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] at position %s" % (name, ' '.join(['%s' % (pos + 1) for pos in np.where(np.diff(idx) < 0)[0]]))) 108 elif np.any(dc == 0): 109 raise ValueError("Duplicate entries in idx for idl[%s] at position %s" % (name, ' '.join(['%s' % (pos + 1) for pos in np.where(np.diff(idx) == 0)[0]]))) 110 if len(dc) == 1: 111 self.idl[name] = range(idx[0], idx[-1] + dc[0], dc[0]) 112 else: 113 self.idl[name] = list(idx) 114 else: 115 raise TypeError('incompatible type for idl[%s].' % (name)) 116 else: 117 for name, sample in sorted(zip(names, samples)): 118 self.idl[name] = range(1, len(sample) + 1) 119 120 if kwargs.get("means") is not None: 121 for name, sample, mean in sorted(zip(names, samples, kwargs.get("means"))): 122 self.shape[name] = len(self.idl[name]) 123 self.N += self.shape[name] 124 self.r_values[name] = mean 125 self.deltas[name] = sample 126 else: 127 for name, sample in sorted(zip(names, samples)): 128 self.shape[name] = len(self.idl[name]) 129 self.N += self.shape[name] 130 if len(sample) != self.shape[name]: 131 raise ValueError('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name])) 132 self.r_values[name] = np.mean(sample) 133 self.deltas[name] = sample - self.r_values[name] 134 self._value += self.shape[name] * self.r_values[name] 135 self._value /= self.N 136 137 self._dvalue = 0.0 138 self.ddvalue = 0.0 139 self.reweighted = False 140 141 self.tag = None 142 143 @property 144 def value(self): 145 return self._value 146 147 @property 148 def dvalue(self): 149 return self._dvalue 150 151 @property 152 def e_names(self): 153 return sorted(set([o.split('|')[0] for o in self.names])) 154 155 @property 156 def cov_names(self): 157 return sorted(set([o for o in self.covobs.keys()])) 158 159 @property 160 def mc_names(self): 161 return sorted(set([o.split('|')[0] for o in self.names if o not in self.cov_names])) 162 163 @property 164 def e_content(self): 165 res = {} 166 for e, e_name in enumerate(self.e_names): 167 res[e_name] = sorted(filter(lambda x: x.startswith(e_name + '|'), self.names)) 168 if e_name in self.names: 169 res[e_name].append(e_name) 170 return res 171 172 @property 173 def covobs(self): 174 return self._covobs 175 176 def gamma_method(self, **kwargs): 177 """Estimate the error and related properties of the Obs. 178 179 Parameters 180 ---------- 181 S : float 182 specifies a custom value for the parameter S (default 2.0). 183 If set to 0 it is assumed that the data exhibits no 184 autocorrelation. In this case the error estimates coincides 185 with the sample standard error. 186 tau_exp : float 187 positive value triggers the critical slowing down analysis 188 (default 0.0). 189 N_sigma : float 190 number of standard deviations from zero until the tail is 191 attached to the autocorrelation function (default 1). 192 fft : bool 193 determines whether the fft algorithm is used for the computation 194 of the autocorrelation function (default True) 195 """ 196 197 e_content = self.e_content 198 self.e_dvalue = {} 199 self.e_ddvalue = {} 200 self.e_tauint = {} 201 self.e_dtauint = {} 202 self.e_windowsize = {} 203 self.e_n_tauint = {} 204 self.e_n_dtauint = {} 205 e_gamma = {} 206 self.e_rho = {} 207 self.e_drho = {} 208 self._dvalue = 0 209 self.ddvalue = 0 210 211 self.S = {} 212 self.tau_exp = {} 213 self.N_sigma = {} 214 215 if kwargs.get('fft') is False: 216 fft = False 217 else: 218 fft = True 219 220 def _parse_kwarg(kwarg_name): 221 if kwarg_name in kwargs: 222 tmp = kwargs.get(kwarg_name) 223 if isinstance(tmp, (int, float)): 224 if tmp < 0: 225 raise Exception(kwarg_name + ' has to be larger or equal to 0.') 226 for e, e_name in enumerate(self.e_names): 227 getattr(self, kwarg_name)[e_name] = tmp 228 else: 229 raise TypeError(kwarg_name + ' is not in proper format.') 230 else: 231 for e, e_name in enumerate(self.e_names): 232 if e_name in getattr(Obs, kwarg_name + '_dict'): 233 getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_dict')[e_name] 234 else: 235 getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_global') 236 237 _parse_kwarg('S') 238 _parse_kwarg('tau_exp') 239 _parse_kwarg('N_sigma') 240 241 for e, e_name in enumerate(self.mc_names): 242 gapsize = _determine_gap(self, e_content, e_name) 243 244 r_length = [] 245 for r_name in e_content[e_name]: 246 if isinstance(self.idl[r_name], range): 247 r_length.append(len(self.idl[r_name]) * self.idl[r_name].step // gapsize) 248 else: 249 r_length.append((self.idl[r_name][-1] - self.idl[r_name][0] + 1) // gapsize) 250 251 e_N = np.sum([self.shape[r_name] for r_name in e_content[e_name]]) 252 w_max = max(r_length) // 2 253 e_gamma[e_name] = np.zeros(w_max) 254 self.e_rho[e_name] = np.zeros(w_max) 255 self.e_drho[e_name] = np.zeros(w_max) 256 257 for r_name in e_content[e_name]: 258 e_gamma[e_name] += self._calc_gamma(self.deltas[r_name], self.idl[r_name], self.shape[r_name], w_max, fft, gapsize) 259 260 gamma_div = np.zeros(w_max) 261 for r_name in e_content[e_name]: 262 gamma_div += self._calc_gamma(np.ones((self.shape[r_name])), self.idl[r_name], self.shape[r_name], w_max, fft, gapsize) 263 gamma_div[gamma_div < 1] = 1.0 264 e_gamma[e_name] /= gamma_div[:w_max] 265 266 if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny: # Prevent division by zero 267 self.e_tauint[e_name] = 0.5 268 self.e_dtauint[e_name] = 0.0 269 self.e_dvalue[e_name] = 0.0 270 self.e_ddvalue[e_name] = 0.0 271 self.e_windowsize[e_name] = 0 272 continue 273 274 self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0] 275 self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:]))) 276 # Make sure no entry of tauint is smaller than 0.5 277 self.e_n_tauint[e_name][self.e_n_tauint[e_name] <= 0.5] = 0.5 + np.finfo(np.float64).eps 278 # hep-lat/0306017 eq. (42) 279 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) 280 self.e_n_dtauint[e_name][0] = 0.0 281 282 def _compute_drho(i): 283 tmp = (self.e_rho[e_name][i + 1:w_max] 284 + 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], 285 self.e_rho[e_name][1:max(1, w_max - 2 * i)]]) 286 - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i]) 287 self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N) 288 289 if self.tau_exp[e_name] > 0: 290 _compute_drho(1) 291 texp = self.tau_exp[e_name] 292 # Critical slowing down analysis 293 if w_max // 2 <= 1: 294 raise Exception("Need at least 8 samples for tau_exp error analysis") 295 for n in range(1, w_max // 2): 296 _compute_drho(n + 1) 297 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: 298 # Bias correction hep-lat/0306017 eq. (49) included 299 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 300 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) 301 # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2 302 self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) 303 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) 304 self.e_windowsize[e_name] = n 305 break 306 else: 307 if self.S[e_name] == 0.0: 308 self.e_tauint[e_name] = 0.5 309 self.e_dtauint[e_name] = 0.0 310 self.e_dvalue[e_name] = np.sqrt(e_gamma[e_name][0] / (e_N - 1)) 311 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt(0.5 / e_N) 312 self.e_windowsize[e_name] = 0 313 else: 314 # Standard automatic windowing procedure 315 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)) 316 g_w = np.exp(- np.arange(1, len(tau) + 1) / tau) - tau / np.sqrt(np.arange(1, len(tau) + 1) * e_N) 317 for n in range(1, w_max): 318 if g_w[n - 1] < 0 or n >= w_max - 1: 319 _compute_drho(n) 320 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) 321 self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n] 322 self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) 323 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) 324 self.e_windowsize[e_name] = n 325 break 326 327 self._dvalue += self.e_dvalue[e_name] ** 2 328 self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2 329 330 for e_name in self.cov_names: 331 self.e_dvalue[e_name] = np.sqrt(self.covobs[e_name].errsq()) 332 self.e_ddvalue[e_name] = 0 333 self._dvalue += self.e_dvalue[e_name]**2 334 335 self._dvalue = np.sqrt(self._dvalue) 336 if self._dvalue == 0.0: 337 self.ddvalue = 0.0 338 else: 339 self.ddvalue = np.sqrt(self.ddvalue) / self._dvalue 340 return 341 342 gm = gamma_method 343 344 def _calc_gamma(self, deltas, idx, shape, w_max, fft, gapsize): 345 """Calculate Gamma_{AA} from the deltas, which are defined on idx. 346 idx is assumed to be a contiguous range (possibly with a stepsize != 1) 347 348 Parameters 349 ---------- 350 deltas : list 351 List of fluctuations 352 idx : list 353 List or range of configurations on which the deltas are defined. 354 shape : int 355 Number of configurations in idx. 356 w_max : int 357 Upper bound for the summation window. 358 fft : bool 359 determines whether the fft algorithm is used for the computation 360 of the autocorrelation function. 361 gapsize : int 362 The target distance between two configurations. If longer distances 363 are found in idx, the data is expanded. 364 """ 365 gamma = np.zeros(w_max) 366 deltas = _expand_deltas(deltas, idx, shape, gapsize) 367 new_shape = len(deltas) 368 if fft: 369 max_gamma = min(new_shape, w_max) 370 # The padding for the fft has to be even 371 padding = new_shape + max_gamma + (new_shape + max_gamma) % 2 372 gamma[:max_gamma] += np.fft.irfft(np.abs(np.fft.rfft(deltas, padding)) ** 2)[:max_gamma] 373 else: 374 for n in range(w_max): 375 if new_shape - n >= 0: 376 gamma[n] += deltas[0:new_shape - n].dot(deltas[n:new_shape]) 377 378 return gamma 379 380 def details(self, ens_content=True): 381 """Output detailed properties of the Obs. 382 383 Parameters 384 ---------- 385 ens_content : bool 386 print details about the ensembles and replica if true. 387 """ 388 if self.tag is not None: 389 print("Description:", self.tag) 390 if not hasattr(self, 'e_dvalue'): 391 print('Result\t %3.8e' % (self.value)) 392 else: 393 if self.value == 0.0: 394 percentage = np.nan 395 else: 396 percentage = np.abs(self._dvalue / self.value) * 100 397 print('Result\t %3.8e +/- %3.8e +/- %3.8e (%3.3f%%)' % (self.value, self._dvalue, self.ddvalue, percentage)) 398 if len(self.e_names) > 1: 399 print(' Ensemble errors:') 400 e_content = self.e_content 401 for e_name in self.mc_names: 402 gap = _determine_gap(self, e_content, e_name) 403 404 if len(self.e_names) > 1: 405 print('', e_name, '\t %3.6e +/- %3.6e' % (self.e_dvalue[e_name], self.e_ddvalue[e_name])) 406 tau_string = " \N{GREEK SMALL LETTER TAU}_int\t " + _format_uncertainty(self.e_tauint[e_name], self.e_dtauint[e_name]) 407 tau_string += f" in units of {gap} config" 408 if gap > 1: 409 tau_string += "s" 410 if self.tau_exp[e_name] > 0: 411 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]) 412 else: 413 tau_string = f"{tau_string: <45}" + '\t(S=%3.2f)' % (self.S[e_name]) 414 print(tau_string) 415 for e_name in self.cov_names: 416 print('', e_name, '\t %3.8e' % (self.e_dvalue[e_name])) 417 if ens_content is True: 418 if len(self.e_names) == 1: 419 print(self.N, 'samples in', len(self.e_names), 'ensemble:') 420 else: 421 print(self.N, 'samples in', len(self.e_names), 'ensembles:') 422 my_string_list = [] 423 for key, value in sorted(self.e_content.items()): 424 if key not in self.covobs: 425 my_string = ' ' + "\u00B7 Ensemble '" + key + "' " 426 if len(value) == 1: 427 my_string += f': {self.shape[value[0]]} configurations' 428 if isinstance(self.idl[value[0]], range): 429 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}' + ')' 430 else: 431 my_string += f' (irregular range from {self.idl[value[0]][0]} to {self.idl[value[0]][-1]})' 432 else: 433 sublist = [] 434 for v in value: 435 my_substring = ' ' + "\u00B7 Replicum '" + v[len(key) + 1:] + "' " 436 my_substring += f': {self.shape[v]} configurations' 437 if isinstance(self.idl[v], range): 438 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}' + ')' 439 else: 440 my_substring += f' (irregular range from {self.idl[v][0]} to {self.idl[v][-1]})' 441 sublist.append(my_substring) 442 443 my_string += '\n' + '\n'.join(sublist) 444 else: 445 my_string = ' ' + "\u00B7 Covobs '" + key + "' " 446 my_string_list.append(my_string) 447 print('\n'.join(my_string_list)) 448 449 def reweight(self, weight): 450 """Reweight the obs with given rewighting factors. 451 452 Parameters 453 ---------- 454 weight : Obs 455 Reweighting factor. An Observable that has to be defined on a superset of the 456 configurations in obs[i].idl for all i. 457 all_configs : bool 458 if True, the reweighted observables are normalized by the average of 459 the reweighting factor on all configurations in weight.idl and not 460 on the configurations in obs[i].idl. Default False. 461 """ 462 return reweight(weight, [self])[0] 463 464 def is_zero_within_error(self, sigma=1): 465 """Checks whether the observable is zero within 'sigma' standard errors. 466 467 Parameters 468 ---------- 469 sigma : int 470 Number of standard errors used for the check. 471 472 Works only properly when the gamma method was run. 473 """ 474 return self.is_zero() or np.abs(self.value) <= sigma * self._dvalue 475 476 def is_zero(self, atol=1e-10): 477 """Checks whether the observable is zero within a given tolerance. 478 479 Parameters 480 ---------- 481 atol : float 482 Absolute tolerance (for details see numpy documentation). 483 """ 484 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()) 485 486 def plot_tauint(self, save=None): 487 """Plot integrated autocorrelation time for each ensemble. 488 489 Parameters 490 ---------- 491 save : str 492 saves the figure to a file named 'save' if. 493 """ 494 if not hasattr(self, 'e_dvalue'): 495 raise Exception('Run the gamma method first.') 496 497 for e, e_name in enumerate(self.mc_names): 498 fig = plt.figure() 499 plt.xlabel(r'$W$') 500 plt.ylabel(r'$\tau_\mathrm{int}$') 501 length = int(len(self.e_n_tauint[e_name])) 502 if self.tau_exp[e_name] > 0: 503 base = self.e_n_tauint[e_name][self.e_windowsize[e_name]] 504 x_help = np.arange(2 * self.tau_exp[e_name]) 505 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 506 x_arr = np.arange(self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]) 507 plt.plot(x_arr, y_help, 'C' + str(e), linewidth=1, ls='--', marker=',') 508 plt.errorbar([self.e_windowsize[e_name] + 2 * self.tau_exp[e_name]], [self.e_tauint[e_name]], 509 yerr=[self.e_dtauint[e_name]], fmt='C' + str(e), linewidth=1, capsize=2, marker='o', mfc=plt.rcParams['axes.facecolor']) 510 xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5 511 label = e_name + r', $\tau_\mathrm{exp}$=' + str(np.around(self.tau_exp[e_name], decimals=2)) 512 else: 513 label = e_name + ', S=' + str(np.around(self.S[e_name], decimals=2)) 514 xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5) 515 516 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) 517 plt.axvline(x=self.e_windowsize[e_name], color='C' + str(e), alpha=0.5, marker=',', ls='--') 518 plt.legend() 519 plt.xlim(-0.5, xmax) 520 ylim = plt.ylim() 521 plt.ylim(bottom=0.0, top=max(1.0, ylim[1])) 522 plt.draw() 523 if save: 524 fig.savefig(save + "_" + str(e)) 525 526 def plot_rho(self, save=None): 527 """Plot normalized autocorrelation function time for each ensemble. 528 529 Parameters 530 ---------- 531 save : str 532 saves the figure to a file named 'save' if. 533 """ 534 if not hasattr(self, 'e_dvalue'): 535 raise Exception('Run the gamma method first.') 536 for e, e_name in enumerate(self.mc_names): 537 fig = plt.figure() 538 plt.xlabel('W') 539 plt.ylabel('rho') 540 length = int(len(self.e_drho[e_name])) 541 plt.errorbar(np.arange(length), self.e_rho[e_name][:length], yerr=self.e_drho[e_name][:], linewidth=1, capsize=2) 542 plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25, ls='--', marker=',') 543 if self.tau_exp[e_name] > 0: 544 plt.plot([self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]], 545 [self.e_rho[e_name][self.e_windowsize[e_name] + 1], 0], 'k-', lw=1) 546 xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5 547 plt.title('Rho ' + e_name + r', tau\_exp=' + str(np.around(self.tau_exp[e_name], decimals=2))) 548 else: 549 xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5) 550 plt.title('Rho ' + e_name + ', S=' + str(np.around(self.S[e_name], decimals=2))) 551 plt.plot([-0.5, xmax], [0, 0], 'k--', lw=1) 552 plt.xlim(-0.5, xmax) 553 plt.draw() 554 if save: 555 fig.savefig(save + "_" + str(e)) 556 557 def plot_rep_dist(self): 558 """Plot replica distribution for each ensemble with more than one replicum.""" 559 if not hasattr(self, 'e_dvalue'): 560 raise Exception('Run the gamma method first.') 561 for e, e_name in enumerate(self.mc_names): 562 if len(self.e_content[e_name]) == 1: 563 print('No replica distribution for a single replicum (', e_name, ')') 564 continue 565 r_length = [] 566 sub_r_mean = 0 567 for r, r_name in enumerate(self.e_content[e_name]): 568 r_length.append(len(self.deltas[r_name])) 569 sub_r_mean += self.shape[r_name] * self.r_values[r_name] 570 e_N = np.sum(r_length) 571 sub_r_mean /= e_N 572 arr = np.zeros(len(self.e_content[e_name])) 573 for r, r_name in enumerate(self.e_content[e_name]): 574 arr[r] = (self.r_values[r_name] - sub_r_mean) / (self.e_dvalue[e_name] * np.sqrt(e_N / self.shape[r_name] - 1)) 575 plt.hist(arr, rwidth=0.8, bins=len(self.e_content[e_name])) 576 plt.title('Replica distribution' + e_name + ' (mean=0, var=1)') 577 plt.draw() 578 579 def plot_history(self, expand=True): 580 """Plot derived Monte Carlo history for each ensemble 581 582 Parameters 583 ---------- 584 expand : bool 585 show expanded history for irregular Monte Carlo chains (default: True). 