n\n\n\n
beta
: beta function
\nbetaincinv
: inverse of the regularized incomplete beta function
\nbetaincc
: complement of the regularized incomplete beta function
\nscipy.stats.beta
: beta distributionNotes
\n\nThe term regularized in the name of this function refers to the\nscaling of the function by the gamma function terms shown in the\nformula. When not qualified as regularized, the name incomplete\nbeta function often refers to just the integral expression,\nwithout the gamma terms. One can use the function
\n\nbeta
from\nscipy.special
to get this \"nonregularized\" incomplete beta\nfunction by multiplying the result ofbetainc(a, b, x)
by\nbeta(a, b)
.References
\n\nExamples
\n\nLet \\( B(a, b) \\) be the
\n\nbeta
function.\n\n\n\n>>> import scipy.special as sc\n
The coefficient in terms of
\n\ngamma
is equal to\n\\( 1/B(a, b) \\). Also, when \\( x=1 \\)\nthe integral is equal to \\( B(a, b) \\).\nTherefore, \\( I_{x=1}(a, b) = 1 \\) for any \\( a, b \\).\n\n\n\n>>> sc.betainc(0.2, 3.5, 1.0)\n1.0\n
It satisfies\n\\( I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b)) \\),\nwhere \\( F \\) is the hypergeometric function
\n\nhyp2f1
:\n\n\n\n>>> a, b, x = 1.4, 3.1, 0.5\n>>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))\n0.8148904036225295\n>>> sc.betainc(a, b, x)\n0.8148904036225296\n
This functions satisfies the relationship\n\\( I_x(a, b) = 1 - I_{1-x}(b, a) \\):
\n\n\n\n\n\n>>> sc.betainc(2.2, 3.1, 0.4)\n0.49339638807619446\n>>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)\n0.49339638807619446\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betaln": {"fullname": "pyerrors.special.betaln", "modulename": "pyerrors.special", "qualname": "betaln", "kind": "function", "doc": "
\n\n
\n- \n
\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/8.17 ↩
\nbetaln(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nbetaln(a, b, out=None)
\n\nNatural logarithm of absolute value of beta function.
\n\nComputes
\n\nln(abs(beta(a, b)))
.Parameters
\n\n\n
\n\n- a, b (array_like):\nPositive, real-valued parameters
\n- out (ndarray, optional):\nOptional output array for function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Value of the betaln function
\nSee Also
\n\n\n\n
gamma
: the gamma function
\nbetainc
: the regularized incomplete beta function
\nbeta
: the beta functionExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import betaln, beta\n
Verify that, for moderate values of
\n\na
andb
,betaln(a, b)
\nis the same aslog(beta(a, b))
:\n\n\n\n>>> betaln(3, 4)\n-4.0943445622221\n
\n\n\n\n>>> np.log(beta(3, 4))\n-4.0943445622221\n
In the following
\n\nbeta(a, b)
underflows to 0, so we can't compute\nthe logarithm of the actual value.\n\n\n\n>>> a = 400\n>>> b = 900\n>>> beta(a, b)\n0.0\n
We can compute the logarithm of
\n\nbeta(a, b)
by usingbetaln
:\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.polygamma": {"fullname": "pyerrors.special.polygamma", "modulename": "pyerrors.special", "qualname": "polygamma", "kind": "function", "doc": "\n>>> betaln(a, b)\n-804.3069951764146\n
Polygamma functions.
\n\nDefined as \\( \\psi^{(n)}(x) \\) where \\( \\psi \\) is the\n
\n\ndigamma
function. See [dlmf]_ for details.Parameters
\n\n\n
\n\n- n (array_like):\nThe order of the derivative of the digamma function; must be\nintegral
\n- x (array_like):\nReal valued input
\nReturns
\n\n\n
\n\n- ndarray: Function results
\nSee Also
\n\n\n\n
digamma
References
\n\n.. [dlmf] NIST, Digital Library of Mathematical Functions,\n https://dlmf.nist.gov/5.15
\n\nExamples
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.psi": {"fullname": "pyerrors.special.psi", "modulename": "pyerrors.special", "qualname": "psi", "kind": "function", "doc": "\n>>> from scipy import special\n>>> x = [2, 3, 25.5]\n>>> special.polygamma(1, x)\narray([ 0.64493407, 0.39493407, 0.03999467])\n>>> special.polygamma(0, x) == special.psi(x)\narray([ True, True, True], dtype=bool)\n
psi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\npsi(z, out=None)
\n\nThe digamma function.
\n\nThe logarithmic derivative of the gamma function evaluated at
\n\nz
.Parameters
\n\n\n
\n\n- z (array_like):\nReal or complex argument.
\n- out (ndarray, optional):\nArray for the computed values of
\npsi
.Returns
\n\n\n
\n\n- digamma (scalar or ndarray):\nComputed values of
\npsi
.Notes
\n\nFor large values not close to the negative real axis,
\n\npsi
is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note thatpsi
has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.References
\n\nExamples
\n\n\n\n\n\n>>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
Verify psi(z) = psi(z + 1) - 1/z:
\n\n\n\n\n\n>>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.digamma": {"fullname": "pyerrors.special.digamma", "modulename": "pyerrors.special", "qualname": "digamma", "kind": "function", "doc": "
\n\n
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\nFredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
\npsi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\npsi(z, out=None)
\n\nThe digamma function.
\n\nThe logarithmic derivative of the gamma function evaluated at
\n\nz
.Parameters
\n\n\n
\n\n- z (array_like):\nReal or complex argument.
\n- out (ndarray, optional):\nArray for the computed values of
\npsi
.Returns
\n\n\n
\n\n- digamma (scalar or ndarray):\nComputed values of
\npsi
.Notes
\n\nFor large values not close to the negative real axis,
\n\npsi
is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note thatpsi
has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.References
\n\nExamples
\n\n\n\n\n\n>>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
Verify psi(z) = psi(z + 1) - 1/z:
\n\n\n\n\n\n>>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gamma": {"fullname": "pyerrors.special.gamma", "modulename": "pyerrors.special", "qualname": "gamma", "kind": "function", "doc": "
\n\n
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\nFredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
\ngamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ngamma(z, out=None)
\n\ngamma function.
\n\nThe gamma function is defined as
\n\n$$\\Gamma(z) = \\int_0^\\infty t^{z-1} e^{-t} dt$$
\n\nfor \\( \\Re(z) > 0 \\) and is extended to the rest of the complex\nplane by analytic continuation. See [dlmf]_ for more details.
\n\nParameters
\n\n\n
\n\n- z (array_like):\nReal or complex valued argument
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the gamma function
\nNotes
\n\nThe gamma function is often referred to as the generalized\nfactorial since \\( \\Gamma(n + 1) = n! \\) for natural numbers\n\\( n \\). More generally it satisfies the recurrence relation\n\\( \\Gamma(z + 1) = z \\cdot \\Gamma(z) \\) for complex \\( z \\),\nwhich, combined with the fact that \\( \\Gamma(1) = 1 \\), implies\nthe above identity for \\( z = n \\).
\n\nReferences
\n\n.. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5.2#E1
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import gamma, factorial\n
\n\n\n\n>>> gamma([0, 0.5, 1, 5])\narray([ inf, 1.77245385, 1. , 24. ])\n
\n\n\n\n>>> z = 2.5 + 1j\n>>> gamma(z)\n(0.77476210455108352+0.70763120437959293j)\n>>> gamma(z+1), z*gamma(z) # Recurrence property\n((1.2292740569981171+2.5438401155000685j),\n (1.2292740569981158+2.5438401155000658j))\n
\n\n\n\n>>> gamma(0.5)**2 # gamma(0.5) = sqrt(pi)\n3.1415926535897927\n
Plot gamma(x) for real x
\n\n\n\n\n\n>>> x = np.linspace(-3.5, 5.5, 2251)\n>>> y = gamma(x)\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaln": {"fullname": "pyerrors.special.gammaln", "modulename": "pyerrors.special", "qualname": "gammaln", "kind": "function", "doc": "\n>>> import matplotlib.pyplot as plt\n>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')\n>>> k = np.arange(1, 7)\n>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,\n... label='(x-1)!, x = 1, 2, ...')\n>>> plt.xlim(-3.5, 5.5)\n>>> plt.ylim(-10, 25)\n>>> plt.grid()\n>>> plt.xlabel('x')\n>>> plt.legend(loc='lower right')\n>>> plt.show()\n
gammaln(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ngammaln(x, out=None)
\n\nLogarithm of the absolute value of the gamma function.
\n\nDefined as
\n\n$$\\ln(\\lvert\\Gamma(x)\\rvert)$$
\n\nwhere \\( \\Gamma \\) is the gamma function. For more details on\nthe gamma function, see [dlmf]_.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nReal argument
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the log of the absolute value of gamma
\nSee Also
\n\n\n\n
gammasgn
: sign of the gamma function
\nloggamma
: principal branch of the logarithm of the gamma functionNotes
\n\nIt is the same function as the Python standard library function\n
\n\nmath.lgamma()
.When used in conjunction with
\n\ngammasgn
, this function is useful\nfor working in logspace on the real axis without having to deal\nwith complex numbers via the relationexp(gammaln(x)) =\ngammasgn(x) * gamma(x)
.For complex-valued log-gamma, use
\n\nloggamma
instead ofgammaln
.References
\n\n.. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> import scipy.special as sc\n
It has two positive zeros.
\n\n\n\n\n\n>>> sc.gammaln([1, 2])\narray([0., 0.])\n
It has poles at nonpositive integers.
\n\n\n\n\n\n>>> sc.gammaln([0, -1, -2, -3, -4])\narray([inf, inf, inf, inf, inf])\n
It asymptotically approaches
\n\nx * log(x)
(Stirling's formula).\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammainc": {"fullname": "pyerrors.special.gammainc", "modulename": "pyerrors.special", "qualname": "gammainc", "kind": "function", "doc": "\n>>> x = np.array([1e10, 1e20, 1e40, 1e80])\n>>> sc.gammaln(x)\narray([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82])\n>>> x * np.log(x)\narray([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])\n
gammainc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ngammainc(a, x, out=None)
\n\nRegularized lower incomplete gamma function.
\n\nIt is defined as
\n\n$$P(a, x) = \\frac{1}{\\Gamma(a)} \\int_0^x t^{a - 1}e^{-t} dt$$
\n\nfor \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
\n\nParameters
\n\n\n
\n\n- a (array_like):\nPositive parameter
\n- x (array_like):\nNonnegative argument
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the lower incomplete gamma function
\nSee Also
\n\n\n\n
gammaincc
: regularized upper incomplete gamma function
\ngammaincinv
: inverse of the regularized lower incomplete gamma function
\ngammainccinv
: inverse of the regularized upper incomplete gamma functionNotes
\n\nThe function satisfies the relation
\n\ngammainc(a, x) +\ngammaincc(a, x) = 1
wheregammaincc
is the regularized upper\nincomplete gamma function.The implementation largely follows that of [boost]_.
\n\nReferences
\n\n.. [dlmf] NIST Digital Library of Mathematical functions\n https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
\n\nExamples
\n\n\n\n\n\n>>> import scipy.special as sc\n
It is the CDF of the gamma distribution, so it starts at 0 and\nmonotonically increases to 1.
\n\n\n\n\n\n>>> sc.gammainc(0.5, [0, 1, 10, 100])\narray([0. , 0.84270079, 0.99999226, 1. ])\n
It is equal to one minus the upper incomplete gamma function.
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaincc": {"fullname": "pyerrors.special.gammaincc", "modulename": "pyerrors.special", "qualname": "gammaincc", "kind": "function", "doc": "\n>>> a, x = 0.5, 0.4\n>>> sc.gammainc(a, x)\n0.6289066304773024\n>>> 1 - sc.gammaincc(a, x)\n0.6289066304773024\n
gammaincc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ngammaincc(a, x, out=None)
\n\nRegularized upper incomplete gamma function.
\n\nIt is defined as
\n\n$$Q(a, x) = \\frac{1}{\\Gamma(a)} \\int_x^\\infty t^{a - 1}e^{-t} dt$$
\n\nfor \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
\n\nParameters
\n\n\n
\n\n- a (array_like):\nPositive parameter
\n- x (array_like):\nNonnegative argument
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the upper incomplete gamma function
\nSee Also
\n\n\n\n
gammainc
: regularized lower incomplete gamma function
\ngammaincinv
: inverse of the regularized lower incomplete gamma function
\ngammainccinv
: inverse of the regularized upper incomplete gamma functionNotes
\n\nThe function satisfies the relation
\n\ngammainc(a, x) +\ngammaincc(a, x) = 1
wheregammainc
is the regularized lower\nincomplete gamma function.The implementation largely follows that of [boost]_.
\n\nReferences
\n\n.. [dlmf] NIST Digital Library of Mathematical functions\n https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
\n\nExamples
\n\n\n\n\n\n>>> import scipy.special as sc\n
It is the survival function of the gamma distribution, so it\nstarts at 1 and monotonically decreases to 0.
\n\n\n\n\n\n>>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000])\narray([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45,\n 0.00000000e+00])\n
It is equal to one minus the lower incomplete gamma function.
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammasgn": {"fullname": "pyerrors.special.gammasgn", "modulename": "pyerrors.special", "qualname": "gammasgn", "kind": "function", "doc": "\n>>> a, x = 0.5, 0.4\n>>> sc.gammaincc(a, x)\n0.37109336952269756\n>>> 1 - sc.gammainc(a, x)\n0.37109336952269756\n
gammasgn(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ngammasgn(x, out=None)
\n\nSign of the gamma function.
\n\nIt is defined as
\n\n$$\\text{gammasgn}(x) =\n\\begin{cases}\n +1 & \\Gamma(x) > 0 \\\n -1 & \\Gamma(x) < 0\n\\end{cases}$$
\n\nwhere \\( \\Gamma \\) is the gamma function; see
\n\ngamma
. This\ndefinition is complete since the gamma function is never zero;\nsee the discussion after [dlmf]_.Parameters
\n\n\n
\n\n- x (array_like):\nReal argument
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Sign of the gamma function
\nSee Also
\n\n\n\n
gamma
: the gamma function
\ngammaln
: log of the absolute value of the gamma function
\nloggamma
: analytic continuation of the log of the gamma functionNotes
\n\nThe gamma function can be computed as
\n\ngammasgn(x) *\nnp.exp(gammaln(x))
.References
\n\n.. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5.2#E1
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> import scipy.special as sc\n
It is 1 for
\n\nx > 0
.\n\n\n\n>>> sc.gammasgn([1, 2, 3, 4])\narray([1., 1., 1., 1.])\n
It alternates between -1 and 1 for negative integers.
\n\n\n\n\n\n>>> sc.gammasgn([-0.5, -1.5, -2.5, -3.5])\narray([-1., 1., -1., 1.])\n
It can be used to compute the gamma function.
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.rgamma": {"fullname": "pyerrors.special.rgamma", "modulename": "pyerrors.special", "qualname": "rgamma", "kind": "function", "doc": "\n>>> x = [1.5, 0.5, -0.5, -1.5]\n>>> sc.gammasgn(x) * np.exp(sc.gammaln(x))\narray([ 0.88622693, 1.77245385, -3.5449077 , 2.3632718 ])\n>>> sc.gamma(x)\narray([ 0.88622693, 1.77245385, -3.5449077 , 2.3632718 ])\n
rgamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nrgamma(z, out=None)
\n\nReciprocal of the gamma function.
\n\nDefined as \\( 1 / \\Gamma(z) \\), where \\( \\Gamma \\) is the\ngamma function. For more on the gamma function see
\n\ngamma
.Parameters
\n\n\n
\n\n- z (array_like):\nReal or complex valued input
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- scalar or ndarray: Function results
\nSee Also
\n\n\n\n
gamma,
,gammaln,
,loggamma
Notes
\n\nThe gamma function has no zeros and has simple poles at\nnonpositive integers, so
\n\nrgamma
is an entire function with zeros\nat the nonpositive integers. See the discussion in [dlmf]_ for\nmore details.References
\n\n.. [dlmf] Nist, Digital Library of Mathematical functions,\n https://dlmf.nist.gov/5.2#i
\n\nExamples
\n\n\n\n\n\n>>> import scipy.special as sc\n
It is the reciprocal of the gamma function.
\n\n\n\n\n\n>>> sc.rgamma([1, 2, 3, 4])\narray([1. , 1. , 0.5 , 0.16666667])\n>>> 1 / sc.gamma([1, 2, 3, 4])\narray([1. , 1. , 0.5 , 0.16666667])\n
It is zero at nonpositive integers.
\n\n\n\n\n\n>>> sc.rgamma([0, -1, -2, -3])\narray([0., 0., 0., 0.])\n
It rapidly underflows to zero along the positive real axis.
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.multigammaln": {"fullname": "pyerrors.special.multigammaln", "modulename": "pyerrors.special", "qualname": "multigammaln", "kind": "function", "doc": "\n>>> sc.rgamma([10, 100, 179])\narray([2.75573192e-006, 1.07151029e-156, 0.00000000e+000])\n
Returns the log of multivariate gamma, also sometimes called the\ngeneralized gamma.
\n\nParameters
\n\n\n
\n\n- a (ndarray):\nThe multivariate gamma is computed for each item of
\na
.- d (int):\nThe dimension of the space of integration.
