pyerrors.obs

   1import warnings
   2import hashlib
   3import pickle
   4import numpy as np
   5import autograd.numpy as anp  # Thinly-wrapped numpy
   6import scipy
   7from autograd import jacobian
   8import matplotlib.pyplot as plt
   9from scipy.stats import skew, skewtest, kurtosis, kurtosistest
  10import numdifftools as nd
  11from itertools import groupby
  12from .covobs import Covobs
  13
  14# Improve print output of numpy.ndarrays containing Obs objects.
  15np.set_printoptions(formatter={'object': lambda x: str(x)})
  16
  17
  18class Obs:
  19    """Class for a general observable.
  20
  21    Instances of Obs are the basic objects of a pyerrors error analysis.
  22    They are initialized with a list which contains arrays of samples for
  23    different ensembles/replica and another list of same length which contains
  24    the names of the ensembles/replica. Mathematical operations can be
  25    performed on instances. The result is another instance of Obs. The error of
  26    an instance can be computed with the gamma_method. Also contains additional
  27    methods for output and visualization of the error calculation.
  28
  29    Attributes
  30    ----------
  31    S_global : float
  32        Standard value for S (default 2.0)
  33    S_dict : dict
  34        Dictionary for S values. If an entry for a given ensemble
  35        exists this overwrites the standard value for that ensemble.
  36    tau_exp_global : float
  37        Standard value for tau_exp (default 0.0)
  38    tau_exp_dict : dict
  39        Dictionary for tau_exp values. If an entry for a given ensemble exists
  40        this overwrites the standard value for that ensemble.
  41    N_sigma_global : float
  42        Standard value for N_sigma (default 1.0)
  43    N_sigma_dict : dict
  44        Dictionary for N_sigma values. If an entry for a given ensemble exists
  45        this overwrites the standard value for that ensemble.
  46    """
  47    __slots__ = ['names', 'shape', 'r_values', 'deltas', 'N', '_value', '_dvalue',
  48                 'ddvalue', 'reweighted', 'S', 'tau_exp', 'N_sigma',
  49                 'e_dvalue', 'e_ddvalue', 'e_tauint', 'e_dtauint',
  50                 'e_windowsize', 'e_rho', 'e_drho', 'e_n_tauint', 'e_n_dtauint',
  51                 'idl', 'tag', '_covobs', '__dict__']
  52
  53    S_global = 2.0
  54    S_dict = {}
  55    tau_exp_global = 0.0
  56    tau_exp_dict = {}
  57    N_sigma_global = 1.0
  58    N_sigma_dict = {}
  59
  60    def __init__(self, samples, names, idl=None, **kwargs):
  61        """ Initialize Obs object.
  62
  63        Parameters
  64        ----------
  65        samples : list
  66            list of numpy arrays containing the Monte Carlo samples
  67        names : list
  68            list of strings labeling the individual samples
  69        idl : list, optional
  70            list of ranges or lists on which the samples are defined
  71        """
  72
  73        if kwargs.get("means") is None and len(samples):
  74            if len(samples) != len(names):
  75                raise ValueError('Length of samples and names incompatible.')
  76            if idl is not None:
  77                if len(idl) != len(names):
  78                    raise ValueError('Length of idl incompatible with samples and names.')
  79            name_length = len(names)
  80            if name_length > 1:
  81                if name_length != len(set(names)):
  82                    raise ValueError('Names are not unique.')
  83                if not all(isinstance(x, str) for x in names):
  84                    raise TypeError('All names have to be strings.')
  85            else:
  86                if not isinstance(names[0], str):
  87                    raise TypeError('All names have to be strings.')
  88            if min(len(x) for x in samples) <= 4:
  89                raise ValueError('Samples have to have at least 5 entries.')
  90
  91        self.names = sorted(names)
  92        self.shape = {}
  93        self.r_values = {}
  94        self.deltas = {}
  95        self._covobs = {}
  96
  97        self._value = 0
  98        self.N = 0
  99        self.idl = {}
 100        if idl is not None:
 101            for name, idx in sorted(zip(names, idl)):
 102                if isinstance(idx, range):
 103                    self.idl[name] = idx
 104                elif isinstance(idx, (list, np.ndarray)):
 105                    dc = np.unique(np.diff(idx))
 106                    if np.any(dc < 0):
 107                        raise ValueError("Unsorted idx for idl[%s]" % (name))
 108                    if len(dc) == 1:
 109                        self.idl[name] = range(idx[0], idx[-1] + dc[0], dc[0])
 110                    else:
 111                        self.idl[name] = list(idx)
 112                else:
 113                    raise TypeError('incompatible type for idl[%s].' % (name))
 114        else:
 115            for name, sample in sorted(zip(names, samples)):
 116                self.idl[name] = range(1, len(sample) + 1)
 117
 118        if kwargs.get("means") is not None:
 119            for name, sample, mean in sorted(zip(names, samples, kwargs.get("means"))):
 120                self.shape[name] = len(self.idl[name])
 121                self.N += self.shape[name]
 122                self.r_values[name] = mean
 123                self.deltas[name] = sample
 124        else:
 125            for name, sample in sorted(zip(names, samples)):
 126                self.shape[name] = len(self.idl[name])
 127                self.N += self.shape[name]
 128                if len(sample) != self.shape[name]:
 129                    raise ValueError('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name]))
 130                self.r_values[name] = np.mean(sample)
 131                self.deltas[name] = sample - self.r_values[name]
 132                self._value += self.shape[name] * self.r_values[name]
 133            self._value /= self.N
 134
 135        self._dvalue = 0.0
 136        self.ddvalue = 0.0
 137        self.reweighted = False
 138
 139        self.tag = None
 140
 141    @property
 142    def value(self):
 143        return self._value
 144
 145    @property
 146    def dvalue(self):
 147        return self._dvalue
 148
 149    @property
 150    def e_names(self):
 151        return sorted(set([o.split('|')[0] for o in self.names]))
 152
 153    @property
 154    def cov_names(self):
 155        return sorted(set([o for o in self.covobs.keys()]))
 156
 157    @property
 158    def mc_names(self):
 159        return sorted(set([o.split('|')[0] for o in self.names if o not in self.cov_names]))
 160
 161    @property
 162    def e_content(self):
 163        res = {}
 164        for e, e_name in enumerate(self.e_names):
 165            res[e_name] = sorted(filter(lambda x: x.startswith(e_name + '|'), self.names))
 166            if e_name in self.names:
 167                res[e_name].append(e_name)
 168        return res
 169
 170    @property
 171    def covobs(self):
 172        return self._covobs
 173
 174    def gamma_method(self, **kwargs):
 175        """Estimate the error and related properties of the Obs.
 176
 177        Parameters
 178        ----------
 179        S : float
 180            specifies a custom value for the parameter S (default 2.0).
 181            If set to 0 it is assumed that the data exhibits no
 182            autocorrelation. In this case the error estimates coincides
 183            with the sample standard error.
 184        tau_exp : float
 185            positive value triggers the critical slowing down analysis
 186            (default 0.0).
 187        N_sigma : float
 188            number of standard deviations from zero until the tail is
 189            attached to the autocorrelation function (default 1).
 190        fft : bool
 191            determines whether the fft algorithm is used for the computation
 192            of the autocorrelation function (default True)
 193        """
 194
 195        e_content = self.e_content
 196        self.e_dvalue = {}
 197        self.e_ddvalue = {}
 198        self.e_tauint = {}
 199        self.e_dtauint = {}
 200        self.e_windowsize = {}
 201        self.e_n_tauint = {}
 202        self.e_n_dtauint = {}
 203        e_gamma = {}
 204        self.e_rho = {}
 205        self.e_drho = {}
 206        self._dvalue = 0
 207        self.ddvalue = 0
 208
 209        self.S = {}
 210        self.tau_exp = {}
 211        self.N_sigma = {}
 212
 213        if kwargs.get('fft') is False:
 214            fft = False
 215        else:
 216            fft = True
 217
 218        def _parse_kwarg(kwarg_name):
 219            if kwarg_name in kwargs:
 220                tmp = kwargs.get(kwarg_name)
 221                if isinstance(tmp, (int, float)):
 222                    if tmp < 0:
 223                        raise Exception(kwarg_name + ' has to be larger or equal to 0.')
 224                    for e, e_name in enumerate(self.e_names):
 225                        getattr(self, kwarg_name)[e_name] = tmp
 226                else:
 227                    raise TypeError(kwarg_name + ' is not in proper format.')
 228            else:
 229                for e, e_name in enumerate(self.e_names):
 230                    if e_name in getattr(Obs, kwarg_name + '_dict'):
 231                        getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_dict')[e_name]
 232                    else:
 233                        getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_global')
 234
 235        _parse_kwarg('S')
 236        _parse_kwarg('tau_exp')
 237        _parse_kwarg('N_sigma')
 238
 239        for e, e_name in enumerate(self.mc_names):
 240            gapsize = _determine_gap(self, e_content, e_name)
 241
 242            r_length = []
 243            for r_name in e_content[e_name]:
 244                if isinstance(self.idl[r_name], range):
 245                    r_length.append(len(self.idl[r_name]) * self.idl[r_name].step // gapsize)
 246                else:
 247                    r_length.append((self.idl[r_name][-1] - self.idl[r_name][0] + 1) // gapsize)
 248
 249            e_N = np.sum([self.shape[r_name] for r_name in e_content[e_name]])
 250            w_max = max(r_length) // 2
 251            e_gamma[e_name] = np.zeros(w_max)
 252            self.e_rho[e_name] = np.zeros(w_max)
 253            self.e_drho[e_name] = np.zeros(w_max)
 254
 255            for r_name in e_content[e_name]:
 256                e_gamma[e_name] += self._calc_gamma(self.deltas[r_name], self.idl[r_name], self.shape[r_name], w_max, fft, gapsize)
 257
 258            gamma_div = np.zeros(w_max)
 259            for r_name in e_content[e_name]:
 260                gamma_div += self._calc_gamma(np.ones((self.shape[r_name])), self.idl[r_name], self.shape[r_name], w_max, fft, gapsize)
 261            gamma_div[gamma_div < 1] = 1.0
 262            e_gamma[e_name] /= gamma_div[:w_max]
 263
 264            if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny:  # Prevent division by zero
 265                self.e_tauint[e_name] = 0.5
 266                self.e_dtauint[e_name] = 0.0
 267                self.e_dvalue[e_name] = 0.0
 268                self.e_ddvalue[e_name] = 0.0
 269                self.e_windowsize[e_name] = 0
 270                continue
 271
 272            self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0]
 273            self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:])))
 274            # Make sure no entry of tauint is smaller than 0.5
 275            self.e_n_tauint[e_name][self.e_n_tauint[e_name] <= 0.5] = 0.5 + np.finfo(np.float64).eps
 276            # hep-lat/0306017 eq. (42)
 277            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)
 278            self.e_n_dtauint[e_name][0] = 0.0
 279
 280            def _compute_drho(i):
 281                tmp = (self.e_rho[e_name][i + 1:w_max]
 282                       + np.concatenate([self.e_rho[e_name][i - 1:None if i - (w_max - 1) // 2 <= 0 else (2 * i - (2 * w_max) // 2):-1],
 283                                         self.e_rho[e_name][1:max(1, w_max - 2 * i)]])
 284                       - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i])
 285                self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N)
 286
 287            if self.tau_exp[e_name] > 0:
 288                _compute_drho(1)
 289                texp = self.tau_exp[e_name]
 290                # Critical slowing down analysis
 291                if w_max // 2 <= 1:
 292                    raise Exception("Need at least 8 samples for tau_exp error analysis")
 293                for n in range(1, w_max // 2):
 294                    _compute_drho(n + 1)
 295                    if (self.e_rho[e_name][n] - self.N_sigma[e_name] * self.e_drho[e_name][n]) < 0 or n >= w_max // 2 - 2:
 296                        # Bias correction hep-lat/0306017 eq. (49) included
 297                        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
 298                        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)
 299                        # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2
 300                        self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
 301                        self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
 302                        self.e_windowsize[e_name] = n
 303                        break
 304            else:
 305                if self.S[e_name] == 0.0:
 306                    self.e_tauint[e_name] = 0.5
 307                    self.e_dtauint[e_name] = 0.0
 308                    self.e_dvalue[e_name] = np.sqrt(e_gamma[e_name][0] / (e_N - 1))
 309                    self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt(0.5 / e_N)
 310                    self.e_windowsize[e_name] = 0
 311                else:
 312                    # Standard automatic windowing procedure
 313                    tau = self.S[e_name] / np.log((2 * self.e_n_tauint[e_name][1:] + 1) / (2 * self.e_n_tauint[e_name][1:] - 1))
 314                    g_w = np.exp(- np.arange(1, len(tau) + 1) / tau) - tau / np.sqrt(np.arange(1, len(tau) + 1) * e_N)
 315                    for n in range(1, w_max):
 316                        if g_w[n - 1] < 0 or n >= w_max - 1:
 317                            _compute_drho(n)
 318                            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)
 319                            self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n]
 320                            self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
 321                            self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
 322                            self.e_windowsize[e_name] = n
 323                            break
 324
 325            self._dvalue += self.e_dvalue[e_name] ** 2
 326            self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2
 327
 328        for e_name in self.cov_names:
 329            self.e_dvalue[e_name] = np.sqrt(self.covobs[e_name].errsq())
 330            self.e_ddvalue[e_name] = 0
 331            self._dvalue += self.e_dvalue[e_name]**2
 332
 333        self._dvalue = np.sqrt(self._dvalue)
 334        if self._dvalue == 0.0:
 335            self.ddvalue = 0.0
 336        else:
 337            self.ddvalue = np.sqrt(self.ddvalue) / self._dvalue
 338        return
 339
 340    gm = gamma_method
 341
 342    def _calc_gamma(self, deltas, idx, shape, w_max, fft, gapsize):
 343        """Calculate Gamma_{AA} from the deltas, which are defined on idx.
 344           idx is assumed to be a contiguous range (possibly with a stepsize != 1)
 345
 346        Parameters
 347        ----------
 348        deltas : list
 349            List of fluctuations
 350        idx : list
 351            List or range of configurations on which the deltas are defined.
 352        shape : int
 353            Number of configurations in idx.
 354        w_max : int
 355            Upper bound for the summation window.
 356        fft : bool
 357            determines whether the fft algorithm is used for the computation
 358            of the autocorrelation function.
 359        gapsize : int
 360            The target distance between two configurations. If longer distances
 361            are found in idx, the data is expanded.
 362        """
 363        gamma = np.zeros(w_max)
 364        deltas = _expand_deltas(deltas, idx, shape, gapsize)
 365        new_shape = len(deltas)
 366        if fft:
 367            max_gamma = min(new_shape, w_max)
 368            # The padding for the fft has to be even
 369            padding = new_shape + max_gamma + (new_shape + max_gamma) % 2
 370            gamma[:max_gamma] += np.fft.irfft(np.abs(np.fft.rfft(deltas, padding)) ** 2)[:max_gamma]
 371        else:
 372            for n in range(w_max):
 373                if new_shape - n >= 0:
 374                    gamma[n] += deltas[0:new_shape - n].dot(deltas[n:new_shape])
 375
 376        return gamma
 377
 378    def details(self, ens_content=True):
 379        """Output detailed properties of the Obs.
 380
 381        Parameters
 382        ----------
 383        ens_content : bool
 384            print details about the ensembles and replica if true.
 385        """
 386        if self.tag is not None:
 387            print("Description:", self.tag)
 388        if not hasattr(self, 'e_dvalue'):
 389            print('Result\t %3.8e' % (self.value))
 390        else:
 391            if self.value == 0.0:
 392                percentage = np.nan
 393            else:
 394                percentage = np.abs(self._dvalue / self.value) * 100
 395            print('Result\t %3.8e +/- %3.8e +/- %3.8e (%3.3f%%)' % (self.value, self._dvalue, self.ddvalue, percentage))
 396            if len(self.e_names) > 1:
 397                print(' Ensemble errors:')
 398            e_content = self.e_content
 399            for e_name in self.mc_names:
 400                gap = _determine_gap(self, e_content, e_name)
 401
 402                if len(self.e_names) > 1:
 403                    print('', e_name, '\t %3.6e +/- %3.6e' % (self.e_dvalue[e_name], self.e_ddvalue[e_name]))
 404                tau_string = " \N{GREEK SMALL LETTER TAU}_int\t " + _format_uncertainty(self.e_tauint[e_name], self.e_dtauint[e_name])
 405                tau_string += f" in units of {gap} config"
 406                if gap > 1:
 407                    tau_string += "s"
 408                if self.tau_exp[e_name] > 0:
 409                    tau_string = f"{tau_string: <45}" + '\t(\N{GREEK SMALL LETTER TAU}_exp=%3.2f, N_\N{GREEK SMALL LETTER SIGMA}=%1.0i)' % (self.tau_exp[e_name], self.N_sigma[e_name])
 410                else:
 411                    tau_string = f"{tau_string: <45}" + '\t(S=%3.2f)' % (self.S[e_name])
 412                print(tau_string)
 413            for e_name in self.cov_names:
 414                print('', e_name, '\t %3.8e' % (self.e_dvalue[e_name]))
 415        if ens_content is True:
 416            if len(self.e_names) == 1:
 417                print(self.N, 'samples in', len(self.e_names), 'ensemble:')
 418            else:
 419                print(self.N, 'samples in', len(self.e_names), 'ensembles:')
 420            my_string_list = []
 421            for key, value in sorted(self.e_content.items()):
 422                if key not in self.covobs:
 423                    my_string = '  ' + "\u00B7 Ensemble '" + key + "' "
 424                    if len(value) == 1:
 425                        my_string += f': {self.shape[value[0]]} configurations'
 426                        if isinstance(self.idl[value[0]], range):
 427                            my_string += f' (from {self.idl[value[0]].start} to {self.idl[value[0]][-1]}' + int(self.idl[value[0]].step != 1) * f' in steps of {self.idl[value[0]].step}' + ')'
 428                        else:
 429                            my_string += f' (irregular range from {self.idl[value[0]][0]} to {self.idl[value[0]][-1]})'
 430                    else:
 431                        sublist = []
 432                        for v in value:
 433                            my_substring = '    ' + "\u00B7 Replicum '" + v[len(key) + 1:] + "' "
 434                            my_substring += f': {self.shape[v]} configurations'
 435                            if isinstance(self.idl[v], range):
 436                                my_substring += f' (from {self.idl[v].start} to {self.idl[v][-1]}' + int(self.idl[v].step != 1) * f' in steps of {self.idl[v].step}' + ')'
 437                            else:
 438                                my_substring += f' (irregular range from {self.idl[v][0]} to {self.idl[v][-1]})'
 439                            sublist.append(my_substring)
 440
 441                        my_string += '\n' + '\n'.join(sublist)
 442                else:
 443                    my_string = '  ' + "\u00B7 Covobs   '" + key + "' "
 444                my_string_list.append(my_string)
 445            print('\n'.join(my_string_list))
 446
 447    def reweight(self, weight):
 448        """Reweight the obs with given rewighting factors.
 449
 450        Parameters
 451        ----------
 452        weight : Obs
 453            Reweighting factor. An Observable that has to be defined on a superset of the
 454            configurations in obs[i].idl for all i.
 455        all_configs : bool
 456            if True, the reweighted observables are normalized by the average of
 457            the reweighting factor on all configurations in weight.idl and not
 458            on the configurations in obs[i].idl. Default False.
 459        """
 460        return reweight(weight, [self])[0]
 461
 462    def is_zero_within_error(self, sigma=1):
 463        """Checks whether the observable is zero within 'sigma' standard errors.
 464
 465        Parameters
 466        ----------
 467        sigma : int
 468            Number of standard errors used for the check.
 469
 470        Works only properly when the gamma method was run.
 471        """
 472        return self.is_zero() or np.abs(self.value) <= sigma * self._dvalue
 473
 474    def is_zero(self, atol=1e-10):
 475        """Checks whether the observable is zero within a given tolerance.
 476
 477        Parameters
 478        ----------
 479        atol : float
 480            Absolute tolerance (for details see numpy documentation).
 481        """
 482        return np.isclose(0.0, self.value, 1e-14, atol) and all(np.allclose(0.0, delta, 1e-14, atol) for delta in self.deltas.values()) and all(np.allclose(0.0, delta.errsq(), 1e-14, atol) for delta in self.covobs.values())
 483
 484    def plot_tauint(self, save=None):
 485        """Plot integrated autocorrelation time for each ensemble.
 486
 487        Parameters
 488        ----------
 489        save : str
 490            saves the figure to a file named 'save' if.
 491        """
 492        if not hasattr(self, 'e_dvalue'):
 493            raise Exception('Run the gamma method first.')
 494
 495        for e, e_name in enumerate(self.mc_names):
 496            fig = plt.figure()
 497            plt.xlabel(r'$W$')
 498            plt.ylabel(r'$\tau_\mathrm{int}$')
 499            length = int(len(self.e_n_tauint[e_name]))
 500            if self.tau_exp[e_name] > 0:
 501                base = self.e_n_tauint[e_name][self.e_windowsize[e_name]]
 502                x_help = np.arange(2 * self.tau_exp[e_name])
 503                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
 504                x_arr = np.arange(self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name])
 505                plt.plot(x_arr, y_help, 'C' + str(e), linewidth=1, ls='--', marker=',')
 506                plt.errorbar([self.e_windowsize[e_name] + 2 * self.tau_exp[e_name]], [self.e_tauint[e_name]],
 507                             yerr=[self.e_dtauint[e_name]], fmt='C' + str(e), linewidth=1, capsize=2, marker='o', mfc=plt.rcParams['axes.facecolor'])
 508                xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
 509                label = e_name + r', $\tau_\mathrm{exp}$=' + str(np.around(self.tau_exp[e_name], decimals=2))
 510            else:
 511                label = e_name + ', S=' + str(np.around(self.S[e_name], decimals=2))
 512                xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)
 513
 514            plt.errorbar(np.arange(length)[:int(xmax) + 1], self.e_n_tauint[e_name][:int(xmax) + 1], yerr=self.e_n_dtauint[e_name][:int(xmax) + 1], linewidth=1, capsize=2, label=label)
 515            plt.axvline(x=self.e_windowsize[e_name], color='C' + str(e), alpha=0.5, marker=',', ls='--')
 516            plt.legend()
 517            plt.xlim(-0.5, xmax)
 518            ylim = plt.ylim()
 519            plt.ylim(bottom=0.0, top=max(1.0, ylim[1]))
 520            plt.draw()
 521            if save:
 522                fig.savefig(save + "_" + str(e))
 523
 524    def plot_rho(self, save=None):
 525        """Plot normalized autocorrelation function time for each ensemble.
 526
 527        Parameters
 528        ----------
 529        save : str
 530            saves the figure to a file named 'save' if.
 531        """
 532        if not hasattr(self, 'e_dvalue'):
 533            raise Exception('Run the gamma method first.')
 534        for e, e_name in enumerate(self.mc_names):
 535            fig = plt.figure()
 536            plt.xlabel('W')
 537            plt.ylabel('rho')
 538            length = int(len(self.e_drho[e_name]))
 539            plt.errorbar(np.arange(length), self.e_rho[e_name][:length], yerr=self.e_drho[e_name][:], linewidth=1, capsize=2)
 540            plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25, ls='--', marker=',')
 541            if self.tau_exp[e_name] > 0:
 542                plt.plot([self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]],
 543                         [self.e_rho[e_name][self.e_windowsize[e_name] + 1], 0], 'k-', lw=1)
 544                xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
 545                plt.title('Rho ' + e_name + r', tau\_exp=' + str(np.around(self.tau_exp[e_name], decimals=2)))
 546            else:
 547                xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)
 548                plt.title('Rho ' + e_name + ', S=' + str(np.around(self.S[e_name], decimals=2)))
 549            plt.plot([-0.5, xmax], [0, 0], 'k--', lw=1)
 550            plt.xlim(-0.5, xmax)
 551            plt.draw()
 552            if save:
 553                fig.savefig(save + "_" + str(e))
 554
 555    def plot_rep_dist(self):
 556        """Plot replica distribution for each ensemble with more than one replicum."""
 557        if not hasattr(self, 'e_dvalue'):
 558            raise Exception('Run the gamma method first.')
 559        for e, e_name in enumerate(self.mc_names):
 560            if len(self.e_content[e_name]) == 1:
 561                print('No replica distribution for a single replicum (', e_name, ')')
 562                continue
 563            r_length = []
 564            sub_r_mean = 0
 565            for r, r_name in enumerate(self.e_content[e_name]):
 566                r_length.append(len(self.deltas[r_name]))
 567                sub_r_mean += self.shape[r_name] * self.r_values[r_name]
 568            e_N = np.sum(r_length)
 569            sub_r_mean /= e_N
 570            arr = np.zeros(len(self.e_content[e_name]))
 571            for r, r_name in enumerate(self.e_content[e_name]):
 572                arr[r] = (self.r_values[r_name] - sub_r_mean) / (self.e_dvalue[e_name] * np.sqrt(e_N / self.shape[r_name] - 1))
 573            plt.hist(arr, rwidth=0.8, bins=len(self.e_content[e_name]))
 574            plt.title('Replica distribution' + e_name + ' (mean=0, var=1)')
 575            plt.draw()
 576
 577    def plot_history(self, expand=True):
 578        """Plot derived Monte Carlo history for each ensemble
 579
 580        Parameters
 581        ----------
 582        expand : bool
 583            show expanded history for irregular Monte Carlo chains (default: True).
 584        """
 585        for e, e_name in enumerate(self.mc_names):
 586            plt.figure()
 587            r_length = []
 588            tmp = []
 589            tmp_expanded = []
 590            for r, r_name in enumerate(self.e_content[e_name]):
 591                tmp.append(self.deltas[r_name] + self.r_values[r_name])
 592                if expand:
 593                    tmp_expanded.append(_expand_deltas(self.deltas[r_name], list(self.idl[r_name]), self.shape[r_name], 1) + self.r_values[r_name])
 594                    r_length.append(len(tmp_expanded[-1]))
 595                else:
 596                    r_length.append(len(tmp[-1]))
 597            e_N = np.sum(r_length)
 598            x = np.arange(e_N)
 599            y_test = np.concatenate(tmp, axis=0)
 600            if expand:
 601                y = np.concatenate(tmp_expanded, axis=0)
 602            else:
 603                y = y_test
 604            plt.errorbar(x, y, fmt='.', markersize=3)
 605            plt.xlim(-0.5, e_N - 0.5)
 606            plt.title(e_name + f'\nskew: {skew(y_test):.3f} (p={skewtest(y_test).pvalue:.3f}), kurtosis: {kurtosis(y_test):.3f} (p={kurtosistest(y_test).pvalue:.3f})')
 607            plt.draw()
 608
 609    def plot_piechart(self, save=None):
 610        """Plot piechart which shows the fractional contribution of each
 611        ensemble to the error and returns a dictionary containing the fractions.
 612
 613        Parameters
 614        ----------
 615        save : str
 616            saves the figure to a file named 'save' if.
 617        """
 618        if not hasattr(self, 'e_dvalue'):
 619            raise Exception('Run the gamma method first.')
 620        if np.isclose(0.0, self._dvalue, atol=1e-15):
 621            raise Exception('Error is 0.0')
 622        labels = self.e_names
 623        sizes = [self.e_dvalue[name] ** 2 for name in labels] / self._dvalue ** 2
 624        fig1, ax1 = plt.subplots()
 625        ax1.pie(sizes, labels=labels, startangle=90, normalize=True)
 626        ax1.axis('equal')
 627        plt.draw()
 628        if save:
 629            fig1.savefig(save)
 630
 631        return dict(zip(labels, sizes))
 632
 633    def dump(self, filename, datatype="json.gz", description="", **kwargs):
 634        """Dump the Obs to a file 'name' of chosen format.
 635
 636        Parameters
 637        ----------
 638        filename : str
 639            name of the file to be saved.
 640        datatype : str
 641            Format of the exported file. Supported formats include
 642            "json.gz" and "pickle"
 643        description : str
 644            Description for output file, only relevant for json.gz format.
 645        path : str
 646            specifies a custom path for the file (default '.')
 647        """
 648        if 'path' in kwargs:
 649            file_name = kwargs.get('path') + '/' + filename
 650        else:
 651            file_name = filename
 652
 653        if datatype == "json.gz":
 654            from .input.json import dump_to_json
 655            dump_to_json([self], file_name, description=description)
 656        elif datatype == "pickle":
 657            with open(file_name + '.p', 'wb') as fb:
 658                pickle.dump(self, fb)
 659        else:
 660            raise Exception("Unknown datatype " + str(datatype))
 661
 662    def export_jackknife(self):
 663        """Export jackknife samples from the Obs
 664
 665        Returns
 666        -------
 667        numpy.ndarray
 668            Returns a numpy array of length N + 1 where N is the number of samples
 669            for the given ensemble and replicum. The zeroth entry of the array contains
 670            the mean value of the Obs, entries 1 to N contain the N jackknife samples
 671            derived from the Obs. The current implementation only works for observables
 672            defined on exactly one ensemble and replicum. The derived jackknife samples
 673            should agree with samples from a full jackknife analysis up to O(1/N).
 674        """
 675
 676        if len(self.names) != 1:
 677            raise Exception("'export_jackknife' is only implemented for Obs defined on one ensemble and replicum.")
 678
 679        name = self.names[0]
 680        full_data = self.deltas[name] + self.r_values[name]
