diff --git a/docs/pyerrors/correlators.html b/docs/pyerrors/correlators.html index afc027f7..a6676e25 100644 --- a/docs/pyerrors/correlators.html +++ b/docs/pyerrors/correlators.html @@ -950,8 +950,8 @@ 734 if len(fitrange) != 2: 735 raise Exception("fitrange has to have exactly two elements [fit_start, fit_stop]") 736 - 737 xs = [x for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None] - 738 ys = [self.content[x][0] for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None] + 737 xs = np.array([x for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]) + 738 ys = np.array([self.content[x][0] for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]) 739 result = least_squares(xs, ys, function, silent=silent, **kwargs) 740 return result 741 @@ -2272,8 +2272,8 @@ 735 if len(fitrange) != 2: 736 raise Exception("fitrange has to have exactly two elements [fit_start, fit_stop]") 737 - 738 xs = [x for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None] - 739 ys = [self.content[x][0] for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None] + 738 xs = np.array([x for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]) + 739 ys = np.array([self.content[x][0] for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]) 740 result = least_squares(xs, ys, function, silent=silent, **kwargs) 741 return result 742 @@ -4095,8 +4095,8 @@ guess for the root finder, only relevant for the root variant 735 if len(fitrange) != 2: 736 raise Exception("fitrange has to have exactly two elements [fit_start, fit_stop]") 737 -738 xs = [x for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None] -739 ys = [self.content[x][0] for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None] +738 xs = np.array([x for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]) +739 ys = np.array([self.content[x][0] for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]) 740 result = least_squares(xs, ys, function, silent=silent, **kwargs) 741 return result diff --git a/docs/pyerrors/fits.html b/docs/pyerrors/fits.html index 5e95633d..c2d1aba5 100644 --- a/docs/pyerrors/fits.html +++ b/docs/pyerrors/fits.html @@ -106,1094 +106,919 @@ - 
   1import gc
-   2from collections.abc import Sequence
-   3import warnings
-   4import numpy as np
-   5import autograd.numpy as anp
-   6import scipy.optimize
-   7import scipy.stats
-   8import matplotlib.pyplot as plt
-   9from matplotlib import gridspec
-  10from scipy.odr import ODR, Model, RealData
-  11import iminuit
-  12from autograd import jacobian as auto_jacobian
-  13from autograd import hessian as auto_hessian
-  14from autograd import elementwise_grad as egrad
-  15from numdifftools import Jacobian as num_jacobian
-  16from numdifftools import Hessian as num_hessian
-  17from .obs import Obs, derived_observable, covariance, cov_Obs
-  18
-  19
-  20class Fit_result(Sequence):
-  21    """Represents fit results.
-  22
-  23    Attributes
-  24    ----------
-  25    fit_parameters : list
-  26        results for the individual fit parameters,
-  27        also accessible via indices.
-  28    chisquare_by_dof : float
-  29        reduced chisquare.
-  30    p_value : float
-  31        p-value of the fit
-  32    t2_p_value : float
-  33        Hotelling t-squared p-value for correlated fits.
-  34    """
-  35
-  36    def __init__(self):
-  37        self.fit_parameters = None
-  38
-  39    def __getitem__(self, idx):
-  40        return self.fit_parameters[idx]
-  41
-  42    def __len__(self):
-  43        return len(self.fit_parameters)
-  44
-  45    def gamma_method(self, **kwargs):
-  46        """Apply the gamma method to all fit parameters"""
-  47        [o.gamma_method(**kwargs) for o in self.fit_parameters]
-  48
-  49    gm = gamma_method
-  50
-  51    def __str__(self):
-  52        my_str = 'Goodness of fit:\n'
-  53        if hasattr(self, 'chisquare_by_dof'):
-  54            my_str += '\u03C7\u00b2/d.o.f. = ' + f'{self.chisquare_by_dof:2.6f}' + '\n'
-  55        elif hasattr(self, 'residual_variance'):
-  56            my_str += 'residual variance = ' + f'{self.residual_variance:2.6f}' + '\n'
-  57        if hasattr(self, 'chisquare_by_expected_chisquare'):
-  58            my_str += '\u03C7\u00b2/\u03C7\u00b2exp  = ' + f'{self.chisquare_by_expected_chisquare:2.6f}' + '\n'
-  59        if hasattr(self, 'p_value'):
-  60            my_str += 'p-value   = ' + f'{self.p_value:2.4f}' + '\n'
-  61        if hasattr(self, 't2_p_value'):
-  62            my_str += 't\u00B2p-value = ' + f'{self.t2_p_value:2.4f}' + '\n'
-  63        my_str += 'Fit parameters:\n'
-  64        for i_par, par in enumerate(self.fit_parameters):
-  65            my_str += str(i_par) + '\t' + ' ' * int(par >= 0) + str(par).rjust(int(par < 0.0)) + '\n'
-  66        return my_str
-  67
-  68    def __repr__(self):
-  69        m = max(map(len, list(self.__dict__.keys()))) + 1
-  70        return '\n'.join([key.rjust(m) + ': ' + repr(value) for key, value in sorted(self.__dict__.items())])
-  71
-  72
-  73def least_squares(x, y, func, priors=None, silent=False, **kwargs):
-  74    r'''Performs a non-linear fit to y = func(x).
-  75        ```
-  76
-  77    Parameters
-  78    ----------
-  79    For an uncombined fit:
-  80
-  81    x : list
-  82        list of floats.
-  83    y : list
-  84        list of Obs.
-  85    func : object
-  86        fit function, has to be of the form
-  87
-  88        ```python
-  89        import autograd.numpy as anp
-  90
-  91        def func(a, x):
-  92            return a[0] + a[1] * x + a[2] * anp.sinh(x)
-  93        ```
-  94
-  95        For multiple x values func can be of the form
-  96
-  97        ```python
-  98        def func(a, x):
-  99            (x1, x2) = x
- 100            return a[0] * x1 ** 2 + a[1] * x2
- 101        ```
- 102        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
- 103        will not work.
- 104
- 105    OR For a combined fit:
- 106
- 107    x : dict
- 108        dict of lists.
- 109    y : dict
- 110        dict of lists of Obs.
- 111    funcs : dict
- 112        dict of objects
- 113        fit functions have to be of the form (here a[0] is the common fit parameter)
- 114        ```python
- 115        import autograd.numpy as anp
- 116        funcs = {"a": func_a,
- 117                "b": func_b}
- 118
- 119        def func_a(a, x):
- 120            return a[1] * anp.exp(-a[0] * x)
- 121
- 122        def func_b(a, x):
- 123            return a[2] * anp.exp(-a[0] * x)
- 124
- 125        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
- 126        will not work.
- 127
- 128    priors : list, optional
- 129        priors has to be a list with an entry for every parameter in the fit. The entries can either be
- 130        Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like
- 131        0.548(23), 500(40) or 0.5(0.4)
- 132    silent : bool, optional
- 133        If true all output to the console is omitted (default False).
- 134    initial_guess : list
- 135        can provide an initial guess for the input parameters. Relevant for
- 136        non-linear fits with many parameters. In case of correlated fits the guess is used to perform
- 137        an uncorrelated fit which then serves as guess for the correlated fit.
- 138    method : str, optional
- 139        can be used to choose an alternative method for the minimization of chisquare.
- 140        The possible methods are the ones which can be used for scipy.optimize.minimize and
- 141        migrad of iminuit. If no method is specified, Levenberg-Marquard is used.
- 142        Reliable alternatives are migrad, Powell and Nelder-Mead.
- 143    tol: float, optional
- 144        can be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence
- 145        to a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly
- 146        invalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values
- 147        The stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
- 148    correlated_fit : bool
- 149        If True, use the full inverse covariance matrix in the definition of the chisquare cost function.
- 150        For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`.
- 151        In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).
- 152        This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
- 153        At the moment this option only works for `prior==None` and when no `method` is given.
- 154    expected_chisquare : bool
- 155        If True estimates the expected chisquare which is
- 156        corrected by effects caused by correlated input data (default False).
- 157    resplot : bool
- 158        If True, a plot which displays fit, data and residuals is generated (default False).
- 159    qqplot : bool
- 160        If True, a quantile-quantile plot of the fit result is generated (default False).
- 161    num_grad : bool
- 162        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
- 163
- 164    Returns
- 165    -------
- 166    output : Fit_result
- 167        Parameters and information on the fitted result.
- 168    '''
- 169    if priors is not None:
- 170        return _prior_fit(x, y, func, priors, silent=silent, **kwargs)
- 171
- 172    elif (type(x) == dict and type(y) == dict and type(func) == dict):
- 173        return _combined_fit(x, y, func, silent=silent, **kwargs)
- 174    elif (type(x) == dict or type(y) == dict or type(func) == dict):
- 175        raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.")
- 176    else:
- 177        return _standard_fit(x, y, func, silent=silent, **kwargs)
- 178
- 179
- 180def total_least_squares(x, y, func, silent=False, **kwargs):
- 181    r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
- 182
- 183    Parameters
- 184    ----------
- 185    x : list
- 186        list of Obs, or a tuple of lists of Obs
- 187    y : list
- 188        list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
- 189    func : object
- 190        func has to be of the form
- 191
- 192        ```python
- 193        import autograd.numpy as anp
- 194
- 195        def func(a, x):
- 196            return a[0] + a[1] * x + a[2] * anp.sinh(x)
- 197        ```
- 198
- 199        For multiple x values func can be of the form
- 200
- 201        ```python
- 202        def func(a, x):
- 203            (x1, x2) = x
- 204            return a[0] * x1 ** 2 + a[1] * x2
- 205        ```
- 206
- 207        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
- 208        will not work.
- 209    silent : bool, optional
- 210        If true all output to the console is omitted (default False).
- 211    initial_guess : list
- 212        can provide an initial guess for the input parameters. Relevant for non-linear
- 213        fits with many parameters.
- 214    expected_chisquare : bool
- 215        If true prints the expected chisquare which is
- 216        corrected by effects caused by correlated input data.
- 217        This can take a while as the full correlation matrix
- 218        has to be calculated (default False).
- 219    num_grad : bool
- 220        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
- 221
- 222    Notes
- 223    -----
- 224    Based on the orthogonal distance regression module of scipy.
- 225
- 226    Returns
- 227    -------
- 228    output : Fit_result
- 229        Parameters and information on the fitted result.
- 230    '''
- 231
- 232    output = Fit_result()
- 233
- 234    output.fit_function = func
- 235
- 236    x = np.array(x)
- 237
- 238    x_shape = x.shape
- 239
- 240    if kwargs.get('num_grad') is True:
- 241        jacobian = num_jacobian
- 242        hessian = num_hessian
- 243    else:
- 244        jacobian = auto_jacobian
- 245        hessian = auto_hessian
- 246
- 247    if not callable(func):
- 248        raise TypeError('func has to be a function.')
- 249
- 250    for i in range(42):
- 251        try:
- 252            func(np.arange(i), x.T[0])
- 253        except TypeError:
- 254            continue
- 255        except IndexError:
- 256            continue
- 257        else:
- 258            break
- 259    else:
- 260        raise RuntimeError("Fit function is not valid.")
- 261
- 262    n_parms = i
- 263    if not silent:
- 264        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
- 265
- 266    x_f = np.vectorize(lambda o: o.value)(x)
- 267    dx_f = np.vectorize(lambda o: o.dvalue)(x)
- 268    y_f = np.array([o.value for o in y])
- 269    dy_f = np.array([o.dvalue for o in y])
- 270
- 271    if np.any(np.asarray(dx_f) <= 0.0):
- 272        raise Exception('No x errors available, run the gamma method first.')
- 273
- 274    if np.any(np.asarray(dy_f) <= 0.0):
- 275        raise Exception('No y errors available, run the gamma method first.')
- 276
- 277    if 'initial_guess' in kwargs:
- 278        x0 = kwargs.get('initial_guess')
- 279        if len(x0) != n_parms:
- 280            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
- 281    else:
- 282        x0 = [1] * n_parms
- 283
- 284    data = RealData(x_f, y_f, sx=dx_f, sy=dy_f)
- 285    model = Model(func)
- 286    odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps)
- 287    odr.set_job(fit_type=0, deriv=1)
- 288    out = odr.run()
- 289
- 290    output.residual_variance = out.res_var
- 291
- 292    output.method = 'ODR'
- 293
- 294    output.message = out.stopreason
- 295
- 296    output.xplus = out.xplus
- 297
- 298    if not silent:
- 299        print('Method: ODR')
- 300        print(*out.stopreason)
- 301        print('Residual variance:', output.residual_variance)
- 302
- 303    if out.info > 3:
- 304        raise Exception('The minimization procedure did not converge.')
- 305
- 306    m = x_f.size
- 307
- 308    def odr_chisquare(p):
- 309        model = func(p[:n_parms], p[n_parms:].reshape(x_shape))
- 310        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2)
- 311        return chisq
- 312
- 313    if kwargs.get('expected_chisquare') is True:
- 314        W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel()))))
- 315
- 316        if kwargs.get('covariance') is not None:
- 317            cov = kwargs.get('covariance')
- 318        else:
- 319            cov = covariance(np.concatenate((y, x.ravel())))
- 320
- 321        number_of_x_parameters = int(m / x_f.shape[-1])
- 322
- 323        old_jac = jacobian(func)(out.beta, out.xplus)
- 324        fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0)))
- 325        fused_row2 = np.concatenate((jacobian(lambda x, y: func(y, x))(out.xplus, out.beta).reshape(x_f.shape[-1], x_f.shape[-1] * number_of_x_parameters), np.identity(number_of_x_parameters * old_jac.shape[0])))
- 326        new_jac = np.concatenate((fused_row1, fused_row2), axis=1)
- 327
- 328        A = W @ new_jac
- 329        P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
- 330        expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W)
- 331        if expected_chisquare <= 0.0:
- 332            warnings.warn("Negative expected_chisquare.", RuntimeWarning)
- 333            expected_chisquare = np.abs(expected_chisquare)
- 334        output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare
- 335        if not silent:
- 336            print('chisquare/expected_chisquare:',
- 337                  output.chisquare_by_expected_chisquare)
- 338
- 339    fitp = out.beta
- 340    try:
- 341        hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel())))
- 342    except TypeError:
- 343        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
- 344
- 345    def odr_chisquare_compact_x(d):
- 346        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
- 347        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((d[n_parms + m:].reshape(x_shape) - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2)
- 348        return chisq
- 349
- 350    jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel())))
- 351
- 352    # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv
- 353    try:
- 354        deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:])
- 355    except np.linalg.LinAlgError:
- 356        raise Exception("Cannot invert hessian matrix.")
- 357
- 358    def odr_chisquare_compact_y(d):
- 359        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
- 360        chisq = anp.sum(((d[n_parms + m:] - model) / dy_f) ** 2) + anp.sum(((x_f - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2)
- 361        return chisq
- 362
- 363    jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f)))
- 364
- 365    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
- 366    try:
- 367        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:])
- 368    except np.linalg.LinAlgError:
- 369        raise Exception("Cannot invert hessian matrix.")
- 370
- 371    result = []
- 372    for i in range(n_parms):
- 373        result.append(derived_observable(lambda my_var, **kwargs: (my_var[0] + np.finfo(np.float64).eps) / (x.ravel()[0].value + np.finfo(np.float64).eps) * out.beta[i], list(x.ravel()) + list(y), man_grad=list(deriv_x[i]) + list(deriv_y[i])))
- 374
- 375    output.fit_parameters = result
- 376
- 377    output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel())))
- 378    output.dof = x.shape[-1] - n_parms
- 379    output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof)
- 380
- 381    return output
- 382
- 383
- 384def _prior_fit(x, y, func, priors, silent=False, **kwargs):
- 385    output = Fit_result()
- 386
- 387    output.fit_function = func
- 388
- 389    x = np.asarray(x)
- 390
- 391    if kwargs.get('num_grad') is True:
- 392        hessian = num_hessian
- 393    else:
- 394        hessian = auto_hessian
- 395
- 396    if not callable(func):
- 397        raise TypeError('func has to be a function.')
- 398
- 399    for i in range(100):
- 400        try:
- 401            func(np.arange(i), 0)
- 402        except TypeError:
- 403            continue
- 404        except IndexError:
- 405            continue
- 406        else:
- 407            break
- 408    else:
- 409        raise RuntimeError("Fit function is not valid.")
- 410
- 411    n_parms = i
- 412
- 413    if n_parms != len(priors):
- 414        raise Exception('Priors does not have the correct length.')
- 415
- 416    def extract_val_and_dval(string):
- 417        split_string = string.split('(')
- 418        if '.' in split_string[0] and '.' not in split_string[1][:-1]:
- 419            factor = 10 ** -len(split_string[0].partition('.')[2])
- 420        else:
- 421            factor = 1
- 422        return float(split_string[0]), float(split_string[1][:-1]) * factor
- 423
- 424    loc_priors = []
- 425    for i_n, i_prior in enumerate(priors):
- 426        if isinstance(i_prior, Obs):
- 427            loc_priors.append(i_prior)
- 428        else:
- 429            loc_val, loc_dval = extract_val_and_dval(i_prior)
- 430            loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}"))
- 431
- 432    output.priors = loc_priors
- 433
- 434    if not silent:
- 435        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
- 436
- 437    y_f = [o.value for o in y]
- 438    dy_f = [o.dvalue for o in y]
- 439
- 440    if np.any(np.asarray(dy_f) <= 0.0):
- 441        raise Exception('No y errors available, run the gamma method first.')
- 442
- 443    p_f = [o.value for o in loc_priors]
- 444    dp_f = [o.dvalue for o in loc_priors]
- 445
- 446    if np.any(np.asarray(dp_f) <= 0.0):
- 447        raise Exception('No prior errors available, run the gamma method first.')
- 448
- 449    if 'initial_guess' in kwargs:
- 450        x0 = kwargs.get('initial_guess')
- 451        if len(x0) != n_parms:
- 452            raise Exception('Initial guess does not have the correct length.')
- 453    else:
- 454        x0 = p_f
- 455
- 456    def chisqfunc(p):
- 457        model = func(p, x)
- 458        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((p_f - p) / dp_f) ** 2)
- 459        return chisq
- 460
- 461    if not silent:
- 462        print('Method: migrad')
- 463
- 464    m = iminuit.Minuit(chisqfunc, x0)
- 465    m.errordef = 1
- 466    m.print_level = 0
- 467    if 'tol' in kwargs:
- 468        m.tol = kwargs.get('tol')
- 469    else:
- 470        m.tol = 1e-4
- 471    m.migrad()
- 472    params = np.asarray(m.values)
- 473
- 474    output.chisquare_by_dof = m.fval / len(x)
- 475
- 476    output.method = 'migrad'
- 477
- 478    if not silent:
- 479        print('chisquare/d.o.f.:', output.chisquare_by_dof)
- 480
- 481    if not m.fmin.is_valid:
- 482        raise Exception('The minimization procedure did not converge.')
- 483
- 484    hess = hessian(chisqfunc)(params)
- 485    hess_inv = np.linalg.pinv(hess)
- 486
- 487    def chisqfunc_compact(d):
- 488        model = func(d[:n_parms], x)
- 489        chisq = anp.sum(((d[n_parms: n_parms + len(x)] - model) / dy_f) ** 2) + anp.sum(((d[n_parms + len(x):] - d[:n_parms]) / dp_f) ** 2)
- 490        return chisq
- 491
- 492    jac_jac = hessian(chisqfunc_compact)(np.concatenate((params, y_f, p_f)))
- 493
- 494    deriv = -hess_inv @ jac_jac[:n_parms, n_parms:]
- 495
- 496    result = []
- 497    for i in range(n_parms):
- 498        result.append(derived_observable(lambda x, **kwargs: (x[0] + np.finfo(np.float64).eps) / (y[0].value + np.finfo(np.float64).eps) * params[i], list(y) + list(loc_priors), man_grad=list(deriv[i])))
- 499
- 500    output.fit_parameters = result
- 501    output.chisquare = chisqfunc(np.asarray(params))
- 502
- 503    if kwargs.get('resplot') is True:
- 504        residual_plot(x, y, func, result)
- 505
- 506    if kwargs.get('qqplot') is True:
- 507        qqplot(x, y, func, result)
- 508
- 509    return output
- 510
- 511
- 512def _standard_fit(x, y, func, silent=False, **kwargs):
- 513    output = Fit_result()
- 514
- 515    output.fit_function = func
- 516
- 517    x = np.asarray(x)
- 518
- 519    if kwargs.get('num_grad') is True:
- 520        jacobian = num_jacobian
- 521        hessian = num_hessian
- 522    else:
- 523        jacobian = auto_jacobian
- 524        hessian = auto_hessian
- 525
- 526    if x.shape[-1] != len(y):
- 527        raise Exception('x and y input have to have the same length')
- 528
- 529    if len(x.shape) > 2:
- 530        raise Exception('Unknown format for x values')
- 531
- 532    if not callable(func):
- 533        raise TypeError('func has to be a function.')
- 534
- 535    for i in range(42):
- 536        try:
- 537            func(np.arange(i), x.T[0])
- 538        except TypeError:
- 539            continue
- 540        except IndexError:
- 541            continue
- 542        else:
- 543            break
- 544    else:
- 545        raise RuntimeError("Fit function is not valid.")
- 546
- 547    n_parms = i
- 548
- 549    if not silent:
- 550        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
- 551
- 552    y_f = [o.value for o in y]
- 553    dy_f = [o.dvalue for o in y]
- 554
- 555    if np.any(np.asarray(dy_f) <= 0.0):
- 556        raise Exception('No y errors available, run the gamma method first.')
- 557
- 558    if 'initial_guess' in kwargs:
- 559        x0 = kwargs.get('initial_guess')
- 560        if len(x0) != n_parms:
- 561            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
- 562    else:
- 563        x0 = [0.1] * n_parms
- 564
- 565    if kwargs.get('correlated_fit') is True:
- 566        corr = covariance(y, correlation=True, **kwargs)
- 567        covdiag = np.diag(1 / np.asarray(dy_f))
- 568        condn = np.linalg.cond(corr)
- 569        if condn > 0.1 / np.finfo(float).eps:
- 570            raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})")
- 571        if condn > 1e13:
- 572            warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning)
- 573        chol = np.linalg.cholesky(corr)
- 574        chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True)
- 575
- 576        def chisqfunc_corr(p):
- 577            model = func(p, x)
- 578            chisq = anp.sum(anp.dot(chol_inv, (y_f - model)) ** 2)
- 579            return chisq
- 580
- 581    def chisqfunc(p):
- 582        model = func(p, x)
- 583        chisq = anp.sum(((y_f - model) / dy_f) ** 2)
- 584        return chisq
- 585
- 586    output.method = kwargs.get('method', 'Levenberg-Marquardt')
- 587    if not silent:
- 588        print('Method:', output.method)
- 589
- 590    if output.method != 'Levenberg-Marquardt':
- 591        if output.method == 'migrad':
- 592            fit_result = iminuit.minimize(chisqfunc, x0, tol=1e-4)  # Stopping criterion 0.002 * tol * errordef
- 593            if kwargs.get('correlated_fit') is True:
- 594                fit_result = iminuit.minimize(chisqfunc_corr, fit_result.x, tol=1e-4)  # Stopping criterion 0.002 * tol * errordef
- 595            output.iterations = fit_result.nfev
- 596        else:
- 597            fit_result = scipy.optimize.minimize(chisqfunc, x0, method=kwargs.get('method'), tol=1e-12)
- 598            if kwargs.get('correlated_fit') is True:
- 599                fit_result = scipy.optimize.minimize(chisqfunc_corr, fit_result.x, method=kwargs.get('method'), tol=1e-12)
- 600            output.iterations = fit_result.nit
- 601
- 602        chisquare = fit_result.fun
- 603
- 604    else:
- 605        if kwargs.get('correlated_fit') is True:
- 606            def chisqfunc_residuals_corr(p):
- 607                model = func(p, x)
- 608                chisq = anp.dot(chol_inv, (y_f - model))
- 609                return chisq
- 610
- 611        def chisqfunc_residuals(p):
- 612            model = func(p, x)
- 613            chisq = ((y_f - model) / dy_f)
- 614            return chisq
- 615
- 616        fit_result = scipy.optimize.least_squares(chisqfunc_residuals, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
- 617        if kwargs.get('correlated_fit') is True:
- 618            fit_result = scipy.optimize.least_squares(chisqfunc_residuals_corr, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
- 619
- 620        chisquare = np.sum(fit_result.fun ** 2)
- 621        if kwargs.get('correlated_fit') is True:
- 622            assert np.isclose(chisquare, chisqfunc_corr(fit_result.x), atol=1e-14)
- 623        else:
- 624            assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14)
- 625
- 626        output.iterations = fit_result.nfev
- 627
- 628    if not fit_result.success:
- 629        raise Exception('The minimization procedure did not converge.')
- 630
- 631    if x.shape[-1] - n_parms > 0:
- 632        output.chisquare_by_dof = chisquare / (x.shape[-1] - n_parms)
- 633    else:
- 634        output.chisquare_by_dof = float('nan')
- 635
- 636    output.message = fit_result.message
- 637    if not silent:
- 638        print(fit_result.message)
- 639        print('chisquare/d.o.f.:', output.chisquare_by_dof)
- 640
- 641    if kwargs.get('expected_chisquare') is True:
- 642        if kwargs.get('correlated_fit') is not True:
- 643            W = np.diag(1 / np.asarray(dy_f))
- 644            cov = covariance(y)
- 645            A = W @ jacobian(func)(fit_result.x, x)
- 646            P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
- 647            expected_chisquare = np.trace((np.identity(x.shape[-1]) - P_phi) @ W @ cov @ W)
- 648            output.chisquare_by_expected_chisquare = chisquare / expected_chisquare
- 649            if not silent:
- 650                print('chisquare/expected_chisquare:',
- 651                      output.chisquare_by_expected_chisquare)
- 652
- 653    fitp = fit_result.x
- 654    try:
- 655        if kwargs.get('correlated_fit') is True:
- 656            hess = hessian(chisqfunc_corr)(fitp)
- 657        else:
- 658            hess = hessian(chisqfunc)(fitp)
- 659    except TypeError:
- 660        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
- 661
- 662    if kwargs.get('correlated_fit') is True:
- 663        def chisqfunc_compact(d):
- 664            model = func(d[:n_parms], x)
- 665            chisq = anp.sum(anp.dot(chol_inv, (d[n_parms:] - model)) ** 2)
- 666            return chisq
- 667
- 668    else:
- 669        def chisqfunc_compact(d):
- 670            model = func(d[:n_parms], x)
- 671            chisq = anp.sum(((d[n_parms:] - model) / dy_f) ** 2)
- 672            return chisq
- 673
- 674    jac_jac = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f)))
- 675
- 676    # Compute hess^{-1} @ jac_jac[:n_parms, n_parms:] using LAPACK dgesv
- 677    try:
- 678        deriv = -scipy.linalg.solve(hess, jac_jac[:n_parms, n_parms:])
- 679    except np.linalg.LinAlgError:
- 680        raise Exception("Cannot invert hessian matrix.")
- 681
- 682    result = []
- 683    for i in range(n_parms):
- 684        result.append(derived_observable(lambda x, **kwargs: (x[0] + np.finfo(np.float64).eps) / (y[0].value + np.finfo(np.float64).eps) * fit_result.x[i], list(y), man_grad=list(deriv[i])))
- 685
- 686    output.fit_parameters = result
- 687
- 688    output.chisquare = chisquare
- 689    output.dof = x.shape[-1] - n_parms
- 690    output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof)
- 691    # Hotelling t-squared p-value for correlated fits.
- 692    if kwargs.get('correlated_fit') is True:
- 693        n_cov = np.min(np.vectorize(lambda x: x.N)(y))
- 694        output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare,
- 695                                                  output.dof, n_cov - output.dof)
- 696
- 697    if kwargs.get('resplot') is True:
- 698        residual_plot(x, y, func, result)
- 699
- 700    if kwargs.get('qqplot') is True:
- 701        qqplot(x, y, func, result)
- 702
- 703    return output
- 704
- 705
- 706def _combined_fit(x, y, func, silent=False, **kwargs):
- 707
- 708    output = Fit_result()
- 709    output.fit_function = func
- 710
- 711    if kwargs.get('num_grad') is True:
- 712        jacobian = num_jacobian
- 713        hessian = num_hessian
- 714    else:
- 715        jacobian = auto_jacobian
- 716        hessian = auto_hessian
- 717
- 718    key_ls = sorted(list(x.keys()))
- 719
- 720    if sorted(list(y.keys())) != key_ls:
- 721        raise Exception('x and y dictionaries do not contain the same keys.')
- 722
- 723    if sorted(list(func.keys())) != key_ls:
- 724        raise Exception('x and func dictionaries do not contain the same keys.')
- 725
- 726    x_all = np.concatenate([np.array(x[key]) for key in key_ls])
- 727    y_all = np.concatenate([np.array(y[key]) for key in key_ls])
- 728
- 729    y_f = [o.value for o in y_all]
- 730    dy_f = [o.dvalue for o in y_all]
- 731
- 732    if len(x_all.shape) > 2:
- 733        raise Exception('Unknown format for x values')
- 734
- 735    if np.any(np.asarray(dy_f) <= 0.0):
- 736        raise Exception('No y errors available, run the gamma method first.')
- 737
- 738    # number of fit parameters
- 739    n_parms_ls = []
- 740    for key in key_ls:
- 741        if not callable(func[key]):
- 742            raise TypeError('func (key=' + key + ') is not a function.')
- 743        if len(x[key]) != len(y[key]):
- 744            raise Exception('x and y input (key=' + key + ') do not have the same length')
- 745        for i in range(100):
- 746            try:
- 747                func[key](np.arange(i), x_all.T[0])
- 748            except TypeError:
- 749                continue
- 750            except IndexError:
- 751                continue
- 752            else:
- 753                break
- 754        else:
- 755            raise RuntimeError("Fit function (key=" + key + ") is not valid.")
- 756        n_parms = i
- 757        n_parms_ls.append(n_parms)
- 758    n_parms = max(n_parms_ls)
- 759    if not silent:
- 760        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
- 761
- 762    if 'initial_guess' in kwargs:
- 763        x0 = kwargs.get('initial_guess')
- 764        if len(x0) != n_parms:
- 765            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
- 766    else:
- 767        x0 = [0.1] * n_parms
- 768
- 769    if kwargs.get('correlated_fit') is True:
- 770        corr = covariance(y_all, correlation=True, **kwargs)
- 771        covdiag = np.diag(1 / np.asarray(dy_f))
- 772        condn = np.linalg.cond(corr)
- 773        if condn > 0.1 / np.finfo(float).eps:
- 774            raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})")
- 775        if condn > 1e13:
- 776            warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning)
- 777        chol = np.linalg.cholesky(corr)
- 778        chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True)
- 779
- 780        def chisqfunc_corr(p):
- 781            model = np.concatenate([np.array(func[key](p, np.asarray(x[key]))) for key in key_ls])
- 782            chisq = anp.sum(anp.dot(chol_inv, (y_f - model)) ** 2)
- 783            return chisq
- 784
- 785    def chisqfunc(p):
- 786        func_list = np.concatenate([[func[k]] * len(x[k]) for k in key_ls])
- 787        model = anp.array([func_list[i](p, x_all[i]) for i in range(len(x_all))])
- 788        chisq = anp.sum(((y_f - model) / dy_f) ** 2)
- 789        return chisq
- 790
- 791    output.method = kwargs.get('method', 'Levenberg-Marquardt')
- 792    if not silent:
- 793        print('Method:', output.method)
- 794
- 795    if output.method != 'Levenberg-Marquardt':
- 796        if output.method == 'migrad':
- 797            tolerance = 1e-4  # default value of 1e-1 set by iminuit can be problematic
- 798            if 'tol' in kwargs:
- 799                tolerance = kwargs.get('tol')
- 800            fit_result = iminuit.minimize(chisqfunc, x0, tol=tolerance)  # Stopping criterion 0.002 * tol * errordef
- 801            if kwargs.get('correlated_fit') is True:
- 802                fit_result = iminuit.minimize(chisqfunc_corr, fit_result.x, tol=tolerance)
- 803            output.iterations = fit_result.nfev
- 804        else:
- 805            tolerance = 1e-12
- 806            if 'tol' in kwargs:
- 807                tolerance = kwargs.get('tol')
- 808            fit_result = scipy.optimize.minimize(chisqfunc, x0, method=kwargs.get('method'), tol=tolerance)
- 809            if kwargs.get('correlated_fit') is True:
- 810                fit_result = scipy.optimize.minimize(chisqfunc_corr, fit_result.x, method=kwargs.get('method'), tol=tolerance)
- 811            output.iterations = fit_result.nit
- 812
- 813        chisquare = fit_result.fun
- 814
- 815    else:
- 816        if kwargs.get('correlated_fit') is True:
- 817            def chisqfunc_residuals_corr(p):
- 818                model = np.concatenate([np.array(func[key](p, np.asarray(x[key]))) for key in key_ls])
- 819                chisq = anp.dot(chol_inv, (y_f - model))
- 820                return chisq
- 821
- 822        def chisqfunc_residuals(p):
- 823            model = np.concatenate([np.array(func[key](p, np.asarray(x[key]))) for key in key_ls])
- 824            chisq = ((y_f - model) / dy_f)
- 825            return chisq
- 826
- 827        if 'tol' in kwargs:
- 828            print('tol cannot be set for Levenberg-Marquardt')
- 829
- 830        fit_result = scipy.optimize.least_squares(chisqfunc_residuals, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
- 831        if kwargs.get('correlated_fit') is True:
- 832            fit_result = scipy.optimize.least_squares(chisqfunc_residuals_corr, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
- 833
- 834        chisquare = np.sum(fit_result.fun ** 2)
- 835        if kwargs.get('correlated_fit') is True:
- 836            assert np.isclose(chisquare, chisqfunc_corr(fit_result.x), atol=1e-14)
- 837        else:
- 838            assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14)
- 839
- 840        output.iterations = fit_result.nfev
- 841
- 842    if not fit_result.success:
- 843        raise Exception('The minimization procedure did not converge.')
- 844
- 845    if x_all.shape[-1] - n_parms > 0:
- 846        output.chisquare = chisquare
- 847        output.dof = x_all.shape[-1] - n_parms
- 848        output.chisquare_by_dof = output.chisquare / output.dof
- 849        output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof)
- 850    else:
- 851        output.chisquare_by_dof = float('nan')
- 852
- 853    output.message = fit_result.message
- 854    if not silent:
- 855        print(fit_result.message)
- 856        print('chisquare/d.o.f.:', output.chisquare_by_dof)
- 857        print('fit parameters', fit_result.x)
- 858
- 859    def prepare_hat_matrix():
- 860        hat_vector = []
- 861        for key in key_ls:
- 862            x_array = np.asarray(x[key])
- 863            if (len(x_array) != 0):
- 864                hat_vector.append(jacobian(func[key])(fit_result.x, x_array))
- 865        hat_vector = [item for sublist in hat_vector for item in sublist]
- 866        return hat_vector
- 867
- 868    if kwargs.get('expected_chisquare') is True:
- 869        if kwargs.get('correlated_fit') is not True:
- 870            W = np.diag(1 / np.asarray(dy_f))
- 871            cov = covariance(y_all)
- 872            hat_vector = prepare_hat_matrix()
- 873            A = W @ hat_vector  # hat_vector = 'jacobian(func)(fit_result.x, x)'
- 874            P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
- 875            expected_chisquare = np.trace((np.identity(x_all.shape[-1]) - P_phi) @ W @ cov @ W)
- 876            output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare
- 877            if not silent:
- 878                print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare)
- 879
- 880    fitp = fit_result.x
- 881    if np.any(np.asarray(dy_f) <= 0.0):
- 882        raise Exception('No y errors available, run the gamma method first.')
- 883
- 884    try:
- 885        if kwargs.get('correlated_fit') is True:
- 886            hess = hessian(chisqfunc_corr)(fitp)
- 887        else:
- 888            hess = hessian(chisqfunc)(fitp)
- 889    except TypeError:
- 890        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
- 891
- 892    if kwargs.get('correlated_fit') is True:
- 893        def chisqfunc_compact(d):
- 894            func_list = np.concatenate([[func[k]] * len(x[k]) for k in key_ls])
- 895            model = anp.array([func_list[i](d[:n_parms], x_all[i]) for i in range(len(x_all))])
- 896            chisq = anp.sum(anp.dot(chol_inv, (d[n_parms:] - model)) ** 2)
- 897            return chisq
- 898    else:
- 899        def chisqfunc_compact(d):
- 900            func_list = np.concatenate([[func[k]] * len(x[k]) for k in key_ls])
- 901            model = anp.array([func_list[i](d[:n_parms], x_all[i]) for i in range(len(x_all))])
- 902            chisq = anp.sum(((d[n_parms:] - model) / dy_f) ** 2)
- 903            return chisq
- 904
- 905    jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f)))
- 906
- 907    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
- 908    try:
- 909        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:])
- 910    except np.linalg.LinAlgError:
- 911        raise Exception("Cannot invert hessian matrix.")
- 912
- 913    result = []
- 914    for i in range(n_parms):
- 915        result.append(derived_observable(lambda x_all, **kwargs: (x_all[0] + np.finfo(np.float64).eps) / (y_all[0].value + np.finfo(np.float64).eps) * fitp[i], list(y_all), man_grad=list(deriv_y[i])))
- 916
- 917    output.fit_parameters = result
- 918
- 919    if kwargs.get('correlated_fit') is True:
- 920        n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all))
- 921        output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare,
- 922                                                  output.dof, n_cov - output.dof)
- 923
- 924    return output
- 925
- 926
- 927def fit_lin(x, y, **kwargs):
- 928    """Performs a linear fit to y = n + m * x and returns two Obs n, m.
