# pyerrors.obs

View Source

    import warnings
    import pickle
    import numpy as np
    import autograd.numpy as anp  # Thinly-wrapped numpy
    from autograd import jacobian
    import matplotlib.pyplot as plt
    import numdifftools as nd
    from itertools import groupby

    class Obs:
        """Class for a general observable.

        Instances of Obs are the basic objects of a pyerrors error analysis.
        They are initialized with a list which contains arrays of samples for
        different ensembles/replica and another list of same length which contains
        the names of the ensembles/replica. Mathematical operations can be
        performed on instances. The result is another instance of Obs. The error of
        an instance can be computed with the gamma_method. Also contains additional
        methods for output and visualization of the error calculation.

        Attributes
        ----------
        S_global : float
            Standard value for S (default 2.0)
        S_dict : dict
            Dictionary for S values. If an entry for a given ensemble
            exists this overwrites the standard value for that ensemble.
        tau_exp_global : float
            Standard value for tau_exp (default 0.0)
        tau_exp_dict :dict
            Dictionary for tau_exp values. If an entry for a given ensemble exists
            this overwrites the standard value for that ensemble.
        N_sigma_global : float
            Standard value for N_sigma (default 1.0)
        """
        __slots__ = ['names', 'shape', 'r_values', 'deltas', 'N', '_value', '_dvalue',
                     'ddvalue', 'reweighted', 'S', 'tau_exp', 'N_sigma',
                     'e_dvalue', 'e_ddvalue', 'e_tauint', 'e_dtauint',
                     'e_windowsize', 'e_rho', 'e_drho', 'e_n_tauint', 'e_n_dtauint',
                     'idl', 'is_merged', 'tag', '__dict__']

        S_global = 2.0
        S_dict = {}
        tau_exp_global = 0.0
        tau_exp_dict = {}
        N_sigma_global = 1.0
        filter_eps = 1e-10

        def __init__(self, samples, names, idl=None, means=None, **kwargs):
            """ Initialize Obs object.

            Parameters
            ----------
            samples : list
                list of numpy arrays containing the Monte Carlo samples
            names : list
                list of strings labeling the indivdual samples
            idl : list, optional
                list of ranges or lists on which the samples are defined
            means : list, optional
                list of mean values for the case that the mean values were
                already subtracted from the samples
            """

            if means is None:
                if len(samples) != len(names):
                    raise Exception('Length of samples and names incompatible.')
                if len(names) != len(set(names)):
                    raise Exception('Names are not unique.')
                if not all(isinstance(x, str) for x in names):
                    raise TypeError('All names have to be strings.')
                if min(len(x) for x in samples) <= 4:
                    raise Exception('Samples have to have at least 4 entries.')

            self.names = sorted(names)
            self.shape = {}
            self.r_values = {}
            self.deltas = {}

            self.idl = {}
            if idl is not None:
                for name, idx in sorted(zip(names, idl)):
                    if isinstance(idx, range):
                        self.idl[name] = idx
                    elif isinstance(idx, (list, np.ndarray)):
                        dc = np.unique(np.diff(idx))
                        if np.any(dc < 0):
                            raise Exception("Unsorted idx for idl[%s]" % (name))
                        if len(dc) == 1:
                            self.idl[name] = range(idx[0], idx[-1] + dc[0], dc[0])
                        else:
                            self.idl[name] = list(idx)
                    else:
                        raise Exception('incompatible type for idl[%s].' % (name))
            else:
                for name, sample in sorted(zip(names, samples)):
                    self.idl[name] = range(1, len(sample) + 1)

            if means is not None:
                for name, sample, mean in sorted(zip(names, samples, means)):
                    self.shape[name] = len(self.idl[name])
                    if len(sample) != self.shape[name]:
                        raise Exception('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name]))
                    self.r_values[name] = mean
                    self.deltas[name] = sample
            else:
                for name, sample in sorted(zip(names, samples)):
                    self.shape[name] = len(self.idl[name])
                    if len(sample) != self.shape[name]:
                        raise Exception('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name]))
                    self.r_values[name] = np.mean(sample)
                    self.deltas[name] = sample - self.r_values[name]
            self.is_merged = False
            self.N = sum(list(self.shape.values()))

            self._value = 0
            if means is None:
                for name in self.names:
                    self._value += self.shape[name] * self.r_values[name]
                self._value /= self.N

            self._dvalue = 0.0
            self.ddvalue = 0.0
            self.reweighted = False

            self.tag = None

        @property
        def value(self):
            return self._value

        @property
        def dvalue(self):
            return self._dvalue

        @property
        def e_names(self):
            return sorted(set([o.split('|')[0] for o in self.names]))

        @property
        def e_content(self):
            res = {}
            for e, e_name in enumerate(self.e_names):
                res[e_name] = sorted(filter(lambda x: x.startswith(e_name + '|'), self.names))
                if e_name in self.names:
                    res[e_name].append(e_name)
            return res

        def gamma_method(self, **kwargs):
            """Calculate the error and related properties of the Obs.

            Parameters
            ----------
            S : float
                specifies a custom value for the parameter S (default 2.0), can be
                a float or an array of floats for different ensembles
            tau_exp : float
                positive value triggers the critical slowing down analysis
                (default 0.0), can be a float or an array of floats for different
                ensembles
            N_sigma : float
                number of standard deviations from zero until the tail is
                attached to the autocorrelation function (default 1)
            fft : bool
                determines whether the fft algorithm is used for the computation
                of the autocorrelation function (default True)
            """

            e_content = self.e_content
            self.e_dvalue = {}
            self.e_ddvalue = {}
            self.e_tauint = {}
            self.e_dtauint = {}
            self.e_windowsize = {}
            self.e_n_tauint = {}
            self.e_n_dtauint = {}
            e_gamma = {}
            self.e_rho = {}
            self.e_drho = {}
            self._dvalue = 0
            self.ddvalue = 0

            self.S = {}
            self.tau_exp = {}

            if kwargs.get('fft') is False:
                fft = False
            else:
                fft = True

            if 'S' in kwargs:
                tmp = kwargs.get('S')
                if isinstance(tmp, list):
                    if len(tmp) != len(self.e_names):
                        raise Exception('Length of S array does not match ensembles.')
                    for e, e_name in enumerate(self.e_names):
                        if tmp[e] <= 0:
                            raise Exception('S has to be larger than 0.')
                        self.S[e_name] = tmp[e]
                else:
                    if isinstance(tmp, (int, float)):
                        if tmp <= 0:
                            raise Exception('S has to be larger than 0.')
                        for e, e_name in enumerate(self.e_names):
                            self.S[e_name] = tmp
                    else:
                        raise TypeError('S is not in proper format.')
            else:
                for e, e_name in enumerate(self.e_names):
                    if e_name in Obs.S_dict:
                        self.S[e_name] = Obs.S_dict[e_name]
                    else:
                        self.S[e_name] = Obs.S_global

            if 'tau_exp' in kwargs:
                tmp = kwargs.get('tau_exp')
                if isinstance(tmp, list):
                    if len(tmp) != len(self.e_names):
                        raise Exception('Length of tau_exp array does not match ensembles.')
                    for e, e_name in enumerate(self.e_names):
                        if tmp[e] < 0:
                            raise Exception('tau_exp smaller than 0.')
                        self.tau_exp[e_name] = tmp[e]
                else:
                    if isinstance(tmp, (int, float)):
                        if tmp < 0:
                            raise Exception('tau_exp smaller than 0.')
                        for e, e_name in enumerate(self.e_names):
                            self.tau_exp[e_name] = tmp
                    else:
                        raise TypeError('tau_exp is not in proper format.')
            else:
                for e, e_name in enumerate(self.e_names):
                    if e_name in Obs.tau_exp_dict:
                        self.tau_exp[e_name] = Obs.tau_exp_dict[e_name]
                    else:
                        self.tau_exp[e_name] = Obs.tau_exp_global

            if 'N_sigma' in kwargs:
                self.N_sigma = kwargs.get('N_sigma')
                if not isinstance(self.N_sigma, (int, float)):
                    raise TypeError('N_sigma is not a number.')
            else:
                self.N_sigma = Obs.N_sigma_global

            for e, e_name in enumerate(self.e_names):

                r_length = []
                for r_name in e_content[e_name]:
                    if isinstance(self.idl[r_name], range):
                        r_length.append(len(self.idl[r_name]))
                    else:
                        r_length.append((self.idl[r_name][-1] - self.idl[r_name][0] + 1))

                e_N = np.sum([self.shape[r_name] for r_name in e_content[e_name]])
                w_max = max(r_length) // 2
                e_gamma[e_name] = np.zeros(w_max)
                self.e_rho[e_name] = np.zeros(w_max)
                self.e_drho[e_name] = np.zeros(w_max)

                for r_name in e_content[e_name]:
                    e_gamma[e_name] += self._calc_gamma(self.deltas[r_name], self.idl[r_name], self.shape[r_name], w_max, fft)

                gamma_div = np.zeros(w_max)
                for r_name in e_content[e_name]:
                    gamma_div += self._calc_gamma(np.ones((self.shape[r_name])), self.idl[r_name], self.shape[r_name], w_max, fft)
                e_gamma[e_name] /= gamma_div[:w_max]

                if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny:  # Prevent division by zero
                    self.e_tauint[e_name] = 0.5
                    self.e_dtauint[e_name] = 0.0
                    self.e_dvalue[e_name] = 0.0
                    self.e_ddvalue[e_name] = 0.0
                    self.e_windowsize[e_name] = 0
                    continue

                self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0]
                self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:])))
                # Make sure no entry of tauint is smaller than 0.5
                self.e_n_tauint[e_name][self.e_n_tauint[e_name] <= 0.5] = 0.5 + np.finfo(np.float64).eps
                # hep-lat/0306017 eq. (42)
                self.e_n_dtauint[e_name] = self.e_n_tauint[e_name] * 2 * np.sqrt(np.abs(np.arange(w_max) + 0.5 - self.e_n_tauint[e_name]) / e_N)
                self.e_n_dtauint[e_name][0] = 0.0

                def _compute_drho(i):
                    tmp = self.e_rho[e_name][i + 1:w_max] + np.concatenate([self.e_rho[e_name][i - 1::-1], self.e_rho[e_name][1:w_max - 2 * i]]) - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i]
                    self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N)

                _compute_drho(1)
                if self.tau_exp[e_name] > 0:
                    texp = self.tau_exp[e_name]
                    # if type(self.idl[e_name]) is range: # scale tau_exp according to step size
                    #    texp /= self.idl[e_name].step
                    # Critical slowing down analysis
                    for n in range(1, w_max // 2):
                        _compute_drho(n + 1)
                        if (self.e_rho[e_name][n] - self.N_sigma * self.e_drho[e_name][n]) < 0 or n >= w_max // 2 - 2:
                            # Bias correction hep-lat/0306017 eq. (49) included
                            self.e_tauint[e_name] = self.e_n_tauint[e_name][n] * (1 + (2 * n + 1) / e_N) / (1 + 1 / e_N) + texp * np.abs(self.e_rho[e_name][n + 1])  # The absolute makes sure, that the tail contribution is always positive
                            self.e_dtauint[e_name] = np.sqrt(self.e_n_dtauint[e_name][n] ** 2 + texp ** 2 * self.e_drho[e_name][n + 1] ** 2)
                            # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2
                            self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
                            self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
                            self.e_windowsize[e_name] = n
                            break
                else:
                    # Standard automatic windowing procedure
                    g_w = self.S[e_name] / np.log((2 * self.e_n_tauint[e_name][1:] + 1) / (2 * self.e_n_tauint[e_name][1:] - 1))
                    g_w = np.exp(- np.arange(1, w_max) / g_w) - g_w / np.sqrt(np.arange(1, w_max) * e_N)
                    for n in range(1, w_max):
                        if n < w_max // 2 - 2:
                            _compute_drho(n + 1)
                        if g_w[n - 1] < 0 or n >= w_max - 1:
                            self.e_tauint[e_name] = self.e_n_tauint[e_name][n] * (1 + (2 * n + 1) / e_N) / (1 + 1 / e_N)  # Bias correction hep-lat/0306017 eq. (49)
                            self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n]
                            self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
                            self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
                            self.e_windowsize[e_name] = n
                            break

                self._dvalue += self.e_dvalue[e_name] ** 2
                self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2

            self._dvalue = np.sqrt(self.dvalue)
            if self._dvalue == 0.0:
                self.ddvalue = 0.0
            else:
                self.ddvalue = np.sqrt(self.ddvalue) / self.dvalue
            return

        def _calc_gamma(self, deltas, idx, shape, w_max, fft):
            """Calculate Gamma_{AA} from the deltas, which are defined on idx.
               idx is assumed to be a contiguous range (possibly with a stepsize != 1)

            Parameters
            ----------
            deltas : list
                List of fluctuations
            idx : list
                List or range of configs on which the deltas are defined.
            shape : int
                Number of configs in idx.
            w_max : int
                Upper bound for the summation window.
            fft : bool
                determines whether the fft algorithm is used for the computation
                of the autocorrelation function.
            """
            gamma = np.zeros(w_max)
            deltas = _expand_deltas(deltas, idx, shape)
            new_shape = len(deltas)
            if fft:
                max_gamma = min(new_shape, w_max)
                # The padding for the fft has to be even
                padding = new_shape + max_gamma + (new_shape + max_gamma) % 2
                gamma[:max_gamma] += np.fft.irfft(np.abs(np.fft.rfft(deltas, padding)) ** 2)[:max_gamma]
            else:
                for n in range(w_max):
                    if new_shape - n >= 0:
                        gamma[n] += deltas[0:new_shape - n].dot(deltas[n:new_shape])

            return gamma

        def print(self, level=1):
            warnings.warn("Method 'print' renamed to 'details'", DeprecationWarning)
            self.details(level > 1)

        def details(self, ens_content=True):
            """Output detailed properties of the Obs."""
            if self.value == 0.0:
                percentage = np.nan
            else:
                percentage = np.abs(self.dvalue / self.value) * 100
            print('Result\t %3.8e +/- %3.8e +/- %3.8e (%3.3f%%)' % (self.value, self.dvalue, self.ddvalue, percentage))
            if hasattr(self, 'e_dvalue'):
                if len(self.e_names) > 1:
                    print(' Ensemble errors:')
                for e_name in self.e_names:
                    if len(self.e_names) > 1:
                        print('', e_name, '\t %3.8e +/- %3.8e' % (self.e_dvalue[e_name], self.e_ddvalue[e_name]))
                    if self.tau_exp[e_name] > 0:
                        print(' t_int\t %3.8e +/- %3.8e tau_exp = %3.2f,  N_sigma = %1.0i' % (self.e_tauint[e_name], self.e_dtauint[e_name], self.tau_exp[e_name], self.N_sigma))
                    else:
                        print(' t_int\t %3.8e +/- %3.8e S = %3.2f' % (self.e_tauint[e_name], self.e_dtauint[e_name], self.S[e_name]))
            if self.tag is not None:
                print("Description:", self.tag)
            if ens_content is True:
                if len(self.e_names) == 1:
                    print(self.N, 'samples in', len(self.e_names), 'ensemble:')
                else:
                    print(self.N, 'samples in', len(self.e_names), 'ensembles:')
                m = max(map(len, list(self.e_content.keys()))) + 1
                print('\n'.join(['  ' + key.rjust(m) + ': ' + str(value) for key, value in sorted(self.e_content.items())]))

        def is_zero_within_error(self, sigma=1):
            """Checks whether the observable is zero within 'sigma' standard errors.