586 """ 587 for e, e_name in enumerate(self.mc_names): 588 plt.figure() 589 r_length = [] 590 tmp = [] 591 tmp_expanded = [] 592 for r, r_name in enumerate(self.e_content[e_name]): 593 tmp.append(self.deltas[r_name] + self.r_values[r_name]) 594 if expand: 595 tmp_expanded.append(_expand_deltas(self.deltas[r_name], list(self.idl[r_name]), self.shape[r_name], 1) + self.r_values[r_name]) 596 r_length.append(len(tmp_expanded[-1])) 597 else: 598 r_length.append(len(tmp[-1])) 599 e_N = np.sum(r_length) 600 x = np.arange(e_N) 601 y_test = np.concatenate(tmp, axis=0) 602 if expand: 603 y = np.concatenate(tmp_expanded, axis=0) 604 else: 605 y = y_test 606 plt.errorbar(x, y, fmt='.', markersize=3) 607 plt.xlim(-0.5, e_N - 0.5) 608 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})') 609 plt.draw() 610 611 def plot_piechart(self, save=None): 612 """Plot piechart which shows the fractional contribution of each 613 ensemble to the error and returns a dictionary containing the fractions. 614 615 Parameters 616 ---------- 617 save : str 618 saves the figure to a file named 'save' if. 619 """ 620 if not hasattr(self, 'e_dvalue'): 621 raise Exception('Run the gamma method first.') 622 if np.isclose(0.0, self._dvalue, atol=1e-15): 623 raise Exception('Error is 0.0') 624 labels = self.e_names 625 sizes = [self.e_dvalue[name] ** 2 for name in labels] / self._dvalue ** 2 626 fig1, ax1 = plt.subplots() 627 ax1.pie(sizes, labels=labels, startangle=90, normalize=True) 628 ax1.axis('equal') 629 plt.draw() 630 if save: 631 fig1.savefig(save) 632 633 return dict(zip(labels, sizes)) 634 635 def dump(self, filename, datatype="json.gz", description="", **kwargs): 636 """Dump the Obs to a file 'name' of chosen format. 637 638 Parameters 639 ---------- 640 filename : str 641 name of the file to be saved. 642 datatype : str 643 Format of the exported file. Supported formats include 644 "json.gz" and "pickle" 645 description : str 646 Description for output file, only relevant for json.gz format. 647 path : str 648 specifies a custom path for the file (default '.') 649 """ 650 if 'path' in kwargs: 651 file_name = kwargs.get('path') + '/' + filename 652 else: 653 file_name = filename 654 655 if datatype == "json.gz": 656 from .input.json import dump_to_json 657 dump_to_json([self], file_name, description=description) 658 elif datatype == "pickle": 659 with open(file_name + '.p', 'wb') as fb: 660 pickle.dump(self, fb) 661 else: 662 raise Exception("Unknown datatype " + str(datatype)) 663 664 def export_jackknife(self): 665 """Export jackknife samples from the Obs 666 667 Returns 668 ------- 669 numpy.ndarray 670 Returns a numpy array of length N + 1 where N is the number of samples 671 for the given ensemble and replicum. The zeroth entry of the array contains 672 the mean value of the Obs, entries 1 to N contain the N jackknife samples 673 derived from the Obs. The current implementation only works for observables 674 defined on exactly one ensemble and replicum. The derived jackknife samples 675 should agree with samples from a full jackknife analysis up to O(1/N). 676 """ 677 678 if len(self.names) != 1: 679 raise Exception("'export_jackknife' is only implemented for Obs defined on one ensemble and replicum.") 680 681 name = self.names[0] 682 full_data = self.deltas[name] + self.r_values[name] 683 n = full_data.size 684 mean = self.value 685 tmp_jacks = np.zeros(n + 1) 686 tmp_jacks[0] = mean 687 tmp_jacks[1:] = (n * mean - full_data) / (n - 1) 688 return tmp_jacks 689 690 def export_bootstrap(self, samples=500, random_numbers=None, save_rng=None): 691 """Export bootstrap samples from the Obs 692 693 Parameters 694 ---------- 695 samples : int 696 Number of bootstrap samples to generate. 697 random_numbers : np.ndarray 698 Array of shape (samples, length) containing the random numbers to generate the bootstrap samples. 699 If not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name. 700 save_rng : str 701 Save the random numbers to a file if a path is specified. 702 703 Returns 704 ------- 705 numpy.ndarray 706 Returns a numpy array of length N + 1 where N is the number of samples 707 for the given ensemble and replicum. The zeroth entry of the array contains 708 the mean value of the Obs, entries 1 to N contain the N import_bootstrap samples 709 derived from the Obs. The current implementation only works for observables 710 defined on exactly one ensemble and replicum. The derived bootstrap samples 711 should agree with samples from a full bootstrap analysis up to O(1/N). 712 """ 713 if len(self.names) != 1: 714 raise Exception("'export_boostrap' is only implemented for Obs defined on one ensemble and replicum.") 715 716 name = self.names[0] 717 length = self.N 718 719 if random_numbers is None: 720 seed = int(hashlib.md5(name.encode()).hexdigest(), 16) & 0xFFFFFFFF 721 rng = np.random.default_rng(seed) 722 random_numbers = rng.integers(0, length, size=(samples, length)) 723 724 if save_rng is not None: 725 np.savetxt(save_rng, random_numbers, fmt='%i') 726 727 proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length 728 ret = np.zeros(samples + 1) 729 ret[0] = self.value 730 ret[1:] = proj @ (self.deltas[name] + self.r_values[name]) 731 return ret 732 733 def __float__(self): 734 return float(self.value) 735 736 def __repr__(self): 737 return 'Obs[' + str(self) + ']' 738 739 def __str__(self): 740 return _format_uncertainty(self.value, self._dvalue) 741 742 def __format__(self, format_type): 743 if format_type == "": 744 significance = 2 745 else: 746 significance = int(float(format_type.replace("+", "").replace("-", ""))) 747 my_str = _format_uncertainty(self.value, self._dvalue, 748 significance=significance) 749 for char in ["+", " "]: 750 if format_type.startswith(char): 751 if my_str[0] != "-": 752 my_str = char + my_str 753 return my_str 754 755 def __hash__(self): 756 hash_tuple = (np.array([self.value]).astype(np.float32).data.tobytes(),) 757 hash_tuple += tuple([o.astype(np.float32).data.tobytes() for o in self.deltas.values()]) 758 hash_tuple += tuple([np.array([o.errsq()]).astype(np.float32).data.tobytes() for o in self.covobs.values()]) 759 hash_tuple += tuple([o.encode() for o in self.names]) 760 m = hashlib.md5() 761 [m.update(o) for o in hash_tuple] 762 return int(m.hexdigest(), 16) & 0xFFFFFFFF 763 764 # Overload comparisons 765 def __lt__(self, other): 766 return self.value < other 767 768 def __le__(self, other): 769 return self.value <= other 770 771 def __gt__(self, other): 772 return self.value > other 773 774 def __ge__(self, other): 775 return self.value >= other 776 777 def __eq__(self, other): 778 if other is None: 779 return False 780 return (self - other).is_zero() 781 782 # Overload math operations 783 def __add__(self, y): 784 if isinstance(y, Obs): 785 return derived_observable(lambda x, **kwargs: x[0] + x[1], [self, y], man_grad=[1, 1]) 786 else: 787 if isinstance(y, np.ndarray): 788 return np.array([self + o for o in y]) 789 elif isinstance(y, complex): 790 return CObs(self, 0) + y 791 elif y.__class__.__name__ in ['Corr', 'CObs']: 792 return NotImplemented 793 else: 794 return derived_observable(lambda x, **kwargs: x[0] + y, [self], man_grad=[1]) 795 796 def __radd__(self, y): 797 return self + y 798 799 def __mul__(self, y): 800 if isinstance(y, Obs): 801 return derived_observable(lambda x, **kwargs: x[0] * x[1], [self, y], man_grad=[y.value, self.value]) 802 else: 803 if isinstance(y, np.ndarray): 804 return np.array([self * o for o in y]) 805 elif isinstance(y, complex): 806 return CObs(self * y.real, self * y.imag) 807 elif y.__class__.__name__ in ['Corr', 'CObs']: 808 return NotImplemented 809 else: 810 return derived_observable(lambda x, **kwargs: x[0] * y, [self], man_grad=[y]) 811 812 def __rmul__(self, y): 813 return self * y 814 815 def __sub__(self, y): 816 if isinstance(y, Obs): 817 return derived_observable(lambda x, **kwargs: x[0] - x[1], [self, y], man_grad=[1, -1]) 818 else: 819 if isinstance(y, np.ndarray): 820 return np.array([self - o for o in y]) 821 elif y.__class__.__name__ in ['Corr', 'CObs']: 822 return NotImplemented 823 else: 824 return derived_observable(lambda x, **kwargs: x[0] - y, [self], man_grad=[1]) 825 826 def __rsub__(self, y): 827 return -1 * (self - y) 828 829 def __pos__(self): 830 return self 831 832 def __neg__(self): 833 return -1 * self 834 835 def __truediv__(self, y): 836 if isinstance(y, Obs): 837 return derived_observable(lambda x, **kwargs: x[0] / x[1], [self, y], man_grad=[1 / y.value, - self.value / y.value ** 2]) 838 else: 839 if isinstance(y, np.ndarray): 840 return np.array([self / o for o in y]) 841 elif y.__class__.__name__ in ['Corr', 'CObs']: 842 return NotImplemented 843 else: 844 return derived_observable(lambda x, **kwargs: x[0] / y, [self], man_grad=[1 / y]) 845 846 def __rtruediv__(self, y): 847 if isinstance(y, Obs): 848 return derived_observable(lambda x, **kwargs: x[0] / x[1], [y, self], man_grad=[1 / self.value, - y.value / self.value ** 2]) 849 else: 850 if isinstance(y, np.ndarray): 851 return np.array([o / self for o in y]) 852 elif y.__class__.__name__ in ['Corr', 'CObs']: 853 return NotImplemented 854 else: 855 return derived_observable(lambda x, **kwargs: y / x[0], [self], man_grad=[-y / self.value ** 2]) 856 857 def __pow__(self, y): 858 if isinstance(y, Obs): 859 return derived_observable(lambda x: x[0] ** x[1], [self, y]) 860 else: 861 return derived_observable(lambda x: x[0] ** y, [self]) 862 863 def __rpow__(self, y): 864 if isinstance(y, Obs): 865 return derived_observable(lambda x: x[0] ** x[1], [y, self]) 866 else: 867 return derived_observable(lambda x: y ** x[0], [self]) 868 869 def __abs__(self): 870 return derived_observable(lambda x: anp.abs(x[0]), [self]) 871 872 # Overload numpy functions 873 def sqrt(self): 874 return derived_observable(lambda x, **kwargs: np.sqrt(x[0]), [self], man_grad=[1 / 2 / np.sqrt(self.value)]) 875 876 def log(self): 877 return derived_observable(lambda x, **kwargs: np.log(x[0]), [self], man_grad=[1 / self.value]) 878 879 def exp(self): 880 return derived_observable(lambda x, **kwargs: np.exp(x[0]), [self], man_grad=[np.exp(self.value)]) 881 882 def sin(self): 883 return derived_observable(lambda x, **kwargs: np.sin(x[0]), [self], man_grad=[np.cos(self.value)]) 884 885 def cos(self): 886 return derived_observable(lambda x, **kwargs: np.cos(x[0]), [self], man_grad=[-np.sin(self.value)]) 887 888 def tan(self): 889 return derived_observable(lambda x, **kwargs: np.tan(x[0]), [self], man_grad=[1 / np.cos(self.value) ** 2]) 890 891 def arcsin(self): 892 return derived_observable(lambda x: anp.arcsin(x[0]), [self]) 893 894 def arccos(self): 895 return derived_observable(lambda x: anp.arccos(x[0]), [self]) 896 897 def arctan(self): 898 return derived_observable(lambda x: anp.arctan(x[0]), [self]) 899 900 def sinh(self): 901 return derived_observable(lambda x, **kwargs: np.sinh(x[0]), [self], man_grad=[np.cosh(self.value)]) 902 903 def cosh(self): 904 return derived_observable(lambda x, **kwargs: np.cosh(x[0]), [self], man_grad=[np.sinh(self.value)]) 905 906 def tanh(self): 907 return derived_observable(lambda x, **kwargs: np.tanh(x[0]), [self], man_grad=[1 / np.cosh(self.value) ** 2]) 908 909 def arcsinh(self): 910 return derived_observable(lambda x: anp.arcsinh(x[0]), [self]) 911 912 def arccosh(self): 913 return derived_observable(lambda x: anp.arccosh(x[0]), [self]) 914 915 def arctanh(self): 916 return derived_observable(lambda x: anp.arctanh(x[0]), [self]) 917 918 919class CObs: 920 """Class for a complex valued observable.""" 921 __slots__ = ['_real', '_imag', 'tag'] 922 923 def __init__(self, real, imag=0.0): 924 self._real = real 925 self._imag = imag 926 self.tag = None 927 928 @property 929 def real(self): 930 return self._real 931 932 @property 933 def imag(self): 934 return self._imag 935 936 def gamma_method(self, **kwargs): 937 """Executes the gamma_method for the real and the imaginary part.""" 938 if isinstance(self.real, Obs): 939 self.real.gamma_method(**kwargs) 940 if isinstance(self.imag, Obs): 941 self.imag.gamma_method(**kwargs) 942 943 def is_zero(self): 944 """Checks whether both real and imaginary part are zero within machine precision.""" 945 return self.real == 0.0 and self.imag == 0.0 946 947 def conjugate(self): 948 return CObs(self.real, -self.imag) 949 950 def __add__(self, other): 951 if isinstance(other, np.ndarray): 952 return other + self 953 elif hasattr(other, 'real') and hasattr(other, 'imag'): 954 return CObs(self.real + other.real, 955 self.imag + other.imag) 956 else: 957 return CObs(self.real + other, self.imag) 958 959 def __radd__(self, y): 960 return self + y 961 962 def __sub__(self, other): 963 if isinstance(other, np.ndarray): 964 return -1 * (other - self) 965 elif hasattr(other, 'real') and hasattr(other, 'imag'): 966 return CObs(self.real - other.real, self.imag - other.imag) 967 else: 968 return CObs(self.real - other, self.imag) 969 970 def __rsub__(self, other): 971 return -1 * (self - other) 972 973 def __mul__(self, other): 974 if isinstance(other, np.ndarray): 975 return other * self 976 elif hasattr(other, 'real') and hasattr(other, 'imag'): 977 if all(isinstance(i, Obs) for i in [self.real, self.imag, other.real, other.imag]): 978 return CObs(derived_observable(lambda x, **kwargs: x[0] * x[1] - x[2] * x[3], 979 [self.real, other.real, self.imag, other.imag], 980 man_grad=[other.real.value, self.real.value, -other.imag.value, -self.imag.value]), 981 derived_observable(lambda x, **kwargs: x[2] * x[1] + x[0] * x[3], 982 [self.real, other.real, self.imag, other.imag], 983 man_grad=[other.imag.value, self.imag.value, other.real.value, self.real.value])) 984 elif getattr(other, 'imag', 0) != 0: 985 return CObs(self.real * other.real - self.imag * other.imag, 986 self.imag * other.real + self.real * other.imag) 987 else: 988 return CObs(self.real * other.real, self.imag * other.real) 989 else: 990 return CObs(self.real * other, self.imag * other) 991 992 def __rmul__(self, other): 993 return self * other 994 995 def __truediv__(self, other): 996 if isinstance(other, np.ndarray): 997 return 1 / (other / self) 998 elif hasattr(other, 'real') and hasattr(other, 'imag'): 999 r = other.real ** 2 + other.imag ** 2 1000 return CObs((self.real * other.real + self.imag * other.imag) / r, (self.imag * other.real - self.real * other.imag) / r) 1001 else: 1002 return CObs(self.real / other, self.imag / other) 1003 1004 def __rtruediv__(self, other): 1005 r = self.real ** 2 + self.imag ** 2 1006 if hasattr(other, 'real') and hasattr(other, 'imag'): 1007 return CObs((self.real * other.real + self.imag * other.imag) / r, (self.real * other.imag - self.imag * other.real) / r) 1008 else: 1009 return CObs(self.real * other / r, -self.imag * other / r) 1010 1011 def __abs__(self): 1012 return np.sqrt(self.real**2 + self.imag**2) 1013 1014 def __pos__(self): 1015 return self 1016 1017 def __neg__(self): 1018 return -1 * self 1019 1020 def __eq__(self, other): 1021 return self.real == other.real and self.imag == other.imag 1022 1023 def __str__(self): 1024 return '(' + str(self.real) + int(self.imag >= 0.0) * '+' + str(self.imag) + 'j)' 1025 1026 def __repr__(self): 1027 return 'CObs[' + str(self) + ']' 1028 1029 def __format__(self, format_type): 1030 if format_type == "": 1031 significance = 2 1032 format_type = "2" 1033 else: 1034 significance = int(float(format_type.replace("+", "").replace("-", ""))) 1035 return f"({self.real:{format_type}}{self.imag:+{significance}}j)" 1036 1037 1038def gamma_method(x, **kwargs): 1039 """Vectorized version of the gamma_method applicable to lists or arrays of Obs. 1040 1041 See docstring of pe.Obs.gamma_method for details. 1042 """ 1043 return np.vectorize(lambda o: o.gm(**kwargs))(x) 1044 1045 1046gm = gamma_method 1047 1048 1049def _format_uncertainty(value, dvalue, significance=2): 1050 """Creates a string of a value and its error in paranthesis notation, e.g., 13.02(45)""" 1051 if dvalue == 0.0 or (not np.isfinite(dvalue)): 1052 return str(value) 1053 if not isinstance(significance, int): 1054 raise TypeError("significance needs to be an integer.") 1055 if significance < 1: 1056 raise ValueError("significance needs to be larger than zero.") 1057 fexp = np.floor(np.log10(dvalue)) 1058 if fexp < 0.0: 1059 return '{:{form}}({:1.0f})'.format(value, dvalue * 10 ** (-fexp + significance - 1), form='.' + str(-int(fexp) + significance - 1) + 'f') 1060 elif fexp == 0.0: 1061 return f"{value:.{significance - 1}f}({dvalue:1.{significance - 1}f})" 1062 else: 1063 return f"{value:.{max(0, int(significance - fexp - 1))}f}({dvalue:2.{max(0, int(significance - fexp - 1))}f})" 1064 1065 1066def _expand_deltas(deltas, idx, shape, gapsize): 1067 """Expand deltas defined on idx to a regular range with spacing gapsize between two 1068 configurations and where holes are filled by 0. 