- 929
- 930    Parameters
- 931    ----------
- 932    x : list
- 933        Can either be a list of floats in which case no xerror is assumed, or
- 934        a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
- 935    y : list
- 936        List of Obs, the dvalues of the Obs are used as yerror for the fit.
- 937
- 938    Returns
- 939    -------
- 940    fit_parameters : list[Obs]
- 941        LIist of fitted observables.
- 942    """
- 943
- 944    def f(a, x):
- 945        y = a[0] + a[1] * x
- 946        return y
- 947
- 948    if all(isinstance(n, Obs) for n in x):
- 949        out = total_least_squares(x, y, f, **kwargs)
- 950        return out.fit_parameters
- 951    elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
- 952        out = least_squares(x, y, f, **kwargs)
- 953        return out.fit_parameters
- 954    else:
- 955        raise Exception('Unsupported types for x')
- 956
- 957
- 958def qqplot(x, o_y, func, p):
- 959    """Generates a quantile-quantile plot of the fit result which can be used to
- 960       check if the residuals of the fit are gaussian distributed.
- 961
- 962    Returns
- 963    -------
- 964    None
- 965    """
- 966
- 967    residuals = []
- 968    for i_x, i_y in zip(x, o_y):
- 969        residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
- 970    residuals = sorted(residuals)
- 971    my_y = [o.value for o in residuals]
- 972    probplot = scipy.stats.probplot(my_y)
- 973    my_x = probplot[0][0]
- 974    plt.figure(figsize=(8, 8 / 1.618))
- 975    plt.errorbar(my_x, my_y, fmt='o')
- 976    fit_start = my_x[0]
- 977    fit_stop = my_x[-1]
- 978    samples = np.arange(fit_start, fit_stop, 0.01)
- 979    plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
- 980    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='-')
- 981
- 982    plt.xlabel('Theoretical quantiles')
- 983    plt.ylabel('Ordered Values')
- 984    plt.legend()
- 985    plt.draw()
- 986
- 987
- 988def residual_plot(x, y, func, fit_res):
- 989    """Generates a plot which compares the fit to the data and displays the corresponding residuals
- 990
- 991    Returns
- 992    -------
- 993    None
- 994    """
- 995    sorted_x = sorted(x)
- 996    xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
- 997    xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
- 998    x_samples = np.arange(xstart, xstop + 0.01, 0.01)
- 999
-1000    plt.figure(figsize=(8, 8 / 1.618))
-1001    gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
-1002    ax0 = plt.subplot(gs[0])
-1003    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')
-1004    ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
-1005    ax0.set_xticklabels([])
-1006    ax0.set_xlim([xstart, xstop])
-1007    ax0.set_xticklabels([])
-1008    ax0.legend()
-1009
-1010    residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], x)) / np.asarray([o.dvalue for o in y])
-1011    ax1 = plt.subplot(gs[1])
-1012    ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
-1013    ax1.tick_params(direction='out')
-1014    ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
-1015    ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
-1016    ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
-1017    ax1.set_xlim([xstart, xstop])
-1018    ax1.set_ylabel('Residuals')
-1019    plt.subplots_adjust(wspace=None, hspace=None)
-1020    plt.draw()
-1021
-1022
-1023def error_band(x, func, beta):
-1024    """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
-1025
-1026    Returns
-1027    -------
-1028    err : np.array(Obs)
-1029        Error band for an array of sample values x
-1030    """
-1031    cov = covariance(beta)
-1032    if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
-1033        warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
-1034
-1035    deriv = []
-1036    for i, item in enumerate(x):
-1037        deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
-1038
-1039    err = []
-1040    for i, item in enumerate(x):
-1041        err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
-1042    err = np.array(err)
-1043
-1044    return err
-1045
-1046
-1047def ks_test(objects=None):
-1048    """Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
-1049
-1050    Parameters
-1051    ----------
-1052    objects : list
-1053        List of fit results to include in the analysis (optional).
-1054
-1055    Returns
-1056    -------
-1057    None
-1058    """
-1059
-1060    if objects is None:
-1061        obs_list = []
-1062        for obj in gc.get_objects():
-1063            if isinstance(obj, Fit_result):
-1064                obs_list.append(obj)
-1065    else:
-1066        obs_list = objects
-1067
-1068    p_values = [o.p_value for o in obs_list]
-1069
-1070    bins = len(p_values)
-1071    x = np.arange(0, 1.001, 0.001)
-1072    plt.plot(x, x, 'k', zorder=1)
-1073    plt.xlim(0, 1)
-1074    plt.ylim(0, 1)
-1075    plt.xlabel('p-value')
-1076    plt.ylabel('Cumulative probability')
-1077    plt.title(str(bins) + ' p-values')
-1078
-1079    n = np.arange(1, bins + 1) / np.float64(bins)
-1080    Xs = np.sort(p_values)
-1081    plt.step(Xs, n)
-1082    diffs = n - Xs
-1083    loc_max_diff = np.argmax(np.abs(diffs))
-1084    loc = Xs[loc_max_diff]
-1085    plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
-1086    plt.draw()
-1087
-1088    print(scipy.stats.kstest(p_values, 'uniform'))
+                        
  1import gc
+  2from collections.abc import Sequence
+  3import warnings
+  4import numpy as np
+  5import autograd.numpy as anp
+  6import scipy.optimize
+  7import scipy.stats
+  8import matplotlib.pyplot as plt
+  9from matplotlib import gridspec
+ 10from scipy.odr import ODR, Model, RealData
+ 11import iminuit
+ 12from autograd import jacobian as auto_jacobian
+ 13from autograd import hessian as auto_hessian
+ 14from autograd import elementwise_grad as egrad
+ 15from numdifftools import Jacobian as num_jacobian
+ 16from numdifftools import Hessian as num_hessian
+ 17from .obs import Obs, derived_observable, covariance, cov_Obs
+ 18
+ 19
+ 20class Fit_result(Sequence):
+ 21    """Represents fit results.
+ 22
+ 23    Attributes
+ 24    ----------
+ 25    fit_parameters : list
+ 26        results for the individual fit parameters,
+ 27        also accessible via indices.
+ 28    chisquare_by_dof : float
+ 29        reduced chisquare.
+ 30    p_value : float
+ 31        p-value of the fit
+ 32    t2_p_value : float
+ 33        Hotelling t-squared p-value for correlated fits.
+ 34    """
+ 35
+ 36    def __init__(self):
+ 37        self.fit_parameters = None
+ 38
+ 39    def __getitem__(self, idx):
+ 40        return self.fit_parameters[idx]
+ 41
+ 42    def __len__(self):
+ 43        return len(self.fit_parameters)
+ 44
+ 45    def gamma_method(self, **kwargs):
+ 46        """Apply the gamma method to all fit parameters"""
+ 47        [o.gamma_method(**kwargs) for o in self.fit_parameters]
+ 48
+ 49    gm = gamma_method
+ 50
+ 51    def __str__(self):
+ 52        my_str = 'Goodness of fit:\n'
+ 53        if hasattr(self, 'chisquare_by_dof'):
+ 54            my_str += '\u03C7\u00b2/d.o.f. = ' + f'{self.chisquare_by_dof:2.6f}' + '\n'
+ 55        elif hasattr(self, 'residual_variance'):
+ 56            my_str += 'residual variance = ' + f'{self.residual_variance:2.6f}' + '\n'
+ 57        if hasattr(self, 'chisquare_by_expected_chisquare'):
+ 58            my_str += '\u03C7\u00b2/\u03C7\u00b2exp  = ' + f'{self.chisquare_by_expected_chisquare:2.6f}' + '\n'
+ 59        if hasattr(self, 'p_value'):
+ 60            my_str += 'p-value   = ' + f'{self.p_value:2.4f}' + '\n'
+ 61        if hasattr(self, 't2_p_value'):
+ 62            my_str += 't\u00B2p-value = ' + f'{self.t2_p_value:2.4f}' + '\n'
+ 63        my_str += 'Fit parameters:\n'
+ 64        for i_par, par in enumerate(self.fit_parameters):
+ 65            my_str += str(i_par) + '\t' + ' ' * int(par >= 0) + str(par).rjust(int(par < 0.0)) + '\n'
+ 66        return my_str
+ 67
+ 68    def __repr__(self):
+ 69        m = max(map(len, list(self.__dict__.keys()))) + 1
+ 70        return '\n'.join([key.rjust(m) + ': ' + repr(value) for key, value in sorted(self.__dict__.items())])
+ 71
+ 72
+ 73def least_squares(x, y, func, priors=None, silent=False, **kwargs):
+ 74    r'''Performs a non-linear fit to y = func(x).
+ 75        ```
+ 76
+ 77    Parameters
+ 78    ----------
+ 79    For an uncombined fit:
+ 80
+ 81    x : list
+ 82        list of floats.
+ 83    y : list
+ 84        list of Obs.
+ 85    func : object
+ 86        fit function, has to be of the form
+ 87
+ 88        ```python
+ 89        import autograd.numpy as anp
+ 90
+ 91        def func(a, x):
+ 92            return a[0] + a[1] * x + a[2] * anp.sinh(x)
+ 93        ```
+ 94
+ 95        For multiple x values func can be of the form
+ 96
+ 97        ```python
+ 98        def func(a, x):
+ 99            (x1, x2) = x
+100            return a[0] * x1 ** 2 + a[1] * x2
+101        ```
+102        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
+103        will not work.
+104
+105    OR For a combined fit:
+106
+107    x : dict
+108        dict of lists.
+109    y : dict
+110        dict of lists of Obs.
+111    funcs : dict
+112        dict of objects
+113        fit functions have to be of the form (here a[0] is the common fit parameter)
+114        ```python
+115        import autograd.numpy as anp
+116        funcs = {"a": func_a,
+117                "b": func_b}
+118
+119        def func_a(a, x):
+120            return a[1] * anp.exp(-a[0] * x)
+121
+122        def func_b(a, x):
+123            return a[2] * anp.exp(-a[0] * x)
+124
+125        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
+126        will not work.
+127
+128    priors : list, optional
+129        priors has to be a list with an entry for every parameter in the fit. The entries can either be
+130        Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like
+131        0.548(23), 500(40) or 0.5(0.4)
+132    silent : bool, optional
+133        If true all output to the console is omitted (default False).
+134    initial_guess : list
+135        can provide an initial guess for the input parameters. Relevant for
+136        non-linear fits with many parameters. In case of correlated fits the guess is used to perform
+137        an uncorrelated fit which then serves as guess for the correlated fit.
+138    method : str, optional
+139        can be used to choose an alternative method for the minimization of chisquare.
+140        The possible methods are the ones which can be used for scipy.optimize.minimize and
+141        migrad of iminuit. If no method is specified, Levenberg-Marquard is used.
+142        Reliable alternatives are migrad, Powell and Nelder-Mead.
+143    tol: float, optional
+144        can be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence
+145        to a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly
+146        invalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values
+147        The stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
+148    correlated_fit : bool
+149        If True, use the full inverse covariance matrix in the definition of the chisquare cost function.
+150        For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`.
+151        In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).
+152        This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
+153        At the moment this option only works for `prior==None` and when no `method` is given.
+154    expected_chisquare : bool
+155        If True estimates the expected chisquare which is
+156        corrected by effects caused by correlated input data (default False).
+157    resplot : bool
+158        If True, a plot which displays fit, data and residuals is generated (default False).
+159    qqplot : bool
+160        If True, a quantile-quantile plot of the fit result is generated (default False).
+161    num_grad : bool
+162        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
+163
+164    Returns
+165    -------
+166    output : Fit_result
+167        Parameters and information on the fitted result.
+168    '''
+169    if priors is not None:
+170        return _prior_fit(x, y, func, priors, silent=silent, **kwargs)
+171    else:
+172        return _combined_fit(x, y, func, silent=silent, **kwargs)
+173
+174
+175def total_least_squares(x, y, func, silent=False, **kwargs):
+176    r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
+177
+178    Parameters
+179    ----------
+180    x : list
+181        list of Obs, or a tuple of lists of Obs
+182    y : list
+183        list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
+184    func : object
+185        func has to be of the form
+186
+187        ```python
+188        import autograd.numpy as anp
+189
+190        def func(a, x):
+191            return a[0] + a[1] * x + a[2] * anp.sinh(x)
+192        ```
+193
+194        For multiple x values func can be of the form
+195
+196        ```python
+197        def func(a, x):
+198            (x1, x2) = x
+199            return a[0] * x1 ** 2 + a[1] * x2
+200        ```
+201
+202        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
+203        will not work.
+204    silent : bool, optional
+205        If true all output to the console is omitted (default False).
+206    initial_guess : list
+207        can provide an initial guess for the input parameters. Relevant for non-linear
+208        fits with many parameters.
+209    expected_chisquare : bool
+210        If true prints the expected chisquare which is
+211        corrected by effects caused by correlated input data.
+212        This can take a while as the full correlation matrix
+213        has to be calculated (default False).
+214    num_grad : bool
+215        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
+216
+217    Notes
+218    -----
+219    Based on the orthogonal distance regression module of scipy.
+220
+221    Returns
+222    -------
+223    output : Fit_result
+224        Parameters and information on the fitted result.
+225    '''
+226
+227    output = Fit_result()
+228
+229    output.fit_function = func
+230
+231    x = np.array(x)
+232
+233    x_shape = x.shape
+234
+235    if kwargs.get('num_grad') is True:
+236        jacobian = num_jacobian
+237        hessian = num_hessian
+238    else:
+239        jacobian = auto_jacobian
+240        hessian = auto_hessian
+241
+242    if not callable(func):
+243        raise TypeError('func has to be a function.')
+244
+245    for i in range(42):
+246        try:
+247            func(np.arange(i), x.T[0])
+248        except TypeError:
+249            continue
+250        except IndexError:
+251            continue
+252        else:
+253            break
+254    else:
+255        raise RuntimeError("Fit function is not valid.")
+256
+257    n_parms = i
+258    if not silent:
+259        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
+260
+261    x_f = np.vectorize(lambda o: o.value)(x)
+262    dx_f = np.vectorize(lambda o: o.dvalue)(x)
+263    y_f = np.array([o.value for o in y])
+264    dy_f = np.array([o.dvalue for o in y])
+265
+266    if np.any(np.asarray(dx_f) <= 0.0):
+267        raise Exception('No x errors available, run the gamma method first.')
+268
+269    if np.any(np.asarray(dy_f) <= 0.0):
+270        raise Exception('No y errors available, run the gamma method first.')
+271
+272    if 'initial_guess' in kwargs:
+273        x0 = kwargs.get('initial_guess')
+274        if len(x0) != n_parms:
+275            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
+276    else:
+277        x0 = [1] * n_parms
+278
+279    data = RealData(x_f, y_f, sx=dx_f, sy=dy_f)
+280    model = Model(func)
+281    odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps)
+282    odr.set_job(fit_type=0, deriv=1)
+283    out = odr.run()
+284
+285    output.residual_variance = out.res_var
+286
+287    output.method = 'ODR'
+288
+289    output.message = out.stopreason
+290
+291    output.xplus = out.xplus
+292
+293    if not silent:
+294        print('Method: ODR')
+295        print(*out.stopreason)
+296        print('Residual variance:', output.residual_variance)
+297
+298    if out.info > 3:
+299        raise Exception('The minimization procedure did not converge.')
+300
+301    m = x_f.size
+302
+303    def odr_chisquare(p):
+304        model = func(p[:n_parms], p[n_parms:].reshape(x_shape))
+305        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2)
+306        return chisq
+307
+308    if kwargs.get('expected_chisquare') is True:
+309        W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel()))))
+310
+311        if kwargs.get('covariance') is not None:
+312            cov = kwargs.get('covariance')
+313        else:
+314            cov = covariance(np.concatenate((y, x.ravel())))
+315
+316        number_of_x_parameters = int(m / x_f.shape[-1])
+317
+318        old_jac = jacobian(func)(out.beta, out.xplus)
+319        fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0)))
+320        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])))
+321        new_jac = np.concatenate((fused_row1, fused_row2), axis=1)
+322
+323        A = W @ new_jac
+324        P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
+325        expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W)
+326        if expected_chisquare <= 0.0:
+327            warnings.warn("Negative expected_chisquare.", RuntimeWarning)
+328            expected_chisquare = np.abs(expected_chisquare)
+329        output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare
+330        if not silent:
+331            print('chisquare/expected_chisquare:',
+332                  output.chisquare_by_expected_chisquare)
+333
+334    fitp = out.beta
+335    try:
+336        hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel())))
+337    except TypeError:
+338        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
+339
+340    def odr_chisquare_compact_x(d):
+341        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
+342        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)
+343        return chisq
+344
+345    jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel())))
+346
+347    # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv
+348    try:
+349        deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:])
+350    except np.linalg.LinAlgError:
+351        raise Exception("Cannot invert hessian matrix.")
+352
+353    def odr_chisquare_compact_y(d):
+354        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
+355        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)
+356        return chisq
+357
+358    jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f)))
+359
+360    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
+361    try:
+362        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:])
+363    except np.linalg.LinAlgError:
+364        raise Exception("Cannot invert hessian matrix.")
+365
+366    result = []
+367    for i in range(n_parms):
+368        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])))
+369
+370    output.fit_parameters = result
+371
+372    output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel())))
+373    output.dof = x.shape[-1] - n_parms
+374    output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof)
+375
+376    return output
+377
+378
+379def _prior_fit(x, y, func, priors, silent=False, **kwargs):
+380    output = Fit_result()
+381
+382    output.fit_function = func
+383
+384    x = np.asarray(x)
+385
+386    if kwargs.get('num_grad') is True:
+387        hessian = num_hessian
+388    else:
+389        hessian = auto_hessian
+390
+391    if not callable(func):
+392        raise TypeError('func has to be a function.')
+393
+394    for i in range(100):
+395        try:
+396            func(np.arange(i), 0)
+397        except TypeError:
+398            continue
+399        except IndexError:
+400            continue
+401        else:
+402            break
+403    else:
+404        raise RuntimeError("Fit function is not valid.")
+405
+406    n_parms = i
+407
+408    if n_parms != len(priors):
+409        raise Exception('Priors does not have the correct length.')
+410
+411    def extract_val_and_dval(string):
+412        split_string = string.split('(')
+413        if '.' in split_string[0] and '.' not in split_string[1][:-1]:
+414            factor = 10 ** -len(split_string[0].partition('.')[2])
+415        else:
+416            factor = 1
+417        return float(split_string[0]), float(split_string[1][:-1]) * factor
+418
+419    loc_priors = []
+420    for i_n, i_prior in enumerate(priors):
+421        if isinstance(i_prior, Obs):
+422            loc_priors.append(i_prior)
+423        else:
+424            loc_val, loc_dval = extract_val_and_dval(i_prior)
+425            loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}"))
+426
+427    output.priors = loc_priors
+428
+429    if not silent:
+430        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
+431
+432    y_f = [o.value for o in y]
+433    dy_f = [o.dvalue for o in y]
+434
+435    if np.any(np.asarray(dy_f) <= 0.0):
+436        raise Exception('No y errors available, run the gamma method first.')
+437
+438    p_f = [o.value for o in loc_priors]
+439    dp_f = [o.dvalue for o in loc_priors]
+440
+441    if np.any(np.asarray(dp_f) <= 0.0):
+442        raise Exception('No prior errors available, run the gamma method first.')
+443
+444    if 'initial_guess' in kwargs:
+445        x0 = kwargs.get('initial_guess')
+446        if len(x0) != n_parms:
+447            raise Exception('Initial guess does not have the correct length.')
+448    else:
+449        x0 = p_f
+450
+451    def chisqfunc(p):
+452        model = func(p, x)
+453        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((p_f - p) / dp_f) ** 2)
+454        return chisq
+455
+456    if not silent:
+457        print('Method: migrad')
+458
+459    m = iminuit.Minuit(chisqfunc, x0)
+460    m.errordef = 1
+461    m.print_level = 0
+462    if 'tol' in kwargs:
+463        m.tol = kwargs.get('tol')
+464    else:
+465        m.tol = 1e-4
+466    m.migrad()
+467    params = np.asarray(m.values)
+468
+469    output.chisquare_by_dof = m.fval / len(x)
+470
+471    output.method = 'migrad'
+472
+473    if not silent:
+474        print('chisquare/d.o.f.:', output.chisquare_by_dof)
+475
+476    if not m.fmin.is_valid:
+477        raise Exception('The minimization procedure did not converge.')
+478
+479    hess = hessian(chisqfunc)(params)
+480    hess_inv = np.linalg.pinv(hess)
+481
+482    def chisqfunc_compact(d):
+483        model = func(d[:n_parms], x)
+484        chisq = anp.sum(((d[n_parms: n_parms + len(x)] - model) / dy_f) ** 2) + anp.sum(((d[n_parms + len(x):] - d[:n_parms]) / dp_f) ** 2)
+485        return chisq
+486
+487    jac_jac = hessian(chisqfunc_compact)(np.concatenate((params, y_f, p_f)))
+488
+489    deriv = -hess_inv @ jac_jac[:n_parms, n_parms:]
+490
+491    result = []
+492    for i in range(n_parms):
+493        result.append(derived_observable(lambda x, **kwargs: (x[0] + np.finfo(np.float64).eps) / (y[0].value + np.finfo(np.float64).eps) * params[i], list(y) + list(loc_priors), man_grad=list(deriv[i])))
+494
+495    output.fit_parameters = result
+496    output.chisquare = chisqfunc(np.asarray(params))
+497
+498    if kwargs.get('resplot') is True:
+499        residual_plot(x, y, func, result)
+500
+501    if kwargs.get('qqplot') is True:
+502        qqplot(x, y, func, result)
+503
+504    return output
+505
+506
+507def _combined_fit(x, y, func, silent=False, **kwargs):
+508
+509    output = Fit_result()
+510
+511    if (type(x) == dict and type(y) == dict and type(func) == dict):
+512        xd = x
+513        yd = y
+514        funcd = func
+515        output.fit_function = func
+516    elif (type(x) == dict or type(y) == dict or type(func) == dict):
+517        raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.")
+518    else:
+519        x = np.asarray(x)
+520        xd = {"": x}
+521        yd = {"": y}
+522        funcd = {"": func}
+523        output.fit_function = func
+524
+525    if kwargs.get('num_grad') is True:
+526        jacobian = num_jacobian
+527        hessian = num_hessian
+528    else:
+529        jacobian = auto_jacobian
+530        hessian = auto_hessian
+531
+532    key_ls = sorted(list(xd.keys()))
+533
+534    if sorted(list(yd.keys())) != key_ls:
+535        raise Exception('x and y dictionaries do not contain the same keys.')
+536
+537    if sorted(list(funcd.keys())) != key_ls:
+538        raise Exception('x and func dictionaries do not contain the same keys.')
+539
+540    x_all = np.concatenate([np.array(xd[key]) for key in key_ls])
+541    y_all = np.concatenate([np.array(yd[key]) for key in key_ls])
+542
+543    y_f = [o.value for o in y_all]
+544    dy_f = [o.dvalue for o in y_all]
+545
+546    if len(x_all.shape) > 2:
+547        raise Exception('Unknown format for x values')
+548
+549    if np.any(np.asarray(dy_f) <= 0.0):
+550        raise Exception('No y errors available, run the gamma method first.')
+551
+552    # number of fit parameters
+553    n_parms_ls = []
+554    for key in key_ls:
+555        if not callable(funcd[key]):
+556            raise TypeError('func (key=' + key + ') is not a function.')
+557        if len(xd[key]) != len(yd[key]):
+558            raise Exception('x and y input (key=' + key + ') do not have the same length')
+559        for i in range(100):
+560            try:
+561                funcd[key](np.arange(i), x_all.T[0])
+562            except TypeError:
+563                continue
+564            except IndexError:
+565                continue
+566            else:
+567                break
+568        else:
+569            raise RuntimeError("Fit function (key=" + key + ") is not valid.")
+570        n_parms = i
+571        n_parms_ls.append(n_parms)
+572    n_parms = max(n_parms_ls)
+573    if not silent:
+574        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
+575
+576    if 'initial_guess' in kwargs:
+577        x0 = kwargs.get('initial_guess')
+578        if len(x0) != n_parms:
+579            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
+580    else:
+581        x0 = [0.1] * n_parms
+582
+583    if kwargs.get('correlated_fit') is True:
+584        corr = covariance(y_all, correlation=True, **kwargs)
+585        covdiag = np.diag(1 / np.asarray(dy_f))
+586        condn = np.linalg.cond(corr)
+587        if condn > 0.1 / np.finfo(float).eps:
+588            raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})")
+589        if condn > 1e13:
+590            warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning)
+591        chol = np.linalg.cholesky(corr)
+592        chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True)
+593
+594        def chisqfunc_corr(p):
+595            model = np.concatenate([np.array(funcd[key](p, np.asarray(xd[key]))).reshape(-1) for key in key_ls])
+596            chisq = anp.sum(anp.dot(chol_inv, (y_f - model)) ** 2)
+597            return chisq
+598
+599    def chisqfunc(p):
+600        func_list = np.concatenate([[funcd[k]] * len(xd[k]) for k in key_ls])
+601        model = anp.array([func_list[i](p, x_all[i]) for i in range(len(x_all))])
+602        chisq = anp.sum(((y_f - model) / dy_f) ** 2)
+603        return chisq
+604
+605    output.method = kwargs.get('method', 'Levenberg-Marquardt')
+606    if not silent:
+607        print('Method:', output.method)
+608
+609    if output.method != 'Levenberg-Marquardt':
+610        if output.method == 'migrad':
+611            tolerance = 1e-4  # default value of 1e-1 set by iminuit can be problematic
+612            if 'tol' in kwargs:
+613                tolerance = kwargs.get('tol')
+614            fit_result = iminuit.minimize(chisqfunc, x0, tol=tolerance)  # Stopping criterion 0.002 * tol * errordef
+615            if kwargs.get('correlated_fit') is True:
+616                fit_result = iminuit.minimize(chisqfunc_corr, fit_result.x, tol=tolerance)
+617            output.iterations = fit_result.nfev
+618        else:
+619            tolerance = 1e-12
+620            if 'tol' in kwargs:
+621                tolerance = kwargs.get('tol')
+622            fit_result = scipy.optimize.minimize(chisqfunc, x0, method=kwargs.get('method'), tol=tolerance)
+623            if kwargs.get('correlated_fit') is True:
+624                fit_result = scipy.optimize.minimize(chisqfunc_corr, fit_result.x, method=kwargs.get('method'), tol=tolerance)
+625            output.iterations = fit_result.nit
+626
+627        chisquare = fit_result.fun
+628
+629    else:
+630        if kwargs.get('correlated_fit') is True:
+631            def chisqfunc_residuals_corr(p):
+632                model = np.concatenate([np.array(funcd[key](p, np.asarray(xd[key]))).reshape(-1) for key in key_ls])
+633                chisq = anp.dot(chol_inv, (y_f - model))
+634                return chisq
+635
+636        def chisqfunc_residuals(p):
+637            model = np.concatenate([np.array(funcd[key](p, np.asarray(xd[key]))).reshape(-1) for key in key_ls])
+638            chisq = ((y_f - model) / dy_f)
+639            return chisq
+640
+641        if 'tol' in kwargs:
+642            print('tol cannot be set for Levenberg-Marquardt')
+643
+644        fit_result = scipy.optimize.least_squares(chisqfunc_residuals, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
+645        if kwargs.get('correlated_fit') is True:
+646            fit_result = scipy.optimize.least_squares(chisqfunc_residuals_corr, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
+647
+648        chisquare = np.sum(fit_result.fun ** 2)
+649        if kwargs.get('correlated_fit') is True:
+650            assert np.isclose(chisquare, chisqfunc_corr(fit_result.x), atol=1e-14)
+651        else:
+652            assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14)
+653
+654        output.iterations = fit_result.nfev
+655
+656    if not fit_result.success:
+657        raise Exception('The minimization procedure did not converge.')
+658
+659    if x_all.shape[-1] - n_parms > 0:
+660        output.chisquare = chisquare
+661        output.dof = x_all.shape[-1] - n_parms
+662        output.chisquare_by_dof = output.chisquare / output.dof
+663        output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof)
+664    else:
+665        output.chisquare_by_dof = float('nan')
+666
+667    output.message = fit_result.message
+668    if not silent:
+669        print(fit_result.message)
+670        print('chisquare/d.o.f.:', output.chisquare_by_dof)
+671        print('fit parameters', fit_result.x)
+672
+673    def prepare_hat_matrix():
+674        hat_vector = []
+675        for key in key_ls:
+676            x_array = np.asarray(xd[key])
+677            if (len(x_array) != 0):
+678                hat_vector.append(jacobian(funcd[key])(fit_result.x, x_array))
+679        hat_vector = [item for sublist in hat_vector for item in sublist]
+680        return hat_vector
+681
+682    if kwargs.get('expected_chisquare') is True:
+683        if kwargs.get('correlated_fit') is not True:
+684            W = np.diag(1 / np.asarray(dy_f))
+685            cov = covariance(y_all)
+686            hat_vector = prepare_hat_matrix()
+687            A = W @ hat_vector
+688            P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
+689            expected_chisquare = np.trace((np.identity(x_all.shape[-1]) - P_phi) @ W @ cov @ W)
+690            output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare
+691            if not silent:
+692                print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare)
+693
+694    fitp = fit_result.x
+695    if np.any(np.asarray(dy_f) <= 0.0):
+696        raise Exception('No y errors available, run the gamma method first.')
+697
+698    try:
+699        if kwargs.get('correlated_fit') is True:
+700            hess = hessian(chisqfunc_corr)(fitp)
+701        else:
+702            hess = hessian(chisqfunc)(fitp)
+703    except TypeError:
+704        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
+705
+706    if kwargs.get('correlated_fit') is True:
+707        def chisqfunc_compact(d):
+708            func_list = np.concatenate([[funcd[k]] * len(xd[k]) for k in key_ls])
+709            model = anp.array([func_list[i](d[:n_parms], x_all[i]) for i in range(len(x_all))])
+710            chisq = anp.sum(anp.dot(chol_inv, (d[n_parms:] - model)) ** 2)
+711            return chisq
+712    else:
+713        def chisqfunc_compact(d):
+714            func_list = np.concatenate([[funcd[k]] * len(xd[k]) for k in key_ls])
+715            model = anp.array([func_list[i](d[:n_parms], x_all[i]) for i in range(len(x_all))])
+716            chisq = anp.sum(((d[n_parms:] - model) / dy_f) ** 2)
+717            return chisq
+718
+719    jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f)))
+720
+721    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
+722    try:
+723        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:])
+724    except np.linalg.LinAlgError:
+725        raise Exception("Cannot invert hessian matrix.")
+726
+727    result = []
+728    for i in range(n_parms):
+729        result.append(derived_observable(lambda x_all, **kwargs: (x_all[0] + np.finfo(np.float64).eps) / (y_all[0].value + np.finfo(np.float64).eps) * fitp[i], list(y_all), man_grad=list(deriv_y[i])))
+730
+731    output.fit_parameters = result
+732
+733    # Hotelling t-squared p-value for correlated fits.
+734    if kwargs.get('correlated_fit') is True:
+735        n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all))
+736        output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare,
+737                                                  output.dof, n_cov - output.dof)
+738
+739    if kwargs.get('resplot') is True:
+740        for key in key_ls:
+741            residual_plot(xd[key], yd[key], funcd[key], result, title=key)
+742
+743    if kwargs.get('qqplot') is True:
+744        for key in key_ls:
+745            qqplot(xd[key], yd[key], funcd[key], result, title=key)
+746
+747    return output
+748
+749
+750def fit_lin(x, y, **kwargs):
+751    """Performs a linear fit to y = n + m * x and returns two Obs n, m.
+752
+753    Parameters
+754    ----------
+755    x : list
+756        Can either be a list of floats in which case no xerror is assumed, or
+757        a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
+758    y : list
+759        List of Obs, the dvalues of the Obs are used as yerror for the fit.
+760
+761    Returns
+762    -------
+763    fit_parameters : list[Obs]
+764        LIist of fitted observables.
+765    """
+766
+767    def f(a, x):
+768        y = a[0] + a[1] * x
+769        return y
+770
+771    if all(isinstance(n, Obs) for n in x):
+772        out = total_least_squares(x, y, f, **kwargs)
+773        return out.fit_parameters
+774    elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
+775        out = least_squares(x, y, f, **kwargs)
+776        return out.fit_parameters
+777    else:
+778        raise Exception('Unsupported types for x')
+779
+780
+781def qqplot(x, o_y, func, p, title=""):
+782    """Generates a quantile-quantile plot of the fit result which can be used to
+783       check if the residuals of the fit are gaussian distributed.
+784
+785    Returns
+786    -------
+787    None
+788    """
+789
+790    residuals = []
+791    for i_x, i_y in zip(x, o_y):
+792        residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
+793    residuals = sorted(residuals)
+794    my_y = [o.value for o in residuals]
+795    probplot = scipy.stats.probplot(my_y)
+796    my_x = probplot[0][0]
+797    plt.figure(figsize=(8, 8 / 1.618))
+798    plt.errorbar(my_x, my_y, fmt='o')
+799    fit_start = my_x[0]
+800    fit_stop = my_x[-1]
+801    samples = np.arange(fit_start, fit_stop, 0.01)
+802    plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
+803    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='-')
+804
+805    plt.xlabel('Theoretical quantiles')
+806    plt.ylabel('Ordered Values')
+807    plt.legend(title=title)
+808    plt.draw()
+809
+810
+811def residual_plot(x, y, func, fit_res, title=""):
+812    """Generates a plot which compares the fit to the data and displays the corresponding residuals
+813
+814    For uncorrelated data the residuals are expected to be distributed ~N(0,1).
+815
+816    Returns
+817    -------
+818    None
+819    """
+820    sorted_x = sorted(x)
+821    xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
+822    xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
+823    x_samples = np.arange(xstart, xstop + 0.01, 0.01)
+824
+825    plt.figure(figsize=(8, 8 / 1.618))
+826    gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
+827    ax0 = plt.subplot(gs[0])
+828    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')
+829    ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
+830    ax0.set_xticklabels([])
+831    ax0.set_xlim([xstart, xstop])
+832    ax0.set_xticklabels([])
+833    ax0.legend(title=title)
+834
+835    residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], x)) / np.asarray([o.dvalue for o in y])
+836    ax1 = plt.subplot(gs[1])
+837    ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
+838    ax1.tick_params(direction='out')
+839    ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
+840    ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
+841    ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
+842    ax1.set_xlim([xstart, xstop])
+843    ax1.set_ylabel('Residuals')
+844    plt.subplots_adjust(wspace=None, hspace=None)
+845    plt.draw()
+846
+847
+848def error_band(x, func, beta):
+849    """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
+850
+851    Returns
+852    -------
+853    err : np.array(Obs)
+854        Error band for an array of sample values x
+855    """
+856    cov = covariance(beta)
+857    if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
+858        warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
+859
+860    deriv = []
+861    for i, item in enumerate(x):
+862        deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
+863
+864    err = []
+865    for i, item in enumerate(x):
+866        err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
+867    err = np.array(err)
+868
+869    return err
+870
+871
+872def ks_test(objects=None):
+873    """Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
+874
+875    Parameters
+876    ----------
+877    objects : list
+878        List of fit results to include in the analysis (optional).
+879
+880    Returns
+881    -------
+882    None
+883    """
+884
+885    if objects is None:
+886        obs_list = []
+887        for obj in gc.get_objects():
+888            if isinstance(obj, Fit_result):
+889                obs_list.append(obj)
+890    else:
+891        obs_list = objects
+892
+893    p_values = [o.p_value for o in obs_list]
+894
+895    bins = len(p_values)
+896    x = np.arange(0, 1.001, 0.001)
+897    plt.plot(x, x, 'k', zorder=1)
+898    plt.xlim(0, 1)
+899    plt.ylim(0, 1)
+900    plt.xlabel('p-value')
+901    plt.ylabel('Cumulative probability')
+902    plt.title(str(bins) + ' p-values')
+903
+904    n = np.arange(1, bins + 1) / np.float64(bins)
+905    Xs = np.sort(p_values)
+906    plt.step(Xs, n)
+907    diffs = n - Xs
+908    loc_max_diff = np.argmax(np.abs(diffs))
+909    loc = Xs[loc_max_diff]
+910    plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
+911    plt.draw()
+912
+913    print(scipy.stats.kstest(p_values, 'uniform'))
 