            Works only properly when the gamma method was run.
            """
            return self.is_zero() or np.abs(self.value) <= sigma * self.dvalue

        def is_zero(self):
            """Checks whether the observable is zero within machine precision."""
            return np.isclose(0.0, self.value) and all(np.allclose(0.0, delta) for delta in self.deltas.values())

        def plot_tauint(self, save=None):
            """Plot integrated autocorrelation time for each ensemble."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')

            fig = plt.figure()
            for e, e_name in enumerate(self.e_names):
                plt.xlabel(r'$W$')
                plt.ylabel(r'$\tau_\mathrm{int}$')
                length = int(len(self.e_n_tauint[e_name]))
                if self.tau_exp[e_name] > 0:
                    base = self.e_n_tauint[e_name][self.e_windowsize[e_name]]
                    x_help = np.arange(2 * self.tau_exp[e_name])
                    y_help = (x_help + 1) * np.abs(self.e_rho[e_name][self.e_windowsize[e_name] + 1]) * (1 - x_help / (2 * (2 * self.tau_exp[e_name] - 1))) + base
                    x_arr = np.arange(self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name])
                    plt.plot(x_arr, y_help, 'C' + str(e), linewidth=1, ls='--', marker=',')
                    plt.errorbar([self.e_windowsize[e_name] + 2 * self.tau_exp[e_name]], [self.e_tauint[e_name]],
                                 yerr=[self.e_dtauint[e_name]], fmt='C' + str(e), linewidth=1, capsize=2, marker='o', mfc=plt.rcParams['axes.facecolor'])
                    xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
                    label = e_name + r', $\tau_\mathrm{exp}$=' + str(np.around(self.tau_exp[e_name], decimals=2))
                else:
                    label = e_name + ', S=' + str(np.around(self.S[e_name], decimals=2))
                    xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)

                plt.errorbar(np.arange(length), self.e_n_tauint[e_name][:], yerr=self.e_n_dtauint[e_name][:], linewidth=1, capsize=2, label=label)
                plt.axvline(x=self.e_windowsize[e_name], color='C' + str(e), alpha=0.5, marker=',', ls='--')
                plt.legend()
                plt.xlim(-0.5, xmax)
                plt.ylim(bottom=0.0)
                plt.draw()
                if save:
                    fig.savefig(save)

        def plot_rho(self):
            """Plot normalized autocorrelation function time for each ensemble."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')
            for e, e_name in enumerate(self.e_names):
                plt.xlabel('W')
                plt.ylabel('rho')
                length = int(len(self.e_drho[e_name]))
                plt.errorbar(np.arange(length), self.e_rho[e_name][:length], yerr=self.e_drho[e_name][:], linewidth=1, capsize=2)
                plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25, ls='--', marker=',')
                if self.tau_exp[e_name] > 0:
                    plt.plot([self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]],
                             [self.e_rho[e_name][self.e_windowsize[e_name] + 1], 0], 'k-', lw=1)
                    xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
                    plt.title('Rho ' + e_name + r', tau\_exp=' + str(np.around(self.tau_exp[e_name], decimals=2)))
                else:
                    xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)
                    plt.title('Rho ' + e_name + ', S=' + str(np.around(self.S[e_name], decimals=2)))
                plt.plot([-0.5, xmax], [0, 0], 'k--', lw=1)
                plt.xlim(-0.5, xmax)
                plt.draw()

        def plot_rep_dist(self):
            """Plot replica distribution for each ensemble with more than one replicum."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')
            for e, e_name in enumerate(self.e_names):
                if len(self.e_content[e_name]) == 1:
                    print('No replica distribution for a single replicum (', e_name, ')')
                    continue
                r_length = []
                sub_r_mean = 0
                for r, r_name in enumerate(self.e_content[e_name]):
                    r_length.append(len(self.deltas[r_name]))
                    sub_r_mean += self.shape[r_name] * self.r_values[r_name]
                e_N = np.sum(r_length)
                sub_r_mean /= e_N
                arr = np.zeros(len(self.e_content[e_name]))
                for r, r_name in enumerate(self.e_content[e_name]):
                    arr[r] = (self.r_values[r_name] - sub_r_mean) / (self.e_dvalue[e_name] * np.sqrt(e_N / self.shape[r_name] - 1))
                plt.hist(arr, rwidth=0.8, bins=len(self.e_content[e_name]))
                plt.title('Replica distribution' + e_name + ' (mean=0, var=1)')
                plt.draw()

        def plot_history(self, expand=True):
            """Plot derived Monte Carlo history for each ensemble."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')

            for e, e_name in enumerate(self.e_names):
                plt.figure()
                r_length = []
                tmp = []
                for r, r_name in enumerate(self.e_content[e_name]):
                    if expand:
                        tmp.append(_expand_deltas(self.deltas[r_name], self.idl[r_name], self.shape[r_name]) + self.r_values[r_name])
                    else:
                        tmp.append(self.deltas[r_name] + self.r_values[r_name])
                    r_length.append(len(tmp[-1]))
                e_N = np.sum(r_length)
                x = np.arange(e_N)
                y = np.concatenate(tmp, axis=0)
                plt.errorbar(x, y, fmt='.', markersize=3)
                plt.xlim(-0.5, e_N - 0.5)
                plt.title(e_name)
                plt.draw()

        def plot_piechart(self):
            """Plot piechart which shows the fractional contribution of each
            ensemble to the error and returns a dictionary containing the fractions."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')
            if self.dvalue == 0.0:
                raise Exception('Error is 0.0')
            labels = self.e_names
            sizes = [i ** 2 for i in list(self.e_dvalue.values())] / self.dvalue ** 2
            fig1, ax1 = plt.subplots()
            ax1.pie(sizes, labels=labels, startangle=90, normalize=True)
            ax1.axis('equal')
            plt.draw()

            return dict(zip(self.e_names, sizes))

        def dump(self, name, **kwargs):
            """Dump the Obs to a pickle file 'name'.

            Parameters
            ----------
            path : str
                specifies a custom path for the file (default '.')
            """
            if 'path' in kwargs:
                file_name = kwargs.get('path') + '/' + name + '.p'
            else:
                file_name = name + '.p'
            with open(file_name, 'wb') as fb:
                pickle.dump(self, fb)

        def __float__(self):
            return float(self.value)

        def __repr__(self):
            return 'Obs[' + str(self) + ']'

        def __str__(self):
            if self.dvalue == 0.0:
                return str(self.value)
            fexp = np.floor(np.log10(self.dvalue))
            if fexp < 0.0:
                return '{:{form}}({:2.0f})'.format(self.value, self.dvalue * 10 ** (-fexp + 1), form='.' + str(-int(fexp) + 1) + 'f')
            elif fexp == 0.0:
                return '{:.1f}({:1.1f})'.format(self.value, self.dvalue)
            else:
                return '{:.0f}({:2.0f})'.format(self.value, self.dvalue)

        # Overload comparisons
        def __lt__(self, other):
            return self.value < other

        def __le__(self, other):
            return self.value <= other

        def __gt__(self, other):
            return self.value > other

        def __ge__(self, other):
            return self.value >= other

        def __eq__(self, other):
            return (self - other).is_zero()

        def __ne__(self, other):
            return not (self - other).is_zero()

        # Overload math operations
        def __add__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x, **kwargs: x[0] + x[1], [self, y], man_grad=[1, 1])
            else:
                if isinstance(y, np.ndarray):
                    return np.array([self + o for o in y])
                elif y.__class__.__name__ == 'Corr':
                    return NotImplemented
                else:
                    return derived_observable(lambda x, **kwargs: x[0] + y, [self], man_grad=[1])

        def __radd__(self, y):
            return self + y

        def __mul__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x, **kwargs: x[0] * x[1], [self, y], man_grad=[y.value, self.value])
            else:
                if isinstance(y, np.ndarray):
                    return np.array([self * o for o in y])
                elif isinstance(y, complex):
                    return CObs(self * y.real, self * y.imag)
                elif y.__class__.__name__ == 'Corr':
                    return NotImplemented
                else:
                    return derived_observable(lambda x, **kwargs: x[0] * y, [self], man_grad=[y])

        def __rmul__(self, y):
            return self * y

        def __sub__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x, **kwargs: x[0] - x[1], [self, y], man_grad=[1, -1])
            else:
                if isinstance(y, np.ndarray):
                    return np.array([self - o for o in y])

                elif y.__class__.__name__ == 'Corr':
                    return NotImplemented

                else:
                    return derived_observable(lambda x, **kwargs: x[0] - y, [self], man_grad=[1])

        def __rsub__(self, y):
            return -1 * (self - y)

        def __neg__(self):
            return -1 * self

        def __truediv__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x, **kwargs: x[0] / x[1], [self, y], man_grad=[1 / y.value, - self.value / y.value ** 2])
            else:
                if isinstance(y, np.ndarray):
                    return np.array([self / o for o in y])
                elif y.__class__.__name__ == 'Corr':
                    return NotImplemented
                else:
                    return derived_observable(lambda x, **kwargs: x[0] / y, [self], man_grad=[1 / y])

        def __rtruediv__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x, **kwargs: x[0] / x[1], [y, self], man_grad=[1 / self.value, - y.value / self.value ** 2])
            else:
                if isinstance(y, np.ndarray):
                    return np.array([o / self for o in y])
                elif y.__class__.__name__ == 'Corr':
                    return NotImplemented
                else:
                    return derived_observable(lambda x, **kwargs: y / x[0], [self], man_grad=[-y / self.value ** 2])

        def __pow__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x: x[0] ** x[1], [self, y])
            else:
                return derived_observable(lambda x: x[0] ** y, [self])

        def __rpow__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x: x[0] ** x[1], [y, self])
            else:
                return derived_observable(lambda x: y ** x[0], [self])

        def __abs__(self):
            return derived_observable(lambda x: anp.abs(x[0]), [self])

        # Overload numpy functions
        def sqrt(self):
            return derived_observable(lambda x, **kwargs: np.sqrt(x[0]), [self], man_grad=[1 / 2 / np.sqrt(self.value)])

        def log(self):
            return derived_observable(lambda x, **kwargs: np.log(x[0]), [self], man_grad=[1 / self.value])

        def exp(self):
            return derived_observable(lambda x, **kwargs: np.exp(x[0]), [self], man_grad=[np.exp(self.value)])

        def sin(self):
            return derived_observable(lambda x, **kwargs: np.sin(x[0]), [self], man_grad=[np.cos(self.value)])

        def cos(self):
            return derived_observable(lambda x, **kwargs: np.cos(x[0]), [self], man_grad=[-np.sin(self.value)])

        def tan(self):
            return derived_observable(lambda x, **kwargs: np.tan(x[0]), [self], man_grad=[1 / np.cos(self.value) ** 2])

        def arcsin(self):
            return derived_observable(lambda x: anp.arcsin(x[0]), [self])

        def arccos(self):
            return derived_observable(lambda x: anp.arccos(x[0]), [self])

        def arctan(self):
            return derived_observable(lambda x: anp.arctan(x[0]), [self])

        def sinh(self):
            return derived_observable(lambda x, **kwargs: np.sinh(x[0]), [self], man_grad=[np.cosh(self.value)])

        def cosh(self):
            return derived_observable(lambda x, **kwargs: np.cosh(x[0]), [self], man_grad=[np.sinh(self.value)])

        def tanh(self):
            return derived_observable(lambda x, **kwargs: np.tanh(x[0]), [self], man_grad=[1 / np.cosh(self.value) ** 2])

        def arcsinh(self):
            return derived_observable(lambda x: anp.arcsinh(x[0]), [self])

        def arccosh(self):
            return derived_observable(lambda x: anp.arccosh(x[0]), [self])

        def arctanh(self):
            return derived_observable(lambda x: anp.arctanh(x[0]), [self])

        def sinc(self):
            return derived_observable(lambda x: anp.sinc(x[0]), [self])

    class CObs:
        """Class for a complex valued observable."""
        __slots__ = ['_real', '_imag', 'tag']

        def __init__(self, real, imag=0.0):
            self._real = real
            self._imag = imag
            self.tag = None

        @property
        def real(self):
            return self._real

        @property
        def imag(self):
            return self._imag

        def gamma_method(self, **kwargs):
            """Executes the gamma_method for the real and the imaginary part."""
            if isinstance(self.real, Obs):
                self.real.gamma_method(**kwargs)
            if isinstance(self.imag, Obs):
                self.imag.gamma_method(**kwargs)

        def is_zero(self):
            """Checks whether both real and imaginary part are zero within machine precision."""
            return self.real == 0.0 and self.imag == 0.0

        def conjugate(self):
            return CObs(self.real, -self.imag)

        def __add__(self, other):
            if isinstance(other, np.ndarray):
                return other + self
            elif hasattr(other, 'real') and hasattr(other, 'imag'):
                return CObs(self.real + other.real,
                            self.imag + other.imag)
            else:
                return CObs(self.real + other, self.imag)

        def __radd__(self, y):
            return self + y

        def __sub__(self, other):
            if isinstance(other, np.ndarray):
                return -1 * (other - self)
            elif hasattr(other, 'real') and hasattr(other, 'imag'):
                return CObs(self.real - other.real, self.imag - other.imag)
            else:
                return CObs(self.real - other, self.imag)

        def __rsub__(self, other):
            return -1 * (self - other)

        def __mul__(self, other):
            if isinstance(other, np.ndarray):
                return other * self
            elif hasattr(other, 'real') and hasattr(other, 'imag'):
                if all(isinstance(i, Obs) for i in [self.real, self.imag, other.real, other.imag]):
                    return CObs(derived_observable(lambda x, **kwargs: x[0] * x[1] - x[2] * x[3],
                                                   [self.real, other.real, self.imag, other.imag],
                                                   man_grad=[other.real.value, self.real.value, -other.imag.value, -self.imag.value]),
                                derived_observable(lambda x, **kwargs: x[2] * x[1] + x[0] * x[3],
                                                   [self.real, other.real, self.imag, other.imag],
                                                   man_grad=[other.imag.value, self.imag.value, other.real.value, self.real.value]))
                elif getattr(other, 'imag', 0) != 0:
                    return CObs(self.real * other.real - self.imag * other.imag,
                                self.imag * other.real + self.real * other.imag)
                else:
                    return CObs(self.real * other.real, self.imag * other.real)
            else:
                return CObs(self.real * other, self.imag * other)

        def __rmul__(self, other):
            return self * other

        def __truediv__(self, other):
            if isinstance(other, np.ndarray):
                return 1 / (other / self)
            elif hasattr(other, 'real') and hasattr(other, 'imag'):
                r = other.real ** 2 + other.imag ** 2
                return CObs((self.real * other.real + self.imag * other.imag) / r, (self.imag * other.real - self.real * other.imag) / r)
            else:
                return CObs(self.real / other, self.imag / other)

        def __rtruediv__(self, other):
            r = self.real ** 2 + self.imag ** 2
            if hasattr(other, 'real') and hasattr(other, 'imag'):
                return CObs((self.real * other.real + self.imag * other.imag) / r, (self.real * other.imag - self.imag * other.real) / r)
            else:
                return CObs(self.real * other / r, -self.imag * other / r)

        def __abs__(self):
            return np.sqrt(self.real**2 + self.imag**2)

        def __neg__(other):
            return -1 * other

        def __eq__(self, other):
            return self.real == other.real and self.imag == other.imag

        def __str__(self):
            return '(' + str(self.real) + int(self.imag >= 0.0) * '+' + str(self.imag) + 'j)'

        def __repr__(self):
            return 'CObs[' + str(self) + ']'

    def _expand_deltas(deltas, idx, shape):
        """Expand deltas defined on idx to a regular, contiguous range, where holes are filled by 0.
           If idx is of type range, the deltas are not changed

        Parameters
        ----------
        deltas : list
            List of fluctuations
        idx : list
            List or range of configs on which the deltas are defined.
        shape : int
            Number of configs in idx.
        """
        if isinstance(idx, range):
            return deltas
        else:
            ret = np.zeros(idx[-1] - idx[0] + 1)
            for i in range(shape):
                ret[idx[i] - idx[0]] = deltas[i]
            return ret

    def _merge_idx(idl):
        """Returns the union of all lists in idl

        Parameters
        ----------
        idl : list
            List of lists or ranges.
        """

        # Use groupby to efficiently check whether all elements of idl are identical
        try:
            g = groupby(idl)
            if next(g, True) and not next(g, False):
                return idl[0]
        except:
            pass

        if np.all([type(idx) is range for idx in idl]):
            if len(set([idx[0] for idx in idl])) == 1:
                idstart = min([idx.start for idx in idl])
                idstop = max([idx.stop for idx in idl])
                idstep = min([idx.step for idx in idl])
                return range(idstart, idstop, idstep)

        return list(set().union(*idl))

    def _expand_deltas_for_merge(deltas, idx, shape, new_idx):
        """Expand deltas defined on idx to the list of configs that is defined by new_idx.
           New, empy entries are filled by 0. If idx and new_idx are of type range, the smallest
           common divisor of the step sizes is used as new step size.

        Parameters
        ----------
        deltas : list
            List of fluctuations
        idx : list
            List or range of configs on which the deltas are defined.
            Has to be a subset of new_idx.
        shape : list
            Number of configs in idx.
        new_idx : list
            List of configs that defines the new range.
        """

        if type(idx) is range and type(new_idx) is range:
            if idx == new_idx:
                return deltas
        ret = np.zeros(new_idx[-1] - new_idx[0] + 1)
        for i in range(shape):
            ret[idx[i] - new_idx[0]] = deltas[i]
        return np.array([ret[new_idx[i] - new_idx[0]] for i in range(len(new_idx))])

    def _filter_zeroes(names, deltas, idl, eps=Obs.filter_eps):
        """Filter out all configurations with vanishing fluctuation such that they do not
           contribute to the error estimate anymore. Returns the new names, deltas and
           idl according to the filtering.
           A fluctuation is considered to be vanishing, if it is smaller than eps times
           the mean of the absolute values of all deltas in one list.