\nReturns
\n\n\n
\n\n- res (ndarray):\nThe values of the log multivariate gamma at the given points
\na
.Notes
\n\nThe formal definition of the multivariate gamma of dimension d for a real\n
\n\na
is$$\\Gamma_d(a) = \\int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA$$
\n\nwith the condition \\( a > (d-1)/2 \\), and \\( A > 0 \\) being the set of\nall the positive definite matrices of dimension
\n\nd
. Note thata
is a\nscalar: the integrand only is multivariate, the argument is not (the\nfunction is defined over a subset of the real set).This can be proven to be equal to the much friendlier equation
\n\n$$\\Gamma_d(a) = \\pi^{d(d-1)/4} \\prod_{i=1}^{d} \\Gamma(a - (i-1)/2).$$
\n\nReferences
\n\nR. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in\nprobability and mathematical statistics).
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import multigammaln, gammaln\n>>> a = 23.5\n>>> d = 10\n>>> multigammaln(a, d)\n454.1488605074416\n
Verify that the result agrees with the logarithm of the equation\nshown above:
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.kn": {"fullname": "pyerrors.special.kn", "modulename": "pyerrors.special", "qualname": "kn", "kind": "function", "doc": "\n>>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()\n454.1488605074416\n
Modified Bessel function of the second kind of integer order n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j0": {"fullname": "pyerrors.special.j0", "modulename": "pyerrors.special", "qualname": "j0", "kind": "function", "doc": "j0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nj0(x, out=None)
\n\nBessel function of the first kind of order 0.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float).
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 0 at
\nx
.See Also
\n\n\n\n
jv
: Bessel function of real order and complex argument.
\nspherical_jn
: spherical Bessel functions.Notes
\n\nThe domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval the following rational approximation is used:
\n\n$$J_0(x) \\approx (w - r_1^2)(w - r_2^2) \\frac{P_3(w)}{Q_8(w)},$$
\n\nwhere \\( w = x^2 \\) and \\( r_1 \\), \\( r_2 \\) are the zeros of\n\\( J_0 \\), and \\( P_3 \\) and \\( Q_8 \\) are polynomials of degrees 3\nand 8, respectively.
\n\nIn the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\nj0
.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn
).References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import j0\n>>> j0(1.)\n0.7651976865579665\n
Calculate the function at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> j0(np.array([-2., 0., 4.]))\narray([ 0.22389078, 1. , -0.39714981])\n
Plot the function from -20 to 20.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y0": {"fullname": "pyerrors.special.y0", "modulename": "pyerrors.special", "qualname": "y0", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\ny0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ny0(x, out=None)
\n\nBessel function of the second kind of order 0.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float).
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 0 at
\nx
.See Also
\n\n\n\n
j0
: Bessel function of the first kind of order 0
\nyv
: Bessel function of the first kindNotes
\n\nThe domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval a rational approximation \\( R(x) \\) is employed to\ncompute,
\n\n$$Y_0(x) = R(x) + \\frac{2 \\log(x) J_0(x)}{\\pi},$$
\n\nwhere \\( J_0 \\) is the Bessel function of the first kind of order 0.
\n\nIn the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\ny0
.References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import y0\n>>> y0(1.)\n0.08825696421567697\n
Calculate at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> y0(np.array([0.5, 2., 3.]))\narray([-0.44451873, 0.51037567, 0.37685001])\n
Plot the function from 0 to 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j1": {"fullname": "pyerrors.special.j1", "modulename": "pyerrors.special", "qualname": "j1", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\nj1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nj1(x, out=None)
\n\nBessel function of the first kind of order 1.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float).
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 1 at
\nx
.See Also
\n\n\n\n
jv
: Bessel function of the first kind
\nspherical_jn
: spherical Bessel functions.Notes
\n\nThe domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 24 term Chebyshev expansion is used. In the second, the\nasymptotic trigonometric representation is employed using two rational\nfunctions of degree 5/5.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\nj1
.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn
).References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import j1\n>>> j1(1.)\n0.44005058574493355\n
Calculate the function at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> j1(np.array([-2., 0., 4.]))\narray([-0.57672481, 0. , -0.06604333])\n
Plot the function from -20 to 20.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y1": {"fullname": "pyerrors.special.y1", "modulename": "pyerrors.special", "qualname": "y1", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\ny1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ny1(x, out=None)
\n\nBessel function of the second kind of order 1.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float).
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 1 at
\nx
.See Also
\n\n\n\n
j1
: Bessel function of the first kind of order 1
\nyn
: Bessel function of the second kind
\nyv
: Bessel function of the second kindNotes
\n\nThe domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 25 term Chebyshev expansion is used, and computing\n\\( J_1 \\) (the Bessel function of the first kind) is required. In the\nsecond, the asymptotic trigonometric representation is employed using two\nrational functions of degree 5/5.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\ny1
.References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import y1\n>>> y1(1.)\n-0.7812128213002888\n
Calculate at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> y1(np.array([0.5, 2., 3.]))\narray([-1.47147239, -0.10703243, 0.32467442])\n
Plot the function from 0 to 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.jn": {"fullname": "pyerrors.special.jn", "modulename": "pyerrors.special", "qualname": "jn", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\njv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\njv(v, z, out=None)
\n\nBessel function of the first kind of real order and complex argument.
\n\nParameters
\n\n\n
\n\n- v (array_like):\nOrder (float).
\n- z (array_like):\nArgument (float or complex).
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- J (scalar or ndarray):\nValue of the Bessel function, \\( J_v(z) \\).
\nSee Also
\n\n\n\n
jve
: \\( J_v \\) with leading exponential behavior stripped off.
\nspherical_jn
: spherical Bessel functions.
\nj0
: faster version of this function for order 0.
\nj1
: faster version of this function for order 1.Notes
\n\nFor positive
\n\nv
values, the computation is carried out using the AMOS\n1zbesj
routine, which exploits the connection to the modified\nBessel function \\( I_v \\),$$J_v(z) = \\exp(v\\pi\\imath/2) I_v(-\\imath z)\\qquad (\\Im z > 0)
\n\nJ_v(z) = \\exp(-v\\pi\\imath/2) I_v(\\imath z)\\qquad (\\Im z < 0)$$
\n\nFor negative
\n\nv
values the formula,$$J_{-v}(z) = J_v(z) \\cos(\\pi v) - Y_v(z) \\sin(\\pi v)$$
\n\nis used, where \\( Y_v(z) \\) is the Bessel function of the second\nkind, computed using the AMOS routine
\n\nzbesy
. Note that the second\nterm is exactly zero for integerv
; to improve accuracy the second\nterm is explicitly omitted forv
values such thatv = floor(v)
.Not to be confused with the spherical Bessel functions (see
\n\nspherical_jn
).References
\n\nExamples
\n\nEvaluate the function of order 0 at one point.
\n\n\n\n\n\n>>> from scipy.special import jv\n>>> jv(0, 1.)\n0.7651976865579666\n
Evaluate the function at one point for different orders.
\n\n\n\n\n\n>>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)\n(0.7651976865579666, 0.44005058574493355, 0.24029783912342725)\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the
\n\nv
parameter:\n\n\n\n>>> jv([0, 1, 1.5], 1.)\narray([0.76519769, 0.44005059, 0.24029784])\n
Evaluate the function at several points for order 0 by providing an\narray for
\n\nz
.\n\n\n\n>>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> jv(0, points)\narray([ 0.22389078, 1. , -0.26005195])\n
If
\n\nz
is an array, the order parameterv
must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:\n\n\n\n>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n\n\n\n>>> jv(orders, points)\narray([[ 0.22389078, 1. , -0.26005195],\n [-0.57672481, 0. , 0.33905896]])\n
Plot the functions of order 0 to 3 from -10 to 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> for i in range(4):\n... ax.plot(x, jv(i, x), label=f'$J_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.yn": {"fullname": "pyerrors.special.yn", "modulename": "pyerrors.special", "qualname": "yn", "kind": "function", "doc": "
\n\n
\n- \n
\nDonald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
\nyn(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nyn(n, x, out=None)
\n\nBessel function of the second kind of integer order and real argument.
\n\nParameters
\n\n\n
\n\n- n (array_like):\nOrder (integer).
\n- x (array_like):\nArgument (float).
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- Y (scalar or ndarray):\nValue of the Bessel function, \\( Y_n(x) \\).
\nSee Also
\n\n\n\n
yv
: For real order and real or complex argument.
\ny0
: faster implementation of this function for order 0
\ny1
: faster implementation of this function for order 1Notes
\n\nWrapper for the Cephes 1 routine
\n\nyn
.The function is evaluated by forward recurrence on
\n\nn
, starting with\nvalues computed by the Cephes routinesy0
andy1
. Ifn = 0
or 1,\nthe routine fory0
ory1
is called directly.References
\n\nExamples
\n\nEvaluate the function of order 0 at one point.
\n\n\n\n\n\n>>> from scipy.special import yn\n>>> yn(0, 1.)\n0.08825696421567697\n
Evaluate the function at one point for different orders.
\n\n\n\n\n\n>>> yn(0, 1.), yn(1, 1.), yn(2, 1.)\n(0.08825696421567697, -0.7812128213002888, -1.6506826068162546)\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the
\n\nv
parameter:\n\n\n\n>>> yn([0, 1, 2], 1.)\narray([ 0.08825696, -0.78121282, -1.65068261])\n
Evaluate the function at several points for order 0 by providing an\narray for
\n\nz
.\n\n\n\n>>> import numpy as np\n>>> points = np.array([0.5, 3., 8.])\n>>> yn(0, points)\narray([-0.44451873, 0.37685001, 0.22352149])\n
If
\n\nz
is an array, the order parameterv
must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:\n\n\n\n>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n\n\n\n>>> yn(orders, points)\narray([[-0.44451873, 0.37685001, 0.22352149],\n [-1.47147239, 0.32467442, -0.15806046]])\n
Plot the functions of order 0 to 3 from 0 to 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> for i in range(4):\n... ax.plot(x, yn(i, x), label=f'$Y_{i!r}$')\n>>> ax.set_ylim(-3, 1)\n>>> ax.legend()\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i0": {"fullname": "pyerrors.special.i0", "modulename": "pyerrors.special", "qualname": "i0", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\ni0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ni0(x, out=None)
\n\nModified Bessel function of order 0.
\n\nDefined as,
\n\n$$I_0(x) = \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{(k!)^2} = J_0(\\imath x),$$
\n\nwhere \\( J_0 \\) is the Bessel function of the first kind of order 0.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float)
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- I (scalar or ndarray):\nValue of the modified Bessel function of order 0 at
\nx
.See Also
\n\n\n\n
iv
: Modified Bessel function of any order
\ni0e
: Exponentially scaled modified Bessel function of order 0Notes
\n\nThe range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\ni0
.References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import i0\n>>> i0(1.)\n1.2660658777520082\n
Calculate at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> i0(np.array([-2., 0., 3.5]))\narray([2.2795853 , 1. , 7.37820343])\n
Plot the function from -10 to 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i1": {"fullname": "pyerrors.special.i1", "modulename": "pyerrors.special", "qualname": "i1", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\ni1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ni1(x, out=None)
\n\nModified Bessel function of order 1.
\n\nDefined as,
\n\n$$I_1(x) = \\frac{1}{2}x \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{k! (k + 1)!}\n = -\\imath J_1(\\imath x),$$
\n\nwhere \\( J_1 \\) is the Bessel function of the first kind of order 1.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float)
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- I (scalar or ndarray):\nValue of the modified Bessel function of order 1 at
\nx
.See Also
\n\n\n\n
iv
: Modified Bessel function of the first kind
\ni1e
: Exponentially scaled modified Bessel function of order 1Notes
\n\nThe range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\ni1
.References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import i1\n>>> i1(1.)\n0.5651591039924851\n
Calculate the function at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> i1(np.array([-2., 0., 6.]))\narray([-1.59063685, 0. , 61.34193678])\n
Plot the function between -10 and 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.iv": {"fullname": "pyerrors.special.iv", "modulename": "pyerrors.special", "qualname": "iv", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\niv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\niv(v, z, out=None)
\n\nModified Bessel function of the first kind of real order.
\n\nParameters
\n\n\n
\n\n- v (array_like):\nOrder. If
\nz
is of real type and negative,v
must be integer\nvalued.- z (array_like of float or complex):\nArgument.
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the modified Bessel function.
\nSee Also
\n\n\n\n
ive
: This function with leading exponential behavior stripped off.
\ni0
: Faster version of this function for order 0.
\ni1
: Faster version of this function for order 1.Notes
\n\nFor real
\n\nz
and \\( v \\in [-50, 50] \\), the evaluation is carried out\nusing Temme's method 1. For larger orders, uniform asymptotic\nexpansions are applied.For complex
\n\nz
and positivev
, the AMOS 2zbesi
routine is\ncalled. It uses a power series for smallz
, the asymptotic expansion\nfor largeabs(z)
, the Miller algorithm normalized by the Wronskian\nand a Neumann series for intermediate magnitudes, and the uniform\nasymptotic expansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large\norders. Backward recurrence is used to generate sequences or reduce\norders when necessary.The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
\n\n$$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
\n\n(valid when the real part of
\n\nz
is positive). For negativev
, the\nformula$$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
\n\nis used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine
\n\nzbesk
.References
\n\nExamples
\n\nEvaluate the function of order 0 at one point.
\n\n\n\n\n\n>>> from scipy.special import iv\n>>> iv(0, 1.)\n1.2660658777520084\n
Evaluate the function at one point for different orders.
\n\n\n\n\n\n>>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.)\n(1.2660658777520084, 0.565159103992485, 0.2935253263474798)\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the
\n\nv
parameter:\n\n\n\n>>> iv([0, 1, 1.5], 1.)\narray([1.26606588, 0.5651591 , 0.29352533])\n
Evaluate the function at several points for order 0 by providing an\narray for
\n\nz
.\n\n\n\n>>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> iv(0, points)\narray([2.2795853 , 1. , 4.88079259])\n
If
\n\nz
is an array, the order parameterv
must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:\n\n\n\n>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n\n\n\n>>> iv(orders, points)\narray([[ 2.2795853 , 1. , 4.88079259],\n [-1.59063685, 0. , 3.95337022]])\n
Plot the functions of order 0 to 3 from -5 to 5.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n... ax.plot(x, iv(i, x), label=f'$I_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.ive": {"fullname": "pyerrors.special.ive", "modulename": "pyerrors.special", "qualname": "ive", "kind": "function", "doc": "
\n\n
\n- \n
\n\nTemme, Journal of Computational Physics, vol 21, 343 (1976) ↩
\n- \n
\nDonald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
\nive(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nive(v, z, out=None)
\n\nExponentially scaled modified Bessel function of the first kind.
\n\nDefined as::
\n\n\n\nive(v, z) = iv(v, z) * exp(-abs(z.real))\n
For imaginary numbers without a real part, returns the unscaled\nBessel function of the first kind
\n\niv
.Parameters
\n\n\n
\n\n- v (array_like of float):\nOrder.
\n- z (array_like of float or complex):\nArgument.
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the exponentially scaled modified Bessel function.
\nSee Also
\n\n\n\n
iv
: Modified Bessel function of the first kind
\ni0e
: Faster implementation of this function for order 0
\ni1e
: Faster implementation of this function for order 1Notes
\n\nFor positive
\n\nv
, the AMOS 1zbesi
routine is called. It uses a\npower series for smallz
, the asymptotic expansion for large\nabs(z)
, the Miller algorithm normalized by the Wronskian and a\nNeumann series for intermediate magnitudes, and the uniform asymptotic\nexpansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large orders.\nBackward recurrence is used to generate sequences or reduce orders when\nnecessary.The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
\n\n$$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
\n\n(valid when the real part of
\n\nz
is positive). For negativev
, the\nformula$$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
\n\nis used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine
\n\nzbesk
.\n\n
ive
is useful for large argumentsz
: for these,iv
easily overflows,\nwhileive
does not due to the exponential scaling.References
\n\nExamples
\n\nIn the following example
\n\niv
returns infinity whereasive
still returns\na finite number.\n\n\n\n>>> from scipy.special import iv, ive\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> iv(3, 1000.), ive(3, 1000.)\n(inf, 0.01256056218254712)\n
Evaluate the function at one point for different orders by\nproviding a list or NumPy array as argument for the
\n\nv
parameter:\n\n\n\n>>> ive([0, 1, 1.5], 1.)\narray([0.46575961, 0.20791042, 0.10798193])\n
Evaluate the function at several points for order 0 by providing an\narray for
\n\nz
.\n\n\n\n>>> points = np.array([-2., 0., 3.])\n>>> ive(0, points)\narray([0.30850832, 1. , 0.24300035])\n
Evaluate the function at several points for different orders by\nproviding arrays for both
\n\nv
forz
. Both arrays have to be\nbroadcastable to the correct shape. To calculate the orders 0, 1\nand 2 for a 1D array of points:\n\n\n\n>>> ive([[0], [1], [2]], points)\narray([[ 0.30850832, 1. , 0.24300035],\n [-0.21526929, 0. , 0.19682671],\n [ 0.09323903, 0. , 0.11178255]])\n
Plot the functions of order 0 to 3 from -5 to 5.