 
 
@@ -1445,13 +1270,8 @@ Hotelling t-squared p-value for correlated fits.
 169    '''
 170    if priors is not None:
 171        return _prior_fit(x, y, func, priors, silent=silent, **kwargs)
-172
-173    elif (type(x) == dict and type(y) == dict and type(func) == dict):
-174        return _combined_fit(x, y, func, silent=silent, **kwargs)
-175    elif (type(x) == dict or type(y) == dict or type(func) == dict):
-176        raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.")
-177    else:
-178        return _standard_fit(x, y, func, silent=silent, **kwargs)
+172    else:
+173        return _combined_fit(x, y, func, silent=silent, **kwargs)
@@ -1567,208 +1387,208 @@ Parameters and information on the fitted result. - 
181def total_least_squares(x, y, func, silent=False, **kwargs):
-182    r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
-183
-184    Parameters
-185    ----------
-186    x : list
-187        list of Obs, or a tuple of lists of Obs
-188    y : list
-189        list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
-190    func : object
-191        func has to be of the form
-192
-193        ```python
-194        import autograd.numpy as anp
-195
-196        def func(a, x):
-197            return a[0] + a[1] * x + a[2] * anp.sinh(x)
-198        ```
-199
-200        For multiple x values func can be of the form
-201
-202        ```python
-203        def func(a, x):
-204            (x1, x2) = x
-205            return a[0] * x1 ** 2 + a[1] * x2
-206        ```
-207
-208        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
-209        will not work.
-210    silent : bool, optional
-211        If true all output to the console is omitted (default False).
-212    initial_guess : list
-213        can provide an initial guess for the input parameters. Relevant for non-linear
-214        fits with many parameters.
-215    expected_chisquare : bool
-216        If true prints the expected chisquare which is
-217        corrected by effects caused by correlated input data.
-218        This can take a while as the full correlation matrix
-219        has to be calculated (default False).
-220    num_grad : bool
-221        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
-222
-223    Notes
-224    -----
-225    Based on the orthogonal distance regression module of scipy.
-226
-227    Returns
-228    -------
-229    output : Fit_result
-230        Parameters and information on the fitted result.
-231    '''
-232
-233    output = Fit_result()
-234
-235    output.fit_function = func
-236
-237    x = np.array(x)
-238
-239    x_shape = x.shape
-240
-241    if kwargs.get('num_grad') is True:
-242        jacobian = num_jacobian
-243        hessian = num_hessian
-244    else:
-245        jacobian = auto_jacobian
-246        hessian = auto_hessian
-247
-248    if not callable(func):
-249        raise TypeError('func has to be a function.')
-250
-251    for i in range(42):
-252        try:
-253            func(np.arange(i), x.T[0])
-254        except TypeError:
-255            continue
-256        except IndexError:
-257            continue
-258        else:
-259            break
-260    else:
-261        raise RuntimeError("Fit function is not valid.")
-262
-263    n_parms = i
-264    if not silent:
-265        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
+            
176def total_least_squares(x, y, func, silent=False, **kwargs):
+177    r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
+178
+179    Parameters
+180    ----------
+181    x : list
+182        list of Obs, or a tuple of lists of Obs
+183    y : list
+184        list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
+185    func : object
+186        func has to be of the form
+187
+188        ```python
+189        import autograd.numpy as anp
+190
+191        def func(a, x):
+192            return a[0] + a[1] * x + a[2] * anp.sinh(x)
+193        ```
+194
+195        For multiple x values func can be of the form
+196
+197        ```python
+198        def func(a, x):
+199            (x1, x2) = x
+200            return a[0] * x1 ** 2 + a[1] * x2
+201        ```
+202
+203        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
+204        will not work.
+205    silent : bool, optional
+206        If true all output to the console is omitted (default False).
+207    initial_guess : list
+208        can provide an initial guess for the input parameters. Relevant for non-linear
+209        fits with many parameters.
+210    expected_chisquare : bool
+211        If true prints the expected chisquare which is
+212        corrected by effects caused by correlated input data.
+213        This can take a while as the full correlation matrix
+214        has to be calculated (default False).
+215    num_grad : bool
+216        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
+217
+218    Notes
+219    -----
+220    Based on the orthogonal distance regression module of scipy.
+221
+222    Returns
+223    -------
+224    output : Fit_result
+225        Parameters and information on the fitted result.
+226    '''
+227
+228    output = Fit_result()
+229
+230    output.fit_function = func
+231
+232    x = np.array(x)
+233
+234    x_shape = x.shape
+235
+236    if kwargs.get('num_grad') is True:
+237        jacobian = num_jacobian
+238        hessian = num_hessian
+239    else:
+240        jacobian = auto_jacobian
+241        hessian = auto_hessian
+242
+243    if not callable(func):
+244        raise TypeError('func has to be a function.')
+245
+246    for i in range(42):
+247        try:
+248            func(np.arange(i), x.T[0])
+249        except TypeError:
+250            continue
+251        except IndexError:
+252            continue
+253        else:
+254            break
+255    else:
+256        raise RuntimeError("Fit function is not valid.")
+257
+258    n_parms = i
+259    if not silent:
+260        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
+261
+262    x_f = np.vectorize(lambda o: o.value)(x)
+263    dx_f = np.vectorize(lambda o: o.dvalue)(x)
+264    y_f = np.array([o.value for o in y])
+265    dy_f = np.array([o.dvalue for o in y])
 266
-267    x_f = np.vectorize(lambda o: o.value)(x)
-268    dx_f = np.vectorize(lambda o: o.dvalue)(x)
-269    y_f = np.array([o.value for o in y])
-270    dy_f = np.array([o.dvalue for o in y])
-271
-272    if np.any(np.asarray(dx_f) <= 0.0):
-273        raise Exception('No x errors available, run the gamma method first.')
-274
-275    if np.any(np.asarray(dy_f) <= 0.0):
-276        raise Exception('No y errors available, run the gamma method first.')
-277
-278    if 'initial_guess' in kwargs:
-279        x0 = kwargs.get('initial_guess')
-280        if len(x0) != n_parms:
-281            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
-282    else:
-283        x0 = [1] * n_parms
-284
-285    data = RealData(x_f, y_f, sx=dx_f, sy=dy_f)
-286    model = Model(func)
-287    odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps)
-288    odr.set_job(fit_type=0, deriv=1)
-289    out = odr.run()
-290
-291    output.residual_variance = out.res_var
-292
-293    output.method = 'ODR'
-294
-295    output.message = out.stopreason
-296
-297    output.xplus = out.xplus
+267    if np.any(np.asarray(dx_f) <= 0.0):
+268        raise Exception('No x errors available, run the gamma method first.')
+269
+270    if np.any(np.asarray(dy_f) <= 0.0):
+271        raise Exception('No y errors available, run the gamma method first.')
+272
+273    if 'initial_guess' in kwargs:
+274        x0 = kwargs.get('initial_guess')
+275        if len(x0) != n_parms:
+276            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
+277    else:
+278        x0 = [1] * n_parms
+279
+280    data = RealData(x_f, y_f, sx=dx_f, sy=dy_f)
+281    model = Model(func)
+282    odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps)
+283    odr.set_job(fit_type=0, deriv=1)
+284    out = odr.run()
+285
+286    output.residual_variance = out.res_var
+287
+288    output.method = 'ODR'
+289
+290    output.message = out.stopreason
+291
+292    output.xplus = out.xplus
+293
+294    if not silent:
+295        print('Method: ODR')
+296        print(*out.stopreason)
+297        print('Residual variance:', output.residual_variance)
 298
-299    if not silent:
-300        print('Method: ODR')
-301        print(*out.stopreason)
-302        print('Residual variance:', output.residual_variance)
+299    if out.info > 3:
+300        raise Exception('The minimization procedure did not converge.')
+301
+302    m = x_f.size
 303
-304    if out.info > 3:
-305        raise Exception('The minimization procedure did not converge.')
-306
-307    m = x_f.size
+304    def odr_chisquare(p):
+305        model = func(p[:n_parms], p[n_parms:].reshape(x_shape))
+306        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2)
+307        return chisq
 308
-309    def odr_chisquare(p):
-310        model = func(p[:n_parms], p[n_parms:].reshape(x_shape))
-311        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2)
-312        return chisq
-313
-314    if kwargs.get('expected_chisquare') is True:
-315        W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel()))))
+309    if kwargs.get('expected_chisquare') is True:
+310        W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel()))))
+311
+312        if kwargs.get('covariance') is not None:
+313            cov = kwargs.get('covariance')
+314        else:
+315            cov = covariance(np.concatenate((y, x.ravel())))
 316
-317        if kwargs.get('covariance') is not None:
-318            cov = kwargs.get('covariance')
-319        else:
-320            cov = covariance(np.concatenate((y, x.ravel())))
-321
-322        number_of_x_parameters = int(m / x_f.shape[-1])
+317        number_of_x_parameters = int(m / x_f.shape[-1])
+318
+319        old_jac = jacobian(func)(out.beta, out.xplus)
+320        fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0)))
+321        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])))
+322        new_jac = np.concatenate((fused_row1, fused_row2), axis=1)
 323
-324        old_jac = jacobian(func)(out.beta, out.xplus)
-325        fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0)))
-326        fused_row2 = np.concatenate((jacobian(lambda x, y: func(y, x))(out.xplus, out.beta).reshape(x_f.shape[-1], x_f.shape[-1] * number_of_x_parameters), np.identity(number_of_x_parameters * old_jac.shape[0])))
-327        new_jac = np.concatenate((fused_row1, fused_row2), axis=1)
-328
-329        A = W @ new_jac
-330        P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
-331        expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W)
-332        if expected_chisquare <= 0.0:
-333            warnings.warn("Negative expected_chisquare.", RuntimeWarning)
-334            expected_chisquare = np.abs(expected_chisquare)
-335        output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare
-336        if not silent:
-337            print('chisquare/expected_chisquare:',
-338                  output.chisquare_by_expected_chisquare)
-339
-340    fitp = out.beta
-341    try:
-342        hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel())))
-343    except TypeError:
-344        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
+324        A = W @ new_jac
+325        P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
+326        expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W)
+327        if expected_chisquare <= 0.0:
+328            warnings.warn("Negative expected_chisquare.", RuntimeWarning)
+329            expected_chisquare = np.abs(expected_chisquare)
+330        output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare
+331        if not silent:
+332            print('chisquare/expected_chisquare:',
+333                  output.chisquare_by_expected_chisquare)
+334
+335    fitp = out.beta
+336    try:
+337        hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel())))
+338    except TypeError:
+339        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
+340
+341    def odr_chisquare_compact_x(d):
+342        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
+343        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)
+344        return chisq
 345
-346    def odr_chisquare_compact_x(d):
-347        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
-348        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((d[n_parms + m:].reshape(x_shape) - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2)
-349        return chisq
-350
-351    jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel())))
-352
-353    # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv
-354    try:
-355        deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:])
-356    except np.linalg.LinAlgError:
-357        raise Exception("Cannot invert hessian matrix.")
+346    jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel())))
+347
+348    # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv
+349    try:
+350        deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:])
+351    except np.linalg.LinAlgError:
+352        raise Exception("Cannot invert hessian matrix.")
+353
+354    def odr_chisquare_compact_y(d):
+355        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
+356        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)
+357        return chisq
 358
-359    def odr_chisquare_compact_y(d):
-360        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
-361        chisq = anp.sum(((d[n_parms + m:] - model) / dy_f) ** 2) + anp.sum(((x_f - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2)
-362        return chisq
-363
-364    jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f)))
-365
-366    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
-367    try:
-368        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:])
-369    except np.linalg.LinAlgError:
-370        raise Exception("Cannot invert hessian matrix.")
-371
-372    result = []
-373    for i in range(n_parms):
-374        result.append(derived_observable(lambda my_var, **kwargs: (my_var[0] + np.finfo(np.float64).eps) / (x.ravel()[0].value + np.finfo(np.float64).eps) * out.beta[i], list(x.ravel()) + list(y), man_grad=list(deriv_x[i]) + list(deriv_y[i])))
-375
-376    output.fit_parameters = result
-377
-378    output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel())))
-379    output.dof = x.shape[-1] - n_parms
-380    output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof)
-381
-382    return output
+359    jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f)))
+360
+361    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
+362    try:
+363        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:])
+364    except np.linalg.LinAlgError:
+365        raise Exception("Cannot invert hessian matrix.")
+366
+367    result = []
+368    for i in range(n_parms):
+369        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])))
+370
+371    output.fit_parameters = result
+372
+373    output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel())))
+374    output.dof = x.shape[-1] - n_parms
+375    output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof)
+376
+377    return output
 

 
 
@@ -1842,35 +1662,35 @@ Parameters and information on the fitted result.
- 
928def fit_lin(x, y, **kwargs):
-929    """Performs a linear fit to y = n + m * x and returns two Obs n, m.
-930
-931    Parameters
-932    ----------
-933    x : list
-934        Can either be a list of floats in which case no xerror is assumed, or
-935        a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
-936    y : list
-937        List of Obs, the dvalues of the Obs are used as yerror for the fit.
-938
-939    Returns
-940    -------
-941    fit_parameters : list[Obs]
-942        LIist of fitted observables.
-943    """
-944
-945    def f(a, x):
-946        y = a[0] + a[1] * x
-947        return y
-948
-949    if all(isinstance(n, Obs) for n in x):
-950        out = total_least_squares(x, y, f, **kwargs)
-951        return out.fit_parameters
-952    elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
-953        out = least_squares(x, y, f, **kwargs)
-954        return out.fit_parameters
-955    else:
-956        raise Exception('Unsupported types for x')
+            
751def fit_lin(x, y, **kwargs):
+752    """Performs a linear fit to y = n + m * x and returns two Obs n, m.
+753
+754    Parameters
+755    ----------
+756    x : list
+757        Can either be a list of floats in which case no xerror is assumed, or
+758        a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
+759    y : list
+760        List of Obs, the dvalues of the Obs are used as yerror for the fit.
+761
+762    Returns
+763    -------
+764    fit_parameters : list[Obs]
+765        LIist of fitted observables.
+766    """
+767
+768    def f(a, x):
+769        y = a[0] + a[1] * x
+770        return y
+771
+772    if all(isinstance(n, Obs) for n in x):
+773        out = total_least_squares(x, y, f, **kwargs)
+774        return out.fit_parameters
+775    elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
+776        out = least_squares(x, y, f, **kwargs)
+777        return out.fit_parameters
+778    else:
+779        raise Exception('Unsupported types for x')
 

 
 
@@ -1901,40 +1721,40 @@ LIist of fitted observables.
 

             
         def
-        qqplot(x, o_y, func, p):
+        qqplot(x, o_y, func, p, title=''):
 
                 
 
     

     
-            
959def qqplot(x, o_y, func, p):
-960    """Generates a quantile-quantile plot of the fit result which can be used to
-961       check if the residuals of the fit are gaussian distributed.
-962
-963    Returns
-964    -------
-965    None
-966    """
-967
-968    residuals = []
-969    for i_x, i_y in zip(x, o_y):
-970        residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
-971    residuals = sorted(residuals)
-972    my_y = [o.value for o in residuals]
-973    probplot = scipy.stats.probplot(my_y)
-974    my_x = probplot[0][0]
-975    plt.figure(figsize=(8, 8 / 1.618))
-976    plt.errorbar(my_x, my_y, fmt='o')
-977    fit_start = my_x[0]
-978    fit_stop = my_x[-1]
-979    samples = np.arange(fit_start, fit_stop, 0.01)
-980    plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
-981    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='-')
-982
-983    plt.xlabel('Theoretical quantiles')
-984    plt.ylabel('Ordered Values')
-985    plt.legend()
-986    plt.draw()
+            
782def qqplot(x, o_y, func, p, title=""):
+783    """Generates a quantile-quantile plot of the fit result which can be used to
+784       check if the residuals of the fit are gaussian distributed.
+785
+786    Returns
+787    -------
+788    None
+789    """
+790
+791    residuals = []
+792    for i_x, i_y in zip(x, o_y):
+793        residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
+794    residuals = sorted(residuals)
+795    my_y = [o.value for o in residuals]
+796    probplot = scipy.stats.probplot(my_y)
+797    my_x = probplot[0][0]
+798    plt.figure(figsize=(8, 8 / 1.618))
+799    plt.errorbar(my_x, my_y, fmt='o')
+800    fit_start = my_x[0]
+801    fit_stop = my_x[-1]
+802    samples = np.arange(fit_start, fit_stop, 0.01)
+803    plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
+804    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='-')
+805
+806    plt.xlabel('Theoretical quantiles')
+807    plt.ylabel('Ordered Values')
+808    plt.legend(title=title)
+809    plt.draw()
 

 
 
@@ -1955,50 +1775,54 @@ LIist of fitted observables.
 

             
         def
-        residual_plot(x, y, func, fit_res):
+        residual_plot(x, y, func, fit_res, title=''):
 
                 
 
     

     
-            
 989def residual_plot(x, y, func, fit_res):
- 990    """Generates a plot which compares the fit to the data and displays the corresponding residuals
- 991
- 992    Returns
- 993    -------
- 994    None
- 995    """
- 996    sorted_x = sorted(x)
- 997    xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
- 998    xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
- 999    x_samples = np.arange(xstart, xstop + 0.01, 0.01)
-1000
-1001    plt.figure(figsize=(8, 8 / 1.618))
-1002    gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
-1003    ax0 = plt.subplot(gs[0])
-1004    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')
-1005    ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
-1006    ax0.set_xticklabels([])
-1007    ax0.set_xlim([xstart, xstop])
-1008    ax0.set_xticklabels([])
-1009    ax0.legend()
-1010
-1011    residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], x)) / np.asarray([o.dvalue for o in y])
-1012    ax1 = plt.subplot(gs[1])
-1013    ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
-1014    ax1.tick_params(direction='out')
-1015    ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
-1016    ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
-1017    ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
-1018    ax1.set_xlim([xstart, xstop])
-1019    ax1.set_ylabel('Residuals')
-1020    plt.subplots_adjust(wspace=None, hspace=None)
-1021    plt.draw()
+            
812def residual_plot(x, y, func, fit_res, title=""):
+813    """Generates a plot which compares the fit to the data and displays the corresponding residuals
+814
+815    For uncorrelated data the residuals are expected to be distributed ~N(0,1).
+816
+817    Returns
+818    -------
+819    None
+820    """
+821    sorted_x = sorted(x)
+822    xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
+823    xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
+824    x_samples = np.arange(xstart, xstop + 0.01, 0.01)
+825
+826    plt.figure(figsize=(8, 8 / 1.618))
+827    gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
+828    ax0 = plt.subplot(gs[0])
+829    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')
+830    ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
+831    ax0.set_xticklabels([])
+832    ax0.set_xlim([xstart, xstop])
+833    ax0.set_xticklabels([])
+834    ax0.legend(title=title)
+835
+836    residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], x)) / np.asarray([o.dvalue for o in y])
+837    ax1 = plt.subplot(gs[1])
+838    ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
+839    ax1.tick_params(direction='out')
+840    ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
+841    ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
+842    ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
+843    ax1.set_xlim([xstart, xstop])
+844    ax1.set_ylabel('Residuals')
+845    plt.subplots_adjust(wspace=None, hspace=None)
+846    plt.draw()
 

 
 
             
Generates a plot which compares the fit to the data and displays the corresponding residuals

 
+
For uncorrelated data the residuals are expected to be distributed ~N(0,1).

+
 
Returns

 
 
    
@@ -2019,28 +1843,28 @@ LIist of fitted observables.
 
     

     
-            
1024def error_band(x, func, beta):
-1025    """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
-1026
-1027    Returns
-1028    -------
-1029    err : np.array(Obs)
-1030        Error band for an array of sample values x
-1031    """
-1032    cov = covariance(beta)
-1033    if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
-1034        warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
-1035
-1036    deriv = []
-1037    for i, item in enumerate(x):
-1038        deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
-1039
-1040    err = []
-1041    for i, item in enumerate(x):
-1042        err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
-1043    err = np.array(err)
-1044
-1045    return err
+            
849def error_band(x, func, beta):
+850    """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
+851
+852    Returns
+853    -------
+854    err : np.array(Obs)
+855        Error band for an array of sample values x
+856    """
+857    cov = covariance(beta)
+858    if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
+859        warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
+860
+861    deriv = []
+862    for i, item in enumerate(x):
+863        deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
+864
+865    err = []
+866    for i, item in enumerate(x):
+867        err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
+868    err = np.array(err)
+869
+870    return err
 

 
 
@@ -2067,48 +1891,48 @@ Error band for an array of sample values x
 
     

     
-            
1048def ks_test(objects=None):
-1049    """Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
-1050
-1051    Parameters
-1052    ----------
-1053    objects : list
-1054        List of fit results to include in the analysis (optional).
-1055
-1056    Returns
-1057    -------
-1058    None
-1059    """
-1060
-1061    if objects is None:
-1062        obs_list = []
-1063        for obj in gc.get_objects():
-1064            if isinstance(obj, Fit_result):
-1065                obs_list.append(obj)
-1066    else:
-1067        obs_list = objects
-1068
-1069    p_values = [o.p_value for o in obs_list]
-1070
-1071    bins = len(p_values)
-1072    x = np.arange(0, 1.001, 0.001)
-1073    plt.plot(x, x, 'k', zorder=1)
-1074    plt.xlim(0, 1)
-1075    plt.ylim(0, 1)
-1076    plt.xlabel('p-value')
-1077    plt.ylabel('Cumulative probability')
-1078    plt.title(str(bins) + ' p-values')
-1079
-1080    n = np.arange(1, bins + 1) / np.float64(bins)
-1081    Xs = np.sort(p_values)
-1082    plt.step(Xs, n)
-1083    diffs = n - Xs
-1084    loc_max_diff = np.argmax(np.abs(diffs))
-1085    loc = Xs[loc_max_diff]
-1086    plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
-1087    plt.draw()
-1088
-1089    print(scipy.stats.kstest(p_values, 'uniform'))
+            
873def ks_test(objects=None):
+874    """Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
+875
+876    Parameters
+877    ----------
+878    objects : list
+879        List of fit results to include in the analysis (optional).
+880
+881    Returns
+882    -------
+883    None
+884    """
+885
+886    if objects is None:
+887        obs_list = []
+888        for obj in gc.get_objects():
+889            if isinstance(obj, Fit_result):
+890                obs_list.append(obj)
+891    else:
+892        obs_list = objects
+893
+894    p_values = [o.p_value for o in obs_list]
+895
+896    bins = len(p_values)
+897    x = np.arange(0, 1.001, 0.001)
+898    plt.plot(x, x, 'k', zorder=1)
+899    plt.xlim(0, 1)
+900    plt.ylim(0, 1)
+901    plt.xlabel('p-value')
+902    plt.ylabel('Cumulative probability')
+903    plt.title(str(bins) + ' p-values')
+904
+905    n = np.arange(1, bins + 1) / np.float64(bins)
+906    Xs = np.sort(p_values)
+907    plt.step(Xs, n)
+908    diffs = n - Xs
+909    loc_max_diff = np.argmax(np.abs(diffs))
+910    loc = Xs[loc_max_diff]
+911    plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
+912    plt.draw()
+913
+914    print(scipy.stats.kstest(p_values, 'uniform'))
 

 
 