        Parameters
        ----------
        names : list
            List of names
        deltas : dict
            Dict lists of fluctuations
        idx : dict
            Dict of lists or ranges of configs on which the deltas are defined.
            Has to be a subset of new_idx.
        eps : float
            Prefactor that enters the filter criterion.
        """
        new_names = []
        new_deltas = {}
        new_idl = {}
        for name in names:
            nd = []
            ni = []
            maxd = np.mean(np.fabs(deltas[name]))
            for i in range(len(deltas[name])):
                if not np.isclose(0.0, deltas[name][i], atol=eps * maxd):
                    nd.append(deltas[name][i])
                    ni.append(idl[name][i])
            if nd:
                new_names.append(name)
                new_deltas[name] = np.array(nd)
                new_idl[name] = ni
        return (new_names, new_deltas, new_idl)

    def derived_observable(func, data, **kwargs):
        """Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.

        Parameters
        ----------
        func : object
            arbitrary function of the form func(data, **kwargs). For the
            automatic differentiation to work, all numpy functions have to have
            the autograd wrapper (use 'import autograd.numpy as anp').
        data : list
            list of Obs, e.g. [obs1, obs2, obs3].
        num_grad : bool
            if True, numerical derivatives are used instead of autograd
            (default False). To control the numerical differentiation the
            kwargs of numdifftools.step_generators.MaxStepGenerator
            can be used.
        man_grad : list
            manually supply a list or an array which contains the jacobian
            of func. Use cautiously, supplying the wrong derivative will
            not be intercepted.

        Notes
        -----
        For simple mathematical operations it can be practical to use anonymous
        functions. For the ratio of two observables one can e.g. use

        new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])
        """

        data = np.asarray(data)
        raveled_data = data.ravel()

        # Workaround for matrix operations containing non Obs data
        for i_data in raveled_data:
            if isinstance(i_data, Obs):
                first_name = i_data.names[0]
                first_shape = i_data.shape[first_name]
                first_idl = i_data.idl[first_name]
                break

        for i in range(len(raveled_data)):
            if isinstance(raveled_data[i], (int, float)):
                raveled_data[i] = Obs([raveled_data[i] + np.zeros(first_shape)], [first_name], idl=[first_idl])

        n_obs = len(raveled_data)
        new_names = sorted(set([y for x in [o.names for o in raveled_data] for y in x]))

        is_merged = len(list(filter(lambda o: o.is_merged is True, raveled_data))) > 0
        reweighted = len(list(filter(lambda o: o.reweighted is True, raveled_data))) > 0
        new_idl_d = {}
        for name in new_names:
            idl = []
            for i_data in raveled_data:
                tmp = i_data.idl.get(name)
                if tmp is not None:
                    idl.append(tmp)
            new_idl_d[name] = _merge_idx(idl)
            if not is_merged:
                is_merged = (1 != len(set([len(idx) for idx in [*idl, new_idl_d[name]]])))

        if data.ndim == 1:
            values = np.array([o.value for o in data])
        else:
            values = np.vectorize(lambda x: x.value)(data)

        new_values = func(values, **kwargs)

        multi = 0
        if isinstance(new_values, np.ndarray):
            multi = 1

        new_r_values = {}
        for name in new_names:
            tmp_values = np.zeros(n_obs)
            for i, item in enumerate(raveled_data):
                tmp = item.r_values.get(name)
                if tmp is None:
                    tmp = item.value
                tmp_values[i] = tmp
            if multi > 0:
                tmp_values = np.array(tmp_values).reshape(data.shape)
            new_r_values[name] = func(tmp_values, **kwargs)

        if 'man_grad' in kwargs:
            deriv = np.asarray(kwargs.get('man_grad'))
            if new_values.shape + data.shape != deriv.shape:
                raise Exception('Manual derivative does not have correct shape.')
        elif kwargs.get('num_grad') is True:
            if multi > 0:
                raise Exception('Multi mode currently not supported for numerical derivative')
            options = {
                'base_step': 0.1,
                'step_ratio': 2.5,
                'num_steps': None,
                'step_nom': None,
                'offset': None,
                'num_extrap': None,
                'use_exact_steps': None,
                'check_num_steps': None,
                'scale': None}
            for key in options.keys():
                kwarg = kwargs.get(key)
                if kwarg is not None:
                    options[key] = kwarg
            tmp_df = nd.Gradient(func, order=4, **{k: v for k, v in options.items() if v is not None})(values, **kwargs)
            if tmp_df.size == 1:
                deriv = np.array([tmp_df.real])
            else:
                deriv = tmp_df.real
        else:
            deriv = jacobian(func)(values, **kwargs)

        final_result = np.zeros(new_values.shape, dtype=object)

        for i_val, new_val in np.ndenumerate(new_values):
            new_deltas = {}
            for j_obs, obs in np.ndenumerate(data):
                for name in obs.names:
                    new_deltas[name] = new_deltas.get(name, 0) + deriv[i_val + j_obs] * _expand_deltas_for_merge(obs.deltas[name], obs.idl[name], obs.shape[name], new_idl_d[name])

            new_samples = []
            new_means = []
            new_idl = []
            if is_merged:
                filtered_names, filtered_deltas, filtered_idl_d = _filter_zeroes(new_names, new_deltas, new_idl_d)
            else:
                filtered_names = new_names
                filtered_deltas = new_deltas
                filtered_idl_d = new_idl_d
            for name in filtered_names:
                new_samples.append(filtered_deltas[name])
                new_means.append(new_r_values[name][i_val])
                new_idl.append(filtered_idl_d[name])
            final_result[i_val] = Obs(new_samples, filtered_names, means=new_means, idl=new_idl)
            final_result[i_val]._value = new_val
            final_result[i_val].is_merged = is_merged
            final_result[i_val].reweighted = reweighted

        if multi == 0:
            final_result = final_result.item()

        return final_result

    def _reduce_deltas(deltas, idx_old, idx_new):
        """Extract deltas defined on idx_old on all configs of idx_new.

        Parameters
        ----------
        deltas : list
            List of fluctuations
        idx_old : list
            List or range of configs on which the deltas are defined
        idx_new : list
            List of configs for which we want to extract the deltas.
            Has to be a subset of idx_old.
        """
        if not len(deltas) == len(idx_old):
            raise Exception('Lenght of deltas and idx_old have to be the same: %d != %d' % (len(deltas), len(idx_old)))
        if type(idx_old) is range and type(idx_new) is range:
            if idx_old == idx_new:
                return deltas
        shape = len(idx_new)
        ret = np.zeros(shape)
        oldpos = 0
        for i in range(shape):
            if oldpos == idx_old[i]:
                raise Exception('idx_old and idx_new do not match!')
            pos = -1
            for j in range(oldpos, len(idx_old)):
                if idx_old[j] == idx_new[i]:
                    pos = j
                    break
            if pos < 0:
                raise Exception('Error in _reduce_deltas: Config %d not in idx_old' % (idx_new[i]))
            ret[i] = deltas[j]
        return np.array(ret)

    def reweight(weight, obs, **kwargs):
        """Reweight a list of observables.

        Parameters
        ----------
        weight : Obs
            Reweighting factor. An Observable that has to be defined on a superset of the
            configurations in obs[i].idl for all i.
        obs : list
            list of Obs, e.g. [obs1, obs2, obs3].
        all_configs : bool
            if True, the reweighted observables are normalized by the average of
            the reweighting factor on all configurations in weight.idl and not
            on the configurations in obs[i].idl.
        """
        result = []
        for i in range(len(obs)):
            if sorted(weight.names) != sorted(obs[i].names):
                raise Exception('Error: Ensembles do not fit')
            for name in weight.names:
                if not set(obs[i].idl[name]).issubset(weight.idl[name]):
                    raise Exception('obs[%d] has to be defined on a subset of the configs in weight.idl[%s]!' % (i, name))
            new_samples = []
            w_deltas = {}
            for name in sorted(weight.names):
                w_deltas[name] = _reduce_deltas(weight.deltas[name], weight.idl[name], obs[i].idl[name])
                new_samples.append((w_deltas[name] + weight.r_values[name]) * (obs[i].deltas[name] + obs[i].r_values[name]))
            tmp_obs = Obs(new_samples, sorted(weight.names), idl=[obs[i].idl[name] for name in sorted(weight.names)])

            if kwargs.get('all_configs'):
                new_weight = weight
            else:
                new_weight = Obs([w_deltas[name] + weight.r_values[name] for name in sorted(weight.names)], sorted(weight.names), idl=[obs[i].idl[name] for name in sorted(weight.names)])

            result.append(derived_observable(lambda x, **kwargs: x[0] / x[1], [tmp_obs, new_weight], **kwargs))
            result[-1].reweighted = True
            result[-1].is_merged = obs[i].is_merged

        return result

    def correlate(obs_a, obs_b):
        """Correlate two observables.

        Parameters
        ----------
        obs_a : Obs
            First observable
        obs_b : Obs
            Second observable

        Keep in mind to only correlate primary observables which have not been reweighted
        yet. The reweighting has to be applied after correlating the observables.
        Currently only works if ensembles are identical. This is not really necessary.
        """

        if sorted(obs_a.names) != sorted(obs_b.names):
            raise Exception('Ensembles do not fit')
        for name in obs_a.names:
            if obs_a.shape[name] != obs_b.shape[name]:
                raise Exception('Shapes of ensemble', name, 'do not fit')
            if obs_a.idl[name] != obs_b.idl[name]:
                raise Exception('idl of ensemble', name, 'do not fit')

        if obs_a.reweighted is True:
            warnings.warn("The first observable is already reweighted.", RuntimeWarning)
        if obs_b.reweighted is True:
            warnings.warn("The second observable is already reweighted.", RuntimeWarning)

        new_samples = []
        new_idl = []
        for name in sorted(obs_a.names):
            new_samples.append((obs_a.deltas[name] + obs_a.r_values[name]) * (obs_b.deltas[name] + obs_b.r_values[name]))
            new_idl.append(obs_a.idl[name])

        o = Obs(new_samples, sorted(obs_a.names), idl=new_idl)
        o.is_merged = obs_a.is_merged or obs_b.is_merged
        o.reweighted = obs_a.reweighted or obs_b.reweighted
        return o

    def covariance(obs1, obs2, correlation=False, **kwargs):
        """Calculates the covariance of two observables.

        covariance(obs, obs) is equal to obs.dvalue ** 2
        The gamma method has to be applied first to both observables.

        If abs(covariance(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance
        is constrained to the maximum value in order to make sure that covariance
        matrices are positive semidefinite.

        Parameters
        ----------
        correlation : bool
            if true the correlation instead of the covariance is
            returned (default False)
        """

        for name in sorted(set(obs1.names + obs2.names)):
            if (obs1.shape.get(name) != obs2.shape.get(name)) and (obs1.shape.get(name) is not None) and (obs2.shape.get(name) is not None):
                raise Exception('Shapes of ensemble', name, 'do not fit')
            if (1 != len(set([len(idx) for idx in [obs1.idl[name], obs2.idl[name], _merge_idx([obs1.idl[name], obs2.idl[name]])]]))):
                raise Exception('Shapes of ensemble', name, 'do not fit')

        if not hasattr(obs1, 'e_names') or not hasattr(obs2, 'e_names'):
            raise Exception('The gamma method has to be applied to both Obs first.')

        dvalue = 0

        for e_name in obs1.e_names:

            if e_name not in obs2.e_names:
                continue

            gamma = 0
            r_length = []
            for r_name in obs1.e_content[e_name]:
                if r_name not in obs2.e_content[e_name]:
                    continue

                r_length.append(len(obs1.deltas[r_name]))

                gamma += np.sum(obs1.deltas[r_name] * obs2.deltas[r_name])

            e_N = np.sum(r_length)

            tau_combined = (obs1.e_tauint[e_name] + obs2.e_tauint[e_name]) / 2
            dvalue += gamma / e_N * (1 + 1 / e_N) / e_N * 2 * tau_combined

        if np.abs(dvalue / obs1.dvalue / obs2.dvalue) > 1.0:
            dvalue = np.sign(dvalue) * obs1.dvalue * obs2.dvalue

        if correlation:
            dvalue = dvalue / obs1.dvalue / obs2.dvalue

        return dvalue

    def covariance2(obs1, obs2, correlation=False, **kwargs):
        """Alternative implementation of the covariance of two observables.

        covariance(obs, obs) is equal to obs.dvalue ** 2
        The gamma method has to be applied first to both observables.

        If abs(covariance(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance
        is constrained to the maximum value in order to make sure that covariance
        matrices are positive semidefinite.

        Keyword arguments
        -----------------
        correlation -- if true the correlation instead of the covariance is
                       returned (default False)
        """

        def expand_deltas(deltas, idx, shape, new_idx):
            """Expand deltas defined on idx to a contiguous range [new_idx[0], new_idx[-1]].
               New, empy entries are filled by 0. If idx and new_idx are of type range, the smallest
               common divisor of the step sizes is used as new step size.

            Parameters
            ----------
            deltas  -- List of fluctuations
            idx     -- List or range of configs on which the deltas are defined.
                       Has to be a subset of new_idx.
            shape   -- Number of configs in idx.
            new_idx -- List of configs that defines the new range.
            """

            if type(idx) is range and type(new_idx) is range:
                if idx == new_idx:
                    return deltas
            ret = np.zeros(new_idx[-1] - new_idx[0] + 1)
            for i in range(shape):
                ret[idx[i] - new_idx[0]] = deltas[i]
            return ret

        def calc_gamma(deltas1, deltas2, idx1, idx2, new_idx, w_max):
            gamma = np.zeros(w_max)
            deltas1 = expand_deltas(deltas1, idx1, len(idx1), new_idx)
            deltas2 = expand_deltas(deltas2, idx2, len(idx2), new_idx)
            new_shape = len(deltas1)
            max_gamma = min(new_shape, w_max)
            # The padding for the fft has to be even
            padding = new_shape + max_gamma + (new_shape + max_gamma) % 2
            gamma[:max_gamma] += (np.fft.irfft(np.fft.rfft(deltas1, padding) * np.conjugate(np.fft.rfft(deltas2, padding)))[:max_gamma] + np.fft.irfft(np.fft.rfft(deltas2, padding) * np.conjugate(np.fft.rfft(deltas1, padding)))[:max_gamma]) / 2.0

            return gamma

        if not hasattr(obs1, 'e_names') or not hasattr(obs2, 'e_names'):
            raise Exception('The gamma method has to be applied to both Obs first.')

        dvalue = 0
        e_gamma = {}
        e_dvalue = {}
        e_n_tauint = {}
        e_rho = {}

        for e_name in obs1.e_names:

            if e_name not in obs2.e_names:
                continue

            idl_d = {}
            r_length = []
            for r_name in obs1.e_content[e_name]:
                if r_name not in obs2.e_content[e_name]:
                    continue
                idl_d[r_name] = _merge_idx([obs1.idl[r_name], obs2.idl[r_name]])
                if isinstance(idl_d[r_name], range):
                    r_length.append(len(idl_d[r_name]))
                else:
                    r_length.append((idl_d[r_name][-1] - idl_d[r_name][0] + 1))

            if not r_length:
                return 0.

            w_max = max(r_length) // 2
            e_gamma[e_name] = np.zeros(w_max)

            for r_name in obs1.e_content[e_name]:
                if r_name not in obs2.e_content[e_name]:
                    continue
                e_gamma[e_name] += calc_gamma(obs1.deltas[r_name], obs2.deltas[r_name], obs1.idl[r_name], obs2.idl[r_name], idl_d[r_name], w_max)

            if np.all(e_gamma[e_name] == 0.0):
                continue

            e_shapes = []
            for r_name in obs1.e_content[e_name]:
                e_shapes.append(obs1.shape[r_name])
            gamma_div = np.zeros(w_max)
            e_N = 0
            for r_name in obs1.e_content[e_name]:
                if r_name not in obs2.e_content[e_name]:
                    continue
                gamma_div += calc_gamma(np.ones(obs1.shape[r_name]), np.ones(obs2.shape[r_name]), obs1.idl[r_name], obs2.idl[r_name], idl_d[r_name], w_max)
                e_N += np.sum(np.ones_like(idl_d[r_name]))
            e_gamma[e_name] /= gamma_div[:w_max]

            e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0]
            e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], e_rho[e_name][1:])))
            # Make sure no entry of tauint is smaller than 0.5
            e_n_tauint[e_name][e_n_tauint[e_name] < 0.5] = 0.500000000001

            window = max(obs1.e_windowsize[e_name], obs2.e_windowsize[e_name])
            # Bias correction hep-lat/0306017 eq. (49)
            e_dvalue[e_name] = 2 * (e_n_tauint[e_name][window] + obs1.tau_exp[e_name] * np.abs(e_rho[e_name][window + 1])) * (1 + (2 * window + 1) / e_N) * e_gamma[e_name][0] / e_N

            dvalue += e_dvalue[e_name]

        if np.abs(dvalue / obs1.dvalue / obs2.dvalue) > 1.0:
            dvalue = np.sign(dvalue) * obs1.dvalue * obs2.dvalue

        if correlation:
            dvalue = dvalue / obs1.dvalue / obs2.dvalue

        return dvalue

    def covariance3(obs1, obs2, correlation=False, **kwargs):
        """Another alternative implementation of the covariance of two observables.

        covariance2(obs, obs) is equal to obs.dvalue ** 2
        Currently only works if ensembles are identical.
        The gamma method has to be applied first to both observables.