 681        n = full_data.size
 682        mean = self.value
 683        tmp_jacks = np.zeros(n + 1)
 684        tmp_jacks[0] = mean
 685        tmp_jacks[1:] = (n * mean - full_data) / (n - 1)
 686        return tmp_jacks
 687
 688    def export_bootstrap(self, samples=500, random_numbers=None, save_rng=None):
 689        """Export bootstrap samples from the Obs
 690
 691        Parameters
 692        ----------
 693        samples : int
 694            Number of bootstrap samples to generate.
 695        random_numbers : np.ndarray
 696            Array of shape (samples, length) containing the random numbers to generate the bootstrap samples.
 697            If not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
 698        save_rng : str
 699            Save the random numbers to a file if a path is specified.
 700
 701        Returns
 702        -------
 703        numpy.ndarray
 704            Returns a numpy array of length N + 1 where N is the number of samples
 705            for the given ensemble and replicum. The zeroth entry of the array contains
 706            the mean value of the Obs, entries 1 to N contain the N import_bootstrap samples
 707            derived from the Obs. The current implementation only works for observables
 708            defined on exactly one ensemble and replicum. The derived bootstrap samples
 709            should agree with samples from a full bootstrap analysis up to O(1/N).
 710        """
 711        if len(self.names) != 1:
 712            raise Exception("'export_boostrap' is only implemented for Obs defined on one ensemble and replicum.")
 713
 714        name = self.names[0]
 715        length = self.N
 716
 717        if random_numbers is None:
 718            seed = int(hashlib.md5(name.encode()).hexdigest(), 16) & 0xFFFFFFFF
 719            rng = np.random.default_rng(seed)
 720            random_numbers = rng.integers(0, length, size=(samples, length))
 721
 722        if save_rng is not None:
 723            np.savetxt(save_rng, random_numbers, fmt='%i')
 724
 725        proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length
 726        ret = np.zeros(samples + 1)
 727        ret[0] = self.value
 728        ret[1:] = proj @ (self.deltas[name] + self.r_values[name])
 729        return ret
 730
 731    def __float__(self):
 732        return float(self.value)
 733
 734    def __repr__(self):
 735        return 'Obs[' + str(self) + ']'
 736
 737    def __str__(self):
 738        return _format_uncertainty(self.value, self._dvalue)
 739
 740    def __format__(self, format_type):
 741        if format_type == "":
 742            significance = 2
 743        else:
 744            significance = int(float(format_type.replace("+", "").replace("-", "")))
 745        my_str = _format_uncertainty(self.value, self._dvalue,
 746                                     significance=significance)
 747        for char in ["+", " "]:
 748            if format_type.startswith(char):
 749                if my_str[0] != "-":
 750                    my_str = char + my_str
 751        return my_str
 752
 753    def __hash__(self):
 754        hash_tuple = (np.array([self.value]).astype(np.float32).data.tobytes(),)
 755        hash_tuple += tuple([o.astype(np.float32).data.tobytes() for o in self.deltas.values()])
 756        hash_tuple += tuple([np.array([o.errsq()]).astype(np.float32).data.tobytes() for o in self.covobs.values()])
 757        hash_tuple += tuple([o.encode() for o in self.names])
 758        m = hashlib.md5()
 759        [m.update(o) for o in hash_tuple]
 760        return int(m.hexdigest(), 16) & 0xFFFFFFFF
 761
 762    # Overload comparisons
 763    def __lt__(self, other):
 764        return self.value < other
 765
 766    def __le__(self, other):
 767        return self.value <= other
 768
 769    def __gt__(self, other):
 770        return self.value > other
 771
 772    def __ge__(self, other):
 773        return self.value >= other
 774
 775    def __eq__(self, other):
 776        if other is None:
 777            return False
 778        return (self - other).is_zero()
 779
 780    # Overload math operations
 781    def __add__(self, y):
 782        if isinstance(y, Obs):
 783            return derived_observable(lambda x, **kwargs: x[0] + x[1], [self, y], man_grad=[1, 1])
 784        else:
 785            if isinstance(y, np.ndarray):
 786                return np.array([self + o for o in y])
 787            elif y.__class__.__name__ in ['Corr', 'CObs']:
 788                return NotImplemented
 789            else:
 790                return derived_observable(lambda x, **kwargs: x[0] + y, [self], man_grad=[1])
 791
 792    def __radd__(self, y):
 793        return self + y
 794
 795    def __mul__(self, y):
 796        if isinstance(y, Obs):
 797            return derived_observable(lambda x, **kwargs: x[0] * x[1], [self, y], man_grad=[y.value, self.value])
 798        else:
 799            if isinstance(y, np.ndarray):
 800                return np.array([self * o for o in y])
 801            elif isinstance(y, complex):
 802                return CObs(self * y.real, self * y.imag)
 803            elif y.__class__.__name__ in ['Corr', 'CObs']:
 804                return NotImplemented
 805            else:
 806                return derived_observable(lambda x, **kwargs: x[0] * y, [self], man_grad=[y])
 807
 808    def __rmul__(self, y):
 809        return self * y
 810
 811    def __sub__(self, y):
 812        if isinstance(y, Obs):
 813            return derived_observable(lambda x, **kwargs: x[0] - x[1], [self, y], man_grad=[1, -1])
 814        else:
 815            if isinstance(y, np.ndarray):
 816                return np.array([self - o for o in y])
 817            elif y.__class__.__name__ in ['Corr', 'CObs']:
 818                return NotImplemented
 819            else:
 820                return derived_observable(lambda x, **kwargs: x[0] - y, [self], man_grad=[1])
 821
 822    def __rsub__(self, y):
 823        return -1 * (self - y)
 824
 825    def __pos__(self):
 826        return self
 827
 828    def __neg__(self):
 829        return -1 * self
 830
 831    def __truediv__(self, y):
 832        if isinstance(y, Obs):
 833            return derived_observable(lambda x, **kwargs: x[0] / x[1], [self, y], man_grad=[1 / y.value, - self.value / y.value ** 2])
 834        else:
 835            if isinstance(y, np.ndarray):
 836                return np.array([self / o for o in y])
 837            elif y.__class__.__name__ in ['Corr', 'CObs']:
 838                return NotImplemented
 839            else:
 840                return derived_observable(lambda x, **kwargs: x[0] / y, [self], man_grad=[1 / y])
 841
 842    def __rtruediv__(self, y):
 843        if isinstance(y, Obs):
 844            return derived_observable(lambda x, **kwargs: x[0] / x[1], [y, self], man_grad=[1 / self.value, - y.value / self.value ** 2])
 845        else:
 846            if isinstance(y, np.ndarray):
 847                return np.array([o / self for o in y])
 848            elif y.__class__.__name__ in ['Corr', 'CObs']:
 849                return NotImplemented
 850            else:
 851                return derived_observable(lambda x, **kwargs: y / x[0], [self], man_grad=[-y / self.value ** 2])
 852
 853    def __pow__(self, y):
 854        if isinstance(y, Obs):
 855            return derived_observable(lambda x: x[0] ** x[1], [self, y])
 856        else:
 857            return derived_observable(lambda x: x[0] ** y, [self])
 858
 859    def __rpow__(self, y):
 860        if isinstance(y, Obs):
 861            return derived_observable(lambda x: x[0] ** x[1], [y, self])
 862        else:
 863            return derived_observable(lambda x: y ** x[0], [self])
 864
 865    def __abs__(self):
 866        return derived_observable(lambda x: anp.abs(x[0]), [self])
 867
 868    # Overload numpy functions
 869    def sqrt(self):
 870        return derived_observable(lambda x, **kwargs: np.sqrt(x[0]), [self], man_grad=[1 / 2 / np.sqrt(self.value)])
 871
 872    def log(self):
 873        return derived_observable(lambda x, **kwargs: np.log(x[0]), [self], man_grad=[1 / self.value])
 874
 875    def exp(self):
 876        return derived_observable(lambda x, **kwargs: np.exp(x[0]), [self], man_grad=[np.exp(self.value)])
 877
 878    def sin(self):
 879        return derived_observable(lambda x, **kwargs: np.sin(x[0]), [self], man_grad=[np.cos(self.value)])
 880
 881    def cos(self):
 882        return derived_observable(lambda x, **kwargs: np.cos(x[0]), [self], man_grad=[-np.sin(self.value)])
 883
 884    def tan(self):
 885        return derived_observable(lambda x, **kwargs: np.tan(x[0]), [self], man_grad=[1 / np.cos(self.value) ** 2])
 886
 887    def arcsin(self):
 888        return derived_observable(lambda x: anp.arcsin(x[0]), [self])
 889
 890    def arccos(self):
 891        return derived_observable(lambda x: anp.arccos(x[0]), [self])
 892
 893    def arctan(self):
 894        return derived_observable(lambda x: anp.arctan(x[0]), [self])
 895
 896    def sinh(self):
 897        return derived_observable(lambda x, **kwargs: np.sinh(x[0]), [self], man_grad=[np.cosh(self.value)])
 898
 899    def cosh(self):
 900        return derived_observable(lambda x, **kwargs: np.cosh(x[0]), [self], man_grad=[np.sinh(self.value)])
 901
 902    def tanh(self):
 903        return derived_observable(lambda x, **kwargs: np.tanh(x[0]), [self], man_grad=[1 / np.cosh(self.value) ** 2])
 904
 905    def arcsinh(self):
 906        return derived_observable(lambda x: anp.arcsinh(x[0]), [self])
 907
 908    def arccosh(self):
 909        return derived_observable(lambda x: anp.arccosh(x[0]), [self])
 910
 911    def arctanh(self):
 912        return derived_observable(lambda x: anp.arctanh(x[0]), [self])
 913
 914
 915class CObs:
 916    """Class for a complex valued observable."""
 917    __slots__ = ['_real', '_imag', 'tag']
 918
 919    def __init__(self, real, imag=0.0):
 920        self._real = real
 921        self._imag = imag
 922        self.tag = None
 923
 924    @property
 925    def real(self):
 926        return self._real
 927
 928    @property
 929    def imag(self):
 930        return self._imag
 931
 932    def gamma_method(self, **kwargs):
 933        """Executes the gamma_method for the real and the imaginary part."""
 934        if isinstance(self.real, Obs):
 935            self.real.gamma_method(**kwargs)
 936        if isinstance(self.imag, Obs):
 937            self.imag.gamma_method(**kwargs)
 938
 939    def is_zero(self):
 940        """Checks whether both real and imaginary part are zero within machine precision."""
 941        return self.real == 0.0 and self.imag == 0.0
 942
 943    def conjugate(self):
 944        return CObs(self.real, -self.imag)
 945
 946    def __add__(self, other):
 947        if isinstance(other, np.ndarray):
 948            return other + self
 949        elif hasattr(other, 'real') and hasattr(other, 'imag'):
 950            return CObs(self.real + other.real,
 951                        self.imag + other.imag)
 952        else:
 953            return CObs(self.real + other, self.imag)
 954
 955    def __radd__(self, y):
 956        return self + y
 957
 958    def __sub__(self, other):
 959        if isinstance(other, np.ndarray):
 960            return -1 * (other - self)
 961        elif hasattr(other, 'real') and hasattr(other, 'imag'):
 962            return CObs(self.real - other.real, self.imag - other.imag)
 963        else:
 964            return CObs(self.real - other, self.imag)
 965
 966    def __rsub__(self, other):
 967        return -1 * (self - other)
 968
 969    def __mul__(self, other):
 970        if isinstance(other, np.ndarray):
 971            return other * self
 972        elif hasattr(other, 'real') and hasattr(other, 'imag'):
 973            if all(isinstance(i, Obs) for i in [self.real, self.imag, other.real, other.imag]):
 974                return CObs(derived_observable(lambda x, **kwargs: x[0] * x[1] - x[2] * x[3],
 975                                               [self.real, other.real, self.imag, other.imag],
 976                                               man_grad=[other.real.value, self.real.value, -other.imag.value, -self.imag.value]),
 977                            derived_observable(lambda x, **kwargs: x[2] * x[1] + x[0] * x[3],
 978                                               [self.real, other.real, self.imag, other.imag],
 979                                               man_grad=[other.imag.value, self.imag.value, other.real.value, self.real.value]))
 980            elif getattr(other, 'imag', 0) != 0:
 981                return CObs(self.real * other.real - self.imag * other.imag,
 982                            self.imag * other.real + self.real * other.imag)
 983            else:
 984                return CObs(self.real * other.real, self.imag * other.real)
 985        else:
 986            return CObs(self.real * other, self.imag * other)
 987
 988    def __rmul__(self, other):
 989        return self * other
 990
 991    def __truediv__(self, other):
 992        if isinstance(other, np.ndarray):
 993            return 1 / (other / self)
 994        elif hasattr(other, 'real') and hasattr(other, 'imag'):
 995            r = other.real ** 2 + other.imag ** 2
 996            return CObs((self.real * other.real + self.imag * other.imag) / r, (self.imag * other.real - self.real * other.imag) / r)
 997        else:
 998            return CObs(self.real / other, self.imag / other)
 999
1000    def __rtruediv__(self, other):
1001        r = self.real ** 2 + self.imag ** 2
1002        if hasattr(other, 'real') and hasattr(other, 'imag'):
1003            return CObs((self.real * other.real + self.imag * other.imag) / r, (self.real * other.imag - self.imag * other.real) / r)
1004        else:
1005            return CObs(self.real * other / r, -self.imag * other / r)
1006
1007    def __abs__(self):
1008        return np.sqrt(self.real**2 + self.imag**2)
1009
1010    def __pos__(self):
1011        return self
1012
1013    def __neg__(self):
1014        return -1 * self
1015
1016    def __eq__(self, other):
1017        return self.real == other.real and self.imag == other.imag
1018
1019    def __str__(self):
1020        return '(' + str(self.real) + int(self.imag >= 0.0) * '+' + str(self.imag) + 'j)'
1021
1022    def __repr__(self):
1023        return 'CObs[' + str(self) + ']'
1024
1025    def __format__(self, format_type):
1026        if format_type == "":
1027            significance = 2
1028            format_type = "2"
1029        else:
1030            significance = int(float(format_type.replace("+", "").replace("-", "")))
1031        return f"({self.real:{format_type}}{self.imag:+{significance}}j)"
1032
1033
1034def gamma_method(x, **kwargs):
1035    """Vectorized version of the gamma_method applicable to lists or arrays of Obs.
1036
1037    See docstring of pe.Obs.gamma_method for details.
1038    """
1039    return np.vectorize(lambda o: o.gm(**kwargs))(x)
1040
1041
1042def gm(x, **kwargs):
1043    """Short version of the vectorized gamma_method.
1044
1045    See docstring of pe.Obs.gamma_method for details
1046    """
1047    return gamma_method(x, **kwargs)
1048
1049
1050def _format_uncertainty(value, dvalue, significance=2):
1051    """Creates a string of a value and its error in paranthesis notation, e.g., 13.02(45)"""
1052    if dvalue == 0.0 or (not np.isfinite(dvalue)):
1053        return str(value)
1054    if not isinstance(significance, int):
1055        raise TypeError("significance needs to be an integer.")
1056    if significance < 1:
1057        raise ValueError("significance needs to be larger than zero.")
1058    fexp = np.floor(np.log10(dvalue))
1059    if fexp < 0.0:
1060        return '{:{form}}({:1.0f})'.format(value, dvalue * 10 ** (-fexp + significance - 1), form='.' + str(-int(fexp) + significance - 1) + 'f')
1061    elif fexp == 0.0:
1062        return f"{value:.{significance - 1}f}({dvalue:1.{significance - 1}f})"
1063    else:
1064        return f"{value:.{max(0, int(significance - fexp - 1))}f}({dvalue:2.{max(0, int(significance - fexp - 1))}f})"
1065
1066
1067def _expand_deltas(deltas, idx, shape, gapsize):
1068    """Expand deltas defined on idx to a regular range with spacing gapsize between two
1069       configurations and where holes are filled by 0.
1070       If idx is of type range, the deltas are not changed if the idx.step == gapsize.
1071
1072    Parameters
1073    ----------
1074    deltas : list
1075        List of fluctuations
1076    idx : list
1077        List or range of configs on which the deltas are defined, has to be sorted in ascending order.
1078    shape : int
1079        Number of configs in idx.
1080    gapsize : int
1081        The target distance between two configurations. If longer distances
1082        are found in idx, the data is expanded.
1083    """
1084    if isinstance(idx, range):
1085        if (idx.step == gapsize):
1086            return deltas
1087    ret = np.zeros((idx[-1] - idx[0] + gapsize) // gapsize)
1088    for i in range(shape):
1089        ret[(idx[i] - idx[0]) // gapsize] = deltas[i]
1090    return ret
1091
1092
1093def _merge_idx(idl):
1094    """Returns the union of all lists in idl as range or sorted list
1095
1096    Parameters
1097    ----------
1098    idl : list
1099        List of lists or ranges.
1100    """
1101
1102    if _check_lists_equal(idl):
1103        return idl[0]
1104
1105    idunion = sorted(set().union(*idl))
1106
1107    # Check whether idunion can be expressed as range
1108    idrange = range(idunion[0], idunion[-1] + 1, idunion[1] - idunion[0])
1109    idtest = [list(idrange), idunion]
1110    if _check_lists_equal(idtest):
1111        return idrange
1112
1113    return idunion
1114
1115
1116def _intersection_idx(idl):
1117    """Returns the intersection of all lists in idl as range or sorted list
1118
1119    Parameters
1120    ----------
1121    idl : list
1122        List of lists or ranges.
1123    """
1124
1125    if _check_lists_equal(idl):
1126        return idl[0]
1127
1128    idinter = sorted(set.intersection(*[set(o) for o in idl]))
1129
1130    # Check whether idinter can be expressed as range
1131    try:
1132        idrange = range(idinter[0], idinter[-1] + 1, idinter[1] - idinter[0])
1133        idtest = [list(idrange), idinter]
1134        if _check_lists_equal(idtest):
1135            return idrange
1136    except IndexError:
1137        pass
1138
1139    return idinter
1140
1141
1142def _expand_deltas_for_merge(deltas, idx, shape, new_idx):
1143    """Expand deltas defined on idx to the list of configs that is defined by new_idx.
1144       New, empty entries are filled by 0. If idx and new_idx are of type range, the smallest
1145       common divisor of the step sizes is used as new step size.
1146
1147    Parameters
1148    ----------
1149    deltas : list
1150        List of fluctuations
1151    idx : list
1152        List or range of configs on which the deltas are defined.
1153        Has to be a subset of new_idx and has to be sorted in ascending order.
1154    shape : list
1155        Number of configs in idx.
1156    new_idx : list
1157        List of configs that defines the new range, has to be sorted in ascending order.
1158    """
1159
1160    if type(idx) is range and type(new_idx) is range:
1161        if idx == new_idx:
1162            return deltas
1163    ret = np.zeros(new_idx[-1] - new_idx[0] + 1)
1164    for i in range(shape):
1165        ret[idx[i] - new_idx[0]] = deltas[i]
1166    return np.array([ret[new_idx[i] - new_idx[0]] for i in range(len(new_idx))]) * len(new_idx) / len(idx)
1167
1168
1169def derived_observable(func, data, array_mode=False, **kwargs):
1170    """Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.
1171
1172    Parameters
1173    ----------
1174    func : object
1175        arbitrary function of the form func(data, **kwargs). For the
1176        automatic differentiation to work, all numpy functions have to have
1177        the autograd wrapper (use 'import autograd.numpy as anp').
1178    data : list
1179        list of Obs, e.g. [obs1, obs2, obs3].
1180    num_grad : bool
1181        if True, numerical derivatives are used instead of autograd
1182        (default False). To control the numerical differentiation the
1183        kwargs of numdifftools.step_generators.MaxStepGenerator
1184        can be used.
1185    man_grad : list
1186        manually supply a list or an array which contains the jacobian
1187        of func. Use cautiously, supplying the wrong derivative will
1188        not be intercepted.
1189
1190    Notes
1191    -----
1192    For simple mathematical operations it can be practical to use anonymous
1193    functions. For the ratio of two observables one can e.g. use
1194
1195    new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])
1196    """
1197
1198    data = np.asarray(data)
1199    raveled_data = data.ravel()
1200
1201    # Workaround for matrix operations containing non Obs data
1202    if not all(isinstance(x, Obs) for x in raveled_data):
1203        for i in range(len(raveled_data)):
1204            if isinstance(raveled_data[i], (int, float)):
1205                raveled_data[i] = cov_Obs(raveled_data[i], 0.0, "###dummy_covobs###")
1206
1207    allcov = {}
1208    for o in raveled_data:
1209        for name in o.cov_names:
1210            if name in allcov:
1211                if not np.allclose(allcov[name], o.covobs[name].cov):
1212                    raise Exception('Inconsistent covariance matrices for %s!' % (name))
1213            else:
1214                allcov[name] = o.covobs[name].cov
1215
1216    n_obs = len(raveled_data)
1217    new_names = sorted(set([y for x in [o.names for o in raveled_data] for y in x]))
1218    new_cov_names = sorted(set([y for x in [o.cov_names for o in raveled_data] for y in x]))
1219    new_sample_names = sorted(set(new_names) - set(new_cov_names))
1220
1221    reweighted = len(list(filter(lambda o: o.reweighted is True, raveled_data))) > 0
1222
1223    if data.ndim == 1:
1224        values = np.array([o.value for o in data])
1225    else:
1226        values = np.vectorize(lambda x: x.value)(data)
1227
1228    new_values = func(values, **kwargs)
1229
1230    multi = int(isinstance(new_values, np.ndarray))
1231
1232    new_r_values = {}
1233    new_idl_d = {}
1234    for name in new_sample_names:
1235        idl = []
1236        tmp_values = np.zeros(n_obs)
1237        for i, item in enumerate(raveled_data):
1238            tmp_values[i] = item.r_values.get(name, item.value)
1239            tmp_idl = item.idl.get(name)
1240            if tmp_idl is not None:
1241                idl.append(tmp_idl)
1242        if multi > 0:
1243            tmp_values = np.array(tmp_values).reshape(data.shape)
1244        new_r_values[name] = func(tmp_values, **kwargs)
1245        new_idl_d[name] = _merge_idx(idl)
1246
1247    if 'man_grad' in kwargs:
1248        deriv = np.asarray(kwargs.get('man_grad'))
1249        if new_values.shape + data.shape != deriv.shape:
1250            raise Exception('Manual derivative does not have correct shape.')
1251    elif kwargs.get('num_grad') is True:
1252        if multi > 0:
1253            raise Exception('Multi mode currently not supported for numerical derivative')
1254        options = {
1255            'base_step': 0.1,
1256            'step_ratio': 2.5}
1257        for key in options.keys():
1258            kwarg = kwargs.get(key)
1259            if kwarg is not None:
1260                options[key] = kwarg
1261        tmp_df = nd.Gradient(func, order=4, **{k: v for k, v in options.items() if v is not None})(values, **kwargs)
1262        if tmp_df.size == 1:
1263            deriv = np.array([tmp_df.real])
1264        else:
1265            deriv = tmp_df.real
1266    else:
1267        deriv = jacobian(func)(values, **kwargs)
1268
1269    final_result = np.zeros(new_values.shape, dtype=object)
1270
1271    if array_mode is True:
1272
1273        class _Zero_grad():
1274            def __init__(self, N):
1275                self.grad = np.zeros((N, 1))
1276
1277        new_covobs_lengths = dict(set([y for x in [[(n, o.covobs[n].N) for n in o.cov_names] for o in raveled_data] for y in x]))
1278        d_extracted = {}
1279        g_extracted = {}
1280        for name in new_sample_names:
1281            d_extracted[name] = []
1282            ens_length = len(new_idl_d[name])
1283            for i_dat, dat in enumerate(data):
1284                d_extracted[name].append(np.array([_expand_deltas_for_merge(o.deltas.get(name, np.zeros(ens_length)), o.idl.get(name, new_idl_d[name]), o.shape.get(name, ens_length), new_idl_d[name]) for o in dat.reshape(np.prod(dat.shape))]).reshape(dat.shape + (ens_length, )))
1285        for name in new_cov_names:
1286            g_extracted[name] = []
1287            zero_grad = _Zero_grad(new_covobs_lengths[name])
1288            for i_dat, dat in enumerate(data):
1289                g_extracted[name].append(np.array([o.covobs.get(name, zero_grad).grad for o in dat.reshape(np.prod(dat.shape))]).reshape(dat.shape + (new_covobs_lengths[name], 1)))
1290
1291    for i_val, new_val in np.ndenumerate(new_values):
1292        new_deltas = {}
1293        new_grad = {}
1294        if array_mode is True:
1295            for name in new_sample_names:
1296                ens_length = d_extracted[name][0].shape[-1]
1297                new_deltas[name] = np.zeros(ens_length)
1298                for i_dat, dat in enumerate(d_extracted[name]):
1299                    new_deltas[name] += np.tensordot(deriv[i_val + (i_dat, )], dat)
1300            for name in new_cov_names:
1301                new_grad[name] = 0
1302                for i_dat, dat in enumerate(g_extracted[name]):
1303                    new_grad[name] += np.tensordot(deriv[i_val + (i_dat, )], dat)
1304        else:
1305            for j_obs, obs in np.ndenumerate(data):
1306                for name in obs.names:
1307                    if name in obs.cov_names:
1308                        new_grad[name] = new_grad.get(name, 0) + deriv[i_val + j_obs] * obs.covobs[name].grad
1309                    else:
1310                        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])
1311
1312        new_covobs = {name: Covobs(0, allcov[name], name, grad=new_grad[name]) for name in new_grad}
1313
1314        if not set(new_covobs.keys()).isdisjoint(new_deltas.keys()):
1315            raise Exception('The same name has been used for deltas and covobs!')
1316        new_samples = []
1317        new_means = []
1318        new_idl = []
1319        new_names_obs = []
1320        for name in new_names:
1321            if name not in new_covobs:
1322                new_samples.append(new_deltas[name])
1323                new_idl.append(new_idl_d[name])
1324                new_means.append(new_r_values[name][i_val])
1325                new_names_obs.append(name)
1326        final_result[i_val] = Obs(new_samples, new_names_obs, means=new_means, idl=new_idl)
1327        for name in new_covobs:
1328            final_result[i_val].names.append(name)
1329        final_result[i_val]._covobs = new_covobs
1330        final_result[i_val]._value = new_val
1331        final_result[i_val].reweighted = reweighted
1332
1333    if multi == 0:
1334        final_result = final_result.item()
1335
1336    return final_result
1337
1338
1339def _reduce_deltas(deltas, idx_old, idx_new):
1340    """Extract deltas defined on idx_old on all configs of idx_new.
1341
1342    Assumes, that idx_old and idx_new are correctly defined idl, i.e., they
1343    are ordered in an ascending order.
1344
1345    Parameters
1346    ----------
1347    deltas : list
1348        List of fluctuations
1349    idx_old : list
1350        List or range of configs on which the deltas are defined
1351    idx_new : list
1352        List of configs for which we want to extract the deltas.
1353        Has to be a subset of idx_old.
1354    """
1355    if not len(deltas) == len(idx_old):
1356        raise Exception('Length of deltas and idx_old have to be the same: %d != %d' % (len(deltas), len(idx_old)))
1357    if type(idx_old) is range and type(idx_new) is range:
1358        if idx_old == idx_new:
1359            return deltas
1360    if _check_lists_equal([idx_old, idx_new]):
1361        return deltas
1362    indices = np.intersect1d(idx_old, idx_new, assume_unique=True, return_indices=True)[1]
1363    if len(indices) < len(idx_new):
1364        raise Exception('Error in _reduce_deltas: Config of idx_new not in idx_old')
1365    return np.array(deltas)[indices]
1366
1367
1368def reweight(weight, obs, **kwargs):
1369    """Reweight a list of observables.
1370
1371    Parameters
1372    ----------
1373    weight : Obs
1374        Reweighting factor. An Observable that has to be defined on a superset of the
1375        configurations in obs[i].idl for all i.
1376    obs : list
1377        list of Obs, e.g. [obs1, obs2, obs3].
1378    all_configs : bool
1379        if True, the reweighted observables are normalized by the average of
1380        the reweighting factor on all configurations in weight.idl and not
1381        on the configurations in obs[i].idl. Default False.
1382    """
1383    result = []
1384    for i in range(len(obs)):
1385        if len(obs[i].cov_names):
1386            raise Exception('Error: Not possible to reweight an Obs that contains covobs!')
1387        if not set(obs[i].names).issubset(weight.names):
1388            raise Exception('Error: Ensembles do not fit')
1389        for name in obs[i].names:
1390            if not set(obs[i].idl[name]).issubset(weight.idl[name]):
1391                raise Exception('obs[%d] has to be defined on a subset of the configs in weight.idl[%s]!' % (i, name))
1392        new_samples = []
1393        w_deltas = {}
1394        for name in sorted(obs[i].names):
1395            w_deltas[name] = _reduce_deltas(weight.deltas[name], weight.idl[name], obs[i].idl[name])
1396            new_samples.append((w_deltas[name] + weight.r_values[name]) * (obs[i].deltas[name] + obs[i].r_values[name]))
1397        tmp_obs = Obs(new_samples, sorted(obs[i].names), idl=[obs[i].idl[name] for name in sorted(obs[i].names)])
1398
1399        if kwargs.get('all_configs'):
1400            new_weight = weight
1401        else:
1402            new_weight = Obs([w_deltas[name] + weight.r_values[name] for name in sorted(obs[i].names)], sorted(obs[i].names), idl=[obs[i].idl[name] for name in sorted(obs[i].names)])
1403
1404        result.append(tmp_obs / new_weight)
1405        result[-1].reweighted = True
1406
1407    return result
1408
1409
1410def correlate(obs_a, obs_b):
1411    """Correlate two observables.
1412
1413    Parameters
1414    ----------
1415    obs_a : Obs
1416        First observable
1417    obs_b : Obs
1418        Second observable
1419
1420    Notes
1421    -----
1422    Keep in mind to only correlate primary observables which have not been reweighted
1423    yet. The reweighting has to be applied after correlating the observables.
1424    Currently only works if ensembles are identical (this is not strictly necessary).
1425    """
1426
1427    if sorted(obs_a.names) != sorted(obs_b.names):
1428        raise Exception(f"Ensembles do not fit {set(sorted(obs_a.names)) ^ set(sorted(obs_b.names))}")
1429    if len(obs_a.cov_names) or len(obs_b.cov_names):
1430        raise Exception('Error: Not possible to correlate Obs that contain covobs!')
1431    for name in obs_a.names:
1432        if obs_a.shape[name] != obs_b.shape[name]:
1433            raise Exception('Shapes of ensemble', name, 'do not fit')
1434        if obs_a.idl[name] != obs_b.idl[name]:
1435            raise Exception('idl of ensemble', name, 'do not fit')
1436
1437    if obs_a.reweighted is True:
1438        warnings.warn("The first observable is already reweighted.", RuntimeWarning)
1439    if obs_b.reweighted is True:
1440        warnings.warn("The second observable is already reweighted.", RuntimeWarning)
1441
1442    new_samples = []
1443    new_idl = []
1444    for name in sorted(obs_a.names):
1445        new_samples.append((obs_a.deltas[name] + obs_a.r_values[name]) * (obs_b.deltas[name] + obs_b.r_values[name]))
1446        new_idl.append(obs_a.idl[name])
1447
1448    o = Obs(new_samples, sorted(obs_a.names), idl=new_idl)
1449    o.reweighted = obs_a.reweighted or obs_b.reweighted
1450    return o
1451
1452