\n\n\n\n\n\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n... ax.plot(x, ive(i, x), label=fr'$I_{i!r}(z)\\cdot e^{{-|z|}}$')\n>>> ax.legend()\n>>> ax.set_xlabel(r"$z$")\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erf": {"fullname": "pyerrors.special.erf", "modulename": "pyerrors.special", "qualname": "erf", "kind": "function", "doc": "
\n\n
\n- \n
\nDonald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
\nerf(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nerf(z, out=None)
\n\nReturns the error function of complex argument.
\n\nIt is defined as
\n\n2/sqrt(pi)*integral(exp(-t**2), t=0..z)
.Parameters
\n\n\n
\n\n- x (ndarray):\nInput array.
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- res (scalar or ndarray):\nThe values of the error function at the given points
\nx
.See Also
\n\n\n\n
erfc,
,erfinv,
,erfcinv,
,wofz,
,erfcx,
,erfi
Notes
\n\nThe cumulative of the unit normal distribution is given by\n
\n\nPhi(z) = 1/2[1 + erf(z/sqrt(2))]
.References
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erf(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erf(x)$')\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfc": {"fullname": "pyerrors.special.erfc", "modulename": "pyerrors.special", "qualname": "erfc", "kind": "function", "doc": "
\n\n
\nerfc(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nerfc(x, out=None)
\n\nComplementary error function,
\n\n1 - erf(x)
.Parameters
\n\n\n
\n\n- x (array_like):\nReal or complex valued argument
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the complementary error function
\nSee Also
\n\n\n\n
erf,
,erfi,
,erfcx,
,dawsn,
,wofz
References
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erfc(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erfc(x)$')\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfinv": {"fullname": "pyerrors.special.erfinv", "modulename": "pyerrors.special", "qualname": "erfinv", "kind": "function", "doc": "
\n\n
\nerfinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nerfinv(y, out=None)
\n\nInverse of the error function.
\n\nComputes the inverse of the error function.
\n\nIn the complex domain, there is no unique complex number w satisfying\nerf(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, -1 < x < 1, there is a unique real\nnumber satisfying erf(erfinv(x)) = x.
\n\nParameters
\n\n\n
\n\n- y (ndarray):\nArgument at which to evaluate. Domain: [-1, 1]
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- erfinv (scalar or ndarray):\nThe inverse of erf of y, element-wise
\nSee Also
\n\n\n\n
erf
: Error function of a complex argument
\nerfc
: Complementary error function,1 - erf(x)
\nerfcinv
: Inverse of the complementary error functionExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfinv, erf\n
\n\n\n\n>>> erfinv(0.5)\n0.4769362762044699\n
\n\n\n\n>>> y = np.linspace(-1.0, 1.0, num=9)\n>>> x = erfinv(y)\n>>> x\narray([ -inf, -0.81341985, -0.47693628, -0.22531206, 0. ,\n 0.22531206, 0.47693628, 0.81341985, inf])\n
Verify that
\n\nerf(erfinv(y))
isy
.\n\n\n\n>>> erf(x)\narray([-1. , -0.75, -0.5 , -0.25, 0. , 0.25, 0.5 , 0.75, 1. ])\n
Plot the function:
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfcinv": {"fullname": "pyerrors.special.erfcinv", "modulename": "pyerrors.special", "qualname": "erfcinv", "kind": "function", "doc": "\n>>> y = np.linspace(-1, 1, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfinv(y)')\n>>> plt.show()\n
erfcinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nerfcinv(y, out=None)
\n\nInverse of the complementary error function.
\n\nComputes the inverse of the complementary error function.
\n\nIn the complex domain, there is no unique complex number w satisfying\nerfc(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, 0 < x < 2, there is a unique real\nnumber satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)).
\n\nIt is related to inverse of the error function by erfcinv(1-x) = erfinv(x)
\n\nParameters
\n\n\n
\n\n- y (ndarray):\nArgument at which to evaluate. Domain: [0, 2]
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- erfcinv (scalar or ndarray):\nThe inverse of erfc of y, element-wise
\nSee Also
\n\n\n\n
erf
: Error function of a complex argument
\nerfc
: Complementary error function,1 - erf(x)
\nerfinv
: Inverse of the error functionExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfcinv\n
\n\n\n\n>>> erfcinv(0.5)\n0.4769362762044699\n
\n\n\n\n>>> y = np.linspace(0.0, 2.0, num=11)\n>>> erfcinv(y)\narray([ inf, 0.9061938 , 0.59511608, 0.37080716, 0.17914345,\n -0. , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,\n -inf])\n
Plot the function:
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logit": {"fullname": "pyerrors.special.logit", "modulename": "pyerrors.special", "qualname": "logit", "kind": "function", "doc": "\n>>> y = np.linspace(0, 2, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfcinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfcinv(y)')\n>>> plt.show()\n
logit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n\"\"\"\nlogit(x, out=None)
\n\nLogit ufunc for ndarrays.
\n\nThe logit function is defined as logit(p) = log(p/(1-p)).\nNote that logit(0) = -inf, logit(1) = inf, and logit(p)\nfor p<0 or p>1 yields nan.
\n\nParameters
\n\n\n
\n\n- x (ndarray):\nThe ndarray to apply logit to element-wise.
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- scalar or ndarray: An ndarray of the same shape as x. Its entries\nare logit of the corresponding entry of x.
\nSee Also
\n\n\n\n
expit
Notes
\n\nAs a ufunc logit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs
\n\nNew in version 0.10.0.
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import logit, expit\n
\n\n\n\n>>> logit([0, 0.25, 0.5, 0.75, 1])\narray([ -inf, -1.09861229, 0. , 1.09861229, inf])\n
\n\n
expit
is the inverse oflogit
:\n\n\n\n>>> expit(logit([0.1, 0.75, 0.999]))\narray([ 0.1 , 0.75 , 0.999])\n
Plot logit(x) for x in [0, 1]:
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.expit": {"fullname": "pyerrors.special.expit", "modulename": "pyerrors.special", "qualname": "expit", "kind": "function", "doc": "\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(0, 1, 501)\n>>> y = logit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.ylim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('logit(x)')\n>>> plt.show()\n
expit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nexpit(x, out=None)
\n\nExpit (a.k.a. logistic sigmoid) ufunc for ndarrays.
\n\nThe expit function, also known as the logistic sigmoid function, is\ndefined as
\n\nexpit(x) = 1/(1+exp(-x))
. It is the inverse of the\nlogit function.Parameters
\n\n\n
\n\n- x (ndarray):\nThe ndarray to apply expit to element-wise.
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: An ndarray of the same shape as x. Its entries\nare
\nexpit
of the corresponding entry of x.See Also
\n\n\n\n
logit
Notes
\n\nAs a ufunc expit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs
\n\nNew in version 0.10.0.
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import expit, logit\n
\n\n\n\n>>> expit([-np.inf, -1.5, 0, 1.5, np.inf])\narray([ 0. , 0.18242552, 0.5 , 0.81757448, 1. ])\n
\n\n
logit
is the inverse ofexpit
:\n\n\n\n>>> logit(expit([-2.5, 0, 3.1, 5.0]))\narray([-2.5, 0. , 3.1, 5. ])\n
Plot expit(x) for x in [-6, 6]:
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logsumexp": {"fullname": "pyerrors.special.logsumexp", "modulename": "pyerrors.special", "qualname": "logsumexp", "kind": "function", "doc": "\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-6, 6, 121)\n>>> y = expit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.xlim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('expit(x)')\n>>> plt.show()\n
Compute the log of the sum of exponentials of input elements.
\n\nParameters
\n\n\n
\n\n- a (array_like):\nInput array.
\n- \n
axis (None or int or tuple of ints, optional):\nAxis or axes over which the sum is taken. By default
\n\naxis
is None,\nand all elements are summed.New in version 0.11.0.
- \n
b (array-like, optional):\nScaling factor for exp(
\n\na
) must be of the same shape asa
or\nbroadcastable toa
. These values may be negative in order to\nimplement subtraction.New in version 0.12.0.
- \n
keepdims (bool, optional):\nIf this is set to True, the axes which are reduced are left in the\nresult as dimensions with size one. With this option, the result\nwill broadcast correctly against the original array.
\n\nNew in version 0.15.0.
- \n
return_sign (bool, optional):\nIf this is set to True, the result will be a pair containing sign\ninformation; if False, results that are negative will be returned\nas NaN. Default is False (no sign information).
\n\nNew in version 0.16.0.
Returns
\n\n\n
\n\n- res (ndarray):\nThe result,
\nnp.log(np.sum(np.exp(a)))
calculated in a numerically\nmore stable way. Ifb
is given thennp.log(np.sum(b*np.exp(a)))
\nis returned. Ifreturn_sign
is True,res
contains the log of\nthe absolute value of the argument.- sgn (ndarray):\nIf
\nreturn_sign
is True, this will be an array of floating-point\nnumbers matching res containing +1, 0, -1 (for real-valued inputs)\nor a complex phase (for complex inputs). This gives the sign of the\nargument of the logarithm inres
.\nIfreturn_sign
is False, only one result is returned.See Also
\n\n\n\n
numpy.logaddexp,
,numpy.logaddexp2
Notes
\n\nNumPy has a logaddexp function which is very similar to
\n\nlogsumexp
, but\nonly handles two arguments.logaddexp.reduce
is similar to this\nfunction, but may be less stable.Examples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import logsumexp\n>>> a = np.arange(10)\n>>> logsumexp(a)\n9.4586297444267107\n>>> np.log(np.sum(np.exp(a)))\n9.4586297444267107\n
With weights
\n\n\n\n\n\n>>> a = np.arange(10)\n>>> b = np.arange(10, 0, -1)\n>>> logsumexp(a, b=b)\n9.9170178533034665\n>>> np.log(np.sum(b*np.exp(a)))\n9.9170178533034647\n
Returning a sign flag
\n\n\n\n\n\n>>> logsumexp([1,2],b=[1,-1],return_sign=True)\n(1.5413248546129181, -1.0)\n
Notice that
\n\nlogsumexp
does not directly support masked arrays. To use it\non a masked array, convert the mask into zero weights:\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.version": {"fullname": "pyerrors.version", "modulename": "pyerrors.version", "kind": "module", "doc": "\n"}}, "docInfo": {"pyerrors": {"qualname": 0, "fullname": 1, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 8336}, "pyerrors.correlators": {"qualname": 0, "fullname": 2, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 3}, "pyerrors.correlators.Corr": {"qualname": 1, "fullname": 3, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 367}, "pyerrors.correlators.Corr.__init__": {"qualname": 3, "fullname": 5, "annotation": 0, "default_value": 0, "signature": 40, "bases": 0, "doc": 100}, "pyerrors.correlators.Corr.tag": {"qualname": 2, "fullname": 4, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 3}, "pyerrors.correlators.Corr.content": {"qualname": 2, 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"default_value", "signature", "bases", "doc"], "ref": "fullname", "documentStore": {"docs": {"pyerrors": {"fullname": "pyerrors", "modulename": "pyerrors", "kind": "module", "doc": "\n>>> a = np.ma.array([np.log(2), 2, np.log(3)],\n... mask=[False, True, False])\n>>> b = (~a.mask).astype(int)\n>>> logsumexp(a.data, b=b), np.log(5)\n1.6094379124341005, 1.6094379124341005\n
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. Comput.Phys.Commun. 288 (2023) 108750.
\n- Ulli Wolff, Monte Carlo errors with less errors. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum).
\n- Alberto Ramos, Automatic differentiation for error analysis of Monte Carlo data. Comput.Phys.Commun. 238 (2019) 19-35.
\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\nInstallation
\n\nInstall the most recent release using pip and pypi:
\n\n\n\n\n\npython -m pip install pyerrors # Fresh install\npython -m pip install -U pyerrors # Update\n
Install the most recent release using conda and conda-forge:
\n\n\n\n\n\nconda install -c conda-forge pyerrors # Fresh install\nconda update -c conda-forge pyerrors # Update\n
Install the current
\n\ndevelop
version:\n\n\n\npython -m pip install -U --no-deps --force-reinstall git+https://github.com/fjosw/pyerrors.git@develop\n
(Also works for any feature branch).
\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
The
\n\ngamma_method
is not automatically called after every intermediate step in order to prevent computational overhead.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
.\nDirect visualizations of the performed fits can be triggered viaresplot=True
orqqplot=True
.For all available options including combined fits to multiple datasets see
\n\npyerrors.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 as 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\n\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 matrix at every timeslice.\nOther dependency (eg. spatial) are not supported.
\n\nThe Corr class can also deal with missing measurements or paddings for fixed boundary conditions.\nThe missing entries are represented via the
\n\nNone
object.Initialization
\n\nA simple correlator can be initialized with a list or a one-dimensional array of
\n\nObs
orCobs
\n\n\n\ncorr11 = pe.Corr([obs1, obs2])\ncorr11 = pe.Corr(np.array([obs1, obs2]))\n
A matrix-valued correlator can either be initialized via a two-dimensional array of
\n\nCorr
objects\n\n\n\nmatrix_corr = pe.Corr(np.array([[corr11, corr12], [corr21, corr22]]))\n
or alternatively via a three-dimensional array of
\n"}, "pyerrors.correlators.Corr.__init__": {"fullname": "pyerrors.correlators.Corr.__init__", "modulename": "pyerrors.correlators", "qualname": "Corr.__init__", "kind": "function", "doc": "Obs
orCObs
of shape (T, N, N) where T is\nthe temporal extent of the correlator and N is the dimension of the matrix.Initialize a Corr object.
\n\nParameters
\n\n\n
\n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "pyerrors.correlators.Corr.tag": {"fullname": "pyerrors.correlators.Corr.tag", "modulename": "pyerrors.correlators", "qualname": "Corr.tag", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.content": {"fullname": "pyerrors.correlators.Corr.content", "modulename": "pyerrors.correlators", "qualname": "Corr.content", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.T": {"fullname": "pyerrors.correlators.Corr.T", "modulename": "pyerrors.correlators", "qualname": "Corr.T", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.prange": {"fullname": "pyerrors.correlators.Corr.prange", "modulename": "pyerrors.correlators", "qualname": "Corr.prange", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.reweighted": {"fullname": "pyerrors.correlators.Corr.reweighted", "modulename": "pyerrors.correlators", "qualname": "Corr.reweighted", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.gamma_method": {"fullname": "pyerrors.correlators.Corr.gamma_method", "modulename": "pyerrors.correlators", "qualname": "Corr.gamma_method", "kind": "function", "doc": "- data_input (list or array):\nlist of Obs or list of arrays of Obs or array of Corrs (see class docstring for details).
\n- padding (list, optional):\nList with two entries where the first labels the padding\nat the front of the correlator and the second the padding\nat the back.
\n- prange (list, optional):\nList containing the first and last timeslice of the plateau\nregion identified for this correlator.
\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.trace": {"fullname": "pyerrors.correlators.Corr.trace", "modulename": "pyerrors.correlators", "qualname": "Corr.trace", "kind": "function", "doc": "Calculates the per-timeslice trace of a correlator matrix.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.matrix_symmetric", "kind": "function", "doc": "Symmetrizes the correlator matrices on every timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.GEVP": {"fullname": "pyerrors.correlators.Corr.GEVP", "modulename": "pyerrors.correlators", "qualname": "Corr.GEVP", "kind": "function", "doc": "Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.
\n\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. (default)
\n- \"Eigenvector\": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.\nThe reference state is identified by its eigenvalue at $t=t_s$.
\n- None: The GEVP is solved only at ts, no sorting is necessary
\n- vector_obs (bool):\nIf True, uncertainties are propagated in the eigenvector computation (default False).
\nOther Parameters
\n\n\n
\n", "signature": "(self, t0, ts=None, sort='Eigenvalue', vector_obs=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Eigenvalue": {"fullname": "pyerrors.correlators.Corr.Eigenvalue", "modulename": "pyerrors.correlators", "qualname": "Corr.Eigenvalue", "kind": "function", "doc": "- state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
\n- method (str):\nMethod used to solve the GEVP.\n
\n\n
- \"eigh\": Use scipy.linalg.eigh to solve the GEVP. (default for vector_obs=False)
\n- \"cholesky\": Use manually implemented solution via the Cholesky decomposition. Automatically chosen if vector_obs==True.
\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', **kwargs):", "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- parity (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:\n - symmetric (default)\n $$\\tilde{\\partial}^2_0 f(x_0) = f(x_0+1)-2f(x_0)+f(x_0-1)$$\n - big_symmetric\n $$\\partial^2_0 f(x_0) = \\frac{f(x_0+2)-2f(x_0)+f(x_0-2)}{4}$$\n - improved\n $$\\partial^2_0 f(x_0) = \\frac{-f(x_0+2) + 16 * f(x_0+1) - 30 * f(x_0) + 16 * f(x_0-1) - f(x_0-2)}{12}$$\n - log\n $$f(x) = \\tilde{\\partial}^2_0 log(f(x_0))+(\\tilde{\\partial}_0 log(f(x_0)))^2$$
\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 periodicity of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.\nsinh : Use anti-periodicity of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.\nSee, e.g., arXiv:1205.5380\narccosh : Uses the explicit form of the symmetrized correlator (not recommended)\nlogsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
\n- guess (float):\nguess for the root finder, only relevant for the root variant
\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,\tfit_key=None,\tylabel=None,\tsave=None,\tauto_gamma=False,\thide_sigma=None,\treferences=None,\ttitle=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.spaghetti_plot": {"fullname": "pyerrors.correlators.Corr.spaghetti_plot", "modulename": "pyerrors.correlators", "qualname": "Corr.spaghetti_plot", "kind": "function", "doc": "- x_range (list):\nlist of two values, determining the range of the x-axis e.g. [4, 8].