        If abs(covariance2(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance
        is constrained to the maximum value in order to make sure that covariance
        matrices are positive semidefinite.

        Keyword arguments
        -----------------
        correlation -- if true the correlation instead of the covariance is
                       returned (default False)
        plot -- if true, the integrated autocorrelation time for each ensemble is
                plotted.
        """

        for name in sorted(set(obs1.names + obs2.names)):
            if (obs1.shape.get(name) != obs2.shape.get(name)) and (obs1.shape.get(name) is not None) and (obs2.shape.get(name) is not None):
                raise Exception('Shapes of ensemble', name, 'do not fit')
            if (1 != len(set([len(idx) for idx in [obs1.idl[name], obs2.idl[name], _merge_idx([obs1.idl[name], obs2.idl[name]])]]))):
                raise Exception('Shapes of ensemble', name, 'do not fit')

        if not hasattr(obs1, 'e_names') or not hasattr(obs2, 'e_names'):
            raise Exception('The gamma method has to be applied to both Obs first.')

        tau_exp = []
        S = []
        for e_name in sorted(set(obs1.e_names + obs2.e_names)):
            t_1 = obs1.tau_exp.get(e_name)
            t_2 = obs2.tau_exp.get(e_name)
            if t_1 is None:
                t_1 = 0
            if t_2 is None:
                t_2 = 0
            tau_exp.append(max(t_1, t_2))
            S_1 = obs1.S.get(e_name)
            S_2 = obs2.S.get(e_name)
            if S_1 is None:
                S_1 = Obs.S_global
            if S_2 is None:
                S_2 = Obs.S_global
            S.append(max(S_1, S_2))

        check_obs = obs1 + obs2
        check_obs.gamma_method(tau_exp=tau_exp, S=S)

        if kwargs.get('plot'):
            check_obs.plot_tauint()
            check_obs.plot_rho()

        cov = (check_obs.dvalue ** 2 - obs1.dvalue ** 2 - obs2.dvalue ** 2) / 2

        if np.abs(cov / obs1.dvalue / obs2.dvalue) > 1.0:
            cov = np.sign(cov) * obs1.dvalue * obs2.dvalue

        if correlation:
            cov = cov / obs1.dvalue / obs2.dvalue

        return cov

    def pseudo_Obs(value, dvalue, name, samples=1000):
        """Generate a pseudo Obs with given value, dvalue and name

        The standard number of samples is a 1000. This can be adjusted.
        """
        if dvalue <= 0.0:
            return Obs([np.zeros(samples) + value], [name])
        else:
            for _ in range(100):
                deltas = [np.random.normal(0.0, dvalue * np.sqrt(samples), samples)]
                deltas -= np.mean(deltas)
                deltas *= dvalue / np.sqrt((np.var(deltas) / samples)) / np.sqrt(1 + 3 / samples)
                deltas += value
                res = Obs(deltas, [name])
                res.gamma_method(S=2, tau_exp=0)
                if abs(res.dvalue - dvalue) < 1e-10 * dvalue:
                    break

            res._value = float(value)

            return res

    def dump_object(obj, name, **kwargs):
        """Dump object into pickle file.

        Parameters
        ----------
        obj : object
            object to be saved in the pickle file
        name : str
            name of the file
        path : str
            specifies a custom path for the file (default '.')
        """
        if 'path' in kwargs:
            file_name = kwargs.get('path') + '/' + name + '.p'
        else:
            file_name = name + '.p'
        with open(file_name, 'wb') as fb:
            pickle.dump(obj, fb)

    def load_object(path):
        """Load object from pickle file. """
        with open(path, 'rb') as file:
            return pickle.load(file)

    def merge_obs(list_of_obs):
        """Combine all observables in list_of_obs into one new observable

        Parameters
        ----------
        list_of_obs : list
            list of the Obs object to be combined

        It is not possible to combine obs which are based on the same replicum
        """
        replist = [item for obs in list_of_obs for item in obs.names]
        if (len(replist) == len(set(replist))) is False:
            raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist)))
        new_dict = {}
        idl_dict = {}
        for o in list_of_obs:
            new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0)
                            for key in set(o.deltas) | set(o.r_values)})
            idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)})

        names = sorted(new_dict.keys())
        o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names])
        o.is_merged = np.any([oi.is_merged for oi in list_of_obs])
        o.reweighted = np.max([oi.reweighted for oi in list_of_obs])
        return o

class Obs:
View Source

    class Obs:
        """Class for a general observable.

        Instances of Obs are the basic objects of a pyerrors error analysis.
        They are initialized with a list which contains arrays of samples for
        different ensembles/replica and another list of same length which contains
        the names of the ensembles/replica. Mathematical operations can be
        performed on instances. The result is another instance of Obs. The error of
        an instance can be computed with the gamma_method. Also contains additional
        methods for output and visualization of the error calculation.

        Attributes
        ----------
        S_global : float
            Standard value for S (default 2.0)
        S_dict : dict
            Dictionary for S values. If an entry for a given ensemble
            exists this overwrites the standard value for that ensemble.
        tau_exp_global : float
            Standard value for tau_exp (default 0.0)
        tau_exp_dict :dict
            Dictionary for tau_exp values. If an entry for a given ensemble exists
            this overwrites the standard value for that ensemble.
        N_sigma_global : float
            Standard value for N_sigma (default 1.0)
        """
        __slots__ = ['names', 'shape', 'r_values', 'deltas', 'N', '_value', '_dvalue',
                     'ddvalue', 'reweighted', 'S', 'tau_exp', 'N_sigma',
                     'e_dvalue', 'e_ddvalue', 'e_tauint', 'e_dtauint',
                     'e_windowsize', 'e_rho', 'e_drho', 'e_n_tauint', 'e_n_dtauint',
                     'idl', 'is_merged', 'tag', '__dict__']

        S_global = 2.0
        S_dict = {}
        tau_exp_global = 0.0
        tau_exp_dict = {}
        N_sigma_global = 1.0
        filter_eps = 1e-10

        def __init__(self, samples, names, idl=None, means=None, **kwargs):
            """ Initialize Obs object.

            Parameters
            ----------
            samples : list
                list of numpy arrays containing the Monte Carlo samples
            names : list
                list of strings labeling the indivdual samples
            idl : list, optional
                list of ranges or lists on which the samples are defined
            means : list, optional
                list of mean values for the case that the mean values were
                already subtracted from the samples
            """

            if means is None:
                if len(samples) != len(names):
                    raise Exception('Length of samples and names incompatible.')
                if len(names) != len(set(names)):
                    raise Exception('Names are not unique.')
                if not all(isinstance(x, str) for x in names):
                    raise TypeError('All names have to be strings.')
                if min(len(x) for x in samples) <= 4:
                    raise Exception('Samples have to have at least 4 entries.')

            self.names = sorted(names)
            self.shape = {}
            self.r_values = {}
            self.deltas = {}

            self.idl = {}
            if idl is not None:
                for name, idx in sorted(zip(names, idl)):
                    if isinstance(idx, range):
                        self.idl[name] = idx
                    elif isinstance(idx, (list, np.ndarray)):
                        dc = np.unique(np.diff(idx))
                        if np.any(dc < 0):
                            raise Exception("Unsorted idx for idl[%s]" % (name))
                        if len(dc) == 1:
                            self.idl[name] = range(idx[0], idx[-1] + dc[0], dc[0])
                        else:
                            self.idl[name] = list(idx)
                    else:
                        raise Exception('incompatible type for idl[%s].' % (name))
            else:
                for name, sample in sorted(zip(names, samples)):
                    self.idl[name] = range(1, len(sample) + 1)

            if means is not None:
                for name, sample, mean in sorted(zip(names, samples, means)):
                    self.shape[name] = len(self.idl[name])
                    if len(sample) != self.shape[name]:
                        raise Exception('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name]))
                    self.r_values[name] = mean
                    self.deltas[name] = sample
            else:
                for name, sample in sorted(zip(names, samples)):
                    self.shape[name] = len(self.idl[name])
                    if len(sample) != self.shape[name]:
                        raise Exception('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name]))
                    self.r_values[name] = np.mean(sample)
                    self.deltas[name] = sample - self.r_values[name]
            self.is_merged = False
            self.N = sum(list(self.shape.values()))

            self._value = 0
            if means is None:
                for name in self.names:
                    self._value += self.shape[name] * self.r_values[name]
                self._value /= self.N

            self._dvalue = 0.0
            self.ddvalue = 0.0
            self.reweighted = False

            self.tag = None

        @property
        def value(self):
            return self._value

        @property
        def dvalue(self):
            return self._dvalue

        @property
        def e_names(self):
            return sorted(set([o.split('|')[0] for o in self.names]))

        @property
        def e_content(self):
            res = {}
            for e, e_name in enumerate(self.e_names):
                res[e_name] = sorted(filter(lambda x: x.startswith(e_name + '|'), self.names))
                if e_name in self.names:
                    res[e_name].append(e_name)
            return res

        def gamma_method(self, **kwargs):
            """Calculate the error and related properties of the Obs.

            Parameters
            ----------
            S : float
                specifies a custom value for the parameter S (default 2.0), can be
                a float or an array of floats for different ensembles
            tau_exp : float
                positive value triggers the critical slowing down analysis
                (default 0.0), can be a float or an array of floats for different
                ensembles
            N_sigma : float
                number of standard deviations from zero until the tail is
                attached to the autocorrelation function (default 1)
            fft : bool
                determines whether the fft algorithm is used for the computation
                of the autocorrelation function (default True)
            """

            e_content = self.e_content
            self.e_dvalue = {}
            self.e_ddvalue = {}
            self.e_tauint = {}
            self.e_dtauint = {}
            self.e_windowsize = {}
            self.e_n_tauint = {}
            self.e_n_dtauint = {}
            e_gamma = {}
            self.e_rho = {}
            self.e_drho = {}
            self._dvalue = 0
            self.ddvalue = 0

            self.S = {}
            self.tau_exp = {}

            if kwargs.get('fft') is False:
                fft = False
            else:
                fft = True

            if 'S' in kwargs:
                tmp = kwargs.get('S')
                if isinstance(tmp, list):
                    if len(tmp) != len(self.e_names):
                        raise Exception('Length of S array does not match ensembles.')
                    for e, e_name in enumerate(self.e_names):
                        if tmp[e] <= 0:
                            raise Exception('S has to be larger than 0.')
                        self.S[e_name] = tmp[e]
                else:
                    if isinstance(tmp, (int, float)):
                        if tmp <= 0:
                            raise Exception('S has to be larger than 0.')
                        for e, e_name in enumerate(self.e_names):
                            self.S[e_name] = tmp
                    else:
                        raise TypeError('S is not in proper format.')
            else:
                for e, e_name in enumerate(self.e_names):
                    if e_name in Obs.S_dict:
                        self.S[e_name] = Obs.S_dict[e_name]
                    else:
                        self.S[e_name] = Obs.S_global

            if 'tau_exp' in kwargs:
                tmp = kwargs.get('tau_exp')
                if isinstance(tmp, list):
                    if len(tmp) != len(self.e_names):
                        raise Exception('Length of tau_exp array does not match ensembles.')
                    for e, e_name in enumerate(self.e_names):
                        if tmp[e] < 0:
                            raise Exception('tau_exp smaller than 0.')
                        self.tau_exp[e_name] = tmp[e]
                else:
                    if isinstance(tmp, (int, float)):
                        if tmp < 0:
                            raise Exception('tau_exp smaller than 0.')
                        for e, e_name in enumerate(self.e_names):
                            self.tau_exp[e_name] = tmp
                    else:
                        raise TypeError('tau_exp is not in proper format.')
            else:
                for e, e_name in enumerate(self.e_names):
                    if e_name in Obs.tau_exp_dict:
                        self.tau_exp[e_name] = Obs.tau_exp_dict[e_name]
                    else:
                        self.tau_exp[e_name] = Obs.tau_exp_global

            if 'N_sigma' in kwargs:
                self.N_sigma = kwargs.get('N_sigma')
                if not isinstance(self.N_sigma, (int, float)):
                    raise TypeError('N_sigma is not a number.')
            else:
                self.N_sigma = Obs.N_sigma_global

            for e, e_name in enumerate(self.e_names):

                r_length = []
                for r_name in e_content[e_name]:
                    if isinstance(self.idl[r_name], range):
                        r_length.append(len(self.idl[r_name]))
                    else:
                        r_length.append((self.idl[r_name][-1] - self.idl[r_name][0] + 1))

                e_N = np.sum([self.shape[r_name] for r_name in e_content[e_name]])
                w_max = max(r_length) // 2
                e_gamma[e_name] = np.zeros(w_max)
                self.e_rho[e_name] = np.zeros(w_max)
                self.e_drho[e_name] = np.zeros(w_max)

                for r_name in e_content[e_name]:
                    e_gamma[e_name] += self._calc_gamma(self.deltas[r_name], self.idl[r_name], self.shape[r_name], w_max, fft)

                gamma_div = np.zeros(w_max)
                for r_name in e_content[e_name]:
                    gamma_div += self._calc_gamma(np.ones((self.shape[r_name])), self.idl[r_name], self.shape[r_name], w_max, fft)
                e_gamma[e_name] /= gamma_div[:w_max]

                if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny:  # Prevent division by zero
                    self.e_tauint[e_name] = 0.5
                    self.e_dtauint[e_name] = 0.0
                    self.e_dvalue[e_name] = 0.0
                    self.e_ddvalue[e_name] = 0.0
                    self.e_windowsize[e_name] = 0
                    continue

                self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0]
                self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:])))
                # Make sure no entry of tauint is smaller than 0.5
                self.e_n_tauint[e_name][self.e_n_tauint[e_name] <= 0.5] = 0.5 + np.finfo(np.float64).eps
                # hep-lat/0306017 eq. (42)
                self.e_n_dtauint[e_name] = self.e_n_tauint[e_name] * 2 * np.sqrt(np.abs(np.arange(w_max) + 0.5 - self.e_n_tauint[e_name]) / e_N)
                self.e_n_dtauint[e_name][0] = 0.0

                def _compute_drho(i):
                    tmp = self.e_rho[e_name][i + 1:w_max] + np.concatenate([self.e_rho[e_name][i - 1::-1], self.e_rho[e_name][1:w_max - 2 * i]]) - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i]
                    self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N)

                _compute_drho(1)
                if self.tau_exp[e_name] > 0:
                    texp = self.tau_exp[e_name]
                    # if type(self.idl[e_name]) is range: # scale tau_exp according to step size
                    #    texp /= self.idl[e_name].step
                    # Critical slowing down analysis
                    for n in range(1, w_max // 2):
                        _compute_drho(n + 1)
                        if (self.e_rho[e_name][n] - self.N_sigma * self.e_drho[e_name][n]) < 0 or n >= w_max // 2 - 2:
                            # Bias correction hep-lat/0306017 eq. (49) included
                            self.e_tauint[e_name] = self.e_n_tauint[e_name][n] * (1 + (2 * n + 1) / e_N) / (1 + 1 / e_N) + texp * np.abs(self.e_rho[e_name][n + 1])  # The absolute makes sure, that the tail contribution is always positive
                            self.e_dtauint[e_name] = np.sqrt(self.e_n_dtauint[e_name][n] ** 2 + texp ** 2 * self.e_drho[e_name][n + 1] ** 2)
                            # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2
                            self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
                            self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
                            self.e_windowsize[e_name] = n
                            break
                else:
                    # Standard automatic windowing procedure
                    g_w = self.S[e_name] / np.log((2 * self.e_n_tauint[e_name][1:] + 1) / (2 * self.e_n_tauint[e_name][1:] - 1))
                    g_w = np.exp(- np.arange(1, w_max) / g_w) - g_w / np.sqrt(np.arange(1, w_max) * e_N)
                    for n in range(1, w_max):
                        if n < w_max // 2 - 2:
                            _compute_drho(n + 1)
                        if g_w[n - 1] < 0 or n >= w_max - 1:
                            self.e_tauint[e_name] = self.e_n_tauint[e_name][n] * (1 + (2 * n + 1) / e_N) / (1 + 1 / e_N)  # Bias correction hep-lat/0306017 eq. (49)
                            self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n]
                            self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
                            self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
                            self.e_windowsize[e_name] = n
                            break

                self._dvalue += self.e_dvalue[e_name] ** 2
                self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2

            self._dvalue = np.sqrt(self.dvalue)
            if self._dvalue == 0.0:
                self.ddvalue = 0.0
            else:
                self.ddvalue = np.sqrt(self.ddvalue) / self.dvalue
            return

        def _calc_gamma(self, deltas, idx, shape, w_max, fft):
            """Calculate Gamma_{AA} from the deltas, which are defined on idx.
               idx is assumed to be a contiguous range (possibly with a stepsize != 1)