diff --git a/docs/search.js b/docs/search.js
index c586ed24..a0631f59 100644
--- a/docs/search.js
+++ b/docs/search.js
@@ -1,6 +1,6 @@
 window.pdocSearch = (function(){
 /** elasticlunr - http://weixsong.github.io * Copyright (C) 2017 Oliver Nightingale * Copyright (C) 2017 Wei Song * MIT Licensed */!function(){function e(e){if(null===e||"object"!=typeof e)return e;var t=e.constructor();for(var n in e)e.hasOwnProperty(n)&&(t[n]=e[n]);return t}var t=function(e){var n=new t.Index;return n.pipeline.add(t.trimmer,t.stopWordFilter,t.stemmer),e&&e.call(n,n),n};t.version="0.9.5",lunr=t,t.utils={},t.utils.warn=function(e){return function(t){e.console&&console.warn&&console.warn(t)}}(this),t.utils.toString=function(e){return void 0===e||null===e?"":e.toString()},t.EventEmitter=function(){this.events={}},t.EventEmitter.prototype.addListener=function(){var e=Array.prototype.slice.call(arguments),t=e.pop(),n=e;if("function"!=typeof t)throw new TypeError("last argument must be a function");n.forEach(function(e){this.hasHandler(e)||(this.events[e]=[]),this.events[e].push(t)},this)},t.EventEmitter.prototype.removeListener=function(e,t){if(this.hasHandler(e)){var n=this.events[e].indexOf(t);-1!==n&&(this.events[e].splice(n,1),0==this.events[e].length&&delete this.events[e])}},t.EventEmitter.prototype.emit=function(e){if(this.hasHandler(e)){var t=Array.prototype.slice.call(arguments,1);this.events[e].forEach(function(e){e.apply(void 0,t)},this)}},t.EventEmitter.prototype.hasHandler=function(e){return e in this.events},t.tokenizer=function(e){if(!arguments.length||null===e||void 0===e)return[];if(Array.isArray(e)){var n=e.filter(function(e){return null===e||void 0===e?!1:!0});n=n.map(function(e){return t.utils.toString(e).toLowerCase()});var i=[];return n.forEach(function(e){var n=e.split(t.tokenizer.seperator);i=i.concat(n)},this),i}return e.toString().trim().toLowerCase().split(t.tokenizer.seperator)},t.tokenizer.defaultSeperator=/[\s\-]+/,t.tokenizer.seperator=t.tokenizer.defaultSeperator,t.tokenizer.setSeperator=function(e){null!==e&&void 0!==e&&"object"==typeof e&&(t.tokenizer.seperator=e)},t.tokenizer.resetSeperator=function(){t.tokenizer.seperator=t.tokenizer.defaultSeperator},t.tokenizer.getSeperator=function(){return t.tokenizer.seperator},t.Pipeline=function(){this._queue=[]},t.Pipeline.registeredFunctions={},t.Pipeline.registerFunction=function(e,n){n in t.Pipeline.registeredFunctions&&t.utils.warn("Overwriting existing registered function: "+n),e.label=n,t.Pipeline.registeredFunctions[n]=e},t.Pipeline.getRegisteredFunction=function(e){return e in t.Pipeline.registeredFunctions!=!0?null:t.Pipeline.registeredFunctions[e]},t.Pipeline.warnIfFunctionNotRegistered=function(e){var n=e.label&&e.label in this.registeredFunctions;n||t.utils.warn("Function is not registered with pipeline. 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configuration"),this.buildDefaultConfig(n)}},t.Configuration.prototype.buildDefaultConfig=function(e){this.reset(),e.forEach(function(e){this.config[e]={boost:1,bool:"OR",expand:!1}},this)},t.Configuration.prototype.buildUserConfig=function(e,n){var i="OR",o=!1;if(this.reset(),"bool"in e&&(i=e.bool||i),"expand"in e&&(o=e.expand||o),"fields"in e)for(var r in e.fields)if(n.indexOf(r)>-1){var s=e.fields[r],u=o;void 0!=s.expand&&(u=s.expand),this.config[r]={boost:s.boost||0===s.boost?s.boost:1,bool:s.bool||i,expand:u}}else t.utils.warn("field name in user configuration not found in index instance fields");else this.addAllFields2UserConfig(i,o,n)},t.Configuration.prototype.addAllFields2UserConfig=function(e,t,n){n.forEach(function(n){this.config[n]={boost:1,bool:e,expand:t}},this)},t.Configuration.prototype.get=function(){return this.config},t.Configuration.prototype.reset=function(){this.config={}},lunr.SortedSet=function(){this.length=0,this.elements=[]},lunr.SortedSet.load=function(e){var t=new this;return t.elements=e,t.length=e.length,t},lunr.SortedSet.prototype.add=function(){var e,t;for(e=0;e1;){if(r===e)return o;e>r&&(t=o),r>e&&(n=o),i=n-t,o=t+Math.floor(i/2),r=this.elements[o]}return r===e?o:-1},lunr.SortedSet.prototype.locationFor=function(e){for(var t=0,n=this.elements.length,i=n-t,o=t+Math.floor(i/2),r=this.elements[o];i>1;)e>r&&(t=o),r>e&&(n=o),i=n-t,o=t+Math.floor(i/2),r=this.elements[o];return r>e?o:e>r?o+1:void 0},lunr.SortedSet.prototype.intersect=function(e){for(var t=new lunr.SortedSet,n=0,i=0,o=this.length,r=e.length,s=this.elements,u=e.elements;;){if(n>o-1||i>r-1)break;s[n]!==u[i]?s[n]u[i]&&i++:(t.add(s[n]),n++,i++)}return t},lunr.SortedSet.prototype.clone=function(){var e=new lunr.SortedSet;return e.elements=this.toArray(),e.length=e.elements.length,e},lunr.SortedSet.prototype.union=function(e){var t,n,i;this.length>=e.length?(t=this,n=e):(t=e,n=this),i=t.clone();for(var o=0,r=n.toArray();oWhat is pyerrors?\n\n
pyerrors is a python package for error computation and propagation of Markov chain Monte Carlo data.\nIt is based on the gamma method arXiv:hep-lat/0306017. Some of its features are:
\n\n
    \n
  • automatic differentiation for exact linear error propagation as suggested in arXiv:1809.01289 (partly based on the autograd package).
    • \n
  • treatment of slow modes in the simulation as suggested in arXiv:1009.5228.
    • \n
  • coherent error propagation for data from different Markov chains.
    • \n
  • non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in arXiv:1809.01289.
    • \n
  • real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...).
    • \n
\n\n
More detailed examples can found in the GitHub repository \"badge\".
\n\n
If you use pyerrors for research that leads to a publication please consider citing:
\n\n
    \n
  • Fabian Joswig, Simon Kuberski, Justus T. Kuhlmann, Jan Neuendorf, pyerrors: a python framework for error analysis of Monte Carlo data. [arXiv:2209.14371 [hep-lat]].
    • \n
  • Ulli Wolff, Monte Carlo errors with less errors. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum).
    • \n
  • Alberto Ramos, Automatic differentiation for error analysis of Monte Carlo data. Comput.Phys.Commun. 238 (2019) 19-35.
    • \n
\n\n
and
\n\n
    \n
  • Stefan Schaefer, Rainer Sommer, Francesco Virotta, Critical slowing down and error analysis in lattice QCD simulations. Nucl.Phys.B 845 (2011) 93-119.
    • \n
\n\n
where applicable.
\n\n
There exist similar publicly available implementations of gamma method error analysis suites in Fortran, Julia and Python.
\n\n
Basic example
\n\n
\n
import numpy as np\nimport pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name']) # Initialize an Obs object\nmy_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object\nmy_new_obs.gamma_method()                     # Estimate the statistical error\nprint(my_new_obs)                             # Print the result to stdout\n> 0.31498(72)\n
\n
\n\n
The Obs class
\n\n
pyerrors introduces a new datatype, Obs, which simplifies error propagation and estimation for auto- and cross-correlated data.\nAn Obs object can be initialized with two arguments, the first is a list containing the samples for an observable from a Monte Carlo chain.\nThe samples can either be provided as python list or as numpy array.\nThe second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble. It is crucial for the correct error propagation that observations from the same Monte Carlo history are labeled with the same name. See Multiple ensembles/replica for details.
\n\n
\n
import pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name'])\n
\n
\n\n
Error propagation
\n\n
When performing mathematical operations on Obs objects the correct error propagation is intrinsically taken care of using a first order Taylor expansion\n$$\\delta_f^i=\\sum_\\alpha \\bar{f}_\\alpha \\delta_\\alpha^i\\,,\\quad \\delta_\\alpha^i=a_\\alpha^i-\\bar{a}_\\alpha\\,,$$\nas introduced in arXiv:hep-lat/0306017.\nThe required derivatives $\\bar{f}_\\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in arXiv:1809.01289.
\n\n
The Obs class is designed such that mathematical numpy functions can be used on Obs just as for regular floats.
\n\n
\n
import numpy as np\nimport pyerrors as pe\n\nmy_obs1 = pe.Obs([samples1], ['ensemble_name'])\nmy_obs2 = pe.Obs([samples2], ['ensemble_name'])\n\nmy_sum = my_obs1 + my_obs2\n\nmy_m_eff = np.log(my_obs1 / my_obs2)\n\niamzero = my_m_eff - my_m_eff\n# Check that value and fluctuations are zero within machine precision\nprint(iamzero == 0.0)\n> True\n
\n
\n\n
Error estimation
\n\n
The error estimation within pyerrors is based on the gamma method introduced in arXiv:hep-lat/0306017.\nAfter having arrived at the derived quantity of interest the gamma_method can be called as detailed in the following example.
\n\n
\n
my_sum.gamma_method()\nprint(my_sum)\n> 1.70(57)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 5.72046658e-01 +/- 7.56746598e-02 (33.650%)\n>  t_int         2.71422900e+00 +/- 6.40320983e-01 S = 2.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
\n
\n\n
We use the following definition of the integrated autocorrelation time established in Madras & Sokal 1988\n$$\\tau_\\mathrm{int}=\\frac{1}{2}+\\sum_{t=1}^{W}\\rho(t)\\geq \\frac{1}{2}\\,.$$\nThe window $W$ is determined via the automatic windowing procedure described in arXiv:hep-lat/0306017.\nThe standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the gamma_method as parameter.
\n\n
\n
my_sum.gamma_method(S=3.0)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.30675201e-01 +/- 1.04585650e-01 (37.099%)\n>  t_int         3.29909703e+00 +/- 9.77310102e-01 S = 3.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
\n
\n\n
The integrated autocorrelation time $\\tau_\\mathrm{int}$ and the autocorrelation function $\\rho(W)$ can be monitored via the methods pyerrors.obs.Obs.plot_tauint and pyerrors.obs.Obs.plot_rho.
\n\n
If the parameter $S$ is set to zero it is assumed that the dataset does not exhibit any autocorrelation and the window size is chosen to be zero.\nIn this case the error estimate is identical to the sample standard error.
\n\n
Exponential tails
\n\n
Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\\rho$ as suggested in arXiv:1009.5228. The longest autocorrelation time in the history, $\\tau_\\mathrm{exp}$, can be passed to the gamma_method as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.
\n\n
\n
my_sum.gamma_method(tau_exp=7.2)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.28097762e-01 +/- 5.79077524e-02 (36.947%)\n>  t_int         3.27218667e+00 +/- 7.99583654e-01 tau_exp = 7.20,  N_sigma = 1\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
\n
\n\n
For the full API see pyerrors.obs.Obs.gamma_method.
\n\n
Multiple ensembles/replica
\n\n
Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their name.
\n\n
\n
obs1 = pe.Obs([samples1], ['ensemble1'])\nobs2 = pe.Obs([samples2], ['ensemble2'])\n\nmy_sum = obs1 + obs2\nmy_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 2 ensembles:\n>   \u00b7 Ensemble 'ensemble1' : 1000 configurations (from 1 to 1000)\n>   \u00b7 Ensemble 'ensemble2' : 500 configurations (from 1 to 500)\n
\n
\n\n
Observables from the same Monte Carlo chain have to be initialized with the same name for correct error propagation. If different names were used in this case the data would be treated as statistically independent resulting in loss of relevant information and a potential over or under estimate of the statistical error.
\n\n
pyerrors identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar | in the name of the data set.
\n\n
\n
obs1 = pe.Obs([samples1], ['ensemble1|r01'])\nobs2 = pe.Obs([samples2], ['ensemble1|r02'])\n\n> my_sum = obs1 + obs2\n> my_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1'\n>     \u00b7 Replicum 'r01' : 1000 configurations (from 1 to 1000)\n>     \u00b7 Replicum 'r02' : 500 configurations (from 1 to 500)\n
\n
\n\n
Error estimation for multiple ensembles
\n\n
In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example.
\n\n
\n
pe.Obs.S_dict['ensemble1'] = 2.5\npe.Obs.tau_exp_dict['ensemble2'] = 8.0\npe.Obs.tau_exp_dict['ensemble3'] = 2.0\n
\n
\n\n
In case the gamma_method is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.\nPassing arguments to the gamma_method still dominates over the dictionaries.
\n\n
Irregular Monte Carlo chains
\n\n
Obs objects defined on irregular Monte Carlo chains can be initialized with the parameter idl.
\n\n
\n
# Observable defined on configurations 20 to 519\nobs1 = pe.Obs([samples1], ['ensemble1'], idl=[range(20, 520)])\nobs1.details()\n> Result         9.98319881e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 20 to 519)\n\n# Observable defined on every second configuration between 5 and 1003\nobs2 = pe.Obs([samples2], ['ensemble1'], idl=[range(5, 1005, 2)])\nobs2.details()\n> Result         9.99100712e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 5 to 1003 in steps of 2)\n\n# Observable defined on configurations 2, 9, 28, 29 and 501\nobs3 = pe.Obs([samples3], ['ensemble1'], idl=[[2, 9, 28, 29, 501]])\nobs3.details()\n> Result         1.01718064e+00\n> 5 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 5 configurations (irregular range)\n
\n
\n\n
Obs objects defined on regular and irregular histories of the same ensemble can be combined with each other and the correct error propagation and estimation is automatically taken care of.
\n\n
Warning: Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.\nMake sure to check the autocorrelation time with e.g. pyerrors.obs.Obs.plot_rho or pyerrors.obs.Obs.plot_tauint.
\n\n
For the full API see pyerrors.obs.Obs.
\n\n
Correlators
\n\n
When one is not interested in single observables but correlation functions, pyerrors offers the Corr class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize a Corr objects one needs to arrange the data as a list of Obs
\n\n
\n
my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0      0.7957(80)\n> 1      0.5156(51)\n> 2      0.3227(33)\n> 3      0.2041(21)\n
\n
\n\n
In case the correlation functions are not defined on the outermost timeslices, for example because of fixed boundary conditions, a padding can be introduced.
\n\n
\n
my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3], padding=[1, 1])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0\n> 1      0.7957(80)\n> 2      0.5156(51)\n> 3      0.3227(33)\n> 4      0.2041(21)\n> 5\n
\n
\n\n
The individual entries of a correlator can be accessed via slicing
\n\n
\n
print(my_corr[3])\n> 0.3227(33)\n
\n
\n\n
Error propagation with the Corr class works very similar to Obs objects. Mathematical operations are overloaded and Corr objects can be computed together with other Corr objects, Obs objects or real numbers and integers.
\n\n
\n
my_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr\n
\n
\n\n
pyerrors provides the user with a set of regularly used methods for the manipulation of correlator objects:
\n\n
    \n
  • Corr.gamma_method applies the gamma method to all entries of the correlator.
    • \n
  • Corr.m_eff to construct effective masses. Various variants for periodic and fixed temporal boundary conditions are available.
    • \n
  • Corr.deriv returns the first derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
    • \n
  • Corr.second_deriv returns the second derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
    • \n
  • Corr.symmetric symmetrizes parity even correlations functions, assuming periodic boundary conditions.
    • \n
  • Corr.anti_symmetric anti-symmetrizes parity odd correlations functions, assuming periodic boundary conditions.
    • \n
  • Corr.T_symmetry averages a correlator with its time symmetry partner, assuming fixed boundary conditions.
    • \n
  • Corr.plateau extracts a plateau value from the correlator in a given range.
    • \n
  • Corr.roll periodically shifts the correlator.
    • \n
  • Corr.reverse reverses the time ordering of the correlator.
    • \n
  • Corr.correlate constructs a disconnected correlation function from the correlator and another Corr or Obs object.
    • \n
  • Corr.reweight reweights the correlator.
    • \n
\n\n
pyerrors can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (see pyerrors.correlators.Corr.GEVP).
\n\n
For the full API see pyerrors.correlators.Corr.
\n\n
Complex valued observables
\n\n
pyerrors can handle complex valued observables via the class pyerrors.obs.CObs.\nCObs are initialized with a real and an imaginary part which both can be Obs valued.
\n\n
\n
my_real_part = pe.Obs([samples1], ['ensemble1'])\nmy_imag_part = pe.Obs([samples2], ['ensemble1'])\n\nmy_cobs = pe.CObs(my_real_part, my_imag_part)\nmy_cobs.gamma_method()\nprint(my_cobs)\n> (0.9959(91)+0.659(28)j)\n
\n
\n\n
Elementary mathematical operations are overloaded and samples are properly propagated as for the Obs class.
\n\n
\n
my_derived_cobs = (my_cobs + my_cobs.conjugate()) / np.abs(my_cobs)\nmy_derived_cobs.gamma_method()\nprint(my_derived_cobs)\n> (1.668(23)+0.0j)\n
\n
\n\n
The Covobs class
\n\n
In many projects, auxiliary data that is not based on Monte Carlo chains enters. Examples are experimentally determined mesons masses which are used to set the scale or renormalization constants. These numbers come with an error that has to be propagated through the analysis. The Covobs class allows to define such quantities in pyerrors. Furthermore, external input might consist of correlated quantities. An example are the parameters of an interpolation formula, which are defined via mean values and a covariance matrix between all parameters. The contribution of the interpolation formula to the error of a derived quantity therefore might depend on the complete covariance matrix.
\n\n
This concept is built into the definition of Covobs. In pyerrors, external input is defined by $M$ mean values, a $M\\times M$ covariance matrix, where $M=1$ is permissible, and a name that uniquely identifies the covariance matrix. Below, we define the pion mass, based on its mean value and error, 134.9768(5). Note, that the square of the error enters cov_Obs, since the second argument of this function is the covariance matrix of the Covobs.
\n\n
\n
import pyerrors.obs as pe\n\nmpi = pe.cov_Obs(134.9768, 0.0005**2, 'pi^0 mass')\nmpi.gamma_method()\nmpi.details()\n> Result         1.34976800e+02 +/- 5.00000000e-04 +/- 0.00000000e+00 (0.000%)\n>  pi^0 mass     5.00000000e-04\n> 0 samples in 1 ensemble:\n>   \u00b7 Covobs   'pi^0 mass'\n
\n
\n\n
The resulting object mpi is an Obs that contains a Covobs. In the following, it may be handled as any other Obs. The contribution of the covariance matrix to the error of an Obs is determined from the $M \\times M$ covariance matrix $\\Sigma$ and the gradient of the Obs with respect to the external quantities, which is the $1\\times M$ Jacobian matrix $J$, via\n$$s = \\sqrt{J^T \\Sigma J}\\,,$$\nwhere the Jacobian is computed for each derived quantity via automatic differentiation.
\n\n
Correlated auxiliary data is defined similarly to above, e.g., via
\n\n
\n
RAP = pe.cov_Obs([16.7457, -19.0475], [[3.49591, -6.07560], [-6.07560, 10.5834]], 'R_AP, 1906.03445, (5.3a)')\nprint(RAP)\n> [Obs[16.7(1.9)], Obs[-19.0(3.3)]]\n
\n
\n\n
where RAP now is a list of two Obs that contains the two correlated parameters.
\n\n
Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the Covobs class allows to quote the derivative of a result with respect to the external quantities. If these derivatives are published together with the result, small shifts in the definition of external quantities, e.g., the definition of the physical point, can be performed a posteriori based on the published information. This may help to compare results of different groups. The gradient of an Obs o with respect to a covariance matrix with the identifying string k may be accessed via
\n\n
\n
o.covobs[k].grad\n
\n
\n\n
Error propagation in iterative algorithms
\n\n
pyerrors supports exact linear error propagation for iterative algorithms like various variants of non-linear least squares fits or root finding. The derivatives required for the error propagation are calculated as described in arXiv:1809.01289.
\n\n
Least squares fits
\n\n
Standard non-linear least square fits with errors on the dependent but not the independent variables can be performed with pyerrors.fits.least_squares. As default solver the Levenberg-Marquardt algorithm implemented in scipy is used.
\n\n
Fit functions have to be of the following form
\n\n
\n
import autograd.numpy as anp\n\ndef func(a, x):\n    return a[1] * anp.exp(-a[0] * x)\n
\n
\n\n
It is important that numerical functions refer to autograd.numpy instead of numpy for the automatic differentiation in iterative algorithms to work properly.
\n\n
Fits can then be performed via
\n\n
\n
fit_result = pe.fits.least_squares(x, y, func)\nprint("\\n", fit_result)\n> Fit with 2 parameters\n> Method: Levenberg-Marquardt\n> `ftol` termination condition is satisfied.\n> chisquare/d.o.f.: 0.9593035785160936\n\n>  Goodness of fit:\n> \u03c7\u00b2/d.o.f. = 0.959304\n> p-value   = 0.5673\n> Fit parameters:\n> 0      0.0548(28)\n> 1      1.933(64)\n
\n
\n\n
where x is a list or numpy.array of floats and y is a list or numpy.array of Obs.
\n\n
Data stored in Corr objects can be fitted directly using the Corr.fit method.
\n\n
\n
my_corr = pe.Corr(y)\nfit_result = my_corr.fit(func, fitrange=[12, 25])\n
\n
\n\n
this can simplify working with absolute fit ranges and takes care of gaps in the data automatically.
\n\n
For fit functions with multiple independent variables the fit function can be of the form
\n\n
\n
def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
\n
\n\n
pyerrors also supports correlated fits which can be triggered via the parameter correlated_fit=True.\nDetails about how the required covariance matrix is estimated can be found in pyerrors.obs.covariance.
\n\n
Direct visualizations of the performed fits can be triggered via resplot=True or qqplot=True. For all available options see pyerrors.fits.least_squares.
\n\n
Total least squares fits
\n\n
pyerrors can also fit data with errors on both the dependent and independent variables using the total least squares method also referred to orthogonal distance regression as implemented in scipy, see pyerrors.fits.least_squares. The syntax is identical to the standard least squares case, the only difference being that x also has to be a list or numpy.array of Obs.
\n\n
For the full API see pyerrors.fits for fits and pyerrors.roots for finding roots of functions.
\n\n
Matrix operations
\n\n
pyerrors provides wrappers for Obs- and CObs-valued matrix operations based on numpy.linalg. The supported functions include:
\n\n
    \n
  • inv for the matrix inverse.
    • \n
  • cholseky for the Cholesky decomposition.
    • \n
  • det for the matrix determinant.
    • \n
  • eigh for eigenvalues and eigenvectors of hermitean matrices.
    • \n
  • eig for eigenvalues of general matrices.
    • \n
  • pinv for the Moore-Penrose pseudoinverse.
    • \n
  • svd for the singular-value-decomposition.
    • \n
\n\n
For the full API see pyerrors.linalg.
\n\n
Export data
\n\n
The preferred exported file format within pyerrors is json.gz. Files written to this format are valid JSON files that have been compressed using gzip. The structure of the content is inspired by the dobs format of the ALPHA collaboration. The aim of the format is to facilitate the storage of data in a self-contained way such that, even years after the creation of the file, it is possible to extract all necessary information:
\n\n
    \n
  • What observables are stored? Possibly: How exactly are they defined.
    • \n
  • How does each single ensemble or external quantity contribute to the error of the observable?
    • \n
  • Who did write the file when and on which machine?
    • \n
\n\n
This can be achieved by storing all information in one single file. The export routines of pyerrors are written such that as much information as possible is written automatically as described in the following example
\n\n
\n
my_obs = pe.Obs([samples], ["test_ensemble"])\nmy_obs.tag = "My observable"\n\npe.input.json.dump_to_json(my_obs, "test_output_file", description="This file contains a test observable")\n# For a single observable one can equivalently use the class method dump\nmy_obs.dump("test_output_file", description="This file contains a test observable")\n\ncheck = pe.input.json.load_json("test_output_file")\n\nprint(my_obs == check)\n> True\n
\n
\n\n
The format also allows to directly write out the content of Corr objects or lists and arrays of Obs objects by passing the desired data to pyerrors.input.json.dump_to_json.
\n\n
json.gz format specification
\n\n
The first entries of the file provide optional auxiliary information:
\n\n
    \n
  • program is a string that indicates which program was used to write the file.
    • \n
  • version is a string that specifies the version of the format.
    • \n
  • who is a string that specifies the user name of the creator of the file.
    • \n
  • date is a string and contains the creation date of the file.
    • \n
  • host is a string and contains the hostname of the machine where the file has been written.
    • \n
  • description contains information on the content of the file. This field is not filled automatically in pyerrors. The user is advised to provide as detailed information as possible in this field. Examples are: Input files of measurements or simulations, LaTeX formulae or references to publications to specify how the observables have been computed, details on the analysis strategy, ... This field may be any valid JSON type. Strings, arrays or objects (equivalent to dicts in python) are well suited to provide information.
    • \n
\n\n
The only necessary entry of the file is the field\n-obsdata, an array that contains the actual data.
\n\n
Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of Obs, list, numpy.ndarray, Corr. All Obs inside a structure (with dimension > 0) have to be defined on the same set of configurations. Different structures, that are represented by entries of the array obsdata, are treated independently. Each entry of the array obsdata has the following required entries:
\n\n
    \n
  • type is a string that specifies the type of the structure. This allows to parse the content to the correct form after reading the file. It is always possible to interpret the content as list of Obs.
    • \n
  • value is an array that contains the mean values of the Obs inside the structure.\nThe following entries are optional:
    • \n
  • layout is a string that specifies the layout of multi-dimensional structures. Examples are \"2, 2\" for a 2x2 dimensional matrix or  \"64, 4, 4\" for a Corr with $T=64$ and 4x4 matrices on each time slices. \"1\" denotes a single Obs. Multi-dimensional structures are stored in row-major format (see below).
    • \n
  • tag is any JSON type. It contains additional information concerning the structure. The tag of an Obs in pyerrors is written here.
    • \n
  • reweighted is a Bool that may be used to specify, whether the Obs in the structure have been reweighted.
    • \n
  • data is an array that contains the data from MC chains. We will define it below.
    • \n
  • cdata is an array that contains the data from external quantities with an error (Covobs in pyerrors). We will define it below.
    • \n
\n\n
The array data contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains:
\n\n
    \n
  • id, a string that contains the name of the ensemble
    • \n
  • replica, an array that contains an entry per replica of the ensemble.
    • \n
\n\n
Each entry of replica contains\nname, a string that contains the name of the replica\ndeltas, an array that contains the actual data.
\n\n
Each entry in deltas corresponds to one configuration of the replica and has $1+N$ many entries. The first entry is an integer that specifies the configuration number that, together with ensemble and replica name, may be used to uniquely identify the configuration on which the data has been obtained. The following N entries specify the deltas, i.e., the deviation of the observable from the mean value on this configuration, of each Obs inside the structure. Multi-dimensional structures are stored in a row-major format. For primary observables, such as correlation functions, $value + delta_i$ matches the primary data obtained on the configuration.
\n\n
The array cdata contains information about the contribution of auxiliary observables, represented by Covobs in pyerrors, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains:
\n\n
    \n
  • id, a string that identifies the covariance matrix
    • \n
  • layout, a string that defines the dimensions of the $M\\times M$ covariance matrix (has to be \"M, M\" or \"1\").
    • \n
  • cov, an array that contains the $M\\times M$ many entries of the covariance matrix, stored in row-major format.
    • \n
  • grad, an array that contains N entries, one for each Obs inside the structure. Each entry itself is an array, that contains the M gradients of the Nth observable  with respect to the quantity that corresponds to the Mth diagonal entry of the covariance matrix.
    • \n
\n\n
A JSON schema that may be used to verify the correctness of a file with respect to the format definition is stored in ./examples/json_schema.json. The schema is a self-descriptive format definition and contains an exemplary file.
\n\n
Julia I/O routines for the json.gz format, compatible with ADerrors.jl, can be found here.
\n"}, "pyerrors.correlators": {"fullname": "pyerrors.correlators", "modulename": "pyerrors.correlators", "kind": "module", "doc": "
\n"}, "pyerrors.correlators.Corr": {"fullname": "pyerrors.correlators.Corr", "modulename": "pyerrors.correlators", "qualname": "Corr", "kind": "class", "doc": "
The class for a correlator (time dependent sequence of pe.Obs).
\n\n
Everything, this class does, can be achieved using lists or arrays of Obs.\nBut it is simply more convenient to have a dedicated object for correlators.\nOne often wants to add or multiply correlators of the same length at every timeslice and it is inconvenient\nto iterate over all timeslices for every operation. This is especially true, when dealing with matrices.
\n\n
The correlator can have two types of content: An Obs at every timeslice OR a GEVP\nmatrix at every timeslice. Other dependency (eg. spatial) are not supported.
\n"}, "pyerrors.correlators.Corr.__init__": {"fullname": "pyerrors.correlators.Corr.__init__", "modulename": "pyerrors.correlators", "qualname": "Corr.__init__", "kind": "function", "doc": "
Initialize a Corr object.
\n\n
Parameters
\n\n
    \n
  • data_input (list or array):\nlist of Obs or list of arrays of Obs or array of Corrs
    • \n
  • padding (list, optional):\nList with two entries where the first labels the padding\nat the front of the correlator and the second the padding\nat the back.
    • \n
  • prange (list, optional):\nList containing the first and last timeslice of the plateau\nregion indentified for this correlator.
    • \n
\n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "pyerrors.correlators.Corr.gamma_method": {"fullname": "pyerrors.correlators.Corr.gamma_method", "modulename": "pyerrors.correlators", "qualname": "Corr.gamma_method", "kind": "function", "doc": "
Apply the gamma method to the content of the Corr.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.gm": {"fullname": "pyerrors.correlators.Corr.gm", "modulename": "pyerrors.correlators", "qualname": "Corr.gm", "kind": "function", "doc": "
Apply the gamma method to the content of the Corr.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.projected": {"fullname": "pyerrors.correlators.Corr.projected", "modulename": "pyerrors.correlators", "qualname": "Corr.projected", "kind": "function", "doc": "
We need to project the Correlator with a Vector to get a single value at each timeslice.
\n\n
The method can use one or two vectors.\nIf two are specified it returns v1@G@v2 (the order might be very important.)\nBy default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to
\n", "signature": "(self, vector_l=None, vector_r=None, normalize=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.item": {"fullname": "pyerrors.correlators.Corr.item", "modulename": "pyerrors.correlators", "qualname": "Corr.item", "kind": "function", "doc": "
Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.
\n\n
Parameters
\n\n
    \n
  • i (int):\nFirst index to be picked.
    • \n
  • j (int):\nSecond index to be picked.
    • \n
\n", "signature": "(self, i, j):", "funcdef": "def"}, "pyerrors.correlators.Corr.plottable": {"fullname": "pyerrors.correlators.Corr.plottable", "modulename": "pyerrors.correlators", "qualname": "Corr.plottable", "kind": "function", "doc": "
Outputs the correlator in a plotable format.
\n\n
Outputs three lists containing the timeslice index, the value on each\ntimeslice and the error on each timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.symmetric": {"fullname": "pyerrors.correlators.Corr.symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.symmetric", "kind": "function", "doc": "
Symmetrize the correlator around x0=0.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.anti_symmetric": {"fullname": "pyerrors.correlators.Corr.anti_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.anti_symmetric", "kind": "function", "doc": "
Anti-symmetrize the correlator around x0=0.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.is_matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.is_matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.is_matrix_symmetric", "kind": "function", "doc": "
Checks whether a correlator matrices is symmetric on every timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.matrix_symmetric", "kind": "function", "doc": "
Symmetrizes the correlator matrices on every timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.GEVP": {"fullname": "pyerrors.correlators.Corr.GEVP", "modulename": "pyerrors.correlators", "qualname": "Corr.GEVP", "kind": "function", "doc": "
Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.
\n\n
The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the\nlargest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing
\n\n
\n
C.GEVP(t0=2)[0]  # Ground state vector(s)\nC.GEVP(t0=2)[:3]  # Vectors for the lowest three states\n
\n
\n\n
Parameters
\n\n
    \n
  • t0 (int):\nThe time t0 for the right hand side of the GEVP according to $G(t)v_i=\\lambda_i G(t_0)v_i$
    • \n
  • ts (int):\nfixed time $G(t_s)v_i=\\lambda_i G(t_0)v_i$ if sort=None.\nIf sort=\"Eigenvector\" it gives a reference point for the sorting method.
    • \n
  • sort (string):\nIf this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.\n
      \n
    • \"Eigenvalue\": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice.
      • \n
    • \"Eigenvector\": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.\nThe reference state is identified by its eigenvalue at $t=t_s$.
      • \n
    • \n
\n\n
Other Parameters
\n\n
    \n
  • state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
    • \n
\n", "signature": "(self, t0, ts=None, sort='Eigenvalue', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Eigenvalue": {"fullname": "pyerrors.correlators.Corr.Eigenvalue", "modulename": "pyerrors.correlators", "qualname": "Corr.Eigenvalue", "kind": "function", "doc": "
Determines the eigenvalue of the GEVP by solving and projecting the correlator
\n\n
Parameters
\n\n
    \n
  • state (int):\nThe state one is interested in ordered by energy. The lowest state is zero.
    • \n
  • All other parameters are identical to the ones of Corr.GEVP.
    • \n
\n", "signature": "(self, t0, ts=None, state=0, sort='Eigenvalue'):", "funcdef": "def"}, "pyerrors.correlators.Corr.Hankel": {"fullname": "pyerrors.correlators.Corr.Hankel", "modulename": "pyerrors.correlators", "qualname": "Corr.Hankel", "kind": "function", "doc": "
Constructs an NxN Hankel matrix
\n\n
C(t) c(t+1) ... c(t+n-1)\nC(t+1) c(t+2) ... c(t+n)\n.................\nC(t+(n-1)) c(t+n) ... c(t+2(n-1))
\n\n
Parameters
\n\n
    \n
  • N (int):\nDimension of the Hankel matrix
    • \n
  • periodic (bool, optional):\ndetermines whether the matrix is extended periodically
    • \n
\n", "signature": "(self, N, periodic=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.roll": {"fullname": "pyerrors.correlators.Corr.roll", "modulename": "pyerrors.correlators", "qualname": "Corr.roll", "kind": "function", "doc": "
Periodically shift the correlator by dt timeslices
\n\n
Parameters
\n\n
    \n
  • dt (int):\nnumber of timeslices
    • \n
\n", "signature": "(self, dt):", "funcdef": "def"}, "pyerrors.correlators.Corr.reverse": {"fullname": "pyerrors.correlators.Corr.reverse", "modulename": "pyerrors.correlators", "qualname": "Corr.reverse", "kind": "function", "doc": "
Reverse the time ordering of the Corr
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.thin": {"fullname": "pyerrors.correlators.Corr.thin", "modulename": "pyerrors.correlators", "qualname": "Corr.thin", "kind": "function", "doc": "
Thin out a correlator to suppress correlations
\n\n
Parameters
\n\n
    \n
  • spacing (int):\nKeep only every 'spacing'th entry of the correlator
    • \n
  • offset (int):\nOffset the equal spacing
    • \n
\n", "signature": "(self, spacing=2, offset=0):", "funcdef": "def"}, "pyerrors.correlators.Corr.correlate": {"fullname": "pyerrors.correlators.Corr.correlate", "modulename": "pyerrors.correlators", "qualname": "Corr.correlate", "kind": "function", "doc": "
Correlate the correlator with another correlator or Obs
\n\n
Parameters
\n\n
    \n
  • partner (Obs or Corr):\npartner to correlate the correlator with.\nCan either be an Obs which is correlated with all entries of the\ncorrelator or a Corr of same length.
    • \n
\n", "signature": "(self, partner):", "funcdef": "def"}, "pyerrors.correlators.Corr.reweight": {"fullname": "pyerrors.correlators.Corr.reweight", "modulename": "pyerrors.correlators", "qualname": "Corr.reweight", "kind": "function", "doc": "
Reweight the correlator.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl.
    • \n
\n", "signature": "(self, weight, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.T_symmetry": {"fullname": "pyerrors.correlators.Corr.T_symmetry", "modulename": "pyerrors.correlators", "qualname": "Corr.T_symmetry", "kind": "function", "doc": "
Return the time symmetry average of the correlator and its partner
\n\n
Parameters
\n\n
    \n
  • partner (Corr):\nTime symmetry partner of the Corr
    • \n
  • partity (int):\nParity quantum number of the correlator, can be +1 or -1
    • \n
\n", "signature": "(self, partner, parity=1):", "funcdef": "def"}, "pyerrors.correlators.Corr.deriv": {"fullname": "pyerrors.correlators.Corr.deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.deriv", "kind": "function", "doc": "
Return the first derivative of the correlator with respect to x0.
\n\n
Parameters
\n\n
    \n
  • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, forward, backward, improved, log, default: symmetric
    • \n
\n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.second_deriv": {"fullname": "pyerrors.correlators.Corr.second_deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.second_deriv", "kind": "function", "doc": "
Return the second derivative of the correlator with respect to x0.
\n\n
Parameters
\n\n
    \n
  • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, improved, log, default: symmetric
    • \n
\n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.m_eff": {"fullname": "pyerrors.correlators.Corr.m_eff", "modulename": "pyerrors.correlators", "qualname": "Corr.m_eff", "kind": "function", "doc": "
Returns the effective mass of the correlator as correlator object
\n\n
Parameters
\n\n
    \n
  • variant (str):\nlog : uses the standard effective mass log(C(t) / C(t+1))\ncosh, periodic : Use periodicitiy of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.\nsinh : Use anti-periodicitiy of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.\nSee, e.g., arXiv:1205.5380\narccosh : Uses the explicit form of the symmetrized correlator (not recommended)\nlogsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
    • \n
  • guess (float):\nguess for the root finder, only relevant for the root variant
    • \n
\n", "signature": "(self, variant='log', guess=1.0):", "funcdef": "def"}, "pyerrors.correlators.Corr.fit": {"fullname": "pyerrors.correlators.Corr.fit", "modulename": "pyerrors.correlators", "qualname": "Corr.fit", "kind": "function", "doc": "
Fits function to the data
\n\n
Parameters
\n\n
    \n
  • function (obj):\nfunction to fit to the data. See fits.least_squares for details.
    • \n
  • fitrange (list):\nTwo element list containing the timeslices on which the fit is supposed to start and stop.\nCaution: This range is inclusive as opposed to standard python indexing.\nfitrange=[4, 6] corresponds to the three entries 4, 5 and 6.\nIf not specified, self.prange or all timeslices are used.
    • \n
  • silent (bool):\nDecides whether output is printed to the standard output.
    • \n
\n", "signature": "(self, function, fitrange=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.plateau": {"fullname": "pyerrors.correlators.Corr.plateau", "modulename": "pyerrors.correlators", "qualname": "Corr.plateau", "kind": "function", "doc": "
Extract a plateau value from a Corr object
\n\n
Parameters
\n\n
    \n
  • plateau_range (list):\nlist with two entries, indicating the first and the last timeslice\nof the plateau region.
    • \n
  • method (str):\nmethod to extract the plateau.\n    'fit' fits a constant to the plateau region\n    'avg', 'average' or 'mean' just average over the given timeslices.
    • \n
  • auto_gamma (bool):\napply gamma_method with default parameters to the Corr. Defaults to None
    • \n
\n", "signature": "(self, plateau_range=None, method='fit', auto_gamma=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.set_prange": {"fullname": "pyerrors.correlators.Corr.set_prange", "modulename": "pyerrors.correlators", "qualname": "Corr.set_prange", "kind": "function", "doc": "
Sets the attribute prange of the Corr object.
\n", "signature": "(self, prange):", "funcdef": "def"}, "pyerrors.correlators.Corr.show": {"fullname": "pyerrors.correlators.Corr.show", "modulename": "pyerrors.correlators", "qualname": "Corr.show", "kind": "function", "doc": "
Plots the correlator using the tag of the correlator as label if available.
\n\n
Parameters
\n\n
    \n
  • x_range (list):\nlist of two values, determining the range of the x-axis e.g. [4, 8].
    • \n
  • comp (Corr or list of Corr):\nCorrelator or list of correlators which are plotted for comparison.\nThe tags of these correlators are used as labels if available.
    • \n
  • logscale (bool):\nSets y-axis to logscale.
    • \n
  • plateau (Obs):\nPlateau value to be visualized in the figure.
    • \n
  • fit_res (Fit_result):\nFit_result object to be visualized.
    • \n
  • ylabel (str):\nLabel for the y-axis.
    • \n
  • save (str):\npath to file in which the figure should be saved.
    • \n
  • auto_gamma (bool):\nApply the gamma method with standard parameters to all correlators and plateau values before plotting.
    • \n
  • hide_sigma (float):\nHides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
    • \n
  • references (list):\nList of floating point values that are displayed as horizontal lines for reference.
    • \n
  • title (string):\nOptional title of the figure.
    • \n
\n", "signature": "(\tself,\tx_range=None,\tcomp=None,\ty_range=None,\tlogscale=False,\tplateau=None,\tfit_res=None,\tylabel=None,\tsave=None,\tauto_gamma=False,\thide_sigma=None,\treferences=None,\ttitle=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.spaghetti_plot": {"fullname": "pyerrors.correlators.Corr.spaghetti_plot", "modulename": "pyerrors.correlators", "qualname": "Corr.spaghetti_plot", "kind": "function", "doc": "
Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.
\n\n
Parameters
\n\n
    \n
  • logscale (bool):\nDetermines whether the scale of the y-axis is logarithmic or standard.
    • \n
\n", "signature": "(self, logscale=True):", "funcdef": "def"}, "pyerrors.correlators.Corr.dump": {"fullname": "pyerrors.correlators.Corr.dump", "modulename": "pyerrors.correlators", "qualname": "Corr.dump", "kind": "function", "doc": "
Dumps the Corr into a file of chosen type
\n\n
Parameters
\n\n
    \n
  • filename (str):\nName of the file to be saved.
    • \n
  • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n", "signature": "(self, filename, datatype='json.gz', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.print": {"fullname": "pyerrors.correlators.Corr.print", "modulename": "pyerrors.correlators", "qualname": "Corr.print", "kind": "function", "doc": "
\n", "signature": "(self, print_range=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.sqrt": {"fullname": "pyerrors.correlators.Corr.sqrt", "modulename": "pyerrors.correlators", "qualname": "Corr.sqrt", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.log": {"fullname": "pyerrors.correlators.Corr.log", "modulename": "pyerrors.correlators", "qualname": "Corr.log", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.exp": {"fullname": "pyerrors.correlators.Corr.exp", "modulename": "pyerrors.correlators", "qualname": "Corr.exp", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sin": {"fullname": "pyerrors.correlators.Corr.sin", "modulename": "pyerrors.correlators", "qualname": "Corr.sin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cos": {"fullname": "pyerrors.correlators.Corr.cos", "modulename": "pyerrors.correlators", "qualname": "Corr.cos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tan": {"fullname": "pyerrors.correlators.Corr.tan", "modulename": "pyerrors.correlators", "qualname": "Corr.tan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sinh": {"fullname": "pyerrors.correlators.Corr.sinh", "modulename": "pyerrors.correlators", "qualname": "Corr.sinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cosh": {"fullname": "pyerrors.correlators.Corr.cosh", "modulename": "pyerrors.correlators", "qualname": "Corr.cosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tanh": {"fullname": "pyerrors.correlators.Corr.tanh", "modulename": "pyerrors.correlators", "qualname": "Corr.tanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsin": {"fullname": "pyerrors.correlators.Corr.arcsin", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccos": {"fullname": "pyerrors.correlators.Corr.arccos", "modulename": "pyerrors.correlators", "qualname": "Corr.arccos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctan": {"fullname": "pyerrors.correlators.Corr.arctan", "modulename": "pyerrors.correlators", "qualname": "Corr.arctan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsinh": {"fullname": "pyerrors.correlators.Corr.arcsinh", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccosh": {"fullname": "pyerrors.correlators.Corr.arccosh", "modulename": "pyerrors.correlators", "qualname": "Corr.arccosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctanh": {"fullname": "pyerrors.correlators.Corr.arctanh", "modulename": "pyerrors.correlators", "qualname": "Corr.arctanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.prune": {"fullname": "pyerrors.correlators.Corr.prune", "modulename": "pyerrors.correlators", "qualname": "Corr.prune", "kind": "function", "doc": "