1453def covariance(obs, visualize=False, correlation=False, smooth=None, **kwargs):
1454    r'''Calculates the error covariance matrix of a set of observables.
1455
1456    WARNING: This function should be used with care, especially for observables with support on multiple
1457             ensembles with differing autocorrelations. See the notes below for details.
1458
1459    The gamma method has to be applied first to all observables.
1460
1461    Parameters
1462    ----------
1463    obs : list or numpy.ndarray
1464        List or one dimensional array of Obs
1465    visualize : bool
1466        If True plots the corresponding normalized correlation matrix (default False).
1467    correlation : bool
1468        If True the correlation matrix instead of the error covariance matrix is returned (default False).
1469    smooth : None or int
1470        If smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue
1471        smoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the
1472        largest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely
1473        small ones.
1474
1475    Notes
1476    -----
1477    The error covariance is defined such that it agrees with the squared standard error for two identical observables
1478    $$\operatorname{cov}(a,a)=\sum_{s=1}^N\delta_a^s\delta_a^s/N^2=\Gamma_{aa}(0)/N=\operatorname{var}(a)/N=\sigma_a^2$$
1479    in the absence of autocorrelation.
1480    The error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite
1481    $$\sum_{i,j}v_i\Gamma_{ij}(0)v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i,j}v_i\delta_i^s\delta_j^s v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i}|v_i\delta_i^s|^2\geq 0\,,$$ for every $v\in\mathbb{R}^M$, while such an identity does not hold for larger windows/lags.
1482    For observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.
1483    $$\tau_{\mathrm{int}, ij}=\sqrt{\tau_{\mathrm{int}, i}\times \tau_{\mathrm{int}, j}}$$
1484    This construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).
1485    '''
1486
1487    length = len(obs)
1488
1489    max_samples = np.max([o.N for o in obs])
1490    if max_samples <= length and not [item for sublist in [o.cov_names for o in obs] for item in sublist]:
1491        warnings.warn(f"The dimension of the covariance matrix ({length}) is larger or equal to the number of samples ({max_samples}). This will result in a rank deficient matrix.", RuntimeWarning)
1492
1493    cov = np.zeros((length, length))
1494    for i in range(length):
1495        for j in range(i, length):
1496            cov[i, j] = _covariance_element(obs[i], obs[j])
1497    cov = cov + cov.T - np.diag(np.diag(cov))
1498
1499    corr = np.diag(1 / np.sqrt(np.diag(cov))) @ cov @ np.diag(1 / np.sqrt(np.diag(cov)))
1500
1501    if isinstance(smooth, int):
1502        corr = _smooth_eigenvalues(corr, smooth)
1503
1504    if visualize:
1505        plt.matshow(corr, vmin=-1, vmax=1)
1506        plt.set_cmap('RdBu')
1507        plt.colorbar()
1508        plt.draw()
1509
1510    if correlation is True:
1511        return corr
1512
1513    errors = [o.dvalue for o in obs]
1514    cov = np.diag(errors) @ corr @ np.diag(errors)
1515
1516    eigenvalues = np.linalg.eigh(cov)[0]
1517    if not np.all(eigenvalues >= 0):
1518        warnings.warn("Covariance matrix is not positive semi-definite (Eigenvalues: " + str(eigenvalues) + ")", RuntimeWarning)
1519
1520    return cov
1521
1522
1523def _smooth_eigenvalues(corr, E):
1524    """Eigenvalue smoothing as described in hep-lat/9412087
1525
1526    corr : np.ndarray
1527        correlation matrix
1528    E : integer
1529        Number of eigenvalues to be left substantially unchanged
1530    """
1531    if not (2 < E < corr.shape[0] - 1):
1532        raise Exception(f"'E' has to be between 2 and the dimension of the correlation matrix minus 1 ({corr.shape[0] - 1}).")
1533    vals, vec = np.linalg.eigh(corr)
1534    lambda_min = np.mean(vals[:-E])
1535    vals[vals < lambda_min] = lambda_min
1536    vals /= np.mean(vals)
1537    return vec @ np.diag(vals) @ vec.T
1538
1539
1540def _covariance_element(obs1, obs2):
1541    """Estimates the covariance of two Obs objects, neglecting autocorrelations."""
1542
1543    def calc_gamma(deltas1, deltas2, idx1, idx2, new_idx):
1544        deltas1 = _reduce_deltas(deltas1, idx1, new_idx)
1545        deltas2 = _reduce_deltas(deltas2, idx2, new_idx)
1546        return np.sum(deltas1 * deltas2)
1547
1548    if set(obs1.names).isdisjoint(set(obs2.names)):
1549        return 0.0
1550
1551    if not hasattr(obs1, 'e_dvalue') or not hasattr(obs2, 'e_dvalue'):
1552        raise Exception('The gamma method has to be applied to both Obs first.')
1553
1554    dvalue = 0.0
1555
1556    for e_name in obs1.mc_names:
1557
1558        if e_name not in obs2.mc_names:
1559            continue
1560
1561        idl_d = {}
1562        for r_name in obs1.e_content[e_name]:
1563            if r_name not in obs2.e_content[e_name]:
1564                continue
1565            idl_d[r_name] = _intersection_idx([obs1.idl[r_name], obs2.idl[r_name]])
1566
1567        gamma = 0.0
1568
1569        for r_name in obs1.e_content[e_name]:
1570            if r_name not in obs2.e_content[e_name]:
1571                continue
1572            if len(idl_d[r_name]) == 0:
1573                continue
1574            gamma += calc_gamma(obs1.deltas[r_name], obs2.deltas[r_name], obs1.idl[r_name], obs2.idl[r_name], idl_d[r_name])
1575
1576        if gamma == 0.0:
1577            continue
1578
1579        gamma_div = 0.0
1580        for r_name in obs1.e_content[e_name]:
1581            if r_name not in obs2.e_content[e_name]:
1582                continue
1583            if len(idl_d[r_name]) == 0:
1584                continue
1585            gamma_div += np.sqrt(calc_gamma(obs1.deltas[r_name], obs1.deltas[r_name], obs1.idl[r_name], obs1.idl[r_name], idl_d[r_name]) * calc_gamma(obs2.deltas[r_name], obs2.deltas[r_name], obs2.idl[r_name], obs2.idl[r_name], idl_d[r_name]))
1586        gamma /= gamma_div
1587
1588        dvalue += gamma
1589
1590    for e_name in obs1.cov_names:
1591
1592        if e_name not in obs2.cov_names:
1593            continue
1594
1595        dvalue += np.dot(np.transpose(obs1.covobs[e_name].grad), np.dot(obs1.covobs[e_name].cov, obs2.covobs[e_name].grad)).item()
1596
1597    return dvalue
1598
1599
1600def import_jackknife(jacks, name, idl=None):
1601    """Imports jackknife samples and returns an Obs
1602
1603    Parameters
1604    ----------
1605    jacks : numpy.ndarray
1606        numpy array containing the mean value as zeroth entry and
1607        the N jackknife samples as first to Nth entry.
1608    name : str
1609        name of the ensemble the samples are defined on.
1610    """
1611    length = len(jacks) - 1
1612    prj = (np.ones((length, length)) - (length - 1) * np.identity(length))
1613    samples = jacks[1:] @ prj
1614    mean = np.mean(samples)
1615    new_obs = Obs([samples - mean], [name], idl=idl, means=[mean])
1616    new_obs._value = jacks[0]
1617    return new_obs
1618
1619
1620def import_bootstrap(boots, name, random_numbers):
1621    """Imports bootstrap samples and returns an Obs
1622
1623    Parameters
1624    ----------
1625    boots : numpy.ndarray
1626        numpy array containing the mean value as zeroth entry and
1627        the N bootstrap samples as first to Nth entry.
1628    name : str
1629        name of the ensemble the samples are defined on.
1630    random_numbers : np.ndarray
1631        Array of shape (samples, length) containing the random numbers to generate the bootstrap samples,
1632        where samples is the number of bootstrap samples and length is the length of the original Monte Carlo
1633        chain to be reconstructed.
1634    """
1635    samples, length = random_numbers.shape
1636    if samples != len(boots) - 1:
1637        raise ValueError("Random numbers do not have the correct shape.")
1638
1639    if samples < length:
1640        raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.")
1641
1642    proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length
1643
1644    samples = scipy.linalg.lstsq(proj, boots[1:])[0]
1645    ret = Obs([samples], [name])
1646    ret._value = boots[0]
1647    return ret
1648
1649
1650def merge_obs(list_of_obs):
1651    """Combine all observables in list_of_obs into one new observable
1652
1653    Parameters
1654    ----------
1655    list_of_obs : list
1656        list of the Obs object to be combined
1657
1658    Notes
1659    -----
1660    It is not possible to combine obs which are based on the same replicum
1661    """
1662    replist = [item for obs in list_of_obs for item in obs.names]
1663    if (len(replist) == len(set(replist))) is False:
1664        raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist)))
1665    if any([len(o.cov_names) for o in list_of_obs]):
1666        raise Exception('Not possible to merge data that contains covobs!')
1667    new_dict = {}
1668    idl_dict = {}
1669    for o in list_of_obs:
1670        new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0)
1671                        for key in set(o.deltas) | set(o.r_values)})
1672        idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)})
1673
1674    names = sorted(new_dict.keys())
1675    o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names])
1676    o.reweighted = np.max([oi.reweighted for oi in list_of_obs])
1677    return o
1678
1679
1680def cov_Obs(means, cov, name, grad=None):
1681    """Create an Obs based on mean(s) and a covariance matrix
1682
1683    Parameters
1684    ----------
1685    mean : list of floats or float
1686        N mean value(s) of the new Obs
1687    cov : list or array
1688        2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
1689    name : str
1690        identifier for the covariance matrix
1691    grad : list or array
1692        Gradient of the Covobs wrt. the means belonging to cov.
1693    """
1694
1695    def covobs_to_obs(co):
1696        """Make an Obs out of a Covobs
1697
1698        Parameters
1699        ----------
1700        co : Covobs
1701            Covobs to be embedded into the Obs
1702        """
1703        o = Obs([], [], means=[])
1704        o._value = co.value
1705        o.names.append(co.name)
1706        o._covobs[co.name] = co
1707        o._dvalue = np.sqrt(co.errsq())
1708        return o
1709
1710    ol = []
1711    if isinstance(means, (float, int)):
1712        means = [means]
1713
1714    for i in range(len(means)):
1715        ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad)))
1716    if ol[0].covobs[name].N != len(means):
1717        raise Exception('You have to provide %d mean values!' % (ol[0].N))
1718    if len(ol) == 1:
1719        return ol[0]
1720    return ol
1721
1722
1723def _determine_gap(o, e_content, e_name):
1724    gaps = []
1725    for r_name in e_content[e_name]:
1726        if isinstance(o.idl[r_name], range):
1727            gaps.append(o.idl[r_name].step)
1728        else:
1729            gaps.append(np.min(np.diff(o.idl[r_name])))
1730
1731    gap = min(gaps)
1732    if not np.all([gi % gap == 0 for gi in gaps]):
1733        raise Exception(f"Replica for ensemble {e_name} do not have a common spacing.", gaps)
1734
1735    return gap
1736
1737
1738def _check_lists_equal(idl):
1739    '''
1740    Use groupby to efficiently check whether all elements of idl are identical.
1741    Returns True if all elements are equal, otherwise False.
1742
1743    Parameters
1744    ----------
1745    idl : list of lists, ranges or np.ndarrays
1746    '''
1747    g = groupby([np.nditer(el) if isinstance(el, np.ndarray) else el for el in idl])
1748    if next(g, True) and not next(g, False):
1749        return True
1750    return False
class Obs:
 19class Obs:
 20    """Class for a general observable.
 21
 22    Instances of Obs are the basic objects of a pyerrors error analysis.
 23    They are initialized with a list which contains arrays of samples for
 24    different ensembles/replica and another list of same length which contains
 25    the names of the ensembles/replica. Mathematical operations can be
 26    performed on instances. The result is another instance of Obs. The error of
 27    an instance can be computed with the gamma_method. Also contains additional
 28    methods for output and visualization of the error calculation.
 29
 30    Attributes
 31    ----------
 32    S_global : float
 33        Standard value for S (default 2.0)
 34    S_dict : dict
 35        Dictionary for S values. If an entry for a given ensemble
 36        exists this overwrites the standard value for that ensemble.
 37    tau_exp_global : float
 38        Standard value for tau_exp (default 0.0)
 39    tau_exp_dict : dict
 40        Dictionary for tau_exp values. If an entry for a given ensemble exists
 41        this overwrites the standard value for that ensemble.
 42    N_sigma_global : float
 43        Standard value for N_sigma (default 1.0)
 44    N_sigma_dict : dict
 45        Dictionary for N_sigma values. If an entry for a given ensemble exists
 46        this overwrites the standard value for that ensemble.
 47    """
 48    __slots__ = ['names', 'shape', 'r_values', 'deltas', 'N', '_value', '_dvalue',
 49                 'ddvalue', 'reweighted', 'S', 'tau_exp', 'N_sigma',
 50                 'e_dvalue', 'e_ddvalue', 'e_tauint', 'e_dtauint',
 51                 'e_windowsize', 'e_rho', 'e_drho', 'e_n_tauint', 'e_n_dtauint',
 52                 'idl', 'tag', '_covobs', '__dict__']
 53
 54    S_global = 2.0
 55    S_dict = {}
 56    tau_exp_global = 0.0
 57    tau_exp_dict = {}
 58    N_sigma_global = 1.0
 59    N_sigma_dict = {}
 60
 61    def __init__(self, samples, names, idl=None, **kwargs):
 62        """ Initialize Obs object.
 63
 64        Parameters
 65        ----------
 66        samples : list
 67            list of numpy arrays containing the Monte Carlo samples
 68        names : list
 69            list of strings labeling the individual samples
 70        idl : list, optional
 71            list of ranges or lists on which the samples are defined
 72        """
 73
 74        if kwargs.get("means") is None and len(samples):
 75            if len(samples) != len(names):
 76                raise ValueError('Length of samples and names incompatible.')
 77            if idl is not None:
 78                if len(idl) != len(names):
 79                    raise ValueError('Length of idl incompatible with samples and names.')
 80            name_length = len(names)
 81            if name_length > 1:
 82                if name_length != len(set(names)):
 83                    raise ValueError('Names are not unique.')
 84                if not all(isinstance(x, str) for x in names):
 85                    raise TypeError('All names have to be strings.')
 86            else:
 87                if not isinstance(names[0], str):
 88                    raise TypeError('All names have to be strings.')
 89            if min(len(x) for x in samples) <= 4:
 90                raise ValueError('Samples have to have at least 5 entries.')
 91
 92        self.names = sorted(names)
 93        self.shape = {}
 94        self.r_values = {}
 95        self.deltas = {}
 96        self._covobs = {}
 97
 98        self._value = 0
 99        self.N = 0
100        self.idl = {}
101        if idl is not None:
102            for name, idx in sorted(zip(names, idl)):
103                if isinstance(idx, range):
104                    self.idl[name] = idx
105                elif isinstance(idx, (list, np.ndarray)):
106                    dc = np.unique(np.diff(idx))
107                    if np.any(dc < 0):
108                        raise ValueError("Unsorted idx for idl[%s]" % (name))
109                    if len(dc) == 1:
110                        self.idl[name] = range(idx[0], idx[-1] + dc[0], dc[0])
111                    else:
112                        self.idl[name] = list(idx)
113                else:
114                    raise TypeError('incompatible type for idl[%s].' % (name))
115        else:
116            for name, sample in sorted(zip(names, samples)):
117                self.idl[name] = range(1, len(sample) + 1)
118
119        if kwargs.get("means") is not None:
120            for name, sample, mean in sorted(zip(names, samples, kwargs.get("means"))):
121                self.shape[name] = len(self.idl[name])
122                self.N += self.shape[name]
123                self.r_values[name] = mean
124                self.deltas[name] = sample
125        else:
126            for name, sample in sorted(zip(names, samples)):
127                self.shape[name] = len(self.idl[name])
128                self.N += self.shape[name]
129                if len(sample) != self.shape[name]:
130                    raise ValueError('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name]))
131                self.r_values[name] = np.mean(sample)
132                self.deltas[name] = sample - self.r_values[name]
133                self._value += self.shape[name] * self.r_values[name]
134            self._value /= self.N
135
136        self._dvalue = 0.0
137        self.ddvalue = 0.0
138        self.reweighted = False
139
140        self.tag = None
141
142    @property
143    def value(self):
144        return self._value
145
146    @property
147    def dvalue(self):
148        return self._dvalue
149
150    @property
151    def e_names(self):
152        return sorted(set([o.split('|')[0] for o in self.names]))
153
154    @property
155    def cov_names(self):
156        return sorted(set([o for o in self.covobs.keys()]))
157
158    @property
159    def mc_names(self):
160        return sorted(set([o.split('|')[0] for o in self.names if o not in self.cov_names]))
161
162    @property
163    def e_content(self):
164        res = {}
165        for e, e_name in enumerate(self.e_names):
166            res[e_name] = sorted(filter(lambda x: x.startswith(e_name + '|'), self.names))
167            if e_name in self.names:
168                res[e_name].append(e_name)
169        return res
170
171    @property
172    def covobs(self):
173        return self._covobs
174
175    def gamma_method(self, **kwargs):
176        """Estimate the error and related properties of the Obs.
177
178        Parameters
179        ----------
180        S : float
181            specifies a custom value for the parameter S (default 2.0).
182            If set to 0 it is assumed that the data exhibits no
183            autocorrelation. In this case the error estimates coincides
184            with the sample standard error.
185        tau_exp : float
186            positive value triggers the critical slowing down analysis
187            (default 0.0).
188        N_sigma : float
189            number of standard deviations from zero until the tail is
190            attached to the autocorrelation function (default 1).
191        fft : bool
192            determines whether the fft algorithm is used for the computation
193            of the autocorrelation function (default True)
194        """
195
196        e_content = self.e_content
197        self.e_dvalue = {}
198        self.e_ddvalue = {}
199        self.e_tauint = {}
200        self.e_dtauint = {}
201        self.e_windowsize = {}
202        self.e_n_tauint = {}
203        self.e_n_dtauint = {}
204        e_gamma = {}
205        self.e_rho = {}
206        self.e_drho = {}
207        self._dvalue = 0
208        self.ddvalue = 0
209
210        self.S = {}
211        self.tau_exp = {}
212        self.N_sigma = {}
213
214        if kwargs.get('fft') is False:
215            fft = False
216        else:
217            fft = True
218
219        def _parse_kwarg(kwarg_name):
220            if kwarg_name in kwargs:
221                tmp = kwargs.get(kwarg_name)
222                if isinstance(tmp, (int, float)):
223                    if tmp < 0:
224                        raise Exception(kwarg_name + ' has to be larger or equal to 0.')
225                    for e, e_name in enumerate(self.e_names):
226                        getattr(self, kwarg_name)[e_name] = tmp
227                else:
228                    raise TypeError(kwarg_name + ' is not in proper format.')
229            else:
230                for e, e_name in enumerate(self.e_names):
231                    if e_name in getattr(Obs, kwarg_name + '_dict'):
232                        getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_dict')[e_name]
233                    else:
234                        getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_global')
235
236        _parse_kwarg('S')
237        _parse_kwarg('tau_exp')
238        _parse_kwarg('N_sigma')
239
240        for e, e_name in enumerate(self.mc_names):
241            gapsize = _determine_gap(self, e_content, e_name)
242
243            r_length = []
244            for r_name in e_content[e_name]:
245                if isinstance(self.idl[r_name], range):
246                    r_length.append(len(self.idl[r_name]) * self.idl[r_name].step // gapsize)
247                else:
248                    r_length.append((self.idl[r_name][-1] - self.idl[r_name][0] + 1) // gapsize)
249
250            e_N = np.sum([self.shape[r_name] for r_name in e_content[e_name]])
251            w_max = max(r_length) // 2
252            e_gamma[e_name] = np.zeros(w_max)
253            self.e_rho[e_name] = np.zeros(w_max)
254            self.e_drho[e_name] = np.zeros(w_max)
255
256            for r_name in e_content[e_name]:
257                e_gamma[e_name] += self._calc_gamma(self.deltas[r_name], self.idl[r_name], self.shape[r_name], w_max, fft, gapsize)
258
259            gamma_div = np.zeros(w_max)
260            for r_name in e_content[e_name]:
261                gamma_div += self._calc_gamma(np.ones((self.shape[r_name])), self.idl[r_name], self.shape[r_name], w_max, fft, gapsize)
262            gamma_div[gamma_div < 1] = 1.0
263            e_gamma[e_name] /= gamma_div[:w_max]
264
265            if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny:  # Prevent division by zero
266                self.e_tauint[e_name] = 0.5
267                self.e_dtauint[e_name] = 0.0
268                self.e_dvalue[e_name] = 0.0
269                self.e_ddvalue[e_name] = 0.0
270                self.e_windowsize[e_name] = 0
271                continue
272
273            self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0]
274            self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:])))
275            # Make sure no entry of tauint is smaller than 0.5
276            self.e_n_tauint[e_name][self.e_n_tauint[e_name] <= 0.5] = 0.5 + np.finfo(np.float64).eps
277            # hep-lat/0306017 eq. (42)
278            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)
279            self.e_n_dtauint[e_name][0] = 0.0
280
281            def _compute_drho(i):
282                tmp = (self.e_rho[e_name][i + 1:w_max]
283                       + np.concatenate([self.e_rho[e_name][i - 1:None if i - (w_max - 1) // 2 <= 0 else (2 * i - (2 * w_max) // 2):-1],
284                                         self.e_rho[e_name][1:max(1, w_max - 2 * i)]])
285                       - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i])
286                self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N)
287
288            if self.tau_exp[e_name] > 0:
289                _compute_drho(1)
290                texp = self.tau_exp[e_name]
291                # Critical slowing down analysis
292                if w_max // 2 <= 1:
293                    raise Exception("Need at least 8 samples for tau_exp error analysis")
294                for n in range(1, w_max // 2):
295                    _compute_drho(n + 1)
296                    if (self.e_rho[e_name][n] - self.N_sigma[e_name] * self.e_drho[e_name][n]) < 0 or n >= w_max // 2 - 2:
297                        # Bias correction hep-lat/0306017 eq. (49) included
298                        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
299                        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)
300                        # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2
301                        self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
302                        self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
303                        self.e_windowsize[e_name] = n
304                        break
305            else:
306                if self.S[e_name] == 0.0:
307                    self.e_tauint[e_name] = 0.5
308                    self.e_dtauint[e_name] = 0.0
309                    self.e_dvalue[e_name] = np.sqrt(e_gamma[e_name][0] / (e_N - 1))
310                    self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt(0.5 / e_N)
311                    self.e_windowsize[e_name] = 0
312                else:
313                    # Standard automatic windowing procedure
314                    tau = self.S[e_name] / np.log((2 * self.e_n_tauint[e_name][1:] + 1) / (2 * self.e_n_tauint[e_name][1:] - 1))
315                    g_w = np.exp(- np.arange(1, len(tau) + 1) / tau) - tau / np.sqrt(np.arange(1, len(tau) + 1) * e_N)
316                    for n in range(1, w_max):
317                        if g_w[n - 1] < 0 or n >= w_max - 1:
318                            _compute_drho(n)
319                            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)
320                            self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n]
321                            self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
322                            self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
323                            self.e_windowsize[e_name] = n
324                            break
325
326            self._dvalue += self.e_dvalue[e_name] ** 2
327            self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2
328
329        for e_name in self.cov_names:
330            self.e_dvalue[e_name] = np.sqrt(self.covobs[e_name].errsq())
331            self.e_ddvalue[e_name] = 0
332            self._dvalue += self.e_dvalue[e_name]**2
333
334        self._dvalue = np.sqrt(self._dvalue)
335        if self._dvalue == 0.0:
336            self.ddvalue = 0.0
337        else:
338            self.ddvalue = np.sqrt(self.ddvalue) / self._dvalue
339        return
340
341    gm = gamma_method
342
343    def _calc_gamma(self, deltas, idx, shape, w_max, fft, gapsize):
344        """Calculate Gamma_{AA} from the deltas, which are defined on idx.
345           idx is assumed to be a contiguous range (possibly with a stepsize != 1)
346
347        Parameters
348        ----------
349        deltas : list
350            List of fluctuations
351        idx : list
352            List or range of configurations on which the deltas are defined.
353        shape : int
354            Number of configurations in idx.
355        w_max : int
356            Upper bound for the summation window.
357        fft : bool
358            determines whether the fft algorithm is used for the computation
359            of the autocorrelation function.
360        gapsize : int
361            The target distance between two configurations. If longer distances
362            are found in idx, the data is expanded.
363        """
364        gamma = np.zeros(w_max)
365        deltas = _expand_deltas(deltas, idx, shape, gapsize)
366        new_shape = len(deltas)
367        if fft:
368            max_gamma = min(new_shape, w_max)
369            # The padding for the fft has to be even
370            padding = new_shape + max_gamma + (new_shape + max_gamma) % 2
371            gamma[:max_gamma] += np.fft.irfft(np.abs(np.fft.rfft(deltas, padding)) ** 2)[:max_gamma]
372        else:
373            for n in range(w_max):
374                if new_shape - n >= 0:
375                    gamma[n] += deltas[0:new_shape - n].dot(deltas[n:new_shape])
376
377        return gamma
378
379    def details(self, ens_content=True):
380        """Output detailed properties of the Obs.
381
382        Parameters
383        ----------
384        ens_content : bool
385            print details about the ensembles and replica if true.
386        """
387        if self.tag is not None:
388            print("Description:", self.tag)
389        if not hasattr(self, 'e_dvalue'):
390            print('Result\t %3.8e' % (self.value))
391        else:
392            if self.value == 0.0:
393                percentage = np.nan
394            else:
395                percentage = np.abs(self._dvalue / self.value) * 100
396            print('Result\t %3.8e +/- %3.8e +/- %3.8e (%3.3f%%)' % (self.value, self._dvalue, self.ddvalue, percentage))
397            if len(self.e_names) > 1:
398                print(' Ensemble errors:')
399            e_content = self.e_content
400            for e_name in self.mc_names:
401                gap = _determine_gap(self, e_content, e_name)
402
403                if len(self.e_names) > 1:
404                    print('', e_name, '\t %3.6e +/- %3.6e' % (self.e_dvalue[e_name], self.e_ddvalue[e_name]))
405                tau_string = " \N{GREEK SMALL LETTER TAU}_int\t " + _format_uncertainty(self.e_tauint[e_name], self.e_dtauint[e_name])
406                tau_string += f" in units of {gap} config"
407                if gap > 1:
408                    tau_string += "s"
409                if self.tau_exp[e_name] > 0:
410                    tau_string = f"{tau_string: <45}" + '\t(\N{GREEK SMALL LETTER TAU}_exp=%3.2f, N_\N{GREEK SMALL LETTER SIGMA}=%1.0i)' % (self.tau_exp[e_name], self.N_sigma[e_name])
411                else:
412                    tau_string = f"{tau_string: <45}" + '\t(S=%3.2f)' % (self.S[e_name])
413                print(tau_string)
414            for e_name in self.cov_names:
415                print('', e_name, '\t %3.8e' % (self.e_dvalue[e_name]))
416        if ens_content is True:
417            if len(self.e_names) == 1:
418                print(self.N, 'samples in', len(self.e_names), 'ensemble:')
419            else:
420                print(self.N, 'samples in', len(self.e_names), 'ensembles:')
421            my_string_list = []
422            for key, value in sorted(self.e_content.items()):
423                if key not in self.covobs:
424                    my_string = '  ' + "\u00B7 Ensemble '" + key + "' "
425                    if len(value) == 1:
426                        my_string += f': {self.shape[value[0]]} configurations'
427                        if isinstance(self.idl[value[0]], range):
428                            my_string += f' (from {self.idl[value[0]].start} to {self.idl[value[0]][-1]}' + int(self.idl[value[0]].step != 1) * f' in steps of {self.idl[value[0]].step}' + ')'
429                        else:
430                            my_string += f' (irregular range from {self.idl[value[0]][0]} to {self.idl[value[0]][-1]})'
431                    else:
432                        sublist = []
433                        for v in value:
434                            my_substring = '    ' + "\u00B7 Replicum '" + v[len(key) + 1:] + "' "
435                            my_substring += f': {self.shape[v]} configurations'
436                            if isinstance(self.idl[v], range):
437                                my_substring += f' (from {self.idl[v].start} to {self.idl[v][-1]}' + int(self.idl[v].step != 1) * f' in steps of {self.idl[v].step}' + ')'
438                            else:
439                                my_substring += f' (irregular range from {self.idl[v][0]} to {self.idl[v][-1]})'
440                            sublist.append(my_substring)
441
442                        my_string += '\n' + '\n'.join(sublist)
443                else:
444                    my_string = '  ' + "\u00B7 Covobs   '" + key + "' "
445                my_string_list.append(my_string)
446            print('\n'.join(my_string_list))
447
448    def reweight(self, weight):
449        """Reweight the obs with given rewighting factors.
450
451        Parameters
452        ----------
453        weight : Obs
454            Reweighting factor. An Observable that has to be defined on a superset of the
455            configurations in obs[i].idl for all i.
456        all_configs : bool
457            if True, the reweighted observables are normalized by the average of
458            the reweighting factor on all configurations in weight.idl and not
459            on the configurations in obs[i].idl. Default False.
460        """