            Parameters
            ----------
            deltas : list
                List of fluctuations
            idx : list
                List or range of configs on which the deltas are defined.
            shape : int
                Number of configs in idx.
            w_max : int
                Upper bound for the summation window.
            fft : bool
                determines whether the fft algorithm is used for the computation
                of the autocorrelation function.
            """
            gamma = np.zeros(w_max)
            deltas = _expand_deltas(deltas, idx, shape)
            new_shape = len(deltas)
            if fft:
                max_gamma = min(new_shape, w_max)
                # The padding for the fft has to be even
                padding = new_shape + max_gamma + (new_shape + max_gamma) % 2
                gamma[:max_gamma] += np.fft.irfft(np.abs(np.fft.rfft(deltas, padding)) ** 2)[:max_gamma]
            else:
                for n in range(w_max):
                    if new_shape - n >= 0:
                        gamma[n] += deltas[0:new_shape - n].dot(deltas[n:new_shape])

            return gamma

        def print(self, level=1):
            warnings.warn("Method 'print' renamed to 'details'", DeprecationWarning)
            self.details(level > 1)

        def details(self, ens_content=True):
            """Output detailed properties of the Obs."""
            if self.value == 0.0:
                percentage = np.nan
            else:
                percentage = np.abs(self.dvalue / self.value) * 100
            print('Result\t %3.8e +/- %3.8e +/- %3.8e (%3.3f%%)' % (self.value, self.dvalue, self.ddvalue, percentage))
            if hasattr(self, 'e_dvalue'):
                if len(self.e_names) > 1:
                    print(' Ensemble errors:')
                for e_name in self.e_names:
                    if len(self.e_names) > 1:
                        print('', e_name, '\t %3.8e +/- %3.8e' % (self.e_dvalue[e_name], self.e_ddvalue[e_name]))
                    if self.tau_exp[e_name] > 0:
                        print(' t_int\t %3.8e +/- %3.8e tau_exp = %3.2f,  N_sigma = %1.0i' % (self.e_tauint[e_name], self.e_dtauint[e_name], self.tau_exp[e_name], self.N_sigma))
                    else:
                        print(' t_int\t %3.8e +/- %3.8e S = %3.2f' % (self.e_tauint[e_name], self.e_dtauint[e_name], self.S[e_name]))
            if self.tag is not None:
                print("Description:", self.tag)
            if ens_content is True:
                if len(self.e_names) == 1:
                    print(self.N, 'samples in', len(self.e_names), 'ensemble:')
                else:
                    print(self.N, 'samples in', len(self.e_names), 'ensembles:')
                m = max(map(len, list(self.e_content.keys()))) + 1
                print('\n'.join(['  ' + key.rjust(m) + ': ' + str(value) for key, value in sorted(self.e_content.items())]))

        def is_zero_within_error(self, sigma=1):
            """Checks whether the observable is zero within 'sigma' standard errors.

            Works only properly when the gamma method was run.
            """
            return self.is_zero() or np.abs(self.value) <= sigma * self.dvalue

        def is_zero(self):
            """Checks whether the observable is zero within machine precision."""
            return np.isclose(0.0, self.value) and all(np.allclose(0.0, delta) for delta in self.deltas.values())

        def plot_tauint(self, save=None):
            """Plot integrated autocorrelation time for each ensemble."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')

            fig = plt.figure()
            for e, e_name in enumerate(self.e_names):
                plt.xlabel(r'$W$')
                plt.ylabel(r'$\tau_\mathrm{int}$')
                length = int(len(self.e_n_tauint[e_name]))
                if self.tau_exp[e_name] > 0:
                    base = self.e_n_tauint[e_name][self.e_windowsize[e_name]]
                    x_help = np.arange(2 * self.tau_exp[e_name])
                    y_help = (x_help + 1) * np.abs(self.e_rho[e_name][self.e_windowsize[e_name] + 1]) * (1 - x_help / (2 * (2 * self.tau_exp[e_name] - 1))) + base
                    x_arr = np.arange(self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name])
                    plt.plot(x_arr, y_help, 'C' + str(e), linewidth=1, ls='--', marker=',')
                    plt.errorbar([self.e_windowsize[e_name] + 2 * self.tau_exp[e_name]], [self.e_tauint[e_name]],
                                 yerr=[self.e_dtauint[e_name]], fmt='C' + str(e), linewidth=1, capsize=2, marker='o', mfc=plt.rcParams['axes.facecolor'])
                    xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
                    label = e_name + r', $\tau_\mathrm{exp}$=' + str(np.around(self.tau_exp[e_name], decimals=2))
                else:
                    label = e_name + ', S=' + str(np.around(self.S[e_name], decimals=2))
                    xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)

                plt.errorbar(np.arange(length), self.e_n_tauint[e_name][:], yerr=self.e_n_dtauint[e_name][:], linewidth=1, capsize=2, label=label)
                plt.axvline(x=self.e_windowsize[e_name], color='C' + str(e), alpha=0.5, marker=',', ls='--')
                plt.legend()
                plt.xlim(-0.5, xmax)
                plt.ylim(bottom=0.0)
                plt.draw()
                if save:
                    fig.savefig(save)

        def plot_rho(self):
            """Plot normalized autocorrelation function time for each ensemble."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')
            for e, e_name in enumerate(self.e_names):
                plt.xlabel('W')
                plt.ylabel('rho')
                length = int(len(self.e_drho[e_name]))
                plt.errorbar(np.arange(length), self.e_rho[e_name][:length], yerr=self.e_drho[e_name][:], linewidth=1, capsize=2)
                plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25, ls='--', marker=',')
                if self.tau_exp[e_name] > 0:
                    plt.plot([self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]],
                             [self.e_rho[e_name][self.e_windowsize[e_name] + 1], 0], 'k-', lw=1)
                    xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
                    plt.title('Rho ' + e_name + r', tau\_exp=' + str(np.around(self.tau_exp[e_name], decimals=2)))
                else:
                    xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)
                    plt.title('Rho ' + e_name + ', S=' + str(np.around(self.S[e_name], decimals=2)))
                plt.plot([-0.5, xmax], [0, 0], 'k--', lw=1)
                plt.xlim(-0.5, xmax)
                plt.draw()

        def plot_rep_dist(self):
            """Plot replica distribution for each ensemble with more than one replicum."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')
            for e, e_name in enumerate(self.e_names):
                if len(self.e_content[e_name]) == 1:
                    print('No replica distribution for a single replicum (', e_name, ')')
                    continue
                r_length = []
                sub_r_mean = 0
                for r, r_name in enumerate(self.e_content[e_name]):
                    r_length.append(len(self.deltas[r_name]))
                    sub_r_mean += self.shape[r_name] * self.r_values[r_name]
                e_N = np.sum(r_length)
                sub_r_mean /= e_N
                arr = np.zeros(len(self.e_content[e_name]))
                for r, r_name in enumerate(self.e_content[e_name]):
                    arr[r] = (self.r_values[r_name] - sub_r_mean) / (self.e_dvalue[e_name] * np.sqrt(e_N / self.shape[r_name] - 1))
                plt.hist(arr, rwidth=0.8, bins=len(self.e_content[e_name]))
                plt.title('Replica distribution' + e_name + ' (mean=0, var=1)')
                plt.draw()

        def plot_history(self, expand=True):
            """Plot derived Monte Carlo history for each ensemble."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')

            for e, e_name in enumerate(self.e_names):
                plt.figure()
                r_length = []
                tmp = []
                for r, r_name in enumerate(self.e_content[e_name]):
                    if expand:
                        tmp.append(_expand_deltas(self.deltas[r_name], self.idl[r_name], self.shape[r_name]) + self.r_values[r_name])
                    else:
                        tmp.append(self.deltas[r_name] + self.r_values[r_name])
                    r_length.append(len(tmp[-1]))
                e_N = np.sum(r_length)
                x = np.arange(e_N)
                y = np.concatenate(tmp, axis=0)
                plt.errorbar(x, y, fmt='.', markersize=3)
                plt.xlim(-0.5, e_N - 0.5)
                plt.title(e_name)
                plt.draw()

        def plot_piechart(self):
            """Plot piechart which shows the fractional contribution of each
            ensemble to the error and returns a dictionary containing the fractions."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')
            if self.dvalue == 0.0:
                raise Exception('Error is 0.0')
            labels = self.e_names
            sizes = [i ** 2 for i in list(self.e_dvalue.values())] / self.dvalue ** 2
            fig1, ax1 = plt.subplots()
            ax1.pie(sizes, labels=labels, startangle=90, normalize=True)
            ax1.axis('equal')
            plt.draw()

            return dict(zip(self.e_names, sizes))

        def dump(self, name, **kwargs):
            """Dump the Obs to a pickle file 'name'.

            Parameters
            ----------
            path : str
                specifies a custom path for the file (default '.')
            """
            if 'path' in kwargs:
                file_name = kwargs.get('path') + '/' + name + '.p'
            else:
                file_name = name + '.p'
            with open(file_name, 'wb') as fb:
                pickle.dump(self, fb)

        def __float__(self):
            return float(self.value)

        def __repr__(self):
            return 'Obs[' + str(self) + ']'

        def __str__(self):
            if self.dvalue == 0.0:
                return str(self.value)
            fexp = np.floor(np.log10(self.dvalue))
            if fexp < 0.0:
                return '{:{form}}({:2.0f})'.format(self.value, self.dvalue * 10 ** (-fexp + 1), form='.' + str(-int(fexp) + 1) + 'f')
            elif fexp == 0.0:
                return '{:.1f}({:1.1f})'.format(self.value, self.dvalue)
            else:
                return '{:.0f}({:2.0f})'.format(self.value, self.dvalue)

        # Overload comparisons
        def __lt__(self, other):
            return self.value < other

        def __le__(self, other):
            return self.value <= other

        def __gt__(self, other):
            return self.value > other

        def __ge__(self, other):
            return self.value >= other

        def __eq__(self, other):
            return (self - other).is_zero()

        def __ne__(self, other):
            return not (self - other).is_zero()

        # Overload math operations
        def __add__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x, **kwargs: x[0] + x[1], [self, y], man_grad=[1, 1])
            else:
                if isinstance(y, np.ndarray):
                    return np.array([self + o for o in y])
                elif y.__class__.__name__ == 'Corr':
                    return NotImplemented
                else:
                    return derived_observable(lambda x, **kwargs: x[0] + y, [self], man_grad=[1])

        def __radd__(self, y):
            return self + y

        def __mul__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x, **kwargs: x[0] * x[1], [self, y], man_grad=[y.value, self.value])
            else:
                if isinstance(y, np.ndarray):
                    return np.array([self * o for o in y])
                elif isinstance(y, complex):
                    return CObs(self * y.real, self * y.imag)
                elif y.__class__.__name__ == 'Corr':
                    return NotImplemented
                else:
                    return derived_observable(lambda x, **kwargs: x[0] * y, [self], man_grad=[y])

        def __rmul__(self, y):
            return self * y

        def __sub__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x, **kwargs: x[0] - x[1], [self, y], man_grad=[1, -1])
            else:
                if isinstance(y, np.ndarray):
                    return np.array([self - o for o in y])

                elif y.__class__.__name__ == 'Corr':
                    return NotImplemented

                else:
                    return derived_observable(lambda x, **kwargs: x[0] - y, [self], man_grad=[1])

        def __rsub__(self, y):
            return -1 * (self - y)

        def __neg__(self):
            return -1 * self

        def __truediv__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x, **kwargs: x[0] / x[1], [self, y], man_grad=[1 / y.value, - self.value / y.value ** 2])
            else:
                if isinstance(y, np.ndarray):
                    return np.array([self / o for o in y])
                elif y.__class__.__name__ == 'Corr':
                    return NotImplemented
                else:
                    return derived_observable(lambda x, **kwargs: x[0] / y, [self], man_grad=[1 / y])

        def __rtruediv__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x, **kwargs: x[0] / x[1], [y, self], man_grad=[1 / self.value, - y.value / self.value ** 2])
            else:
                if isinstance(y, np.ndarray):
                    return np.array([o / self for o in y])
                elif y.__class__.__name__ == 'Corr':
                    return NotImplemented
                else:
                    return derived_observable(lambda x, **kwargs: y / x[0], [self], man_grad=[-y / self.value ** 2])

        def __pow__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x: x[0] ** x[1], [self, y])
            else:
                return derived_observable(lambda x: x[0] ** y, [self])

        def __rpow__(self, y):
            if isinstance(y, Obs):
                return derived_observable(lambda x: x[0] ** x[1], [y, self])
            else:
                return derived_observable(lambda x: y ** x[0], [self])

        def __abs__(self):
            return derived_observable(lambda x: anp.abs(x[0]), [self])

        # Overload numpy functions
        def sqrt(self):
            return derived_observable(lambda x, **kwargs: np.sqrt(x[0]), [self], man_grad=[1 / 2 / np.sqrt(self.value)])

        def log(self):
            return derived_observable(lambda x, **kwargs: np.log(x[0]), [self], man_grad=[1 / self.value])

        def exp(self):
            return derived_observable(lambda x, **kwargs: np.exp(x[0]), [self], man_grad=[np.exp(self.value)])

        def sin(self):
            return derived_observable(lambda x, **kwargs: np.sin(x[0]), [self], man_grad=[np.cos(self.value)])

        def cos(self):
            return derived_observable(lambda x, **kwargs: np.cos(x[0]), [self], man_grad=[-np.sin(self.value)])

        def tan(self):
            return derived_observable(lambda x, **kwargs: np.tan(x[0]), [self], man_grad=[1 / np.cos(self.value) ** 2])

        def arcsin(self):
            return derived_observable(lambda x: anp.arcsin(x[0]), [self])

        def arccos(self):
            return derived_observable(lambda x: anp.arccos(x[0]), [self])

        def arctan(self):
            return derived_observable(lambda x: anp.arctan(x[0]), [self])

        def sinh(self):
            return derived_observable(lambda x, **kwargs: np.sinh(x[0]), [self], man_grad=[np.cosh(self.value)])

        def cosh(self):
            return derived_observable(lambda x, **kwargs: np.cosh(x[0]), [self], man_grad=[np.sinh(self.value)])

        def tanh(self):
            return derived_observable(lambda x, **kwargs: np.tanh(x[0]), [self], man_grad=[1 / np.cosh(self.value) ** 2])

        def arcsinh(self):
            return derived_observable(lambda x: anp.arcsinh(x[0]), [self])

        def arccosh(self):
            return derived_observable(lambda x: anp.arccosh(x[0]), [self])

        def arctanh(self):
            return derived_observable(lambda x: anp.arctanh(x[0]), [self])

        def sinc(self):
            return derived_observable(lambda x: anp.sinc(x[0]), [self])

Class for a general observable.

Instances of Obs are the basic objects of a pyerrors error analysis. They are initialized with a list which contains arrays of samples for different ensembles/replica and another list of same length which contains the names of the ensembles/replica. Mathematical operations can be performed on instances. The result is another instance of Obs. The error of an instance can be computed with the gamma_method. Also contains additional methods for output and visualization of the error calculation.

###### Attributes

- S_global (float): Standard value for S (default 2.0)
- S_dict (dict): Dictionary for S values. If an entry for a given ensemble exists this overwrites the standard value for that ensemble.
- tau_exp_global (float): Standard value for tau_exp (default 0.0)
- tau_exp_dict (dict): Dictionary for tau_exp values. If an entry for a given ensemble exists this overwrites the standard value for that ensemble.
- N_sigma_global (float): Standard value for N_sigma (default 1.0)

Obs(samples, names, idl=None, means=None, **kwargs)
View Source

        def __init__(self, samples, names, idl=None, means=None, **kwargs):
            """ Initialize Obs object.