1069 If idx is of type range, the deltas are not changed if the idx.step == gapsize. 1070 1071 Parameters 1072 ---------- 1073 deltas : list 1074 List of fluctuations 1075 idx : list 1076 List or range of configs on which the deltas are defined, has to be sorted in ascending order. 1077 shape : int 1078 Number of configs in idx. 1079 gapsize : int 1080 The target distance between two configurations. If longer distances 1081 are found in idx, the data is expanded. 1082 """ 1083 if isinstance(idx, range): 1084 if (idx.step == gapsize): 1085 return deltas 1086 ret = np.zeros((idx[-1] - idx[0] + gapsize) // gapsize) 1087 for i in range(shape): 1088 ret[(idx[i] - idx[0]) // gapsize] = deltas[i] 1089 return ret 1090 1091 1092def _merge_idx(idl): 1093 """Returns the union of all lists in idl as range or sorted list 1094 1095 Parameters 1096 ---------- 1097 idl : list 1098 List of lists or ranges. 1099 """ 1100 1101 if _check_lists_equal(idl): 1102 return idl[0] 1103 1104 idunion = sorted(set().union(*idl)) 1105 1106 # Check whether idunion can be expressed as range 1107 idrange = range(idunion[0], idunion[-1] + 1, idunion[1] - idunion[0]) 1108 idtest = [list(idrange), idunion] 1109 if _check_lists_equal(idtest): 1110 return idrange 1111 1112 return idunion 1113 1114 1115def _intersection_idx(idl): 1116 """Returns the intersection of all lists in idl as range or sorted list 1117 1118 Parameters 1119 ---------- 1120 idl : list 1121 List of lists or ranges. 1122 """ 1123 1124 if _check_lists_equal(idl): 1125 return idl[0] 1126 1127 idinter = sorted(set.intersection(*[set(o) for o in idl])) 1128 1129 # Check whether idinter can be expressed as range 1130 try: 1131 idrange = range(idinter[0], idinter[-1] + 1, idinter[1] - idinter[0]) 1132 idtest = [list(idrange), idinter] 1133 if _check_lists_equal(idtest): 1134 return idrange 1135 except IndexError: 1136 pass 1137 1138 return idinter 1139 1140 1141def _expand_deltas_for_merge(deltas, idx, shape, new_idx, scalefactor): 1142 """Expand deltas defined on idx to the list of configs that is defined by new_idx. 1143 New, empty entries are filled by 0. If idx and new_idx are of type range, the smallest 1144 common divisor of the step sizes is used as new step size. 1145 1146 Parameters 1147 ---------- 1148 deltas : list 1149 List of fluctuations 1150 idx : list 1151 List or range of configs on which the deltas are defined. 1152 Has to be a subset of new_idx and has to be sorted in ascending order. 1153 shape : list 1154 Number of configs in idx. 1155 new_idx : list 1156 List of configs that defines the new range, has to be sorted in ascending order. 1157 scalefactor : float 1158 An additional scaling factor that can be applied to scale the fluctuations, 1159 e.g., when Obs with differing numbers of replica are merged. 1160 """ 1161 if type(idx) is range and type(new_idx) is range: 1162 if idx == new_idx: 1163 if scalefactor == 1: 1164 return deltas 1165 else: 1166 return deltas * scalefactor 1167 ret = np.zeros(new_idx[-1] - new_idx[0] + 1) 1168 for i in range(shape): 1169 ret[idx[i] - new_idx[0]] = deltas[i] 1170 return np.array([ret[new_idx[i] - new_idx[0]] for i in range(len(new_idx))]) * len(new_idx) / len(idx) * scalefactor 1171 1172 1173def derived_observable(func, data, array_mode=False, **kwargs): 1174 """Construct a derived Obs according to func(data, **kwargs) using automatic differentiation. 1175 1176 Parameters 1177 ---------- 1178 func : object 1179 arbitrary function of the form func(data, **kwargs). For the 1180 automatic differentiation to work, all numpy functions have to have 1181 the autograd wrapper (use 'import autograd.numpy as anp'). 1182 data : list 1183 list of Obs, e.g. [obs1, obs2, obs3]. 1184 num_grad : bool 1185 if True, numerical derivatives are used instead of autograd 1186 (default False). To control the numerical differentiation the 1187 kwargs of numdifftools.step_generators.MaxStepGenerator 1188 can be used. 1189 man_grad : list 1190 manually supply a list or an array which contains the jacobian 1191 of func. Use cautiously, supplying the wrong derivative will 1192 not be intercepted. 1193 1194 Notes 1195 ----- 1196 For simple mathematical operations it can be practical to use anonymous 1197 functions. For the ratio of two observables one can e.g. use 1198 1199 new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2]) 1200 """ 1201 1202 data = np.asarray(data) 1203 raveled_data = data.ravel() 1204 1205 # Workaround for matrix operations containing non Obs data 1206 if not all(isinstance(x, Obs) for x in raveled_data): 1207 for i in range(len(raveled_data)): 1208 if isinstance(raveled_data[i], (int, float)): 1209 raveled_data[i] = cov_Obs(raveled_data[i], 0.0, "###dummy_covobs###") 1210 1211 allcov = {} 1212 for o in raveled_data: 1213 for name in o.cov_names: 1214 if name in allcov: 1215 if not np.allclose(allcov[name], o.covobs[name].cov): 1216 raise Exception('Inconsistent covariance matrices for %s!' % (name)) 1217 else: 1218 allcov[name] = o.covobs[name].cov 1219 1220 n_obs = len(raveled_data) 1221 new_names = sorted(set([y for x in [o.names for o in raveled_data] for y in x])) 1222 new_cov_names = sorted(set([y for x in [o.cov_names for o in raveled_data] for y in x])) 1223 new_sample_names = sorted(set(new_names) - set(new_cov_names)) 1224 1225 reweighted = len(list(filter(lambda o: o.reweighted is True, raveled_data))) > 0 1226 1227 if data.ndim == 1: 1228 values = np.array([o.value for o in data]) 1229 else: 1230 values = np.vectorize(lambda x: x.value)(data) 1231 1232 new_values = func(values, **kwargs) 1233 1234 multi = int(isinstance(new_values, np.ndarray)) 1235 1236 new_r_values = {} 1237 new_idl_d = {} 1238 for name in new_sample_names: 1239 idl = [] 1240 tmp_values = np.zeros(n_obs) 1241 for i, item in enumerate(raveled_data): 1242 tmp_values[i] = item.r_values.get(name, item.value) 1243 tmp_idl = item.idl.get(name) 1244 if tmp_idl is not None: 1245 idl.append(tmp_idl) 1246 if multi > 0: 1247 tmp_values = np.array(tmp_values).reshape(data.shape) 1248 new_r_values[name] = func(tmp_values, **kwargs) 1249 new_idl_d[name] = _merge_idx(idl) 1250 1251 def _compute_scalefactor_missing_rep(obs): 1252 """ 1253 Computes the scale factor that is to be multiplied with the deltas 1254 in the case where Obs with different subsets of replica are merged. 1255 Returns a dictionary with the scale factor for each Monte Carlo name. 1256 1257 Parameters 1258 ---------- 1259 obs : Obs 1260 The observable corresponding to the deltas that are to be scaled 1261 """ 1262 scalef_d = {} 1263 for mc_name in obs.mc_names: 1264 mc_idl_d = [name for name in obs.idl if name.startswith(mc_name + '|')] 1265 new_mc_idl_d = [name for name in new_idl_d if name.startswith(mc_name + '|')] 1266 if len(mc_idl_d) > 0 and len(mc_idl_d) < len(new_mc_idl_d): 1267 scalef_d[mc_name] = sum([len(new_idl_d[name]) for name in new_mc_idl_d]) / sum([len(new_idl_d[name]) for name in mc_idl_d]) 1268 return scalef_d 1269 1270 if 'man_grad' in kwargs: 1271 deriv = np.asarray(kwargs.get('man_grad')) 1272 if new_values.shape + data.shape != deriv.shape: 1273 raise Exception('Manual derivative does not have correct shape.') 1274 elif kwargs.get('num_grad') is True: 1275 if multi > 0: 1276 raise Exception('Multi mode currently not supported for numerical derivative') 1277 options = { 1278 'base_step': 0.1, 1279 'step_ratio': 2.5} 1280 for key in options.keys(): 1281 kwarg = kwargs.get(key) 1282 if kwarg is not None: 1283 options[key] = kwarg 1284 tmp_df = nd.Gradient(func, order=4, **{k: v for k, v in options.items() if v is not None})(values, **kwargs) 1285 if tmp_df.size == 1: 1286 deriv = np.array([tmp_df.real]) 1287 else: 1288 deriv = tmp_df.real 1289 else: 1290 deriv = jacobian(func)(values, **kwargs) 1291 1292 final_result = np.zeros(new_values.shape, dtype=object) 1293 1294 if array_mode is True: 1295 1296 class _Zero_grad(): 1297 def __init__(self, N): 1298 self.grad = np.zeros((N, 1)) 1299 1300 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])) 1301 d_extracted = {} 1302 g_extracted = {} 1303 for name in new_sample_names: 1304 d_extracted[name] = [] 1305 ens_length = len(new_idl_d[name]) 1306 for i_dat, dat in enumerate(data): 1307 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], _compute_scalefactor_missing_rep(o).get(name.split('|')[0], 1)) for o in dat.reshape(np.prod(dat.shape))]).reshape(dat.shape + (ens_length, ))) 1308 for name in new_cov_names: 1309 g_extracted[name] = [] 1310 zero_grad = _Zero_grad(new_covobs_lengths[name]) 1311 for i_dat, dat in enumerate(data): 1312 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))) 1313 1314 for i_val, new_val in np.ndenumerate(new_values): 1315 new_deltas = {} 1316 new_grad = {} 1317 if array_mode is True: 1318 for name in new_sample_names: 1319 ens_length = d_extracted[name][0].shape[-1] 1320 new_deltas[name] = np.zeros(ens_length) 1321 for i_dat, dat in enumerate(d_extracted[name]): 1322 new_deltas[name] += np.tensordot(deriv[i_val + (i_dat, )], dat) 1323 for name in new_cov_names: 1324 new_grad[name] = 0 1325 for i_dat, dat in enumerate(g_extracted[name]): 1326 new_grad[name] += np.tensordot(deriv[i_val + (i_dat, )], dat) 1327 else: 1328 for j_obs, obs in np.ndenumerate(data): 1329 scalef_d = _compute_scalefactor_missing_rep(obs) 1330 for name in obs.names: 1331 if name in obs.cov_names: 1332 new_grad[name] = new_grad.get(name, 0) + deriv[i_val + j_obs] * obs.covobs[name].grad 1333 else: 1334 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], scalef_d.get(name.split('|')[0], 1)) 1335 1336 new_covobs = {name: Covobs(0, allcov[name], name, grad=new_grad[name]) for name in new_grad} 1337 1338 if not set(new_covobs.keys()).isdisjoint(new_deltas.keys()): 1339 raise Exception('The same name has been used for deltas and covobs!') 1340 new_samples = [] 1341 new_means = [] 1342 new_idl = [] 1343 new_names_obs = [] 1344 for name in new_names: 1345 if name not in new_covobs: 1346 new_samples.append(new_deltas[name]) 1347 new_idl.append(new_idl_d[name]) 1348 new_means.append(new_r_values[name][i_val]) 1349 new_names_obs.append(name) 1350 final_result[i_val] = Obs(new_samples, new_names_obs, means=new_means, idl=new_idl) 1351 for name in new_covobs: 1352 final_result[i_val].names.append(name) 1353 final_result[i_val]._covobs = new_covobs 1354 final_result[i_val]._value = new_val 1355 final_result[i_val].reweighted = reweighted 1356 1357 if multi == 0: 1358 final_result = final_result.item() 1359 1360 return final_result 1361 1362 1363def _reduce_deltas(deltas, idx_old, idx_new): 1364 """Extract deltas defined on idx_old on all configs of idx_new. 1365 1366 Assumes, that idx_old and idx_new are correctly defined idl, i.e., they 1367 are ordered in an ascending order. 1368 1369 Parameters 1370 ---------- 1371 deltas : list 1372 List of fluctuations 1373 idx_old : list 1374 List or range of configs on which the deltas are defined 1375 idx_new : list 1376 List of configs for which we want to extract the deltas. 1377 Has to be a subset of idx_old. 1378 """ 1379 if not len(deltas) == len(idx_old): 1380 raise Exception('Length of deltas and idx_old have to be the same: %d != %d' % (len(deltas), len(idx_old))) 1381 if type(idx_old) is range and type(idx_new) is range: 1382 if idx_old == idx_new: 1383 return deltas 1384 if _check_lists_equal([idx_old, idx_new]): 1385 return deltas 1386 indices = np.intersect1d(idx_old, idx_new, assume_unique=True, return_indices=True)[1] 1387 if len(indices) < len(idx_new): 1388 raise Exception('Error in _reduce_deltas: Config of idx_new not in idx_old') 1389 return np.array(deltas)[indices] 1390 1391 1392def reweight(weight, obs, **kwargs): 1393 """Reweight a list of observables. 1394 1395 Parameters 1396 ---------- 1397 weight : Obs 1398 Reweighting factor. An Observable that has to be defined on a superset of the 1399 configurations in obs[i].idl for all i. 1400 obs : list 1401 list of Obs, e.g. [obs1, obs2, obs3]. 1402 all_configs : bool 1403 if True, the reweighted observables are normalized by the average of 1404 the reweighting factor on all configurations in weight.idl and not 1405 on the configurations in obs[i].idl. Default False. 1406 """ 1407 result = [] 1408 for i in range(len(obs)): 1409 if len(obs[i].cov_names): 1410 raise Exception('Error: Not possible to reweight an Obs that contains covobs!') 1411 if not set(obs[i].names).issubset(weight.names): 1412 raise Exception('Error: Ensembles do not fit') 1413 for name in obs[i].names: 1414 if not set(obs[i].idl[name]).issubset(weight.idl[name]): 1415 raise Exception('obs[%d] has to be defined on a subset of the configs in weight.idl[%s]!' % (i, name)) 1416 new_samples = [] 1417 w_deltas = {} 1418 for name in sorted(obs[i].names): 1419 w_deltas[name] = _reduce_deltas(weight.deltas[name], weight.idl[name], obs[i].idl[name]) 1420 new_samples.append((w_deltas[name] + weight.r_values[name]) * (obs[i].deltas[name] + obs[i].r_values[name])) 1421 tmp_obs = Obs(new_samples, sorted(obs[i].names), idl=[obs[i].idl[name] for name in sorted(obs[i].names)]) 1422 1423 if kwargs.get('all_configs'): 1424 new_weight = weight 1425 else: 1426 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)]) 1427 1428 result.append(tmp_obs / new_weight) 1429 result[-1].reweighted = True 1430 1431 return result 1432 1433 1434def correlate(obs_a, obs_b): 1435 """Correlate two observables. 1436 1437 Parameters 1438 ---------- 1439 obs_a : Obs 1440 First observable 1441 obs_b : Obs 1442 Second observable 1443 1444 Notes 1445 ----- 1446 Keep in mind to only correlate primary observables which have not been reweighted 1447 yet. The reweighting has to be applied after correlating the observables. 1448 Currently only works if ensembles are identical (this is not strictly necessary). 1449 """ 1450 1451 if sorted(obs_a.names) != sorted(obs_b.names): 1452 raise Exception(f"Ensembles do not fit {set(sorted(obs_a.names)) ^ set(sorted(obs_b.names))}") 1453 if len(obs_a.cov_names) or len(obs_b.cov_names): 1454 raise Exception('Error: Not possible to correlate Obs that contain covobs!') 1455 for name in obs_a.names: 1456 if obs_a.shape[name] != obs_b.shape[name]: 1457 raise Exception('Shapes of ensemble', name, 'do not fit') 1458 if obs_a.idl[name] != obs_b.idl[name]: 1459 raise Exception('idl of ensemble', name, 'do not fit') 1460 1461 if obs_a.reweighted is True: 1462 warnings.warn("The first observable is already reweighted.", RuntimeWarning) 1463 if obs_b.reweighted is True: 1464 warnings.warn("The second observable is already reweighted.", RuntimeWarning) 1465 1466 new_samples = [] 1467 new_idl = [] 1468 for name in sorted(obs_a.names): 1469 new_samples.append((obs_a.deltas[name] + obs_a.r_values[name]) * (obs_b.deltas[name] + obs_b.r_values[name])) 1470 new_idl.append(obs_a.idl[name]) 1471 1472 o = Obs(new_samples, sorted(obs_a.names), idl=new_idl) 1473 o.reweighted = obs_a.reweighted or obs_b.reweighted 1474 return o 1475 1476 1477def covariance(obs, visualize=False, correlation=False, smooth=None, **kwargs): 1478 r'''Calculates the error covariance matrix of a set of observables. 1479 1480 WARNING: This function should be used with care, especially for observables with support on multiple 1481 ensembles with differing autocorrelations. See the notes below for details. 1482 1483 The gamma method has to be applied first to all observables. 1484 1485 Parameters 1486 ---------- 1487 obs : list or numpy.ndarray 1488 List or one dimensional array of Obs 1489 visualize : bool 1490 If True plots the corresponding normalized correlation matrix (default False). 1491 correlation : bool 1492 If True the correlation matrix instead of the error covariance matrix is returned (default False). 1493 smooth : None or int 1494 If smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue 1495 smoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the 1496 largest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely 1497 small ones. 1498 1499 Notes 1500 ----- 1501 The error covariance is defined such that it agrees with the squared standard error for two identical observables 1502 $$\operatorname{cov}(a,a)=\sum_{s=1}^N\delta_a^s\delta_a^s/N^2=\Gamma_{aa}(0)/N=\operatorname{var}(a)/N=\sigma_a^2$$ 1503 in the absence of autocorrelation. 1504 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 1505 $$\sum_{i,j}v_i\Gamma_{ij}(0)v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i,j}v_i\delta_i^s\delta_j^s v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i}|v_i\delta_i^s|^2\geq 0\,,$$ for every $v\in\mathbb{R}^M$, while such an identity does not hold for larger windows/lags. 1506 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. 1507 $$\tau_{\mathrm{int}, ij}=\sqrt{\tau_{\mathrm{int}, i}\times \tau_{\mathrm{int}, j}}$$ 1508 This construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors). 