461        return reweight(weight, [self])[0]
462
463    def is_zero_within_error(self, sigma=1):
464        """Checks whether the observable is zero within 'sigma' standard errors.
465
466        Parameters
467        ----------
468        sigma : int
469            Number of standard errors used for the check.
470
471        Works only properly when the gamma method was run.
472        """
473        return self.is_zero() or np.abs(self.value) <= sigma * self._dvalue
474
475    def is_zero(self, atol=1e-10):
476        """Checks whether the observable is zero within a given tolerance.
477
478        Parameters
479        ----------
480        atol : float
481            Absolute tolerance (for details see numpy documentation).
482        """
483        return np.isclose(0.0, self.value, 1e-14, atol) and all(np.allclose(0.0, delta, 1e-14, atol) for delta in self.deltas.values()) and all(np.allclose(0.0, delta.errsq(), 1e-14, atol) for delta in self.covobs.values())
484
485    def plot_tauint(self, save=None):
486        """Plot integrated autocorrelation time for each ensemble.
487
488        Parameters
489        ----------
490        save : str
491            saves the figure to a file named 'save' if.
492        """
493        if not hasattr(self, 'e_dvalue'):
494            raise Exception('Run the gamma method first.')
495
496        for e, e_name in enumerate(self.mc_names):
497            fig = plt.figure()
498            plt.xlabel(r'$W$')
499            plt.ylabel(r'$\tau_\mathrm{int}$')
500            length = int(len(self.e_n_tauint[e_name]))
501            if self.tau_exp[e_name] > 0:
502                base = self.e_n_tauint[e_name][self.e_windowsize[e_name]]
503                x_help = np.arange(2 * self.tau_exp[e_name])
504                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
505                x_arr = np.arange(self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name])
506                plt.plot(x_arr, y_help, 'C' + str(e), linewidth=1, ls='--', marker=',')
507                plt.errorbar([self.e_windowsize[e_name] + 2 * self.tau_exp[e_name]], [self.e_tauint[e_name]],
508                             yerr=[self.e_dtauint[e_name]], fmt='C' + str(e), linewidth=1, capsize=2, marker='o', mfc=plt.rcParams['axes.facecolor'])
509                xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
510                label = e_name + r', $\tau_\mathrm{exp}$=' + str(np.around(self.tau_exp[e_name], decimals=2))
511            else:
512                label = e_name + ', S=' + str(np.around(self.S[e_name], decimals=2))
513                xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)
514
515            plt.errorbar(np.arange(length)[:int(xmax) + 1], self.e_n_tauint[e_name][:int(xmax) + 1], yerr=self.e_n_dtauint[e_name][:int(xmax) + 1], linewidth=1, capsize=2, label=label)
516            plt.axvline(x=self.e_windowsize[e_name], color='C' + str(e), alpha=0.5, marker=',', ls='--')
517            plt.legend()
518            plt.xlim(-0.5, xmax)
519            ylim = plt.ylim()
520            plt.ylim(bottom=0.0, top=max(1.0, ylim[1]))
521            plt.draw()
522            if save:
523                fig.savefig(save + "_" + str(e))
524
525    def plot_rho(self, save=None):
526        """Plot normalized autocorrelation function time for each ensemble.
527
528        Parameters
529        ----------
530        save : str
531            saves the figure to a file named 'save' if.
532        """
533        if not hasattr(self, 'e_dvalue'):
534            raise Exception('Run the gamma method first.')
535        for e, e_name in enumerate(self.mc_names):
536            fig = plt.figure()
537            plt.xlabel('W')
538            plt.ylabel('rho')
539            length = int(len(self.e_drho[e_name]))
540            plt.errorbar(np.arange(length), self.e_rho[e_name][:length], yerr=self.e_drho[e_name][:], linewidth=1, capsize=2)
541            plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25, ls='--', marker=',')
542            if self.tau_exp[e_name] > 0:
543                plt.plot([self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]],
544                         [self.e_rho[e_name][self.e_windowsize[e_name] + 1], 0], 'k-', lw=1)
545                xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
546                plt.title('Rho ' + e_name + r', tau\_exp=' + str(np.around(self.tau_exp[e_name], decimals=2)))
547            else:
548                xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)
549                plt.title('Rho ' + e_name + ', S=' + str(np.around(self.S[e_name], decimals=2)))
550            plt.plot([-0.5, xmax], [0, 0], 'k--', lw=1)
551            plt.xlim(-0.5, xmax)
552            plt.draw()
553            if save:
554                fig.savefig(save + "_" + str(e))
555
556    def plot_rep_dist(self):
557        """Plot replica distribution for each ensemble with more than one replicum."""
558        if not hasattr(self, 'e_dvalue'):
559            raise Exception('Run the gamma method first.')
560        for e, e_name in enumerate(self.mc_names):
561            if len(self.e_content[e_name]) == 1:
562                print('No replica distribution for a single replicum (', e_name, ')')
563                continue
564            r_length = []
565            sub_r_mean = 0
566            for r, r_name in enumerate(self.e_content[e_name]):
567                r_length.append(len(self.deltas[r_name]))
568                sub_r_mean += self.shape[r_name] * self.r_values[r_name]
569            e_N = np.sum(r_length)
570            sub_r_mean /= e_N
571            arr = np.zeros(len(self.e_content[e_name]))
572            for r, r_name in enumerate(self.e_content[e_name]):
573                arr[r] = (self.r_values[r_name] - sub_r_mean) / (self.e_dvalue[e_name] * np.sqrt(e_N / self.shape[r_name] - 1))
574            plt.hist(arr, rwidth=0.8, bins=len(self.e_content[e_name]))
575            plt.title('Replica distribution' + e_name + ' (mean=0, var=1)')
576            plt.draw()
577
578    def plot_history(self, expand=True):
579        """Plot derived Monte Carlo history for each ensemble
580
581        Parameters
582        ----------
583        expand : bool
584            show expanded history for irregular Monte Carlo chains (default: True).
585        """
586        for e, e_name in enumerate(self.mc_names):
587            plt.figure()
588            r_length = []
589            tmp = []
590            tmp_expanded = []
591            for r, r_name in enumerate(self.e_content[e_name]):
592                tmp.append(self.deltas[r_name] + self.r_values[r_name])
593                if expand:
594                    tmp_expanded.append(_expand_deltas(self.deltas[r_name], list(self.idl[r_name]), self.shape[r_name], 1) + self.r_values[r_name])
595                    r_length.append(len(tmp_expanded[-1]))
596                else:
597                    r_length.append(len(tmp[-1]))
598            e_N = np.sum(r_length)
599            x = np.arange(e_N)
600            y_test = np.concatenate(tmp, axis=0)
601            if expand:
602                y = np.concatenate(tmp_expanded, axis=0)
603            else:
604                y = y_test
605            plt.errorbar(x, y, fmt='.', markersize=3)
606            plt.xlim(-0.5, e_N - 0.5)
607            plt.title(e_name + f'\nskew: {skew(y_test):.3f} (p={skewtest(y_test).pvalue:.3f}), kurtosis: {kurtosis(y_test):.3f} (p={kurtosistest(y_test).pvalue:.3f})')
608            plt.draw()
609
610    def plot_piechart(self, save=None):
611        """Plot piechart which shows the fractional contribution of each
612        ensemble to the error and returns a dictionary containing the fractions.
613
614        Parameters
615        ----------
616        save : str
617            saves the figure to a file named 'save' if.
618        """
619        if not hasattr(self, 'e_dvalue'):
620            raise Exception('Run the gamma method first.')
621        if np.isclose(0.0, self._dvalue, atol=1e-15):
622            raise Exception('Error is 0.0')
623        labels = self.e_names
624        sizes = [self.e_dvalue[name] ** 2 for name in labels] / self._dvalue ** 2
625        fig1, ax1 = plt.subplots()
626        ax1.pie(sizes, labels=labels, startangle=90, normalize=True)
627        ax1.axis('equal')
628        plt.draw()
629        if save:
630            fig1.savefig(save)
631
632        return dict(zip(labels, sizes))
633
634    def dump(self, filename, datatype="json.gz", description="", **kwargs):
635        """Dump the Obs to a file 'name' of chosen format.
636
637        Parameters
638        ----------
639        filename : str
640            name of the file to be saved.
641        datatype : str
642            Format of the exported file. Supported formats include
643            "json.gz" and "pickle"
644        description : str
645            Description for output file, only relevant for json.gz format.
646        path : str
647            specifies a custom path for the file (default '.')
648        """
649        if 'path' in kwargs:
650            file_name = kwargs.get('path') + '/' + filename
651        else:
652            file_name = filename
653
654        if datatype == "json.gz":
655            from .input.json import dump_to_json
656            dump_to_json([self], file_name, description=description)
657        elif datatype == "pickle":
658            with open(file_name + '.p', 'wb') as fb:
659                pickle.dump(self, fb)
660        else:
661            raise Exception("Unknown datatype " + str(datatype))
662
663    def export_jackknife(self):
664        """Export jackknife samples from the Obs
665
666        Returns
667        -------
668        numpy.ndarray
669            Returns a numpy array of length N + 1 where N is the number of samples
670            for the given ensemble and replicum. The zeroth entry of the array contains
671            the mean value of the Obs, entries 1 to N contain the N jackknife samples
672            derived from the Obs. The current implementation only works for observables
673            defined on exactly one ensemble and replicum. The derived jackknife samples
674            should agree with samples from a full jackknife analysis up to O(1/N).
675        """
676
677        if len(self.names) != 1:
678            raise Exception("'export_jackknife' is only implemented for Obs defined on one ensemble and replicum.")
679
680        name = self.names[0]
681        full_data = self.deltas[name] + self.r_values[name]
682        n = full_data.size
683        mean = self.value
684        tmp_jacks = np.zeros(n + 1)
685        tmp_jacks[0] = mean
686        tmp_jacks[1:] = (n * mean - full_data) / (n - 1)
687        return tmp_jacks
688
689    def export_bootstrap(self, samples=500, random_numbers=None, save_rng=None):
690        """Export bootstrap samples from the Obs
691
692        Parameters
693        ----------
694        samples : int
695            Number of bootstrap samples to generate.
696        random_numbers : np.ndarray
697            Array of shape (samples, length) containing the random numbers to generate the bootstrap samples.
698            If not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
699        save_rng : str
700            Save the random numbers to a file if a path is specified.
701
702        Returns
703        -------
704        numpy.ndarray
705            Returns a numpy array of length N + 1 where N is the number of samples
706            for the given ensemble and replicum. The zeroth entry of the array contains
707            the mean value of the Obs, entries 1 to N contain the N import_bootstrap samples
708            derived from the Obs. The current implementation only works for observables
709            defined on exactly one ensemble and replicum. The derived bootstrap samples
710            should agree with samples from a full bootstrap analysis up to O(1/N).
711        """
712        if len(self.names) != 1:
713            raise Exception("'export_boostrap' is only implemented for Obs defined on one ensemble and replicum.")
714
715        name = self.names[0]
716        length = self.N
717
718        if random_numbers is None:
719            seed = int(hashlib.md5(name.encode()).hexdigest(), 16) & 0xFFFFFFFF
720            rng = np.random.default_rng(seed)
721            random_numbers = rng.integers(0, length, size=(samples, length))
722
723        if save_rng is not None:
724            np.savetxt(save_rng, random_numbers, fmt='%i')
725
726        proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length
727        ret = np.zeros(samples + 1)
728        ret[0] = self.value
729        ret[1:] = proj @ (self.deltas[name] + self.r_values[name])
730        return ret
731
732    def __float__(self):
733        return float(self.value)
734
735    def __repr__(self):
736        return 'Obs[' + str(self) + ']'
737
738    def __str__(self):
739        return _format_uncertainty(self.value, self._dvalue)
740
741    def __format__(self, format_type):
742        if format_type == "":
743            significance = 2
744        else:
745            significance = int(float(format_type.replace("+", "").replace("-", "")))
746        my_str = _format_uncertainty(self.value, self._dvalue,
747                                     significance=significance)
748        for char in ["+", " "]:
749            if format_type.startswith(char):
750                if my_str[0] != "-":
751                    my_str = char + my_str
752        return my_str
753
754    def __hash__(self):
755        hash_tuple = (np.array([self.value]).astype(np.float32).data.tobytes(),)
756        hash_tuple += tuple([o.astype(np.float32).data.tobytes() for o in self.deltas.values()])
757        hash_tuple += tuple([np.array([o.errsq()]).astype(np.float32).data.tobytes() for o in self.covobs.values()])
758        hash_tuple += tuple([o.encode() for o in self.names])
759        m = hashlib.md5()
760        [m.update(o) for o in hash_tuple]
761        return int(m.hexdigest(), 16) & 0xFFFFFFFF
762
763    # Overload comparisons
764    def __lt__(self, other):
765        return self.value < other
766
767    def __le__(self, other):
768        return self.value <= other
769
770    def __gt__(self, other):
771        return self.value > other
772
773    def __ge__(self, other):
774        return self.value >= other
775
776    def __eq__(self, other):
777        if other is None:
778            return False
779        return (self - other).is_zero()
780
781    # Overload math operations
782    def __add__(self, y):
783        if isinstance(y, Obs):
784            return derived_observable(lambda x, **kwargs: x[0] + x[1], [self, y], man_grad=[1, 1])
785        else:
786            if isinstance(y, np.ndarray):
787                return np.array([self + o for o in y])
788            elif y.__class__.__name__ in ['Corr', 'CObs']:
789                return NotImplemented
790            else:
791                return derived_observable(lambda x, **kwargs: x[0] + y, [self], man_grad=[1])
792
793    def __radd__(self, y):
794        return self + y
795
796    def __mul__(self, y):
797        if isinstance(y, Obs):
798            return derived_observable(lambda x, **kwargs: x[0] * x[1], [self, y], man_grad=[y.value, self.value])
799        else:
800            if isinstance(y, np.ndarray):
801                return np.array([self * o for o in y])
802            elif isinstance(y, complex):
803                return CObs(self * y.real, self * y.imag)
804            elif y.__class__.__name__ in ['Corr', 'CObs']:
805                return NotImplemented
806            else:
807                return derived_observable(lambda x, **kwargs: x[0] * y, [self], man_grad=[y])
808
809    def __rmul__(self, y):
810        return self * y
811
812    def __sub__(self, y):
813        if isinstance(y, Obs):
814            return derived_observable(lambda x, **kwargs: x[0] - x[1], [self, y], man_grad=[1, -1])
815        else:
816            if isinstance(y, np.ndarray):
817                return np.array([self - o for o in y])
818            elif y.__class__.__name__ in ['Corr', 'CObs']:
819                return NotImplemented
820            else:
821                return derived_observable(lambda x, **kwargs: x[0] - y, [self], man_grad=[1])
822
823    def __rsub__(self, y):
824        return -1 * (self - y)
825
826    def __pos__(self):
827        return self
828
829    def __neg__(self):
830        return -1 * self
831
832    def __truediv__(self, y):
833        if isinstance(y, Obs):
834            return derived_observable(lambda x, **kwargs: x[0] / x[1], [self, y], man_grad=[1 / y.value, - self.value / y.value ** 2])
835        else:
836            if isinstance(y, np.ndarray):
837                return np.array([self / o for o in y])
838            elif y.__class__.__name__ in ['Corr', 'CObs']:
839                return NotImplemented
840            else:
841                return derived_observable(lambda x, **kwargs: x[0] / y, [self], man_grad=[1 / y])
842
843    def __rtruediv__(self, y):
844        if isinstance(y, Obs):
845            return derived_observable(lambda x, **kwargs: x[0] / x[1], [y, self], man_grad=[1 / self.value, - y.value / self.value ** 2])
846        else:
847            if isinstance(y, np.ndarray):
848                return np.array([o / self for o in y])
849            elif y.__class__.__name__ in ['Corr', 'CObs']:
850                return NotImplemented
851            else:
852                return derived_observable(lambda x, **kwargs: y / x[0], [self], man_grad=[-y / self.value ** 2])
853
854    def __pow__(self, y):
855        if isinstance(y, Obs):
856            return derived_observable(lambda x: x[0] ** x[1], [self, y])
857        else:
858            return derived_observable(lambda x: x[0] ** y, [self])
859
860    def __rpow__(self, y):
861        if isinstance(y, Obs):
862            return derived_observable(lambda x: x[0] ** x[1], [y, self])
863        else:
864            return derived_observable(lambda x: y ** x[0], [self])
865
866    def __abs__(self):
867        return derived_observable(lambda x: anp.abs(x[0]), [self])
868
869    # Overload numpy functions
870    def sqrt(self):
871        return derived_observable(lambda x, **kwargs: np.sqrt(x[0]), [self], man_grad=[1 / 2 / np.sqrt(self.value)])
872
873    def log(self):
874        return derived_observable(lambda x, **kwargs: np.log(x[0]), [self], man_grad=[1 / self.value])
875
876    def exp(self):
877        return derived_observable(lambda x, **kwargs: np.exp(x[0]), [self], man_grad=[np.exp(self.value)])
878
879    def sin(self):
880        return derived_observable(lambda x, **kwargs: np.sin(x[0]), [self], man_grad=[np.cos(self.value)])
881
882    def cos(self):
883        return derived_observable(lambda x, **kwargs: np.cos(x[0]), [self], man_grad=[-np.sin(self.value)])
884
885    def tan(self):
886        return derived_observable(lambda x, **kwargs: np.tan(x[0]), [self], man_grad=[1 / np.cos(self.value) ** 2])
887
888    def arcsin(self):
889        return derived_observable(lambda x: anp.arcsin(x[0]), [self])
890
891    def arccos(self):
892        return derived_observable(lambda x: anp.arccos(x[0]), [self])
893
894    def arctan(self):
895        return derived_observable(lambda x: anp.arctan(x[0]), [self])
896
897    def sinh(self):
898        return derived_observable(lambda x, **kwargs: np.sinh(x[0]), [self], man_grad=[np.cosh(self.value)])
899
900    def cosh(self):
901        return derived_observable(lambda x, **kwargs: np.cosh(x[0]), [self], man_grad=[np.sinh(self.value)])
902
903    def tanh(self):
904        return derived_observable(lambda x, **kwargs: np.tanh(x[0]), [self], man_grad=[1 / np.cosh(self.value) ** 2])
905
906    def arcsinh(self):
907        return derived_observable(lambda x: anp.arcsinh(x[0]), [self])
908
909    def arccosh(self):
910        return derived_observable(lambda x: anp.arccosh(x[0]), [self])
911
912    def arctanh(self):
913        return derived_observable(lambda x: anp.arctanh(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)
  • N_sigma_dict (dict): Dictionary for N_sigma values. If an entry for a given ensemble exists this overwrites the standard value for that ensemble.
Obs(samples, names, idl=None, **kwargs)
 61    def __init__(self, samples, names, idl=None, **kwargs):
 62        """ Initialize Obs object.
 63
 64        Parameters
 65        ----------
 66        samples : list
 67            list of numpy arrays containing the Monte Carlo samples
 68        names : list
 69            list of strings labeling the individual samples
 70        idl : list, optional
 71            list of ranges or lists on which the samples are defined
 72        """
 73
 74        if kwargs.get("means") is None and len(samples):
 75            if len(samples) != len(names):
 76                raise ValueError('Length of samples and names incompatible.')
 77            if idl is not None:
 78                if len(idl) != len(names):
 79                    raise ValueError('Length of idl incompatible with samples and names.')
 80            name_length = len(names)
 81            if name_length > 1:
 82                if name_length != len(set(names)):
 83                    raise ValueError('Names are not unique.')
 84                if not all(isinstance(x, str) for x in names):
 85                    raise TypeError('All names have to be strings.')
 86            else:
 87                if not isinstance(names[0], str):
 88                    raise TypeError('All names have to be strings.')
 89            if min(len(x) for x in samples) <= 4:
 90                raise ValueError('Samples have to have at least 5 entries.')
 91
 92        self.names = sorted(names)
 93        self.shape = {}
 94        self.r_values = {}
 95        self.deltas = {}
 96        self._covobs = {}
 97
 98        self._value = 0
 99        self.N = 0
100        self.idl = {}
101        if idl is not None:
102            for name, idx in sorted(zip(names, idl)):
103                if isinstance(idx, range):
104                    self.idl[name] = idx
105                elif isinstance(idx, (list, np.ndarray)):
106                    dc = np.unique(np.diff(idx))
107                    if np.any(dc < 0):
108                        raise ValueError("Unsorted idx for idl[%s]" % (name))
109                    if len(dc) == 1:
110                        self.idl[name] = range(idx[0], idx[-1] + dc[0], dc[0])
111                    else:
112                        self.idl[name] = list(idx)
113                else:
114                    raise TypeError('incompatible type for idl[%s].' % (name))
115        else:
116            for name, sample in sorted(zip(names, samples)):
117                self.idl[name] = range(1, len(sample) + 1)
118
119        if kwargs.get("means") is not None:
120            for name, sample, mean in sorted(zip(names, samples, kwargs.get("means"))):
121                self.shape[name] = len(self.idl[name])
122                self.N += self.shape[name]
123                self.r_values[name] = mean
124                self.deltas[name] = sample
125        else:
126            for name, sample in sorted(zip(names, samples)):
127                self.shape[name] = len(self.idl[name])
128                self.N += self.shape[name]
129                if len(sample) != self.shape[name]:
130                    raise ValueError('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name]))
131                self.r_values[name] = np.mean(sample)
132                self.deltas[name] = sample - self.r_values[name]
133                self._value += self.shape[name] * self.r_values[name]
134            self._value /= self.N
135
136        self._dvalue = 0.0
137        self.ddvalue = 0.0
138        self.reweighted = False
139
140        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 individual samples
  • idl (list, optional): list of ranges or lists on which the samples are defined
S_global = 2.0
S_dict = {}
tau_exp_global = 0.0
tau_exp_dict = {}
N_sigma_global = 1.0
N_sigma_dict = {}
names
shape
r_values
deltas
N
idl
ddvalue
reweighted
tag
value
dvalue
e_names
cov_names
mc_names
e_content
covobs
def gamma_method(self, **kwargs):
175    def gamma_method(self, **kwargs):
176        """Estimate the error and related properties of the Obs.
177
178        Parameters
179        ----------
180        S : float
181            specifies a custom value for the parameter S (default 2.0).
182            If set to 0 it is assumed that the data exhibits no
183            autocorrelation. In this case the error estimates coincides
184            with the sample standard error.
185        tau_exp : float
186            positive value triggers the critical slowing down analysis
187            (default 0.0).
188        N_sigma : float
189            number of standard deviations from zero until the tail is
190            attached to the autocorrelation function (default 1).
191        fft : bool
192            determines whether the fft algorithm is used for the computation
193            of the autocorrelation function (default True)
194        """
195
196        e_content = self.e_content
197        self.e_dvalue = {}
198        self.e_ddvalue = {}
199        self.e_tauint = {}
200        self.e_dtauint = {}
201        self.e_windowsize = {}
202        self.e_n_tauint = {}
203        self.e_n_dtauint = {}
204        e_gamma = {}
205        self.e_rho = {}
206        self.e_drho = {}
207        self._dvalue = 0
208        self.ddvalue = 0
209
210        self.S = {}
211        self.tau_exp = {}
212        self.N_sigma = {}
213
214        if kwargs.get('fft') is False:
215            fft = False
216        else:
217            fft = True
218
219        def _parse_kwarg(kwarg_name):
220            if kwarg_name in kwargs:
221                tmp = kwargs.get(kwarg_name)
222                if isinstance(tmp, (int, float)):
223                    if tmp < 0:
224                        raise Exception(kwarg_name + ' has to be larger or equal to 0.')
225                    for e, e_name in enumerate(self.e_names):
226                        getattr(self, kwarg_name)[e_name] = tmp
227                else:
228                    raise TypeError(kwarg_name + ' is not in proper format.')
229            else:
230                for e, e_name in enumerate(self.e_names):
231                    if e_name in getattr(Obs, kwarg_name + '_dict'):
232                        getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_dict')[e_name]
233                    else:
234                        getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_global')
235
236        _parse_kwarg('S')
237        _parse_kwarg('tau_exp')
238        _parse_kwarg('N_sigma')
239
240        for e, e_name in enumerate(self.mc_names):
241            gapsize = _determine_gap(self, e_content, e_name)
242
243            r_length = []
244            for r_name in e_content[e_name]:
245                if isinstance(self.idl[r_name], range):
246                    r_length.append(len(self.idl[r_name]) * self.idl[r_name].step // gapsize)
247                else:
248                    r_length.append((self.idl[r_name][-1] - self.idl[r_name][0] + 1) // gapsize)
249
250            e_N = np.sum([self.shape[r_name] for r_name in e_content[e_name]])
251            w_max = max(r_length) // 2
252            e_gamma[e_name] = np.zeros(w_max)
253            self.e_rho[e_name] = np.zeros(w_max)
254            self.e_drho[e_name] = np.zeros(w_max)
255
256            for r_name in e_content[e_name]:
257                e_gamma[e_name] += self._calc_gamma(self.deltas[r_name], self.idl[r_name], self.shape[r_name], w_max, fft, gapsize)
258
259            gamma_div = np.zeros(w_max)
260            for r_name in e_content[e_name]:
261                gamma_div += self._calc_gamma(np.ones((self.shape[r_name])), self.idl[r_name], self.shape[r_name], w_max, fft, gapsize)
262            gamma_div[gamma_div < 1] = 1.0
263            e_gamma[e_name] /= gamma_div[:w_max]
264
265            if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny:  # Prevent division by zero
266                self.e_tauint[e_name] = 0.5
267                self.e_dtauint[e_name] = 0.0
268                self.e_dvalue[e_name] = 0.0
269                self.e_ddvalue[e_name] = 0.0
270                self.e_windowsize[e_name] = 0
271                continue
272
273            self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0]
274            self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:])))
275            # Make sure no entry of tauint is smaller than 0.5
276            self.e_n_tauint[e_name][self.e_n_tauint[e_name] <= 0.5] = 0.5 + np.finfo(np.float64).eps
277            # hep-lat/0306017 eq. (42)
278            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)
279            self.e_n_dtauint[e_name][0] = 0.0
280
281            def _compute_drho(i):
282                tmp = (self.e_rho[e_name][i + 1:w_max]
283                       + np.concatenate([self.e_rho[e_name][i - 1:None if i - (w_max - 1) // 2 <= 0 else (2 * i - (2 * w_max) // 2):-1],
284                                         self.e_rho[e_name][1:max(1, w_max - 2 * i)]])
285                       - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i])
286                self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N)
287
288            if self.tau_exp[e_name] > 0:
289                _compute_drho(1)
290                texp = self.tau_exp[e_name]
291                # Critical slowing down analysis
292                if w_max // 2 <= 1:
293                    raise Exception("Need at least 8 samples for tau_exp error analysis")
294                for n in range(1, w_max // 2):
295                    _compute_drho(n + 1)
296                    if (self.e_rho[e_name][n] - self.N_sigma[e_name] * self.e_drho[e_name][n]) < 0 or n >= w_max // 2 - 2:
297                        # Bias correction hep-lat/0306017 eq. (49) included
298                        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
299                        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)
300                        # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2
301                        self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
302                        self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
303                        self.e_windowsize[e_name] = n
304                        break
305            else:
306                if self.S[e_name] == 0.0:
307                    self.e_tauint[e_name] = 0.5
308                    self.e_dtauint[e_name] = 0.0
309                    self.e_dvalue[e_name] = np.sqrt(e_gamma[e_name][0] / (e_N - 1))
310                    self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt(0.5 / e_N)
311                    self.e_windowsize[e_name] = 0
312                else:
313                    # Standard automatic windowing procedure
314                    tau = self.S[e_name] / np.log((2 * self.e_n_tauint[e_name][1:] + 1) / (2 * self.e_n_tauint[e_name][1:] - 1))
315                    g_w = np.exp(- np.arange(1, len(tau) + 1) / tau) - tau / np.sqrt(np.arange(1, len(tau) + 1) * e_N)
316                    for n in range(1, w_max):
317                        if g_w[n - 1] < 0 or n >= w_max - 1:
318                            _compute_drho(n)
319                            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)
320                            self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n]
321                            self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
322                            self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
323                            self.e_windowsize[e_name] = n
324                            break
325
326            self._dvalue += self.e_dvalue[e_name] ** 2
327            self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2
328
329        for e_name in self.cov_names:
330            self.e_dvalue[e_name] = np.sqrt(self.covobs[e_name].errsq())
331            self.e_ddvalue[e_name] = 0
332            self._dvalue += self.e_dvalue[e_name]**2
333
334        self._dvalue = np.sqrt(self._dvalue)
335        if self._dvalue == 0.0:
336            self.ddvalue = 0.0
337        else:
338            self.ddvalue = np.sqrt(self.ddvalue) / self._dvalue
339        return