            Parameters
            ----------
            samples : list
                list of numpy arrays containing the Monte Carlo samples
            names : list
                list of strings labeling the indivdual samples
            idl : list, optional
                list of ranges or lists on which the samples are defined
            means : list, optional
                list of mean values for the case that the mean values were
                already subtracted from the samples
            """

            if means is None:
                if len(samples) != len(names):
                    raise Exception('Length of samples and names incompatible.')
                if len(names) != len(set(names)):
                    raise Exception('Names are not unique.')
                if not all(isinstance(x, str) for x in names):
                    raise TypeError('All names have to be strings.')
                if min(len(x) for x in samples) <= 4:
                    raise Exception('Samples have to have at least 4 entries.')

            self.names = sorted(names)
            self.shape = {}
            self.r_values = {}
            self.deltas = {}

            self.idl = {}
            if idl is not None:
                for name, idx in sorted(zip(names, idl)):
                    if isinstance(idx, range):
                        self.idl[name] = idx
                    elif isinstance(idx, (list, np.ndarray)):
                        dc = np.unique(np.diff(idx))
                        if np.any(dc < 0):
                            raise Exception("Unsorted idx for idl[%s]" % (name))
                        if len(dc) == 1:
                            self.idl[name] = range(idx[0], idx[-1] + dc[0], dc[0])
                        else:
                            self.idl[name] = list(idx)
                    else:
                        raise Exception('incompatible type for idl[%s].' % (name))
            else:
                for name, sample in sorted(zip(names, samples)):
                    self.idl[name] = range(1, len(sample) + 1)

            if means is not None:
                for name, sample, mean in sorted(zip(names, samples, means)):
                    self.shape[name] = len(self.idl[name])
                    if len(sample) != self.shape[name]:
                        raise Exception('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name]))
                    self.r_values[name] = mean
                    self.deltas[name] = sample
            else:
                for name, sample in sorted(zip(names, samples)):
                    self.shape[name] = len(self.idl[name])
                    if len(sample) != self.shape[name]:
                        raise Exception('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name]))
                    self.r_values[name] = np.mean(sample)
                    self.deltas[name] = sample - self.r_values[name]
            self.is_merged = False
            self.N = sum(list(self.shape.values()))

            self._value = 0
            if means is None:
                for name in self.names:
                    self._value += self.shape[name] * self.r_values[name]
                self._value /= self.N

            self._dvalue = 0.0
            self.ddvalue = 0.0
            self.reweighted = False

            self.tag = None

Initialize Obs object.

###### Parameters

- samples (list): list of numpy arrays containing the Monte Carlo samples
- names (list): list of strings labeling the indivdual samples
- idl (list, optional): list of ranges or lists on which the samples are defined
- means (list, optional): list of mean values for the case that the mean values were already subtracted from the samples

S_global = 2.0
S_dict = {}
tau_exp_global = 0.0
tau_exp_dict = {}
N_sigma_global = 1.0
filter_eps = 1e-10
names
shape
r_values
deltas
idl
is_merged
N
ddvalue
reweighted
tag
value
dvalue
e_names
e_content
def gamma_method(self, **kwargs):
View Source

        def gamma_method(self, **kwargs):
            """Calculate the error and related properties of the Obs.

            Parameters
            ----------
            S : float
                specifies a custom value for the parameter S (default 2.0), can be
                a float or an array of floats for different ensembles
            tau_exp : float
                positive value triggers the critical slowing down analysis
                (default 0.0), can be a float or an array of floats for different
                ensembles
            N_sigma : float
                number of standard deviations from zero until the tail is
                attached to the autocorrelation function (default 1)
            fft : bool
                determines whether the fft algorithm is used for the computation
                of the autocorrelation function (default True)
            """

            e_content = self.e_content
            self.e_dvalue = {}
            self.e_ddvalue = {}
            self.e_tauint = {}
            self.e_dtauint = {}
            self.e_windowsize = {}
            self.e_n_tauint = {}
            self.e_n_dtauint = {}
            e_gamma = {}
            self.e_rho = {}
            self.e_drho = {}
            self._dvalue = 0
            self.ddvalue = 0

            self.S = {}
            self.tau_exp = {}

            if kwargs.get('fft') is False:
                fft = False
            else:
                fft = True

            if 'S' in kwargs:
                tmp = kwargs.get('S')
                if isinstance(tmp, list):
                    if len(tmp) != len(self.e_names):
                        raise Exception('Length of S array does not match ensembles.')
                    for e, e_name in enumerate(self.e_names):
                        if tmp[e] <= 0:
                            raise Exception('S has to be larger than 0.')
                        self.S[e_name] = tmp[e]
                else:
                    if isinstance(tmp, (int, float)):
                        if tmp <= 0:
                            raise Exception('S has to be larger than 0.')
                        for e, e_name in enumerate(self.e_names):
                            self.S[e_name] = tmp
                    else:
                        raise TypeError('S is not in proper format.')
            else:
                for e, e_name in enumerate(self.e_names):
                    if e_name in Obs.S_dict:
                        self.S[e_name] = Obs.S_dict[e_name]
                    else:
                        self.S[e_name] = Obs.S_global

            if 'tau_exp' in kwargs:
                tmp = kwargs.get('tau_exp')
                if isinstance(tmp, list):
                    if len(tmp) != len(self.e_names):
                        raise Exception('Length of tau_exp array does not match ensembles.')
                    for e, e_name in enumerate(self.e_names):
                        if tmp[e] < 0:
                            raise Exception('tau_exp smaller than 0.')
                        self.tau_exp[e_name] = tmp[e]
                else:
                    if isinstance(tmp, (int, float)):
                        if tmp < 0:
                            raise Exception('tau_exp smaller than 0.')
                        for e, e_name in enumerate(self.e_names):
                            self.tau_exp[e_name] = tmp
                    else:
                        raise TypeError('tau_exp is not in proper format.')
            else:
                for e, e_name in enumerate(self.e_names):
                    if e_name in Obs.tau_exp_dict:
                        self.tau_exp[e_name] = Obs.tau_exp_dict[e_name]
                    else:
                        self.tau_exp[e_name] = Obs.tau_exp_global

            if 'N_sigma' in kwargs:
                self.N_sigma = kwargs.get('N_sigma')
                if not isinstance(self.N_sigma, (int, float)):
                    raise TypeError('N_sigma is not a number.')
            else:
                self.N_sigma = Obs.N_sigma_global

            for e, e_name in enumerate(self.e_names):

                r_length = []
                for r_name in e_content[e_name]:
                    if isinstance(self.idl[r_name], range):
                        r_length.append(len(self.idl[r_name]))
                    else:
                        r_length.append((self.idl[r_name][-1] - self.idl[r_name][0] + 1))

                e_N = np.sum([self.shape[r_name] for r_name in e_content[e_name]])
                w_max = max(r_length) // 2
                e_gamma[e_name] = np.zeros(w_max)
                self.e_rho[e_name] = np.zeros(w_max)
                self.e_drho[e_name] = np.zeros(w_max)

                for r_name in e_content[e_name]:
                    e_gamma[e_name] += self._calc_gamma(self.deltas[r_name], self.idl[r_name], self.shape[r_name], w_max, fft)

                gamma_div = np.zeros(w_max)
                for r_name in e_content[e_name]:
                    gamma_div += self._calc_gamma(np.ones((self.shape[r_name])), self.idl[r_name], self.shape[r_name], w_max, fft)
                e_gamma[e_name] /= gamma_div[:w_max]

                if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny:  # Prevent division by zero
                    self.e_tauint[e_name] = 0.5
                    self.e_dtauint[e_name] = 0.0
                    self.e_dvalue[e_name] = 0.0
                    self.e_ddvalue[e_name] = 0.0
                    self.e_windowsize[e_name] = 0
                    continue

                self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0]
                self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:])))
                # Make sure no entry of tauint is smaller than 0.5
                self.e_n_tauint[e_name][self.e_n_tauint[e_name] <= 0.5] = 0.5 + np.finfo(np.float64).eps
                # hep-lat/0306017 eq. (42)
                self.e_n_dtauint[e_name] = self.e_n_tauint[e_name] * 2 * np.sqrt(np.abs(np.arange(w_max) + 0.5 - self.e_n_tauint[e_name]) / e_N)
                self.e_n_dtauint[e_name][0] = 0.0

                def _compute_drho(i):
                    tmp = self.e_rho[e_name][i + 1:w_max] + np.concatenate([self.e_rho[e_name][i - 1::-1], self.e_rho[e_name][1:w_max - 2 * i]]) - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i]
                    self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N)

                _compute_drho(1)
                if self.tau_exp[e_name] > 0:
                    texp = self.tau_exp[e_name]
                    # if type(self.idl[e_name]) is range: # scale tau_exp according to step size
                    #    texp /= self.idl[e_name].step
                    # Critical slowing down analysis
                    for n in range(1, w_max // 2):
                        _compute_drho(n + 1)
                        if (self.e_rho[e_name][n] - self.N_sigma * self.e_drho[e_name][n]) < 0 or n >= w_max // 2 - 2:
                            # Bias correction hep-lat/0306017 eq. (49) included
                            self.e_tauint[e_name] = self.e_n_tauint[e_name][n] * (1 + (2 * n + 1) / e_N) / (1 + 1 / e_N) + texp * np.abs(self.e_rho[e_name][n + 1])  # The absolute makes sure, that the tail contribution is always positive
                            self.e_dtauint[e_name] = np.sqrt(self.e_n_dtauint[e_name][n] ** 2 + texp ** 2 * self.e_drho[e_name][n + 1] ** 2)
                            # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2
                            self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
                            self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
                            self.e_windowsize[e_name] = n
                            break
                else:
                    # Standard automatic windowing procedure
                    g_w = self.S[e_name] / np.log((2 * self.e_n_tauint[e_name][1:] + 1) / (2 * self.e_n_tauint[e_name][1:] - 1))
                    g_w = np.exp(- np.arange(1, w_max) / g_w) - g_w / np.sqrt(np.arange(1, w_max) * e_N)
                    for n in range(1, w_max):
                        if n < w_max // 2 - 2:
                            _compute_drho(n + 1)
                        if g_w[n - 1] < 0 or n >= w_max - 1:
                            self.e_tauint[e_name] = self.e_n_tauint[e_name][n] * (1 + (2 * n + 1) / e_N) / (1 + 1 / e_N)  # Bias correction hep-lat/0306017 eq. (49)
                            self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n]
                            self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
                            self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
                            self.e_windowsize[e_name] = n
                            break

                self._dvalue += self.e_dvalue[e_name] ** 2
                self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2

            self._dvalue = np.sqrt(self.dvalue)
            if self._dvalue == 0.0:
                self.ddvalue = 0.0
            else:
                self.ddvalue = np.sqrt(self.ddvalue) / self.dvalue
            return

Calculate the error and related properties of the Obs.

###### Parameters

- S (float): specifies a custom value for the parameter S (default 2.0), can be a float or an array of floats for different ensembles
- tau_exp (float): positive value triggers the critical slowing down analysis (default 0.0), can be a float or an array of floats for different ensembles
- N_sigma (float): number of standard deviations from zero until the tail is attached to the autocorrelation function (default 1)
- fft (bool): determines whether the fft algorithm is used for the computation of the autocorrelation function (default True)

def print(self, level=1):
View Source

        def print(self, level=1):
            warnings.warn("Method 'print' renamed to 'details'", DeprecationWarning)
            self.details(level > 1)

def details(self, ens_content=True):
View Source

        def details(self, ens_content=True):
            """Output detailed properties of the Obs."""
            if self.value == 0.0:
                percentage = np.nan
            else:
                percentage = np.abs(self.dvalue / self.value) * 100
            print('Result\t %3.8e +/- %3.8e +/- %3.8e (%3.3f%%)' % (self.value, self.dvalue, self.ddvalue, percentage))
            if hasattr(self, 'e_dvalue'):
                if len(self.e_names) > 1:
                    print(' Ensemble errors:')
                for e_name in self.e_names:
                    if len(self.e_names) > 1:
                        print('', e_name, '\t %3.8e +/- %3.8e' % (self.e_dvalue[e_name], self.e_ddvalue[e_name]))
                    if self.tau_exp[e_name] > 0:
                        print(' t_int\t %3.8e +/- %3.8e tau_exp = %3.2f,  N_sigma = %1.0i' % (self.e_tauint[e_name], self.e_dtauint[e_name], self.tau_exp[e_name], self.N_sigma))
                    else:
                        print(' t_int\t %3.8e +/- %3.8e S = %3.2f' % (self.e_tauint[e_name], self.e_dtauint[e_name], self.S[e_name]))
            if self.tag is not None:
                print("Description:", self.tag)
            if ens_content is True:
                if len(self.e_names) == 1:
                    print(self.N, 'samples in', len(self.e_names), 'ensemble:')
                else:
                    print(self.N, 'samples in', len(self.e_names), 'ensembles:')
                m = max(map(len, list(self.e_content.keys()))) + 1
                print('\n'.join(['  ' + key.rjust(m) + ': ' + str(value) for key, value in sorted(self.e_content.items())]))

Output detailed properties of the Obs.

def is_zero_within_error(self, sigma=1):
View Source

        def is_zero_within_error(self, sigma=1):
            """Checks whether the observable is zero within 'sigma' standard errors.

            Works only properly when the gamma method was run.
            """
            return self.is_zero() or np.abs(self.value) <= sigma * self.dvalue

Checks whether the observable is zero within 'sigma' standard errors.

Works only properly when the gamma method was run.

def is_zero(self):
View Source

        def is_zero(self):
            """Checks whether the observable is zero within machine precision."""
            return np.isclose(0.0, self.value) and all(np.allclose(0.0, delta) for delta in self.deltas.values())

Checks whether the observable is zero within machine precision.

def plot_tauint(self, save=None):
View Source

        def plot_tauint(self, save=None):
            """Plot integrated autocorrelation time for each ensemble."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')

            fig = plt.figure()
            for e, e_name in enumerate(self.e_names):
                plt.xlabel(r'$W$')
                plt.ylabel(r'$\tau_\mathrm{int}$')
                length = int(len(self.e_n_tauint[e_name]))
                if self.tau_exp[e_name] > 0:
                    base = self.e_n_tauint[e_name][self.e_windowsize[e_name]]
                    x_help = np.arange(2 * self.tau_exp[e_name])
                    y_help = (x_help + 1) * np.abs(self.e_rho[e_name][self.e_windowsize[e_name] + 1]) * (1 - x_help / (2 * (2 * self.tau_exp[e_name] - 1))) + base
                    x_arr = np.arange(self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name])
                    plt.plot(x_arr, y_help, 'C' + str(e), linewidth=1, ls='--', marker=',')
                    plt.errorbar([self.e_windowsize[e_name] + 2 * self.tau_exp[e_name]], [self.e_tauint[e_name]],
                                 yerr=[self.e_dtauint[e_name]], fmt='C' + str(e), linewidth=1, capsize=2, marker='o', mfc=plt.rcParams['axes.facecolor'])
                    xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
                    label = e_name + r', $\tau_\mathrm{exp}$=' + str(np.around(self.tau_exp[e_name], decimals=2))
                else:
                    label = e_name + ', S=' + str(np.around(self.S[e_name], decimals=2))
                    xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)

                plt.errorbar(np.arange(length), self.e_n_tauint[e_name][:], yerr=self.e_n_dtauint[e_name][:], linewidth=1, capsize=2, label=label)
                plt.axvline(x=self.e_windowsize[e_name], color='C' + str(e), alpha=0.5, marker=',', ls='--')
                plt.legend()
                plt.xlim(-0.5, xmax)
                plt.ylim(bottom=0.0)
                plt.draw()
                if save:
                    fig.savefig(save)

Plot integrated autocorrelation time for each ensemble.

def plot_rho(self):
View Source

        def plot_rho(self):
            """Plot normalized autocorrelation function time for each ensemble."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')
            for e, e_name in enumerate(self.e_names):
                plt.xlabel('W')
                plt.ylabel('rho')
                length = int(len(self.e_drho[e_name]))
                plt.errorbar(np.arange(length), self.e_rho[e_name][:length], yerr=self.e_drho[e_name][:], linewidth=1, capsize=2)
                plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25, ls='--', marker=',')
                if self.tau_exp[e_name] > 0:
                    plt.plot([self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]],
                             [self.e_rho[e_name][self.e_windowsize[e_name] + 1], 0], 'k-', lw=1)
                    xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
                    plt.title('Rho ' + e_name + r', tau\_exp=' + str(np.around(self.tau_exp[e_name], decimals=2)))
                else:
                    xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)
                    plt.title('Rho ' + e_name + ', S=' + str(np.around(self.S[e_name], decimals=2)))
                plt.plot([-0.5, xmax], [0, 0], 'k--', lw=1)
                plt.xlim(-0.5, xmax)
                plt.draw()

Plot normalized autocorrelation function time for each ensemble.

def plot_rep_dist(self):
View Source

        def plot_rep_dist(self):
            """Plot replica distribution for each ensemble with more than one replicum."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')
            for e, e_name in enumerate(self.e_names):
                if len(self.e_content[e_name]) == 1:
                    print('No replica distribution for a single replicum (', e_name, ')')
                    continue
                r_length = []
                sub_r_mean = 0
                for r, r_name in enumerate(self.e_content[e_name]):
                    r_length.append(len(self.deltas[r_name]))
                    sub_r_mean += self.shape[r_name] * self.r_values[r_name]
                e_N = np.sum(r_length)
                sub_r_mean /= e_N
                arr = np.zeros(len(self.e_content[e_name]))
                for r, r_name in enumerate(self.e_content[e_name]):
                    arr[r] = (self.r_values[r_name] - sub_r_mean) / (self.e_dvalue[e_name] * np.sqrt(e_N / self.shape[r_name] - 1))
                plt.hist(arr, rwidth=0.8, bins=len(self.e_content[e_name]))
                plt.title('Replica distribution' + e_name + ' (mean=0, var=1)')
                plt.draw()

Plot replica distribution for each ensemble with more than one replicum.

def plot_history(self, expand=True):
View Source

        def plot_history(self, expand=True):
            """Plot derived Monte Carlo history for each ensemble."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')

            for e, e_name in enumerate(self.e_names):
                plt.figure()
                r_length = []
                tmp = []
                for r, r_name in enumerate(self.e_content[e_name]):
                    if expand:
                        tmp.append(_expand_deltas(self.deltas[r_name], self.idl[r_name], self.shape[r_name]) + self.r_values[r_name])
                    else:
                        tmp.append(self.deltas[r_name] + self.r_values[r_name])
                    r_length.append(len(tmp[-1]))
                e_N = np.sum(r_length)
                x = np.arange(e_N)
                y = np.concatenate(tmp, axis=0)
                plt.errorbar(x, y, fmt='.', markersize=3)
                plt.xlim(-0.5, e_N - 0.5)
                plt.title(e_name)
                plt.draw()

Plot derived Monte Carlo history for each ensemble.

def plot_piechart(self):
View Source

        def plot_piechart(self):
            """Plot piechart which shows the fractional contribution of each
            ensemble to the error and returns a dictionary containing the fractions."""
            if not hasattr(self, 'e_names'):
                raise Exception('Run the gamma method first.')
            if self.dvalue == 0.0:
                raise Exception('Error is 0.0')
            labels = self.e_names
            sizes = [i ** 2 for i in list(self.e_dvalue.values())] / self.dvalue ** 2
            fig1, ax1 = plt.subplots()
            ax1.pie(sizes, labels=labels, startangle=90, normalize=True)
            ax1.axis('equal')
            plt.draw()

            return dict(zip(self.e_names, sizes))

Plot piechart which shows the fractional contribution of each ensemble to the error and returns a dictionary containing the fractions.

def dump(self, name, **kwargs):
View Source

        def dump(self, name, **kwargs):
            """Dump the Obs to a pickle file 'name'.