\n- comp (Corr or list of Corr):\nCorrelator or list of correlators which are plotted for comparison.\nThe tags of these correlators are used as labels if available.
\n- logscale (bool):\nSets y-axis to logscale.
\n- plateau (Obs):\nPlateau value to be visualized in the figure.
\n- fit_res (Fit_result):\nFit_result object to be visualized.
\n- fit_key (str):\nKey for the fit function in Fit_result.fit_function (for combined fits).
\n- ylabel (str):\nLabel for the y-axis.
\n- save (str):\npath to file in which the figure should be saved.
\n- auto_gamma (bool):\nApply the gamma method with standard parameters to all correlators and plateau values before plotting.
\n- hide_sigma (float):\nHides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
\n- references (list):\nList of floating point values that are displayed as horizontal lines for reference.
\n- title (string):\nOptional title of the figure.
\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.real": {"fullname": "pyerrors.correlators.Corr.real", "modulename": "pyerrors.correlators", "qualname": "Corr.real", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.imag": {"fullname": "pyerrors.correlators.Corr.imag", "modulename": "pyerrors.correlators", "qualname": "Corr.imag", "kind": "variable", "doc": "\n"}, "pyerrors.correlators.Corr.prune": {"fullname": "pyerrors.correlators.Corr.prune", "modulename": "pyerrors.correlators", "qualname": "Corr.prune", "kind": "function", "doc": "- 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.correlators.Corr.N": {"fullname": "pyerrors.correlators.Corr.N", "modulename": "pyerrors.correlators", "qualname": "Corr.N", "kind": "variable", "doc": "\n"}, "pyerrors.covobs": {"fullname": "pyerrors.covobs", "modulename": "pyerrors.covobs", "kind": "module", "doc": "\n"}, "pyerrors.covobs.Covobs": {"fullname": "pyerrors.covobs.Covobs", "modulename": "pyerrors.covobs", "qualname": "Covobs", "kind": "class", "doc": "\n"}, "pyerrors.covobs.Covobs.__init__": {"fullname": "pyerrors.covobs.Covobs.__init__", "modulename": "pyerrors.covobs", "qualname": "Covobs.__init__", "kind": "function", "doc": "Initialize Covobs object.
\n\nParameters
\n\n\n
\n", "signature": "(mean, cov, name, pos=None, grad=None)"}, "pyerrors.covobs.Covobs.name": {"fullname": "pyerrors.covobs.Covobs.name", "modulename": "pyerrors.covobs", "qualname": "Covobs.name", "kind": "variable", "doc": "\n"}, "pyerrors.covobs.Covobs.value": {"fullname": "pyerrors.covobs.Covobs.value", "modulename": "pyerrors.covobs", "qualname": "Covobs.value", "kind": "variable", "doc": "\n"}, "pyerrors.covobs.Covobs.errsq": {"fullname": "pyerrors.covobs.Covobs.errsq", "modulename": "pyerrors.covobs", "qualname": "Covobs.errsq", "kind": "function", "doc": "- 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.covobs.Covobs.cov": {"fullname": "pyerrors.covobs.Covobs.cov", "modulename": "pyerrors.covobs", "qualname": "Covobs.cov", "kind": "variable", "doc": "\n"}, "pyerrors.covobs.Covobs.grad": {"fullname": "pyerrors.covobs.Covobs.grad", "modulename": "pyerrors.covobs", "qualname": "Covobs.grad", "kind": "variable", "doc": "\n"}, "pyerrors.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "\n"}, "pyerrors.dirac.gammaX": {"fullname": "pyerrors.dirac.gammaX", "modulename": "pyerrors.dirac", "qualname": "gammaX", "kind": "variable", "doc": "\n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+0.j, 0.+1.j],\n [ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, -0.-1.j, 0.+0.j, 0.+0.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaY": {"fullname": "pyerrors.dirac.gammaY", "modulename": "pyerrors.dirac", "qualname": "gammaY", "kind": "variable", "doc": "\n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j],\n [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [-1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaZ": {"fullname": "pyerrors.dirac.gammaZ", "modulename": "pyerrors.dirac", "qualname": "gammaZ", "kind": "variable", "doc": "\n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -0.-1.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+1.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaT": {"fullname": "pyerrors.dirac.gammaT", "modulename": "pyerrors.dirac", "qualname": "gammaT", "kind": "variable", "doc": "\n", "default_value": "array([[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n [1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gamma": {"fullname": "pyerrors.dirac.gamma", "modulename": "pyerrors.dirac", "qualname": "gamma", "kind": "variable", "doc": "\n", "default_value": "array([[[ 0.+0.j, 0.+0.j, 0.+0.j, 0.+1.j],\n [ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, -0.-1.j, 0.+0.j, 0.+0.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j],\n [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [-1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -0.-1.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+1.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n [ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]]])"}, "pyerrors.dirac.gamma5": {"fullname": "pyerrors.dirac.gamma5", "modulename": "pyerrors.dirac", "qualname": "gamma5", "kind": "variable", "doc": "\n", "default_value": "array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, -1.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j]])"}, "pyerrors.dirac.identity": {"fullname": "pyerrors.dirac.identity", "modulename": "pyerrors.dirac", "qualname": "identity", "kind": "variable", "doc": "\n", "default_value": "array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]])"}, "pyerrors.dirac.epsilon_tensor": {"fullname": "pyerrors.dirac.epsilon_tensor", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor", "kind": "function", "doc": "Rank-3 epsilon tensor
\n\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.fit_parameters": {"fullname": "pyerrors.fits.Fit_result.fit_parameters", "modulename": "pyerrors.fits", "qualname": "Fit_result.fit_parameters", "kind": "variable", "doc": "\n"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "- 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 (dict or list, optional):\npriors can either be a dictionary with integer keys and the corresponding priors as values or\na list with an entry for every parameter in the fit. The entries can either be\nObs (e.g. results from a previous fit) or strings containing a value and an error formatted like\n0.548(23), 500(40) or 0.5(0.4)
\n- silent (bool, optional):\nIf true all output to the console is omitted (default False).
\n- initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for\nnon-linear fits with many parameters. In case of correlated fits the guess is used to perform\nan uncorrelated fit which then serves as guess for the correlated fit.
\n- method (str, optional):\ncan be used to choose an alternative method for the minimization of chisquare.\nThe possible methods are the ones which can be used for scipy.optimize.minimize and\nmigrad of iminuit. If no method is specified, Levenberg-Marquard is used.\nReliable alternatives are migrad, Powell and Nelder-Mead.
\n- tol (float, optional):\ncan be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence\nto a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly\ninvalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values\nThe stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
\n- correlated_fit (bool):\nIf True, use the full inverse covariance matrix in the definition of the chisquare cost function.\nFor details about how the covariance matrix is estimated see
\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).- inv_chol_cov_matrix [array,list], optional: array: shape = (no of y values) X (no of y values)\nlist: for an uncombined fit: [\"\"]\nfor a combined fit: list of keys belonging to the corr_matrix saved in the array, must be the same as the keys of the y dict in alphabetical order\nIf correlated_fit=True is set as well, can provide an inverse covariance matrix (y errors, dy_f included!) of your own choosing for a correlated fit.\nThe matrix must be a lower triangular matrix constructed from a Cholesky decomposition: The function invert_corr_cov_cholesky(corr, inverrdiag) can be\nused to construct it from a correlation matrix (corr) and the errors dy_f of the data points (inverrdiag = np.diag(1 / np.asarray(dy_f))). For the correct\nordering the correlation matrix (corr) can be sorted via the function sort_corr(corr, kl, yd) where kl is the list of keys and yd the y dict.
\n- expected_chisquare (bool):\nIf True estimates the expected chisquare which is\ncorrected by effects caused by correlated input data (default False).
\n- resplot (bool):\nIf True, a plot which displays fit, data and residuals is generated (default False).
\n- qqplot (bool):\nIf True, a quantile-quantile plot of the fit result is generated (default False).
\n- num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
\nReturns
\n\n\n
\n\n- output (Fit_result):\nParameters and information on the fitted result.
\nExamples
\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": "\n>>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set\n>>> import numpy as np\n>>> from scipy.stats import norm\n>>> from scipy.linalg import cholesky\n>>> import pyerrors as pe\n>>> # generating the random data set\n>>> num_samples = 400\n>>> N = 3\n>>> x = np.arange(N)\n>>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers\n>>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers\n>>> r = r1 = r2 = np.zeros((N, N))\n>>> y = {}\n>>> for i in range(N):\n>>> for j in range(N):\n>>> r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix\n>>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors\n>>> for i in range(N):\n>>> for j in range(N):\n>>> r[i, j] *= errl[i] * errl[j] # element in covariance matrix\n>>> c = cholesky(r, lower=True)\n>>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined\n>>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built\n>>> x_dict = {}\n>>> y_dict = {}\n>>> chol_inv_dict = {}\n>>> data = []\n>>> for key in y.keys():\n>>> x_dict[key] = x\n>>> for i in range(N):\n>>> data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data\n>>> [o.gamma_method() for o in data]\n>>> corr = pe.covariance(data, correlation=True)\n>>> inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data]))\n>>> chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below\n>>> y_dict = {'a': data[:3], 'b': data[3:]}\n>>> # common fit parameter p[0] in combined fit\n>>> def fit1(p, x):\n>>> return p[0] + p[1] * x\n>>> def fit2(p, x):\n>>> return p[0] + p[2] * x\n>>> fitf_dict = {'a': fit1, 'b':fit2}\n>>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']])\nFit with 3 parameters\nMethod: Levenberg-Marquardt\n`ftol` termination condition is satisfied.\nchisquare/d.o.f.: 0.5388013574561786 # random\nfit parameters [1.11897846 0.96361162 0.92325319] # random\n
Performs 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, title=''):", "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\nFor uncorrelated data the residuals are expected to be distributed ~N(0,1).
\n\nReturns
\n\n\n
\n", "signature": "(x, y, func, fit_res, title=''):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "- 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- 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, 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- 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": "(fname, full_output=False, gz=True, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.create_dobs_string": {"fullname": "pyerrors.input.dobs.create_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_dobs_string", "kind": "function", "doc": "- 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_hd5": {"fullname": "pyerrors.input.hadrons.read_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_hd5", "kind": "function", "doc": "- None
\nRead hadrons hdf5 file and extract entry based on attributes.
\n\nParameters
\n\n\n
\n\n- filestem (str):\nFull namestem of the files to read, including the full path.
\n- ens_id (str):\nname of the ensemble, required for internal bookkeeping
\n- group (str):\nlabel of the group to be extracted.
\n- \n
attrs (dict or int):\nDictionary containing the attributes. For example
\n\n\n\n\n\nattrs = {"gamma_snk": "Gamma5",\n "gamma_src": "Gamma5"}\n
Alternatively an integer can be specified to identify the sub group.\nThis is discouraged as the order in the file is not guaranteed.
- idl (range):\nIf specified only configurations in the given range are read in.
\n- part (str):\nstring specifying whether to extract the real part ('real'),\nthe imaginary part ('imag') or a complex correlator ('complex').\nDefault 'real'.
\nReturns
\n\n\n
\n", "signature": "(filestem, ens_id, group, attrs=None, idl=None, part='real'):", "funcdef": "def"}, "pyerrors.input.hadrons.read_meson_hd5": {"fullname": "pyerrors.input.hadrons.read_meson_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_meson_hd5", "kind": "function", "doc": "- corr (Corr):\nCorrelator of the source sink combination in question.
\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 sink and source (gamma_snk, gamma_src).\n(\"Gamma5\", \"Gamma5\") corresponds to the pseudoscalar pseudoscalar\ntwo-point function. The gammas argument dominateds over meson.
\n- idl (range):\nIf specified only configurations in the given range are read in.
\nReturns
\n\n\n
\n", "signature": "(path, filestem, ens_id, meson='meson_0', idl=None, gammas=None):", "funcdef": "def"}, "pyerrors.input.hadrons.extract_t0_hd5": {"fullname": "pyerrors.input.hadrons.extract_t0_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "extract_t0_hd5", "kind": "function", "doc": "- corr (Corr):\nCorrelator of the source sink combination in question.
\nRead hadrons FlowObservables hdf5 file and extract t0
\n\nParameters
\n\n\n
\n", "signature": "(\tpath,\tfilestem,\tens_id,\tobs='Clover energy density',\tfit_range=5,\tidl=None,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.hadrons.read_DistillationContraction_hd5": {"fullname": "pyerrors.input.hadrons.read_DistillationContraction_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_DistillationContraction_hd5", "kind": "function", "doc": "- path (str):\npath to the files to read
\n- filestem (str):\nnamestem of the files to read
\n- ens_id (str):\nname of the ensemble, required for internal bookkeeping
\n- obs (str):\nlabel of the observable from which t0 should be extracted.\nOptions: 'Clover energy density' and 'Plaquette energy density'
\n- fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
\n- idl (range):\nIf specified only configurations in the given range are read in.
\n- plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
\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>>> import numpy as np\n>>> np.ndarray(shape=(2,2), dtype=float, order='F')\narray([[0.0e+000, 0.0e+000], # random\n [ nan, 2.5e-323]])\n
Second mode:
\n\n\n\n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "\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.\nDict keys that are not JSON-serializable such as floats are converted to strings.
\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.fit_t0": {"fullname": "pyerrors.input.misc.fit_t0", "modulename": "pyerrors.input.misc", "qualname": "fit_t0", "kind": "function", "doc": "- data (Obs / list / Corr):\nRead data
\n- or
\n- data (dict):\nRead data and meta-data
\nCompute the root of (flow-based) data based on a dictionary that contains\nthe necessary information in key-value pairs a la (flow time: observable at flow time).
\n\nIt is assumed that the data is monotonically increasing and passes zero from below.\nNo exception is thrown if this is not the case (several roots, no monotonic increase).\nAn exception is thrown if no root can be found in the data.
\n\nA linear fit in the vicinity of the root is performed to exctract the root from the\ntwo fit parameters.
\n\nParameters
\n\n\n
\n\n- t2E_dict (dict):\nDictionary with pairs of (flow time: observable at flow time) where the flow times\nare of type float and the observables of type Obs.
\n- fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit.
\n- plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data. (Default: False)
\n- observable (str):\nKeyword to identify the observable to print the correct ylabel (if plot_fit is True)\nfor the observables 't0' and 'w0'. No y label is printed otherwise. (Default: 't0')
\nReturns
\n\n\n
\n", "signature": "(t2E_dict, fit_range, plot_fit=False, observable='t0'):", "funcdef": "def"}, "pyerrors.input.misc.read_pbp": {"fullname": "pyerrors.input.misc.read_pbp", "modulename": "pyerrors.input.misc", "qualname": "read_pbp", "kind": "function", "doc": "- root (Obs):\nThe root of the data series.
\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/a^2 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- c (where c=0.3 by default)\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- postfix (str):\nPostfix of measurement file (Default: ms)
\n- c (float):\nConstant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1.
\n- r_start (list):\nlist which contains the first config to be read for each replicum.
\n- r_stop (list):\nlist which contains the last config to be read for each replicum.
\n- r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
\n- plaquette (bool):\nIf true extract the plaquette estimate of t0 instead.
\n- names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
\n- files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
\n- plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
\n- assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
\nReturns
\n\n\n
\n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_w0": {"fullname": "pyerrors.input.openQCD.extract_w0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_w0", "kind": "function", "doc": "- t0 (Obs):\nExtracted t0
\nExtract w0/a from given .ms.dat files. Returns w0 as Obs.
\n\nIt is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t d(t^2
\n\n)/dt - (where c=0.3 by default)\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- postfix (str):\nPostfix of measurement file (Default: ms)
\n- c (float):\nConstant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1.
\n- r_start (list):\nlist which contains the first config to be read for each replicum.
\n- r_stop (list):\nlist which contains the last config to be read for each replicum.
\n- r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
\n- plaquette (bool):\nIf true extract the plaquette estimate of w0 instead.
\n- names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
\n- files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
\n- plot_fit (bool):\nIf true, the fit for the extraction of w0 is shown together with the data.
\n- assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
\nReturns
\n\n\n
\n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop": {"fullname": "pyerrors.input.openQCD.read_qtop", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop", "kind": "function", "doc": "- w0 (Obs):\nExtracted w0
\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.
\n- files (List[str]): A list of files to read data from.
\n- idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given.