Estimate the error and related properties of the Obs.

Parameters
  • S (float): specifies a custom value for the parameter S (default 2.0). If set to 0 it is assumed that the data exhibits no autocorrelation. In this case the error estimates coincides with the sample standard error.
  • tau_exp (float): positive value triggers the critical slowing down analysis (default 0.0).
  • 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 gm(self, **kwargs):
175    def gamma_method(self, **kwargs):
176        """Estimate the error and related properties of the Obs.
177
178        Parameters
179        ----------
180        S : float
181            specifies a custom value for the parameter S (default 2.0).
182            If set to 0 it is assumed that the data exhibits no
183            autocorrelation. In this case the error estimates coincides
184            with the sample standard error.
185        tau_exp : float
186            positive value triggers the critical slowing down analysis
187            (default 0.0).
188        N_sigma : float
189            number of standard deviations from zero until the tail is
190            attached to the autocorrelation function (default 1).
191        fft : bool
192            determines whether the fft algorithm is used for the computation
193            of the autocorrelation function (default True)
194        """
195
196        e_content = self.e_content
197        self.e_dvalue = {}
198        self.e_ddvalue = {}
199        self.e_tauint = {}
200        self.e_dtauint = {}
201        self.e_windowsize = {}
202        self.e_n_tauint = {}
203        self.e_n_dtauint = {}
204        e_gamma = {}
205        self.e_rho = {}
206        self.e_drho = {}
207        self._dvalue = 0
208        self.ddvalue = 0
209
210        self.S = {}
211        self.tau_exp = {}
212        self.N_sigma = {}
213
214        if kwargs.get('fft') is False:
215            fft = False
216        else:
217            fft = True
218
219        def _parse_kwarg(kwarg_name):
220            if kwarg_name in kwargs:
221                tmp = kwargs.get(kwarg_name)
222                if isinstance(tmp, (int, float)):
223                    if tmp < 0:
224                        raise Exception(kwarg_name + ' has to be larger or equal to 0.')
225                    for e, e_name in enumerate(self.e_names):
226                        getattr(self, kwarg_name)[e_name] = tmp
227                else:
228                    raise TypeError(kwarg_name + ' is not in proper format.')
229            else:
230                for e, e_name in enumerate(self.e_names):
231                    if e_name in getattr(Obs, kwarg_name + '_dict'):
232                        getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_dict')[e_name]
233                    else:
234                        getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_global')
235
236        _parse_kwarg('S')
237        _parse_kwarg('tau_exp')
238        _parse_kwarg('N_sigma')
239
240        for e, e_name in enumerate(self.mc_names):
241            gapsize = _determine_gap(self, e_content, e_name)
242
243            r_length = []
244            for r_name in e_content[e_name]:
245                if isinstance(self.idl[r_name], range):
246                    r_length.append(len(self.idl[r_name]) * self.idl[r_name].step // gapsize)
247                else:
248                    r_length.append((self.idl[r_name][-1] - self.idl[r_name][0] + 1) // gapsize)
249
250            e_N = np.sum([self.shape[r_name] for r_name in e_content[e_name]])
251            w_max = max(r_length) // 2
252            e_gamma[e_name] = np.zeros(w_max)
253            self.e_rho[e_name] = np.zeros(w_max)
254            self.e_drho[e_name] = np.zeros(w_max)
255
256            for r_name in e_content[e_name]:
257                e_gamma[e_name] += self._calc_gamma(self.deltas[r_name], self.idl[r_name], self.shape[r_name], w_max, fft, gapsize)
258
259            gamma_div = np.zeros(w_max)
260            for r_name in e_content[e_name]:
261                gamma_div += self._calc_gamma(np.ones((self.shape[r_name])), self.idl[r_name], self.shape[r_name], w_max, fft, gapsize)
262            gamma_div[gamma_div < 1] = 1.0
263            e_gamma[e_name] /= gamma_div[:w_max]
264
265            if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny:  # Prevent division by zero
266                self.e_tauint[e_name] = 0.5
267                self.e_dtauint[e_name] = 0.0
268                self.e_dvalue[e_name] = 0.0
269                self.e_ddvalue[e_name] = 0.0
270                self.e_windowsize[e_name] = 0
271                continue
272
273            self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0]
274            self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:])))
275            # Make sure no entry of tauint is smaller than 0.5
276            self.e_n_tauint[e_name][self.e_n_tauint[e_name] <= 0.5] = 0.5 + np.finfo(np.float64).eps
277            # hep-lat/0306017 eq. (42)
278            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)
279            self.e_n_dtauint[e_name][0] = 0.0
280
281            def _compute_drho(i):
282                tmp = (self.e_rho[e_name][i + 1:w_max]
283                       + np.concatenate([self.e_rho[e_name][i - 1:None if i - (w_max - 1) // 2 <= 0 else (2 * i - (2 * w_max) // 2):-1],
284                                         self.e_rho[e_name][1:max(1, w_max - 2 * i)]])
285                       - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i])
286                self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N)
287
288            if self.tau_exp[e_name] > 0:
289                _compute_drho(1)
290                texp = self.tau_exp[e_name]
291                # Critical slowing down analysis
292                if w_max // 2 <= 1:
293                    raise Exception("Need at least 8 samples for tau_exp error analysis")
294                for n in range(1, w_max // 2):
295                    _compute_drho(n + 1)
296                    if (self.e_rho[e_name][n] - self.N_sigma[e_name] * self.e_drho[e_name][n]) < 0 or n >= w_max // 2 - 2:
297                        # Bias correction hep-lat/0306017 eq. (49) included
298                        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
299                        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)
300                        # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2
301                        self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
302                        self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
303                        self.e_windowsize[e_name] = n
304                        break
305            else:
306                if self.S[e_name] == 0.0:
307                    self.e_tauint[e_name] = 0.5
308                    self.e_dtauint[e_name] = 0.0
309                    self.e_dvalue[e_name] = np.sqrt(e_gamma[e_name][0] / (e_N - 1))
310                    self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt(0.5 / e_N)
311                    self.e_windowsize[e_name] = 0
312                else:
313                    # Standard automatic windowing procedure
314                    tau = self.S[e_name] / np.log((2 * self.e_n_tauint[e_name][1:] + 1) / (2 * self.e_n_tauint[e_name][1:] - 1))
315                    g_w = np.exp(- np.arange(1, len(tau) + 1) / tau) - tau / np.sqrt(np.arange(1, len(tau) + 1) * e_N)
316                    for n in range(1, w_max):
317                        if g_w[n - 1] < 0 or n >= w_max - 1:
318                            _compute_drho(n)
319                            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)
320                            self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n]
321                            self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N)
322                            self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N)
323                            self.e_windowsize[e_name] = n
324                            break
325
326            self._dvalue += self.e_dvalue[e_name] ** 2
327            self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2
328
329        for e_name in self.cov_names:
330            self.e_dvalue[e_name] = np.sqrt(self.covobs[e_name].errsq())
331            self.e_ddvalue[e_name] = 0
332            self._dvalue += self.e_dvalue[e_name]**2
333
334        self._dvalue = np.sqrt(self._dvalue)
335        if self._dvalue == 0.0:
336            self.ddvalue = 0.0
337        else:
338            self.ddvalue = np.sqrt(self.ddvalue) / self._dvalue
339        return