1509 ''' 1510 1511 length = len(obs) 1512 1513 max_samples = np.max([o.N for o in obs]) 1514 if max_samples <= length and not [item for sublist in [o.cov_names for o in obs] for item in sublist]: 1515 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) 1516 1517 cov = np.zeros((length, length)) 1518 for i in range(length): 1519 for j in range(i, length): 1520 cov[i, j] = _covariance_element(obs[i], obs[j]) 1521 cov = cov + cov.T - np.diag(np.diag(cov)) 1522 1523 corr = np.diag(1 / np.sqrt(np.diag(cov))) @ cov @ np.diag(1 / np.sqrt(np.diag(cov))) 1524 1525 if isinstance(smooth, int): 1526 corr = _smooth_eigenvalues(corr, smooth) 1527 1528 if visualize: 1529 plt.matshow(corr, vmin=-1, vmax=1) 1530 plt.set_cmap('RdBu') 1531 plt.colorbar() 1532 plt.draw() 1533 1534 if correlation is True: 1535 return corr 1536 1537 errors = [o.dvalue for o in obs] 1538 cov = np.diag(errors) @ corr @ np.diag(errors) 1539 1540 eigenvalues = np.linalg.eigh(cov)[0] 1541 if not np.all(eigenvalues >= 0): 1542 warnings.warn("Covariance matrix is not positive semi-definite (Eigenvalues: " + str(eigenvalues) + ")", RuntimeWarning) 1543 1544 return cov 1545 1546 1547def _smooth_eigenvalues(corr, E): 1548 """Eigenvalue smoothing as described in hep-lat/9412087 1549 1550 corr : np.ndarray 1551 correlation matrix 1552 E : integer 1553 Number of eigenvalues to be left substantially unchanged 1554 """ 1555 if not (2 < E < corr.shape[0] - 1): 1556 raise Exception(f"'E' has to be between 2 and the dimension of the correlation matrix minus 1 ({corr.shape[0] - 1}).") 1557 vals, vec = np.linalg.eigh(corr) 1558 lambda_min = np.mean(vals[:-E]) 1559 vals[vals < lambda_min] = lambda_min 1560 vals /= np.mean(vals) 1561 return vec @ np.diag(vals) @ vec.T 1562 1563 1564def _covariance_element(obs1, obs2): 1565 """Estimates the covariance of two Obs objects, neglecting autocorrelations.""" 1566 1567 def calc_gamma(deltas1, deltas2, idx1, idx2, new_idx): 1568 deltas1 = _reduce_deltas(deltas1, idx1, new_idx) 1569 deltas2 = _reduce_deltas(deltas2, idx2, new_idx) 1570 return np.sum(deltas1 * deltas2) 1571 1572 if set(obs1.names).isdisjoint(set(obs2.names)): 1573 return 0.0 1574 1575 if not hasattr(obs1, 'e_dvalue') or not hasattr(obs2, 'e_dvalue'): 1576 raise Exception('The gamma method has to be applied to both Obs first.') 1577 1578 dvalue = 0.0 1579 1580 for e_name in obs1.mc_names: 1581 1582 if e_name not in obs2.mc_names: 1583 continue 1584 1585 idl_d = {} 1586 for r_name in obs1.e_content[e_name]: 1587 if r_name not in obs2.e_content[e_name]: 1588 continue 1589 idl_d[r_name] = _intersection_idx([obs1.idl[r_name], obs2.idl[r_name]]) 1590 1591 gamma = 0.0 1592 1593 for r_name in obs1.e_content[e_name]: 1594 if r_name not in obs2.e_content[e_name]: 1595 continue 1596 if len(idl_d[r_name]) == 0: 1597 continue 1598 gamma += calc_gamma(obs1.deltas[r_name], obs2.deltas[r_name], obs1.idl[r_name], obs2.idl[r_name], idl_d[r_name]) 1599 1600 if gamma == 0.0: 1601 continue 1602 1603 gamma_div = 0.0 1604 for r_name in obs1.e_content[e_name]: 1605 if r_name not in obs2.e_content[e_name]: 1606 continue 1607 if len(idl_d[r_name]) == 0: 1608 continue 1609 gamma_div += np.sqrt(calc_gamma(obs1.deltas[r_name], obs1.deltas[r_name], obs1.idl[r_name], obs1.idl[r_name], idl_d[r_name]) * calc_gamma(obs2.deltas[r_name], obs2.deltas[r_name], obs2.idl[r_name], obs2.idl[r_name], idl_d[r_name])) 1610 gamma /= gamma_div 1611 1612 dvalue += gamma 1613 1614 for e_name in obs1.cov_names: 1615 1616 if e_name not in obs2.cov_names: 1617 continue 1618 1619 dvalue += np.dot(np.transpose(obs1.covobs[e_name].grad), np.dot(obs1.covobs[e_name].cov, obs2.covobs[e_name].grad)).item() 1620 1621 return dvalue 1622 1623 1624def import_jackknife(jacks, name, idl=None): 1625 """Imports jackknife samples and returns an Obs 1626 1627 Parameters 1628 ---------- 1629 jacks : numpy.ndarray 1630 numpy array containing the mean value as zeroth entry and 1631 the N jackknife samples as first to Nth entry. 1632 name : str 1633 name of the ensemble the samples are defined on. 1634 """ 1635 length = len(jacks) - 1 1636 prj = (np.ones((length, length)) - (length - 1) * np.identity(length)) 1637 samples = jacks[1:] @ prj 1638 mean = np.mean(samples) 1639 new_obs = Obs([samples - mean], [name], idl=idl, means=[mean]) 1640 new_obs._value = jacks[0] 1641 return new_obs 1642 1643 1644def import_bootstrap(boots, name, random_numbers): 1645 """Imports bootstrap samples and returns an Obs 1646 1647 Parameters 1648 ---------- 1649 boots : numpy.ndarray 1650 numpy array containing the mean value as zeroth entry and 1651 the N bootstrap samples as first to Nth entry. 1652 name : str 1653 name of the ensemble the samples are defined on. 1654 random_numbers : np.ndarray 1655 Array of shape (samples, length) containing the random numbers to generate the bootstrap samples, 1656 where samples is the number of bootstrap samples and length is the length of the original Monte Carlo 1657 chain to be reconstructed. 1658 """ 1659 samples, length = random_numbers.shape 1660 if samples != len(boots) - 1: 1661 raise ValueError("Random numbers do not have the correct shape.") 1662 1663 if samples < length: 1664 raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.") 1665 1666 proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length 1667 1668 samples = scipy.linalg.lstsq(proj, boots[1:])[0] 1669 ret = Obs([samples], [name]) 1670 ret._value = boots[0] 1671 return ret 1672 1673 1674def merge_obs(list_of_obs): 1675 """Combine all observables in list_of_obs into one new observable 1676 1677 Parameters 1678 ---------- 1679 list_of_obs : list 1680 list of the Obs object to be combined 1681 1682 Notes 1683 ----- 1684 It is not possible to combine obs which are based on the same replicum 1685 """ 1686 replist = [item for obs in list_of_obs for item in obs.names] 1687 if (len(replist) == len(set(replist))) is False: 1688 raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist))) 1689 if any([len(o.cov_names) for o in list_of_obs]): 1690 raise Exception('Not possible to merge data that contains covobs!') 1691 new_dict = {} 1692 idl_dict = {} 1693 for o in list_of_obs: 1694 new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0) 1695 for key in set(o.deltas) | set(o.r_values)}) 1696 idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)}) 1697 1698 names = sorted(new_dict.keys()) 1699 o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names]) 1700 o.reweighted = np.max([oi.reweighted for oi in list_of_obs]) 1701 return o 1702 1703 1704def cov_Obs(means, cov, name, grad=None): 1705 """Create an Obs based on mean(s) and a covariance matrix 1706 1707 Parameters 1708 ---------- 1709 mean : list of floats or float 1710 N mean value(s) of the new Obs 1711 cov : list or array 1712 2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance 1713 name : str 1714 identifier for the covariance matrix 1715 grad : list or array 1716 Gradient of the Covobs wrt. the means belonging to cov. 1717 """ 1718 1719 def covobs_to_obs(co): 1720 """Make an Obs out of a Covobs 1721 1722 Parameters 1723 ---------- 1724 co : Covobs 1725 Covobs to be embedded into the Obs 1726 """ 1727 o = Obs([], [], means=[]) 1728 o._value = co.value 1729 o.names.append(co.name) 1730 o._covobs[co.name] = co 1731 o._dvalue = np.sqrt(co.errsq()) 1732 return o 1733 1734 ol = [] 1735 if isinstance(means, (float, int)): 1736 means = [means] 1737 1738 for i in range(len(means)): 1739 ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad))) 1740 if ol[0].covobs[name].N != len(means): 1741 raise Exception('You have to provide %d mean values!' % (ol[0].N)) 1742 if len(ol) == 1: 1743 return ol[0] 1744 return ol 1745 1746 1747def _determine_gap(o, e_content, e_name): 1748 gaps = [] 1749 for r_name in e_content[e_name]: 1750 if isinstance(o.idl[r_name], range): 1751 gaps.append(o.idl[r_name].step) 1752 else: 1753 gaps.append(np.min(np.diff(o.idl[r_name]))) 1754 1755 gap = min(gaps) 1756 if not np.all([gi % gap == 0 for gi in gaps]): 1757 raise Exception(f"Replica for ensemble {e_name} do not have a common spacing.", gaps) 1758 1759 return gap 1760 1761 1762def _check_lists_equal(idl): 1763 ''' 1764 Use groupby to efficiently check whether all elements of idl are identical. 1765 Returns True if all elements are equal, otherwise False. 1766 1767 Parameters 1768 ---------- 1769 idl : list of lists, ranges or np.ndarrays 1770 ''' 1771 g = groupby([np.nditer(el) if isinstance(el, np.ndarray) else el for el in idl]) 1772 if next(g, True) and not next(g, False): 1773 return True 1774 return False
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] at position %s" % (name, ' '.join(['%s' % (pos + 1) for pos in np.where(np.diff(idx) < 0)[0]]))) 109 elif np.any(dc == 0): 110 raise ValueError("Duplicate entries in idx for idl[%s] at position %s" % (name, ' '.join(['%s' % (pos + 1) for pos in np.where(np.diff(idx) == 0)[0]]))) 111 if len(dc) == 1: 112 self.idl[name] = range(idx[0], idx[-1] + dc[0], dc[0]) 113 else: 114 self.idl[name] = list(idx) 115 else: 116 raise TypeError('incompatible type for idl[%s].' % (name)) 117 else: 118 for name, sample in sorted(zip(names, samples)): 119 self.idl[name] = range(1, len(sample) + 1) 120 121 if kwargs.get("means") is not None: 122 for name, sample, mean in sorted(zip(names, samples, kwargs.get("means"))): 123 self.shape[name] = len(self.idl[name]) 124 self.N += self.shape[name] 125 self.r_values[name] = mean 126 self.deltas[name] = sample 127 else: 128 for name, sample in sorted(zip(names, samples)): 129 self.shape[name] = len(self.idl[name]) 130 self.N += self.shape[name] 131 if len(sample) != self.shape[name]: 132 raise ValueError('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name])) 133 self.r_values[name] = np.mean(sample) 134 self.deltas[name] = sample - self.r_values[name] 135 self._value += self.shape[name] * self.r_values[name] 136 self._value /= self.N 137 138 self._dvalue = 0.0 139 self.ddvalue = 0.0 140 self.reweighted = False 141 142 self.tag = None 143 144 @property 145 def value(self): 146 return self._value 147 148 @property 149 def dvalue(self): 150 return self._dvalue 151 152 @property 153 def e_names(self): 154 return sorted(set([o.split('|')[0] for o in self.names])) 155 156 @property 157 def cov_names(self): 158 return sorted(set([o for o in self.covobs.keys()])) 159 160 @property 161 def mc_names(self): 162 return sorted(set([o.split('|')[0] for o in self.names if o not in self.cov_names])) 163 164 @property 165 def e_content(self): 166 res = {} 167 for e, e_name in enumerate(self.e_names): 168 res[e_name] = sorted(filter(lambda x: x.startswith(e_name + '|'), self.names)) 169 if e_name in self.names: 170 res[e_name].append(e_name) 171 return res 172 173 @property 174 def covobs(self): 175 return self._covobs 176 177 def gamma_method(self, **kwargs): 178 """Estimate the error and related properties of the Obs. 179 180 Parameters 181 ---------- 182 S : float 183 specifies a custom value for the parameter S (default 2.0). 184 If set to 0 it is assumed that the data exhibits no 185 autocorrelation. In this case the error estimates coincides 186 with the sample standard error. 187 tau_exp : float 188 positive value triggers the critical slowing down analysis 189 (default 0.0). 190 N_sigma : float 191 number of standard deviations from zero until the tail is 192 attached to the autocorrelation function (default 1). 193 fft : bool 194 determines whether the fft algorithm is used for the computation 195 of the autocorrelation function (default True) 196 """ 197 198 e_content = self.e_content 199 self.e_dvalue = {} 200 self.e_ddvalue = {} 201 self.e_tauint = {} 202 self.e_dtauint = {} 203 self.e_windowsize = {} 204 self.e_n_tauint = {} 205 self.e_n_dtauint = {} 206 e_gamma = {} 207 self.e_rho = {} 208 self.e_drho = {} 209 self._dvalue = 0 210 self.ddvalue = 0 211 212 self.S = {} 213 self.tau_exp = {} 214 self.N_sigma = {} 215 216 if kwargs.get('fft') is False: 217 fft = False 218 else: 219 fft = True 220 221 def _parse_kwarg(kwarg_name): 222 if kwarg_name in kwargs: 223 tmp = kwargs.get(kwarg_name) 224 if isinstance(tmp, (int, float)): 225 if tmp < 0: 226 raise Exception(kwarg_name + ' has to be larger or equal to 0.') 227 for e, e_name in enumerate(self.e_names): 228 getattr(self, kwarg_name)[e_name] = tmp 229 else: 230 raise TypeError(kwarg_name + ' is not in proper format.') 231 else: 232 for e, e_name in enumerate(self.e_names): 233 if e_name in getattr(Obs, kwarg_name + '_dict'): 234 getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_dict')[e_name] 235 else: 236 getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_global') 237 238 _parse_kwarg('S') 239 _parse_kwarg('tau_exp') 240 _parse_kwarg('N_sigma') 241 242 for e, e_name in enumerate(self.mc_names): 243 gapsize = _determine_gap(self, e_content, e_name) 244 245 r_length = [] 246 for r_name in e_content[e_name]: 247 if isinstance(self.idl[r_name], range): 248 r_length.append(len(self.idl[r_name]) * self.idl[r_name].step // gapsize) 249 else: 250 r_length.append((self.idl[r_name][-1] - self.idl[r_name][0] + 1) // gapsize) 251 252 e_N = np.sum([self.shape[r_name] for r_name in e_content[e_name]]) 253 w_max = max(r_length) // 2 254 e_gamma[e_name] = np.zeros(w_max) 255 self.e_rho[e_name] = np.zeros(w_max) 256 self.e_drho[e_name] = np.zeros(w_max) 257 258 for r_name in e_content[e_name]: 259 e_gamma[e_name] += self._calc_gamma(self.deltas[r_name], self.idl[r_name], self.shape[r_name], w_max, fft, gapsize) 260 261 gamma_div = np.zeros(w_max) 262 for r_name in e_content[e_name]: 263 gamma_div += self._calc_gamma(np.ones((self.shape[r_name])), self.idl[r_name], self.shape[r_name], w_max, fft, gapsize) 264 gamma_div[gamma_div < 1] = 1.0 265 e_gamma[e_name] /= gamma_div[:w_max] 266 267 if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny: # Prevent division by zero 268 self.e_tauint[e_name] = 0.5 269 self.e_dtauint[e_name] = 0.0 270 self.e_dvalue[e_name] = 0.0 271 self.e_ddvalue[e_name] = 0.0 272 self.e_windowsize[e_name] = 0 273 continue 274 275 self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0] 276 self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:]))) 277 # Make sure no entry of tauint is smaller than 0.5 278 self.e_n_tauint[e_name][self.e_n_tauint[e_name] <= 0.5] = 0.5 + np.finfo(np.float64).eps 279 # hep-lat/0306017 eq. (42) 280 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) 281 self.e_n_dtauint[e_name][0] = 0.0 282 283 def _compute_drho(i): 284 tmp = (self.e_rho[e_name][i + 1:w_max] 285 + 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], 286 self.e_rho[e_name][1:max(1, w_max - 2 * i)]]) 287 - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i]) 288 self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N) 289 290 if self.tau_exp[e_name] > 0: 291 _compute_drho(1) 292 texp = self.tau_exp[e_name] 293 # Critical slowing down analysis 294 if w_max // 2 <= 1: 295 raise Exception("Need at least 8 samples for tau_exp error analysis") 296 for n in range(1, w_max // 2): 297 _compute_drho(n + 1) 298 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: 299 # Bias correction hep-lat/0306017 eq. (49) included 300 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 301 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) 302 # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2 303 self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) 304 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) 305 self.e_windowsize[e_name] = n 306 break 307 else: 308 if self.S[e_name] == 0.0: 309 self.e_tauint[e_name] = 0.5 310 self.e_dtauint[e_name] = 0.0 311 self.e_dvalue[e_name] = np.sqrt(e_gamma[e_name][0] / (e_N - 1)) 312 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt(0.5 / e_N) 313 self.e_windowsize[e_name] = 0 314 else: 315 # Standard automatic windowing procedure 316 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)) 317 g_w = np.exp(- np.arange(1, len(tau) + 1) / tau) - tau / np.sqrt(np.arange(1, len(tau) + 1) * e_N) 318 for n in range(1, w_max): 319 if g_w[n - 1] < 0 or n >= w_max - 1: 320 _compute_drho(n) 321 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) 322 self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n] 323 self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) 324 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) 325 self.e_windowsize[e_name] = n 326 break 327 328 self._dvalue += self.e_dvalue[e_name] ** 2 329 self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2 330 331 for e_name in self.cov_names: 332 self.e_dvalue[e_name] = np.sqrt(self.covobs[e_name].errsq()) 333 self.e_ddvalue[e_name] = 0 334 self._dvalue += self.e_dvalue[e_name]**2 335 336 self._dvalue = np.sqrt(self._dvalue) 337 if self._dvalue == 0.0: 338 self.ddvalue = 0.0 339 else: 340 self.ddvalue = np.sqrt(self.ddvalue) / self._dvalue 341 return 342 343 gm = gamma_method 344 345 def _calc_gamma(self, deltas, idx, shape, w_max, fft, gapsize): 346 """Calculate Gamma_{AA} from the deltas, which are defined on idx. 347 idx is assumed to be a contiguous range (possibly with a stepsize != 1) 348 349 Parameters 350 ---------- 351 deltas : list 352 List of fluctuations 353 idx : list 354 List or range of configurations on which the deltas are defined. 355 shape : int 356 Number of configurations in idx. 357 w_max : int 358 Upper bound for the summation window. 359 fft : bool 360 determines whether the fft algorithm is used for the computation 361 of the autocorrelation function. 362 gapsize : int 363 The target distance between two configurations. If longer distances 364 are found in idx, the data is expanded. 