\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.sep": {"fullname": "pyerrors.input.sfcf.sep", "modulename": "pyerrors.input.sfcf", "qualname": "sep", "kind": "variable", "doc": "\n", "default_value": "'/'"}, "pyerrors.input.sfcf.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "- data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
\nRead sfcf files 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',\tsilent=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.sfcf.read_sfcf_multi": {"fullname": "pyerrors.input.sfcf.read_sfcf_multi", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf_multi", "kind": "function", "doc": "- result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
\nRead sfcf files 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_list (list[str]):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
\n- corr_type_list (list[str]):\nType of correlation function to read. Can be\n
\n\n
- 'bi' for boundary-inner
\n- 'bb' for boundary-boundary
\n- 'bib' for boundary-inner-boundary
\n- noffset_list (list[int]):\nOffset of the source (only relevant when wavefunctions are used)
\n- wf_list (int):\nID of wave function
\n- wf2_list (list[int]):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
\n- im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
\n- names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
\n- ens_name (str):\nreplaces the name of the ensemble
\n- version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
\n- cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
\n- replica (list):\nlist of replica to be read, default is all
\n- files (list[list[int]]):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
\n- check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
\nReturns
\n\n\n
\n", "signature": "(\tpath,\tprefix,\tname_list,\tquarks_list=['.*'],\tcorr_type_list=['bi'],\tnoffset_list=[0],\twf_list=[0],\twf2_list=[0],\tversion='1.0c',\tcfg_separator='n',\tsilent=False,\tkeyed_out=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.utils": {"fullname": "pyerrors.input.utils", "modulename": "pyerrors.input.utils", "kind": "module", "doc": "- result (dict[list[Obs]]):\ndict with one of the following properties:\nif keyed_out:\n dict[key] = list[Obs]\n where key has the form name/quarks/offset/wf/wf2\nif not keyed_out:\n dict[name][quarks][offset][wf][wf2] = list[Obs]
\nUtilities for the input
\n"}, "pyerrors.input.utils.sort_names": {"fullname": "pyerrors.input.utils.sort_names", "modulename": "pyerrors.input.utils", "qualname": "sort_names", "kind": "function", "doc": "Sorts a list of names of replika with searches for
\n\nr
andid
in the replikum string.\nIf this search fails, a fallback method is used,\nwhere the strings are simply compared and the first diffeing numeral is used for differentiation.Parameters
\n\n\n
\n\n- ll (list):\nlist to sort
\nReturns
\n\n\n
\n", "signature": "(ll):", "funcdef": "def"}, "pyerrors.input.utils.check_idl": {"fullname": "pyerrors.input.utils.check_idl", "modulename": "pyerrors.input.utils", "qualname": "check_idl", "kind": "function", "doc": "- ll (list):\nsorted list
\nChecks 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.input.utils.check_params": {"fullname": "pyerrors.input.utils.check_params", "modulename": "pyerrors.input.utils", "qualname": "check_params", "kind": "function", "doc": "- miss_str (str):\nstring with integers of which idls are missing
\nCheck if, for sfcf, the parameter hashes at the end of the parameter files are in fact the expected one.
\n\nParameters
\n\n\n
\n\n- path (str):\nmeasurement path, same as for sfcf read method
\n- param_hash (str):\nexpected parameter hash
\n- prefix (str):\ndata prefix to find the appropriate replicum folders in path
\n- param_prefix (str):\nprefix of the parameter file. Defaults to 'parameters_'
\nReturns
\n\n\n
\n", "signature": "(path, param_hash, prefix, param_prefix='parameters_'):", "funcdef": "def"}, "pyerrors.integrate": {"fullname": "pyerrors.integrate", "modulename": "pyerrors.integrate", "kind": "module", "doc": "\n"}, "pyerrors.integrate.quad": {"fullname": "pyerrors.integrate.quad", "modulename": "pyerrors.integrate", "qualname": "quad", "kind": "function", "doc": "- nums (dict):\ndictionary of faulty parameter files sorted by the replica paths
\nPerforms a (one-dimensional) numeric integration of f(p, x) from a to b.
\n\nThe integration is performed using scipy.integrate.quad().\nAll parameters that can be passed to scipy.integrate.quad may also be passed to this function.\nThe output is the same as for scipy.integrate.quad, the first element being an Obs.
\n\nParameters
\n\n\n
\n\n- \n
func (object):\nfunction to integrate, has to be of the form
\n\n\n\n\n\nimport autograd.numpy as anp\n\ndef func(p, x):\n return p[0] + p[1] * x + p[2] * anp.sinh(x)\n
where x is the integration variable.
- p (list of floats or Obs):\nparameters of the function func.
\n- a (float or Obs):\nLower limit of integration (use -numpy.inf for -infinity).
\n- b (float or Obs):\nUpper limit of integration (use -numpy.inf for -infinity).
\n- All parameters of scipy.integrate.quad
\nReturns
\n\n\n
\n", "signature": "(func, p, a, b, **kwargs):", "funcdef": "def"}, "pyerrors.linalg": {"fullname": "pyerrors.linalg", "modulename": "pyerrors.linalg", "kind": "module", "doc": "\n"}, "pyerrors.linalg.matmul": {"fullname": "pyerrors.linalg.matmul", "modulename": "pyerrors.linalg", "qualname": "matmul", "kind": "function", "doc": "- y (Obs):\nThe integral of func from
\na
tob
.- abserr (float):\nAn estimate of the absolute error in the result.
\n- infodict (dict):\nA dictionary containing additional information.\nRun scipy.integrate.quad_explain() for more information.
\n- message: A convergence message.
\n- explain: Appended only with 'cos' or 'sin' weighting and infinite\nintegration limits, it contains an explanation of the codes in\ninfodict['ierlst']
\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.eigv": {"fullname": "pyerrors.linalg.eigv", "modulename": "pyerrors.linalg", "qualname": "eigv", "kind": "function", "doc": "Computes the eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.pinv": {"fullname": "pyerrors.linalg.pinv", "modulename": "pyerrors.linalg", "qualname": "pinv", "kind": "function", "doc": "Computes the Moore-Penrose pseudoinverse of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.svd": {"fullname": "pyerrors.linalg.svd", "modulename": "pyerrors.linalg", "qualname": "svd", "kind": "function", "doc": "Computes the singular value decomposition of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.misc": {"fullname": "pyerrors.misc", "modulename": "pyerrors.misc", "kind": "module", "doc": "\n"}, "pyerrors.misc.print_config": {"fullname": "pyerrors.misc.print_config", "modulename": "pyerrors.misc", "qualname": "print_config", "kind": "function", "doc": "Print information about version of python, pyerrors and dependencies.
\n", "signature": "():", "funcdef": "def"}, "pyerrors.misc.errorbar": {"fullname": "pyerrors.misc.errorbar", "modulename": "pyerrors.misc", "qualname": "errorbar", "kind": "function", "doc": "pyerrors wrapper for the errorbars method of matplotlib
\n\nParameters
\n\n\n
\n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.10.14/x64/lib/python3.10/site-packages/matplotlib/pyplot.py'>,\t**kwargs):", "funcdef": "def"}, "pyerrors.misc.dump_object": {"fullname": "pyerrors.misc.dump_object", "modulename": "pyerrors.misc", "qualname": "dump_object", "kind": "function", "doc": "- 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.S_global": {"fullname": "pyerrors.obs.Obs.S_global", "modulename": "pyerrors.obs", "qualname": "Obs.S_global", "kind": "variable", "doc": "\n", "default_value": "2.0"}, "pyerrors.obs.Obs.S_dict": {"fullname": "pyerrors.obs.Obs.S_dict", "modulename": "pyerrors.obs", "qualname": "Obs.S_dict", "kind": "variable", "doc": "\n", "default_value": "{}"}, "pyerrors.obs.Obs.tau_exp_global": {"fullname": "pyerrors.obs.Obs.tau_exp_global", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_global", "kind": "variable", "doc": "\n", "default_value": "0.0"}, "pyerrors.obs.Obs.tau_exp_dict": {"fullname": "pyerrors.obs.Obs.tau_exp_dict", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_dict", "kind": "variable", "doc": "\n", "default_value": "{}"}, "pyerrors.obs.Obs.N_sigma_global": {"fullname": "pyerrors.obs.Obs.N_sigma_global", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_global", "kind": "variable", "doc": "\n", "default_value": "1.0"}, "pyerrors.obs.Obs.N_sigma_dict": {"fullname": "pyerrors.obs.Obs.N_sigma_dict", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_dict", "kind": "variable", "doc": "\n", "default_value": "{}"}, "pyerrors.obs.Obs.names": {"fullname": "pyerrors.obs.Obs.names", "modulename": "pyerrors.obs", "qualname": "Obs.names", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.shape": {"fullname": "pyerrors.obs.Obs.shape", "modulename": "pyerrors.obs", "qualname": "Obs.shape", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.r_values": {"fullname": "pyerrors.obs.Obs.r_values", "modulename": "pyerrors.obs", "qualname": "Obs.r_values", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.deltas": {"fullname": "pyerrors.obs.Obs.deltas", "modulename": "pyerrors.obs", "qualname": "Obs.deltas", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.N": {"fullname": "pyerrors.obs.Obs.N", "modulename": "pyerrors.obs", "qualname": "Obs.N", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.idl": {"fullname": "pyerrors.obs.Obs.idl", "modulename": "pyerrors.obs", "qualname": "Obs.idl", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.ddvalue": {"fullname": "pyerrors.obs.Obs.ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.ddvalue", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.reweighted": {"fullname": "pyerrors.obs.Obs.reweighted", "modulename": "pyerrors.obs", "qualname": "Obs.reweighted", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.tag": {"fullname": "pyerrors.obs.Obs.tag", "modulename": "pyerrors.obs", "qualname": "Obs.tag", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.value": {"fullname": "pyerrors.obs.Obs.value", "modulename": "pyerrors.obs", "qualname": "Obs.value", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.dvalue": {"fullname": "pyerrors.obs.Obs.dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.dvalue", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_names": {"fullname": "pyerrors.obs.Obs.e_names", "modulename": "pyerrors.obs", "qualname": "Obs.e_names", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.cov_names": {"fullname": "pyerrors.obs.Obs.cov_names", "modulename": "pyerrors.obs", "qualname": "Obs.cov_names", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.mc_names": {"fullname": "pyerrors.obs.Obs.mc_names", "modulename": "pyerrors.obs", "qualname": "Obs.mc_names", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_content": {"fullname": "pyerrors.obs.Obs.e_content", "modulename": "pyerrors.obs", "qualname": "Obs.e_content", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.covobs": {"fullname": "pyerrors.obs.Obs.covobs", "modulename": "pyerrors.obs", "qualname": "Obs.covobs", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.gamma_method": {"fullname": "pyerrors.obs.Obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "Obs.gamma_method", "kind": "function", "doc": "- 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.export_bootstrap": {"fullname": "pyerrors.obs.Obs.export_bootstrap", "modulename": "pyerrors.obs", "qualname": "Obs.export_bootstrap", "kind": "function", "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).
\nExport bootstrap samples from the Obs
\n\nParameters
\n\n\n
\n\n- samples (int):\nNumber of bootstrap samples to generate.
\n- random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples.\nIf not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
\n- save_rng (str):\nSave the random numbers to a file if a path is specified.
\nReturns
\n\n\n
\n", "signature": "(self, samples=500, random_numbers=None, save_rng=None):", "funcdef": "def"}, "pyerrors.obs.Obs.sqrt": {"fullname": "pyerrors.obs.Obs.sqrt", "modulename": "pyerrors.obs", "qualname": "Obs.sqrt", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.log": {"fullname": "pyerrors.obs.Obs.log", "modulename": "pyerrors.obs", "qualname": "Obs.log", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.exp": {"fullname": "pyerrors.obs.Obs.exp", "modulename": "pyerrors.obs", "qualname": "Obs.exp", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sin": {"fullname": "pyerrors.obs.Obs.sin", "modulename": "pyerrors.obs", "qualname": "Obs.sin", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cos": {"fullname": "pyerrors.obs.Obs.cos", "modulename": "pyerrors.obs", "qualname": "Obs.cos", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tan": {"fullname": "pyerrors.obs.Obs.tan", "modulename": "pyerrors.obs", "qualname": "Obs.tan", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsin": {"fullname": "pyerrors.obs.Obs.arcsin", "modulename": "pyerrors.obs", "qualname": "Obs.arcsin", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccos": {"fullname": "pyerrors.obs.Obs.arccos", "modulename": "pyerrors.obs", "qualname": "Obs.arccos", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctan": {"fullname": "pyerrors.obs.Obs.arctan", "modulename": "pyerrors.obs", "qualname": "Obs.arctan", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sinh": {"fullname": "pyerrors.obs.Obs.sinh", "modulename": "pyerrors.obs", "qualname": "Obs.sinh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cosh": {"fullname": "pyerrors.obs.Obs.cosh", "modulename": "pyerrors.obs", "qualname": "Obs.cosh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tanh": {"fullname": "pyerrors.obs.Obs.tanh", "modulename": "pyerrors.obs", "qualname": "Obs.tanh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsinh": {"fullname": "pyerrors.obs.Obs.arcsinh", "modulename": "pyerrors.obs", "qualname": "Obs.arcsinh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccosh": {"fullname": "pyerrors.obs.Obs.arccosh", "modulename": "pyerrors.obs", "qualname": "Obs.arccosh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctanh": {"fullname": "pyerrors.obs.Obs.arctanh", "modulename": "pyerrors.obs", "qualname": "Obs.arctanh", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.N_sigma": {"fullname": "pyerrors.obs.Obs.N_sigma", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.S": {"fullname": "pyerrors.obs.Obs.S", "modulename": "pyerrors.obs", "qualname": "Obs.S", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_ddvalue": {"fullname": "pyerrors.obs.Obs.e_ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_ddvalue", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_drho": {"fullname": "pyerrors.obs.Obs.e_drho", "modulename": "pyerrors.obs", "qualname": "Obs.e_drho", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_dtauint": {"fullname": "pyerrors.obs.Obs.e_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_dtauint", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_dvalue": {"fullname": "pyerrors.obs.Obs.e_dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_dvalue", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_n_dtauint": {"fullname": "pyerrors.obs.Obs.e_n_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_dtauint", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_n_tauint": {"fullname": "pyerrors.obs.Obs.e_n_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_tauint", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_rho": {"fullname": "pyerrors.obs.Obs.e_rho", "modulename": "pyerrors.obs", "qualname": "Obs.e_rho", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_tauint": {"fullname": "pyerrors.obs.Obs.e_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_tauint", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.e_windowsize": {"fullname": "pyerrors.obs.Obs.e_windowsize", "modulename": "pyerrors.obs", "qualname": "Obs.e_windowsize", "kind": "variable", "doc": "\n"}, "pyerrors.obs.Obs.tau_exp": {"fullname": "pyerrors.obs.Obs.tau_exp", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp", "kind": "variable", "doc": "\n"}, "pyerrors.obs.CObs": {"fullname": "pyerrors.obs.CObs", "modulename": "pyerrors.obs", "qualname": "CObs", "kind": "class", "doc": "- numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N import_bootstrap samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived bootstrap samples\nshould agree with samples from a full bootstrap analysis up to O(1/N).
\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.tag": {"fullname": "pyerrors.obs.CObs.tag", "modulename": "pyerrors.obs", "qualname": "CObs.tag", "kind": "variable", "doc": "\n"}, "pyerrors.obs.CObs.real": {"fullname": "pyerrors.obs.CObs.real", "modulename": "pyerrors.obs", "qualname": "CObs.real", "kind": "variable", "doc": "\n"}, "pyerrors.obs.CObs.imag": {"fullname": "pyerrors.obs.CObs.imag", "modulename": "pyerrors.obs", "qualname": "CObs.imag", "kind": "variable", "doc": "\n"}, "pyerrors.obs.CObs.gamma_method": {"fullname": "pyerrors.obs.CObs.gamma_method", "modulename": "pyerrors.obs", "qualname": "CObs.gamma_method", "kind": "function", "doc": "Executes the gamma_method for the real and the imaginary part.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.CObs.is_zero": {"fullname": "pyerrors.obs.CObs.is_zero", "modulename": "pyerrors.obs", "qualname": "CObs.is_zero", "kind": "function", "doc": "Checks whether both real and imaginary part are zero within machine precision.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs.conjugate": {"fullname": "pyerrors.obs.CObs.conjugate", "modulename": "pyerrors.obs", "qualname": "CObs.conjugate", "kind": "function", "doc": "\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.gamma_method": {"fullname": "pyerrors.obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "gamma_method", "kind": "function", "doc": "Vectorized version of the gamma_method applicable to lists or arrays of Obs.
\n\nSee docstring of pe.Obs.gamma_method for details.
\n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.gm": {"fullname": "pyerrors.obs.gm", "modulename": "pyerrors.obs", "qualname": "gm", "kind": "function", "doc": "Vectorized version of the gamma_method applicable to lists or arrays of Obs.
\n\nSee docstring of pe.Obs.gamma_method for details.
\n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.derived_observable": {"fullname": "pyerrors.obs.derived_observable", "modulename": "pyerrors.obs", "qualname": "derived_observable", "kind": "function", "doc": "Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.
\n\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.invert_corr_cov_cholesky": {"fullname": "pyerrors.obs.invert_corr_cov_cholesky", "modulename": "pyerrors.obs", "qualname": "invert_corr_cov_cholesky", "kind": "function", "doc": "Constructs a lower triangular matrix
\n\nchol
via the Cholesky decomposition of the correlation matrixcorr
\n and then returns the inverse covariance matrixchol_inv
as a lower triangular matrix by solvingchol * x = inverrdiag
.Parameters
\n\n\n
\n", "signature": "(corr, inverrdiag):", "funcdef": "def"}, "pyerrors.obs.sort_corr": {"fullname": "pyerrors.obs.sort_corr", "modulename": "pyerrors.obs", "qualname": "sort_corr", "kind": "function", "doc": "- corr (np.ndarray):\ncorrelation matrix
\n- inverrdiag (np.ndarray):\ndiagonal matrix, the entries are the inverse errors of the data points considered
\nReorders a correlation matrix to match the alphabetical order of its underlying y data.