Estimate the error and related properties of the Obs.

Parameters
  • S (float): specifies a custom value for the parameter S (default 2.0). If set to 0 it is assumed that the data exhibits no autocorrelation. In this case the error estimates coincides with the sample standard error.
  • tau_exp (float): positive value triggers the critical slowing down analysis (default 0.0).
  • 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 details(self, ens_content=True):
379    def details(self, ens_content=True):
380        """Output detailed properties of the Obs.
381
382        Parameters
383        ----------
384        ens_content : bool
385            print details about the ensembles and replica if true.
386        """
387        if self.tag is not None:
388            print("Description:", self.tag)
389        if not hasattr(self, 'e_dvalue'):
390            print('Result\t %3.8e' % (self.value))
391        else:
392            if self.value == 0.0:
393                percentage = np.nan
394            else:
395                percentage = np.abs(self._dvalue / self.value) * 100
396            print('Result\t %3.8e +/- %3.8e +/- %3.8e (%3.3f%%)' % (self.value, self._dvalue, self.ddvalue, percentage))
397            if len(self.e_names) > 1:
398                print(' Ensemble errors:')
399            e_content = self.e_content
400            for e_name in self.mc_names:
401                gap = _determine_gap(self, e_content, e_name)
402
403                if len(self.e_names) > 1:
404                    print('', e_name, '\t %3.6e +/- %3.6e' % (self.e_dvalue[e_name], self.e_ddvalue[e_name]))
405                tau_string = " \N{GREEK SMALL LETTER TAU}_int\t " + _format_uncertainty(self.e_tauint[e_name], self.e_dtauint[e_name])
406                tau_string += f" in units of {gap} config"
407                if gap > 1:
408                    tau_string += "s"
409                if self.tau_exp[e_name] > 0:
410                    tau_string = f"{tau_string: <45}" + '\t(\N{GREEK SMALL LETTER TAU}_exp=%3.2f, N_\N{GREEK SMALL LETTER SIGMA}=%1.0i)' % (self.tau_exp[e_name], self.N_sigma[e_name])
411                else:
412                    tau_string = f"{tau_string: <45}" + '\t(S=%3.2f)' % (self.S[e_name])
413                print(tau_string)
414            for e_name in self.cov_names:
415                print('', e_name, '\t %3.8e' % (self.e_dvalue[e_name]))
416        if ens_content is True:
417            if len(self.e_names) == 1:
418                print(self.N, 'samples in', len(self.e_names), 'ensemble:')
419            else:
420                print(self.N, 'samples in', len(self.e_names), 'ensembles:')
421            my_string_list = []
422            for key, value in sorted(self.e_content.items()):
423                if key not in self.covobs:
424                    my_string = '  ' + "\u00B7 Ensemble '" + key + "' "
425                    if len(value) == 1:
426                        my_string += f': {self.shape[value[0]]} configurations'
427                        if isinstance(self.idl[value[0]], range):
428                            my_string += f' (from {self.idl[value[0]].start} to {self.idl[value[0]][-1]}' + int(self.idl[value[0]].step != 1) * f' in steps of {self.idl[value[0]].step}' + ')'
429                        else:
430                            my_string += f' (irregular range from {self.idl[value[0]][0]} to {self.idl[value[0]][-1]})'
431                    else:
432                        sublist = []
433                        for v in value:
434                            my_substring = '    ' + "\u00B7 Replicum '" + v[len(key) + 1:] + "' "
435                            my_substring += f': {self.shape[v]} configurations'
436                            if isinstance(self.idl[v], range):
437                                my_substring += f' (from {self.idl[v].start} to {self.idl[v][-1]}' + int(self.idl[v].step != 1) * f' in steps of {self.idl[v].step}' + ')'
438                            else:
439                                my_substring += f' (irregular range from {self.idl[v][0]} to {self.idl[v][-1]})'
440                            sublist.append(my_substring)
441
442                        my_string += '\n' + '\n'.join(sublist)
443                else:
444                    my_string = '  ' + "\u00B7 Covobs   '" + key + "' "
445                my_string_list.append(my_string)
446            print('\n'.join(my_string_list))

Output detailed properties of the Obs.

Parameters
  • ens_content (bool): print details about the ensembles and replica if true.
def reweight(self, weight):
448    def reweight(self, weight):
449        """Reweight the obs with given rewighting factors.
450
451        Parameters
452        ----------
453        weight : Obs
454            Reweighting factor. An Observable that has to be defined on a superset of the
455            configurations in obs[i].idl for all i.
456        all_configs : bool
457            if True, the reweighted observables are normalized by the average of
458            the reweighting factor on all configurations in weight.idl and not
459            on the configurations in obs[i].idl. Default False.
460        """
461        return reweight(weight, [self])[0]

Reweight the obs with given rewighting factors.