Project large correlation matrix to lowest states
\n\n
This method can be used to reduce the size of an (N x N) correlation matrix\nto (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise\nis still small.
\n\n
Parameters
\n\n
    \n
  • Ntrunc (int):\nRank of the target matrix.
    • \n
  • tproj (int):\nTime where the eigenvectors are evaluated, corresponds to ts in the GEVP method.\nThe default value is 3.
    • \n
  • t0proj (int):\nTime where the correlation matrix is inverted. Choosing t0proj=1 is strongly\ndiscouraged for O(a) improved theories, since the correctness of the procedure\ncannot be granted in this case. The default value is 2.
    • \n
  • basematrix (Corr):\nCorrelation matrix that is used to determine the eigenvectors of the\nlowest states based on a GEVP. basematrix is taken to be the Corr itself if\nis is not specified.
    • \n
\n\n
Notes
\n\n
We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving\nthe GEVP $$C(t) v_n(t, t_0) = \\lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \\equiv t_\\mathrm{proj}$\nand $t_0 \\equiv t_{0, \\mathrm{proj}}$. The target matrix is projected onto the subspace of the\nresulting eigenvectors $v_n, n=1,\\dots,N_\\mathrm{trunc}$ via\n$$G^\\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large\ncorrelation matrix and to remove some noise that is added by irrelevant operators.\nThis may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated\nbound $t_0 \\leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.
\n", "signature": "(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):", "funcdef": "def"}, "pyerrors.covobs": {"fullname": "pyerrors.covobs", "modulename": "pyerrors.covobs", "kind": "module", "doc": "
\n"}, "pyerrors.covobs.Covobs": {"fullname": "pyerrors.covobs.Covobs", "modulename": "pyerrors.covobs", "qualname": "Covobs", "kind": "class", "doc": "
\n"}, "pyerrors.covobs.Covobs.__init__": {"fullname": "pyerrors.covobs.Covobs.__init__", "modulename": "pyerrors.covobs", "qualname": "Covobs.__init__", "kind": "function", "doc": "
Initialize Covobs object.
\n\n
Parameters
\n\n
    \n
  • mean (float):\nMean value of the new Obs
    • \n
  • cov (list or array):\n2d Covariance matrix or 1d diagonal entries
    • \n
  • name (str):\nidentifier for the covariance matrix
    • \n
  • pos (int):\nPosition of the variance belonging to mean in cov.\nIs taken to be 1 if cov is 0-dimensional
    • \n
  • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
    • \n
\n", "signature": "(mean, cov, name, pos=None, grad=None)"}, "pyerrors.covobs.Covobs.errsq": {"fullname": "pyerrors.covobs.Covobs.errsq", "modulename": "pyerrors.covobs", "qualname": "Covobs.errsq", "kind": "function", "doc": "
Return the variance (= square of the error) of the Covobs
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "
\n"}, "pyerrors.dirac.epsilon_tensor": {"fullname": "pyerrors.dirac.epsilon_tensor", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor", "kind": "function", "doc": "
Rank-3 epsilon tensor
\n\n
Based on https://codegolf.stackexchange.com/a/160375
\n\n
Returns
\n\n
    \n
  • elem (int):\nElement (i,j,k) of the epsilon tensor of rank 3
    • \n
\n", "signature": "(i, j, k):", "funcdef": "def"}, "pyerrors.dirac.epsilon_tensor_rank4": {"fullname": "pyerrors.dirac.epsilon_tensor_rank4", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor_rank4", "kind": "function", "doc": "
Rank-4 epsilon tensor
\n\n
Extension of https://codegolf.stackexchange.com/a/160375
\n\n
Returns
\n\n
    \n
  • elem (int):\nElement (i,j,k,o) of the epsilon tensor of rank 4
    • \n
\n", "signature": "(i, j, k, o):", "funcdef": "def"}, "pyerrors.dirac.Grid_gamma": {"fullname": "pyerrors.dirac.Grid_gamma", "modulename": "pyerrors.dirac", "qualname": "Grid_gamma", "kind": "function", "doc": "
Returns gamma matrix in Grid labeling.
\n", "signature": "(gamma_tag):", "funcdef": "def"}, "pyerrors.fits": {"fullname": "pyerrors.fits", "modulename": "pyerrors.fits", "kind": "module", "doc": "
\n"}, "pyerrors.fits.Fit_result": {"fullname": "pyerrors.fits.Fit_result", "modulename": "pyerrors.fits", "qualname": "Fit_result", "kind": "class", "doc": "
Represents fit results.
\n\n
Attributes
\n\n
    \n
  • fit_parameters (list):\nresults for the individual fit parameters,\nalso accessible via indices.
    • \n
  • chisquare_by_dof (float):\nreduced chisquare.
    • \n
  • p_value (float):\np-value of the fit
    • \n
  • t2_p_value (float):\nHotelling t-squared p-value for correlated fits.
    • \n
\n", "bases": "collections.abc.Sequence"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "
Apply the gamma method to all fit parameters
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.Fit_result.gm": {"fullname": "pyerrors.fits.Fit_result.gm", "modulename": "pyerrors.fits", "qualname": "Fit_result.gm", "kind": "function", "doc": "
Apply the gamma method to all fit parameters
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.least_squares": {"fullname": "pyerrors.fits.least_squares", "modulename": "pyerrors.fits", "qualname": "least_squares", "kind": "function", "doc": "
Performs a non-linear fit to y = func(x).\n    ```
\n\n
Parameters
\n\n
    \n
  • For an uncombined fit:
    • \n
  • x (list):\nlist of floats.
    • \n
  • y (list):\nlist of Obs.
    • \n
  • func (object):\nfit function, has to be of the form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n   return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
    \n
    \n\n
    For multiple x values func can be of the form
    \n\n
    \n
    def func(a, x):\n   (x1, x2) = x\n   return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • OR For a combined fit:
    • \n
  • x (dict):\ndict of lists.
    • \n
  • y (dict):\ndict of lists of Obs.
    • \n
  • funcs (dict):\ndict of objects\nfit functions have to be of the form (here a[0] is the common fit parameter)\n```python\nimport autograd.numpy as anp\nfuncs = {\"a\": func_a,\n       \"b\": func_b}
    \n\n
    def func_a(a, x):\n   return a[1] * anp.exp(-a[0] * x)
    \n\n
    def func_b(a, x):\n   return a[2] * anp.exp(-a[0] * x)
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • priors (list, optional):\npriors has to be a list with an entry for every parameter in the fit. The entries can either be\nObs (e.g. results from a previous fit) or strings containing a value and an error formatted like\n0.548(23), 500(40) or 0.5(0.4)
    • \n
  • silent (bool, optional):\nIf true all output to the console is omitted (default False).
    • \n
  • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for\nnon-linear fits with many parameters. In case of correlated fits the guess is used to perform\nan uncorrelated fit which then serves as guess for the correlated fit.
    • \n
  • method (str, optional):\ncan be used to choose an alternative method for the minimization of chisquare.\nThe possible methods are the ones which can be used for scipy.optimize.minimize and\nmigrad of iminuit. If no method is specified, Levenberg-Marquard is used.\nReliable alternatives are migrad, Powell and Nelder-Mead.
    • \n
  • tol (float, optional):\ncan be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence\nto a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly\ninvalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values\nThe stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
    • \n
  • correlated_fit (bool):\nIf True, use the full inverse covariance matrix in the definition of the chisquare cost function.\nFor details about how the covariance matrix is estimated see pyerrors.obs.covariance.\nIn practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).\nThis procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).\nAt the moment this option only works for prior==None and when no method is given.
    • \n
  • expected_chisquare (bool):\nIf True estimates the expected chisquare which is\ncorrected by effects caused by correlated input data (default False).
    • \n
  • resplot (bool):\nIf True, a plot which displays fit, data and residuals is generated (default False).
    • \n
  • qqplot (bool):\nIf True, a quantile-quantile plot of the fit result is generated (default False).
    • \n
  • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
    • \n
\n\n
Returns
\n\n
    \n
  • output (Fit_result):\nParameters and information on the fitted result.
    • \n
\n", "signature": "(x, y, func, priors=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.total_least_squares": {"fullname": "pyerrors.fits.total_least_squares", "modulename": "pyerrors.fits", "qualname": "total_least_squares", "kind": "function", "doc": "
Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
\n\n
Parameters
\n\n
    \n
  • x (list):\nlist of Obs, or a tuple of lists of Obs
    • \n
  • y (list):\nlist of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
    • \n
  • func (object):\nfunc has to be of the form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n   return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
    \n
    \n\n
    For multiple x values func can be of the form
    \n\n
    \n
    def func(a, x):\n   (x1, x2) = x\n   return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • silent (bool, optional):\nIf true all output to the console is omitted (default False).
    • \n
  • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for non-linear\nfits with many parameters.
    • \n
  • expected_chisquare (bool):\nIf true prints the expected chisquare which is\ncorrected by effects caused by correlated input data.\nThis can take a while as the full correlation matrix\nhas to be calculated (default False).
    • \n
  • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
    • \n
\n\n
Notes
\n\n
Based on the orthogonal distance regression module of scipy.
\n\n
Returns
\n\n
    \n
  • output (Fit_result):\nParameters and information on the fitted result.
    • \n
\n", "signature": "(x, y, func, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.fit_lin": {"fullname": "pyerrors.fits.fit_lin", "modulename": "pyerrors.fits", "qualname": "fit_lin", "kind": "function", "doc": "
Performs a linear fit to y = n + m * x and returns two Obs n, m.
\n\n
Parameters
\n\n
    \n
  • x (list):\nCan either be a list of floats in which case no xerror is assumed, or\na list of Obs, where the dvalues of the Obs are used as xerror for the fit.
    • \n
  • y (list):\nList of Obs, the dvalues of the Obs are used as yerror for the fit.
    • \n
\n\n
Returns
\n\n
    \n
  • fit_parameters (list[Obs]):\nLIist of fitted observables.
    • \n
\n", "signature": "(x, y, **kwargs):", "funcdef": "def"}, "pyerrors.fits.qqplot": {"fullname": "pyerrors.fits.qqplot", "modulename": "pyerrors.fits", "qualname": "qqplot", "kind": "function", "doc": "
Generates a quantile-quantile plot of the fit result which can be used to\n   check if the residuals of the fit are gaussian distributed.
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(x, o_y, func, p):", "funcdef": "def"}, "pyerrors.fits.residual_plot": {"fullname": "pyerrors.fits.residual_plot", "modulename": "pyerrors.fits", "qualname": "residual_plot", "kind": "function", "doc": "
Generates a plot which compares the fit to the data and displays the corresponding residuals
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(x, y, func, fit_res):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "
Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
\n\n
Returns
\n\n
    \n
  • err (np.array(Obs)):\nError band for an array of sample values x
    • \n
\n", "signature": "(x, func, beta):", "funcdef": "def"}, "pyerrors.fits.ks_test": {"fullname": "pyerrors.fits.ks_test", "modulename": "pyerrors.fits", "qualname": "ks_test", "kind": "function", "doc": "
Performs a Kolmogorov\u2013Smirnov test for the p-values of all fit object.
\n\n
Parameters
\n\n
    \n
  • objects (list):\nList of fit results to include in the analysis (optional).
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(objects=None):", "funcdef": "def"}, "pyerrors.input": {"fullname": "pyerrors.input", "modulename": "pyerrors.input", "kind": "module", "doc": "
pyerrors includes an input submodule in which input routines and parsers for the output of various numerical programs are contained.
\n\n
Jackknife samples
\n\n
For comparison with other analysis workflows pyerrors can also generate jackknife samples from an Obs object or import jackknife samples into an Obs object.\nSee pyerrors.obs.Obs.export_jackknife and pyerrors.obs.import_jackknife for details.
\n"}, "pyerrors.input.bdio": {"fullname": "pyerrors.input.bdio", "modulename": "pyerrors.input.bdio", "kind": "module", "doc": "
\n"}, "pyerrors.input.bdio.read_ADerrors": {"fullname": "pyerrors.input.bdio.read_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "read_ADerrors", "kind": "function", "doc": "
Extract generic MCMC data from a bdio file
\n\n
read_ADerrors requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path -- path to the bdio file
    • \n
  • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
\n\n
Returns
\n\n
    \n
  • data (List[Obs]):\nExtracted data
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.write_ADerrors": {"fullname": "pyerrors.input.bdio.write_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "write_ADerrors", "kind": "function", "doc": "
Write Obs to a bdio file according to ADerrors conventions
\n\n
read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path -- path to the bdio file
    • \n
  • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
\n\n
Returns
\n\n
    \n
  • success (int):\nreturns 0 is successful
    • \n
\n", "signature": "(obs_list, file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_mesons": {"fullname": "pyerrors.input.bdio.read_mesons", "modulename": "pyerrors.input.bdio", "qualname": "read_mesons", "kind": "function", "doc": "
Extract mesons data from a bdio file and return it as a dictionary
\n\n
The dictionary can be accessed with a tuple consisting of (type, source_position, kappa1, kappa2)
\n\n
read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path (str):\npath to the bdio file
    • \n
  • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
  • start (int):\nThe first configuration to be read (default 1)
    • \n
  • stop (int):\nThe last configuration to be read (default None)
    • \n
  • step (int):\nFixed step size between two measurements (default 1)
    • \n
  • alternative_ensemble_name (str):\nManually overwrite ensemble name
    • \n
\n\n
Returns
\n\n
    \n
  • data (dict):\nExtracted meson data
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_dSdm": {"fullname": "pyerrors.input.bdio.read_dSdm", "modulename": "pyerrors.input.bdio", "qualname": "read_dSdm", "kind": "function", "doc": "
Extract dSdm data from a bdio file and return it as a dictionary
\n\n
The dictionary can be accessed with a tuple consisting of (type, kappa)
\n\n
read_dSdm requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path (str):\npath to the bdio file
    • \n
  • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
  • start (int):\nThe first configuration to be read (default 1)
    • \n
  • stop (int):\nThe last configuration to be read (default None)
    • \n
  • step (int):\nFixed step size between two measurements (default 1)
    • \n
  • alternative_ensemble_name (str):\nManually overwrite ensemble name
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.dobs": {"fullname": "pyerrors.input.dobs", "modulename": "pyerrors.input.dobs", "kind": "module", "doc": "
\n"}, "pyerrors.input.dobs.create_pobs_string": {"fullname": "pyerrors.input.dobs.create_pobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_pobs_string", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to an xml string\naccording to the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
    • \n
\n\n
Returns
\n\n
    \n
  • xml_str (str):\nXML formatted string of the input data
    • \n
\n", "signature": "(obsl, name, spec='', origin='', symbol=[], enstag=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_pobs": {"fullname": "pyerrors.input.dobs.write_pobs", "modulename": "pyerrors.input.dobs", "qualname": "write_pobs", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
    • \n
  • gz (bool):\nIf True, the output is a gzipped xml. If False, the output is an xml file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(\tobsl,\tfname,\tname,\tspec='',\torigin='',\tsymbol=[],\tenstag=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_pobs": {"fullname": "pyerrors.input.dobs.read_pobs", "modulename": "pyerrors.input.dobs", "qualname": "read_pobs", "kind": "function", "doc": "
Import a list of Obs from an xml.gz file in the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • separatior_insertion (str or int):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nNone (default): Replica names remain unchanged.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(fname, full_output=False, gz=True, separator_insertion=None):", "funcdef": "def"}, "pyerrors.input.dobs.import_dobs_string": {"fullname": "pyerrors.input.dobs.import_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "import_dobs_string", "kind": "function", "doc": "
Import a list of Obs from a string in the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • content (str):\nXML string containing the data
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(content, full_output=False, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_dobs": {"fullname": "pyerrors.input.dobs.read_dobs", "modulename": "pyerrors.input.dobs", "qualname": "read_dobs", "kind": "function", "doc": "
Import a list of Obs from an xml.gz file in the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes XML file.
    • \n
  • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(fname, full_output=False, gz=True, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.create_dobs_string": {"fullname": "pyerrors.input.dobs.create_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_dobs_string", "kind": "function", "doc": "
Generate the string for the export of a list of Obs or structures containing Obs\nto a .xml.gz file according to the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically. The separator |is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • who (str):\nProvide the name of the person that exports the data.
    • \n
  • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
    • \n
\n\n
Returns
\n\n
    \n
  • xml_str (str):\nXML string generated from the data
    • \n
\n", "signature": "(\tobsl,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_dobs": {"fullname": "pyerrors.input.dobs.write_dobs", "modulename": "pyerrors.input.dobs", "qualname": "write_dobs", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • who (str):\nProvide the name of the person that exports the data.
    • \n
  • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
    • \n
  • gz (bool):\nIf True, the output is a gzipped XML. If False, the output is a XML file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(\tobsl,\tfname,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.hadrons": {"fullname": "pyerrors.input.hadrons", "modulename": "pyerrors.input.hadrons", "kind": "module", "doc": "
\n"}, "pyerrors.input.hadrons.read_meson_hd5": {"fullname": "pyerrors.input.hadrons.read_meson_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_meson_hd5", "kind": "function", "doc": "
Read hadrons meson hdf5 file and extract the meson labeled 'meson'
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • meson (str):\nlabel of the meson to be extracted, standard value meson_0 which\ncorresponds to the pseudoscalar pseudoscalar two-point function.
    • \n
  • gammas (tuple of strings):\nInstrad of a meson label one can also provide a tuple of two strings\nindicating the gamma matrices at source and sink.\n(\"Gamma5\", \"Gamma5\") corresponds to the pseudoscalar pseudoscalar\ntwo-point function. The gammas argument dominateds over meson.
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • corr (Corr):\nCorrelator of the source sink combination in question.
    • \n
\n", "signature": "(path, filestem, ens_id, meson='meson_0', idl=None, gammas=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_DistillationContraction_hd5": {"fullname": "pyerrors.input.hadrons.read_DistillationContraction_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_DistillationContraction_hd5", "kind": "function", "doc": "
Read hadrons DistillationContraction hdf5 files in given directory structure
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the directories to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • diagrams (list):\nList of strings of the diagrams to extract, e.g. [\"direct\", \"box\", \"cross\"].
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result (dict):\nextracted DistillationContration data
    • \n
\n", "signature": "(path, ens_id, diagrams=['direct'], idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.Npr_matrix": {"fullname": "pyerrors.input.hadrons.Npr_matrix", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix", "kind": "class", "doc": "
ndarray(shape, dtype=float, buffer=None, offset=0,\n        strides=None, order=None)
\n\n
An array object represents a multidimensional, homogeneous array\nof fixed-size items.  An associated data-type object describes the\nformat of each element in the array (its byte-order, how many bytes it\noccupies in memory, whether it is an integer, a floating point number,\nor something else, etc.)
\n\n
Arrays should be constructed using array, zeros or empty (refer\nto the See Also section below).  The parameters given here refer to\na low-level method (ndarray(...)) for instantiating an array.
\n\n
For more information, refer to the numpy module and examine the\nmethods and attributes of an array.
\n\n
Parameters
\n\n
    \n
  • (for the __new__ method; see Notes below)
    • \n
  • shape (tuple of ints):\nShape of created array.
    • \n
  • dtype (data-type, optional):\nAny object that can be interpreted as a numpy data type.
    • \n
  • buffer (object exposing buffer interface, optional):\nUsed to fill the array with data.
    • \n
  • offset (int, optional):\nOffset of array data in buffer.
    • \n
  • strides (tuple of ints, optional):\nStrides of data in memory.
    • \n
  • order ({'C', 'F'}, optional):\nRow-major (C-style) or column-major (Fortran-style) order.
    • \n
\n\n
Attributes
\n\n
    \n
  • T (ndarray):\nTranspose of the array.
    • \n
  • data (buffer):\nThe array's elements, in memory.
    • \n
  • dtype (dtype object):\nDescribes the format of the elements in the array.
    • \n
  • flags (dict):\nDictionary containing information related to memory use, e.g.,\n'C_CONTIGUOUS', 'OWNDATA', 'WRITEABLE', etc.
    • \n
  • flat (numpy.flatiter object):\nFlattened version of the array as an iterator.  The iterator\nallows assignments, e.g., x.flat = 3 (See ndarray.flat for\nassignment examples; TODO).
    • \n
  • imag (ndarray):\nImaginary part of the array.
    • \n
  • real (ndarray):\nReal part of the array.
    • \n
  • size (int):\nNumber of elements in the array.
    • \n
  • itemsize (int):\nThe memory use of each array element in bytes.
    • \n
  • nbytes (int):\nThe total number of bytes required to store the array data,\ni.e., itemsize * size.
    • \n
  • ndim (int):\nThe array's number of dimensions.
    • \n
  • shape (tuple of ints):\nShape of the array.
    • \n
  • strides (tuple of ints):\nThe step-size required to move from one element to the next in\nmemory. For example, a contiguous (3, 4) array of type\nint16 in C-order has strides (8, 2).  This implies that\nto move from element to element in memory requires jumps of 2 bytes.\nTo move from row-to-row, one needs to jump 8 bytes at a time\n(2 * 4).
    • \n
  • ctypes (ctypes object):\nClass containing properties of the array needed for interaction\nwith ctypes.
    • \n
  • base (ndarray):\nIf the array is a view into another array, that array is its base\n(unless that array is also a view).  The base array is where the\narray data is actually stored.
    • \n
\n\n
See Also
\n\n
array:  Construct an array. 
\nzeros:  Create an array, each element of which is zero. 
\nempty:  Create an array, but leave its allocated memory unchanged (i.e.,\nit contains \"garbage\"). 
\ndtype:  Create a data-type. 
\nnumpy.typing.NDArray:  An ndarray alias :term:generic <generic type>\nw.r.t. its dtype.type <numpy.dtype.type>.  
\n\n
Notes
\n\n
There are two modes of creating an array using __new__:
\n\n
    \n
  1. If buffer is None, then only shape, dtype, and order\nare used.
    2. \n
  2. If buffer is an object exposing the buffer interface, then\nall keywords are interpreted.
    3. \n
\n\n
No __init__ method is needed because the array is fully initialized\nafter the __new__ method.
\n\n
Examples
\n\n
These examples illustrate the low-level ndarray constructor.  Refer\nto the See Also section above for easier ways of constructing an\nndarray.
\n\n
First mode, buffer is None:
\n\n
\n
>>> np.ndarray(shape=(2,2), dtype=float, order='F')\narray([[0.0e+000, 0.0e+000], # random\n       [     nan, 2.5e-323]])\n
\n
\n\n
Second mode:
\n\n
\n
>>> np.ndarray((2,), buffer=np.array([1,2,3]),\n...            offset=np.int_().itemsize,\n...            dtype=int) # offset = 1*itemsize, i.e. skip first element\narray([2, 3])\n
\n
\n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "
Gamma_5 hermitean conjugate
\n\n
Uses the fact that the propagator is gamma5 hermitean, so just the\nin and out momenta of the propagator are exchanged.
\n"}, "pyerrors.input.hadrons.read_ExternalLeg_hd5": {"fullname": "pyerrors.input.hadrons.read_ExternalLeg_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_ExternalLeg_hd5", "kind": "function", "doc": "
Read hadrons ExternalLeg hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result (Npr_matrix):\nread Cobs-matrix
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Bilinear_hd5": {"fullname": "pyerrors.input.hadrons.read_Bilinear_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Bilinear_hd5", "kind": "function", "doc": "
Read hadrons Bilinear hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result_dict (dict[Npr_matrix]):\nextracted Bilinears
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Fourquark_hd5": {"fullname": "pyerrors.input.hadrons.read_Fourquark_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Fourquark_hd5", "kind": "function", "doc": "
Read hadrons FourquarkFullyConnected hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
  • vertices (list):\nVertex functions to be extracted.
    • \n
\n\n
Returns
\n\n
    \n
  • result_dict (dict):\nextracted fourquark matrizes
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None, vertices=['VA', 'AV']):", "funcdef": "def"}, "pyerrors.input.json": {"fullname": "pyerrors.input.json", "modulename": "pyerrors.input.json", "kind": "module", "doc": "
\n"}, "pyerrors.input.json.create_json_string": {"fullname": "pyerrors.input.json.create_json_string", "modulename": "pyerrors.input.json", "qualname": "create_json_string", "kind": "function", "doc": "
Generate the string for the export of a list of Obs or structures containing Obs\nto a .json(.gz) file
\n\n
Parameters
\n\n
    \n
  • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
\n\n
Returns
\n\n
    \n
  • json_string (str):\nString for export to .json(.gz) file
    • \n
\n", "signature": "(ol, description='', indent=1):", "funcdef": "def"}, "pyerrors.input.json.dump_to_json": {"fullname": "pyerrors.input.json.dump_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_to_json", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .json(.gz) file
\n\n
Parameters
\n\n
    \n
  • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
  • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
    • \n
\n\n
Returns
\n\n
    \n
  • Null
    • \n
\n", "signature": "(ol, fname, description='', indent=1, gz=True):", "funcdef": "def"}, "pyerrors.input.json.import_json_string": {"fullname": "pyerrors.input.json.import_json_string", "modulename": "pyerrors.input.json", "qualname": "import_json_string", "kind": "function", "doc": "
Reconstruct a list of Obs or structures containing Obs from a json string.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\n
Parameters
\n\n
    \n
  • json_string (str):\njson string containing the data.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nreconstructed list of observables from the json string
    • \n
  • or
    • \n
  • result (Obs):\nonly one observable if the list only has one entry
    • \n
  • or
    • \n
  • result (dict):\nif full_output=True
    • \n
\n", "signature": "(json_string, verbose=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.load_json": {"fullname": "pyerrors.input.json.load_json", "modulename": "pyerrors.input.json", "qualname": "load_json", "kind": "function", "doc": "
Import a list of Obs or structures containing Obs from a .json(.gz) file.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nreconstructed list of observables from the json string
    • \n
  • or
    • \n
  • result (Obs):\nonly one observable if the list only has one entry
    • \n
  • or
    • \n
  • result (dict):\nif full_output=True
    • \n
\n", "signature": "(fname, verbose=True, gz=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.dump_dict_to_json": {"fullname": "pyerrors.input.json.dump_dict_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_dict_to_json", "kind": "function", "doc": "
Export a dict of Obs or structures containing Obs to a .json(.gz) file
\n\n
Parameters
\n\n
    \n
  • od (dict):\nDict of JSON valid structures and objects that will be exported.\nAt the moment, these objects can be either of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
  • reps (str):\nSpecify the structure of the placeholder in exported dict to be reps[0-9]+.
    • \n
  • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(od, fname, description='', indent=1, reps='DICTOBS', gz=True):", "funcdef": "def"}, "pyerrors.input.json.load_json_dict": {"fullname": "pyerrors.input.json.load_json_dict", "modulename": "pyerrors.input.json", "qualname": "load_json_dict", "kind": "function", "doc": "
Import a dict of Obs or structures containing Obs from a .json(.gz) file.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
  • reps (str):\nSpecify the structure of the placeholder in imported dict to be reps[0-9]+.
    • \n
\n\n
Returns
\n\n
    \n
  • data (Obs / list / Corr):\nRead data
    • \n
  • or
    • \n
  • data (dict):\nRead data and meta-data
    • \n
\n", "signature": "(fname, verbose=True, gz=True, full_output=False, reps='DICTOBS'):", "funcdef": "def"}, "pyerrors.input.misc": {"fullname": "pyerrors.input.misc", "modulename": "pyerrors.input.misc", "kind": "module", "doc": "
\n"}, "pyerrors.input.misc.read_pbp": {"fullname": "pyerrors.input.misc.read_pbp", "modulename": "pyerrors.input.misc", "qualname": "read_pbp", "kind": "function", "doc": "
Read pbp format from given folder structure.
\n\n
Parameters
\n\n
    \n
  • r_start (list):\nlist which contains the first config to be read for each replicum
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nlist of observables read
    • \n
\n", "signature": "(path, prefix, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD": {"fullname": "pyerrors.input.openQCD", "modulename": "pyerrors.input.openQCD", "kind": "module", "doc": "
\n"}, "pyerrors.input.openQCD.read_rwms": {"fullname": "pyerrors.input.openQCD.read_rwms", "modulename": "pyerrors.input.openQCD", "qualname": "read_rwms", "kind": "function", "doc": "
Read rwms format from given folder structure. Returns a list of length nrw
\n\n
Parameters
\n\n
    \n
  • path (str):\npath that contains the data files
    • \n
  • prefix (str):\nall files in path that start with prefix are considered as input files.\nMay be used together postfix to consider only special file endings.\nPrefix is ignored, if the keyword 'files' is used.
    • \n
  • version (str):\nversion of openQCD, default 2.0
    • \n
  • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum
    • \n
  • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.ms1' for openQCD-files
    • \n
  • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
    • \n
  • print_err (bool):\nPrint additional information that is useful for debugging.
    • \n
\n\n
Returns
\n\n
    \n
  • rwms (Obs):\nReweighting factors read
    • \n
\n", "signature": "(path, prefix, version='2.0', names=None, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_t0": {"fullname": "pyerrors.input.openQCD.extract_t0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_t0", "kind": "function", "doc": "
Extract t0 from given .ms.dat files. Returns t0 as Obs.
\n\n
It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t^2 - 0.3\nis fitted with a linear function\nfrom which the exact root is extracted.
\n\n
It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to .ms.dat files
    • \n
  • prefix (str):\nEnsemble prefix
    • \n
  • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
    • \n
  • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
    • \n
  • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
    • \n
  • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
    • \n
  • plaquette (bool):\nIf true extract the plaquette estimate of t0 instead.
    • \n
  • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
    • \n
  • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
    • \n
  • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
    • \n
  • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
    • \n
\n\n
Returns
\n\n
    \n
  • t0 (Obs):\nExtracted t0
    • \n
\n", "signature": "(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop": {"fullname": "pyerrors.input.openQCD.read_qtop", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop", "kind": "function", "doc": "
Read the topologial charge based on openQCD gradient flow measurements.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • version (str):\nEither openQCD or sfqcd, depending on the data.
    • \n
  • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
    • \n
  • integer_charge (bool):\nIf True, the charge is rounded towards the nearest integer on each config.
    • \n
\n\n
Returns
\n\n
    \n
  • result (Obs):\nRead topological charge
    • \n
\n", "signature": "(path, prefix, c, dtr_cnfg=1, version='openQCD', **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_gf_coupling": {"fullname": "pyerrors.input.openQCD.read_gf_coupling", "modulename": "pyerrors.input.openQCD", "qualname": "read_gf_coupling", "kind": "function", "doc": "
Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.
\n\n
Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used.
    • \n
\n", "signature": "(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.qtop_projection": {"fullname": "pyerrors.input.openQCD.qtop_projection", "modulename": "pyerrors.input.openQCD", "qualname": "qtop_projection", "kind": "function", "doc": "
Returns the projection to the topological charge sector defined by target.
\n\n
Parameters
\n\n
    \n
  • path (Obs):\nTopological charge.
    • \n
  • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
    • \n
\n\n
Returns
\n\n
    \n
  • reto (Obs):\nprojection to the topological charge sector defined by target
    • \n
\n", "signature": "(qtop, target=0):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop_sector": {"fullname": "pyerrors.input.openQCD.read_qtop_sector", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop_sector", "kind": "function", "doc": "
Constructs reweighting factors to a specified topological sector.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L
    • \n
  • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of trajectories\nbetween two configs.\nif it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • version (str):\nversion string of the openQCD (sfqcd) version used to create\nthe ensemble. Default is 2.0. May also be set to sfqcd.
    • \n
  • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
    • \n
  • r_start (list):\noffset of the first ensemble, making it easier to match\nlater on with other Obs
    • \n
  • r_stop (list):\nlast configurations that need to be read (per replicum)
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
    • \n
\n\n
Returns
\n\n
    \n
  • reto (Obs):\nprojection to the topological charge sector defined by target
    • \n
\n", "signature": "(path, prefix, c, target=0, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_ms5_xsf": {"fullname": "pyerrors.input.openQCD.read_ms5_xsf", "modulename": "pyerrors.input.openQCD", "qualname": "read_ms5_xsf", "kind": "function", "doc": "
Read data from files in the specified directory with the specified prefix and quark combination extension, and return a Corr object containing the data.
\n\n
Parameters
\n\n
    \n
  • path (str):\nThe directory to search for the files in.
    • \n
  • prefix (str):\nThe prefix to match the files against.
    • \n
  • qc (str):\nThe quark combination extension to match the files against.
    • \n
  • corr (str):\nThe correlator to extract data for.
    • \n
  • sep (str, optional):\nThe separator to use when parsing the replika names.
    • \n
  • **kwargs: Additional keyword arguments. The following keyword arguments are recognized:
    \n\n
      \n
    • names (List[str]): A list of names to use for the replicas.
      • \n
    • \n
\n\n
Returns
\n\n
    \n
  • Corr: A complex valued Corr object containing the data read from the files. In case of boudary to bulk correlators.
    • \n
  • or
    • \n
  • CObs: A complex valued CObs object containing the data read from the files. In case of boudary to boundary correlators.
    • \n
\n\n
Raises
\n\n
    \n
  • FileNotFoundError: If no files matching the specified prefix and quark combination extension are found in the specified directory.
    • \n
  • IOError: If there is an error reading a file.
    • \n
  • struct.error: If there is an error unpacking binary data.
    • \n
\n", "signature": "(path, prefix, qc, corr, sep='r', **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas": {"fullname": "pyerrors.input.pandas", "modulename": "pyerrors.input.pandas", "kind": "module", "doc": "
\n"}, "pyerrors.input.pandas.to_sql": {"fullname": "pyerrors.input.pandas.to_sql", "modulename": "pyerrors.input.pandas", "qualname": "to_sql", "kind": "function", "doc": "
Write DataFrame including Obs or Corr valued columns to sqlite database.
\n\n
Parameters
\n\n
    \n
  • df (pandas.DataFrame):\nDataframe to be written to the database.
    • \n
  • table_name (str):\nName of the table in the database.
    • \n
  • db (str):\nPath to the sqlite database.
    • \n
  • if exists (str):\nHow to behave if table already exists. Options 'fail', 'replace', 'append'.
    • \n
  • gz (bool):\nIf True the json strings are gzipped.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(df, table_name, db, if_exists='fail', gz=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.read_sql": {"fullname": "pyerrors.input.pandas.read_sql", "modulename": "pyerrors.input.pandas", "qualname": "read_sql", "kind": "function", "doc": "
Execute SQL query on sqlite database and obtain DataFrame including Obs or Corr valued columns.
\n\n
Parameters
\n\n
    \n
  • sql (str):\nSQL query to be executed.
    • \n
  • db (str):\nPath to the sqlite database.
    • \n
  • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
    • \n
\n\n
Returns
\n\n
    \n
  • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
    • \n
\n", "signature": "(sql, db, auto_gamma=False, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.dump_df": {"fullname": "pyerrors.input.pandas.dump_df", "modulename": "pyerrors.input.pandas", "qualname": "dump_df", "kind": "function", "doc": "
Exports a pandas DataFrame containing Obs valued columns to a (gzipped) csv file.
\n\n
Before making use of pandas to_csv functionality Obs objects are serialized via the standardized\njson format of pyerrors.
\n\n
Parameters
\n\n
    \n
  • df (pandas.DataFrame):\nDataframe to be dumped to a file.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • gz (bool):\nIf True, the output is a gzipped csv file. If False, the output is a csv file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(df, fname, gz=True):", "funcdef": "def"}, "pyerrors.input.pandas.load_df": {"fullname": "pyerrors.input.pandas.load_df", "modulename": "pyerrors.input.pandas", "qualname": "load_df", "kind": "function", "doc": "
Imports a pandas DataFrame from a csv.(gz) file in which Obs objects are serialized as json strings.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
\n\n
Returns
\n\n
    \n
  • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
    • \n
\n", "signature": "(fname, auto_gamma=False, gz=True):", "funcdef": "def"}, "pyerrors.input.sfcf": {"fullname": "pyerrors.input.sfcf", "modulename": "pyerrors.input.sfcf", "kind": "module", "doc": "
\n"}, "pyerrors.input.sfcf.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "
Read sfcf c format from given folder structure.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to the sfcf files.
    • \n
  • prefix (str):\nPrefix of the sfcf files.
    • \n
  • name (str):\nName of the correlation function to read.
    • \n
  • quarks (str):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
    • \n
  • corr_type (str):\nType of correlation function to read. Can be\n
      \n
    • 'bi' for boundary-inner
      • \n
    • 'bb' for boundary-boundary
      • \n
    • 'bib' for boundary-inner-boundary
      • \n
    • \n
  • noffset (int):\nOffset of the source (only relevant when wavefunctions are used)
    • \n
  • wf (int):\nID of wave function
    • \n
  • wf2 (int):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
    • \n
  • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
    • \n
  • ens_name (str):\nreplaces the name of the ensemble
    • \n
  • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
    • \n
  • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
    • \n
  • replica (list):\nlist of replica to be read, default is all
    • \n
  • files (list):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
    • \n
  • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
    • \n
\n", "signature": "(\tpath,\tprefix,\tname,\tquarks='.*',\tcorr_type='bi',\tnoffset=0,\twf=0,\twf2=0,\tversion='1.0c',\tcfg_separator='n',\t**kwargs):", "funcdef": "def"}, "pyerrors.input.utils": {"fullname": "pyerrors.input.utils", "modulename": "pyerrors.input.utils", "kind": "module", "doc": "
Utilities for the input
\n"}, "pyerrors.input.utils.check_idl": {"fullname": "pyerrors.input.utils.check_idl", "modulename": "pyerrors.input.utils", "qualname": "check_idl", "kind": "function", "doc": "
Checks if list of configurations is contained in an idl
\n\n
Parameters
\n\n
    \n
  • idl (range or list):\nidl of the current replicum
    • \n
  • che (list):\nlist of configurations to be checked against
    • \n
\n\n
Returns
\n\n
    \n
  • miss_str (str):\nstring with integers of which idls are missing
    • \n
\n", "signature": "(idl, che):", "funcdef": "def"}, "pyerrors.linalg": {"fullname": "pyerrors.linalg", "modulename": "pyerrors.linalg", "kind": "module", "doc": "
\n"}, "pyerrors.linalg.matmul": {"fullname": "pyerrors.linalg.matmul", "modulename": "pyerrors.linalg", "qualname": "matmul", "kind": "function", "doc": "
Matrix multiply all operands.
\n\n
Parameters
\n\n
    \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
  • This implementation is faster compared to standard multiplication via the @ operator.
    • \n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.jack_matmul": {"fullname": "pyerrors.linalg.jack_matmul", "modulename": "pyerrors.linalg", "qualname": "jack_matmul", "kind": "function", "doc": "
Matrix multiply both operands making use of the jackknife approximation.
\n\n
Parameters
\n\n
    \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
  • For large matrices this is considerably faster compared to matmul.
    • \n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.einsum": {"fullname": "pyerrors.linalg.einsum", "modulename": "pyerrors.linalg", "qualname": "einsum", "kind": "function", "doc": "
Wrapper for numpy.einsum
\n\n
Parameters
\n\n
    \n
  • subscripts (str):\nSubscripts for summation (see numpy documentation for details)
    • \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
\n", "signature": "(subscripts, *operands):", "funcdef": "def"}, "pyerrors.linalg.inv": {"fullname": "pyerrors.linalg.inv", "modulename": "pyerrors.linalg", "qualname": "inv", "kind": "function", "doc": "
Inverse of Obs or CObs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.cholesky": {"fullname": "pyerrors.linalg.cholesky", "modulename": "pyerrors.linalg", "qualname": "cholesky", "kind": "function", "doc": "