365 """ 366 gamma = np.zeros(w_max) 367 deltas = _expand_deltas(deltas, idx, shape, gapsize) 368 new_shape = len(deltas) 369 if fft: 370 max_gamma = min(new_shape, w_max) 371 # The padding for the fft has to be even 372 padding = new_shape + max_gamma + (new_shape + max_gamma) % 2 373 gamma[:max_gamma] += np.fft.irfft(np.abs(np.fft.rfft(deltas, padding)) ** 2)[:max_gamma] 374 else: 375 for n in range(w_max): 376 if new_shape - n >= 0: 377 gamma[n] += deltas[0:new_shape - n].dot(deltas[n:new_shape]) 378 379 return gamma 380 381 def details(self, ens_content=True): 382 """Output detailed properties of the Obs. 383 384 Parameters 385 ---------- 386 ens_content : bool 387 print details about the ensembles and replica if true. 388 """ 389 if self.tag is not None: 390 print("Description:", self.tag) 391 if not hasattr(self, 'e_dvalue'): 392 print('Result\t %3.8e' % (self.value)) 393 else: 394 if self.value == 0.0: 395 percentage = np.nan 396 else: 397 percentage = np.abs(self._dvalue / self.value) * 100 398 print('Result\t %3.8e +/- %3.8e +/- %3.8e (%3.3f%%)' % (self.value, self._dvalue, self.ddvalue, percentage)) 399 if len(self.e_names) > 1: 400 print(' Ensemble errors:') 401 e_content = self.e_content 402 for e_name in self.mc_names: 403 gap = _determine_gap(self, e_content, e_name) 404 405 if len(self.e_names) > 1: 406 print('', e_name, '\t %3.6e +/- %3.6e' % (self.e_dvalue[e_name], self.e_ddvalue[e_name])) 407 tau_string = " \N{GREEK SMALL LETTER TAU}_int\t " + _format_uncertainty(self.e_tauint[e_name], self.e_dtauint[e_name]) 408 tau_string += f" in units of {gap} config" 409 if gap > 1: 410 tau_string += "s" 411 if self.tau_exp[e_name] > 0: 412 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]) 413 else: 414 tau_string = f"{tau_string: <45}" + '\t(S=%3.2f)' % (self.S[e_name]) 415 print(tau_string) 416 for e_name in self.cov_names: 417 print('', e_name, '\t %3.8e' % (self.e_dvalue[e_name])) 418 if ens_content is True: 419 if len(self.e_names) == 1: 420 print(self.N, 'samples in', len(self.e_names), 'ensemble:') 421 else: 422 print(self.N, 'samples in', len(self.e_names), 'ensembles:') 423 my_string_list = [] 424 for key, value in sorted(self.e_content.items()): 425 if key not in self.covobs: 426 my_string = ' ' + "\u00B7 Ensemble '" + key + "' " 427 if len(value) == 1: 428 my_string += f': {self.shape[value[0]]} configurations' 429 if isinstance(self.idl[value[0]], range): 430 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}' + ')' 431 else: 432 my_string += f' (irregular range from {self.idl[value[0]][0]} to {self.idl[value[0]][-1]})' 433 else: 434 sublist = [] 435 for v in value: 436 my_substring = ' ' + "\u00B7 Replicum '" + v[len(key) + 1:] + "' " 437 my_substring += f': {self.shape[v]} configurations' 438 if isinstance(self.idl[v], range): 439 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}' + ')' 440 else: 441 my_substring += f' (irregular range from {self.idl[v][0]} to {self.idl[v][-1]})' 442 sublist.append(my_substring) 443 444 my_string += '\n' + '\n'.join(sublist) 445 else: 446 my_string = ' ' + "\u00B7 Covobs '" + key + "' " 447 my_string_list.append(my_string) 448 print('\n'.join(my_string_list)) 449 450 def reweight(self, weight): 451 """Reweight the obs with given rewighting factors. 452 453 Parameters 454 ---------- 455 weight : Obs 456 Reweighting factor. An Observable that has to be defined on a superset of the 457 configurations in obs[i].idl for all i. 458 all_configs : bool 459 if True, the reweighted observables are normalized by the average of 460 the reweighting factor on all configurations in weight.idl and not 461 on the configurations in obs[i].idl. Default False. 462 """ 463 return reweight(weight, [self])[0] 464 465 def is_zero_within_error(self, sigma=1): 466 """Checks whether the observable is zero within 'sigma' standard errors. 467 468 Parameters 469 ---------- 470 sigma : int 471 Number of standard errors used for the check. 472 473 Works only properly when the gamma method was run. 474 """ 475 return self.is_zero() or np.abs(self.value) <= sigma * self._dvalue 476 477 def is_zero(self, atol=1e-10): 478 """Checks whether the observable is zero within a given tolerance. 479 480 Parameters 481 ---------- 482 atol : float 483 Absolute tolerance (for details see numpy documentation). 484 """ 485 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()) 486 487 def plot_tauint(self, save=None): 488 """Plot integrated autocorrelation time for each ensemble. 489 490 Parameters 491 ---------- 492 save : str 493 saves the figure to a file named 'save' if. 494 """ 495 if not hasattr(self, 'e_dvalue'): 496 raise Exception('Run the gamma method first.') 497 498 for e, e_name in enumerate(self.mc_names): 499 fig = plt.figure() 500 plt.xlabel(r'$W$') 501 plt.ylabel(r'$\tau_\mathrm{int}$') 502 length = int(len(self.e_n_tauint[e_name])) 503 if self.tau_exp[e_name] > 0: 504 base = self.e_n_tauint[e_name][self.e_windowsize[e_name]] 505 x_help = np.arange(2 * self.tau_exp[e_name]) 506 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 507 x_arr = np.arange(self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]) 508 plt.plot(x_arr, y_help, 'C' + str(e), linewidth=1, ls='--', marker=',') 509 plt.errorbar([self.e_windowsize[e_name] + 2 * self.tau_exp[e_name]], [self.e_tauint[e_name]], 510 yerr=[self.e_dtauint[e_name]], fmt='C' + str(e), linewidth=1, capsize=2, marker='o', mfc=plt.rcParams['axes.facecolor']) 511 xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5 512 label = e_name + r', $\tau_\mathrm{exp}$=' + str(np.around(self.tau_exp[e_name], decimals=2)) 513 else: 514 label = e_name + ', S=' + str(np.around(self.S[e_name], decimals=2)) 515 xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5) 516 517 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) 518 plt.axvline(x=self.e_windowsize[e_name], color='C' + str(e), alpha=0.5, marker=',', ls='--') 519 plt.legend() 520 plt.xlim(-0.5, xmax) 521 ylim = plt.ylim() 522 plt.ylim(bottom=0.0, top=max(1.0, ylim[1])) 523 plt.draw() 524 if save: 525 fig.savefig(save + "_" + str(e)) 526 527 def plot_rho(self, save=None): 528 """Plot normalized autocorrelation function time for each ensemble. 529 530 Parameters 531 ---------- 532 save : str 533 saves the figure to a file named 'save' if. 534 """ 535 if not hasattr(self, 'e_dvalue'): 536 raise Exception('Run the gamma method first.') 537 for e, e_name in enumerate(self.mc_names): 538 fig = plt.figure() 539 plt.xlabel('W') 540 plt.ylabel('rho') 541 length = int(len(self.e_drho[e_name])) 542 plt.errorbar(np.arange(length), self.e_rho[e_name][:length], yerr=self.e_drho[e_name][:], linewidth=1, capsize=2) 543 plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25, ls='--', marker=',') 544 if self.tau_exp[e_name] > 0: 545 plt.plot([self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]], 546 [self.e_rho[e_name][self.e_windowsize[e_name] + 1], 0], 'k-', lw=1) 547 xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5 548 plt.title('Rho ' + e_name + r', tau\_exp=' + str(np.around(self.tau_exp[e_name], decimals=2))) 549 else: 550 xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5) 551 plt.title('Rho ' + e_name + ', S=' + str(np.around(self.S[e_name], decimals=2))) 552 plt.plot([-0.5, xmax], [0, 0], 'k--', lw=1) 553 plt.xlim(-0.5, xmax) 554 plt.draw() 555 if save: 556 fig.savefig(save + "_" + str(e)) 557 558 def plot_rep_dist(self): 559 """Plot replica distribution for each ensemble with more than one replicum.""" 560 if not hasattr(self, 'e_dvalue'): 561 raise Exception('Run the gamma method first.') 562 for e, e_name in enumerate(self.mc_names): 563 if len(self.e_content[e_name]) == 1: 564 print('No replica distribution for a single replicum (', e_name, ')') 565 continue 566 r_length = [] 567 sub_r_mean = 0 568 for r, r_name in enumerate(self.e_content[e_name]): 569 r_length.append(len(self.deltas[r_name])) 570 sub_r_mean += self.shape[r_name] * self.r_values[r_name] 571 e_N = np.sum(r_length) 572 sub_r_mean /= e_N 573 arr = np.zeros(len(self.e_content[e_name])) 574 for r, r_name in enumerate(self.e_content[e_name]): 575 arr[r] = (self.r_values[r_name] - sub_r_mean) / (self.e_dvalue[e_name] * np.sqrt(e_N / self.shape[r_name] - 1)) 576 plt.hist(arr, rwidth=0.8, bins=len(self.e_content[e_name])) 577 plt.title('Replica distribution' + e_name + ' (mean=0, var=1)') 578 plt.draw() 579 580 def plot_history(self, expand=True): 581 """Plot derived Monte Carlo history for each ensemble 582 583 Parameters 584 ---------- 585 expand : bool 586 show expanded history for irregular Monte Carlo chains (default: True). 587 """ 588 for e, e_name in enumerate(self.mc_names): 589 plt.figure() 590 r_length = [] 591 tmp = [] 592 tmp_expanded = [] 593 for r, r_name in enumerate(self.e_content[e_name]): 594 tmp.append(self.deltas[r_name] + self.r_values[r_name]) 595 if expand: 596 tmp_expanded.append(_expand_deltas(self.deltas[r_name], list(self.idl[r_name]), self.shape[r_name], 1) + self.r_values[r_name]) 597 r_length.append(len(tmp_expanded[-1])) 598 else: 599 r_length.append(len(tmp[-1])) 600 e_N = np.sum(r_length) 601 x = np.arange(e_N) 602 y_test = np.concatenate(tmp, axis=0) 603 if expand: 604 y = np.concatenate(tmp_expanded, axis=0) 605 else: 606 y = y_test 607 plt.errorbar(x, y, fmt='.', markersize=3) 608 plt.xlim(-0.5, e_N - 0.5) 609 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})') 610 plt.draw() 611 612 def plot_piechart(self, save=None): 613 """Plot piechart which shows the fractional contribution of each 614 ensemble to the error and returns a dictionary containing the fractions. 615 616 Parameters 617 ---------- 618 save : str 619 saves the figure to a file named 'save' if. 620 """ 621 if not hasattr(self, 'e_dvalue'): 622 raise Exception('Run the gamma method first.') 623 if np.isclose(0.0, self._dvalue, atol=1e-15): 624 raise Exception('Error is 0.0') 625 labels = self.e_names 626 sizes = [self.e_dvalue[name] ** 2 for name in labels] / self._dvalue ** 2 627 fig1, ax1 = plt.subplots() 628 ax1.pie(sizes, labels=labels, startangle=90, normalize=True) 629 ax1.axis('equal') 630 plt.draw() 631 if save: 632 fig1.savefig(save) 633 634 return dict(zip(labels, sizes)) 635 636 def dump(self, filename, datatype="json.gz", description="", **kwargs): 637 """Dump the Obs to a file 'name' of chosen format. 638 639 Parameters 640 ---------- 641 filename : str 642 name of the file to be saved. 643 datatype : str 644 Format of the exported file. Supported formats include 645 "json.gz" and "pickle" 646 description : str 647 Description for output file, only relevant for json.gz format. 648 path : str 649 specifies a custom path for the file (default '.') 650 """ 651 if 'path' in kwargs: 652 file_name = kwargs.get('path') + '/' + filename 653 else: 654 file_name = filename 655 656 if datatype == "json.gz": 657 from .input.json import dump_to_json 658 dump_to_json([self], file_name, description=description) 659 elif datatype == "pickle": 660 with open(file_name + '.p', 'wb') as fb: 661 pickle.dump(self, fb) 662 else: 663 raise Exception("Unknown datatype " + str(datatype)) 664 665 def export_jackknife(self): 666 """Export jackknife samples from the Obs 667 668 Returns 669 ------- 670 numpy.ndarray 671 Returns a numpy array of length N + 1 where N is the number of samples 672 for the given ensemble and replicum. The zeroth entry of the array contains 673 the mean value of the Obs, entries 1 to N contain the N jackknife samples 674 derived from the Obs. The current implementation only works for observables 675 defined on exactly one ensemble and replicum. The derived jackknife samples 676 should agree with samples from a full jackknife analysis up to O(1/N). 677 """ 678 679 if len(self.names) != 1: 680 raise Exception("'export_jackknife' is only implemented for Obs defined on one ensemble and replicum.") 681 682 name = self.names[0] 683 full_data = self.deltas[name] + self.r_values[name] 684 n = full_data.size 685 mean = self.value 686 tmp_jacks = np.zeros(n + 1) 687 tmp_jacks[0] = mean 688 tmp_jacks[1:] = (n * mean - full_data) / (n - 1) 689 return tmp_jacks 690 691 def export_bootstrap(self, samples=500, random_numbers=None, save_rng=None): 692 """Export bootstrap samples from the Obs 693 694 Parameters 695 ---------- 696 samples : int 697 Number of bootstrap samples to generate. 698 random_numbers : np.ndarray 699 Array of shape (samples, length) containing the random numbers to generate the bootstrap samples. 700 If not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name. 701 save_rng : str 702 Save the random numbers to a file if a path is specified. 703 704 Returns 705 ------- 706 numpy.ndarray 707 Returns a numpy array of length N + 1 where N is the number of samples 708 for the given ensemble and replicum. The zeroth entry of the array contains 709 the mean value of the Obs, entries 1 to N contain the N import_bootstrap samples 710 derived from the Obs. The current implementation only works for observables 711 defined on exactly one ensemble and replicum. The derived bootstrap samples 712 should agree with samples from a full bootstrap analysis up to O(1/N). 713 """ 714 if len(self.names) != 1: 715 raise Exception("'export_boostrap' is only implemented for Obs defined on one ensemble and replicum.") 716 717 name = self.names[0] 718 length = self.N 719 720 if random_numbers is None: 721 seed = int(hashlib.md5(name.encode()).hexdigest(), 16) & 0xFFFFFFFF 722 rng = np.random.default_rng(seed) 723 random_numbers = rng.integers(0, length, size=(samples, length)) 724 725 if save_rng is not None: 726 np.savetxt(save_rng, random_numbers, fmt='%i') 727 728 proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length 729 ret = np.zeros(samples + 1) 730 ret[0] = self.value 731 ret[1:] = proj @ (self.deltas[name] + self.r_values[name]) 732 return ret 733 734 def __float__(self): 735 return float(self.value) 736 737 def __repr__(self): 738 return 'Obs[' + str(self) + ']' 739 740 def __str__(self): 741 return _format_uncertainty(self.value, self._dvalue) 742 743 def __format__(self, format_type): 744 if format_type == "": 745 significance = 2 746 else: 747 significance = int(float(format_type.replace("+", "").replace("-", ""))) 748 my_str = _format_uncertainty(self.value, self._dvalue, 749 significance=significance) 750 for char in ["+", " "]: 751 if format_type.startswith(char): 752 if my_str[0] != "-": 753 my_str = char + my_str 754 return my_str 755 756 def __hash__(self): 757 hash_tuple = (np.array([self.value]).astype(np.float32).data.tobytes(),) 758 hash_tuple += tuple([o.astype(np.float32).data.tobytes() for o in self.deltas.values()]) 759 hash_tuple += tuple([np.array([o.errsq()]).astype(np.float32).data.tobytes() for o in self.covobs.values()]) 760 hash_tuple += tuple([o.encode() for o in self.names]) 761 m = hashlib.md5() 762 [m.update(o) for o in hash_tuple] 763 return int(m.hexdigest(), 16) & 0xFFFFFFFF 764 765 # Overload comparisons 766 def __lt__(self, other): 767 return self.value < other 768 769 def __le__(self, other): 770 return self.value <= other 771 772 def __gt__(self, other): 773 return self.value > other 774 775 def __ge__(self, other): 776 return self.value >= other 777 778 def __eq__(self, other): 779 if other is None: 780 return False 781 return (self - other).is_zero() 782 783 # Overload math operations 784 def __add__(self, y): 785 if isinstance(y, Obs): 786 return derived_observable(lambda x, **kwargs: x[0] + x[1], [self, y], man_grad=[1, 1]) 787 else: 788 if isinstance(y, np.ndarray): 789 return np.array([self + o for o in y]) 790 elif isinstance(y, complex): 791 return CObs(self, 0) + y 792 elif y.__class__.__name__ in ['Corr', 'CObs']: 793 return NotImplemented 794 else: 795 return derived_observable(lambda x, **kwargs: x[0] + y, [self], man_grad=[1]) 796 797 def __radd__(self, y): 798 return self + y 799 800 def __mul__(self, y): 801 if isinstance(y, Obs): 802 return derived_observable(lambda x, **kwargs: x[0] * x[1], [self, y], man_grad=[y.value, self.value]) 803 else: 804 if isinstance(y, np.ndarray): 805 return np.array([self * o for o in y]) 806 elif isinstance(y, complex): 807 return CObs(self * y.real, self * y.imag) 808 elif y.__class__.__name__ in ['Corr', 'CObs']: 809 return NotImplemented 810 else: 811 return derived_observable(lambda x, **kwargs: x[0] * y, [self], man_grad=[y]) 812 813 def __rmul__(self, y): 814 return self * y 815 816 def __sub__(self, y): 817 if isinstance(y, Obs): 818 return derived_observable(lambda x, **kwargs: x[0] - x[1], [self, y], man_grad=[1, -1]) 819 else: 820 if isinstance(y, np.ndarray): 821 return np.array([self - o for o in y]) 822 elif y.__class__.__name__ in ['Corr', 'CObs']: 823 return NotImplemented 824 else: 825 return derived_observable(lambda x, **kwargs: x[0] - y, [self], man_grad=[1]) 826 827 def __rsub__(self, y): 828 return -1 * (self - y) 829 830 def __pos__(self): 831 return self 832 833 def __neg__(self): 834 return -1 * self 835 836 def __truediv__(self, y): 837 if isinstance(y, Obs): 838 return derived_observable(lambda x, **kwargs: x[0] / x[1], [self, y], man_grad=[1 / y.value, - self.value / y.value ** 2]) 839 else: 840 if isinstance(y, np.ndarray): 841 return np.array([self / o for o in y]) 842 elif y.__class__.