\n\nThe ordering of the input correlation matrix
\n\ncorr
is given by the list of keyskl
.\nThe input dictionaryyd
(with the same keyskl
) must contain the corresponding y data\nthat the correlation matrix is based on.\nThis function sorts the list of keyskl
alphabetically and sorts the matrixcorr
\naccording to this alphabetical order such that the sorted matrixcorr_sorted
corresponds\nto the y datayd
when arranged in an alphabetical order by its keys.Parameters
\n\n\n
\n\n- corr (np.ndarray):\nA square correlation matrix constructed using the order of the y data specified by
\nkl
.\nThe dimensions ofcorr
should match the total number of y data points inyd
combined.- kl (list of str):\nA list of keys that denotes the order in which the y data from
\nyd
was used to build the\ninput correlation matrixcorr
.- yd (dict of list):\nA dictionary where each key corresponds to a unique identifier, and its value is a list of\ny data points. The total number of y data points across all keys must match the dimensions\nof
\ncorr
. The lists in the dictionary can be lists of Obs.Returns
\n\n\n
\n\n- np.ndarray: A new, sorted correlation matrix that corresponds to the y data from
\nyd
when arranged alphabetically by its keys.Example
\n\n\n\n", "signature": "(corr, kl, yd):", "funcdef": "def"}, "pyerrors.obs.import_jackknife": {"fullname": "pyerrors.obs.import_jackknife", "modulename": "pyerrors.obs", "qualname": "import_jackknife", "kind": "function", "doc": "\n>>> import numpy as np\n>>> import pyerrors as pe\n>>> corr = np.array([[1, 0.2, 0.3], [0.2, 1, 0.4], [0.3, 0.4, 1]])\n>>> kl = ['b', 'a']\n>>> yd = {'a': [1, 2], 'b': [3]}\n>>> sorted_corr = pe.obs.sort_corr(corr, kl, yd)\n>>> print(sorted_corr)\narray([[1. , 0.3, 0.4],\n [0.3, 1. , 0.2],\n [0.4, 0.2, 1. ]])\n
Imports jackknife samples and returns an Obs
\n\nParameters
\n\n\n
\n", "signature": "(jacks, name, idl=None):", "funcdef": "def"}, "pyerrors.obs.import_bootstrap": {"fullname": "pyerrors.obs.import_bootstrap", "modulename": "pyerrors.obs", "qualname": "import_bootstrap", "kind": "function", "doc": "- 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.
\nImports bootstrap samples and returns an Obs
\n\nParameters
\n\n\n
\n", "signature": "(boots, name, random_numbers):", "funcdef": "def"}, "pyerrors.obs.merge_obs": {"fullname": "pyerrors.obs.merge_obs", "modulename": "pyerrors.obs", "qualname": "merge_obs", "kind": "function", "doc": "- boots (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N bootstrap samples as first to Nth entry.
\n- name (str):\nname of the ensemble the samples are defined on.
\n- random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples,\nwhere samples is the number of bootstrap samples and length is the length of the original Monte Carlo\nchain to be reconstructed.
\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
\n", "signature": "(d, func, guess=1.0, **kwargs):", "funcdef": "def"}, "pyerrors.special": {"fullname": "pyerrors.special", "modulename": "pyerrors.special", "kind": "module", "doc": "\n"}, "pyerrors.special.beta": {"fullname": "pyerrors.special.beta", "modulename": "pyerrors.special", "qualname": "beta", "kind": "function", "doc": "- res (Obs):\n
\nObs
valued root of the function.beta(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nbeta(a, b, out=None)
\n\nBeta function.
\n\nThis function is defined in 1 as
\n\n$$B(a, b) = \\int_0^1 t^{a-1}(1-t)^{b-1}dt\n = \\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)},$$
\n\nwhere \\( \\Gamma \\) is the gamma function.
\n\nParameters
\n\n\n
\n\n- a, b (array_like):\nReal-valued arguments
\n- out (ndarray, optional):\nOptional output array for the function result
\nReturns
\n\n\n
\n\n- scalar or ndarray: Value of the beta function
\nSee Also
\n\n\n\n
gamma
: the gamma function
\nbetainc
: the regularized incomplete beta function
\nbetaln
: the natural logarithm of the absolute\nvalue of the beta functionReferences
\n\nExamples
\n\n\n\n\n\n>>> import scipy.special as sc\n
The beta function relates to the gamma function by the\ndefinition given above:
\n\n\n\n\n\n>>> sc.beta(2, 3)\n0.08333333333333333\n>>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3)\n0.08333333333333333\n
As this relationship demonstrates, the beta function\nis symmetric:
\n\n\n\n\n\n>>> sc.beta(1.7, 2.4)\n0.16567527689031739\n>>> sc.beta(2.4, 1.7)\n0.16567527689031739\n
This function satisfies \\( B(1, b) = 1/b \\):
\n\n\n\n\n\n>>> sc.beta(1, 4)\n0.25\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betainc": {"fullname": "pyerrors.special.betainc", "modulename": "pyerrors.special", "qualname": "betainc", "kind": "function", "doc": "
\n\n
\n- \n
\nNIST Digital Library of Mathematical Functions,\nEq. 5.12.1. https://dlmf.nist.gov/5.12 ↩
\nbetainc(x1, x2, x3, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nbetainc(a, b, x, out=None)
\n\nRegularized incomplete beta function.
\n\nComputes the regularized incomplete beta function, defined as 1:
\n\n$$I_x(a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^x\nt^{a-1}(1-t)^{b-1}dt,$$
\n\nfor \\( 0 \\leq x \\leq 1 \\).
\n\nThis function is the cumulative distribution function for the beta\ndistribution; its range is [0, 1].
\n\nParameters
\n\n\n
\n\n- a, b (array_like):\nPositive, real-valued parameters
\n- x (array_like):\nReal-valued such that \\( 0 \\leq x \\leq 1 \\),\nthe upper limit of integration
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Value of the regularized incomplete beta function
\nSee Also
\n\n\n\n
beta
: beta function
\nbetaincinv
: inverse of the regularized incomplete beta function
\nbetaincc
: complement of the regularized incomplete beta function
\nscipy.stats.beta
: beta distributionNotes
\n\nThe term regularized in the name of this function refers to the\nscaling of the function by the gamma function terms shown in the\nformula. When not qualified as regularized, the name incomplete\nbeta function often refers to just the integral expression,\nwithout the gamma terms. One can use the function
\n\nbeta
from\nscipy.special
to get this \"nonregularized\" incomplete beta\nfunction by multiplying the result ofbetainc(a, b, x)
by\nbeta(a, b)
.References
\n\nExamples
\n\nLet \\( B(a, b) \\) be the
\n\nbeta
function.\n\n\n\n>>> import scipy.special as sc\n
The coefficient in terms of
\n\ngamma
is equal to\n\\( 1/B(a, b) \\). Also, when \\( x=1 \\)\nthe integral is equal to \\( B(a, b) \\).\nTherefore, \\( I_{x=1}(a, b) = 1 \\) for any \\( a, b \\).\n\n\n\n>>> sc.betainc(0.2, 3.5, 1.0)\n1.0\n
It satisfies\n\\( I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b)) \\),\nwhere \\( F \\) is the hypergeometric function
\n\nhyp2f1
:\n\n\n\n>>> a, b, x = 1.4, 3.1, 0.5\n>>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))\n0.8148904036225295\n>>> sc.betainc(a, b, x)\n0.8148904036225296\n
This functions satisfies the relationship\n\\( I_x(a, b) = 1 - I_{1-x}(b, a) \\):
\n\n\n\n\n\n>>> sc.betainc(2.2, 3.1, 0.4)\n0.49339638807619446\n>>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)\n0.49339638807619446\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betaln": {"fullname": "pyerrors.special.betaln", "modulename": "pyerrors.special", "qualname": "betaln", "kind": "function", "doc": "
\n\n
\n- \n
\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/8.17 ↩
\nbetaln(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nbetaln(a, b, out=None)
\n\nNatural logarithm of absolute value of beta function.
\n\nComputes
\n\nln(abs(beta(a, b)))
.Parameters
\n\n\n
\n\n- a, b (array_like):\nPositive, real-valued parameters
\n- out (ndarray, optional):\nOptional output array for function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Value of the betaln function
\nSee Also
\n\n\n\n
gamma
: the gamma function
\nbetainc
: the regularized incomplete beta function
\nbeta
: the beta functionExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import betaln, beta\n
Verify that, for moderate values of
\n\na
andb
,betaln(a, b)
\nis the same aslog(beta(a, b))
:\n\n\n\n>>> betaln(3, 4)\n-4.0943445622221\n
\n\n\n\n>>> np.log(beta(3, 4))\n-4.0943445622221\n
In the following
\n\nbeta(a, b)
underflows to 0, so we can't compute\nthe logarithm of the actual value.\n\n\n\n>>> a = 400\n>>> b = 900\n>>> beta(a, b)\n0.0\n
We can compute the logarithm of
\n\nbeta(a, b)
by usingbetaln
:\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.polygamma": {"fullname": "pyerrors.special.polygamma", "modulename": "pyerrors.special", "qualname": "polygamma", "kind": "function", "doc": "\n>>> betaln(a, b)\n-804.3069951764146\n
Polygamma functions.
\n\nDefined as \\( \\psi^{(n)}(x) \\) where \\( \\psi \\) is the\n
\n\ndigamma
function. See [dlmf]_ for details.Parameters
\n\n\n
\n\n- n (array_like):\nThe order of the derivative of the digamma function; must be\nintegral
\n- x (array_like):\nReal valued input
\nReturns
\n\n\n
\n\n- ndarray: Function results
\nSee Also
\n\n\n\n
digamma
References
\n\n.. [dlmf] NIST, Digital Library of Mathematical Functions,\n https://dlmf.nist.gov/5.15
\n\nExamples
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.psi": {"fullname": "pyerrors.special.psi", "modulename": "pyerrors.special", "qualname": "psi", "kind": "function", "doc": "\n>>> from scipy import special\n>>> x = [2, 3, 25.5]\n>>> special.polygamma(1, x)\narray([ 0.64493407, 0.39493407, 0.03999467])\n>>> special.polygamma(0, x) == special.psi(x)\narray([ True, True, True], dtype=bool)\n
psi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\npsi(z, out=None)
\n\nThe digamma function.
\n\nThe logarithmic derivative of the gamma function evaluated at
\n\nz
.Parameters
\n\n\n
\n\n- z (array_like):\nReal or complex argument.
\n- out (ndarray, optional):\nArray for the computed values of
\npsi
.Returns
\n\n\n
\n\n- digamma (scalar or ndarray):\nComputed values of
\npsi
.Notes
\n\nFor large values not close to the negative real axis,
\n\npsi
is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note thatpsi
has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.References
\n\nExamples
\n\n\n\n\n\n>>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
Verify psi(z) = psi(z + 1) - 1/z:
\n\n\n\n\n\n>>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.digamma": {"fullname": "pyerrors.special.digamma", "modulename": "pyerrors.special", "qualname": "digamma", "kind": "function", "doc": "
\n\n
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\nFredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
\npsi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\npsi(z, out=None)
\n\nThe digamma function.
\n\nThe logarithmic derivative of the gamma function evaluated at
\n\nz
.Parameters
\n\n\n
\n\n- z (array_like):\nReal or complex argument.
\n- out (ndarray, optional):\nArray for the computed values of
\npsi
.Returns
\n\n\n
\n\n- digamma (scalar or ndarray):\nComputed values of
\npsi
.Notes
\n\nFor large values not close to the negative real axis,
\n\npsi
is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note thatpsi
has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.References
\n\nExamples
\n\n\n\n\n\n>>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
Verify psi(z) = psi(z + 1) - 1/z:
\n\n\n\n\n\n>>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gamma": {"fullname": "pyerrors.special.gamma", "modulename": "pyerrors.special", "qualname": "gamma", "kind": "function", "doc": "
\n\n
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\n\nNIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩
\n- \n
\nFredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩
\ngamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ngamma(z, out=None)
\n\ngamma function.
\n\nThe gamma function is defined as
\n\n$$\\Gamma(z) = \\int_0^\\infty t^{z-1} e^{-t} dt$$
\n\nfor \\( \\Re(z) > 0 \\) and is extended to the rest of the complex\nplane by analytic continuation. See [dlmf]_ for more details.
\n\nParameters
\n\n\n
\n\n- z (array_like):\nReal or complex valued argument
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the gamma function
\nNotes
\n\nThe gamma function is often referred to as the generalized\nfactorial since \\( \\Gamma(n + 1) = n! \\) for natural numbers\n\\( n \\). More generally it satisfies the recurrence relation\n\\( \\Gamma(z + 1) = z \\cdot \\Gamma(z) \\) for complex \\( z \\),\nwhich, combined with the fact that \\( \\Gamma(1) = 1 \\), implies\nthe above identity for \\( z = n \\).
\n\nReferences
\n\n.. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5.2#E1
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import gamma, factorial\n
\n\n\n\n>>> gamma([0, 0.5, 1, 5])\narray([ inf, 1.77245385, 1. , 24. ])\n
\n\n\n\n>>> z = 2.5 + 1j\n>>> gamma(z)\n(0.77476210455108352+0.70763120437959293j)\n>>> gamma(z+1), z*gamma(z) # Recurrence property\n((1.2292740569981171+2.5438401155000685j),\n (1.2292740569981158+2.5438401155000658j))\n
\n\n\n\n>>> gamma(0.5)**2 # gamma(0.5) = sqrt(pi)\n3.1415926535897927\n
Plot gamma(x) for real x
\n\n\n\n\n\n>>> x = np.linspace(-3.5, 5.5, 2251)\n>>> y = gamma(x)\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaln": {"fullname": "pyerrors.special.gammaln", "modulename": "pyerrors.special", "qualname": "gammaln", "kind": "function", "doc": "\n>>> import matplotlib.pyplot as plt\n>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')\n>>> k = np.arange(1, 7)\n>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,\n... label='(x-1)!, x = 1, 2, ...')\n>>> plt.xlim(-3.5, 5.5)\n>>> plt.ylim(-10, 25)\n>>> plt.grid()\n>>> plt.xlabel('x')\n>>> plt.legend(loc='lower right')\n>>> plt.show()\n
gammaln(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ngammaln(x, out=None)
\n\nLogarithm of the absolute value of the gamma function.
\n\nDefined as
\n\n$$\\ln(\\lvert\\Gamma(x)\\rvert)$$
\n\nwhere \\( \\Gamma \\) is the gamma function. For more details on\nthe gamma function, see [dlmf]_.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nReal argument
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the log of the absolute value of gamma
\nSee Also
\n\n\n\n
gammasgn
: sign of the gamma function
\nloggamma
: principal branch of the logarithm of the gamma functionNotes
\n\nIt is the same function as the Python standard library function\n
\n\nmath.lgamma()
.When used in conjunction with
\n\ngammasgn
, this function is useful\nfor working in logspace on the real axis without having to deal\nwith complex numbers via the relationexp(gammaln(x)) =\ngammasgn(x) * gamma(x)
.For complex-valued log-gamma, use
\n\nloggamma
instead ofgammaln
.References
\n\n.. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> import scipy.special as sc\n
It has two positive zeros.
\n\n\n\n\n\n>>> sc.gammaln([1, 2])\narray([0., 0.])\n
It has poles at nonpositive integers.
\n\n\n\n\n\n>>> sc.gammaln([0, -1, -2, -3, -4])\narray([inf, inf, inf, inf, inf])\n
It asymptotically approaches
\n\nx * log(x)
(Stirling's formula).\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammainc": {"fullname": "pyerrors.special.gammainc", "modulename": "pyerrors.special", "qualname": "gammainc", "kind": "function", "doc": "\n>>> x = np.array([1e10, 1e20, 1e40, 1e80])\n>>> sc.gammaln(x)\narray([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82])\n>>> x * np.log(x)\narray([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])\n
gammainc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ngammainc(a, x, out=None)
\n\nRegularized lower incomplete gamma function.
\n\nIt is defined as
\n\n$$P(a, x) = \\frac{1}{\\Gamma(a)} \\int_0^x t^{a - 1}e^{-t} dt$$
\n\nfor \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
\n\nParameters
\n\n\n
\n\n- a (array_like):\nPositive parameter
\n- x (array_like):\nNonnegative argument
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the lower incomplete gamma function
\nSee Also
\n\n\n\n
gammaincc
: regularized upper incomplete gamma function
\ngammaincinv
: inverse of the regularized lower incomplete gamma function
\ngammainccinv
: inverse of the regularized upper incomplete gamma functionNotes
\n\nThe function satisfies the relation
\n\ngammainc(a, x) +\ngammaincc(a, x) = 1
wheregammaincc
is the regularized upper\nincomplete gamma function.The implementation largely follows that of [boost]_.
\n\nReferences
\n\n.. [dlmf] NIST Digital Library of Mathematical functions\n https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
\n\nExamples
\n\n\n\n\n\n>>> import scipy.special as sc\n
It is the CDF of the gamma distribution, so it starts at 0 and\nmonotonically increases to 1.
\n\n\n\n\n\n>>> sc.gammainc(0.5, [0, 1, 10, 100])\narray([0. , 0.84270079, 0.99999226, 1. ])\n
It is equal to one minus the upper incomplete gamma function.