Cholesky decomposition of Obs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.det": {"fullname": "pyerrors.linalg.det", "modulename": "pyerrors.linalg", "qualname": "det", "kind": "function", "doc": "
Determinant of Obs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.eigh": {"fullname": "pyerrors.linalg.eigh", "modulename": "pyerrors.linalg", "qualname": "eigh", "kind": "function", "doc": "
Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eig": {"fullname": "pyerrors.linalg.eig", "modulename": "pyerrors.linalg", "qualname": "eig", "kind": "function", "doc": "
Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.pinv": {"fullname": "pyerrors.linalg.pinv", "modulename": "pyerrors.linalg", "qualname": "pinv", "kind": "function", "doc": "
Computes the Moore-Penrose pseudoinverse of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.svd": {"fullname": "pyerrors.linalg.svd", "modulename": "pyerrors.linalg", "qualname": "svd", "kind": "function", "doc": "
Computes the singular value decomposition of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.misc": {"fullname": "pyerrors.misc", "modulename": "pyerrors.misc", "kind": "module", "doc": "
\n"}, "pyerrors.misc.errorbar": {"fullname": "pyerrors.misc.errorbar", "modulename": "pyerrors.misc", "qualname": "errorbar", "kind": "function", "doc": "
pyerrors wrapper for the errorbars method of matplotlib
\n\n
Parameters
\n\n
    \n
  • x (list):\nA list of x-values which can be Obs.
    • \n
  • y (list):\nA list of y-values which can be Obs.
    • \n
  • axes ((matplotlib.pyplot.axes)):\nThe axes to plot on. default is plt.
    • \n
\n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.10.10/x64/lib/python3.10/site-packages/matplotlib/pyplot.py'>,\t**kwargs):", "funcdef": "def"}, "pyerrors.misc.dump_object": {"fullname": "pyerrors.misc.dump_object", "modulename": "pyerrors.misc", "qualname": "dump_object", "kind": "function", "doc": "
Dump object into pickle file.
\n\n
Parameters
\n\n
    \n
  • obj (object):\nobject to be saved in the pickle file
    • \n
  • name (str):\nname of the file
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(obj, name, **kwargs):", "funcdef": "def"}, "pyerrors.misc.load_object": {"fullname": "pyerrors.misc.load_object", "modulename": "pyerrors.misc", "qualname": "load_object", "kind": "function", "doc": "
Load object from pickle file.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the file
    • \n
\n\n
Returns
\n\n
    \n
  • object (Obs):\nLoaded Object
    • \n
\n", "signature": "(path):", "funcdef": "def"}, "pyerrors.misc.pseudo_Obs": {"fullname": "pyerrors.misc.pseudo_Obs", "modulename": "pyerrors.misc", "qualname": "pseudo_Obs", "kind": "function", "doc": "
Generate an Obs object with given value, dvalue and name for test purposes
\n\n
Parameters
\n\n
    \n
  • value (float):\ncentral value of the Obs to be generated.
    • \n
  • dvalue (float):\nerror of the Obs to be generated.
    • \n
  • name (str):\nname of the ensemble for which the Obs is to be generated.
    • \n
  • samples (int):\nnumber of samples for the Obs (default 1000).
    • \n
\n\n
Returns
\n\n
    \n
  • res (Obs):\nGenerated Observable
    • \n
\n", "signature": "(value, dvalue, name, samples=1000):", "funcdef": "def"}, "pyerrors.misc.gen_correlated_data": {"fullname": "pyerrors.misc.gen_correlated_data", "modulename": "pyerrors.misc", "qualname": "gen_correlated_data", "kind": "function", "doc": "
Generate observables with given covariance and autocorrelation times.
\n\n
Parameters
\n\n
    \n
  • means (list):\nlist containing the mean value of each observable.
    • \n
  • cov (numpy.ndarray):\ncovariance matrix for the data to be generated.
    • \n
  • name (str):\nensemble name for the data to be geneated.
    • \n
  • tau (float or list):\ncan either be a real number or a list with an entry for\nevery dataset.
    • \n
  • samples (int):\nnumber of samples to be generated for each observable.
    • \n
\n\n
Returns
\n\n
    \n
  • corr_obs (list[Obs]):\nGenerated observable list
    • \n
\n", "signature": "(means, cov, name, tau=0.5, samples=1000):", "funcdef": "def"}, "pyerrors.mpm": {"fullname": "pyerrors.mpm", "modulename": "pyerrors.mpm", "kind": "module", "doc": "
\n"}, "pyerrors.mpm.matrix_pencil_method": {"fullname": "pyerrors.mpm.matrix_pencil_method", "modulename": "pyerrors.mpm", "qualname": "matrix_pencil_method", "kind": "function", "doc": "
Matrix pencil method to extract k energy levels from data
\n\n
Implementation of the matrix pencil method based on\neq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)
\n\n
Parameters
\n\n
    \n
  • data (list):\ncan be a list of Obs for the analysis of a single correlator, or a list of lists\nof Obs if several correlators are to analyzed at once.
    • \n
  • k (int):\nNumber of states to extract (default 1).
    • \n
  • p (int):\nmatrix pencil parameter which filters noise. The optimal value is expected between\nlen(data)/3 and 2*len(data)/3. The computation is more expensive the closer p is\nto len(data)/2 but could possibly suppress more noise (default len(data)//2).
    • \n
\n\n
Returns
\n\n
    \n
  • energy_levels (list[Obs]):\nExtracted energy levels
    • \n
\n", "signature": "(corrs, k=1, p=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs": {"fullname": "pyerrors.obs", "modulename": "pyerrors.obs", "kind": "module", "doc": "
\n"}, "pyerrors.obs.Obs": {"fullname": "pyerrors.obs.Obs", "modulename": "pyerrors.obs", "qualname": "Obs", "kind": "class", "doc": "
Class for a general observable.
\n\n
Instances of Obs are the basic objects of a pyerrors error analysis.\nThey are initialized with a list which contains arrays of samples for\ndifferent ensembles/replica and another list of same length which contains\nthe names of the ensembles/replica. Mathematical operations can be\nperformed on instances. The result is another instance of Obs. The error of\nan instance can be computed with the gamma_method. Also contains additional\nmethods for output and visualization of the error calculation.
\n\n
Attributes
\n\n
    \n
  • S_global (float):\nStandard value for S (default 2.0)
    • \n
  • S_dict (dict):\nDictionary for S values. If an entry for a given ensemble\nexists this overwrites the standard value for that ensemble.
    • \n
  • tau_exp_global (float):\nStandard value for tau_exp (default 0.0)
    • \n
  • tau_exp_dict (dict):\nDictionary for tau_exp values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
    • \n
  • N_sigma_global (float):\nStandard value for N_sigma (default 1.0)
    • \n
  • N_sigma_dict (dict):\nDictionary for N_sigma values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
    • \n
\n"}, "pyerrors.obs.Obs.__init__": {"fullname": "pyerrors.obs.Obs.__init__", "modulename": "pyerrors.obs", "qualname": "Obs.__init__", "kind": "function", "doc": "
Initialize Obs object.
\n\n
Parameters
\n\n
    \n
  • samples (list):\nlist of numpy arrays containing the Monte Carlo samples
    • \n
  • names (list):\nlist of strings labeling the individual samples
    • \n
  • idl (list, optional):\nlist of ranges or lists on which the samples are defined
    • \n
\n", "signature": "(samples, names, idl=None, **kwargs)"}, "pyerrors.obs.Obs.gamma_method": {"fullname": "pyerrors.obs.Obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "Obs.gamma_method", "kind": "function", "doc": "
Estimate the error and related properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
    • \n
  • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
    • \n
  • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
    • \n
  • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
    • \n
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.gm": {"fullname": "pyerrors.obs.Obs.gm", "modulename": "pyerrors.obs", "qualname": "Obs.gm", "kind": "function", "doc": "
Estimate the error and related properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
    • \n
  • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
    • \n
  • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
    • \n
  • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
    • \n
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.details": {"fullname": "pyerrors.obs.Obs.details", "modulename": "pyerrors.obs", "qualname": "Obs.details", "kind": "function", "doc": "
Output detailed properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • ens_content (bool):\nprint details about the ensembles and replica if true.
    • \n
\n", "signature": "(self, ens_content=True):", "funcdef": "def"}, "pyerrors.obs.Obs.reweight": {"fullname": "pyerrors.obs.Obs.reweight", "modulename": "pyerrors.obs", "qualname": "Obs.reweight", "kind": "function", "doc": "
Reweight the obs with given rewighting factors.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
    • \n
\n", "signature": "(self, weight):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero_within_error": {"fullname": "pyerrors.obs.Obs.is_zero_within_error", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero_within_error", "kind": "function", "doc": "
Checks whether the observable is zero within 'sigma' standard errors.
\n\n
Parameters
\n\n
    \n
  • sigma (int):\nNumber of standard errors used for the check.
    • \n
  • Works only properly when the gamma method was run.
    • \n
\n", "signature": "(self, sigma=1):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero": {"fullname": "pyerrors.obs.Obs.is_zero", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero", "kind": "function", "doc": "
Checks whether the observable is zero within a given tolerance.
\n\n
Parameters
\n\n
    \n
  • atol (float):\nAbsolute tolerance (for details see numpy documentation).
    • \n
\n", "signature": "(self, atol=1e-10):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_tauint": {"fullname": "pyerrors.obs.Obs.plot_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.plot_tauint", "kind": "function", "doc": "
Plot integrated autocorrelation time for each ensemble.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rho": {"fullname": "pyerrors.obs.Obs.plot_rho", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rho", "kind": "function", "doc": "
Plot normalized autocorrelation function time for each ensemble.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rep_dist": {"fullname": "pyerrors.obs.Obs.plot_rep_dist", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rep_dist", "kind": "function", "doc": "
Plot replica distribution for each ensemble with more than one replicum.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_history": {"fullname": "pyerrors.obs.Obs.plot_history", "modulename": "pyerrors.obs", "qualname": "Obs.plot_history", "kind": "function", "doc": "
Plot derived Monte Carlo history for each ensemble
\n\n
Parameters
\n\n
    \n
  • expand (bool):\nshow expanded history for irregular Monte Carlo chains (default: True).
    • \n
\n", "signature": "(self, expand=True):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_piechart": {"fullname": "pyerrors.obs.Obs.plot_piechart", "modulename": "pyerrors.obs", "qualname": "Obs.plot_piechart", "kind": "function", "doc": "
Plot piechart which shows the fractional contribution of each\nensemble to the error and returns a dictionary containing the fractions.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.dump": {"fullname": "pyerrors.obs.Obs.dump", "modulename": "pyerrors.obs", "qualname": "Obs.dump", "kind": "function", "doc": "
Dump the Obs to a file 'name' of chosen format.
\n\n
Parameters
\n\n
    \n
  • filename (str):\nname of the file to be saved.
    • \n
  • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
    • \n
  • description (str):\nDescription for output file, only relevant for json.gz format.
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n", "signature": "(self, filename, datatype='json.gz', description='', **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.export_jackknife": {"fullname": "pyerrors.obs.Obs.export_jackknife", "modulename": "pyerrors.obs", "qualname": "Obs.export_jackknife", "kind": "function", "doc": "
Export jackknife samples from the Obs
\n\n
Returns
\n\n
    \n
  • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N jackknife samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived jackknife samples\nshould agree with samples from a full jackknife analysis up to O(1/N).
    • \n
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sqrt": {"fullname": "pyerrors.obs.Obs.sqrt", "modulename": "pyerrors.obs", "qualname": "Obs.sqrt", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.log": {"fullname": "pyerrors.obs.Obs.log", "modulename": "pyerrors.obs", "qualname": "Obs.log", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.exp": {"fullname": "pyerrors.obs.Obs.exp", "modulename": "pyerrors.obs", "qualname": "Obs.exp", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sin": {"fullname": "pyerrors.obs.Obs.sin", "modulename": "pyerrors.obs", "qualname": "Obs.sin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cos": {"fullname": "pyerrors.obs.Obs.cos", "modulename": "pyerrors.obs", "qualname": "Obs.cos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tan": {"fullname": "pyerrors.obs.Obs.tan", "modulename": "pyerrors.obs", "qualname": "Obs.tan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsin": {"fullname": "pyerrors.obs.Obs.arcsin", "modulename": "pyerrors.obs", "qualname": "Obs.arcsin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccos": {"fullname": "pyerrors.obs.Obs.arccos", "modulename": "pyerrors.obs", "qualname": "Obs.arccos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctan": {"fullname": "pyerrors.obs.Obs.arctan", "modulename": "pyerrors.obs", "qualname": "Obs.arctan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sinh": {"fullname": "pyerrors.obs.Obs.sinh", "modulename": "pyerrors.obs", "qualname": "Obs.sinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cosh": {"fullname": "pyerrors.obs.Obs.cosh", "modulename": "pyerrors.obs", "qualname": "Obs.cosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tanh": {"fullname": "pyerrors.obs.Obs.tanh", "modulename": "pyerrors.obs", "qualname": "Obs.tanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsinh": {"fullname": "pyerrors.obs.Obs.arcsinh", "modulename": "pyerrors.obs", "qualname": "Obs.arcsinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccosh": {"fullname": "pyerrors.obs.Obs.arccosh", "modulename": "pyerrors.obs", "qualname": "Obs.arccosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctanh": {"fullname": "pyerrors.obs.Obs.arctanh", "modulename": "pyerrors.obs", "qualname": "Obs.arctanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs": {"fullname": "pyerrors.obs.CObs", "modulename": "pyerrors.obs", "qualname": "CObs", "kind": "class", "doc": "
Class for a complex valued observable.
\n"}, "pyerrors.obs.CObs.__init__": {"fullname": "pyerrors.obs.CObs.__init__", "modulename": "pyerrors.obs", "qualname": "CObs.__init__", "kind": "function", "doc": "
\n", "signature": "(real, imag=0.0)"}, "pyerrors.obs.CObs.gamma_method": {"fullname": "pyerrors.obs.CObs.gamma_method", "modulename": "pyerrors.obs", "qualname": "CObs.gamma_method", "kind": "function", "doc": "
Executes the gamma_method for the real and the imaginary part.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.CObs.is_zero": {"fullname": "pyerrors.obs.CObs.is_zero", "modulename": "pyerrors.obs", "qualname": "CObs.is_zero", "kind": "function", "doc": "
Checks whether both real and imaginary part are zero within machine precision.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs.conjugate": {"fullname": "pyerrors.obs.CObs.conjugate", "modulename": "pyerrors.obs", "qualname": "CObs.conjugate", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.derived_observable": {"fullname": "pyerrors.obs.derived_observable", "modulename": "pyerrors.obs", "qualname": "derived_observable", "kind": "function", "doc": "
Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.
\n\n
Parameters
\n\n
    \n
  • func (object):\narbitrary function of the form func(data, **kwargs). For the\nautomatic differentiation to work, all numpy functions have to have\nthe autograd wrapper (use 'import autograd.numpy as anp').
    • \n
  • data (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
    • \n
  • num_grad (bool):\nif True, numerical derivatives are used instead of autograd\n(default False). To control the numerical differentiation the\nkwargs of numdifftools.step_generators.MaxStepGenerator\ncan be used.
    • \n
  • man_grad (list):\nmanually supply a list or an array which contains the jacobian\nof func. Use cautiously, supplying the wrong derivative will\nnot be intercepted.
    • \n
\n\n
Notes
\n\n
For simple mathematical operations it can be practical to use anonymous\nfunctions. For the ratio of two observables one can e.g. use
\n\n
new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])
\n", "signature": "(func, data, array_mode=False, **kwargs):", "funcdef": "def"}, "pyerrors.obs.reweight": {"fullname": "pyerrors.obs.reweight", "modulename": "pyerrors.obs", "qualname": "reweight", "kind": "function", "doc": "
Reweight a list of observables.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • obs (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
    • \n
\n", "signature": "(weight, obs, **kwargs):", "funcdef": "def"}, "pyerrors.obs.correlate": {"fullname": "pyerrors.obs.correlate", "modulename": "pyerrors.obs", "qualname": "correlate", "kind": "function", "doc": "
Correlate two observables.
\n\n
Parameters
\n\n
    \n
  • obs_a (Obs):\nFirst observable
    • \n
  • obs_b (Obs):\nSecond observable
    • \n
\n\n
Notes
\n\n
Keep in mind to only correlate primary observables which have not been reweighted\nyet. The reweighting has to be applied after correlating the observables.\nCurrently only works if ensembles are identical (this is not strictly necessary).
\n", "signature": "(obs_a, obs_b):", "funcdef": "def"}, "pyerrors.obs.covariance": {"fullname": "pyerrors.obs.covariance", "modulename": "pyerrors.obs", "qualname": "covariance", "kind": "function", "doc": "
Calculates the error covariance matrix of a set of observables.
\n\n
WARNING: This function should be used with care, especially for observables with support on multiple\n         ensembles with differing autocorrelations. See the notes below for details.
\n\n
The gamma method has to be applied first to all observables.
\n\n
Parameters
\n\n
    \n
  • obs (list or numpy.ndarray):\nList or one dimensional array of Obs
    • \n
  • visualize (bool):\nIf True plots the corresponding normalized correlation matrix (default False).
    • \n
  • correlation (bool):\nIf True the correlation matrix instead of the error covariance matrix is returned (default False).
    • \n
  • smooth (None or int):\nIf smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue\nsmoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the\nlargest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely\nsmall ones.
    • \n
\n\n
Notes
\n\n
The error covariance is defined such that it agrees with the squared standard error for two identical observables\n$$\\operatorname{cov}(a,a)=\\sum_{s=1}^N\\delta_a^s\\delta_a^s/N^2=\\Gamma_{aa}(0)/N=\\operatorname{var}(a)/N=\\sigma_a^2$$\nin the absence of autocorrelation.\nThe error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite\n$$\\sum_{i,j}v_i\\Gamma_{ij}(0)v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i,j}v_i\\delta_i^s\\delta_j^s v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i}|v_i\\delta_i^s|^2\\geq 0\\,,$$ for every $v\\in\\mathbb{R}^M$, while such an identity does not hold for larger windows/lags.\nFor observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.\n$$\\tau_{\\mathrm{int}, ij}=\\sqrt{\\tau_{\\mathrm{int}, i}\\times \\tau_{\\mathrm{int}, j}}$$\nThis construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).
\n", "signature": "(obs, visualize=False, correlation=False, smooth=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs.import_jackknife": {"fullname": "pyerrors.obs.import_jackknife", "modulename": "pyerrors.obs", "qualname": "import_jackknife", "kind": "function", "doc": "
Imports jackknife samples and returns an Obs
\n\n
Parameters
\n\n
    \n
  • jacks (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N jackknife samples as first to Nth entry.
    • \n
  • name (str):\nname of the ensemble the samples are defined on.
    • \n
\n", "signature": "(jacks, name, idl=None):", "funcdef": "def"}, "pyerrors.obs.merge_obs": {"fullname": "pyerrors.obs.merge_obs", "modulename": "pyerrors.obs", "qualname": "merge_obs", "kind": "function", "doc": "
Combine all observables in list_of_obs into one new observable
\n\n
Parameters
\n\n
    \n
  • list_of_obs (list):\nlist of the Obs object to be combined
    • \n
\n\n
Notes
\n\n
It is not possible to combine obs which are based on the same replicum
\n", "signature": "(list_of_obs):", "funcdef": "def"}, "pyerrors.obs.cov_Obs": {"fullname": "pyerrors.obs.cov_Obs", "modulename": "pyerrors.obs", "qualname": "cov_Obs", "kind": "function", "doc": "
Create an Obs based on mean(s) and a covariance matrix
\n\n
Parameters
\n\n
    \n
  • mean (list of floats or float):\nN mean value(s) of the new Obs
    • \n
  • cov (list or array):\n2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
    • \n
  • name (str):\nidentifier for the covariance matrix
    • \n
  • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
    • \n
\n", "signature": "(means, cov, name, grad=None):", "funcdef": "def"}, "pyerrors.roots": {"fullname": "pyerrors.roots", "modulename": "pyerrors.roots", "kind": "module", "doc": "
\n"}, "pyerrors.roots.find_root": {"fullname": "pyerrors.roots.find_root", "modulename": "pyerrors.roots", "qualname": "find_root", "kind": "function", "doc": "
Finds the root of the function func(x, d) where d is an Obs.
\n\n
Parameters
\n\n
    \n
  • d (Obs):\nObs passed to the function.
    • \n
  • func (object):\nFunction to be minimized. Any numpy functions have to use the autograd.numpy wrapper.\nExample:
    \n\n
    \n
    import autograd.numpy as anp\ndef root_func(x, d):\n   return anp.exp(-x ** 2) - d\n
    \n
    • \n
  • guess (float):\nInitial guess for the minimization.
    • \n
\n\n
Returns
\n\n
    \n
  • res (Obs):\nObs valued root of the function.
    • \n
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What is pyerrors?
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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:
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  • automatic differentiation for exact linear error propagation as suggested in arXiv:1809.01289 (partly based on the autograd package).
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  • treatment of slow modes in the simulation as suggested in arXiv:1009.5228.
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  • coherent error propagation for data from different Markov chains.
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  • non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in arXiv:1809.01289.
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  • 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...).
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More detailed examples can found in the GitHub repository \"badge\".
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If you use pyerrors for research that leads to a publication please consider citing:
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  • Fabian Joswig, Simon Kuberski, Justus T. Kuhlmann, Jan Neuendorf, pyerrors: a python framework for error analysis of Monte Carlo data. [arXiv:2209.14371 [hep-lat]].
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  • Ulli Wolff, Monte Carlo errors with less errors. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum).
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  • Alberto Ramos, Automatic differentiation for error analysis of Monte Carlo data. Comput.Phys.Commun. 238 (2019) 19-35.
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and
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  • Stefan Schaefer, Rainer Sommer, Francesco Virotta, Critical slowing down and error analysis in lattice QCD simulations. Nucl.Phys.B 845 (2011) 93-119.
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where applicable.
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There exist similar publicly available implementations of gamma method error analysis suites in Fortran, Julia and Python.
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Basic example
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import numpy as np\nimport pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name']) # Initialize an Obs object\nmy_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object\nmy_new_obs.gamma_method()                     # Estimate the statistical error\nprint(my_new_obs)                             # Print the result to stdout\n> 0.31498(72)\n
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The Obs class
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pyerrors introduces a new datatype, Obs, which simplifies error propagation and estimation for auto- and cross-correlated data.\nAn Obs object can be initialized with two arguments, the first is a list containing the samples for an observable from a Monte Carlo chain.\nThe samples can either be provided as python list or as numpy array.\nThe second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble. It is crucial for the correct error propagation that observations from the same Monte Carlo history are labeled with the same name. See Multiple ensembles/replica for details.
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import pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name'])\n
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Error propagation
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When performing mathematical operations on Obs objects the correct error propagation is intrinsically taken care of using a first order Taylor expansion\n$$\\delta_f^i=\\sum_\\alpha \\bar{f}_\\alpha \\delta_\\alpha^i\\,,\\quad \\delta_\\alpha^i=a_\\alpha^i-\\bar{a}_\\alpha\\,,$$\nas introduced in arXiv:hep-lat/0306017.\nThe required derivatives $\\bar{f}_\\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in arXiv:1809.01289.
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The Obs class is designed such that mathematical numpy functions can be used on Obs just as for regular floats.
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import numpy as np\nimport pyerrors as pe\n\nmy_obs1 = pe.Obs([samples1], ['ensemble_name'])\nmy_obs2 = pe.Obs([samples2], ['ensemble_name'])\n\nmy_sum = my_obs1 + my_obs2\n\nmy_m_eff = np.log(my_obs1 / my_obs2)\n\niamzero = my_m_eff - my_m_eff\n# Check that value and fluctuations are zero within machine precision\nprint(iamzero == 0.0)\n> True\n
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Error estimation
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The error estimation within pyerrors is based on the gamma method introduced in arXiv:hep-lat/0306017.\nAfter having arrived at the derived quantity of interest the gamma_method can be called as detailed in the following example.
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my_sum.gamma_method()\nprint(my_sum)\n> 1.70(57)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 5.72046658e-01 +/- 7.56746598e-02 (33.650%)\n>  t_int         2.71422900e+00 +/- 6.40320983e-01 S = 2.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
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We use the following definition of the integrated autocorrelation time established in Madras & Sokal 1988\n$$\\tau_\\mathrm{int}=\\frac{1}{2}+\\sum_{t=1}^{W}\\rho(t)\\geq \\frac{1}{2}\\,.$$\nThe window $W$ is determined via the automatic windowing procedure described in arXiv:hep-lat/0306017.\nThe standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the gamma_method as parameter.
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my_sum.gamma_method(S=3.0)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.30675201e-01 +/- 1.04585650e-01 (37.099%)\n>  t_int         3.29909703e+00 +/- 9.77310102e-01 S = 3.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
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The integrated autocorrelation time $\\tau_\\mathrm{int}$ and the autocorrelation function $\\rho(W)$ can be monitored via the methods pyerrors.obs.Obs.plot_tauint and pyerrors.obs.Obs.plot_rho.
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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.
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Exponential tails
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Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\\rho$ as suggested in arXiv:1009.5228. The longest autocorrelation time in the history, $\\tau_\\mathrm{exp}$, can be passed to the gamma_method as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.
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my_sum.gamma_method(tau_exp=7.2)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.28097762e-01 +/- 5.79077524e-02 (36.947%)\n>  t_int         3.27218667e+00 +/- 7.99583654e-01 tau_exp = 7.20,  N_sigma = 1\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
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For the full API see pyerrors.obs.Obs.gamma_method.
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Multiple ensembles/replica
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Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their name.
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obs1 = pe.Obs([samples1], ['ensemble1'])\nobs2 = pe.Obs([samples2], ['ensemble2'])\n\nmy_sum = obs1 + obs2\nmy_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 2 ensembles:\n>   \u00b7 Ensemble 'ensemble1' : 1000 configurations (from 1 to 1000)\n>   \u00b7 Ensemble 'ensemble2' : 500 configurations (from 1 to 500)\n
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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.
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pyerrors identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar | in the name of the data set.
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obs1 = pe.Obs([samples1], ['ensemble1|r01'])\nobs2 = pe.Obs([samples2], ['ensemble1|r02'])\n\n> my_sum = obs1 + obs2\n> my_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1'\n>     \u00b7 Replicum 'r01' : 1000 configurations (from 1 to 1000)\n>     \u00b7 Replicum 'r02' : 500 configurations (from 1 to 500)\n
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Error estimation for multiple ensembles
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In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example.
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pe.Obs.S_dict['ensemble1'] = 2.5\npe.Obs.tau_exp_dict['ensemble2'] = 8.0\npe.Obs.tau_exp_dict['ensemble3'] = 2.0\n
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In case the gamma_method is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.\nPassing arguments to the gamma_method still dominates over the dictionaries.
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Irregular Monte Carlo chains
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Obs objects defined on irregular Monte Carlo chains can be initialized with the parameter idl.
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# 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
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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.
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Warning: Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.\nMake sure to check the autocorrelation time with e.g. pyerrors.obs.Obs.plot_rho or pyerrors.obs.Obs.plot_tauint.
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For the full API see pyerrors.obs.Obs.
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Correlators
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When one is not interested in single observables but correlation functions, pyerrors offers the Corr class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize a Corr objects one needs to arrange the data as a list of Obs
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my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0      0.7957(80)\n> 1      0.5156(51)\n> 2      0.3227(33)\n> 3      0.2041(21)\n
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In case the correlation functions are not defined on the outermost timeslices, for example because of fixed boundary conditions, a padding can be introduced.
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my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3], padding=[1, 1])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0\n> 1      0.7957(80)\n> 2      0.5156(51)\n> 3      0.3227(33)\n> 4      0.2041(21)\n> 5\n
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The individual entries of a correlator can be accessed via slicing
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print(my_corr[3])\n> 0.3227(33)\n
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Error propagation with the Corr class works very similar to Obs objects. Mathematical operations are overloaded and Corr objects can be computed together with other Corr objects, Obs objects or real numbers and integers.
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my_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr\n
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pyerrors provides the user with a set of regularly used methods for the manipulation of correlator objects:
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  • Corr.gamma_method applies the gamma method to all entries of the correlator.
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  • Corr.m_eff to construct effective masses. Various variants for periodic and fixed temporal boundary conditions are available.
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  • Corr.deriv returns the first derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
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  • Corr.second_deriv returns the second derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
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  • Corr.symmetric symmetrizes parity even correlations functions, assuming periodic boundary conditions.
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  • Corr.anti_symmetric anti-symmetrizes parity odd correlations functions, assuming periodic boundary conditions.
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  • Corr.T_symmetry averages a correlator with its time symmetry partner, assuming fixed boundary conditions.
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  • Corr.plateau extracts a plateau value from the correlator in a given range.
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  • Corr.roll periodically shifts the correlator.
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  • Corr.reverse reverses the time ordering of the correlator.
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  • Corr.correlate constructs a disconnected correlation function from the correlator and another Corr or Obs object.
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  • Corr.reweight reweights the correlator.
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pyerrors can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (see pyerrors.correlators.Corr.GEVP).
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For the full API see pyerrors.correlators.Corr.
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Complex valued observables
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pyerrors can handle complex valued observables via the class pyerrors.obs.CObs.\nCObs are initialized with a real and an imaginary part which both can be Obs valued.
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my_real_part = pe.Obs([samples1], ['ensemble1'])\nmy_imag_part = pe.Obs([samples2], ['ensemble1'])\n\nmy_cobs = pe.CObs(my_real_part, my_imag_part)\nmy_cobs.gamma_method()\nprint(my_cobs)\n> (0.9959(91)+0.659(28)j)\n
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Elementary mathematical operations are overloaded and samples are properly propagated as for the Obs class.
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my_derived_cobs = (my_cobs + my_cobs.conjugate()) / np.abs(my_cobs)\nmy_derived_cobs.gamma_method()\nprint(my_derived_cobs)\n> (1.668(23)+0.0j)\n
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The Covobs class
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In many projects, auxiliary data that is not based on Monte Carlo chains enters. Examples are experimentally determined mesons masses which are used to set the scale or renormalization constants. These numbers come with an error that has to be propagated through the analysis. The Covobs class allows to define such quantities in pyerrors. Furthermore, external input might consist of correlated quantities. An example are the parameters of an interpolation formula, which are defined via mean values and a covariance matrix between all parameters. The contribution of the interpolation formula to the error of a derived quantity therefore might depend on the complete covariance matrix.
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This concept is built into the definition of Covobs. In pyerrors, external input is defined by $M$ mean values, a $M\\times M$ covariance matrix, where $M=1$ is permissible, and a name that uniquely identifies the covariance matrix. Below, we define the pion mass, based on its mean value and error, 134.9768(5). Note, that the square of the error enters cov_Obs, since the second argument of this function is the covariance matrix of the Covobs.
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import pyerrors.obs as pe\n\nmpi = pe.cov_Obs(134.9768, 0.0005**2, 'pi^0 mass')\nmpi.gamma_method()\nmpi.details()\n> Result         1.34976800e+02 +/- 5.00000000e-04 +/- 0.00000000e+00 (0.000%)\n>  pi^0 mass     5.00000000e-04\n> 0 samples in 1 ensemble:\n>   \u00b7 Covobs   'pi^0 mass'\n
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The resulting object mpi is an Obs that contains a Covobs. In the following, it may be handled as any other Obs. The contribution of the covariance matrix to the error of an Obs is determined from the $M \\times M$ covariance matrix $\\Sigma$ and the gradient of the Obs with respect to the external quantities, which is the $1\\times M$ Jacobian matrix $J$, via\n$$s = \\sqrt{J^T \\Sigma J}\\,,$$\nwhere the Jacobian is computed for each derived quantity via automatic differentiation.
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Correlated auxiliary data is defined similarly to above, e.g., via
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RAP = pe.cov_Obs([16.7457, -19.0475], [[3.49591, -6.07560], [-6.07560, 10.5834]], 'R_AP, 1906.03445, (5.3a)')\nprint(RAP)\n> [Obs[16.7(1.9)], Obs[-19.0(3.3)]]\n
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where RAP now is a list of two Obs that contains the two correlated parameters.
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Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the Covobs class allows to quote the derivative of a result with respect to the external quantities. If these derivatives are published together with the result, small shifts in the definition of external quantities, e.g., the definition of the physical point, can be performed a posteriori based on the published information. This may help to compare results of different groups. The gradient of an Obs o with respect to a covariance matrix with the identifying string k may be accessed via
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o.covobs[k].grad\n
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Error propagation in iterative algorithms
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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.
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Least squares fits
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Standard non-linear least square fits with errors on the dependent but not the independent variables can be performed with pyerrors.fits.least_squares. As default solver the Levenberg-Marquardt algorithm implemented in scipy is used.
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Fit functions have to be of the following form
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import autograd.numpy as anp\n\ndef func(a, x):\n    return a[1] * anp.exp(-a[0] * x)\n
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It is important that numerical functions refer to autograd.numpy instead of numpy for the automatic differentiation in iterative algorithms to work properly.
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Fits can then be performed via
\n\n
\n
fit_result = pe.fits.least_squares(x, y, func)\nprint("\\n", fit_result)\n> Fit with 2 parameters\n> Method: Levenberg-Marquardt\n> `ftol` termination condition is satisfied.\n> chisquare/d.o.f.: 0.9593035785160936\n\n>  Goodness of fit:\n> \u03c7\u00b2/d.o.f. = 0.959304\n> p-value   = 0.5673\n> Fit parameters:\n> 0      0.0548(28)\n> 1      1.933(64)\n
\n
\n\n
where x is a list or numpy.array of floats and y is a list or numpy.array of Obs.
\n\n
Data stored in Corr objects can be fitted directly using the Corr.fit method.
\n\n
\n
my_corr = pe.Corr(y)\nfit_result = my_corr.fit(func, fitrange=[12, 25])\n
\n
\n\n
this can simplify working with absolute fit ranges and takes care of gaps in the data automatically.
\n\n
For fit functions with multiple independent variables the fit function can be of the form
\n\n
\n
def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
\n
\n\n
pyerrors also supports correlated fits which can be triggered via the parameter correlated_fit=True.\nDetails about how the required covariance matrix is estimated can be found in pyerrors.obs.covariance.
\n\n
Direct visualizations of the performed fits can be triggered via resplot=True or qqplot=True. For all available options see pyerrors.fits.least_squares.
\n\n
Total least squares fits
\n\n
pyerrors can also fit data with errors on both the dependent and independent variables using the total least squares method also referred to orthogonal distance regression as implemented in scipy, see pyerrors.fits.least_squares. The syntax is identical to the standard least squares case, the only difference being that x also has to be a list or numpy.array of Obs.
\n\n
For the full API see pyerrors.fits for fits and pyerrors.roots for finding roots of functions.
\n\n
Matrix operations
\n\n
pyerrors provides wrappers for Obs- and CObs-valued matrix operations based on numpy.linalg. The supported functions include:
\n\n
    \n
  • inv for the matrix inverse.
    • \n
  • cholseky for the Cholesky decomposition.
    • \n
  • det for the matrix determinant.
    • \n
  • eigh for eigenvalues and eigenvectors of hermitean matrices.
    • \n
  • eig for eigenvalues of general matrices.
    • \n
  • pinv for the Moore-Penrose pseudoinverse.
    • \n
  • svd for the singular-value-decomposition.
    • \n
\n\n
For the full API see pyerrors.linalg.
\n\n
Export data
\n\n
The preferred exported file format within pyerrors is json.gz. Files written to this format are valid JSON files that have been compressed using gzip. The structure of the content is inspired by the dobs format of the ALPHA collaboration. The aim of the format is to facilitate the storage of data in a self-contained way such that, even years after the creation of the file, it is possible to extract all necessary information:
\n\n
    \n
  • What observables are stored? Possibly: How exactly are they defined.
    • \n
  • How does each single ensemble or external quantity contribute to the error of the observable?
    • \n
  • Who did write the file when and on which machine?
    • \n
\n\n
This can be achieved by storing all information in one single file. The export routines of pyerrors are written such that as much information as possible is written automatically as described in the following example
\n\n
\n
my_obs = pe.Obs([samples], ["test_ensemble"])\nmy_obs.tag = "My observable"\n\npe.input.json.dump_to_json(my_obs, "test_output_file", description="This file contains a test observable")\n# For a single observable one can equivalently use the class method dump\nmy_obs.dump("test_output_file", description="This file contains a test observable")\n\ncheck = pe.input.json.load_json("test_output_file")\n\nprint(my_obs == check)\n> True\n
\n
\n\n
The format also allows to directly write out the content of Corr objects or lists and arrays of Obs objects by passing the desired data to pyerrors.input.json.dump_to_json.
\n\n
json.gz format specification
\n\n
The first entries of the file provide optional auxiliary information:
\n\n
    \n
  • program is a string that indicates which program was used to write the file.
    • \n
  • version is a string that specifies the version of the format.
    • \n
  • who is a string that specifies the user name of the creator of the file.
    • \n
  • date is a string and contains the creation date of the file.
    • \n
  • host is a string and contains the hostname of the machine where the file has been written.
    • \n
  • description contains information on the content of the file. This field is not filled automatically in pyerrors. The user is advised to provide as detailed information as possible in this field. Examples are: Input files of measurements or simulations, LaTeX formulae or references to publications to specify how the observables have been computed, details on the analysis strategy, ... This field may be any valid JSON type. Strings, arrays or objects (equivalent to dicts in python) are well suited to provide information.
    • \n
\n\n
The only necessary entry of the file is the field\n-obsdata, an array that contains the actual data.
\n\n
Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of Obs, list, numpy.ndarray, Corr. All Obs inside a structure (with dimension > 0) have to be defined on the same set of configurations. Different structures, that are represented by entries of the array obsdata, are treated independently. Each entry of the array obsdata has the following required entries:
\n\n
    \n
  • type is a string that specifies the type of the structure. This allows to parse the content to the correct form after reading the file. It is always possible to interpret the content as list of Obs.
    • \n
  • value is an array that contains the mean values of the Obs inside the structure.\nThe following entries are optional:
    • \n
  • layout is a string that specifies the layout of multi-dimensional structures. Examples are \"2, 2\" for a 2x2 dimensional matrix or  \"64, 4, 4\" for a Corr with $T=64$ and 4x4 matrices on each time slices. \"1\" denotes a single Obs. Multi-dimensional structures are stored in row-major format (see below).
    • \n
  • tag is any JSON type. It contains additional information concerning the structure. The tag of an Obs in pyerrors is written here.
    • \n
  • reweighted is a Bool that may be used to specify, whether the Obs in the structure have been reweighted.
    • \n
  • data is an array that contains the data from MC chains. We will define it below.
    • \n
  • cdata is an array that contains the data from external quantities with an error (Covobs in pyerrors). We will define it below.
    • \n
\n\n
The array data contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains:
\n\n
    \n
  • id, a string that contains the name of the ensemble
    • \n
  • replica, an array that contains an entry per replica of the ensemble.
    • \n
\n\n
Each entry of replica contains\nname, a string that contains the name of the replica\ndeltas, an array that contains the actual data.
\n\n
Each entry in deltas corresponds to one configuration of the replica and has $1+N$ many entries. The first entry is an integer that specifies the configuration number that, together with ensemble and replica name, may be used to uniquely identify the configuration on which the data has been obtained. The following N entries specify the deltas, i.e., the deviation of the observable from the mean value on this configuration, of each Obs inside the structure. Multi-dimensional structures are stored in a row-major format. For primary observables, such as correlation functions, $value + delta_i$ matches the primary data obtained on the configuration.
\n\n
The array cdata contains information about the contribution of auxiliary observables, represented by Covobs in pyerrors, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains:
\n\n
    \n
  • id, a string that identifies the covariance matrix
    • \n
  • layout, a string that defines the dimensions of the $M\\times M$ covariance matrix (has to be \"M, M\" or \"1\").
    • \n
  • cov, an array that contains the $M\\times M$ many entries of the covariance matrix, stored in row-major format.
    • \n
  • grad, an array that contains N entries, one for each Obs inside the structure. Each entry itself is an array, that contains the M gradients of the Nth observable  with respect to the quantity that corresponds to the Mth diagonal entry of the covariance matrix.
    • \n
\n\n
A JSON schema that may be used to verify the correctness of a file with respect to the format definition is stored in ./examples/json_schema.json. The schema is a self-descriptive format definition and contains an exemplary file.
\n\n
Julia I/O routines for the json.gz format, compatible with ADerrors.jl, can be found here.
\n"}, "pyerrors.correlators": {"fullname": "pyerrors.correlators", "modulename": "pyerrors.correlators", "kind": "module", "doc": "
\n"}, "pyerrors.correlators.Corr": {"fullname": "pyerrors.correlators.Corr", "modulename": "pyerrors.correlators", "qualname": "Corr", "kind": "class", "doc": "
The class for a correlator (time dependent sequence of pe.Obs).
\n\n
Everything, this class does, can be achieved using lists or arrays of Obs.\nBut it is simply more convenient to have a dedicated object for correlators.\nOne often wants to add or multiply correlators of the same length at every timeslice and it is inconvenient\nto iterate over all timeslices for every operation. This is especially true, when dealing with matrices.
\n\n
The correlator can have two types of content: An Obs at every timeslice OR a GEVP\nmatrix at every timeslice. Other dependency (eg. spatial) are not supported.
\n"}, "pyerrors.correlators.Corr.__init__": {"fullname": "pyerrors.correlators.Corr.__init__", "modulename": "pyerrors.correlators", "qualname": "Corr.__init__", "kind": "function", "doc": "
Initialize a Corr object.
\n\n
Parameters
\n\n
    \n
  • data_input (list or array):\nlist of Obs or list of arrays of Obs or array of Corrs
    • \n
  • padding (list, optional):\nList with two entries where the first labels the padding\nat the front of the correlator and the second the padding\nat the back.
    • \n
  • prange (list, optional):\nList containing the first and last timeslice of the plateau\nregion indentified for this correlator.
    • \n
\n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "pyerrors.correlators.Corr.gamma_method": {"fullname": "pyerrors.correlators.Corr.gamma_method", "modulename": "pyerrors.correlators", "qualname": "Corr.gamma_method", "kind": "function", "doc": "
Apply the gamma method to the content of the Corr.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.gm": {"fullname": "pyerrors.correlators.Corr.gm", "modulename": "pyerrors.correlators", "qualname": "Corr.gm", "kind": "function", "doc": "
Apply the gamma method to the content of the Corr.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.projected": {"fullname": "pyerrors.correlators.Corr.projected", "modulename": "pyerrors.correlators", "qualname": "Corr.projected", "kind": "function", "doc": "
We need to project the Correlator with a Vector to get a single value at each timeslice.
\n\n
The method can use one or two vectors.\nIf two are specified it returns v1@G@v2 (the order might be very important.)\nBy default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to
\n", "signature": "(self, vector_l=None, vector_r=None, normalize=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.item": {"fullname": "pyerrors.correlators.Corr.item", "modulename": "pyerrors.correlators", "qualname": "Corr.item", "kind": "function", "doc": "
Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.
\n\n
Parameters
\n\n
    \n
  • i (int):\nFirst index to be picked.
    • \n
  • j (int):\nSecond index to be picked.
    • \n
\n", "signature": "(self, i, j):", "funcdef": "def"}, "pyerrors.correlators.Corr.plottable": {"fullname": "pyerrors.correlators.Corr.plottable", "modulename": "pyerrors.correlators", "qualname": "Corr.plottable", "kind": "function", "doc": "
Outputs the correlator in a plotable format.
\n\n
Outputs three lists containing the timeslice index, the value on each\ntimeslice and the error on each timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.symmetric": {"fullname": "pyerrors.correlators.Corr.symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.symmetric", "kind": "function", "doc": "
Symmetrize the correlator around x0=0.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.anti_symmetric": {"fullname": "pyerrors.correlators.Corr.anti_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.anti_symmetric", "kind": "function", "doc": "
Anti-symmetrize the correlator around x0=0.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.is_matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.is_matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.is_matrix_symmetric", "kind": "function", "doc": "
Checks whether a correlator matrices is symmetric on every timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.matrix_symmetric", "kind": "function", "doc": "
Symmetrizes the correlator matrices on every timeslice.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.GEVP": {"fullname": "pyerrors.correlators.Corr.GEVP", "modulename": "pyerrors.correlators", "qualname": "Corr.GEVP", "kind": "function", "doc": "
Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.
\n\n
The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the\nlargest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing
\n\n
\n
C.GEVP(t0=2)[0]  # Ground state vector(s)\nC.GEVP(t0=2)[:3]  # Vectors for the lowest three states\n
\n
\n\n
Parameters
\n\n
    \n
  • t0 (int):\nThe time t0 for the right hand side of the GEVP according to $G(t)v_i=\\lambda_i G(t_0)v_i$
    • \n
  • ts (int):\nfixed time $G(t_s)v_i=\\lambda_i G(t_0)v_i$ if sort=None.\nIf sort=\"Eigenvector\" it gives a reference point for the sorting method.
    • \n
  • sort (string):\nIf this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.\n
      \n
    • \"Eigenvalue\": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice.
      • \n
    • \"Eigenvector\": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.\nThe reference state is identified by its eigenvalue at $t=t_s$.
      • \n
    • \n
\n\n
Other Parameters
\n\n
    \n
  • state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
    • \n
\n", "signature": "(self, t0, ts=None, sort='Eigenvalue', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Eigenvalue": {"fullname": "pyerrors.correlators.Corr.Eigenvalue", "modulename": "pyerrors.correlators", "qualname": "Corr.Eigenvalue", "kind": "function", "doc": "
Determines the eigenvalue of the GEVP by solving and projecting the correlator
\n\n
Parameters
\n\n
    \n
  • state (int):\nThe state one is interested in ordered by energy. The lowest state is zero.
    • \n
  • All other parameters are identical to the ones of Corr.GEVP.
    • \n
\n", "signature": "(self, t0, ts=None, state=0, sort='Eigenvalue'):", "funcdef": "def"}, "pyerrors.correlators.Corr.Hankel": {"fullname": "pyerrors.correlators.Corr.Hankel", "modulename": "pyerrors.correlators", "qualname": "Corr.Hankel", "kind": "function", "doc": "
Constructs an NxN Hankel matrix
\n\n
C(t) c(t+1) ... c(t+n-1)\nC(t+1) c(t+2) ... c(t+n)\n.................\nC(t+(n-1)) c(t+n) ... c(t+2(n-1))
\n\n
Parameters
\n\n
    \n
  • N (int):\nDimension of the Hankel matrix
    • \n
  • periodic (bool, optional):\ndetermines whether the matrix is extended periodically
    • \n
\n", "signature": "(self, N, periodic=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.roll": {"fullname": "pyerrors.correlators.Corr.roll", "modulename": "pyerrors.correlators", "qualname": "Corr.roll", "kind": "function", "doc": "
Periodically shift the correlator by dt timeslices
\n\n
Parameters
\n\n
    \n
  • dt (int):\nnumber of timeslices
    • \n
\n", "signature": "(self, dt):", "funcdef": "def"}, "pyerrors.correlators.Corr.reverse": {"fullname": "pyerrors.correlators.Corr.reverse", "modulename": "pyerrors.correlators", "qualname": "Corr.reverse", "kind": "function", "doc": "
Reverse the time ordering of the Corr
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.thin": {"fullname": "pyerrors.correlators.Corr.thin", "modulename": "pyerrors.correlators", "qualname": "Corr.thin", "kind": "function", "doc": "
Thin out a correlator to suppress correlations
\n\n
Parameters
\n\n
    \n
  • spacing (int):\nKeep only every 'spacing'th entry of the correlator
    • \n
  • offset (int):\nOffset the equal spacing
    • \n
\n", "signature": "(self, spacing=2, offset=0):", "funcdef": "def"}, "pyerrors.correlators.Corr.correlate": {"fullname": "pyerrors.correlators.Corr.correlate", "modulename": "pyerrors.correlators", "qualname": "Corr.correlate", "kind": "function", "doc": "
Correlate the correlator with another correlator or Obs
\n\n
Parameters
\n\n
    \n
  • partner (Obs or Corr):\npartner to correlate the correlator with.\nCan either be an Obs which is correlated with all entries of the\ncorrelator or a Corr of same length.
    • \n
\n", "signature": "(self, partner):", "funcdef": "def"}, "pyerrors.correlators.Corr.reweight": {"fullname": "pyerrors.correlators.Corr.reweight", "modulename": "pyerrors.correlators", "qualname": "Corr.reweight", "kind": "function", "doc": "
Reweight the correlator.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl.
    • \n
\n", "signature": "(self, weight, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.T_symmetry": {"fullname": "pyerrors.correlators.Corr.T_symmetry", "modulename": "pyerrors.correlators", "qualname": "Corr.T_symmetry", "kind": "function", "doc": "
Return the time symmetry average of the correlator and its partner
\n\n
Parameters
\n\n
    \n
  • partner (Corr):\nTime symmetry partner of the Corr
    • \n
  • partity (int):\nParity quantum number of the correlator, can be +1 or -1
    • \n
\n", "signature": "(self, partner, parity=1):", "funcdef": "def"}, "pyerrors.correlators.Corr.deriv": {"fullname": "pyerrors.correlators.Corr.deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.deriv", "kind": "function", "doc": "
Return the first derivative of the correlator with respect to x0.
\n\n
Parameters
\n\n
    \n
  • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, forward, backward, improved, log, default: symmetric
    • \n
\n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.second_deriv": {"fullname": "pyerrors.correlators.Corr.second_deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.second_deriv", "kind": "function", "doc": "
Return the second derivative of the correlator with respect to x0.
\n\n
Parameters
\n\n
    \n
  • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, improved, log, default: symmetric
    • \n
\n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.m_eff": {"fullname": "pyerrors.correlators.Corr.m_eff", "modulename": "pyerrors.correlators", "qualname": "Corr.m_eff", "kind": "function", "doc": "
Returns the effective mass of the correlator as correlator object
\n\n
Parameters
\n\n
    \n
  • variant (str):\nlog : uses the standard effective mass log(C(t) / C(t+1))\ncosh, periodic : Use periodicitiy of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.\nsinh : Use anti-periodicitiy of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.\nSee, e.g., arXiv:1205.5380\narccosh : Uses the explicit form of the symmetrized correlator (not recommended)\nlogsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
    • \n
  • guess (float):\nguess for the root finder, only relevant for the root variant
    • \n
\n", "signature": "(self, variant='log', guess=1.0):", "funcdef": "def"}, "pyerrors.correlators.Corr.fit": {"fullname": "pyerrors.correlators.Corr.fit", "modulename": "pyerrors.correlators", "qualname": "Corr.fit", "kind": "function", "doc": "
Fits function to the data
\n\n
Parameters
\n\n
    \n
  • function (obj):\nfunction to fit to the data. See fits.least_squares for details.
    • \n
  • fitrange (list):\nTwo element list containing the timeslices on which the fit is supposed to start and stop.\nCaution: This range is inclusive as opposed to standard python indexing.\nfitrange=[4, 6] corresponds to the three entries 4, 5 and 6.\nIf not specified, self.prange or all timeslices are used.
    • \n
  • silent (bool):\nDecides whether output is printed to the standard output.
    • \n
\n", "signature": "(self, function, fitrange=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.plateau": {"fullname": "pyerrors.correlators.Corr.plateau", "modulename": "pyerrors.correlators", "qualname": "Corr.plateau", "kind": "function", "doc": "
Extract a plateau value from a Corr object
\n\n
Parameters
\n\n
    \n
  • plateau_range (list):\nlist with two entries, indicating the first and the last timeslice\nof the plateau region.
    • \n
  • method (str):\nmethod to extract the plateau.\n    'fit' fits a constant to the plateau region\n    'avg', 'average' or 'mean' just average over the given timeslices.
    • \n
  • auto_gamma (bool):\napply gamma_method with default parameters to the Corr. Defaults to None
    • \n
\n", "signature": "(self, plateau_range=None, method='fit', auto_gamma=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.set_prange": {"fullname": "pyerrors.correlators.Corr.set_prange", "modulename": "pyerrors.correlators", "qualname": "Corr.set_prange", "kind": "function", "doc": "
Sets the attribute prange of the Corr object.
\n", "signature": "(self, prange):", "funcdef": "def"}, "pyerrors.correlators.Corr.show": {"fullname": "pyerrors.correlators.Corr.show", "modulename": "pyerrors.correlators", "qualname": "Corr.show", "kind": "function", "doc": "
Plots the correlator using the tag of the correlator as label if available.
\n\n
Parameters
\n\n
    \n
  • x_range (list):\nlist of two values, determining the range of the x-axis e.g. [4, 8].
    • \n
  • comp (Corr or list of Corr):\nCorrelator or list of correlators which are plotted for comparison.\nThe tags of these correlators are used as labels if available.
    • \n
  • logscale (bool):\nSets y-axis to logscale.
    • \n
  • plateau (Obs):\nPlateau value to be visualized in the figure.
    • \n
  • fit_res (Fit_result):\nFit_result object to be visualized.
    • \n
  • ylabel (str):\nLabel for the y-axis.
    • \n
  • save (str):\npath to file in which the figure should be saved.
    • \n
  • auto_gamma (bool):\nApply the gamma method with standard parameters to all correlators and plateau values before plotting.
    • \n
  • hide_sigma (float):\nHides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
    • \n
  • references (list):\nList of floating point values that are displayed as horizontal lines for reference.
    • \n
  • title (string):\nOptional title of the figure.
    • \n
\n", "signature": "(\tself,\tx_range=None,\tcomp=None,\ty_range=None,\tlogscale=False,\tplateau=None,\tfit_res=None,\tylabel=None,\tsave=None,\tauto_gamma=False,\thide_sigma=None,\treferences=None,\ttitle=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.spaghetti_plot": {"fullname": "pyerrors.correlators.Corr.spaghetti_plot", "modulename": "pyerrors.correlators", "qualname": "Corr.spaghetti_plot", "kind": "function", "doc": "
Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.
\n\n
Parameters
\n\n
    \n
  • logscale (bool):\nDetermines whether the scale of the y-axis is logarithmic or standard.
    • \n
\n", "signature": "(self, logscale=True):", "funcdef": "def"}, "pyerrors.correlators.Corr.dump": {"fullname": "pyerrors.correlators.Corr.dump", "modulename": "pyerrors.correlators", "qualname": "Corr.dump", "kind": "function", "doc": "
Dumps the Corr into a file of chosen type
\n\n
Parameters
\n\n
    \n
  • filename (str):\nName of the file to be saved.
    • \n
  • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n", "signature": "(self, filename, datatype='json.gz', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.print": {"fullname": "pyerrors.correlators.Corr.print", "modulename": "pyerrors.correlators", "qualname": "Corr.print", "kind": "function", "doc": "
\n", "signature": "(self, print_range=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.sqrt": {"fullname": "pyerrors.correlators.Corr.sqrt", "modulename": "pyerrors.correlators", "qualname": "Corr.sqrt", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.log": {"fullname": "pyerrors.correlators.Corr.log", "modulename": "pyerrors.correlators", "qualname": "Corr.log", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.exp": {"fullname": "pyerrors.correlators.Corr.exp", "modulename": "pyerrors.correlators", "qualname": "Corr.exp", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sin": {"fullname": "pyerrors.correlators.Corr.sin", "modulename": "pyerrors.correlators", "qualname": "Corr.sin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cos": {"fullname": "pyerrors.correlators.Corr.cos", "modulename": "pyerrors.correlators", "qualname": "Corr.cos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tan": {"fullname": "pyerrors.correlators.Corr.tan", "modulename": "pyerrors.correlators", "qualname": "Corr.tan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sinh": {"fullname": "pyerrors.correlators.Corr.sinh", "modulename": "pyerrors.correlators", "qualname": "Corr.sinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cosh": {"fullname": "pyerrors.correlators.Corr.cosh", "modulename": "pyerrors.correlators", "qualname": "Corr.cosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tanh": {"fullname": "pyerrors.correlators.Corr.tanh", "modulename": "pyerrors.correlators", "qualname": "Corr.tanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsin": {"fullname": "pyerrors.correlators.Corr.arcsin", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccos": {"fullname": "pyerrors.correlators.Corr.arccos", "modulename": "pyerrors.correlators", "qualname": "Corr.arccos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctan": {"fullname": "pyerrors.correlators.Corr.arctan", "modulename": "pyerrors.correlators", "qualname": "Corr.arctan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsinh": {"fullname": "pyerrors.correlators.Corr.arcsinh", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccosh": {"fullname": "pyerrors.correlators.Corr.arccosh", "modulename": "pyerrors.correlators", "qualname": "Corr.arccosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctanh": {"fullname": "pyerrors.correlators.Corr.arctanh", "modulename": "pyerrors.correlators", "qualname": "Corr.arctanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.prune": {"fullname": "pyerrors.correlators.Corr.prune", "modulename": "pyerrors.correlators", "qualname": "Corr.prune", "kind": "function", "doc": "
Project large correlation matrix to lowest states
\n\n
This method can be used to reduce the size of an (N x N) correlation matrix\nto (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise\nis still small.
\n\n
Parameters
\n\n
    \n
  • Ntrunc (int):\nRank of the target matrix.
    • \n
  • tproj (int):\nTime where the eigenvectors are evaluated, corresponds to ts in the GEVP method.\nThe default value is 3.
    • \n
  • t0proj (int):\nTime where the correlation matrix is inverted. Choosing t0proj=1 is strongly\ndiscouraged for O(a) improved theories, since the correctness of the procedure\ncannot be granted in this case. The default value is 2.
    • \n
  • basematrix (Corr):\nCorrelation matrix that is used to determine the eigenvectors of the\nlowest states based on a GEVP. basematrix is taken to be the Corr itself if\nis is not specified.
    • \n
\n\n
Notes
\n\n
We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving\nthe GEVP $$C(t) v_n(t, t_0) = \\lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \\equiv t_\\mathrm{proj}$\nand $t_0 \\equiv t_{0, \\mathrm{proj}}$. The target matrix is projected onto the subspace of the\nresulting eigenvectors $v_n, n=1,\\dots,N_\\mathrm{trunc}$ via\n$$G^\\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large\ncorrelation matrix and to remove some noise that is added by irrelevant operators.\nThis may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated\nbound $t_0 \\leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.
\n", "signature": "(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):", "funcdef": "def"}, "pyerrors.covobs": {"fullname": "pyerrors.covobs", "modulename": "pyerrors.covobs", "kind": "module", "doc": "
\n"}, "pyerrors.covobs.Covobs": {"fullname": "pyerrors.covobs.Covobs", "modulename": "pyerrors.covobs", "qualname": "Covobs", "kind": "class", "doc": "
\n"}, "pyerrors.covobs.Covobs.__init__": {"fullname": "pyerrors.covobs.Covobs.__init__", "modulename": "pyerrors.covobs", "qualname": "Covobs.__init__", "kind": "function", "doc": "
Initialize Covobs object.
\n\n
Parameters
\n\n
    \n
  • mean (float):\nMean value of the new Obs
    • \n
  • cov (list or array):\n2d Covariance matrix or 1d diagonal entries
    • \n
  • name (str):\nidentifier for the covariance matrix
    • \n
  • pos (int):\nPosition of the variance belonging to mean in cov.\nIs taken to be 1 if cov is 0-dimensional
    • \n
  • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
    • \n
\n", "signature": "(mean, cov, name, pos=None, grad=None)"}, "pyerrors.covobs.Covobs.errsq": {"fullname": "pyerrors.covobs.Covobs.errsq", "modulename": "pyerrors.covobs", "qualname": "Covobs.errsq", "kind": "function", "doc": "
Return the variance (= square of the error) of the Covobs
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "
\n"}, "pyerrors.dirac.epsilon_tensor": {"fullname": "pyerrors.dirac.epsilon_tensor", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor", "kind": "function", "doc": "
Rank-3 epsilon tensor
\n\n
Based on https://codegolf.stackexchange.com/a/160375
\n\n
Returns
\n\n
    \n
  • elem (int):\nElement (i,j,k) of the epsilon tensor of rank 3
    • \n
\n", "signature": "(i, j, k):", "funcdef": "def"}, "pyerrors.dirac.epsilon_tensor_rank4": {"fullname": "pyerrors.dirac.epsilon_tensor_rank4", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor_rank4", "kind": "function", "doc": "
Rank-4 epsilon tensor
\n\n
Extension of https://codegolf.stackexchange.com/a/160375
\n\n
Returns
\n\n
    \n
  • elem (int):\nElement (i,j,k,o) of the epsilon tensor of rank 4
    • \n
\n", "signature": "(i, j, k, o):", "funcdef": "def"}, "pyerrors.dirac.Grid_gamma": {"fullname": "pyerrors.dirac.Grid_gamma", "modulename": "pyerrors.dirac", "qualname": "Grid_gamma", "kind": "function", "doc": "
Returns gamma matrix in Grid labeling.
\n", "signature": "(gamma_tag):", "funcdef": "def"}, "pyerrors.fits": {"fullname": "pyerrors.fits", "modulename": "pyerrors.fits", "kind": "module", "doc": "
\n"}, "pyerrors.fits.Fit_result": {"fullname": "pyerrors.fits.Fit_result", "modulename": "pyerrors.fits", "qualname": "Fit_result", "kind": "class", "doc": "
Represents fit results.
\n\n
Attributes
\n\n
    \n
  • fit_parameters (list):\nresults for the individual fit parameters,\nalso accessible via indices.
    • \n
  • chisquare_by_dof (float):\nreduced chisquare.
    • \n
  • p_value (float):\np-value of the fit
    • \n
  • t2_p_value (float):\nHotelling t-squared p-value for correlated fits.
    • \n
\n", "bases": "collections.abc.Sequence"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "
Apply the gamma method to all fit parameters
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.Fit_result.gm": {"fullname": "pyerrors.fits.Fit_result.gm", "modulename": "pyerrors.fits", "qualname": "Fit_result.gm", "kind": "function", "doc": "
Apply the gamma method to all fit parameters
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.least_squares": {"fullname": "pyerrors.fits.least_squares", "modulename": "pyerrors.fits", "qualname": "least_squares", "kind": "function", "doc": "
Performs a non-linear fit to y = func(x).\n    ```
\n\n
Parameters
\n\n
    \n
  • For an uncombined fit:
    • \n
  • x (list):\nlist of floats.
    • \n
  • y (list):\nlist of Obs.
    • \n
  • func (object):\nfit function, has to be of the form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n   return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
    \n
    \n\n
    For multiple x values func can be of the form
    \n\n
    \n
    def func(a, x):\n   (x1, x2) = x\n   return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • OR For a combined fit:
    • \n
  • x (dict):\ndict of lists.
    • \n
  • y (dict):\ndict of lists of Obs.
    • \n
  • funcs (dict):\ndict of objects\nfit functions have to be of the form (here a[0] is the common fit parameter)\n```python\nimport autograd.numpy as anp\nfuncs = {\"a\": func_a,\n       \"b\": func_b}
    \n\n
    def func_a(a, x):\n   return a[1] * anp.exp(-a[0] * x)
    \n\n
    def func_b(a, x):\n   return a[2] * anp.exp(-a[0] * x)
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • priors (list, optional):\npriors has to be a list with an entry for every parameter in the fit. The entries can either be\nObs (e.g. results from a previous fit) or strings containing a value and an error formatted like\n0.548(23), 500(40) or 0.5(0.4)
    • \n
  • silent (bool, optional):\nIf true all output to the console is omitted (default False).
    • \n
  • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for\nnon-linear fits with many parameters. In case of correlated fits the guess is used to perform\nan uncorrelated fit which then serves as guess for the correlated fit.
    • \n
  • method (str, optional):\ncan be used to choose an alternative method for the minimization of chisquare.\nThe possible methods are the ones which can be used for scipy.optimize.minimize and\nmigrad of iminuit. If no method is specified, Levenberg-Marquard is used.\nReliable alternatives are migrad, Powell and Nelder-Mead.
    • \n
  • tol (float, optional):\ncan be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence\nto a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly\ninvalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values\nThe stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
    • \n
  • correlated_fit (bool):\nIf True, use the full inverse covariance matrix in the definition of the chisquare cost function.\nFor details about how the covariance matrix is estimated see pyerrors.obs.covariance.\nIn practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).\nThis procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).\nAt the moment this option only works for prior==None and when no method is given.
    • \n
  • expected_chisquare (bool):\nIf True estimates the expected chisquare which is\ncorrected by effects caused by correlated input data (default False).
    • \n
  • resplot (bool):\nIf True, a plot which displays fit, data and residuals is generated (default False).
    • \n
  • qqplot (bool):\nIf True, a quantile-quantile plot of the fit result is generated (default False).
    • \n
  • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
    • \n
\n\n
Returns
\n\n
    \n
  • output (Fit_result):\nParameters and information on the fitted result.
    • \n
\n", "signature": "(x, y, func, priors=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.total_least_squares": {"fullname": "pyerrors.fits.total_least_squares", "modulename": "pyerrors.fits", "qualname": "total_least_squares", "kind": "function", "doc": "
Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
\n\n
Parameters
\n\n
    \n
  • x (list):\nlist of Obs, or a tuple of lists of Obs
    • \n
  • y (list):\nlist of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
    • \n
  • func (object):\nfunc has to be of the form
    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n   return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
    \n
    \n\n
    For multiple x values func can be of the form
    \n\n
    \n
    def func(a, x):\n   (x1, x2) = x\n   return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n
    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.
    • \n
  • silent (bool, optional):\nIf true all output to the console is omitted (default False).
    • \n
  • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for non-linear\nfits with many parameters.
    • \n
  • expected_chisquare (bool):\nIf true prints the expected chisquare which is\ncorrected by effects caused by correlated input data.\nThis can take a while as the full correlation matrix\nhas to be calculated (default False).
    • \n
  • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
    • \n
\n\n
Notes
\n\n
Based on the orthogonal distance regression module of scipy.
\n\n
Returns
\n\n
    \n
  • output (Fit_result):\nParameters and information on the fitted result.
    • \n
\n", "signature": "(x, y, func, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.fit_lin": {"fullname": "pyerrors.fits.fit_lin", "modulename": "pyerrors.fits", "qualname": "fit_lin", "kind": "function", "doc": "
Performs a linear fit to y = n + m * x and returns two Obs n, m.
\n\n
Parameters
\n\n
    \n
  • x (list):\nCan either be a list of floats in which case no xerror is assumed, or\na list of Obs, where the dvalues of the Obs are used as xerror for the fit.
    • \n
  • y (list):\nList of Obs, the dvalues of the Obs are used as yerror for the fit.
    • \n
\n\n
Returns
\n\n
    \n
  • fit_parameters (list[Obs]):\nLIist of fitted observables.
    • \n
\n", "signature": "(x, y, **kwargs):", "funcdef": "def"}, "pyerrors.fits.qqplot": {"fullname": "pyerrors.fits.qqplot", "modulename": "pyerrors.fits", "qualname": "qqplot", "kind": "function", "doc": "
Generates a quantile-quantile plot of the fit result which can be used to\n   check if the residuals of the fit are gaussian distributed.
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(x, o_y, func, p, title=''):", "funcdef": "def"}, "pyerrors.fits.residual_plot": {"fullname": "pyerrors.fits.residual_plot", "modulename": "pyerrors.fits", "qualname": "residual_plot", "kind": "function", "doc": "
Generates a plot which compares the fit to the data and displays the corresponding residuals
\n\n
For uncorrelated data the residuals are expected to be distributed ~N(0,1).
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(x, y, func, fit_res, title=''):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "
Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
\n\n
Returns
\n\n
    \n
  • err (np.array(Obs)):\nError band for an array of sample values x
    • \n
\n", "signature": "(x, func, beta):", "funcdef": "def"}, "pyerrors.fits.ks_test": {"fullname": "pyerrors.fits.ks_test", "modulename": "pyerrors.fits", "qualname": "ks_test", "kind": "function", "doc": "
Performs a Kolmogorov\u2013Smirnov test for the p-values of all fit object.
\n\n
Parameters
\n\n
    \n
  • objects (list):\nList of fit results to include in the analysis (optional).
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(objects=None):", "funcdef": "def"}, "pyerrors.input": {"fullname": "pyerrors.input", "modulename": "pyerrors.input", "kind": "module", "doc": "
pyerrors includes an input submodule in which input routines and parsers for the output of various numerical programs are contained.
\n\n
Jackknife samples
\n\n
For comparison with other analysis workflows pyerrors can also generate jackknife samples from an Obs object or import jackknife samples into an Obs object.\nSee pyerrors.obs.Obs.export_jackknife and pyerrors.obs.import_jackknife for details.
\n"}, "pyerrors.input.bdio": {"fullname": "pyerrors.input.bdio", "modulename": "pyerrors.input.bdio", "kind": "module", "doc": "
\n"}, "pyerrors.input.bdio.read_ADerrors": {"fullname": "pyerrors.input.bdio.read_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "read_ADerrors", "kind": "function", "doc": "
Extract generic MCMC data from a bdio file
\n\n
read_ADerrors requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path -- path to the bdio file
    • \n
  • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
\n\n
Returns
\n\n
    \n
  • data (List[Obs]):\nExtracted data
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.write_ADerrors": {"fullname": "pyerrors.input.bdio.write_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "write_ADerrors", "kind": "function", "doc": "
Write Obs to a bdio file according to ADerrors conventions
\n\n
read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path -- path to the bdio file
    • \n
  • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
\n\n
Returns
\n\n
    \n
  • success (int):\nreturns 0 is successful
    • \n
\n", "signature": "(obs_list, file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_mesons": {"fullname": "pyerrors.input.bdio.read_mesons", "modulename": "pyerrors.input.bdio", "qualname": "read_mesons", "kind": "function", "doc": "
Extract mesons data from a bdio file and return it as a dictionary
\n\n
The dictionary can be accessed with a tuple consisting of (type, source_position, kappa1, kappa2)
\n\n
read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path (str):\npath to the bdio file
    • \n
  • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
  • start (int):\nThe first configuration to be read (default 1)
    • \n
  • stop (int):\nThe last configuration to be read (default None)
    • \n
  • step (int):\nFixed step size between two measurements (default 1)
    • \n
  • alternative_ensemble_name (str):\nManually overwrite ensemble name
    • \n
\n\n
Returns
\n\n
    \n
  • data (dict):\nExtracted meson data
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_dSdm": {"fullname": "pyerrors.input.bdio.read_dSdm", "modulename": "pyerrors.input.bdio", "qualname": "read_dSdm", "kind": "function", "doc": "
Extract dSdm data from a bdio file and return it as a dictionary
\n\n
The dictionary can be accessed with a tuple consisting of (type, kappa)
\n\n
read_dSdm requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to
\n\n
all:                bdio.o $(LIBDIR)\n            gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n            cp $(BUILDDIR)/libbdio.so $(LIBDIR)/
\n\n
Parameters
\n\n
    \n
  • file_path (str):\npath to the bdio file
    • \n
  • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
    • \n
  • start (int):\nThe first configuration to be read (default 1)
    • \n
  • stop (int):\nThe last configuration to be read (default None)
    • \n
  • step (int):\nFixed step size between two measurements (default 1)
    • \n
  • alternative_ensemble_name (str):\nManually overwrite ensemble name
    • \n
\n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.dobs": {"fullname": "pyerrors.input.dobs", "modulename": "pyerrors.input.dobs", "kind": "module", "doc": "
\n"}, "pyerrors.input.dobs.create_pobs_string": {"fullname": "pyerrors.input.dobs.create_pobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_pobs_string", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to an xml string\naccording to the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
    • \n
\n\n
Returns
\n\n
    \n
  • xml_str (str):\nXML formatted string of the input data
    • \n
\n", "signature": "(obsl, name, spec='', origin='', symbol=[], enstag=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_pobs": {"fullname": "pyerrors.input.dobs.write_pobs", "modulename": "pyerrors.input.dobs", "qualname": "write_pobs", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
    • \n
  • gz (bool):\nIf True, the output is a gzipped xml. If False, the output is an xml file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(\tobsl,\tfname,\tname,\tspec='',\torigin='',\tsymbol=[],\tenstag=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_pobs": {"fullname": "pyerrors.input.dobs.read_pobs", "modulename": "pyerrors.input.dobs", "qualname": "read_pobs", "kind": "function", "doc": "
Import a list of Obs from an xml.gz file in the Zeuthen pobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • separatior_insertion (str or int):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nNone (default): Replica names remain unchanged.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(fname, full_output=False, gz=True, separator_insertion=None):", "funcdef": "def"}, "pyerrors.input.dobs.import_dobs_string": {"fullname": "pyerrors.input.dobs.import_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "import_dobs_string", "kind": "function", "doc": "
Import a list of Obs from a string in the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • content (str):\nXML string containing the data
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(content, full_output=False, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_dobs": {"fullname": "pyerrors.input.dobs.read_dobs", "modulename": "pyerrors.input.dobs", "qualname": "read_dobs", "kind": "function", "doc": "
Import a list of Obs from an xml.gz file in the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes XML file.
    • \n
  • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
    • \n
\n\n
Returns
\n\n
    \n
  • res (list[Obs]):\nImported data
    • \n
  • or
    • \n
  • res (dict):\nImported data and meta-data
    • \n
\n", "signature": "(fname, full_output=False, gz=True, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.create_dobs_string": {"fullname": "pyerrors.input.dobs.create_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_dobs_string", "kind": "function", "doc": "
Generate the string for the export of a list of Obs or structures containing Obs\nto a .xml.gz file according to the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically. The separator |is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • who (str):\nProvide the name of the person that exports the data.
    • \n
  • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
    • \n
\n\n
Returns
\n\n
    \n
  • xml_str (str):\nXML string generated from the data
    • \n
\n", "signature": "(\tobsl,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_dobs": {"fullname": "pyerrors.input.dobs.write_dobs", "modulename": "pyerrors.input.dobs", "qualname": "write_dobs", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen dobs format.
\n\n
Tags are not written or recovered automatically. The separator | is removed from the replica names.
\n\n
Parameters
\n\n
    \n
  • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • name (str):\nThe name of the observable.
    • \n
  • spec (str):\nOptional string that describes the contents of the file.
    • \n
  • origin (str):\nSpecify where the data has its origin.
    • \n
  • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
    • \n
  • who (str):\nProvide the name of the person that exports the data.
    • \n
  • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
    • \n
  • gz (bool):\nIf True, the output is a gzipped XML. If False, the output is a XML file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(\tobsl,\tfname,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.hadrons": {"fullname": "pyerrors.input.hadrons", "modulename": "pyerrors.input.hadrons", "kind": "module", "doc": "
\n"}, "pyerrors.input.hadrons.read_meson_hd5": {"fullname": "pyerrors.input.hadrons.read_meson_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_meson_hd5", "kind": "function", "doc": "
Read hadrons meson hdf5 file and extract the meson labeled 'meson'
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • meson (str):\nlabel of the meson to be extracted, standard value meson_0 which\ncorresponds to the pseudoscalar pseudoscalar two-point function.
    • \n
  • gammas (tuple of strings):\nInstrad of a meson label one can also provide a tuple of two strings\nindicating the gamma matrices at source and sink.\n(\"Gamma5\", \"Gamma5\") corresponds to the pseudoscalar pseudoscalar\ntwo-point function. The gammas argument dominateds over meson.
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • corr (Corr):\nCorrelator of the source sink combination in question.
    • \n
\n", "signature": "(path, filestem, ens_id, meson='meson_0', idl=None, gammas=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_DistillationContraction_hd5": {"fullname": "pyerrors.input.hadrons.read_DistillationContraction_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_DistillationContraction_hd5", "kind": "function", "doc": "
Read hadrons DistillationContraction hdf5 files in given directory structure
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the directories to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • diagrams (list):\nList of strings of the diagrams to extract, e.g. [\"direct\", \"box\", \"cross\"].
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result (dict):\nextracted DistillationContration data
    • \n
\n", "signature": "(path, ens_id, diagrams=['direct'], idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.Npr_matrix": {"fullname": "pyerrors.input.hadrons.Npr_matrix", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix", "kind": "class", "doc": "
ndarray(shape, dtype=float, buffer=None, offset=0,\n        strides=None, order=None)
\n\n
An array object represents a multidimensional, homogeneous array\nof fixed-size items.  An associated data-type object describes the\nformat of each element in the array (its byte-order, how many bytes it\noccupies in memory, whether it is an integer, a floating point number,\nor something else, etc.)
\n\n
Arrays should be constructed using array, zeros or empty (refer\nto the See Also section below).  The parameters given here refer to\na low-level method (ndarray(...)) for instantiating an array.
\n\n
For more information, refer to the numpy module and examine the\nmethods and attributes of an array.
\n\n
Parameters
\n\n
    \n
  • (for the __new__ method; see Notes below)
    • \n
  • shape (tuple of ints):\nShape of created array.
    • \n
  • dtype (data-type, optional):\nAny object that can be interpreted as a numpy data type.
    • \n
  • buffer (object exposing buffer interface, optional):\nUsed to fill the array with data.
    • \n
  • offset (int, optional):\nOffset of array data in buffer.
    • \n
  • strides (tuple of ints, optional):\nStrides of data in memory.
    • \n
  • order ({'C', 'F'}, optional):\nRow-major (C-style) or column-major (Fortran-style) order.
    • \n
\n\n
Attributes
\n\n
    \n
  • T (ndarray):\nTranspose of the array.
    • \n
  • data (buffer):\nThe array's elements, in memory.
    • \n
  • dtype (dtype object):\nDescribes the format of the elements in the array.
    • \n
  • flags (dict):\nDictionary containing information related to memory use, e.g.,\n'C_CONTIGUOUS', 'OWNDATA', 'WRITEABLE', etc.
    • \n
  • flat (numpy.flatiter object):\nFlattened version of the array as an iterator.  The iterator\nallows assignments, e.g., x.flat = 3 (See ndarray.flat for\nassignment examples; TODO).
    • \n
  • imag (ndarray):\nImaginary part of the array.
    • \n
  • real (ndarray):\nReal part of the array.
    • \n
  • size (int):\nNumber of elements in the array.
    • \n
  • itemsize (int):\nThe memory use of each array element in bytes.
    • \n
  • nbytes (int):\nThe total number of bytes required to store the array data,\ni.e., itemsize * size.
    • \n
  • ndim (int):\nThe array's number of dimensions.
    • \n
  • shape (tuple of ints):\nShape of the array.
    • \n