            Parameters
            ----------
            path : str
                specifies a custom path for the file (default '.')
            """
            if 'path' in kwargs:
                file_name = kwargs.get('path') + '/' + name + '.p'
            else:
                file_name = name + '.p'
            with open(file_name, 'wb') as fb:
                pickle.dump(self, fb)

Dump the Obs to a pickle file 'name'.

###### Parameters

- path (str): specifies a custom path for the file (default '.')

def sqrt(self):
View Source

        def sqrt(self):
            return derived_observable(lambda x, **kwargs: np.sqrt(x[0]), [self], man_grad=[1 / 2 / np.sqrt(self.value)])

def log(self):
View Source

        def log(self):
            return derived_observable(lambda x, **kwargs: np.log(x[0]), [self], man_grad=[1 / self.value])

def exp(self):
View Source

        def exp(self):
            return derived_observable(lambda x, **kwargs: np.exp(x[0]), [self], man_grad=[np.exp(self.value)])

def sin(self):
View Source

        def sin(self):
            return derived_observable(lambda x, **kwargs: np.sin(x[0]), [self], man_grad=[np.cos(self.value)])

def cos(self):
View Source

        def cos(self):
            return derived_observable(lambda x, **kwargs: np.cos(x[0]), [self], man_grad=[-np.sin(self.value)])

def tan(self):
View Source

        def tan(self):
            return derived_observable(lambda x, **kwargs: np.tan(x[0]), [self], man_grad=[1 / np.cos(self.value) ** 2])

def arcsin(self):
View Source

        def arcsin(self):
            return derived_observable(lambda x: anp.arcsin(x[0]), [self])

def arccos(self):
View Source

        def arccos(self):
            return derived_observable(lambda x: anp.arccos(x[0]), [self])

def arctan(self):
View Source

        def arctan(self):
            return derived_observable(lambda x: anp.arctan(x[0]), [self])

def sinh(self):
View Source

        def sinh(self):
            return derived_observable(lambda x, **kwargs: np.sinh(x[0]), [self], man_grad=[np.cosh(self.value)])

def cosh(self):
View Source

        def cosh(self):
            return derived_observable(lambda x, **kwargs: np.cosh(x[0]), [self], man_grad=[np.sinh(self.value)])

def tanh(self):
View Source

        def tanh(self):
            return derived_observable(lambda x, **kwargs: np.tanh(x[0]), [self], man_grad=[1 / np.cosh(self.value) ** 2])

def arcsinh(self):
View Source

        def arcsinh(self):
            return derived_observable(lambda x: anp.arcsinh(x[0]), [self])

def arccosh(self):
View Source

        def arccosh(self):
            return derived_observable(lambda x: anp.arccosh(x[0]), [self])

def arctanh(self):
View Source

        def arctanh(self):
            return derived_observable(lambda x: anp.arctanh(x[0]), [self])

def sinc(self):
View Source

        def sinc(self):
            return derived_observable(lambda x: anp.sinc(x[0]), [self])

N_sigma
S
e_ddvalue
e_drho
e_dtauint
e_dvalue
e_n_dtauint
e_n_tauint
e_rho
e_tauint
e_windowsize
tau_exp
class CObs:
View Source

    class CObs:
        """Class for a complex valued observable."""
        __slots__ = ['_real', '_imag', 'tag']

        def __init__(self, real, imag=0.0):
            self._real = real
            self._imag = imag
            self.tag = None

        @property
        def real(self):
            return self._real

        @property
        def imag(self):
            return self._imag

        def gamma_method(self, **kwargs):
            """Executes the gamma_method for the real and the imaginary part."""
            if isinstance(self.real, Obs):
                self.real.gamma_method(**kwargs)
            if isinstance(self.imag, Obs):
                self.imag.gamma_method(**kwargs)

        def is_zero(self):
            """Checks whether both real and imaginary part are zero within machine precision."""
            return self.real == 0.0 and self.imag == 0.0

        def conjugate(self):
            return CObs(self.real, -self.imag)

        def __add__(self, other):
            if isinstance(other, np.ndarray):
                return other + self
            elif hasattr(other, 'real') and hasattr(other, 'imag'):
                return CObs(self.real + other.real,
                            self.imag + other.imag)
            else:
                return CObs(self.real + other, self.imag)

        def __radd__(self, y):
            return self + y

        def __sub__(self, other):
            if isinstance(other, np.ndarray):
                return -1 * (other - self)
            elif hasattr(other, 'real') and hasattr(other, 'imag'):
                return CObs(self.real - other.real, self.imag - other.imag)
            else:
                return CObs(self.real - other, self.imag)

        def __rsub__(self, other):
            return -1 * (self - other)

        def __mul__(self, other):
            if isinstance(other, np.ndarray):
                return other * self
            elif hasattr(other, 'real') and hasattr(other, 'imag'):
                if all(isinstance(i, Obs) for i in [self.real, self.imag, other.real, other.imag]):
                    return CObs(derived_observable(lambda x, **kwargs: x[0] * x[1] - x[2] * x[3],
                                                   [self.real, other.real, self.imag, other.imag],
                                                   man_grad=[other.real.value, self.real.value, -other.imag.value, -self.imag.value]),
                                derived_observable(lambda x, **kwargs: x[2] * x[1] + x[0] * x[3],
                                                   [self.real, other.real, self.imag, other.imag],
                                                   man_grad=[other.imag.value, self.imag.value, other.real.value, self.real.value]))
                elif getattr(other, 'imag', 0) != 0:
                    return CObs(self.real * other.real - self.imag * other.imag,
                                self.imag * other.real + self.real * other.imag)
                else:
                    return CObs(self.real * other.real, self.imag * other.real)
            else:
                return CObs(self.real * other, self.imag * other)

        def __rmul__(self, other):
            return self * other

        def __truediv__(self, other):
            if isinstance(other, np.ndarray):
                return 1 / (other / self)
            elif hasattr(other, 'real') and hasattr(other, 'imag'):
                r = other.real ** 2 + other.imag ** 2
                return CObs((self.real * other.real + self.imag * other.imag) / r, (self.imag * other.real - self.real * other.imag) / r)
            else:
                return CObs(self.real / other, self.imag / other)

        def __rtruediv__(self, other):
            r = self.real ** 2 + self.imag ** 2
            if hasattr(other, 'real') and hasattr(other, 'imag'):
                return CObs((self.real * other.real + self.imag * other.imag) / r, (self.real * other.imag - self.imag * other.real) / r)
            else:
                return CObs(self.real * other / r, -self.imag * other / r)

        def __abs__(self):
            return np.sqrt(self.real**2 + self.imag**2)

        def __neg__(other):
            return -1 * other

        def __eq__(self, other):
            return self.real == other.real and self.imag == other.imag

        def __str__(self):
            return '(' + str(self.real) + int(self.imag >= 0.0) * '+' + str(self.imag) + 'j)'

        def __repr__(self):
            return 'CObs[' + str(self) + ']'

Class for a complex valued observable.

CObs(real, imag=0.0)
View Source

        def __init__(self, real, imag=0.0):
            self._real = real
            self._imag = imag
            self.tag = None

tag
real
imag
def gamma_method(self, **kwargs):
View Source

        def gamma_method(self, **kwargs):
            """Executes the gamma_method for the real and the imaginary part."""
            if isinstance(self.real, Obs):
                self.real.gamma_method(**kwargs)
            if isinstance(self.imag, Obs):
                self.imag.gamma_method(**kwargs)

Executes the gamma_method for the real and the imaginary part.

def is_zero(self):
View Source

        def is_zero(self):
            """Checks whether both real and imaginary part are zero within machine precision."""
            return self.real == 0.0 and self.imag == 0.0

Checks whether both real and imaginary part are zero within machine precision.

def conjugate(self):
View Source

        def conjugate(self):
            return CObs(self.real, -self.imag)

def derived_observable(func, data, **kwargs):
View Source

    def derived_observable(func, data, **kwargs):
        """Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.

        Parameters
        ----------
        func : object
            arbitrary function of the form func(data, **kwargs). For the
            automatic differentiation to work, all numpy functions have to have
            the autograd wrapper (use 'import autograd.numpy as anp').
        data : list
            list of Obs, e.g. [obs1, obs2, obs3].
        num_grad : bool
            if True, numerical derivatives are used instead of autograd
            (default False). To control the numerical differentiation the
            kwargs of numdifftools.step_generators.MaxStepGenerator
            can be used.
        man_grad : list
            manually supply a list or an array which contains the jacobian
            of func. Use cautiously, supplying the wrong derivative will
            not be intercepted.

        Notes
        -----
        For simple mathematical operations it can be practical to use anonymous
        functions. For the ratio of two observables one can e.g. use

        new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])
        """

        data = np.asarray(data)
        raveled_data = data.ravel()

        # Workaround for matrix operations containing non Obs data
        for i_data in raveled_data:
            if isinstance(i_data, Obs):
                first_name = i_data.names[0]
                first_shape = i_data.shape[first_name]
                first_idl = i_data.idl[first_name]
                break

        for i in range(len(raveled_data)):
            if isinstance(raveled_data[i], (int, float)):
                raveled_data[i] = Obs([raveled_data[i] + np.zeros(first_shape)], [first_name], idl=[first_idl])

        n_obs = len(raveled_data)
        new_names = sorted(set([y for x in [o.names for o in raveled_data] for y in x]))

        is_merged = len(list(filter(lambda o: o.is_merged is True, raveled_data))) > 0
        reweighted = len(list(filter(lambda o: o.reweighted is True, raveled_data))) > 0
        new_idl_d = {}
        for name in new_names:
            idl = []
            for i_data in raveled_data:
                tmp = i_data.idl.get(name)
                if tmp is not None:
                    idl.append(tmp)
            new_idl_d[name] = _merge_idx(idl)
            if not is_merged:
                is_merged = (1 != len(set([len(idx) for idx in [*idl, new_idl_d[name]]])))

        if data.ndim == 1:
            values = np.array([o.value for o in data])
        else:
            values = np.vectorize(lambda x: x.value)(data)

        new_values = func(values, **kwargs)

        multi = 0
        if isinstance(new_values, np.ndarray):
            multi = 1

        new_r_values = {}
        for name in new_names:
            tmp_values = np.zeros(n_obs)
            for i, item in enumerate(raveled_data):
                tmp = item.r_values.get(name)
                if tmp is None:
                    tmp = item.value
                tmp_values[i] = tmp
            if multi > 0:
                tmp_values = np.array(tmp_values).reshape(data.shape)
            new_r_values[name] = func(tmp_values, **kwargs)

        if 'man_grad' in kwargs:
            deriv = np.asarray(kwargs.get('man_grad'))
            if new_values.shape + data.shape != deriv.shape:
                raise Exception('Manual derivative does not have correct shape.')
        elif kwargs.get('num_grad') is True:
            if multi > 0:
                raise Exception('Multi mode currently not supported for numerical derivative')
            options = {
                'base_step': 0.1,
                'step_ratio': 2.5,
                'num_steps': None,
                'step_nom': None,
                'offset': None,
                'num_extrap': None,
                'use_exact_steps': None,
                'check_num_steps': None,
                'scale': None}
            for key in options.keys():
                kwarg = kwargs.get(key)
                if kwarg is not None:
                    options[key] = kwarg
            tmp_df = nd.Gradient(func, order=4, **{k: v for k, v in options.items() if v is not None})(values, **kwargs)
            if tmp_df.size == 1:
                deriv = np.array([tmp_df.real])
            else:
                deriv = tmp_df.real
        else:
            deriv = jacobian(func)(values, **kwargs)

        final_result = np.zeros(new_values.shape, dtype=object)

        for i_val, new_val in np.ndenumerate(new_values):
            new_deltas = {}
            for j_obs, obs in np.ndenumerate(data):
                for name in obs.names:
                    new_deltas[name] = new_deltas.get(name, 0) + deriv[i_val + j_obs] * _expand_deltas_for_merge(obs.deltas[name], obs.idl[name], obs.shape[name], new_idl_d[name])

            new_samples = []
            new_means = []
            new_idl = []
            if is_merged:
                filtered_names, filtered_deltas, filtered_idl_d = _filter_zeroes(new_names, new_deltas, new_idl_d)
            else:
                filtered_names = new_names
                filtered_deltas = new_deltas
                filtered_idl_d = new_idl_d
            for name in filtered_names:
                new_samples.append(filtered_deltas[name])
                new_means.append(new_r_values[name][i_val])
                new_idl.append(filtered_idl_d[name])
            final_result[i_val] = Obs(new_samples, filtered_names, means=new_means, idl=new_idl)
            final_result[i_val]._value = new_val
            final_result[i_val].is_merged = is_merged
            final_result[i_val].reweighted = reweighted

        if multi == 0:
            final_result = final_result.item()

        return final_result

Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.

###### Parameters

- func (object): arbitrary function of the form func(data, **kwargs). For the automatic differentiation to work, all numpy functions have to have the autograd wrapper (use 'import autograd.numpy as anp').
- data (list): list of Obs, e.g. [obs1, obs2, obs3].
- num_grad (bool): if True, numerical derivatives are used instead of autograd (default False). To control the numerical differentiation the kwargs of numdifftools.step_generators.MaxStepGenerator can be used.
- man_grad (list): manually supply a list or an array which contains the jacobian of func. Use cautiously, supplying the wrong derivative will not be intercepted.

###### Notes

For simple mathematical operations it can be practical to use anonymous functions. For the ratio of two observables one can e.g. use

new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])

def reweight(weight, obs, **kwargs):
View Source

    def reweight(weight, obs, **kwargs):
        """Reweight a list of observables.