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaincc": {"fullname": "pyerrors.special.gammaincc", "modulename": "pyerrors.special", "qualname": "gammaincc", "kind": "function", "doc": "\n>>> a, x = 0.5, 0.4\n>>> sc.gammainc(a, x)\n0.6289066304773024\n>>> 1 - sc.gammaincc(a, x)\n0.6289066304773024\n
gammaincc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ngammaincc(a, x, out=None)
\n\nRegularized upper incomplete gamma function.
\n\nIt is defined as
\n\n$$Q(a, x) = \\frac{1}{\\Gamma(a)} \\int_x^\\infty t^{a - 1}e^{-t} dt$$
\n\nfor \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.
\n\nParameters
\n\n\n
\n\n- a (array_like):\nPositive parameter
\n- x (array_like):\nNonnegative argument
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the upper incomplete gamma function
\nSee Also
\n\n\n\n
gammainc
: regularized lower incomplete gamma function
\ngammaincinv
: inverse of the regularized lower incomplete gamma function
\ngammainccinv
: inverse of the regularized upper incomplete gamma functionNotes
\n\nThe function satisfies the relation
\n\ngammainc(a, x) +\ngammaincc(a, x) = 1
wheregammainc
is the regularized lower\nincomplete gamma function.The implementation largely follows that of [boost]_.
\n\nReferences
\n\n.. [dlmf] NIST Digital Library of Mathematical functions\n https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html
\n\nExamples
\n\n\n\n\n\n>>> import scipy.special as sc\n
It is the survival function of the gamma distribution, so it\nstarts at 1 and monotonically decreases to 0.
\n\n\n\n\n\n>>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000])\narray([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45,\n 0.00000000e+00])\n
It is equal to one minus the lower incomplete gamma function.
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammasgn": {"fullname": "pyerrors.special.gammasgn", "modulename": "pyerrors.special", "qualname": "gammasgn", "kind": "function", "doc": "\n>>> a, x = 0.5, 0.4\n>>> sc.gammaincc(a, x)\n0.37109336952269756\n>>> 1 - sc.gammainc(a, x)\n0.37109336952269756\n
gammasgn(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ngammasgn(x, out=None)
\n\nSign of the gamma function.
\n\nIt is defined as
\n\n$$\\text{gammasgn}(x) =\n\\begin{cases}\n +1 & \\Gamma(x) > 0 \\\n -1 & \\Gamma(x) < 0\n\\end{cases}$$
\n\nwhere \\( \\Gamma \\) is the gamma function; see
\n\ngamma
. This\ndefinition is complete since the gamma function is never zero;\nsee the discussion after [dlmf]_.Parameters
\n\n\n
\n\n- x (array_like):\nReal argument
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Sign of the gamma function
\nSee Also
\n\n\n\n
gamma
: the gamma function
\ngammaln
: log of the absolute value of the gamma function
\nloggamma
: analytic continuation of the log of the gamma functionNotes
\n\nThe gamma function can be computed as
\n\ngammasgn(x) *\nnp.exp(gammaln(x))
.References
\n\n.. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5.2#E1
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> import scipy.special as sc\n
It is 1 for
\n\nx > 0
.\n\n\n\n>>> sc.gammasgn([1, 2, 3, 4])\narray([1., 1., 1., 1.])\n
It alternates between -1 and 1 for negative integers.
\n\n\n\n\n\n>>> sc.gammasgn([-0.5, -1.5, -2.5, -3.5])\narray([-1., 1., -1., 1.])\n
It can be used to compute the gamma function.
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.rgamma": {"fullname": "pyerrors.special.rgamma", "modulename": "pyerrors.special", "qualname": "rgamma", "kind": "function", "doc": "\n>>> x = [1.5, 0.5, -0.5, -1.5]\n>>> sc.gammasgn(x) * np.exp(sc.gammaln(x))\narray([ 0.88622693, 1.77245385, -3.5449077 , 2.3632718 ])\n>>> sc.gamma(x)\narray([ 0.88622693, 1.77245385, -3.5449077 , 2.3632718 ])\n
rgamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nrgamma(z, out=None)
\n\nReciprocal of the gamma function.
\n\nDefined as \\( 1 / \\Gamma(z) \\), where \\( \\Gamma \\) is the\ngamma function. For more on the gamma function see
\n\ngamma
.Parameters
\n\n\n
\n\n- z (array_like):\nReal or complex valued input
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- scalar or ndarray: Function results
\nSee Also
\n\n\n\n
gamma,
,gammaln,
,loggamma
Notes
\n\nThe gamma function has no zeros and has simple poles at\nnonpositive integers, so
\n\nrgamma
is an entire function with zeros\nat the nonpositive integers. See the discussion in [dlmf]_ for\nmore details.References
\n\n.. [dlmf] Nist, Digital Library of Mathematical functions,\n https://dlmf.nist.gov/5.2#i
\n\nExamples
\n\n\n\n\n\n>>> import scipy.special as sc\n
It is the reciprocal of the gamma function.
\n\n\n\n\n\n>>> sc.rgamma([1, 2, 3, 4])\narray([1. , 1. , 0.5 , 0.16666667])\n>>> 1 / sc.gamma([1, 2, 3, 4])\narray([1. , 1. , 0.5 , 0.16666667])\n
It is zero at nonpositive integers.
\n\n\n\n\n\n>>> sc.rgamma([0, -1, -2, -3])\narray([0., 0., 0., 0.])\n
It rapidly underflows to zero along the positive real axis.
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.multigammaln": {"fullname": "pyerrors.special.multigammaln", "modulename": "pyerrors.special", "qualname": "multigammaln", "kind": "function", "doc": "\n>>> sc.rgamma([10, 100, 179])\narray([2.75573192e-006, 1.07151029e-156, 0.00000000e+000])\n
Returns the log of multivariate gamma, also sometimes called the\ngeneralized gamma.
\n\nParameters
\n\n\n
\n\n- a (ndarray):\nThe multivariate gamma is computed for each item of
\na
.- d (int):\nThe dimension of the space of integration.
\nReturns
\n\n\n
\n\n- res (ndarray):\nThe values of the log multivariate gamma at the given points
\na
.Notes
\n\nThe formal definition of the multivariate gamma of dimension d for a real\n
\n\na
is$$\\Gamma_d(a) = \\int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA$$
\n\nwith the condition \\( a > (d-1)/2 \\), and \\( A > 0 \\) being the set of\nall the positive definite matrices of dimension
\n\nd
. Note thata
is a\nscalar: the integrand only is multivariate, the argument is not (the\nfunction is defined over a subset of the real set).This can be proven to be equal to the much friendlier equation
\n\n$$\\Gamma_d(a) = \\pi^{d(d-1)/4} \\prod_{i=1}^{d} \\Gamma(a - (i-1)/2).$$
\n\nReferences
\n\nR. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in\nprobability and mathematical statistics).
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import multigammaln, gammaln\n>>> a = 23.5\n>>> d = 10\n>>> multigammaln(a, d)\n454.1488605074416\n
Verify that the result agrees with the logarithm of the equation\nshown above:
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.kn": {"fullname": "pyerrors.special.kn", "modulename": "pyerrors.special", "qualname": "kn", "kind": "function", "doc": "\n>>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()\n454.1488605074416\n
Modified Bessel function of the second kind of integer order n
\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j0": {"fullname": "pyerrors.special.j0", "modulename": "pyerrors.special", "qualname": "j0", "kind": "function", "doc": "j0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nj0(x, out=None)
\n\nBessel function of the first kind of order 0.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float).
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 0 at
\nx
.See Also
\n\n\n\n
jv
: Bessel function of real order and complex argument.
\nspherical_jn
: spherical Bessel functions.Notes
\n\nThe domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval the following rational approximation is used:
\n\n$$J_0(x) \\approx (w - r_1^2)(w - r_2^2) \\frac{P_3(w)}{Q_8(w)},$$
\n\nwhere \\( w = x^2 \\) and \\( r_1 \\), \\( r_2 \\) are the zeros of\n\\( J_0 \\), and \\( P_3 \\) and \\( Q_8 \\) are polynomials of degrees 3\nand 8, respectively.
\n\nIn the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\nj0
.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn
).References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import j0\n>>> j0(1.)\n0.7651976865579665\n
Calculate the function at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> j0(np.array([-2., 0., 4.]))\narray([ 0.22389078, 1. , -0.39714981])\n
Plot the function from -20 to 20.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y0": {"fullname": "pyerrors.special.y0", "modulename": "pyerrors.special", "qualname": "y0", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\ny0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ny0(x, out=None)
\n\nBessel function of the second kind of order 0.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float).
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 0 at
\nx
.See Also
\n\n\n\n
j0
: Bessel function of the first kind of order 0
\nyv
: Bessel function of the first kindNotes
\n\nThe domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval a rational approximation \\( R(x) \\) is employed to\ncompute,
\n\n$$Y_0(x) = R(x) + \\frac{2 \\log(x) J_0(x)}{\\pi},$$
\n\nwhere \\( J_0 \\) is the Bessel function of the first kind of order 0.
\n\nIn the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\ny0
.References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import y0\n>>> y0(1.)\n0.08825696421567697\n
Calculate at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> y0(np.array([0.5, 2., 3.]))\narray([-0.44451873, 0.51037567, 0.37685001])\n
Plot the function from 0 to 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j1": {"fullname": "pyerrors.special.j1", "modulename": "pyerrors.special", "qualname": "j1", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\nj1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nj1(x, out=None)
\n\nBessel function of the first kind of order 1.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float).
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 1 at
\nx
.See Also
\n\n\n\n
jv
: Bessel function of the first kind
\nspherical_jn
: spherical Bessel functions.Notes
\n\nThe domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 24 term Chebyshev expansion is used. In the second, the\nasymptotic trigonometric representation is employed using two rational\nfunctions of degree 5/5.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\nj1
.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn
).References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import j1\n>>> j1(1.)\n0.44005058574493355\n
Calculate the function at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> j1(np.array([-2., 0., 4.]))\narray([-0.57672481, 0. , -0.06604333])\n
Plot the function from -20 to 20.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y1": {"fullname": "pyerrors.special.y1", "modulename": "pyerrors.special", "qualname": "y1", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\ny1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ny1(x, out=None)
\n\nBessel function of the second kind of order 1.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float).
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 1 at
\nx
.See Also
\n\n\n\n
j1
: Bessel function of the first kind of order 1
\nyn
: Bessel function of the second kind
\nyv
: Bessel function of the second kindNotes
\n\nThe domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 25 term Chebyshev expansion is used, and computing\n\\( J_1 \\) (the Bessel function of the first kind) is required. In the\nsecond, the asymptotic trigonometric representation is employed using two\nrational functions of degree 5/5.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\ny1
.References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import y1\n>>> y1(1.)\n-0.7812128213002888\n
Calculate at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> y1(np.array([0.5, 2., 3.]))\narray([-1.47147239, -0.10703243, 0.32467442])\n
Plot the function from 0 to 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.jn": {"fullname": "pyerrors.special.jn", "modulename": "pyerrors.special", "qualname": "jn", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\njv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\njv(v, z, out=None)
\n\nBessel function of the first kind of real order and complex argument.
\n\nParameters
\n\n\n
\n\n- v (array_like):\nOrder (float).
\n- z (array_like):\nArgument (float or complex).
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- J (scalar or ndarray):\nValue of the Bessel function, \\( J_v(z) \\).
\nSee Also
\n\n\n\n
jve
: \\( J_v \\) with leading exponential behavior stripped off.
\nspherical_jn
: spherical Bessel functions.
\nj0
: faster version of this function for order 0.
\nj1
: faster version of this function for order 1.Notes
\n\nFor positive
\n\nv
values, the computation is carried out using the AMOS\n1zbesj
routine, which exploits the connection to the modified\nBessel function \\( I_v \\),$$J_v(z) = \\exp(v\\pi\\imath/2) I_v(-\\imath z)\\qquad (\\Im z > 0)
\n\nJ_v(z) = \\exp(-v\\pi\\imath/2) I_v(\\imath z)\\qquad (\\Im z < 0)$$
\n\nFor negative
\n\nv
values the formula,$$J_{-v}(z) = J_v(z) \\cos(\\pi v) - Y_v(z) \\sin(\\pi v)$$
\n\nis used, where \\( Y_v(z) \\) is the Bessel function of the second\nkind, computed using the AMOS routine
\n\nzbesy
. Note that the second\nterm is exactly zero for integerv
; to improve accuracy the second\nterm is explicitly omitted forv
values such thatv = floor(v)
.Not to be confused with the spherical Bessel functions (see
\n\nspherical_jn
).References
\n\nExamples
\n\nEvaluate the function of order 0 at one point.
\n\n\n\n\n\n>>> from scipy.special import jv\n>>> jv(0, 1.)\n0.7651976865579666\n
Evaluate the function at one point for different orders.
\n\n\n\n\n\n>>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)\n(0.7651976865579666, 0.44005058574493355, 0.24029783912342725)\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the
\n\nv
parameter:\n\n\n\n>>> jv([0, 1, 1.5], 1.)\narray([0.76519769, 0.44005059, 0.24029784])\n
Evaluate the function at several points for order 0 by providing an\narray for
\n\nz
.\n\n\n\n>>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> jv(0, points)\narray([ 0.22389078, 1. , -0.26005195])\n
If
\n\nz
is an array, the order parameterv
must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:\n\n\n\n>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n\n\n\n>>> jv(orders, points)\narray([[ 0.22389078, 1. , -0.26005195],\n [-0.57672481, 0. , 0.33905896]])\n
Plot the functions of order 0 to 3 from -10 to 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> for i in range(4):\n... ax.plot(x, jv(i, x), label=f'$J_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.yn": {"fullname": "pyerrors.special.yn", "modulename": "pyerrors.special", "qualname": "yn", "kind": "function", "doc": "
\n\n
\n- \n
\nDonald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
\nyn(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nyn(n, x, out=None)
\n\nBessel function of the second kind of integer order and real argument.
\n\nParameters
\n\n\n
\n\n- n (array_like):\nOrder (integer).
\n- x (array_like):\nArgument (float).
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- Y (scalar or ndarray):\nValue of the Bessel function, \\( Y_n(x) \\).
\nSee Also
\n\n\n\n
yv
: For real order and real or complex argument.
\ny0
: faster implementation of this function for order 0
\ny1
: faster implementation of this function for order 1Notes
\n\nWrapper for the Cephes 1 routine
\n\nyn
.The function is evaluated by forward recurrence on
\n\nn
, starting with\nvalues computed by the Cephes routinesy0
andy1
. Ifn = 0
or 1,\nthe routine fory0
ory1
is called directly.References
\n\nExamples
\n\nEvaluate the function of order 0 at one point.
\n\n\n\n\n\n>>> from scipy.special import yn\n>>> yn(0, 1.)\n0.08825696421567697\n
Evaluate the function at one point for different orders.
\n\n\n\n\n\n>>> yn(0, 1.), yn(1, 1.), yn(2, 1.)\n(0.08825696421567697, -0.7812128213002888, -1.6506826068162546)\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the
\n\nv
parameter:\n\n\n\n>>> yn([0, 1, 2], 1.)\narray([ 0.08825696, -0.78121282, -1.65068261])\n
Evaluate the function at several points for order 0 by providing an\narray for
\n\nz
.\n\n\n\n>>> import numpy as np\n>>> points = np.array([0.5, 3., 8.])\n>>> yn(0, points)\narray([-0.44451873, 0.37685001, 0.22352149])\n
If
\n\nz
is an array, the order parameterv
must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:\n\n\n\n>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n\n\n\n>>> yn(orders, points)\narray([[-0.44451873, 0.37685001, 0.22352149],\n [-1.47147239, 0.32467442, -0.15806046]])\n
Plot the functions of order 0 to 3 from 0 to 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> for i in range(4):\n... ax.plot(x, yn(i, x), label=f'$Y_{i!r}$')\n>>> ax.set_ylim(-3, 1)\n>>> ax.legend()\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i0": {"fullname": "pyerrors.special.i0", "modulename": "pyerrors.special", "qualname": "i0", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\ni0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ni0(x, out=None)
\n\nModified Bessel function of order 0.
\n\nDefined as,
\n\n$$I_0(x) = \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{(k!)^2} = J_0(\\imath x),$$
\n\nwhere \\( J_0 \\) is the Bessel function of the first kind of order 0.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float)
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- I (scalar or ndarray):\nValue of the modified Bessel function of order 0 at
\nx
.See Also
\n\n\n\n
iv
: Modified Bessel function of any order
\ni0e
: Exponentially scaled modified Bessel function of order 0Notes
\n\nThe range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\ni0
.References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import i0\n>>> i0(1.)\n1.2660658777520082\n
Calculate at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> i0(np.array([-2., 0., 3.5]))\narray([2.2795853 , 1. , 7.37820343])\n
Plot the function from -10 to 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i1": {"fullname": "pyerrors.special.i1", "modulename": "pyerrors.special", "qualname": "i1", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\ni1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\ni1(x, out=None)
\n\nModified Bessel function of order 1.
\n\nDefined as,
\n\n$$I_1(x) = \\frac{1}{2}x \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{k! (k + 1)!}\n = -\\imath J_1(\\imath x),$$
\n\nwhere \\( J_1 \\) is the Bessel function of the first kind of order 1.
\n\nParameters
\n\n\n
\n\n- x (array_like):\nArgument (float)
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- I (scalar or ndarray):\nValue of the modified Bessel function of order 1 at
\nx
.See Also
\n\n\n\n
iv
: Modified Bessel function of the first kind
\ni1e
: Exponentially scaled modified Bessel function of order 1Notes
\n\nThe range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.