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.
  • 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. Default False.
def is_zero_within_error(self, sigma=1):
463    def is_zero_within_error(self, sigma=1):
464        """Checks whether the observable is zero within 'sigma' standard errors.
465
466        Parameters
467        ----------
468        sigma : int
469            Number of standard errors used for the check.
470
471        Works only properly when the gamma method was run.
472        """
473        return self.is_zero() or np.abs(self.value) <= sigma * self._dvalue

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

Parameters
  • sigma (int): Number of standard errors used for the check.
  • Works only properly when the gamma method was run.
def is_zero(self, atol=1e-10):
475    def is_zero(self, atol=1e-10):
476        """Checks whether the observable is zero within a given tolerance.
477
478        Parameters
479        ----------
480        atol : float
481            Absolute tolerance (for details see numpy documentation).
482        """
483        return np.isclose(0.0, self.value, 1e-14, atol) and all(np.allclose(0.0, delta, 1e-14, atol) for delta in self.deltas.values()) and all(np.allclose(0.0, delta.errsq(), 1e-14, atol) for delta in self.covobs.values())

Checks whether the observable is zero within a given tolerance.

Parameters
  • atol (float): Absolute tolerance (for details see numpy documentation).
def plot_tauint(self, save=None):
485    def plot_tauint(self, save=None):
486        """Plot integrated autocorrelation time for each ensemble.
487
488        Parameters
489        ----------
490        save : str
491            saves the figure to a file named 'save' if.
492        """
493        if not hasattr(self, 'e_dvalue'):
494            raise Exception('Run the gamma method first.')
495
496        for e, e_name in enumerate(self.mc_names):
497            fig = plt.figure()
498            plt.xlabel(r'$W$')
499            plt.ylabel(r'$\tau_\mathrm{int}$')
500            length = int(len(self.e_n_tauint[e_name]))
501            if self.tau_exp[e_name] > 0:
502                base = self.e_n_tauint[e_name][self.e_windowsize[e_name]]
503                x_help = np.arange(2 * self.tau_exp[e_name])
504                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
505                x_arr = np.arange(self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name])
506                plt.plot(x_arr, y_help, 'C' + str(e), linewidth=1, ls='--', marker=',')
507                plt.errorbar([self.e_windowsize[e_name] + 2 * self.tau_exp[e_name]], [self.e_tauint[e_name]],
508                             yerr=[self.e_dtauint[e_name]], fmt='C' + str(e), linewidth=1, capsize=2, marker='o', mfc=plt.rcParams['axes.facecolor'])
509                xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
510                label = e_name + r', $\tau_\mathrm{exp}$=' + str(np.around(self.tau_exp[e_name], decimals=2))
511            else:
512                label = e_name + ', S=' + str(np.around(self.S[e_name], decimals=2))
513                xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)
514
515            plt.errorbar(np.arange(length)[:int(xmax) + 1], self.e_n_tauint[e_name][:int(xmax) + 1], yerr=self.e_n_dtauint[e_name][:int(xmax) + 1], linewidth=1, capsize=2, label=label)
516            plt.axvline(x=self.e_windowsize[e_name], color='C' + str(e), alpha=0.5, marker=',', ls='--')
517            plt.legend()
518            plt.xlim(-0.5, xmax)
519            ylim = plt.ylim()
520            plt.ylim(bottom=0.0, top=max(1.0, ylim[1]))
521            plt.draw()
522            if save:
523                fig.savefig(save + "_" + str(e))

Plot integrated autocorrelation time for each ensemble.

Parameters
  • save (str): saves the figure to a file named 'save' if.
def plot_rho(self, save=None):
525    def plot_rho(self, save=None):
526        """Plot normalized autocorrelation function time for each ensemble.
527
528        Parameters
529        ----------
530        save : str
531            saves the figure to a file named 'save' if.
532        """
533        if not hasattr(self, 'e_dvalue'):
534            raise Exception('Run the gamma method first.')
535        for e, e_name in enumerate(self.mc_names):
536            fig = plt.figure()
537            plt.xlabel('W')
538            plt.ylabel('rho')
539            length = int(len(self.e_drho[e_name]))
540            plt.errorbar(np.arange(length), self.e_rho[e_name][:length], yerr=self.e_drho[e_name][:], linewidth=1, capsize=2)
541            plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25, ls='--', marker=',')
542            if self.tau_exp[e_name] > 0:
543                plt.plot([self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]],
544                         [self.e_rho[e_name][self.e_windowsize[e_name] + 1], 0], 'k-', lw=1)
545                xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5
546                plt.title('Rho ' + e_name + r', tau\_exp=' + str(np.around(self.tau_exp[e_name], decimals=2)))
547            else:
548                xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5)
549                plt.title('Rho ' + e_name + ', S=' + str(np.around(self.S[e_name], decimals=2)))
550            plt.plot([-0.5, xmax], [0, 0], 'k--', lw=1)
551            plt.xlim(-0.5, xmax)
552            plt.draw()
553            if save:
554                fig.savefig(save + "_" + str(e))

Plot normalized autocorrelation function time for each ensemble.

Parameters
  • save (str): saves the figure to a file named 'save' if.
def plot_rep_dist(self):
556    def plot_rep_dist(self):
557        """Plot replica distribution for each ensemble with more than one replicum."""
558        if not hasattr(self, 'e_dvalue'):
559            raise Exception('Run the gamma method first.')
560        for e, e_name in enumerate(self.mc_names):
561            if len(self.e_content[e_name]) == 1:
562                print('No replica distribution for a single replicum (', e_name, ')')
563                continue
564            r_length = []
565            sub_r_mean = 0
566            for r, r_name in enumerate(self.e_content[e_name]):
567                r_length.append(len(self.deltas[r_name]))
568                sub_r_mean += self.shape[r_name] * self.r_values[r_name]
569            e_N = np.sum(r_length)
570            sub_r_mean /= e_N
571            arr = np.zeros(len(self.e_content[e_name]))
572            for r, r_name in enumerate(self.e_content[e_name]):
573                arr[r] = (self.r_values[r_name] - sub_r_mean) / (self.e_dvalue[e_name] * np.sqrt(e_N / self.shape[r_name] - 1))
574            plt.hist(arr, rwidth=0.8, bins=len(self.e_content[e_name]))
575            plt.title('Replica distribution' + e_name + ' (mean=0, var=1)')
576            plt.draw()

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

def plot_history(self, expand=True):
578    def plot_history(self, expand=True):
579        """Plot derived Monte Carlo history for each ensemble
580
581        Parameters
582        ----------
583        expand : bool
584            show expanded history for irregular Monte Carlo chains (default: True).
585        """
586        for e, e_name in enumerate(self.mc_names):
587            plt.figure()
588            r_length = []
589            tmp = []
590            tmp_expanded = []
591            for r, r_name in enumerate(self.e_content[e_name]):
592                tmp.append(self.deltas[r_name] + self.r_values[r_name])
593                if expand:
594                    tmp_expanded.append(_expand_deltas(self.deltas[r_name], list(self.idl[r_name]), self.shape[r_name], 1) + self.r_values[r_name])
595                    r_length.append(len(tmp_expanded[-1]))
596                else:
597                    r_length.append(len(tmp[-1]))
598            e_N = np.sum(r_length)
599            x = np.arange(e_N)
600            y_test = np.concatenate(tmp, axis=0)
601            if expand:
602                y = np.concatenate(tmp_expanded, axis=0)
603            else:
604                y = y_test
605            plt.errorbar(x, y, fmt='.', markersize=3)
606            plt.xlim(-0.5, e_N - 0.5)
607            plt.title(e_name + f'\nskew: {skew(y_test):.3f} (p={skewtest(y_test).pvalue:.3f}), kurtosis: {kurtosis(y_test):.3f} (p={kurtosistest(y_test).pvalue:.3f})')
608            plt.draw()

Plot derived Monte Carlo history for each ensemble

Parameters
  • expand (bool): show expanded history for irregular Monte Carlo chains (default: True).
def plot_piechart(self, save=None):
610    def plot_piechart(self, save=None):
611        """Plot piechart which shows the fractional contribution of each
612        ensemble to the error and returns a dictionary containing the fractions.
613
614        Parameters
615        ----------
616        save : str
617            saves the figure to a file named 'save' if.
618        """
619        if not hasattr(self, 'e_dvalue'):
620            raise Exception('Run the gamma method first.')
621        if np.isclose(0.0, self._dvalue, atol=1e-15):
622            raise Exception('Error is 0.0')
623        labels = self.e_names
624        sizes = [self.e_dvalue[name] ** 2 for name in labels] / self._dvalue ** 2
625        fig1, ax1 = plt.subplots()
626        ax1.pie(sizes, labels=labels, startangle=90, normalize=True)
627        ax1.axis('equal')
628        plt.draw()
629        if save:
630            fig1.savefig(save)
631
632        return dict(zip(labels, sizes))

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

Parameters
  • save (str): saves the figure to a file named 'save' if.
def dump(self, filename, datatype='json.gz', description='', **kwargs):
634    def dump(self, filename, datatype="json.gz", description="", **kwargs):
635        """Dump the Obs to a file 'name' of chosen format.
636
637        Parameters
638        ----------
639        filename : str
640            name of the file to be saved.
641        datatype : str
642            Format of the exported file. Supported formats include
643            "json.gz" and "pickle"
644        description : str
645            Description for output file, only relevant for json.gz format.
646        path : str
647            specifies a custom path for the file (default '.')
648        """
649        if 'path' in kwargs:
650            file_name = kwargs.get('path') + '/' + filename
651        else:
652            file_name = filename
653
654        if datatype == "json.gz":
655            from .input.json import dump_to_json
656            dump_to_json([self], file_name, description=description)
657        elif datatype == "pickle":
658            with open(file_name + '.p', 'wb') as fb:
659                pickle.dump(self, fb)
660        else:
661            raise Exception("Unknown datatype " + str(datatype))

Dump the Obs to a file 'name' of chosen format.

Parameters
  • filename (str): name of the file to be saved.
  • datatype (str): Format of the exported file. Supported formats include "json.gz" and "pickle"
  • description (str): Description for output file, only relevant for json.gz format.
  • path (str): specifies a custom path for the file (default '.')
def export_jackknife(self):
663    def export_jackknife(self):
664        """Export jackknife samples from the Obs
665
666        Returns
667        -------
668        numpy.ndarray
669            Returns a numpy array of length N + 1 where N is the number of samples
670            for the given ensemble and replicum. The zeroth entry of the array contains
671            the mean value of the Obs, entries 1 to N contain the N jackknife samples
672            derived from the Obs. The current implementation only works for observables
673            defined on exactly one ensemble and replicum. The derived jackknife samples
674            should agree with samples from a full jackknife analysis up to O(1/N).
675        """
676
677        if len(self.names) != 1:
678            raise Exception("'export_jackknife' is only implemented for Obs defined on one ensemble and replicum.")
679
680        name = self.names[0]
681        full_data = self.deltas[name] + self.r_values[name]
682        n = full_data.size
683        mean = self.value
684        tmp_jacks = np.zeros(n + 1)
685        tmp_jacks[0] = mean
686        tmp_jacks[1:] = (n * mean - full_data) / (n - 1)
687        return tmp_jacks

Export jackknife samples from the Obs

Returns
  • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples for the given ensemble and replicum. The zeroth entry of the array contains the mean value of the Obs, entries 1 to N contain the N jackknife samples derived from the Obs. The current implementation only works for observables defined on exactly one ensemble and replicum. The derived jackknife samples should agree with samples from a full jackknife analysis up to O(1/N).
def export_bootstrap(self, samples=500, random_numbers=None, save_rng=None):
689    def export_bootstrap(self, samples=500, random_numbers=None, save_rng=None):
690        """Export bootstrap samples from the Obs
691
692        Parameters
693        ----------
694        samples : int
695            Number of bootstrap samples to generate.
696        random_numbers : np.ndarray
697            Array of shape (samples, length) containing the random numbers to generate the bootstrap samples.
698            If not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
699        save_rng : str
700            Save the random numbers to a file if a path is specified.
701
702        Returns
703        -------
704        numpy.ndarray
705            Returns a numpy array of length N + 1 where N is the number of samples
706            for the given ensemble and replicum. The zeroth entry of the array contains
707            the mean value of the Obs, entries 1 to N contain the N import_bootstrap samples
708            derived from the Obs. The current implementation only works for observables
709            defined on exactly one ensemble and replicum. The derived bootstrap samples
710            should agree with samples from a full bootstrap analysis up to O(1/N).
711        """
712        if len(self.names) != 1:
713            raise Exception("'export_boostrap' is only implemented for Obs defined on one ensemble and replicum.")
714
715        name = self.names[0]
716        length = self.N
717
718        if random_numbers is None:
719            seed = int(hashlib.md5(name.encode()).hexdigest(), 16) & 0xFFFFFFFF
720            rng = np.random.default_rng(seed)
721            random_numbers = rng.integers(0, length, size=(samples, length))
722
723        if save_rng is not None:
724            np.savetxt(save_rng, random_numbers, fmt='%i')
725
726        proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length
727        ret = np.zeros(samples + 1)
728        ret[0] = self.value
729        ret[1:] = proj @ (self.deltas[name] + self.r_values[name])
730        return ret

Export bootstrap samples from the Obs

Parameters
  • samples (int): Number of bootstrap samples to generate.
  • random_numbers (np.ndarray): Array of shape (samples, length) containing the random numbers to generate the bootstrap samples. If not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
  • save_rng (str): Save the random numbers to a file if a path is specified.
Returns
  • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples for the given ensemble and replicum. The zeroth entry of the array contains the mean value of the Obs, entries 1 to N contain the N import_bootstrap samples derived from the Obs. The current implementation only works for observables defined on exactly one ensemble and replicum. The derived bootstrap samples should agree with samples from a full bootstrap analysis up to O(1/N).
def sqrt(self):
870    def sqrt(self):
871        return derived_observable(lambda x, **kwargs: np.sqrt(x[0]), [self], man_grad=[1 / 2 / np.sqrt(self.value)])
def log(self):
873    def log(self):
874        return derived_observable(lambda x, **kwargs: np.log(x[0]), [self], man_grad=[1 / self.value])
def exp(self):
876    def exp(self):
877        return derived_observable(lambda x, **kwargs: np.exp(x[0]), [self], man_grad=[np.exp(self.value)])
def sin(self):
879    def sin(self):
880        return derived_observable(lambda x, **kwargs: np.sin(x[0]), [self], man_grad=[np.cos(self.value)])
def cos(self):
882    def cos(self):
883        return derived_observable(lambda x, **kwargs: np.cos(x[0]), [self], man_grad=[-np.sin(self.value)])
def tan(self):
885    def tan(self):
886        return derived_observable(lambda x, **kwargs: np.tan(x[0]), [self], man_grad=[1 / np.cos(self.value) ** 2])
def arcsin(self):
888    def arcsin(self):
889        return derived_observable(lambda x: anp.arcsin(x[0]), [self])
def arccos(self):
891    def arccos(self):
892        return derived_observable(lambda x: anp.arccos(x[0]), [self])
def arctan(self):
894    def arctan(self):
895        return derived_observable(lambda x: anp.arctan(x[0]), [self])
def sinh(self):
897    def sinh(self):
898        return derived_observable(lambda x, **kwargs: np.sinh(x[0]), [self], man_grad=[np.cosh(self.value)])
def cosh(self):
900    def cosh(self):
901        return derived_observable(lambda x, **kwargs: np.cosh(x[0]), [self], man_grad=[np.sinh(self.value)])
def tanh(self):
903    def tanh(self):
904        return derived_observable(lambda x, **kwargs: np.tanh(x[0]), [self], man_grad=[1 / np.cosh(self.value) ** 2])
def arcsinh(self):
906    def arcsinh(self):
907        return derived_observable(lambda x: anp.arcsinh(x[0]), [self])
def arccosh(self):
909    def arccosh(self):
910        return derived_observable(lambda x: anp.arccosh(x[0]), [self])
def arctanh(self):
912    def arctanh(self):
913        return derived_observable(lambda x: anp.arctanh(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:
 916class CObs:
 917    """Class for a complex valued observable."""
 918    __slots__ = ['_real', '_imag', 'tag']
 919
 920    def __init__(self, real, imag=0.0):
 921        self._real = real
 922        self._imag = imag
 923        self.tag = None
 924
 925    @property
 926    def real(self):
 927        return self._real
 928
 929    @property
 930    def imag(self):
 931        return self._imag
 932
 933    def gamma_method(self, **kwargs):
 934        """Executes the gamma_method for the real and the imaginary part."""
 935        if isinstance(self.real, Obs):
 936            self.real.gamma_method(**kwargs)
 937        if isinstance(self.imag, Obs):
 938            self.imag.gamma_method(**kwargs)
 939
 940    def is_zero(self):
 941        """Checks whether both real and imaginary part are zero within machine precision."""
 942        return self.real == 0.0 and self.imag == 0.0
 943
 944    def conjugate(self):
 945        return CObs(self.real, -self.imag)
 946
 947    def __add__(self, other):
 948        if isinstance(other, np.ndarray):
 949            return other + self
 950        elif hasattr(other, 'real') and hasattr(other, 'imag'):
 951            return CObs(self.real + other.real,
 952                        self.imag + other.imag)
 953        else:
 954            return CObs(self.real + other, self.imag)
 955
 956    def __radd__(self, y):
 957        return self + y
 958
 959    def __sub__(self, other):
 960        if isinstance(other, np.ndarray):
 961            return -1 * (other - self)
 962        elif hasattr(other, 'real') and hasattr(other, 'imag'):
 963            return CObs(self.real - other.real, self.imag - other.imag)
 964        else:
 965            return CObs(self.real - other, self.imag)
 966
 967    def __rsub__(self, other):
 968        return -1 * (self - other)
 969
 970    def __mul__(self, other):
 971        if isinstance(other, np.ndarray):
 972            return other * self
 973        elif hasattr(other, 'real') and hasattr(other, 'imag'):
 974            if all(isinstance(i, Obs) for i in [self.real, self.imag, other.real, other.imag]):
 975                return CObs(derived_observable(lambda x, **kwargs: x[0] * x[1] - x[2] * x[3],
 976                                               [self.real, other.real, self.imag, other.imag],
 977                                               man_grad=[other.real.value, self.real.value, -other.imag.value, -self.imag.value]),
 978                            derived_observable(lambda x, **kwargs: x[2] * x[1] + x[0] * x[3],
 979                                               [self.real, other.real, self.imag, other.imag],
 980                                               man_grad=[other.imag.value, self.imag.value, other.real.value, self.real.value]))
 981            elif getattr(other, 'imag', 0) != 0:
 982                return CObs(self.real * other.real - self.imag * other.imag,
 983                            self.imag * other.real + self.real * other.imag)
 984            else:
 985                return CObs(self.real * other.real, self.imag * other.real)
 986        else:
 987            return CObs(self.real * other, self.imag * other)
 988
 989    def __rmul__(self, other):
 990        return self * other
 991
 992    def __truediv__(self, other):
 993        if isinstance(other, np.ndarray):
 994            return 1 / (other / self)
 995        elif hasattr(other, 'real') and hasattr(other, 'imag'):
 996            r = other.real ** 2 + other.imag ** 2
 997            return CObs((self.real * other.real + self.imag * other.imag) / r, (self.imag * other.real - self.real * other.imag) / r)
 998        else:
 999            return CObs(self.real / other, self.imag / other)
1000
1001    def __rtruediv__(self, other):
1002        r = self.real ** 2 + self.imag ** 2
1003        if hasattr(other, 'real') and hasattr(other, 'imag'):
1004            return CObs((self.real * other.real + self.imag * other.imag) / r, (self.real * other.imag - self.imag * other.real) / r)
1005        else:
1006            return CObs(self.real * other / r, -self.imag * other / r)
1007
1008    def __abs__(self):
1009        return np.sqrt(self.real**2 + self.imag**2)
1010
1011    def __pos__(self):
1012        return self
1013
1014    def __neg__(self):
1015        return -1 * self
1016
1017    def __eq__(self, other):
1018        return self.real == other.real and self.imag == other.imag
1019
1020    def __str__(self):
1021        return '(' + str(self.real) + int(self.imag >= 0.0) * '+' + str(self.imag) + 'j)'
1022
1023    def __repr__(self):
1024        return 'CObs[' + str(self) + ']'
1025
1026    def __format__(self, format_type):
1027        if format_type == "":
1028            significance = 2
1029            format_type = "2"
1030        else:
1031            significance = int(float(format_type.replace("+", "").replace("-", "")))
1032        return f"({self.real:{format_type}}{self.imag:+{significance}}j)"

Class for a complex valued observable.

CObs(real, imag=0.0)
920    def __init__(self, real, imag=0.0):
921        self._real = real
922        self._imag = imag
923        self.tag = None
tag
real
imag
def gamma_method(self, **kwargs):
933    def gamma_method(self, **kwargs):
934        """Executes the gamma_method for the real and the imaginary part."""
935        if isinstance(self.real, Obs):
936            self.real.gamma_method(**kwargs)
937        if isinstance(self.imag, Obs):
938            self.imag.gamma_method(**kwargs)

Executes the gamma_method for the real and the imaginary part.

def is_zero(self):
940    def is_zero(self):
941        """Checks whether both real and imaginary part are zero within machine precision."""
942        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):
944    def conjugate(self):
945        return CObs(self.real, -self.imag)
def gamma_method(x, **kwargs):
1035def gamma_method(x, **kwargs):
1036    """Vectorized version of the gamma_method applicable to lists or arrays of Obs.
1037
1038    See docstring of pe.Obs.gamma_method for details.
1039    """
1040    return np.vectorize(lambda o: o.gm(**kwargs))(x)

Vectorized version of the gamma_method applicable to lists or arrays of Obs.