  • strides (tuple of ints):\nThe step-size required to move from one element to the next in\nmemory. For example, a contiguous (3, 4) array of type\nint16 in C-order has strides (8, 2).  This implies that\nto move from element to element in memory requires jumps of 2 bytes.\nTo move from row-to-row, one needs to jump 8 bytes at a time\n(2 * 4).
    • \n
  • ctypes (ctypes object):\nClass containing properties of the array needed for interaction\nwith ctypes.
    • \n
  • base (ndarray):\nIf the array is a view into another array, that array is its base\n(unless that array is also a view).  The base array is where the\narray data is actually stored.
    • \n
\n\n
See Also
\n\n
array:  Construct an array. 
\nzeros:  Create an array, each element of which is zero. 
\nempty:  Create an array, but leave its allocated memory unchanged (i.e.,\nit contains \"garbage\"). 
\ndtype:  Create a data-type. 
\nnumpy.typing.NDArray:  An ndarray alias :term:generic <generic type>\nw.r.t. its dtype.type <numpy.dtype.type>.  
\n\n
Notes
\n\n
There are two modes of creating an array using __new__:
\n\n
    \n
  1. If buffer is None, then only shape, dtype, and order\nare used.
    2. \n
  2. If buffer is an object exposing the buffer interface, then\nall keywords are interpreted.
    3. \n
\n\n
No __init__ method is needed because the array is fully initialized\nafter the __new__ method.
\n\n
Examples
\n\n
These examples illustrate the low-level ndarray constructor.  Refer\nto the See Also section above for easier ways of constructing an\nndarray.
\n\n
First mode, buffer is None:
\n\n
\n
>>> np.ndarray(shape=(2,2), dtype=float, order='F')\narray([[0.0e+000, 0.0e+000], # random\n       [     nan, 2.5e-323]])\n
\n
\n\n
Second mode:
\n\n
\n
>>> np.ndarray((2,), buffer=np.array([1,2,3]),\n...            offset=np.int_().itemsize,\n...            dtype=int) # offset = 1*itemsize, i.e. skip first element\narray([2, 3])\n
\n
\n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "
Gamma_5 hermitean conjugate
\n\n
Uses the fact that the propagator is gamma5 hermitean, so just the\nin and out momenta of the propagator are exchanged.
\n"}, "pyerrors.input.hadrons.read_ExternalLeg_hd5": {"fullname": "pyerrors.input.hadrons.read_ExternalLeg_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_ExternalLeg_hd5", "kind": "function", "doc": "
Read hadrons ExternalLeg hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result (Npr_matrix):\nread Cobs-matrix
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Bilinear_hd5": {"fullname": "pyerrors.input.hadrons.read_Bilinear_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Bilinear_hd5", "kind": "function", "doc": "
Read hadrons Bilinear hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
\n\n
Returns
\n\n
    \n
  • result_dict (dict[Npr_matrix]):\nextracted Bilinears
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Fourquark_hd5": {"fullname": "pyerrors.input.hadrons.read_Fourquark_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Fourquark_hd5", "kind": "function", "doc": "
Read hadrons FourquarkFullyConnected hdf5 file and output an array of CObs
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the files to read
    • \n
  • filestem (str):\nnamestem of the files to read
    • \n
  • ens_id (str):\nname of the ensemble, required for internal bookkeeping
    • \n
  • idl (range):\nIf specified only configurations in the given range are read in.
    • \n
  • vertices (list):\nVertex functions to be extracted.
    • \n
\n\n
Returns
\n\n
    \n
  • result_dict (dict):\nextracted fourquark matrizes
    • \n
\n", "signature": "(path, filestem, ens_id, idl=None, vertices=['VA', 'AV']):", "funcdef": "def"}, "pyerrors.input.json": {"fullname": "pyerrors.input.json", "modulename": "pyerrors.input.json", "kind": "module", "doc": "
\n"}, "pyerrors.input.json.create_json_string": {"fullname": "pyerrors.input.json.create_json_string", "modulename": "pyerrors.input.json", "qualname": "create_json_string", "kind": "function", "doc": "
Generate the string for the export of a list of Obs or structures containing Obs\nto a .json(.gz) file
\n\n
Parameters
\n\n
    \n
  • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
\n\n
Returns
\n\n
    \n
  • json_string (str):\nString for export to .json(.gz) file
    • \n
\n", "signature": "(ol, description='', indent=1):", "funcdef": "def"}, "pyerrors.input.json.dump_to_json": {"fullname": "pyerrors.input.json.dump_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_to_json", "kind": "function", "doc": "
Export a list of Obs or structures containing Obs to a .json(.gz) file
\n\n
Parameters
\n\n
    \n
  • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
  • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
    • \n
\n\n
Returns
\n\n
    \n
  • Null
    • \n
\n", "signature": "(ol, fname, description='', indent=1, gz=True):", "funcdef": "def"}, "pyerrors.input.json.import_json_string": {"fullname": "pyerrors.input.json.import_json_string", "modulename": "pyerrors.input.json", "qualname": "import_json_string", "kind": "function", "doc": "
Reconstruct a list of Obs or structures containing Obs from a json string.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\n
Parameters
\n\n
    \n
  • json_string (str):\njson string containing the data.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nreconstructed list of observables from the json string
    • \n
  • or
    • \n
  • result (Obs):\nonly one observable if the list only has one entry
    • \n
  • or
    • \n
  • result (dict):\nif full_output=True
    • \n
\n", "signature": "(json_string, verbose=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.load_json": {"fullname": "pyerrors.input.json.load_json", "modulename": "pyerrors.input.json", "qualname": "load_json", "kind": "function", "doc": "
Import a list of Obs or structures containing Obs from a .json(.gz) file.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nreconstructed list of observables from the json string
    • \n
  • or
    • \n
  • result (Obs):\nonly one observable if the list only has one entry
    • \n
  • or
    • \n
  • result (dict):\nif full_output=True
    • \n
\n", "signature": "(fname, verbose=True, gz=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.dump_dict_to_json": {"fullname": "pyerrors.input.json.dump_dict_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_dict_to_json", "kind": "function", "doc": "
Export a dict of Obs or structures containing Obs to a .json(.gz) file
\n\n
Parameters
\n\n
    \n
  • od (dict):\nDict of JSON valid structures and objects that will be exported.\nAt the moment, these objects can be either of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • description (str):\nOptional string that describes the contents of the json file.
    • \n
  • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
    • \n
  • reps (str):\nSpecify the structure of the placeholder in exported dict to be reps[0-9]+.
    • \n
  • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(od, fname, description='', indent=1, reps='DICTOBS', gz=True):", "funcdef": "def"}, "pyerrors.input.json.load_json_dict": {"fullname": "pyerrors.input.json.load_json_dict", "modulename": "pyerrors.input.json", "qualname": "load_json_dict", "kind": "function", "doc": "
Import a dict of Obs or structures containing Obs from a .json(.gz) file.
\n\n
The following structures are supported: Obs, list, numpy.ndarray, Corr
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • verbose (bool):\nPrint additional information that was written to the file.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
  • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
    • \n
  • reps (str):\nSpecify the structure of the placeholder in imported dict to be reps[0-9]+.
    • \n
\n\n
Returns
\n\n
    \n
  • data (Obs / list / Corr):\nRead data
    • \n
  • or
    • \n
  • data (dict):\nRead data and meta-data
    • \n
\n", "signature": "(fname, verbose=True, gz=True, full_output=False, reps='DICTOBS'):", "funcdef": "def"}, "pyerrors.input.misc": {"fullname": "pyerrors.input.misc", "modulename": "pyerrors.input.misc", "kind": "module", "doc": "
\n"}, "pyerrors.input.misc.read_pbp": {"fullname": "pyerrors.input.misc.read_pbp", "modulename": "pyerrors.input.misc", "qualname": "read_pbp", "kind": "function", "doc": "
Read pbp format from given folder structure.
\n\n
Parameters
\n\n
    \n
  • r_start (list):\nlist which contains the first config to be read for each replicum
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nlist of observables read
    • \n
\n", "signature": "(path, prefix, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD": {"fullname": "pyerrors.input.openQCD", "modulename": "pyerrors.input.openQCD", "kind": "module", "doc": "
\n"}, "pyerrors.input.openQCD.read_rwms": {"fullname": "pyerrors.input.openQCD.read_rwms", "modulename": "pyerrors.input.openQCD", "qualname": "read_rwms", "kind": "function", "doc": "
Read rwms format from given folder structure. Returns a list of length nrw
\n\n
Parameters
\n\n
    \n
  • path (str):\npath that contains the data files
    • \n
  • prefix (str):\nall files in path that start with prefix are considered as input files.\nMay be used together postfix to consider only special file endings.\nPrefix is ignored, if the keyword 'files' is used.
    • \n
  • version (str):\nversion of openQCD, default 2.0
    • \n
  • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum
    • \n
  • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.ms1' for openQCD-files
    • \n
  • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
    • \n
  • print_err (bool):\nPrint additional information that is useful for debugging.
    • \n
\n\n
Returns
\n\n
    \n
  • rwms (Obs):\nReweighting factors read
    • \n
\n", "signature": "(path, prefix, version='2.0', names=None, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_t0": {"fullname": "pyerrors.input.openQCD.extract_t0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_t0", "kind": "function", "doc": "
Extract t0 from given .ms.dat files. Returns t0 as Obs.
\n\n
It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t^2 - 0.3\nis fitted with a linear function\nfrom which the exact root is extracted.
\n\n
It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to .ms.dat files
    • \n
  • prefix (str):\nEnsemble prefix
    • \n
  • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
    • \n
  • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
    • \n
  • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
    • \n
  • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
    • \n
  • plaquette (bool):\nIf true extract the plaquette estimate of t0 instead.
    • \n
  • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
    • \n
  • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
    • \n
  • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
    • \n
  • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
    • \n
\n\n
Returns
\n\n
    \n
  • t0 (Obs):\nExtracted t0
    • \n
\n", "signature": "(path, prefix, dtr_read, xmin, spatial_extent, fit_range=5, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop": {"fullname": "pyerrors.input.openQCD.read_qtop", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop", "kind": "function", "doc": "
Read the topologial charge based on openQCD gradient flow measurements.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • version (str):\nEither openQCD or sfqcd, depending on the data.
    • \n
  • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
    • \n
  • integer_charge (bool):\nIf True, the charge is rounded towards the nearest integer on each config.
    • \n
\n\n
Returns
\n\n
    \n
  • result (Obs):\nRead topological charge
    • \n
\n", "signature": "(path, prefix, c, dtr_cnfg=1, version='openQCD', **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_gf_coupling": {"fullname": "pyerrors.input.openQCD.read_gf_coupling", "modulename": "pyerrors.input.openQCD", "qualname": "read_gf_coupling", "kind": "function", "doc": "
Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.
\n\n
Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • r_start (list):\nlist which contains the first config to be read for each replicum.
    • \n
  • r_stop (list):\nlist which contains the last config to be read for each replicum.
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
    • \n
  • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used.
    • \n
\n", "signature": "(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.qtop_projection": {"fullname": "pyerrors.input.openQCD.qtop_projection", "modulename": "pyerrors.input.openQCD", "qualname": "qtop_projection", "kind": "function", "doc": "
Returns the projection to the topological charge sector defined by target.
\n\n
Parameters
\n\n
    \n
  • path (Obs):\nTopological charge.
    • \n
  • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
    • \n
\n\n
Returns
\n\n
    \n
  • reto (Obs):\nprojection to the topological charge sector defined by target
    • \n
\n", "signature": "(qtop, target=0):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop_sector": {"fullname": "pyerrors.input.openQCD.read_qtop_sector", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop_sector", "kind": "function", "doc": "
Constructs reweighting factors to a specified topological sector.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath of the measurement files
    • \n
  • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat
    • \n
  • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L
    • \n
  • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
    • \n
  • dtr_cnfg (int):\n(optional) parameter that specifies the number of trajectories\nbetween two configs.\nif it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
    • \n
  • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
    • \n
  • version (str):\nversion string of the openQCD (sfqcd) version used to create\nthe ensemble. Default is 2.0. May also be set to sfqcd.
    • \n
  • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
    • \n
  • r_start (list):\noffset of the first ensemble, making it easier to match\nlater on with other Obs
    • \n
  • r_stop (list):\nlast configurations that need to be read (per replicum)
    • \n
  • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
    • \n
  • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
    • \n
\n\n
Returns
\n\n
    \n
  • reto (Obs):\nprojection to the topological charge sector defined by target
    • \n
\n", "signature": "(path, prefix, c, target=0, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_ms5_xsf": {"fullname": "pyerrors.input.openQCD.read_ms5_xsf", "modulename": "pyerrors.input.openQCD", "qualname": "read_ms5_xsf", "kind": "function", "doc": "
Read data from files in the specified directory with the specified prefix and quark combination extension, and return a Corr object containing the data.
\n\n
Parameters
\n\n
    \n
  • path (str):\nThe directory to search for the files in.
    • \n
  • prefix (str):\nThe prefix to match the files against.
    • \n
  • qc (str):\nThe quark combination extension to match the files against.
    • \n
  • corr (str):\nThe correlator to extract data for.
    • \n
  • sep (str, optional):\nThe separator to use when parsing the replika names.
    • \n
  • **kwargs: Additional keyword arguments. The following keyword arguments are recognized:
    \n\n
      \n
    • names (List[str]): A list of names to use for the replicas.
      • \n
    • \n
\n\n
Returns
\n\n
    \n
  • Corr: A complex valued Corr object containing the data read from the files. In case of boudary to bulk correlators.
    • \n
  • or
    • \n
  • CObs: A complex valued CObs object containing the data read from the files. In case of boudary to boundary correlators.
    • \n
\n\n
Raises
\n\n
    \n
  • FileNotFoundError: If no files matching the specified prefix and quark combination extension are found in the specified directory.
    • \n
  • IOError: If there is an error reading a file.
    • \n
  • struct.error: If there is an error unpacking binary data.
    • \n
\n", "signature": "(path, prefix, qc, corr, sep='r', **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas": {"fullname": "pyerrors.input.pandas", "modulename": "pyerrors.input.pandas", "kind": "module", "doc": "
\n"}, "pyerrors.input.pandas.to_sql": {"fullname": "pyerrors.input.pandas.to_sql", "modulename": "pyerrors.input.pandas", "qualname": "to_sql", "kind": "function", "doc": "
Write DataFrame including Obs or Corr valued columns to sqlite database.
\n\n
Parameters
\n\n
    \n
  • df (pandas.DataFrame):\nDataframe to be written to the database.
    • \n
  • table_name (str):\nName of the table in the database.
    • \n
  • db (str):\nPath to the sqlite database.
    • \n
  • if exists (str):\nHow to behave if table already exists. Options 'fail', 'replace', 'append'.
    • \n
  • gz (bool):\nIf True the json strings are gzipped.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(df, table_name, db, if_exists='fail', gz=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.read_sql": {"fullname": "pyerrors.input.pandas.read_sql", "modulename": "pyerrors.input.pandas", "qualname": "read_sql", "kind": "function", "doc": "
Execute SQL query on sqlite database and obtain DataFrame including Obs or Corr valued columns.
\n\n
Parameters
\n\n
    \n
  • sql (str):\nSQL query to be executed.
    • \n
  • db (str):\nPath to the sqlite database.
    • \n
  • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
    • \n
\n\n
Returns
\n\n
    \n
  • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
    • \n
\n", "signature": "(sql, db, auto_gamma=False, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.dump_df": {"fullname": "pyerrors.input.pandas.dump_df", "modulename": "pyerrors.input.pandas", "qualname": "dump_df", "kind": "function", "doc": "
Exports a pandas DataFrame containing Obs valued columns to a (gzipped) csv file.
\n\n
Before making use of pandas to_csv functionality Obs objects are serialized via the standardized\njson format of pyerrors.
\n\n
Parameters
\n\n
    \n
  • df (pandas.DataFrame):\nDataframe to be dumped to a file.
    • \n
  • fname (str):\nFilename of the output file.
    • \n
  • gz (bool):\nIf True, the output is a gzipped csv file. If False, the output is a csv file.
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(df, fname, gz=True):", "funcdef": "def"}, "pyerrors.input.pandas.load_df": {"fullname": "pyerrors.input.pandas.load_df", "modulename": "pyerrors.input.pandas", "qualname": "load_df", "kind": "function", "doc": "
Imports a pandas DataFrame from a csv.(gz) file in which Obs objects are serialized as json strings.
\n\n
Parameters
\n\n
    \n
  • fname (str):\nFilename of the input file.
    • \n
  • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
    • \n
  • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
    • \n
\n\n
Returns
\n\n
    \n
  • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
    • \n
\n", "signature": "(fname, auto_gamma=False, gz=True):", "funcdef": "def"}, "pyerrors.input.sfcf": {"fullname": "pyerrors.input.sfcf", "modulename": "pyerrors.input.sfcf", "kind": "module", "doc": "
\n"}, "pyerrors.input.sfcf.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "
Read sfcf c format from given folder structure.
\n\n
Parameters
\n\n
    \n
  • path (str):\nPath to the sfcf files.
    • \n
  • prefix (str):\nPrefix of the sfcf files.
    • \n
  • name (str):\nName of the correlation function to read.
    • \n
  • quarks (str):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
    • \n
  • corr_type (str):\nType of correlation function to read. Can be\n
      \n
    • 'bi' for boundary-inner
      • \n
    • 'bb' for boundary-boundary
      • \n
    • 'bib' for boundary-inner-boundary
      • \n
    • \n
  • noffset (int):\nOffset of the source (only relevant when wavefunctions are used)
    • \n
  • wf (int):\nID of wave function
    • \n
  • wf2 (int):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
    • \n
  • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
    • \n
  • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
    • \n
  • ens_name (str):\nreplaces the name of the ensemble
    • \n
  • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
    • \n
  • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
    • \n
  • replica (list):\nlist of replica to be read, default is all
    • \n
  • files (list):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
    • \n
  • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
    • \n
\n\n
Returns
\n\n
    \n
  • result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
    • \n
\n", "signature": "(\tpath,\tprefix,\tname,\tquarks='.*',\tcorr_type='bi',\tnoffset=0,\twf=0,\twf2=0,\tversion='1.0c',\tcfg_separator='n',\t**kwargs):", "funcdef": "def"}, "pyerrors.input.utils": {"fullname": "pyerrors.input.utils", "modulename": "pyerrors.input.utils", "kind": "module", "doc": "
Utilities for the input
\n"}, "pyerrors.input.utils.check_idl": {"fullname": "pyerrors.input.utils.check_idl", "modulename": "pyerrors.input.utils", "qualname": "check_idl", "kind": "function", "doc": "
Checks if list of configurations is contained in an idl
\n\n
Parameters
\n\n
    \n
  • idl (range or list):\nidl of the current replicum
    • \n
  • che (list):\nlist of configurations to be checked against
    • \n
\n\n
Returns
\n\n
    \n
  • miss_str (str):\nstring with integers of which idls are missing
    • \n
\n", "signature": "(idl, che):", "funcdef": "def"}, "pyerrors.linalg": {"fullname": "pyerrors.linalg", "modulename": "pyerrors.linalg", "kind": "module", "doc": "
\n"}, "pyerrors.linalg.matmul": {"fullname": "pyerrors.linalg.matmul", "modulename": "pyerrors.linalg", "qualname": "matmul", "kind": "function", "doc": "
Matrix multiply all operands.
\n\n
Parameters
\n\n
    \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
  • This implementation is faster compared to standard multiplication via the @ operator.
    • \n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.jack_matmul": {"fullname": "pyerrors.linalg.jack_matmul", "modulename": "pyerrors.linalg", "qualname": "jack_matmul", "kind": "function", "doc": "
Matrix multiply both operands making use of the jackknife approximation.
\n\n
Parameters
\n\n
    \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
  • For large matrices this is considerably faster compared to matmul.
    • \n
\n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.einsum": {"fullname": "pyerrors.linalg.einsum", "modulename": "pyerrors.linalg", "qualname": "einsum", "kind": "function", "doc": "
Wrapper for numpy.einsum
\n\n
Parameters
\n\n
    \n
  • subscripts (str):\nSubscripts for summation (see numpy documentation for details)
    • \n
  • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
    • \n
\n", "signature": "(subscripts, *operands):", "funcdef": "def"}, "pyerrors.linalg.inv": {"fullname": "pyerrors.linalg.inv", "modulename": "pyerrors.linalg", "qualname": "inv", "kind": "function", "doc": "
Inverse of Obs or CObs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.cholesky": {"fullname": "pyerrors.linalg.cholesky", "modulename": "pyerrors.linalg", "qualname": "cholesky", "kind": "function", "doc": "
Cholesky decomposition of Obs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.det": {"fullname": "pyerrors.linalg.det", "modulename": "pyerrors.linalg", "qualname": "det", "kind": "function", "doc": "
Determinant of Obs valued matrices.
\n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.eigh": {"fullname": "pyerrors.linalg.eigh", "modulename": "pyerrors.linalg", "qualname": "eigh", "kind": "function", "doc": "
Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eig": {"fullname": "pyerrors.linalg.eig", "modulename": "pyerrors.linalg", "qualname": "eig", "kind": "function", "doc": "
Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.pinv": {"fullname": "pyerrors.linalg.pinv", "modulename": "pyerrors.linalg", "qualname": "pinv", "kind": "function", "doc": "
Computes the Moore-Penrose pseudoinverse of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.svd": {"fullname": "pyerrors.linalg.svd", "modulename": "pyerrors.linalg", "qualname": "svd", "kind": "function", "doc": "
Computes the singular value decomposition of a matrix of Obs.
\n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.misc": {"fullname": "pyerrors.misc", "modulename": "pyerrors.misc", "kind": "module", "doc": "
\n"}, "pyerrors.misc.errorbar": {"fullname": "pyerrors.misc.errorbar", "modulename": "pyerrors.misc", "qualname": "errorbar", "kind": "function", "doc": "
pyerrors wrapper for the errorbars method of matplotlib
\n\n
Parameters
\n\n
    \n
  • x (list):\nA list of x-values which can be Obs.
    • \n
  • y (list):\nA list of y-values which can be Obs.
    • \n
  • axes ((matplotlib.pyplot.axes)):\nThe axes to plot on. default is plt.
    • \n
\n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.10.10/x64/lib/python3.10/site-packages/matplotlib/pyplot.py'>,\t**kwargs):", "funcdef": "def"}, "pyerrors.misc.dump_object": {"fullname": "pyerrors.misc.dump_object", "modulename": "pyerrors.misc", "qualname": "dump_object", "kind": "function", "doc": "
Dump object into pickle file.
\n\n
Parameters
\n\n
    \n
  • obj (object):\nobject to be saved in the pickle file
    • \n
  • name (str):\nname of the file
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n\n
Returns
\n\n
    \n
  • None
    • \n
\n", "signature": "(obj, name, **kwargs):", "funcdef": "def"}, "pyerrors.misc.load_object": {"fullname": "pyerrors.misc.load_object", "modulename": "pyerrors.misc", "qualname": "load_object", "kind": "function", "doc": "
Load object from pickle file.
\n\n
Parameters
\n\n
    \n
  • path (str):\npath to the file
    • \n
\n\n
Returns
\n\n
    \n
  • object (Obs):\nLoaded Object
    • \n
\n", "signature": "(path):", "funcdef": "def"}, "pyerrors.misc.pseudo_Obs": {"fullname": "pyerrors.misc.pseudo_Obs", "modulename": "pyerrors.misc", "qualname": "pseudo_Obs", "kind": "function", "doc": "
Generate an Obs object with given value, dvalue and name for test purposes
\n\n
Parameters
\n\n
    \n
  • value (float):\ncentral value of the Obs to be generated.
    • \n
  • dvalue (float):\nerror of the Obs to be generated.
    • \n
  • name (str):\nname of the ensemble for which the Obs is to be generated.
    • \n
  • samples (int):\nnumber of samples for the Obs (default 1000).
    • \n
\n\n
Returns
\n\n
    \n
  • res (Obs):\nGenerated Observable
    • \n
\n", "signature": "(value, dvalue, name, samples=1000):", "funcdef": "def"}, "pyerrors.misc.gen_correlated_data": {"fullname": "pyerrors.misc.gen_correlated_data", "modulename": "pyerrors.misc", "qualname": "gen_correlated_data", "kind": "function", "doc": "
Generate observables with given covariance and autocorrelation times.
\n\n
Parameters
\n\n
    \n
  • means (list):\nlist containing the mean value of each observable.
    • \n
  • cov (numpy.ndarray):\ncovariance matrix for the data to be generated.
    • \n
  • name (str):\nensemble name for the data to be geneated.
    • \n
  • tau (float or list):\ncan either be a real number or a list with an entry for\nevery dataset.
    • \n
  • samples (int):\nnumber of samples to be generated for each observable.
    • \n
\n\n
Returns
\n\n
    \n
  • corr_obs (list[Obs]):\nGenerated observable list
    • \n
\n", "signature": "(means, cov, name, tau=0.5, samples=1000):", "funcdef": "def"}, "pyerrors.mpm": {"fullname": "pyerrors.mpm", "modulename": "pyerrors.mpm", "kind": "module", "doc": "
\n"}, "pyerrors.mpm.matrix_pencil_method": {"fullname": "pyerrors.mpm.matrix_pencil_method", "modulename": "pyerrors.mpm", "qualname": "matrix_pencil_method", "kind": "function", "doc": "
Matrix pencil method to extract k energy levels from data
\n\n
Implementation of the matrix pencil method based on\neq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)
\n\n
Parameters
\n\n
    \n
  • data (list):\ncan be a list of Obs for the analysis of a single correlator, or a list of lists\nof Obs if several correlators are to analyzed at once.
    • \n
  • k (int):\nNumber of states to extract (default 1).
    • \n
  • p (int):\nmatrix pencil parameter which filters noise. The optimal value is expected between\nlen(data)/3 and 2*len(data)/3. The computation is more expensive the closer p is\nto len(data)/2 but could possibly suppress more noise (default len(data)//2).
    • \n
\n\n
Returns
\n\n
    \n
  • energy_levels (list[Obs]):\nExtracted energy levels
    • \n
\n", "signature": "(corrs, k=1, p=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs": {"fullname": "pyerrors.obs", "modulename": "pyerrors.obs", "kind": "module", "doc": "
\n"}, "pyerrors.obs.Obs": {"fullname": "pyerrors.obs.Obs", "modulename": "pyerrors.obs", "qualname": "Obs", "kind": "class", "doc": "
Class for a general observable.
\n\n
Instances of Obs are the basic objects of a pyerrors error analysis.\nThey are initialized with a list which contains arrays of samples for\ndifferent ensembles/replica and another list of same length which contains\nthe names of the ensembles/replica. Mathematical operations can be\nperformed on instances. The result is another instance of Obs. The error of\nan instance can be computed with the gamma_method. Also contains additional\nmethods for output and visualization of the error calculation.
\n\n
Attributes
\n\n
    \n
  • S_global (float):\nStandard value for S (default 2.0)
    • \n
  • S_dict (dict):\nDictionary for S values. If an entry for a given ensemble\nexists this overwrites the standard value for that ensemble.
    • \n
  • tau_exp_global (float):\nStandard value for tau_exp (default 0.0)
    • \n
  • tau_exp_dict (dict):\nDictionary for tau_exp values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
    • \n
  • N_sigma_global (float):\nStandard value for N_sigma (default 1.0)
    • \n
  • N_sigma_dict (dict):\nDictionary for N_sigma values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
    • \n
\n"}, "pyerrors.obs.Obs.__init__": {"fullname": "pyerrors.obs.Obs.__init__", "modulename": "pyerrors.obs", "qualname": "Obs.__init__", "kind": "function", "doc": "
Initialize Obs object.
\n\n
Parameters
\n\n
    \n
  • samples (list):\nlist of numpy arrays containing the Monte Carlo samples
    • \n
  • names (list):\nlist of strings labeling the individual samples
    • \n
  • idl (list, optional):\nlist of ranges or lists on which the samples are defined
    • \n
\n", "signature": "(samples, names, idl=None, **kwargs)"}, "pyerrors.obs.Obs.gamma_method": {"fullname": "pyerrors.obs.Obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "Obs.gamma_method", "kind": "function", "doc": "
Estimate the error and related properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
    • \n
  • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
    • \n
  • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
    • \n
  • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
    • \n
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.gm": {"fullname": "pyerrors.obs.Obs.gm", "modulename": "pyerrors.obs", "qualname": "Obs.gm", "kind": "function", "doc": "
Estimate the error and related properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
    • \n
  • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
    • \n
  • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
    • \n
  • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
    • \n
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.details": {"fullname": "pyerrors.obs.Obs.details", "modulename": "pyerrors.obs", "qualname": "Obs.details", "kind": "function", "doc": "
Output detailed properties of the Obs.
\n\n
Parameters
\n\n
    \n
  • ens_content (bool):\nprint details about the ensembles and replica if true.
    • \n
\n", "signature": "(self, ens_content=True):", "funcdef": "def"}, "pyerrors.obs.Obs.reweight": {"fullname": "pyerrors.obs.Obs.reweight", "modulename": "pyerrors.obs", "qualname": "Obs.reweight", "kind": "function", "doc": "
Reweight the obs with given rewighting factors.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
    • \n
\n", "signature": "(self, weight):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero_within_error": {"fullname": "pyerrors.obs.Obs.is_zero_within_error", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero_within_error", "kind": "function", "doc": "
Checks whether the observable is zero within 'sigma' standard errors.
\n\n
Parameters
\n\n
    \n
  • sigma (int):\nNumber of standard errors used for the check.
    • \n
  • Works only properly when the gamma method was run.
    • \n
\n", "signature": "(self, sigma=1):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero": {"fullname": "pyerrors.obs.Obs.is_zero", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero", "kind": "function", "doc": "
Checks whether the observable is zero within a given tolerance.
\n\n
Parameters
\n\n
    \n
  • atol (float):\nAbsolute tolerance (for details see numpy documentation).
    • \n
\n", "signature": "(self, atol=1e-10):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_tauint": {"fullname": "pyerrors.obs.Obs.plot_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.plot_tauint", "kind": "function", "doc": "
Plot integrated autocorrelation time for each ensemble.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rho": {"fullname": "pyerrors.obs.Obs.plot_rho", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rho", "kind": "function", "doc": "
Plot normalized autocorrelation function time for each ensemble.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rep_dist": {"fullname": "pyerrors.obs.Obs.plot_rep_dist", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rep_dist", "kind": "function", "doc": "
Plot replica distribution for each ensemble with more than one replicum.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_history": {"fullname": "pyerrors.obs.Obs.plot_history", "modulename": "pyerrors.obs", "qualname": "Obs.plot_history", "kind": "function", "doc": "
Plot derived Monte Carlo history for each ensemble
\n\n
Parameters
\n\n
    \n
  • expand (bool):\nshow expanded history for irregular Monte Carlo chains (default: True).
    • \n
\n", "signature": "(self, expand=True):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_piechart": {"fullname": "pyerrors.obs.Obs.plot_piechart", "modulename": "pyerrors.obs", "qualname": "Obs.plot_piechart", "kind": "function", "doc": "
Plot piechart which shows the fractional contribution of each\nensemble to the error and returns a dictionary containing the fractions.
\n\n
Parameters
\n\n
    \n
  • save (str):\nsaves the figure to a file named 'save' if.
    • \n
\n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.dump": {"fullname": "pyerrors.obs.Obs.dump", "modulename": "pyerrors.obs", "qualname": "Obs.dump", "kind": "function", "doc": "
Dump the Obs to a file 'name' of chosen format.
\n\n
Parameters
\n\n
    \n
  • filename (str):\nname of the file to be saved.
    • \n
  • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
    • \n
  • description (str):\nDescription for output file, only relevant for json.gz format.
    • \n
  • path (str):\nspecifies a custom path for the file (default '.')
    • \n
\n", "signature": "(self, filename, datatype='json.gz', description='', **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.export_jackknife": {"fullname": "pyerrors.obs.Obs.export_jackknife", "modulename": "pyerrors.obs", "qualname": "Obs.export_jackknife", "kind": "function", "doc": "
Export jackknife samples from the Obs
\n\n
Returns
\n\n
    \n
  • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N jackknife samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived jackknife samples\nshould agree with samples from a full jackknife analysis up to O(1/N).
    • \n
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sqrt": {"fullname": "pyerrors.obs.Obs.sqrt", "modulename": "pyerrors.obs", "qualname": "Obs.sqrt", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.log": {"fullname": "pyerrors.obs.Obs.log", "modulename": "pyerrors.obs", "qualname": "Obs.log", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.exp": {"fullname": "pyerrors.obs.Obs.exp", "modulename": "pyerrors.obs", "qualname": "Obs.exp", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sin": {"fullname": "pyerrors.obs.Obs.sin", "modulename": "pyerrors.obs", "qualname": "Obs.sin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cos": {"fullname": "pyerrors.obs.Obs.cos", "modulename": "pyerrors.obs", "qualname": "Obs.cos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tan": {"fullname": "pyerrors.obs.Obs.tan", "modulename": "pyerrors.obs", "qualname": "Obs.tan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsin": {"fullname": "pyerrors.obs.Obs.arcsin", "modulename": "pyerrors.obs", "qualname": "Obs.arcsin", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccos": {"fullname": "pyerrors.obs.Obs.arccos", "modulename": "pyerrors.obs", "qualname": "Obs.arccos", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctan": {"fullname": "pyerrors.obs.Obs.arctan", "modulename": "pyerrors.obs", "qualname": "Obs.arctan", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sinh": {"fullname": "pyerrors.obs.Obs.sinh", "modulename": "pyerrors.obs", "qualname": "Obs.sinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cosh": {"fullname": "pyerrors.obs.Obs.cosh", "modulename": "pyerrors.obs", "qualname": "Obs.cosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tanh": {"fullname": "pyerrors.obs.Obs.tanh", "modulename": "pyerrors.obs", "qualname": "Obs.tanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsinh": {"fullname": "pyerrors.obs.Obs.arcsinh", "modulename": "pyerrors.obs", "qualname": "Obs.arcsinh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccosh": {"fullname": "pyerrors.obs.Obs.arccosh", "modulename": "pyerrors.obs", "qualname": "Obs.arccosh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctanh": {"fullname": "pyerrors.obs.Obs.arctanh", "modulename": "pyerrors.obs", "qualname": "Obs.arctanh", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs": {"fullname": "pyerrors.obs.CObs", "modulename": "pyerrors.obs", "qualname": "CObs", "kind": "class", "doc": "
Class for a complex valued observable.
\n"}, "pyerrors.obs.CObs.__init__": {"fullname": "pyerrors.obs.CObs.__init__", "modulename": "pyerrors.obs", "qualname": "CObs.__init__", "kind": "function", "doc": "
\n", "signature": "(real, imag=0.0)"}, "pyerrors.obs.CObs.gamma_method": {"fullname": "pyerrors.obs.CObs.gamma_method", "modulename": "pyerrors.obs", "qualname": "CObs.gamma_method", "kind": "function", "doc": "
Executes the gamma_method for the real and the imaginary part.
\n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.CObs.is_zero": {"fullname": "pyerrors.obs.CObs.is_zero", "modulename": "pyerrors.obs", "qualname": "CObs.is_zero", "kind": "function", "doc": "
Checks whether both real and imaginary part are zero within machine precision.
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs.conjugate": {"fullname": "pyerrors.obs.CObs.conjugate", "modulename": "pyerrors.obs", "qualname": "CObs.conjugate", "kind": "function", "doc": "
\n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.derived_observable": {"fullname": "pyerrors.obs.derived_observable", "modulename": "pyerrors.obs", "qualname": "derived_observable", "kind": "function", "doc": "
Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.
\n\n
Parameters
\n\n
    \n
  • func (object):\narbitrary function of the form func(data, **kwargs). For the\nautomatic differentiation to work, all numpy functions have to have\nthe autograd wrapper (use 'import autograd.numpy as anp').
    • \n
  • data (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
    • \n
  • num_grad (bool):\nif True, numerical derivatives are used instead of autograd\n(default False). To control the numerical differentiation the\nkwargs of numdifftools.step_generators.MaxStepGenerator\ncan be used.
    • \n
  • man_grad (list):\nmanually supply a list or an array which contains the jacobian\nof func. Use cautiously, supplying the wrong derivative will\nnot be intercepted.
    • \n
\n\n
Notes
\n\n
For simple mathematical operations it can be practical to use anonymous\nfunctions. For the ratio of two observables one can e.g. use
\n\n
new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])
\n", "signature": "(func, data, array_mode=False, **kwargs):", "funcdef": "def"}, "pyerrors.obs.reweight": {"fullname": "pyerrors.obs.reweight", "modulename": "pyerrors.obs", "qualname": "reweight", "kind": "function", "doc": "
Reweight a list of observables.
\n\n
Parameters
\n\n
    \n
  • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
    • \n
  • obs (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
    • \n
  • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
    • \n
\n", "signature": "(weight, obs, **kwargs):", "funcdef": "def"}, "pyerrors.obs.correlate": {"fullname": "pyerrors.obs.correlate", "modulename": "pyerrors.obs", "qualname": "correlate", "kind": "function", "doc": "
Correlate two observables.
\n\n
Parameters
\n\n
    \n
  • obs_a (Obs):\nFirst observable
    • \n
  • obs_b (Obs):\nSecond observable
    • \n
\n\n
Notes
\n\n
Keep in mind to only correlate primary observables which have not been reweighted\nyet. The reweighting has to be applied after correlating the observables.\nCurrently only works if ensembles are identical (this is not strictly necessary).
\n", "signature": "(obs_a, obs_b):", "funcdef": "def"}, "pyerrors.obs.covariance": {"fullname": "pyerrors.obs.covariance", "modulename": "pyerrors.obs", "qualname": "covariance", "kind": "function", "doc": "
Calculates the error covariance matrix of a set of observables.
\n\n
WARNING: This function should be used with care, especially for observables with support on multiple\n         ensembles with differing autocorrelations. See the notes below for details.
\n\n
The gamma method has to be applied first to all observables.
\n\n
Parameters
\n\n
    \n
  • obs (list or numpy.ndarray):\nList or one dimensional array of Obs
    • \n
  • visualize (bool):\nIf True plots the corresponding normalized correlation matrix (default False).
    • \n
  • correlation (bool):\nIf True the correlation matrix instead of the error covariance matrix is returned (default False).
    • \n
  • smooth (None or int):\nIf smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue\nsmoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the\nlargest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely\nsmall ones.
    • \n
\n\n
Notes
\n\n
The error covariance is defined such that it agrees with the squared standard error for two identical observables\n$$\\operatorname{cov}(a,a)=\\sum_{s=1}^N\\delta_a^s\\delta_a^s/N^2=\\Gamma_{aa}(0)/N=\\operatorname{var}(a)/N=\\sigma_a^2$$\nin the absence of autocorrelation.\nThe error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite\n$$\\sum_{i,j}v_i\\Gamma_{ij}(0)v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i,j}v_i\\delta_i^s\\delta_j^s v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i}|v_i\\delta_i^s|^2\\geq 0\\,,$$ for every $v\\in\\mathbb{R}^M$, while such an identity does not hold for larger windows/lags.\nFor observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.\n$$\\tau_{\\mathrm{int}, ij}=\\sqrt{\\tau_{\\mathrm{int}, i}\\times \\tau_{\\mathrm{int}, j}}$$\nThis construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).
\n", "signature": "(obs, visualize=False, correlation=False, smooth=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs.import_jackknife": {"fullname": "pyerrors.obs.import_jackknife", "modulename": "pyerrors.obs", "qualname": "import_jackknife", "kind": "function", "doc": "
Imports jackknife samples and returns an Obs
\n\n
Parameters
\n\n
    \n
  • jacks (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N jackknife samples as first to Nth entry.
    • \n
  • name (str):\nname of the ensemble the samples are defined on.
    • \n
\n", "signature": "(jacks, name, idl=None):", "funcdef": "def"}, "pyerrors.obs.merge_obs": {"fullname": "pyerrors.obs.merge_obs", "modulename": "pyerrors.obs", "qualname": "merge_obs", "kind": "function", "doc": "
Combine all observables in list_of_obs into one new observable
\n\n
Parameters
\n\n
    \n
  • list_of_obs (list):\nlist of the Obs object to be combined
    • \n
\n\n
Notes
\n\n
It is not possible to combine obs which are based on the same replicum
\n", "signature": "(list_of_obs):", "funcdef": "def"}, "pyerrors.obs.cov_Obs": {"fullname": "pyerrors.obs.cov_Obs", "modulename": "pyerrors.obs", "qualname": "cov_Obs", "kind": "function", "doc": "
Create an Obs based on mean(s) and a covariance matrix
\n\n
Parameters
\n\n
    \n
  • mean (list of floats or float):\nN mean value(s) of the new Obs
    • \n
  • cov (list or array):\n2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
    • \n
  • name (str):\nidentifier for the covariance matrix
    • \n
  • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
    • \n
\n", "signature": "(means, cov, name, grad=None):", "funcdef": "def"}, "pyerrors.roots": {"fullname": "pyerrors.roots", "modulename": "pyerrors.roots", "kind": "module", "doc": "
\n"}, "pyerrors.roots.find_root": {"fullname": "pyerrors.roots.find_root", "modulename": "pyerrors.roots", "qualname": "find_root", "kind": "function", "doc": "
Finds the root of the function func(x, d) where d is an Obs.
\n\n
Parameters
\n\n
    \n
  • d (Obs):\nObs passed to the function.
    • \n
  • func (object):\nFunction to be minimized. Any numpy functions have to use the autograd.numpy wrapper.\nExample:
    \n\n
    \n
    import autograd.numpy as anp\ndef root_func(x, d):\n   return anp.exp(-x ** 2) - d\n
    \n
    • \n
  • guess (float):\nInitial guess for the minimization.
    • \n
\n\n
Returns
\n\n
    \n
  • res (Obs):\nObs valued root of the function.
    • \n
\n", "signature": "(d, func, guess=1.0, **kwargs):", "funcdef": "def"}, "pyerrors.version": {"fullname": "pyerrors.version", "modulename": "pyerrors.version", "kind": "module", "doc": "
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