__name__ in ['Corr', 'CObs']: 843 return NotImplemented 844 else: 845 return derived_observable(lambda x, **kwargs: x[0] / y, [self], man_grad=[1 / y]) 846 847 def __rtruediv__(self, y): 848 if isinstance(y, Obs): 849 return derived_observable(lambda x, **kwargs: x[0] / x[1], [y, self], man_grad=[1 / self.value, - y.value / self.value ** 2]) 850 else: 851 if isinstance(y, np.ndarray): 852 return np.array([o / self for o in y]) 853 elif y.__class__.__name__ in ['Corr', 'CObs']: 854 return NotImplemented 855 else: 856 return derived_observable(lambda x, **kwargs: y / x[0], [self], man_grad=[-y / self.value ** 2]) 857 858 def __pow__(self, y): 859 if isinstance(y, Obs): 860 return derived_observable(lambda x: x[0] ** x[1], [self, y]) 861 else: 862 return derived_observable(lambda x: x[0] ** y, [self]) 863 864 def __rpow__(self, y): 865 if isinstance(y, Obs): 866 return derived_observable(lambda x: x[0] ** x[1], [y, self]) 867 else: 868 return derived_observable(lambda x: y ** x[0], [self]) 869 870 def __abs__(self): 871 return derived_observable(lambda x: anp.abs(x[0]), [self]) 872 873 # Overload numpy functions 874 def sqrt(self): 875 return derived_observable(lambda x, **kwargs: np.sqrt(x[0]), [self], man_grad=[1 / 2 / np.sqrt(self.value)]) 876 877 def log(self): 878 return derived_observable(lambda x, **kwargs: np.log(x[0]), [self], man_grad=[1 / self.value]) 879 880 def exp(self): 881 return derived_observable(lambda x, **kwargs: np.exp(x[0]), [self], man_grad=[np.exp(self.value)]) 882 883 def sin(self): 884 return derived_observable(lambda x, **kwargs: np.sin(x[0]), [self], man_grad=[np.cos(self.value)]) 885 886 def cos(self): 887 return derived_observable(lambda x, **kwargs: np.cos(x[0]), [self], man_grad=[-np.sin(self.value)]) 888 889 def tan(self): 890 return derived_observable(lambda x, **kwargs: np.tan(x[0]), [self], man_grad=[1 / np.cos(self.value) ** 2]) 891 892 def arcsin(self): 893 return derived_observable(lambda x: anp.arcsin(x[0]), [self]) 894 895 def arccos(self): 896 return derived_observable(lambda x: anp.arccos(x[0]), [self]) 897 898 def arctan(self): 899 return derived_observable(lambda x: anp.arctan(x[0]), [self]) 900 901 def sinh(self): 902 return derived_observable(lambda x, **kwargs: np.sinh(x[0]), [self], man_grad=[np.cosh(self.value)]) 903 904 def cosh(self): 905 return derived_observable(lambda x, **kwargs: np.cosh(x[0]), [self], man_grad=[np.sinh(self.value)]) 906 907 def tanh(self): 908 return derived_observable(lambda x, **kwargs: np.tanh(x[0]), [self], man_grad=[1 / np.cosh(self.value) ** 2]) 909 910 def arcsinh(self): 911 return derived_observable(lambda x: anp.arcsinh(x[0]), [self]) 912 913 def arccosh(self): 914 return derived_observable(lambda x: anp.arccosh(x[0]), [self]) 915 916 def arctanh(self): 917 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] at position %s" % (name, ' '.join(['%s' % (pos + 1) for pos in np.where(np.diff(idx) < 0)[0]]))) 109 elif np.any(dc == 0): 110 raise ValueError("Duplicate entries in idx for idl[%s] at position %s" % (name, ' '.join(['%s' % (pos + 1) for pos in np.where(np.diff(idx) == 0)[0]]))) 111 if len(dc) == 1: 112 self.idl[name] = range(idx[0], idx[-1] + dc[0], dc[0]) 113 else: 114 self.idl[name] = list(idx) 115 else: 116 raise TypeError('incompatible type for idl[%s].' % (name)) 117 else: 118 for name, sample in sorted(zip(names, samples)): 119 self.idl[name] = range(1, len(sample) + 1) 120 121 if kwargs.get("means") is not None: 122 for name, sample, mean in sorted(zip(names, samples, kwargs.get("means"))): 123 self.shape[name] = len(self.idl[name]) 124 self.N += self.shape[name] 125 self.r_values[name] = mean 126 self.deltas[name] = sample 127 else: 128 for name, sample in sorted(zip(names, samples)): 129 self.shape[name] = len(self.idl[name]) 130 self.N += self.shape[name] 131 if len(sample) != self.shape[name]: 132 raise ValueError('Incompatible samples and idx for %s: %d vs. %d' % (name, len(sample), self.shape[name])) 133 self.r_values[name] = np.mean(sample) 134 self.deltas[name] = sample - self.r_values[name] 135 self._value += self.shape[name] * self.r_values[name] 136 self._value /= self.N 137 138 self._dvalue = 0.0 139 self.ddvalue = 0.0 140 self.reweighted = False 141 142 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
def
gamma_method(self, **kwargs):
177 def gamma_method(self, **kwargs): 178 """Estimate the error and related properties of the Obs. 179 180 Parameters 181 ---------- 182 S : float 183 specifies a custom value for the parameter S (default 2.0). 184 If set to 0 it is assumed that the data exhibits no 185 autocorrelation. In this case the error estimates coincides 186 with the sample standard error. 187 tau_exp : float 188 positive value triggers the critical slowing down analysis 189 (default 0.0). 190 N_sigma : float 191 number of standard deviations from zero until the tail is 192 attached to the autocorrelation function (default 1). 193 fft : bool 194 determines whether the fft algorithm is used for the computation 195 of the autocorrelation function (default True) 196 """ 197 198 e_content = self.e_content 199 self.e_dvalue = {} 200 self.e_ddvalue = {} 201 self.e_tauint = {} 202 self.e_dtauint = {} 203 self.e_windowsize = {} 204 self.e_n_tauint = {} 205 self.e_n_dtauint = {} 206 e_gamma = {} 207 self.e_rho = {} 208 self.e_drho = {} 209 self._dvalue = 0 210 self.ddvalue = 0 211 212 self.S = {} 213 self.tau_exp = {} 214 self.N_sigma = {} 215 216 if kwargs.get('fft') is False: 217 fft = False 218 else: 219 fft = True 220 221 def _parse_kwarg(kwarg_name): 222 if kwarg_name in kwargs: 223 tmp = kwargs.get(kwarg_name) 224 if isinstance(tmp, (int, float)): 225 if tmp < 0: 226 raise Exception(kwarg_name + ' has to be larger or equal to 0.') 227 for e, e_name in enumerate(self.e_names): 228 getattr(self, kwarg_name)[e_name] = tmp 229 else: 230 raise TypeError(kwarg_name + ' is not in proper format.') 231 else: 232 for e, e_name in enumerate(self.e_names): 233 if e_name in getattr(Obs, kwarg_name + '_dict'): 234 getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_dict')[e_name] 235 else: 236 getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_global') 237 238 _parse_kwarg('S') 239 _parse_kwarg('tau_exp') 240 _parse_kwarg('N_sigma') 241 242 for e, e_name in enumerate(self.mc_names): 243 gapsize = _determine_gap(self, e_content, e_name) 244 245 r_length = [] 246 for r_name in e_content[e_name]: 247 if isinstance(self.idl[r_name], range): 248 r_length.append(len(self.idl[r_name]) * self.idl[r_name].step // gapsize) 249 else: 250 r_length.append((self.idl[r_name][-1] - self.idl[r_name][0] + 1) // gapsize) 251 252 e_N = np.sum([self.shape[r_name] for r_name in e_content[e_name]]) 253 w_max = max(r_length) // 2 254 e_gamma[e_name] = np.zeros(w_max) 255 self.e_rho[e_name] = np.zeros(w_max) 256 self.e_drho[e_name] = np.zeros(w_max) 257 258 for r_name in e_content[e_name]: 259 e_gamma[e_name] += self._calc_gamma(self.deltas[r_name], self.idl[r_name], self.shape[r_name], w_max, fft, gapsize) 260 261 gamma_div = np.zeros(w_max) 262 for r_name in e_content[e_name]: 263 gamma_div += self._calc_gamma(np.ones((self.shape[r_name])), self.idl[r_name], self.shape[r_name], w_max, fft, gapsize) 264 gamma_div[gamma_div < 1] = 1.0 265 e_gamma[e_name] /= gamma_div[:w_max] 266 267 if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny: # Prevent division by zero 268 self.e_tauint[e_name] = 0.5 269 self.e_dtauint[e_name] = 0.0 270 self.e_dvalue[e_name] = 0.0 271 self.e_ddvalue[e_name] = 0.0 272 self.e_windowsize[e_name] = 0 273 continue 274 275 self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0] 276 self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:]))) 277 # Make sure no entry of tauint is smaller than 0.5 278 self.e_n_tauint[e_name][self.e_n_tauint[e_name] <= 0.5] = 0.5 + np.finfo(np.float64).eps 279 # hep-lat/0306017 eq. (42) 280 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) 281 self.e_n_dtauint[e_name][0] = 0.0 282 283 def _compute_drho(i): 284 tmp = (self.e_rho[e_name][i + 1:w_max] 285 + 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], 286 self.e_rho[e_name][1:max(1, w_max - 2 * i)]]) 287 - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i]) 288 self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N) 289 290 if self.tau_exp[e_name] > 0: 291 _compute_drho(1) 292 texp = self.tau_exp[e_name] 293 # Critical slowing down analysis 294 if w_max // 2 <= 1: 295 raise Exception("Need at least 8 samples for tau_exp error analysis") 296 for n in range(1, w_max // 2): 297 _compute_drho(n + 1) 298 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: 299 # Bias correction hep-lat/0306017 eq. (49) included 300 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 301 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) 302 # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2 303 self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) 304 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) 305 self.e_windowsize[e_name] = n 306 break 307 else: 308 if self.S[e_name] == 0.0: 309 self.e_tauint[e_name] = 0.5 310 self.e_dtauint[e_name] = 0.0 311 self.e_dvalue[e_name] = np.sqrt(e_gamma[e_name][0] / (e_N - 1)) 312 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt(0.5 / e_N) 313 self.e_windowsize[e_name] = 0 314 else: 315 # Standard automatic windowing procedure 316 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)) 317 g_w = np.exp(- np.arange(1, len(tau) + 1) / tau) - tau / np.sqrt(np.arange(1, len(tau) + 1) * e_N) 318 for n in range(1, w_max): 319 if g_w[n - 1] < 0 or n >= w_max - 1: 320 _compute_drho(n) 321 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) 322 self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n] 323 self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) 324 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) 325 self.e_windowsize[e_name] = n 326 break 327 328 self._dvalue += self.e_dvalue[e_name] ** 2 329 self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2 330 331 for e_name in self.cov_names: 332 self.e_dvalue[e_name] = np.sqrt(self.covobs[e_name].errsq()) 333 self.e_ddvalue[e_name] = 0 334 self._dvalue += self.e_dvalue[e_name]**2 335 336 self._dvalue = np.sqrt(self._dvalue) 337 if self._dvalue == 0.0: 338 self.ddvalue = 0.0 339 else: 340 self.ddvalue = np.sqrt(self.ddvalue) / self._dvalue 341 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):
177 def gamma_method(self, **kwargs): 178 """Estimate the error and related properties of the Obs. 179 180 Parameters 181 ---------- 182 S : float 183 specifies a custom value for the parameter S (default 2.0). 184 If set to 0 it is assumed that the data exhibits no 185 autocorrelation. In this case the error estimates coincides 186 with the sample standard error. 187 tau_exp : float 188 positive value triggers the critical slowing down analysis 189 (default 0.0). 190 N_sigma : float 191 number of standard deviations from zero until the tail is 192 attached to the autocorrelation function (default 1). 193 fft : bool 194 determines whether the fft algorithm is used for the computation 195 of the autocorrelation function (default True) 196 """ 197 198 e_content = self.e_content 199 self.e_dvalue = {} 200 self.e_ddvalue = {} 201 self.e_tauint = {} 202 self.e_dtauint = {} 203 self.e_windowsize = {} 204 self.e_n_tauint = {} 205 self.e_n_dtauint = {} 206 e_gamma = {} 207 self.e_rho = {} 208 self.e_drho = {} 209 self._dvalue = 0 210 self.ddvalue = 0 211 212 self.S = {} 213 self.tau_exp = {} 214 self.N_sigma = {} 215 216 if kwargs.get('fft') is False: 217 fft = False 218 else: 219 fft = True 220 221 def _parse_kwarg(kwarg_name): 222 if kwarg_name in kwargs: 223 tmp = kwargs.get(kwarg_name) 224 if isinstance(tmp, (int, float)): 225 if tmp < 0: 226 raise Exception(kwarg_name + ' has to be larger or equal to 0.') 227 for e, e_name in enumerate(self.e_names): 228 getattr(self, kwarg_name)[e_name] = tmp 229 else: 230 raise TypeError(kwarg_name + ' is not in proper format.') 231 else: 232 for e, e_name in enumerate(self.e_names): 233 if e_name in getattr(Obs, kwarg_name + '_dict'): 234 getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_dict')[e_name] 235 else: 236 getattr(self, kwarg_name)[e_name] = getattr(Obs, kwarg_name + '_global') 237 238 _parse_kwarg('S') 239 _parse_kwarg('tau_exp') 240 _parse_kwarg('N_sigma') 241 242 for e, e_name in enumerate(self.mc_names): 243 gapsize = _determine_gap(self, e_content, e_name) 244 245 r_length = [] 246 for r_name in e_content[e_name]: 247 if isinstance(self.idl[r_name], range): 248 r_length.append(len(self.idl[r_name]) * self.idl[r_name].step // gapsize) 249 else: 250 r_length.append((self.idl[r_name][-1] - self.idl[r_name][0] + 1) // gapsize) 251 252 e_N = np.sum([self.shape[r_name] for r_name in e_content[e_name]]) 253 w_max = max(r_length) // 2 254 e_gamma[e_name] = np.zeros(w_max) 255 self.e_rho[e_name] = np.zeros(w_max) 256 self.e_drho[e_name] = np.zeros(w_max) 257 258 for r_name in e_content[e_name]: 259 e_gamma[e_name] += self._calc_gamma(self.deltas[r_name], self.idl[r_name], self.shape[r_name], w_max, fft, gapsize) 260 261 gamma_div = np.zeros(w_max) 262 for r_name in e_content[e_name]: 263 gamma_div += self._calc_gamma(np.ones((self.shape[r_name])), self.idl[r_name], self.shape[r_name], w_max, fft, gapsize) 264 gamma_div[gamma_div < 1] = 1.0 265 e_gamma[e_name] /= gamma_div[:w_max] 266 267 if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny: # Prevent division by zero 268 self.e_tauint[e_name] = 0.5 269 self.e_dtauint[e_name] = 0.0 270 self.e_dvalue[e_name] = 0.0 271 self.e_ddvalue[e_name] = 0.0 272 self.e_windowsize[e_name] = 0 273 continue 274 275 self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0] 276 self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:]))) 277 # Make sure no entry of tauint is smaller than 0.5 278 self.e_n_tauint[e_name][self.e_n_tauint[e_name] <= 0.5] = 0.5 + np.finfo(np.float64).eps 279 # hep-lat/0306017 eq. (42) 280 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) 281 self.e_n_dtauint[e_name][0] = 0.0 282 283 def _compute_drho(i): 284 tmp = (self.e_rho[e_name][i + 1:w_max] 285 + 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], 286 self.e_rho[e_name][1:max(1, w_max - 2 * i)]]) 287 - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i]) 288 self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N) 289 290 if self.tau_exp[e_name] > 0: 291 _compute_drho(1) 292 texp = self.tau_exp[e_name] 293 # Critical slowing down analysis 294 if w_max // 2 <= 1: 295 raise Exception("Need at least 8 samples for tau_exp error analysis") 296 for n in range(1, w_max // 2): 297 _compute_drho(n + 1) 298 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: 299 # Bias correction hep-lat/0306017 eq. (49) included 300 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 301 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) 302 # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2 303 self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) 304 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) 305 self.e_windowsize[e_name] = n 306 break 307 else: 308 if self.S[e_name] == 0.0: 309 self.e_tauint[e_name] = 0.5 310 self.e_dtauint[e_name] = 0.0 311 self.e_dvalue[e_name] = np.sqrt(e_gamma[e_name][0] / (e_N - 1)) 312 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt(0.5 / e_N) 313 self.e_windowsize[e_name] = 0 314 else: 315 # Standard automatic windowing procedure 316 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)) 317 g_w = np.exp(- np.arange(1, len(tau) + 1) / tau) - tau / np.sqrt(np.arange(1, len(tau) + 1) * e_N) 318 for n in range(1, w_max): 319 if g_w[n - 1] < 0 or n >= w_max - 1: 320 _compute_drho(n) 321 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) 322 self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n] 323 self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) 324 self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) 325 self.e_windowsize[e_name] = n 326 break 327 328 self._dvalue += self.e_dvalue[e_name] ** 2 329 self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2 330 331 for e_name in self.cov_names: 332 self.e_dvalue[e_name] = np.sqrt(self.covobs[e_name].errsq()) 333 self.e_ddvalue[e_name] = 0 334 self._dvalue += self.e_dvalue[e_name]**2 335 336 self._dvalue = np.sqrt(self._dvalue) 337 if self._dvalue == 0.0: 338 self.ddvalue = 0.0 339 else: 340 self.ddvalue = np.sqrt(self.ddvalue) / self._dvalue 341 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):