See docstring of pe.Obs.gamma_method for details.

def gm(x, **kwargs):
1043def gm(x, **kwargs):
1044    """Short version of the vectorized gamma_method.
1045
1046    See docstring of pe.Obs.gamma_method for details
1047    """
1048    return gamma_method(x, **kwargs)

Short version of the vectorized gamma_method.

See docstring of pe.Obs.gamma_method for details

def derived_observable(func, data, array_mode=False, **kwargs):
1170def derived_observable(func, data, array_mode=False, **kwargs):
1171    """Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.
1172
1173    Parameters
1174    ----------
1175    func : object
1176        arbitrary function of the form func(data, **kwargs). For the
1177        automatic differentiation to work, all numpy functions have to have
1178        the autograd wrapper (use 'import autograd.numpy as anp').
1179    data : list
1180        list of Obs, e.g. [obs1, obs2, obs3].
1181    num_grad : bool
1182        if True, numerical derivatives are used instead of autograd
1183        (default False). To control the numerical differentiation the
1184        kwargs of numdifftools.step_generators.MaxStepGenerator
1185        can be used.
1186    man_grad : list
1187        manually supply a list or an array which contains the jacobian
1188        of func. Use cautiously, supplying the wrong derivative will
1189        not be intercepted.
1190
1191    Notes
1192    -----
1193    For simple mathematical operations it can be practical to use anonymous
1194    functions. For the ratio of two observables one can e.g. use
1195
1196    new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])
1197    """
1198
1199    data = np.asarray(data)
1200    raveled_data = data.ravel()
1201
1202    # Workaround for matrix operations containing non Obs data
1203    if not all(isinstance(x, Obs) for x in raveled_data):
1204        for i in range(len(raveled_data)):
1205            if isinstance(raveled_data[i], (int, float)):
1206                raveled_data[i] = cov_Obs(raveled_data[i], 0.0, "###dummy_covobs###")
1207
1208    allcov = {}
1209    for o in raveled_data:
1210        for name in o.cov_names:
1211            if name in allcov:
1212                if not np.allclose(allcov[name], o.covobs[name].cov):
1213                    raise Exception('Inconsistent covariance matrices for %s!' % (name))
1214            else:
1215                allcov[name] = o.covobs[name].cov
1216
1217    n_obs = len(raveled_data)
1218    new_names = sorted(set([y for x in [o.names for o in raveled_data] for y in x]))
1219    new_cov_names = sorted(set([y for x in [o.cov_names for o in raveled_data] for y in x]))
1220    new_sample_names = sorted(set(new_names) - set(new_cov_names))
1221
1222    reweighted = len(list(filter(lambda o: o.reweighted is True, raveled_data))) > 0
1223
1224    if data.ndim == 1:
1225        values = np.array([o.value for o in data])
1226    else:
1227        values = np.vectorize(lambda x: x.value)(data)
1228
1229    new_values = func(values, **kwargs)
1230
1231    multi = int(isinstance(new_values, np.ndarray))
1232
1233    new_r_values = {}
1234    new_idl_d = {}
1235    for name in new_sample_names:
1236        idl = []
1237        tmp_values = np.zeros(n_obs)
1238        for i, item in enumerate(raveled_data):
1239            tmp_values[i] = item.r_values.get(name, item.value)
1240            tmp_idl = item.idl.get(name)
1241            if tmp_idl is not None:
1242                idl.append(tmp_idl)
1243        if multi > 0:
1244            tmp_values = np.array(tmp_values).reshape(data.shape)
1245        new_r_values[name] = func(tmp_values, **kwargs)
1246        new_idl_d[name] = _merge_idx(idl)
1247
1248    if 'man_grad' in kwargs:
1249        deriv = np.asarray(kwargs.get('man_grad'))
1250        if new_values.shape + data.shape != deriv.shape:
1251            raise Exception('Manual derivative does not have correct shape.')
1252    elif kwargs.get('num_grad') is True:
1253        if multi > 0:
1254            raise Exception('Multi mode currently not supported for numerical derivative')
1255        options = {
1256            'base_step': 0.1,
1257            'step_ratio': 2.5}
1258        for key in options.keys():
1259            kwarg = kwargs.get(key)
1260            if kwarg is not None:
1261                options[key] = kwarg
1262        tmp_df = nd.Gradient(func, order=4, **{k: v for k, v in options.items() if v is not None})(values, **kwargs)
1263        if tmp_df.size == 1:
1264            deriv = np.array([tmp_df.real])
1265        else:
1266            deriv = tmp_df.real
1267    else:
1268        deriv = jacobian(func)(values, **kwargs)
1269
1270    final_result = np.zeros(new_values.shape, dtype=object)
1271
1272    if array_mode is True:
1273
1274        class _Zero_grad():
1275            def __init__(self, N):
1276                self.grad = np.zeros((N, 1))
1277
1278        new_covobs_lengths = dict(set([y for x in [[(n, o.covobs[n].N) for n in o.cov_names] for o in raveled_data] for y in x]))
1279        d_extracted = {}
1280        g_extracted = {}
1281        for name in new_sample_names:
1282            d_extracted[name] = []
1283            ens_length = len(new_idl_d[name])
1284            for i_dat, dat in enumerate(data):
1285                d_extracted[name].append(np.array([_expand_deltas_for_merge(o.deltas.get(name, np.zeros(ens_length)), o.idl.get(name, new_idl_d[name]), o.shape.get(name, ens_length), new_idl_d[name]) for o in dat.reshape(np.prod(dat.shape))]).reshape(dat.shape + (ens_length, )))
1286        for name in new_cov_names:
1287            g_extracted[name] = []
1288            zero_grad = _Zero_grad(new_covobs_lengths[name])
1289            for i_dat, dat in enumerate(data):
1290                g_extracted[name].append(np.array([o.covobs.get(name, zero_grad).grad for o in dat.reshape(np.prod(dat.shape))]).reshape(dat.shape + (new_covobs_lengths[name], 1)))
1291
1292    for i_val, new_val in np.ndenumerate(new_values):
1293        new_deltas = {}
1294        new_grad = {}
1295        if array_mode is True:
1296            for name in new_sample_names:
1297                ens_length = d_extracted[name][0].shape[-1]
1298                new_deltas[name] = np.zeros(ens_length)
1299                for i_dat, dat in enumerate(d_extracted[name]):
1300                    new_deltas[name] += np.tensordot(deriv[i_val + (i_dat, )], dat)
1301            for name in new_cov_names:
1302                new_grad[name] = 0
1303                for i_dat, dat in enumerate(g_extracted[name]):
1304                    new_grad[name] += np.tensordot(deriv[i_val + (i_dat, )], dat)
1305        else:
1306            for j_obs, obs in np.ndenumerate(data):
1307                for name in obs.names:
1308                    if name in obs.cov_names:
1309                        new_grad[name] = new_grad.get(name, 0) + deriv[i_val + j_obs] * obs.covobs[name].grad
1310                    else:
1311                        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])
1312
1313        new_covobs = {name: Covobs(0, allcov[name], name, grad=new_grad[name]) for name in new_grad}
1314
1315        if not set(new_covobs.keys()).isdisjoint(new_deltas.keys()):
1316            raise Exception('The same name has been used for deltas and covobs!')
1317        new_samples = []
1318        new_means = []
1319        new_idl = []
1320        new_names_obs = []
1321        for name in new_names:
1322            if name not in new_covobs:
1323                new_samples.append(new_deltas[name])
1324                new_idl.append(new_idl_d[name])
1325                new_means.append(new_r_values[name][i_val])
1326                new_names_obs.append(name)
1327        final_result[i_val] = Obs(new_samples, new_names_obs, means=new_means, idl=new_idl)
1328        for name in new_covobs:
1329            final_result[i_val].names.append(name)
1330        final_result[i_val]._covobs = new_covobs
1331        final_result[i_val]._value = new_val
1332        final_result[i_val].reweighted = reweighted
1333
1334    if multi == 0:
1335        final_result = final_result.item()
1336
1337    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):
1369def reweight(weight, obs, **kwargs):
1370    """Reweight a list of observables.
1371
1372    Parameters
1373    ----------
1374    weight : Obs
1375        Reweighting factor. An Observable that has to be defined on a superset of the
1376        configurations in obs[i].idl for all i.
1377    obs : list
1378        list of Obs, e.g. [obs1, obs2, obs3].
1379    all_configs : bool
1380        if True, the reweighted observables are normalized by the average of
1381        the reweighting factor on all configurations in weight.idl and not
1382        on the configurations in obs[i].idl. Default False.
1383    """
1384    result = []
1385    for i in range(len(obs)):
1386        if len(obs[i].cov_names):
1387            raise Exception('Error: Not possible to reweight an Obs that contains covobs!')
1388        if not set(obs[i].names).issubset(weight.names):
1389            raise Exception('Error: Ensembles do not fit')
1390        for name in obs[i].names:
1391            if not set(obs[i].idl[name]).issubset(weight.idl[name]):
1392                raise Exception('obs[%d] has to be defined on a subset of the configs in weight.idl[%s]!' % (i, name))
1393        new_samples = []
1394        w_deltas = {}
1395        for name in sorted(obs[i].names):
1396            w_deltas[name] = _reduce_deltas(weight.deltas[name], weight.idl[name], obs[i].idl[name])
1397            new_samples.append((w_deltas[name] + weight.r_values[name]) * (obs[i].deltas[name] + obs[i].r_values[name]))
1398        tmp_obs = Obs(new_samples, sorted(obs[i].names), idl=[obs[i].idl[name] for name in sorted(obs[i].names)])
1399
1400        if kwargs.get('all_configs'):
1401            new_weight = weight
1402        else:
1403            new_weight = Obs([w_deltas[name] + weight.r_values[name] for name in sorted(obs[i].names)], sorted(obs[i].names), idl=[obs[i].idl[name] for name in sorted(obs[i].names)])
1404
1405        result.append(tmp_obs / new_weight)
1406        result[-1].reweighted = True
1407
1408    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. Default False.
def correlate(obs_a, obs_b):
1411def correlate(obs_a, obs_b):
1412    """Correlate two observables.
1413
1414    Parameters
1415    ----------
1416    obs_a : Obs
1417        First observable
1418    obs_b : Obs
1419        Second observable
1420
1421    Notes
1422    -----
1423    Keep in mind to only correlate primary observables which have not been reweighted
1424    yet. The reweighting has to be applied after correlating the observables.
1425    Currently only works if ensembles are identical (this is not strictly necessary).
1426    """
1427
1428    if sorted(obs_a.names) != sorted(obs_b.names):
1429        raise Exception(f"Ensembles do not fit {set(sorted(obs_a.names)) ^ set(sorted(obs_b.names))}")
1430    if len(obs_a.cov_names) or len(obs_b.cov_names):
1431        raise Exception('Error: Not possible to correlate Obs that contain covobs!')
1432    for name in obs_a.names:
1433        if obs_a.shape[name] != obs_b.shape[name]:
1434            raise Exception('Shapes of ensemble', name, 'do not fit')
1435        if obs_a.idl[name] != obs_b.idl[name]:
1436            raise Exception('idl of ensemble', name, 'do not fit')
1437
1438    if obs_a.reweighted is True:
1439        warnings.warn("The first observable is already reweighted.", RuntimeWarning)
1440    if obs_b.reweighted is True:
1441        warnings.warn("The second observable is already reweighted.", RuntimeWarning)
1442
1443    new_samples = []
1444    new_idl = []
1445    for name in sorted(obs_a.names):
1446        new_samples.append((obs_a.deltas[name] + obs_a.r_values[name]) * (obs_b.deltas[name] + obs_b.r_values[name]))
1447        new_idl.append(obs_a.idl[name])
1448
1449    o = Obs(new_samples, sorted(obs_a.names), idl=new_idl)
1450    o.reweighted = obs_a.reweighted or obs_b.reweighted
1451    return o

Correlate two observables.

Parameters
  • obs_a (Obs): First observable
  • obs_b (Obs): Second observable
Notes

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 strictly necessary).

def covariance(obs, visualize=False, correlation=False, smooth=None, **kwargs):
1454def covariance(obs, visualize=False, correlation=False, smooth=None, **kwargs):
1455    r'''Calculates the error covariance matrix of a set of observables.
1456
1457    WARNING: This function should be used with care, especially for observables with support on multiple
1458             ensembles with differing autocorrelations. See the notes below for details.
1459
1460    The gamma method has to be applied first to all observables.
1461
1462    Parameters
1463    ----------
1464    obs : list or numpy.ndarray
1465        List or one dimensional array of Obs
1466    visualize : bool
1467        If True plots the corresponding normalized correlation matrix (default False).
1468    correlation : bool
1469        If True the correlation matrix instead of the error covariance matrix is returned (default False).
1470    smooth : None or int
1471        If smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue
1472        smoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the
1473        largest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely
1474        small ones.
1475
1476    Notes
1477    -----
1478    The error covariance is defined such that it agrees with the squared standard error for two identical observables
1479    $$\operatorname{cov}(a,a)=\sum_{s=1}^N\delta_a^s\delta_a^s/N^2=\Gamma_{aa}(0)/N=\operatorname{var}(a)/N=\sigma_a^2$$
1480    in the absence of autocorrelation.
1481    The error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite
1482    $$\sum_{i,j}v_i\Gamma_{ij}(0)v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i,j}v_i\delta_i^s\delta_j^s v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i}|v_i\delta_i^s|^2\geq 0\,,$$ for every $v\in\mathbb{R}^M$, while such an identity does not hold for larger windows/lags.
1483    For observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.
1484    $$\tau_{\mathrm{int}, ij}=\sqrt{\tau_{\mathrm{int}, i}\times \tau_{\mathrm{int}, j}}$$
1485    This construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).
1486    '''
1487
1488    length = len(obs)
1489
1490    max_samples = np.max([o.N for o in obs])
1491    if max_samples <= length and not [item for sublist in [o.cov_names for o in obs] for item in sublist]:
1492        warnings.warn(f"The dimension of the covariance matrix ({length}) is larger or equal to the number of samples ({max_samples}). This will result in a rank deficient matrix.", RuntimeWarning)
1493
1494    cov = np.zeros((length, length))
1495    for i in range(length):
1496        for j in range(i, length):
1497            cov[i, j] = _covariance_element(obs[i], obs[j])
1498    cov = cov + cov.T - np.diag(np.diag(cov))
1499
1500    corr = np.diag(1 / np.sqrt(np.diag(cov))) @ cov @ np.diag(1 / np.sqrt(np.diag(cov)))
1501
1502    if isinstance(smooth, int):
1503        corr = _smooth_eigenvalues(corr, smooth)
1504
1505    if visualize:
1506        plt.matshow(corr, vmin=-1, vmax=1)
1507        plt.set_cmap('RdBu')
1508        plt.colorbar()
1509        plt.draw()
1510
1511    if correlation is True:
1512        return corr
1513
1514    errors = [o.dvalue for o in obs]
1515    cov = np.diag(errors) @ corr @ np.diag(errors)
1516
1517    eigenvalues = np.linalg.eigh(cov)[0]
1518    if not np.all(eigenvalues >= 0):
1519        warnings.warn("Covariance matrix is not positive semi-definite (Eigenvalues: " + str(eigenvalues) + ")", RuntimeWarning)
1520
1521    return cov

Calculates the error covariance matrix of a set of observables.

WARNING: This function should be used with care, especially for observables with support on multiple ensembles with differing autocorrelations. See the notes below for details.

The gamma method has to be applied first to all observables.

Parameters
  • obs (list or numpy.ndarray): List or one dimensional array of Obs
  • visualize (bool): If True plots the corresponding normalized correlation matrix (default False).
  • correlation (bool): If True the correlation matrix instead of the error covariance matrix is returned (default False).
  • smooth (None or int): If smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue smoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the largest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely small ones.
Notes

The error covariance is defined such that it agrees with the squared standard error for two identical observables $$\operatorname{cov}(a,a)=\sum_{s=1}^N\delta_a^s\delta_a^s/N^2=\Gamma_{aa}(0)/N=\operatorname{var}(a)/N=\sigma_a^2$$ in the absence of autocorrelation. The error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite $$\sum_{i,j}v_i\Gamma_{ij}(0)v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i,j}v_i\delta_i^s\delta_j^s v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i}|v_i\delta_i^s|^2\geq 0\,,$$ for every $v\in\mathbb{R}^M$, while such an identity does not hold for larger windows/lags. For observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements. $$\tau_{\mathrm{int}, ij}=\sqrt{\tau_{\mathrm{int}, i}\times \tau_{\mathrm{int}, j}}$$ This construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).

def import_jackknife(jacks, name, idl=None):
1601def import_jackknife(jacks, name, idl=None):
1602    """Imports jackknife samples and returns an Obs
1603
1604    Parameters
1605    ----------
1606    jacks : numpy.ndarray
1607        numpy array containing the mean value as zeroth entry and
1608        the N jackknife samples as first to Nth entry.
1609    name : str
1610        name of the ensemble the samples are defined on.
1611    """
1612    length = len(jacks) - 1
1613    prj = (np.ones((length, length)) - (length - 1) * np.identity(length))
1614    samples = jacks[1:] @ prj
1615    mean = np.mean(samples)
1616    new_obs = Obs([samples - mean], [name], idl=idl, means=[mean])
1617    new_obs._value = jacks[0]
1618    return new_obs

Imports jackknife samples and returns an Obs

Parameters
  • jacks (numpy.ndarray): numpy array containing the mean value as zeroth entry and the N jackknife samples as first to Nth entry.
  • name (str): name of the ensemble the samples are defined on.
def import_bootstrap(boots, name, random_numbers):
1621def import_bootstrap(boots, name, random_numbers):
1622    """Imports bootstrap samples and returns an Obs
1623
1624    Parameters
1625    ----------
1626    boots : numpy.ndarray
1627        numpy array containing the mean value as zeroth entry and
1628        the N bootstrap samples as first to Nth entry.
1629    name : str
1630        name of the ensemble the samples are defined on.
1631    random_numbers : np.ndarray
1632        Array of shape (samples, length) containing the random numbers to generate the bootstrap samples,
1633        where samples is the number of bootstrap samples and length is the length of the original Monte Carlo
1634        chain to be reconstructed.
1635    """
1636    samples, length = random_numbers.shape
1637    if samples != len(boots) - 1:
1638        raise ValueError("Random numbers do not have the correct shape.")
1639
1640    if samples < length:
1641        raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.")
1642
1643    proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length
1644
1645    samples = scipy.linalg.lstsq(proj, boots[1:])[0]
1646    ret = Obs([samples], [name])
1647    ret._value = boots[0]
1648    return ret

Imports bootstrap samples and returns an Obs

Parameters
  • boots (numpy.ndarray): numpy array containing the mean value as zeroth entry and the N bootstrap samples as first to Nth entry.
  • name (str): name of the ensemble the samples are defined on.
  • random_numbers (np.ndarray): Array of shape (samples, length) containing the random numbers to generate the bootstrap samples, where samples is the number of bootstrap samples and length is the length of the original Monte Carlo chain to be reconstructed.
def merge_obs(list_of_obs):
1651def merge_obs(list_of_obs):
1652    """Combine all observables in list_of_obs into one new observable
1653
1654    Parameters
1655    ----------
1656    list_of_obs : list
1657        list of the Obs object to be combined
1658
1659    Notes
1660    -----
1661    It is not possible to combine obs which are based on the same replicum
1662    """
1663    replist = [item for obs in list_of_obs for item in obs.names]
1664    if (len(replist) == len(set(replist))) is False:
1665        raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist)))
1666    if any([len(o.cov_names) for o in list_of_obs]):
1667        raise Exception('Not possible to merge data that contains covobs!')
1668    new_dict = {}
1669    idl_dict = {}
1670    for o in list_of_obs:
1671        new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0)
1672                        for key in set(o.deltas) | set(o.r_values)})
1673        idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)})
1674
1675    names = sorted(new_dict.keys())
1676    o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names])
1677    o.reweighted = np.max([oi.reweighted for oi in list_of_obs])
1678    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
Notes

It is not possible to combine obs which are based on the same replicum

def cov_Obs(means, cov, name, grad=None):
1681def cov_Obs(means, cov, name, grad=None):
1682    """Create an Obs based on mean(s) and a covariance matrix
1683
1684    Parameters
1685    ----------
1686    mean : list of floats or float
1687        N mean value(s) of the new Obs
1688    cov : list or array
1689        2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
1690    name : str
1691        identifier for the covariance matrix
1692    grad : list or array
1693        Gradient of the Covobs wrt. the means belonging to cov.
1694    """
1695
1696    def covobs_to_obs(co):
1697        """Make an Obs out of a Covobs
1698
1699        Parameters
1700        ----------
1701        co : Covobs
1702            Covobs to be embedded into the Obs
1703        """
1704        o = Obs([], [], means=[])
1705        o._value = co.value
1706        o.names.append(co.name)
1707        o._covobs[co.name] = co
1708        o._dvalue = np.sqrt(co.errsq())
1709        return o
1710
1711    ol = []
1712    if isinstance(means, (float, int)):
1713        means = [means]
1714
1715    for i in range(len(means)):
1716        ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad)))
1717    if ol[0].covobs[name].N != len(means):
1718        raise Exception('You have to provide %d mean values!' % (ol[0].N))
1719    if len(ol) == 1:
1720        return ol[0]
1721    return ol

Create an Obs based on mean(s) and a covariance matrix

Parameters
  • mean (list of floats or float): N mean value(s) of the new Obs
  • cov (list or array): 2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
  • name (str): identifier for the covariance matrix
  • grad (list or array): Gradient of the Covobs wrt. the means belonging to cov.