\n\nThis function is a wrapper for the Cephes 1 routine
\n\ni1
.References
\n\nExamples
\n\nCalculate the function at one point:
\n\n\n\n\n\n>>> from scipy.special import i1\n>>> i1(1.)\n0.5651591039924851\n
Calculate the function at several points:
\n\n\n\n\n\n>>> import numpy as np\n>>> i1(np.array([-2., 0., 6.]))\narray([-1.59063685, 0. , 61.34193678])\n
Plot the function between -10 and 10.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.iv": {"fullname": "pyerrors.special.iv", "modulename": "pyerrors.special", "qualname": "iv", "kind": "function", "doc": "
\n\n
\n- \n
\nCephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩
\niv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\niv(v, z, out=None)
\n\nModified Bessel function of the first kind of real order.
\n\nParameters
\n\n\n
\n\n- v (array_like):\nOrder. If
\nz
is of real type and negative,v
must be integer\nvalued.- z (array_like of float or complex):\nArgument.
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the modified Bessel function.
\nSee Also
\n\n\n\n
ive
: This function with leading exponential behavior stripped off.
\ni0
: Faster version of this function for order 0.
\ni1
: Faster version of this function for order 1.Notes
\n\nFor real
\n\nz
and \\( v \\in [-50, 50] \\), the evaluation is carried out\nusing Temme's method 1. For larger orders, uniform asymptotic\nexpansions are applied.For complex
\n\nz
and positivev
, the AMOS 2zbesi
routine is\ncalled. It uses a power series for smallz
, the asymptotic expansion\nfor largeabs(z)
, the Miller algorithm normalized by the Wronskian\nand a Neumann series for intermediate magnitudes, and the uniform\nasymptotic expansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large\norders. Backward recurrence is used to generate sequences or reduce\norders when necessary.The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
\n\n$$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
\n\n(valid when the real part of
\n\nz
is positive). For negativev
, the\nformula$$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
\n\nis used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine
\n\nzbesk
.References
\n\nExamples
\n\nEvaluate the function of order 0 at one point.
\n\n\n\n\n\n>>> from scipy.special import iv\n>>> iv(0, 1.)\n1.2660658777520084\n
Evaluate the function at one point for different orders.
\n\n\n\n\n\n>>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.)\n(1.2660658777520084, 0.565159103992485, 0.2935253263474798)\n
The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the
\n\nv
parameter:\n\n\n\n>>> iv([0, 1, 1.5], 1.)\narray([1.26606588, 0.5651591 , 0.29352533])\n
Evaluate the function at several points for order 0 by providing an\narray for
\n\nz
.\n\n\n\n>>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> iv(0, points)\narray([2.2795853 , 1. , 4.88079259])\n
If
\n\nz
is an array, the order parameterv
must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:\n\n\n\n>>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
\n\n\n\n>>> iv(orders, points)\narray([[ 2.2795853 , 1. , 4.88079259],\n [-1.59063685, 0. , 3.95337022]])\n
Plot the functions of order 0 to 3 from -5 to 5.
\n\n\n\n\n\n>>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n... ax.plot(x, iv(i, x), label=f'$I_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.ive": {"fullname": "pyerrors.special.ive", "modulename": "pyerrors.special", "qualname": "ive", "kind": "function", "doc": "
\n\n
\n- \n
\n\nTemme, Journal of Computational Physics, vol 21, 343 (1976) ↩
\n- \n
\nDonald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
\nive(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nive(v, z, out=None)
\n\nExponentially scaled modified Bessel function of the first kind.
\n\nDefined as::
\n\n\n\nive(v, z) = iv(v, z) * exp(-abs(z.real))\n
For imaginary numbers without a real part, returns the unscaled\nBessel function of the first kind
\n\niv
.Parameters
\n\n\n
\n\n- v (array_like of float):\nOrder.
\n- z (array_like of float or complex):\nArgument.
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the exponentially scaled modified Bessel function.
\nSee Also
\n\n\n\n
iv
: Modified Bessel function of the first kind
\ni0e
: Faster implementation of this function for order 0
\ni1e
: Faster implementation of this function for order 1Notes
\n\nFor positive
\n\nv
, the AMOS 1zbesi
routine is called. It uses a\npower series for smallz
, the asymptotic expansion for large\nabs(z)
, the Miller algorithm normalized by the Wronskian and a\nNeumann series for intermediate magnitudes, and the uniform asymptotic\nexpansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large orders.\nBackward recurrence is used to generate sequences or reduce orders when\nnecessary.The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,
\n\n$$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$
\n\n(valid when the real part of
\n\nz
is positive). For negativev
, the\nformula$$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$
\n\nis used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine
\n\nzbesk
.\n\n
ive
is useful for large argumentsz
: for these,iv
easily overflows,\nwhileive
does not due to the exponential scaling.References
\n\nExamples
\n\nIn the following example
\n\niv
returns infinity whereasive
still returns\na finite number.\n\n\n\n>>> from scipy.special import iv, ive\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> iv(3, 1000.), ive(3, 1000.)\n(inf, 0.01256056218254712)\n
Evaluate the function at one point for different orders by\nproviding a list or NumPy array as argument for the
\n\nv
parameter:\n\n\n\n>>> ive([0, 1, 1.5], 1.)\narray([0.46575961, 0.20791042, 0.10798193])\n
Evaluate the function at several points for order 0 by providing an\narray for
\n\nz
.\n\n\n\n>>> points = np.array([-2., 0., 3.])\n>>> ive(0, points)\narray([0.30850832, 1. , 0.24300035])\n
Evaluate the function at several points for different orders by\nproviding arrays for both
\n\nv
forz
. Both arrays have to be\nbroadcastable to the correct shape. To calculate the orders 0, 1\nand 2 for a 1D array of points:\n\n\n\n>>> ive([[0], [1], [2]], points)\narray([[ 0.30850832, 1. , 0.24300035],\n [-0.21526929, 0. , 0.19682671],\n [ 0.09323903, 0. , 0.11178255]])\n
Plot the functions of order 0 to 3 from -5 to 5.
\n\n\n\n\n\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n... ax.plot(x, ive(i, x), label=fr'$I_{i!r}(z)\\cdot e^{{-|z|}}$')\n>>> ax.legend()\n>>> ax.set_xlabel(r"$z$")\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erf": {"fullname": "pyerrors.special.erf", "modulename": "pyerrors.special", "qualname": "erf", "kind": "function", "doc": "
\n\n
\n- \n
\nDonald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩
\nerf(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nerf(z, out=None)
\n\nReturns the error function of complex argument.
\n\nIt is defined as
\n\n2/sqrt(pi)*integral(exp(-t**2), t=0..z)
.Parameters
\n\n\n
\n\n- x (ndarray):\nInput array.
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- res (scalar or ndarray):\nThe values of the error function at the given points
\nx
.See Also
\n\n\n\n
erfc,
,erfinv,
,erfcinv,
,wofz,
,erfcx,
,erfi
Notes
\n\nThe cumulative of the unit normal distribution is given by\n
\n\nPhi(z) = 1/2[1 + erf(z/sqrt(2))]
.References
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erf(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erf(x)$')\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfc": {"fullname": "pyerrors.special.erfc", "modulename": "pyerrors.special", "qualname": "erfc", "kind": "function", "doc": "
\n\n
\nerfc(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nerfc(x, out=None)
\n\nComplementary error function,
\n\n1 - erf(x)
.Parameters
\n\n\n
\n\n- x (array_like):\nReal or complex valued argument
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- scalar or ndarray: Values of the complementary error function
\nSee Also
\n\n\n\n
erf,
,erfi,
,erfcx,
,dawsn,
,wofz
References
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erfc(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erfc(x)$')\n>>> plt.show()\n
\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfinv": {"fullname": "pyerrors.special.erfinv", "modulename": "pyerrors.special", "qualname": "erfinv", "kind": "function", "doc": "
\n\n
\nerfinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nerfinv(y, out=None)
\n\nInverse of the error function.
\n\nComputes the inverse of the error function.
\n\nIn the complex domain, there is no unique complex number w satisfying\nerf(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, -1 < x < 1, there is a unique real\nnumber satisfying erf(erfinv(x)) = x.
\n\nParameters
\n\n\n
\n\n- y (ndarray):\nArgument at which to evaluate. Domain: [-1, 1]
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- erfinv (scalar or ndarray):\nThe inverse of erf of y, element-wise
\nSee Also
\n\n\n\n
erf
: Error function of a complex argument
\nerfc
: Complementary error function,1 - erf(x)
\nerfcinv
: Inverse of the complementary error functionExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfinv, erf\n
\n\n\n\n>>> erfinv(0.5)\n0.4769362762044699\n
\n\n\n\n>>> y = np.linspace(-1.0, 1.0, num=9)\n>>> x = erfinv(y)\n>>> x\narray([ -inf, -0.81341985, -0.47693628, -0.22531206, 0. ,\n 0.22531206, 0.47693628, 0.81341985, inf])\n
Verify that
\n\nerf(erfinv(y))
isy
.\n\n\n\n>>> erf(x)\narray([-1. , -0.75, -0.5 , -0.25, 0. , 0.25, 0.5 , 0.75, 1. ])\n
Plot the function:
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfcinv": {"fullname": "pyerrors.special.erfcinv", "modulename": "pyerrors.special", "qualname": "erfcinv", "kind": "function", "doc": "\n>>> y = np.linspace(-1, 1, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfinv(y)')\n>>> plt.show()\n
erfcinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nerfcinv(y, out=None)
\n\nInverse of the complementary error function.
\n\nComputes the inverse of the complementary error function.
\n\nIn the complex domain, there is no unique complex number w satisfying\nerfc(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, 0 < x < 2, there is a unique real\nnumber satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)).
\n\nIt is related to inverse of the error function by erfcinv(1-x) = erfinv(x)
\n\nParameters
\n\n\n
\n\n- y (ndarray):\nArgument at which to evaluate. Domain: [0, 2]
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- erfcinv (scalar or ndarray):\nThe inverse of erfc of y, element-wise
\nSee Also
\n\n\n\n
erf
: Error function of a complex argument
\nerfc
: Complementary error function,1 - erf(x)
\nerfinv
: Inverse of the error functionExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfcinv\n
\n\n\n\n>>> erfcinv(0.5)\n0.4769362762044699\n
\n\n\n\n>>> y = np.linspace(0.0, 2.0, num=11)\n>>> erfcinv(y)\narray([ inf, 0.9061938 , 0.59511608, 0.37080716, 0.17914345,\n -0. , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,\n -inf])\n
Plot the function:
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logit": {"fullname": "pyerrors.special.logit", "modulename": "pyerrors.special", "qualname": "logit", "kind": "function", "doc": "\n>>> y = np.linspace(0, 2, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfcinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfcinv(y)')\n>>> plt.show()\n
logit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\n\"\"\"\nlogit(x, out=None)
\n\nLogit ufunc for ndarrays.
\n\nThe logit function is defined as logit(p) = log(p/(1-p)).\nNote that logit(0) = -inf, logit(1) = inf, and logit(p)\nfor p<0 or p>1 yields nan.
\n\nParameters
\n\n\n
\n\n- x (ndarray):\nThe ndarray to apply logit to element-wise.
\n- out (ndarray, optional):\nOptional output array for the function results
\nReturns
\n\n\n
\n\n- scalar or ndarray: An ndarray of the same shape as x. Its entries\nare logit of the corresponding entry of x.
\nSee Also
\n\n\n\n
expit
Notes
\n\nAs a ufunc logit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs
\n\nNew in version 0.10.0.
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import logit, expit\n
\n\n\n\n>>> logit([0, 0.25, 0.5, 0.75, 1])\narray([ -inf, -1.09861229, 0. , 1.09861229, inf])\n
\n\n
expit
is the inverse oflogit
:\n\n\n\n>>> expit(logit([0.1, 0.75, 0.999]))\narray([ 0.1 , 0.75 , 0.999])\n
Plot logit(x) for x in [0, 1]:
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.expit": {"fullname": "pyerrors.special.expit", "modulename": "pyerrors.special", "qualname": "expit", "kind": "function", "doc": "\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(0, 1, 501)\n>>> y = logit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.ylim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('logit(x)')\n>>> plt.show()\n
expit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
\n\nexpit(x, out=None)
\n\nExpit (a.k.a. logistic sigmoid) ufunc for ndarrays.
\n\nThe expit function, also known as the logistic sigmoid function, is\ndefined as
\n\nexpit(x) = 1/(1+exp(-x))
. It is the inverse of the\nlogit function.Parameters
\n\n\n
\n\n- x (ndarray):\nThe ndarray to apply expit to element-wise.
\n- out (ndarray, optional):\nOptional output array for the function values
\nReturns
\n\n\n
\n\n- scalar or ndarray: An ndarray of the same shape as x. Its entries\nare
\nexpit
of the corresponding entry of x.See Also
\n\n\n\n
logit
Notes
\n\nAs a ufunc expit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs
\n\nNew in version 0.10.0.
\n\nExamples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import expit, logit\n
\n\n\n\n>>> expit([-np.inf, -1.5, 0, 1.5, np.inf])\narray([ 0. , 0.18242552, 0.5 , 0.81757448, 1. ])\n
\n\n
logit
is the inverse ofexpit
:\n\n\n\n>>> logit(expit([-2.5, 0, 3.1, 5.0]))\narray([-2.5, 0. , 3.1, 5. ])\n
Plot expit(x) for x in [-6, 6]:
\n\n\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logsumexp": {"fullname": "pyerrors.special.logsumexp", "modulename": "pyerrors.special", "qualname": "logsumexp", "kind": "function", "doc": "\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-6, 6, 121)\n>>> y = expit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.xlim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('expit(x)')\n>>> plt.show()\n
Compute the log of the sum of exponentials of input elements.
\n\nParameters
\n\n\n
\n\n- a (array_like):\nInput array.
\n- \n
axis (None or int or tuple of ints, optional):\nAxis or axes over which the sum is taken. By default
\n\naxis
is None,\nand all elements are summed.New in version 0.11.0.
- \n
b (array-like, optional):\nScaling factor for exp(
\n\na
) must be of the same shape asa
or\nbroadcastable toa
. These values may be negative in order to\nimplement subtraction.New in version 0.12.0.
- \n
keepdims (bool, optional):\nIf this is set to True, the axes which are reduced are left in the\nresult as dimensions with size one. With this option, the result\nwill broadcast correctly against the original array.
\n\nNew in version 0.15.0.
- \n
return_sign (bool, optional):\nIf this is set to True, the result will be a pair containing sign\ninformation; if False, results that are negative will be returned\nas NaN. Default is False (no sign information).
\n\nNew in version 0.16.0.
Returns
\n\n\n
\n\n- res (ndarray):\nThe result,
\nnp.log(np.sum(np.exp(a)))
calculated in a numerically\nmore stable way. Ifb
is given thennp.log(np.sum(b*np.exp(a)))
\nis returned. Ifreturn_sign
is True,res
contains the log of\nthe absolute value of the argument.- sgn (ndarray):\nIf
\nreturn_sign
is True, this will be an array of floating-point\nnumbers matching res containing +1, 0, -1 (for real-valued inputs)\nor a complex phase (for complex inputs). This gives the sign of the\nargument of the logarithm inres
.\nIfreturn_sign
is False, only one result is returned.See Also
\n\n\n\n
numpy.logaddexp,
,numpy.logaddexp2
Notes
\n\nNumPy has a logaddexp function which is very similar to
\n\nlogsumexp
, but\nonly handles two arguments.logaddexp.reduce
is similar to this\nfunction, but may be less stable.Examples
\n\n\n\n\n\n>>> import numpy as np\n>>> from scipy.special import logsumexp\n>>> a = np.arange(10)\n>>> logsumexp(a)\n9.4586297444267107\n>>> np.log(np.sum(np.exp(a)))\n9.4586297444267107\n
With weights
\n\n\n\n\n\n>>> a = np.arange(10)\n>>> b = np.arange(10, 0, -1)\n>>> logsumexp(a, b=b)\n9.9170178533034665\n>>> np.log(np.sum(b*np.exp(a)))\n9.9170178533034647\n
Returning a sign flag
\n\n\n\n\n\n>>> logsumexp([1,2],b=[1,-1],return_sign=True)\n(1.5413248546129181, -1.0)\n
Notice that
\n\nlogsumexp
does not directly support masked arrays. To use it\non a masked array, convert the mask into zero weights:\n\n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.version": {"fullname": "pyerrors.version", "modulename": "pyerrors.version", "kind": "module", "doc": "\n"}}, "docInfo": {"pyerrors": {"qualname": 0, "fullname": 1, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 8336}, "pyerrors.correlators": {"qualname": 0, "fullname": 2, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 3}, "pyerrors.correlators.Corr": {"qualname": 1, "fullname": 3, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 367}, "pyerrors.correlators.Corr.__init__": {"qualname": 3, "fullname": 5, "annotation": 0, "default_value": 0, "signature": 40, "bases": 0, "doc": 100}, "pyerrors.correlators.Corr.tag": {"qualname": 2, "fullname": 4, "annotation": 0, "default_value": 0, "signature": 0, "bases": 0, "doc": 3}, "pyerrors.correlators.Corr.content": {"qualname": 2, 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