381 def details(self, ens_content=True): 382 """Output detailed properties of the Obs. 383 384 Parameters 385 ---------- 386 ens_content : bool 387 print details about the ensembles and replica if true. 388 """ 389 if self.tag is not None: 390 print("Description:", self.tag) 391 if not hasattr(self, 'e_dvalue'): 392 print('Result\t %3.8e' % (self.value)) 393 else: 394 if self.value == 0.0: 395 percentage = np.nan 396 else: 397 percentage = np.abs(self._dvalue / self.value) * 100 398 print('Result\t %3.8e +/- %3.8e +/- %3.8e (%3.3f%%)' % (self.value, self._dvalue, self.ddvalue, percentage)) 399 if len(self.e_names) > 1: 400 print(' Ensemble errors:') 401 e_content = self.e_content 402 for e_name in self.mc_names: 403 gap = _determine_gap(self, e_content, e_name) 404 405 if len(self.e_names) > 1: 406 print('', e_name, '\t %3.6e +/- %3.6e' % (self.e_dvalue[e_name], self.e_ddvalue[e_name])) 407 tau_string = " \N{GREEK SMALL LETTER TAU}_int\t " + _format_uncertainty(self.e_tauint[e_name], self.e_dtauint[e_name]) 408 tau_string += f" in units of {gap} config" 409 if gap > 1: 410 tau_string += "s" 411 if self.tau_exp[e_name] > 0: 412 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]) 413 else: 414 tau_string = f"{tau_string: <45}" + '\t(S=%3.2f)' % (self.S[e_name]) 415 print(tau_string) 416 for e_name in self.cov_names: 417 print('', e_name, '\t %3.8e' % (self.e_dvalue[e_name])) 418 if ens_content is True: 419 if len(self.e_names) == 1: 420 print(self.N, 'samples in', len(self.e_names), 'ensemble:') 421 else: 422 print(self.N, 'samples in', len(self.e_names), 'ensembles:') 423 my_string_list = [] 424 for key, value in sorted(self.e_content.items()): 425 if key not in self.covobs: 426 my_string = ' ' + "\u00B7 Ensemble '" + key + "' " 427 if len(value) == 1: 428 my_string += f': {self.shape[value[0]]} configurations' 429 if isinstance(self.idl[value[0]], range): 430 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}' + ')' 431 else: 432 my_string += f' (irregular range from {self.idl[value[0]][0]} to {self.idl[value[0]][-1]})' 433 else: 434 sublist = [] 435 for v in value: 436 my_substring = ' ' + "\u00B7 Replicum '" + v[len(key) + 1:] + "' " 437 my_substring += f': {self.shape[v]} configurations' 438 if isinstance(self.idl[v], range): 439 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}' + ')' 440 else: 441 my_substring += f' (irregular range from {self.idl[v][0]} to {self.idl[v][-1]})' 442 sublist.append(my_substring) 443 444 my_string += '\n' + '\n'.join(sublist) 445 else: 446 my_string = ' ' + "\u00B7 Covobs '" + key + "' " 447 my_string_list.append(my_string) 448 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):
450 def reweight(self, weight): 451 """Reweight the obs with given rewighting factors. 452 453 Parameters 454 ---------- 455 weight : Obs 456 Reweighting factor. An Observable that has to be defined on a superset of the 457 configurations in obs[i].idl for all i. 458 all_configs : bool 459 if True, the reweighted observables are normalized by the average of 460 the reweighting factor on all configurations in weight.idl and not 461 on the configurations in obs[i].idl. Default False. 462 """ 463 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):
465 def is_zero_within_error(self, sigma=1): 466 """Checks whether the observable is zero within 'sigma' standard errors. 467 468 Parameters 469 ---------- 470 sigma : int 471 Number of standard errors used for the check. 472 473 Works only properly when the gamma method was run. 474 """ 475 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):
477 def is_zero(self, atol=1e-10): 478 """Checks whether the observable is zero within a given tolerance. 479 480 Parameters 481 ---------- 482 atol : float 483 Absolute tolerance (for details see numpy documentation). 484 """ 485 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):
487 def plot_tauint(self, save=None): 488 """Plot integrated autocorrelation time for each ensemble. 489 490 Parameters 491 ---------- 492 save : str 493 saves the figure to a file named 'save' if. 494 """ 495 if not hasattr(self, 'e_dvalue'): 496 raise Exception('Run the gamma method first.') 497 498 for e, e_name in enumerate(self.mc_names): 499 fig = plt.figure() 500 plt.xlabel(r'$W$') 501 plt.ylabel(r'$\tau_\mathrm{int}$') 502 length = int(len(self.e_n_tauint[e_name])) 503 if self.tau_exp[e_name] > 0: 504 base = self.e_n_tauint[e_name][self.e_windowsize[e_name]] 505 x_help = np.arange(2 * self.tau_exp[e_name]) 506 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 507 x_arr = np.arange(self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]) 508 plt.plot(x_arr, y_help, 'C' + str(e), linewidth=1, ls='--', marker=',') 509 plt.errorbar([self.e_windowsize[e_name] + 2 * self.tau_exp[e_name]], [self.e_tauint[e_name]], 510 yerr=[self.e_dtauint[e_name]], fmt='C' + str(e), linewidth=1, capsize=2, marker='o', mfc=plt.rcParams['axes.facecolor']) 511 xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5 512 label = e_name + r', $\tau_\mathrm{exp}$=' + str(np.around(self.tau_exp[e_name], decimals=2)) 513 else: 514 label = e_name + ', S=' + str(np.around(self.S[e_name], decimals=2)) 515 xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5) 516 517 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) 518 plt.axvline(x=self.e_windowsize[e_name], color='C' + str(e), alpha=0.5, marker=',', ls='--') 519 plt.legend() 520 plt.xlim(-0.5, xmax) 521 ylim = plt.ylim() 522 plt.ylim(bottom=0.0, top=max(1.0, ylim[1])) 523 plt.draw() 524 if save: 525 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):
527 def plot_rho(self, save=None): 528 """Plot normalized autocorrelation function time for each ensemble. 529 530 Parameters 531 ---------- 532 save : str 533 saves the figure to a file named 'save' if. 534 """ 535 if not hasattr(self, 'e_dvalue'): 536 raise Exception('Run the gamma method first.') 537 for e, e_name in enumerate(self.mc_names): 538 fig = plt.figure() 539 plt.xlabel('W') 540 plt.ylabel('rho') 541 length = int(len(self.e_drho[e_name])) 542 plt.errorbar(np.arange(length), self.e_rho[e_name][:length], yerr=self.e_drho[e_name][:], linewidth=1, capsize=2) 543 plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25, ls='--', marker=',') 544 if self.tau_exp[e_name] > 0: 545 plt.plot([self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]], 546 [self.e_rho[e_name][self.e_windowsize[e_name] + 1], 0], 'k-', lw=1) 547 xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5 548 plt.title('Rho ' + e_name + r', tau\_exp=' + str(np.around(self.tau_exp[e_name], decimals=2))) 549 else: 550 xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5) 551 plt.title('Rho ' + e_name + ', S=' + str(np.around(self.S[e_name], decimals=2))) 552 plt.plot([-0.5, xmax], [0, 0], 'k--', lw=1) 553 plt.xlim(-0.5, xmax) 554 plt.draw() 555 if save: 556 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):
558 def plot_rep_dist(self): 559 """Plot replica distribution for each ensemble with more than one replicum.""" 560 if not hasattr(self, 'e_dvalue'): 561 raise Exception('Run the gamma method first.') 562 for e, e_name in enumerate(self.mc_names): 563 if len(self.e_content[e_name]) == 1: 564 print('No replica distribution for a single replicum (', e_name, ')') 565 continue 566 r_length = [] 567 sub_r_mean = 0 568 for r, r_name in enumerate(self.e_content[e_name]): 569 r_length.append(len(self.deltas[r_name])) 570 sub_r_mean += self.shape[r_name] * self.r_values[r_name] 571 e_N = np.sum(r_length) 572 sub_r_mean /= e_N 573 arr = np.zeros(len(self.e_content[e_name])) 574 for r, r_name in enumerate(self.e_content[e_name]): 575 arr[r] = (self.r_values[r_name] - sub_r_mean) / (self.e_dvalue[e_name] * np.sqrt(e_N / self.shape[r_name] - 1)) 576 plt.hist(arr, rwidth=0.8, bins=len(self.e_content[e_name])) 577 plt.title('Replica distribution' + e_name + ' (mean=0, var=1)') 578 plt.draw()
Plot replica distribution for each ensemble with more than one replicum.
def
plot_history(self, expand=True):
580 def plot_history(self, expand=True): 581 """Plot derived Monte Carlo history for each ensemble 582 583 Parameters 584 ---------- 585 expand : bool 586 show expanded history for irregular Monte Carlo chains (default: True). 587 """ 588 for e, e_name in enumerate(self.mc_names): 589 plt.figure() 590 r_length = [] 591 tmp = [] 592 tmp_expanded = [] 593 for r, r_name in enumerate(self.e_content[e_name]): 594 tmp.append(self.deltas[r_name] + self.r_values[r_name]) 595 if expand: 596 tmp_expanded.append(_expand_deltas(self.deltas[r_name], list(self.idl[r_name]), self.shape[r_name], 1) + self.r_values[r_name]) 597 r_length.append(len(tmp_expanded[-1])) 598 else: 599 r_length.append(len(tmp[-1])) 600 e_N = np.sum(r_length) 601 x = np.arange(e_N) 602 y_test = np.concatenate(tmp, axis=0) 603 if expand: 604 y = np.concatenate(tmp_expanded, axis=0) 605 else: 606 y = y_test 607 plt.errorbar(x, y, fmt='.', markersize=3) 608 plt.xlim(-0.5, e_N - 0.5) 609 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})') 610 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):
612 def plot_piechart(self, save=None): 613 """Plot piechart which shows the fractional contribution of each 614 ensemble to the error and returns a dictionary containing the fractions. 615 616 Parameters 617 ---------- 618 save : str 619 saves the figure to a file named 'save' if. 620 """ 621 if not hasattr(self, 'e_dvalue'): 622 raise Exception('Run the gamma method first.') 623 if np.isclose(0.0, self._dvalue, atol=1e-15): 624 raise Exception('Error is 0.0') 625 labels = self.e_names 626 sizes = [self.e_dvalue[name] ** 2 for name in labels] / self._dvalue ** 2 627 fig1, ax1 = plt.subplots() 628 ax1.pie(sizes, labels=labels, startangle=90, normalize=True) 629 ax1.axis('equal') 630 plt.draw() 631 if save: 632 fig1.savefig(save) 633 634 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):
636 def dump(self, filename, datatype="json.gz", description="", **kwargs): 637 """Dump the Obs to a file 'name' of chosen format. 638 639 Parameters 640 ---------- 641 filename : str 642 name of the file to be saved. 643 datatype : str 644 Format of the exported file. Supported formats include 645 "json.gz" and "pickle" 646 description : str 647 Description for output file, only relevant for json.gz format. 648 path : str 649 specifies a custom path for the file (default '.') 650 """ 651 if 'path' in kwargs: 652 file_name = kwargs.get('path') + '/' + filename 653 else: 654 file_name = filename 655 656 if datatype == "json.gz": 657 from .input.json import dump_to_json 658 dump_to_json([self], file_name, description=description) 659 elif datatype == "pickle": 660 with open(file_name + '.p', 'wb') as fb: 661 pickle.dump(self, fb) 662 else: 663 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):
665 def export_jackknife(self): 666 """Export jackknife samples from the Obs 667 668 Returns 669 ------- 670 numpy.ndarray 671 Returns a numpy array of length N + 1 where N is the number of samples 672 for the given ensemble and replicum. The zeroth entry of the array contains 673 the mean value of the Obs, entries 1 to N contain the N jackknife samples 674 derived from the Obs. The current implementation only works for observables 675 defined on exactly one ensemble and replicum. The derived jackknife samples 676 should agree with samples from a full jackknife analysis up to O(1/N). 677 """ 678 679 if len(self.names) != 1: 680 raise Exception("'export_jackknife' is only implemented for Obs defined on one ensemble and replicum.") 681 682 name = self.names[0] 683 full_data = self.deltas[name] + self.r_values[name] 684 n = full_data.size 685 mean = self.value 686 tmp_jacks = np.zeros(n + 1) 687 tmp_jacks[0] = mean 688 tmp_jacks[1:] = (n * mean - full_data) / (n - 1) 689 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):
691 def export_bootstrap(self, samples=500, random_numbers=None, save_rng=None): 692 """Export bootstrap samples from the Obs 693 694 Parameters 695 ---------- 696 samples : int 697 Number of bootstrap samples to generate. 698 random_numbers : np.ndarray 699 Array of shape (samples, length) containing the random numbers to generate the bootstrap samples. 700 If not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name. 701 save_rng : str 702 Save the random numbers to a file if a path is specified. 703 704 Returns 705 ------- 706 numpy.ndarray 707 Returns a numpy array of length N + 1 where N is the number of samples 708 for the given ensemble and replicum. The zeroth entry of the array contains 709 the mean value of the Obs, entries 1 to N contain the N import_bootstrap samples 710 derived from the Obs. The current implementation only works for observables 711 defined on exactly one ensemble and replicum. The derived bootstrap samples 712 should agree with samples from a full bootstrap analysis up to O(1/N). 713 """ 714 if len(self.names) != 1: 715 raise Exception("'export_boostrap' is only implemented for Obs defined on one ensemble and replicum.") 716 717 name = self.names[0] 718 length = self.N 719 720 if random_numbers is None: 721 seed = int(hashlib.md5(name.encode()).hexdigest(), 16) & 0xFFFFFFFF 722 rng = np.random.default_rng(seed) 723 random_numbers = rng.integers(0, length, size=(samples, length)) 724 725 if save_rng is not None: 726 np.savetxt(save_rng, random_numbers, fmt='%i') 727 728 proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length 729 ret = np.zeros(samples + 1) 730 ret[0] = self.value 731 ret[1:] = proj @ (self.deltas[name] + self.r_values[name]) 732 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).
class
CObs:
920class CObs: 921 """Class for a complex valued observable.""" 922 __slots__ = ['_real', '_imag', 'tag'] 923 924 def __init__(self, real, imag=0.0): 925 self._real = real 926 self._imag = imag 927 self.tag = None 928 929 @property 930 def real(self): 931 return self._real 932 933 @property 934 def imag(self): 935 return self._imag 936 937 def gamma_method(self, **kwargs): 938 """Executes the gamma_method for the real and the imaginary part.""" 939 if isinstance(self.real, Obs): 940 self.real.gamma_method(**kwargs) 941 if isinstance(self.imag, Obs): 942 self.imag.gamma_method(**kwargs) 943 944 def is_zero(self): 945 """Checks whether both real and imaginary part are zero within machine precision.""" 946 return self.real == 0.0 and self.imag == 0.0 947 948 def conjugate(self): 949 return CObs(self.real, -self.imag) 950 951 def __add__(self, other): 952 if isinstance(other, np.ndarray): 953 return other + self 954 elif hasattr(other, 'real') and hasattr(other, 'imag'): 955 return CObs(self.real + other.real, 956 self.imag + other.imag) 957 else: 958 return CObs(self.real + other, self.imag) 959 960 def __radd__(self, y): 961 return self + y 962 963 def __sub__(self, other): 964 if isinstance(other, np.ndarray): 965 return -1 * (other - self) 966 elif hasattr(other, 'real') and hasattr(other, 'imag'): 967 return CObs(self.real - other.real, self.imag - other.imag) 968 else: 969 return CObs(self.real - other, self.imag) 970 971 def __rsub__(self, other): 972 return -1 * (self - other) 973 974 def __mul__(self, other): 975 if isinstance(other, np.ndarray): 976 return other * self 977 elif hasattr(other, 'real') and hasattr(other, 'imag'): 978 if all(isinstance(i, Obs) for i in [self.real, self.imag, other.real, other.imag]): 979 return CObs(derived_observable(lambda x, **kwargs: x[0] * x[1] - x[2] * x[3], 980 [self.real, other.real, self.imag, other.imag], 981 man_grad=[other.real.value, self.real.value, -other.imag.value, -self.imag.value]), 982 derived_observable(lambda x, **kwargs: x[2] * x[1] + x[0] * x[3], 983 [self.real, other.real, self.imag, other.imag], 984 man_grad=[other.imag.value, self.imag.value, other.real.value, self.real.value])) 985 elif getattr(other, 'imag', 0) != 0: 986 return CObs(self.real * other.real - self.imag * other.imag, 987 self.imag * other.real + self.real * other.imag) 988 else: 989 return CObs(self.real * other.real, self.imag * other.real) 990 else: 991 return CObs(self.real * other, self.imag * other) 992 993 def __rmul__(self, other): 994 return self * other 995 996 def __truediv__(self, other): 997 if isinstance(other, np.ndarray): 998 return 1 / (other / self) 999 elif hasattr(other, 'real') and hasattr(other, 'imag'): 1000 r = other.real ** 2 + other.imag ** 2 1001 return CObs((self.real * other.real + self.imag * other.imag) / r, (self.imag * other.real - self.real * other.imag) / r) 1002 else: 1003 return CObs(self.real / other, self.imag / other) 1004 1005 def __rtruediv__(self, other): 1006 r = self.real ** 2 + self.imag ** 2 1007 if hasattr(other, 'real') and hasattr(other, 'imag'): 1008 return CObs((self.real * other.real + self.imag * other.imag) / r, (self.real * other.imag - self.imag * other.real) / r) 1009 else: 1010 return CObs(self.real * other / r, -self.imag * other / r) 1011 1012 def __abs__(self): 1013 return np.sqrt(self.real**2 + self.imag**2) 1014 1015 def __pos__(self): 1016 return self 1017 1018 def __neg__(self): 1019 return -1 * self 1020 1021 def __eq__(self, other): 1022 return self.real == other.real and self.imag == other.imag 1023 1024 def __str__(self): 1025 return '(' + str(self.real) + int(self.imag >= 0.0) * '+' + str(self.imag) + 'j)' 1026 1027 def __repr__(self): 1028 return 'CObs[' + str(self) + ']' 1029 1030 def __format__(self, format_type): 1031 if format_type == "": 1032 significance = 2 1033 format_type = "2" 1034 else: 1035 significance = int(float(format_type.replace("+", "").replace("-", ""))) 1036 return f"({self.real:{format_type}}{self.imag:+{significance}}j)"
Class for a complex valued observable.
def
gamma_method(self, **kwargs):
937 def gamma_method(self, **kwargs): 938 """Executes the gamma_method for the real and the imaginary part.""" 939 if isinstance(self.real, Obs): 940 self.real.gamma_method(**kwargs) 941 if isinstance(self.imag, Obs): 942 self.imag.gamma_method(**kwargs)
Executes the gamma_method for the real and the imaginary part.