        Parameters
        ----------
        weight : Obs
            Reweighting factor. An Observable that has to be defined on a superset of the
            configurations in obs[i].idl for all i.
        obs : list
            list of Obs, e.g. [obs1, obs2, obs3].
        all_configs : bool
            if True, the reweighted observables are normalized by the average of
            the reweighting factor on all configurations in weight.idl and not
            on the configurations in obs[i].idl.
        """
        result = []
        for i in range(len(obs)):
            if sorted(weight.names) != sorted(obs[i].names):
                raise Exception('Error: Ensembles do not fit')
            for name in weight.names:
                if not set(obs[i].idl[name]).issubset(weight.idl[name]):
                    raise Exception('obs[%d] has to be defined on a subset of the configs in weight.idl[%s]!' % (i, name))
            new_samples = []
            w_deltas = {}
            for name in sorted(weight.names):
                w_deltas[name] = _reduce_deltas(weight.deltas[name], weight.idl[name], obs[i].idl[name])
                new_samples.append((w_deltas[name] + weight.r_values[name]) * (obs[i].deltas[name] + obs[i].r_values[name]))
            tmp_obs = Obs(new_samples, sorted(weight.names), idl=[obs[i].idl[name] for name in sorted(weight.names)])

            if kwargs.get('all_configs'):
                new_weight = weight
            else:
                new_weight = Obs([w_deltas[name] + weight.r_values[name] for name in sorted(weight.names)], sorted(weight.names), idl=[obs[i].idl[name] for name in sorted(weight.names)])

            result.append(derived_observable(lambda x, **kwargs: x[0] / x[1], [tmp_obs, new_weight], **kwargs))
            result[-1].reweighted = True
            result[-1].is_merged = obs[i].is_merged

        return result

Reweight a list of observables.

###### Parameters

- weight (Obs): Reweighting factor. An Observable that has to be defined on a superset of the configurations in obs[i].idl for all i.
- obs (list): list of Obs, e.g. [obs1, obs2, obs3].
- all_configs (bool): if True, the reweighted observables are normalized by the average of the reweighting factor on all configurations in weight.idl and not on the configurations in obs[i].idl.

def correlate(obs_a, obs_b):
View Source

    def correlate(obs_a, obs_b):
        """Correlate two observables.

        Parameters
        ----------
        obs_a : Obs
            First observable
        obs_b : Obs
            Second observable

        Keep in mind to only correlate primary observables which have not been reweighted
        yet. The reweighting has to be applied after correlating the observables.
        Currently only works if ensembles are identical. This is not really necessary.
        """

        if sorted(obs_a.names) != sorted(obs_b.names):
            raise Exception('Ensembles do not fit')
        for name in obs_a.names:
            if obs_a.shape[name] != obs_b.shape[name]:
                raise Exception('Shapes of ensemble', name, 'do not fit')
            if obs_a.idl[name] != obs_b.idl[name]:
                raise Exception('idl of ensemble', name, 'do not fit')

        if obs_a.reweighted is True:
            warnings.warn("The first observable is already reweighted.", RuntimeWarning)
        if obs_b.reweighted is True:
            warnings.warn("The second observable is already reweighted.", RuntimeWarning)

        new_samples = []
        new_idl = []
        for name in sorted(obs_a.names):
            new_samples.append((obs_a.deltas[name] + obs_a.r_values[name]) * (obs_b.deltas[name] + obs_b.r_values[name]))
            new_idl.append(obs_a.idl[name])

        o = Obs(new_samples, sorted(obs_a.names), idl=new_idl)
        o.is_merged = obs_a.is_merged or obs_b.is_merged
        o.reweighted = obs_a.reweighted or obs_b.reweighted
        return o

Correlate two observables.

###### Parameters

- obs_a (Obs): First observable
- obs_b (Obs): Second observable
- Keep in mind to only correlate primary observables which have not been reweighted
- yet. The reweighting has to be applied after correlating the observables.
- Currently only works if ensembles are identical. This is not really necessary.

def covariance(obs1, obs2, correlation=False, **kwargs):
View Source

    def covariance(obs1, obs2, correlation=False, **kwargs):
        """Calculates the covariance of two observables.

        covariance(obs, obs) is equal to obs.dvalue ** 2
        The gamma method has to be applied first to both observables.

        If abs(covariance(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance
        is constrained to the maximum value in order to make sure that covariance
        matrices are positive semidefinite.

        Parameters
        ----------
        correlation : bool
            if true the correlation instead of the covariance is
            returned (default False)
        """

        for name in sorted(set(obs1.names + obs2.names)):
            if (obs1.shape.get(name) != obs2.shape.get(name)) and (obs1.shape.get(name) is not None) and (obs2.shape.get(name) is not None):
                raise Exception('Shapes of ensemble', name, 'do not fit')
            if (1 != len(set([len(idx) for idx in [obs1.idl[name], obs2.idl[name], _merge_idx([obs1.idl[name], obs2.idl[name]])]]))):
                raise Exception('Shapes of ensemble', name, 'do not fit')

        if not hasattr(obs1, 'e_names') or not hasattr(obs2, 'e_names'):
            raise Exception('The gamma method has to be applied to both Obs first.')

        dvalue = 0

        for e_name in obs1.e_names:

            if e_name not in obs2.e_names:
                continue

            gamma = 0
            r_length = []
            for r_name in obs1.e_content[e_name]:
                if r_name not in obs2.e_content[e_name]:
                    continue

                r_length.append(len(obs1.deltas[r_name]))

                gamma += np.sum(obs1.deltas[r_name] * obs2.deltas[r_name])

            e_N = np.sum(r_length)

            tau_combined = (obs1.e_tauint[e_name] + obs2.e_tauint[e_name]) / 2
            dvalue += gamma / e_N * (1 + 1 / e_N) / e_N * 2 * tau_combined

        if np.abs(dvalue / obs1.dvalue / obs2.dvalue) > 1.0:
            dvalue = np.sign(dvalue) * obs1.dvalue * obs2.dvalue

        if correlation:
            dvalue = dvalue / obs1.dvalue / obs2.dvalue

        return dvalue

Calculates the covariance of two observables.

covariance(obs, obs) is equal to obs.dvalue ** 2 The gamma method has to be applied first to both observables.

If abs(covariance(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance is constrained to the maximum value in order to make sure that covariance matrices are positive semidefinite.

###### Parameters

- correlation (bool): if true the correlation instead of the covariance is returned (default False)

def covariance2(obs1, obs2, correlation=False, **kwargs):
View Source

    def covariance2(obs1, obs2, correlation=False, **kwargs):
        """Alternative implementation of the covariance of two observables.

        covariance(obs, obs) is equal to obs.dvalue ** 2
        The gamma method has to be applied first to both observables.

        If abs(covariance(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance
        is constrained to the maximum value in order to make sure that covariance
        matrices are positive semidefinite.

        Keyword arguments
        -----------------
        correlation -- if true the correlation instead of the covariance is
                       returned (default False)
        """

        def expand_deltas(deltas, idx, shape, new_idx):
            """Expand deltas defined on idx to a contiguous range [new_idx[0], new_idx[-1]].
               New, empy entries are filled by 0. If idx and new_idx are of type range, the smallest
               common divisor of the step sizes is used as new step size.

            Parameters
            ----------
            deltas  -- List of fluctuations
            idx     -- List or range of configs on which the deltas are defined.
                       Has to be a subset of new_idx.
            shape   -- Number of configs in idx.
            new_idx -- List of configs that defines the new range.
            """

            if type(idx) is range and type(new_idx) is range:
                if idx == new_idx:
                    return deltas
            ret = np.zeros(new_idx[-1] - new_idx[0] + 1)
            for i in range(shape):
                ret[idx[i] - new_idx[0]] = deltas[i]
            return ret

        def calc_gamma(deltas1, deltas2, idx1, idx2, new_idx, w_max):
            gamma = np.zeros(w_max)
            deltas1 = expand_deltas(deltas1, idx1, len(idx1), new_idx)
            deltas2 = expand_deltas(deltas2, idx2, len(idx2), new_idx)
            new_shape = len(deltas1)
            max_gamma = min(new_shape, w_max)
            # The padding for the fft has to be even
            padding = new_shape + max_gamma + (new_shape + max_gamma) % 2
            gamma[:max_gamma] += (np.fft.irfft(np.fft.rfft(deltas1, padding) * np.conjugate(np.fft.rfft(deltas2, padding)))[:max_gamma] + np.fft.irfft(np.fft.rfft(deltas2, padding) * np.conjugate(np.fft.rfft(deltas1, padding)))[:max_gamma]) / 2.0

            return gamma

        if not hasattr(obs1, 'e_names') or not hasattr(obs2, 'e_names'):
            raise Exception('The gamma method has to be applied to both Obs first.')

        dvalue = 0
        e_gamma = {}
        e_dvalue = {}
        e_n_tauint = {}
        e_rho = {}

        for e_name in obs1.e_names:

            if e_name not in obs2.e_names:
                continue

            idl_d = {}
            r_length = []
            for r_name in obs1.e_content[e_name]:
                if r_name not in obs2.e_content[e_name]:
                    continue
                idl_d[r_name] = _merge_idx([obs1.idl[r_name], obs2.idl[r_name]])
                if isinstance(idl_d[r_name], range):
                    r_length.append(len(idl_d[r_name]))
                else:
                    r_length.append((idl_d[r_name][-1] - idl_d[r_name][0] + 1))

            if not r_length:
                return 0.

            w_max = max(r_length) // 2
            e_gamma[e_name] = np.zeros(w_max)

            for r_name in obs1.e_content[e_name]:
                if r_name not in obs2.e_content[e_name]:
                    continue
                e_gamma[e_name] += calc_gamma(obs1.deltas[r_name], obs2.deltas[r_name], obs1.idl[r_name], obs2.idl[r_name], idl_d[r_name], w_max)

            if np.all(e_gamma[e_name] == 0.0):
                continue

            e_shapes = []
            for r_name in obs1.e_content[e_name]:
                e_shapes.append(obs1.shape[r_name])
            gamma_div = np.zeros(w_max)
            e_N = 0
            for r_name in obs1.e_content[e_name]:
                if r_name not in obs2.e_content[e_name]:
                    continue
                gamma_div += calc_gamma(np.ones(obs1.shape[r_name]), np.ones(obs2.shape[r_name]), obs1.idl[r_name], obs2.idl[r_name], idl_d[r_name], w_max)
                e_N += np.sum(np.ones_like(idl_d[r_name]))
            e_gamma[e_name] /= gamma_div[:w_max]

            e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0]
            e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], e_rho[e_name][1:])))
            # Make sure no entry of tauint is smaller than 0.5
            e_n_tauint[e_name][e_n_tauint[e_name] < 0.5] = 0.500000000001

            window = max(obs1.e_windowsize[e_name], obs2.e_windowsize[e_name])
            # Bias correction hep-lat/0306017 eq. (49)
            e_dvalue[e_name] = 2 * (e_n_tauint[e_name][window] + obs1.tau_exp[e_name] * np.abs(e_rho[e_name][window + 1])) * (1 + (2 * window + 1) / e_N) * e_gamma[e_name][0] / e_N

            dvalue += e_dvalue[e_name]

        if np.abs(dvalue / obs1.dvalue / obs2.dvalue) > 1.0:
            dvalue = np.sign(dvalue) * obs1.dvalue * obs2.dvalue

        if correlation:
            dvalue = dvalue / obs1.dvalue / obs2.dvalue

        return dvalue

Alternative implementation of the covariance of two observables.

covariance(obs, obs) is equal to obs.dvalue ** 2 The gamma method has to be applied first to both observables.

If abs(covariance(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance is constrained to the maximum value in order to make sure that covariance matrices are positive semidefinite.

###### Keyword arguments

correlation -- if true the correlation instead of the covariance is returned (default False)

def covariance3(obs1, obs2, correlation=False, **kwargs):
View Source

    def covariance3(obs1, obs2, correlation=False, **kwargs):
        """Another alternative implementation of the covariance of two observables.

        covariance2(obs, obs) is equal to obs.dvalue ** 2
        Currently only works if ensembles are identical.
        The gamma method has to be applied first to both observables.

        If abs(covariance2(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance
        is constrained to the maximum value in order to make sure that covariance
        matrices are positive semidefinite.

        Keyword arguments
        -----------------
        correlation -- if true the correlation instead of the covariance is
                       returned (default False)
        plot -- if true, the integrated autocorrelation time for each ensemble is
                plotted.
        """

        for name in sorted(set(obs1.names + obs2.names)):
            if (obs1.shape.get(name) != obs2.shape.get(name)) and (obs1.shape.get(name) is not None) and (obs2.shape.get(name) is not None):
                raise Exception('Shapes of ensemble', name, 'do not fit')
            if (1 != len(set([len(idx) for idx in [obs1.idl[name], obs2.idl[name], _merge_idx([obs1.idl[name], obs2.idl[name]])]]))):
                raise Exception('Shapes of ensemble', name, 'do not fit')

        if not hasattr(obs1, 'e_names') or not hasattr(obs2, 'e_names'):
            raise Exception('The gamma method has to be applied to both Obs first.')

        tau_exp = []
        S = []
        for e_name in sorted(set(obs1.e_names + obs2.e_names)):
            t_1 = obs1.tau_exp.get(e_name)
            t_2 = obs2.tau_exp.get(e_name)
            if t_1 is None:
                t_1 = 0
            if t_2 is None:
                t_2 = 0
            tau_exp.append(max(t_1, t_2))
            S_1 = obs1.S.get(e_name)
            S_2 = obs2.S.get(e_name)
            if S_1 is None:
                S_1 = Obs.S_global
            if S_2 is None:
                S_2 = Obs.S_global
            S.append(max(S_1, S_2))

        check_obs = obs1 + obs2
        check_obs.gamma_method(tau_exp=tau_exp, S=S)

        if kwargs.get('plot'):
            check_obs.plot_tauint()
            check_obs.plot_rho()

        cov = (check_obs.dvalue ** 2 - obs1.dvalue ** 2 - obs2.dvalue ** 2) / 2

        if np.abs(cov / obs1.dvalue / obs2.dvalue) > 1.0:
            cov = np.sign(cov) * obs1.dvalue * obs2.dvalue

        if correlation:
            cov = cov / obs1.dvalue / obs2.dvalue

        return cov

Another alternative implementation of the covariance of two observables.

covariance2(obs, obs) is equal to obs.dvalue ** 2 Currently only works if ensembles are identical. The gamma method has to be applied first to both observables.

If abs(covariance2(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance is constrained to the maximum value in order to make sure that covariance matrices are positive semidefinite.

###### Keyword arguments

correlation -- if true the correlation instead of the covariance is returned (default False) plot -- if true, the integrated autocorrelation time for each ensemble is plotted.

def pseudo_Obs(value, dvalue, name, samples=1000):
View Source

    def pseudo_Obs(value, dvalue, name, samples=1000):
        """Generate a pseudo Obs with given value, dvalue and name

        The standard number of samples is a 1000. This can be adjusted.
        """
        if dvalue <= 0.0:
            return Obs([np.zeros(samples) + value], [name])
        else:
            for _ in range(100):
                deltas = [np.random.normal(0.0, dvalue * np.sqrt(samples), samples)]
                deltas -= np.mean(deltas)
                deltas *= dvalue / np.sqrt((np.var(deltas) / samples)) / np.sqrt(1 + 3 / samples)
                deltas += value
                res = Obs(deltas, [name])
                res.gamma_method(S=2, tau_exp=0)
                if abs(res.dvalue - dvalue) < 1e-10 * dvalue:
                    break

            res._value = float(value)

            return res

Generate a pseudo Obs with given value, dvalue and name

The standard number of samples is a 1000. This can be adjusted.

def dump_object(obj, name, **kwargs):
View Source

    def dump_object(obj, name, **kwargs):
        """Dump object into pickle file.

        Parameters
        ----------
        obj : object
            object to be saved in the pickle file
        name : str
            name of the file
        path : str
            specifies a custom path for the file (default '.')
        """
        if 'path' in kwargs:
            file_name = kwargs.get('path') + '/' + name + '.p'
        else:
            file_name = name + '.p'
        with open(file_name, 'wb') as fb:
            pickle.dump(obj, fb)

Dump object into pickle file.

###### Parameters

- obj (object): object to be saved in the pickle file
- name (str): name of the file
- path (str): specifies a custom path for the file (default '.')

def load_object(path):
View Source

    def load_object(path):
        """Load object from pickle file. """
        with open(path, 'rb') as file:
            return pickle.load(file)

Load object from pickle file.

def merge_obs(list_of_obs):
View Source

    def merge_obs(list_of_obs):
        """Combine all observables in list_of_obs into one new observable

        Parameters
        ----------
        list_of_obs : list
            list of the Obs object to be combined

        It is not possible to combine obs which are based on the same replicum
        """
        replist = [item for obs in list_of_obs for item in obs.names]
        if (len(replist) == len(set(replist))) is False:
            raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist)))
        new_dict = {}
        idl_dict = {}
        for o in list_of_obs:
            new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0)
                            for key in set(o.deltas) | set(o.r_values)})
            idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)})

        names = sorted(new_dict.keys())
        o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names])
        o.is_merged = np.any([oi.is_merged for oi in list_of_obs])
        o.reweighted = np.max([oi.reweighted for oi in list_of_obs])
        return o

Combine all observables in list_of_obs into one new observable

###### Parameters

- list_of_obs (list): list of the Obs object to be combined
- It is not possible to combine obs which are based on the same replicum