def
gamma_method(x, **kwargs):
1039def gamma_method(x, **kwargs): 1040 """Vectorized version of the gamma_method applicable to lists or arrays of Obs. 1041 1042 See docstring of pe.Obs.gamma_method for details. 1043 """ 1044 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):
1039def gamma_method(x, **kwargs): 1040 """Vectorized version of the gamma_method applicable to lists or arrays of Obs. 1041 1042 See docstring of pe.Obs.gamma_method for details. 1043 """ 1044 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
derived_observable(func, data, array_mode=False, **kwargs):
1174def derived_observable(func, data, array_mode=False, **kwargs): 1175 """Construct a derived Obs according to func(data, **kwargs) using automatic differentiation. 1176 1177 Parameters 1178 ---------- 1179 func : object 1180 arbitrary function of the form func(data, **kwargs). For the 1181 automatic differentiation to work, all numpy functions have to have 1182 the autograd wrapper (use 'import autograd.numpy as anp'). 1183 data : list 1184 list of Obs, e.g. [obs1, obs2, obs3]. 1185 num_grad : bool 1186 if True, numerical derivatives are used instead of autograd 1187 (default False). To control the numerical differentiation the 1188 kwargs of numdifftools.step_generators.MaxStepGenerator 1189 can be used. 1190 man_grad : list 1191 manually supply a list or an array which contains the jacobian 1192 of func. Use cautiously, supplying the wrong derivative will 1193 not be intercepted. 1194 1195 Notes 1196 ----- 1197 For simple mathematical operations it can be practical to use anonymous 1198 functions. For the ratio of two observables one can e.g. use 1199 1200 new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2]) 1201 """ 1202 1203 data = np.asarray(data) 1204 raveled_data = data.ravel() 1205 1206 # Workaround for matrix operations containing non Obs data 1207 if not all(isinstance(x, Obs) for x in raveled_data): 1208 for i in range(len(raveled_data)): 1209 if isinstance(raveled_data[i], (int, float)): 1210 raveled_data[i] = cov_Obs(raveled_data[i], 0.0, "###dummy_covobs###") 1211 1212 allcov = {} 1213 for o in raveled_data: 1214 for name in o.cov_names: 1215 if name in allcov: 1216 if not np.allclose(allcov[name], o.covobs[name].cov): 1217 raise Exception('Inconsistent covariance matrices for %s!' % (name)) 1218 else: 1219 allcov[name] = o.covobs[name].cov 1220 1221 n_obs = len(raveled_data) 1222 new_names = sorted(set([y for x in [o.names for o in raveled_data] for y in x])) 1223 new_cov_names = sorted(set([y for x in [o.cov_names for o in raveled_data] for y in x])) 1224 new_sample_names = sorted(set(new_names) - set(new_cov_names)) 1225 1226 reweighted = len(list(filter(lambda o: o.reweighted is True, raveled_data))) > 0 1227 1228 if data.ndim == 1: 1229 values = np.array([o.value for o in data]) 1230 else: 1231 values = np.vectorize(lambda x: x.value)(data) 1232 1233 new_values = func(values, **kwargs) 1234 1235 multi = int(isinstance(new_values, np.ndarray)) 1236 1237 new_r_values = {} 1238 new_idl_d = {} 1239 for name in new_sample_names: 1240 idl = [] 1241 tmp_values = np.zeros(n_obs) 1242 for i, item in enumerate(raveled_data): 1243 tmp_values[i] = item.r_values.get(name, item.value) 1244 tmp_idl = item.idl.get(name) 1245 if tmp_idl is not None: 1246 idl.append(tmp_idl) 1247 if multi > 0: 1248 tmp_values = np.array(tmp_values).reshape(data.shape) 1249 new_r_values[name] = func(tmp_values, **kwargs) 1250 new_idl_d[name] = _merge_idx(idl) 1251 1252 def _compute_scalefactor_missing_rep(obs): 1253 """ 1254 Computes the scale factor that is to be multiplied with the deltas 1255 in the case where Obs with different subsets of replica are merged. 1256 Returns a dictionary with the scale factor for each Monte Carlo name. 1257 1258 Parameters 1259 ---------- 1260 obs : Obs 1261 The observable corresponding to the deltas that are to be scaled 1262 """ 1263 scalef_d = {} 1264 for mc_name in obs.mc_names: 1265 mc_idl_d = [name for name in obs.idl if name.startswith(mc_name + '|')] 1266 new_mc_idl_d = [name for name in new_idl_d if name.startswith(mc_name + '|')] 1267 if len(mc_idl_d) > 0 and len(mc_idl_d) < len(new_mc_idl_d): 1268 scalef_d[mc_name] = sum([len(new_idl_d[name]) for name in new_mc_idl_d]) / sum([len(new_idl_d[name]) for name in mc_idl_d]) 1269 return scalef_d 1270 1271 if 'man_grad' in kwargs: 1272 deriv = np.asarray(kwargs.get('man_grad')) 1273 if new_values.shape + data.shape != deriv.shape: 1274 raise Exception('Manual derivative does not have correct shape.') 1275 elif kwargs.get('num_grad') is True: 1276 if multi > 0: 1277 raise Exception('Multi mode currently not supported for numerical derivative') 1278 options = { 1279 'base_step': 0.1, 1280 'step_ratio': 2.5} 1281 for key in options.keys(): 1282 kwarg = kwargs.get(key) 1283 if kwarg is not None: 1284 options[key] = kwarg 1285 tmp_df = nd.Gradient(func, order=4, **{k: v for k, v in options.items() if v is not None})(values, **kwargs) 1286 if tmp_df.size == 1: 1287 deriv = np.array([tmp_df.real]) 1288 else: 1289 deriv = tmp_df.real 1290 else: 1291 deriv = jacobian(func)(values, **kwargs) 1292 1293 final_result = np.zeros(new_values.shape, dtype=object) 1294 1295 if array_mode is True: 1296 1297 class _Zero_grad(): 1298 def __init__(self, N): 1299 self.grad = np.zeros((N, 1)) 1300 1301 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])) 1302 d_extracted = {} 1303 g_extracted = {} 1304 for name in new_sample_names: 1305 d_extracted[name] = [] 1306 ens_length = len(new_idl_d[name]) 1307 for i_dat, dat in enumerate(data): 1308 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], _compute_scalefactor_missing_rep(o).get(name.split('|')[0], 1)) for o in dat.reshape(np.prod(dat.shape))]).reshape(dat.shape + (ens_length, ))) 1309 for name in new_cov_names: 1310 g_extracted[name] = [] 1311 zero_grad = _Zero_grad(new_covobs_lengths[name]) 1312 for i_dat, dat in enumerate(data): 1313 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))) 1314 1315 for i_val, new_val in np.ndenumerate(new_values): 1316 new_deltas = {} 1317 new_grad = {} 1318 if array_mode is True: 1319 for name in new_sample_names: 1320 ens_length = d_extracted[name][0].shape[-1] 1321 new_deltas[name] = np.zeros(ens_length) 1322 for i_dat, dat in enumerate(d_extracted[name]): 1323 new_deltas[name] += np.tensordot(deriv[i_val + (i_dat, )], dat) 1324 for name in new_cov_names: 1325 new_grad[name] = 0 1326 for i_dat, dat in enumerate(g_extracted[name]): 1327 new_grad[name] += np.tensordot(deriv[i_val + (i_dat, )], dat) 1328 else: 1329 for j_obs, obs in np.ndenumerate(data): 1330 scalef_d = _compute_scalefactor_missing_rep(obs) 1331 for name in obs.names: 1332 if name in obs.cov_names: 1333 new_grad[name] = new_grad.get(name, 0) + deriv[i_val + j_obs] * obs.covobs[name].grad 1334 else: 1335 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], scalef_d.get(name.split('|')[0], 1)) 1336 1337 new_covobs = {name: Covobs(0, allcov[name], name, grad=new_grad[name]) for name in new_grad} 1338 1339 if not set(new_covobs.keys()).isdisjoint(new_deltas.keys()): 1340 raise Exception('The same name has been used for deltas and covobs!') 1341 new_samples = [] 1342 new_means = [] 1343 new_idl = [] 1344 new_names_obs = [] 1345 for name in new_names: 1346 if name not in new_covobs: 1347 new_samples.append(new_deltas[name]) 1348 new_idl.append(new_idl_d[name]) 1349 new_means.append(new_r_values[name][i_val]) 1350 new_names_obs.append(name) 1351 final_result[i_val] = Obs(new_samples, new_names_obs, means=new_means, idl=new_idl) 1352 for name in new_covobs: 1353 final_result[i_val].names.append(name) 1354 final_result[i_val]._covobs = new_covobs 1355 final_result[i_val]._value = new_val 1356 final_result[i_val].reweighted = reweighted 1357 1358 if multi == 0: 1359 final_result = final_result.item() 1360 1361 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):
1393def reweight(weight, obs, **kwargs): 1394 """Reweight a list of observables. 1395 1396 Parameters 1397 ---------- 1398 weight : Obs 1399 Reweighting factor. An Observable that has to be defined on a superset of the 1400 configurations in obs[i].idl for all i. 1401 obs : list 1402 list of Obs, e.g. [obs1, obs2, obs3]. 1403 all_configs : bool 1404 if True, the reweighted observables are normalized by the average of 1405 the reweighting factor on all configurations in weight.idl and not 1406 on the configurations in obs[i].idl. Default False. 1407 """ 1408 result = [] 1409 for i in range(len(obs)): 1410 if len(obs[i].cov_names): 1411 raise Exception('Error: Not possible to reweight an Obs that contains covobs!') 1412 if not set(obs[i].names).issubset(weight.names): 1413 raise Exception('Error: Ensembles do not fit') 1414 for name in obs[i].names: 1415 if not set(obs[i].idl[name]).issubset(weight.idl[name]): 1416 raise Exception('obs[%d] has to be defined on a subset of the configs in weight.idl[%s]!' % (i, name)) 1417 new_samples = [] 1418 w_deltas = {} 1419 for name in sorted(obs[i].names): 1420 w_deltas[name] = _reduce_deltas(weight.deltas[name], weight.idl[name], obs[i].idl[name]) 1421 new_samples.append((w_deltas[name] + weight.r_values[name]) * (obs[i].deltas[name] + obs[i].r_values[name])) 1422 tmp_obs = Obs(new_samples, sorted(obs[i].names), idl=[obs[i].idl[name] for name in sorted(obs[i].names)]) 1423 1424 if kwargs.get('all_configs'): 1425 new_weight = weight 1426 else: 1427 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)]) 1428 1429 result.append(tmp_obs / new_weight) 1430 result[-1].reweighted = True 1431 1432 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):
1435def correlate(obs_a, obs_b): 1436 """Correlate two observables. 1437 1438 Parameters 1439 ---------- 1440 obs_a : Obs 1441 First observable 1442 obs_b : Obs 1443 Second observable 1444 1445 Notes 1446 ----- 1447 Keep in mind to only correlate primary observables which have not been reweighted 1448 yet. The reweighting has to be applied after correlating the observables. 1449 Currently only works if ensembles are identical (this is not strictly necessary). 1450 """ 1451 1452 if sorted(obs_a.names) != sorted(obs_b.names): 1453 raise Exception(f"Ensembles do not fit {set(sorted(obs_a.names)) ^ set(sorted(obs_b.names))}") 1454 if len(obs_a.cov_names) or len(obs_b.cov_names): 1455 raise Exception('Error: Not possible to correlate Obs that contain covobs!') 1456 for name in obs_a.names: 1457 if obs_a.shape[name] != obs_b.shape[name]: 1458 raise Exception('Shapes of ensemble', name, 'do not fit') 1459 if obs_a.idl[name] != obs_b.idl[name]: 1460 raise Exception('idl of ensemble', name, 'do not fit') 1461 1462 if obs_a.reweighted is True: 1463 warnings.warn("The first observable is already reweighted.", RuntimeWarning) 1464 if obs_b.reweighted is True: 1465 warnings.warn("The second observable is already reweighted.", RuntimeWarning) 1466 1467 new_samples = [] 1468 new_idl = [] 1469 for name in sorted(obs_a.names): 1470 new_samples.append((obs_a.deltas[name] + obs_a.r_values[name]) * (obs_b.deltas[name] + obs_b.r_values[name])) 1471 new_idl.append(obs_a.idl[name]) 1472 1473 o = Obs(new_samples, sorted(obs_a.names), idl=new_idl) 1474 o.reweighted = obs_a.reweighted or obs_b.reweighted 1475 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):
1478def covariance(obs, visualize=False, correlation=False, smooth=None, **kwargs): 1479 r'''Calculates the error covariance matrix of a set of observables. 1480 1481 WARNING: This function should be used with care, especially for observables with support on multiple 1482 ensembles with differing autocorrelations. See the notes below for details. 1483 1484 The gamma method has to be applied first to all observables. 1485 1486 Parameters 1487 ---------- 1488 obs : list or numpy.ndarray 1489 List or one dimensional array of Obs 1490 visualize : bool 1491 If True plots the corresponding normalized correlation matrix (default False). 1492 correlation : bool 1493 If True the correlation matrix instead of the error covariance matrix is returned (default False). 1494 smooth : None or int 1495 If smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue 1496 smoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the 1497 largest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely 1498 small ones. 1499 1500 Notes 1501 ----- 1502 The error covariance is defined such that it agrees with the squared standard error for two identical observables 1503 $$\operatorname{cov}(a,a)=\sum_{s=1}^N\delta_a^s\delta_a^s/N^2=\Gamma_{aa}(0)/N=\operatorname{var}(a)/N=\sigma_a^2$$ 1504 in the absence of autocorrelation. 1505 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 1506 $$\sum_{i,j}v_i\Gamma_{ij}(0)v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i,j}v_i\delta_i^s\delta_j^s v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i}|v_i\delta_i^s|^2\geq 0\,,$$ for every $v\in\mathbb{R}^M$, while such an identity does not hold for larger windows/lags. 1507 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. 1508 $$\tau_{\mathrm{int}, ij}=\sqrt{\tau_{\mathrm{int}, i}\times \tau_{\mathrm{int}, j}}$$ 1509 This construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors). 1510 ''' 1511 1512 length = len(obs) 1513 1514 max_samples = np.max([o.N for o in obs]) 1515 if max_samples <= length and not [item for sublist in [o.cov_names for o in obs] for item in sublist]: 1516 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) 1517 1518 cov = np.zeros((length, length)) 1519 for i in range(length): 1520 for j in range(i, length): 1521 cov[i, j] = _covariance_element(obs[i], obs[j]) 1522 cov = cov + cov.T - np.diag(np.diag(cov)) 1523 1524 corr = np.diag(1 / np.sqrt(np.diag(cov))) @ cov @ np.diag(1 / np.sqrt(np.diag(cov))) 1525 1526 if isinstance(smooth, int): 1527 corr = _smooth_eigenvalues(corr, smooth) 1528 1529 if visualize: 1530 plt.matshow(corr, vmin=-1, vmax=1) 1531 plt.set_cmap('RdBu') 1532 plt.colorbar() 1533 plt.draw() 1534 1535 if correlation is True: 1536 return corr 1537 1538 errors = [o.dvalue for o in obs] 1539 cov = np.diag(errors) @ corr @ np.diag(errors) 1540 1541 eigenvalues = np.linalg.eigh(cov)[0] 1542 if not np.all(eigenvalues >= 0): 1543 warnings.warn("Covariance matrix is not positive semi-definite (Eigenvalues: " + str(eigenvalues) + ")", RuntimeWarning) 1544 1545 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):
1625def import_jackknife(jacks, name, idl=None): 1626 """Imports jackknife samples and returns an Obs 1627 1628 Parameters 1629 ---------- 1630 jacks : numpy.ndarray 1631 numpy array containing the mean value as zeroth entry and 1632 the N jackknife samples as first to Nth entry. 1633 name : str 1634 name of the ensemble the samples are defined on. 1635 """ 1636 length = len(jacks) - 1 1637 prj = (np.ones((length, length)) - (length - 1) * np.identity(length)) 1638 samples = jacks[1:] @ prj 1639 mean = np.mean(samples) 1640 new_obs = Obs([samples - mean], [name], idl=idl, means=[mean]) 1641 new_obs._value = jacks[0] 1642 return new_obs
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):
1645def import_bootstrap(boots, name, random_numbers): 1646 """Imports bootstrap samples and returns an Obs 1647 1648 Parameters 1649 ---------- 1650 boots : numpy.ndarray 1651 numpy array containing the mean value as zeroth entry and 1652 the N bootstrap samples as first to Nth entry. 1653 name : str 1654 name of the ensemble the samples are defined on. 1655 random_numbers : np.ndarray 1656 Array of shape (samples, length) containing the random numbers to generate the bootstrap samples, 1657 where samples is the number of bootstrap samples and length is the length of the original Monte Carlo 1658 chain to be reconstructed. 1659 """ 1660 samples, length = random_numbers.shape 1661 if samples != len(boots) - 1: 1662 raise ValueError("Random numbers do not have the correct shape.") 1663 1664 if samples < length: 1665 raise ValueError("Obs can't be reconstructed if there are fewer bootstrap samples than Monte Carlo data points.") 1666 1667 proj = np.vstack([np.bincount(o, minlength=length) for o in random_numbers]) / length 1668 1669 samples = scipy.linalg.lstsq(proj, boots[1:])[0] 1670 ret = Obs([samples], [name]) 1671 ret._value = boots[0] 1672 return ret
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):
1675def merge_obs(list_of_obs): 1676 """Combine all observables in list_of_obs into one new observable 1677 1678 Parameters 1679 ---------- 1680 list_of_obs : list 1681 list of the Obs object to be combined 1682 1683 Notes 1684 ----- 1685 It is not possible to combine obs which are based on the same replicum 1686 """ 1687 replist = [item for obs in list_of_obs for item in obs.names] 1688 if (len(replist) == len(set(replist))) is False: 1689 raise Exception('list_of_obs contains duplicate replica: %s' % (str(replist))) 1690 if any([len(o.cov_names) for o in list_of_obs]): 1691 raise Exception('Not possible to merge data that contains covobs!') 1692 new_dict = {} 1693 idl_dict = {} 1694 for o in list_of_obs: 1695 new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0) 1696 for key in set(o.deltas) | set(o.r_values)}) 1697 idl_dict.update({key: o.idl.get(key, 0) for key in set(o.deltas)}) 1698 1699 names = sorted(new_dict.keys()) 1700 o = Obs([new_dict[name] for name in names], names, idl=[idl_dict[name] for name in names]) 1701 o.reweighted = np.max([oi.reweighted for oi in list_of_obs]) 1702 return o
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):
1705def cov_Obs(means, cov, name, grad=None): 1706 """Create an Obs based on mean(s) and a covariance matrix 1707 1708 Parameters 1709 ---------- 1710 mean : list of floats or float 1711 N mean value(s) of the new Obs 1712 cov : list or array 1713 2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance 1714 name : str 1715 identifier for the covariance matrix 1716 grad : list or array 1717 Gradient of the Covobs wrt. the means belonging to cov. 1718 """ 1719 1720 def covobs_to_obs(co): 1721 """Make an Obs out of a Covobs 1722 1723 Parameters 1724 ---------- 1725 co : Covobs 1726 Covobs to be embedded into the Obs 1727 """ 1728 o = Obs([], [], means=[]) 1729 o._value = co.value 1730 o.names.append(co.name) 1731 o._covobs[co.name] = co 1732 o._dvalue = np.sqrt(co.errsq()) 1733 return o 1734 1735 ol = [] 1736 if isinstance(means, (float, int)): 1737 means = [means] 1738 1739 for i in range(len(means)): 1740 ol.append(covobs_to_obs(Covobs(means[i], cov, name, pos=i, grad=grad))) 1741 if ol[0].covobs[name].N != len(means): 1742 raise Exception('You have to provide %d mean values!' % (ol[0].N)) 1743 if len(ol) == 1: 1744 return ol[0] 1745 return ol
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.