pyerrors.fits
1import gc 2from collections.abc import Sequence 3import warnings 4import numpy as np 5import autograd.numpy as anp 6import scipy.optimize 7import scipy.stats 8import matplotlib.pyplot as plt 9from matplotlib import gridspec 10from scipy.odr import ODR, Model, RealData 11from scipy.stats import chi2 12import iminuit 13from autograd import jacobian 14from autograd import elementwise_grad as egrad 15from .obs import Obs, derived_observable, covariance, cov_Obs 16 17 18class Fit_result(Sequence): 19 """Represents fit results. 20 21 Attributes 22 ---------- 23 fit_parameters : list 24 results for the individual fit parameters, 25 also accessible via indices. 26 """ 27 28 def __init__(self): 29 self.fit_parameters = None 30 31 def __getitem__(self, idx): 32 return self.fit_parameters[idx] 33 34 def __len__(self): 35 return len(self.fit_parameters) 36 37 def gamma_method(self): 38 """Apply the gamma method to all fit parameters""" 39 [o.gamma_method() for o in self.fit_parameters] 40 41 def __str__(self): 42 my_str = 'Goodness of fit:\n' 43 if hasattr(self, 'chisquare_by_dof'): 44 my_str += '\u03C7\u00b2/d.o.f. = ' + f'{self.chisquare_by_dof:2.6f}' + '\n' 45 elif hasattr(self, 'residual_variance'): 46 my_str += 'residual variance = ' + f'{self.residual_variance:2.6f}' + '\n' 47 if hasattr(self, 'chisquare_by_expected_chisquare'): 48 my_str += '\u03C7\u00b2/\u03C7\u00b2exp = ' + f'{self.chisquare_by_expected_chisquare:2.6f}' + '\n' 49 if hasattr(self, 'p_value'): 50 my_str += 'p-value = ' + f'{self.p_value:2.4f}' + '\n' 51 my_str += 'Fit parameters:\n' 52 for i_par, par in enumerate(self.fit_parameters): 53 my_str += str(i_par) + '\t' + ' ' * int(par >= 0) + str(par).rjust(int(par < 0.0)) + '\n' 54 return my_str 55 56 def __repr__(self): 57 m = max(map(len, list(self.__dict__.keys()))) + 1 58 return '\n'.join([key.rjust(m) + ': ' + repr(value) for key, value in sorted(self.__dict__.items())]) 59 60 61def least_squares(x, y, func, priors=None, silent=False, **kwargs): 62 r'''Performs a non-linear fit to y = func(x). 63 64 Parameters 65 ---------- 66 x : list 67 list of floats. 68 y : list 69 list of Obs. 70 func : object 71 fit function, has to be of the form 72 73 ```python 74 import autograd.numpy as anp 75 76 def func(a, x): 77 return a[0] + a[1] * x + a[2] * anp.sinh(x) 78 ``` 79 80 For multiple x values func can be of the form 81 82 ```python 83 def func(a, x): 84 (x1, x2) = x 85 return a[0] * x1 ** 2 + a[1] * x2 86 ``` 87 88 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation 89 will not work. 90 priors : list, optional 91 priors has to be a list with an entry for every parameter in the fit. The entries can either be 92 Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like 93 0.548(23), 500(40) or 0.5(0.4) 94 silent : bool, optional 95 If true all output to the console is omitted (default False). 96 initial_guess : list 97 can provide an initial guess for the input parameters. Relevant for 98 non-linear fits with many parameters. 99 method : str, optional 100 can be used to choose an alternative method for the minimization of chisquare. 101 The possible methods are the ones which can be used for scipy.optimize.minimize and 102 migrad of iminuit. If no method is specified, Levenberg-Marquard is used. 103 Reliable alternatives are migrad, Powell and Nelder-Mead. 104 correlated_fit : bool 105 If True, use the full inverse covariance matrix in the definition of the chisquare cost function. 106 For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`. 107 In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix). 108 This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning). 109 At the moment this option only works for `prior==None` and when no `method` is given. 110 expected_chisquare : bool 111 If True estimates the expected chisquare which is 112 corrected by effects caused by correlated input data (default False). 113 resplot : bool 114 If True, a plot which displays fit, data and residuals is generated (default False). 115 qqplot : bool 116 If True, a quantile-quantile plot of the fit result is generated (default False). 117 ''' 118 if priors is not None: 119 return _prior_fit(x, y, func, priors, silent=silent, **kwargs) 120 else: 121 return _standard_fit(x, y, func, silent=silent, **kwargs) 122 123 124def total_least_squares(x, y, func, silent=False, **kwargs): 125 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. 126 127 Parameters 128 ---------- 129 x : list 130 list of Obs, or a tuple of lists of Obs 131 y : list 132 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. 133 func : object 134 func has to be of the form 135 136 ```python 137 import autograd.numpy as anp 138 139 def func(a, x): 140 return a[0] + a[1] * x + a[2] * anp.sinh(x) 141 ``` 142 143 For multiple x values func can be of the form 144 145 ```python 146 def func(a, x): 147 (x1, x2) = x 148 return a[0] * x1 ** 2 + a[1] * x2 149 ``` 150 151 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation 152 will not work. 153 silent : bool, optional 154 If true all output to the console is omitted (default False). 155 initial_guess : list 156 can provide an initial guess for the input parameters. Relevant for non-linear 157 fits with many parameters. 158 expected_chisquare : bool 159 If true prints the expected chisquare which is 160 corrected by effects caused by correlated input data. 161 This can take a while as the full correlation matrix 162 has to be calculated (default False). 163 164 Notes 165 ----- 166 Based on the orthogonal distance regression module of scipy 167 ''' 168 169 output = Fit_result() 170 171 output.fit_function = func 172 173 x = np.array(x) 174 175 x_shape = x.shape 176 177 if not callable(func): 178 raise TypeError('func has to be a function.') 179 180 for i in range(25): 181 try: 182 func(np.arange(i), x.T[0]) 183 except Exception: 184 pass 185 else: 186 break 187 188 n_parms = i 189 if not silent: 190 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) 191 192 x_f = np.vectorize(lambda o: o.value)(x) 193 dx_f = np.vectorize(lambda o: o.dvalue)(x) 194 y_f = np.array([o.value for o in y]) 195 dy_f = np.array([o.dvalue for o in y]) 196 197 if np.any(np.asarray(dx_f) <= 0.0): 198 raise Exception('No x errors available, run the gamma method first.') 199 200 if np.any(np.asarray(dy_f) <= 0.0): 201 raise Exception('No y errors available, run the gamma method first.') 202 203 if 'initial_guess' in kwargs: 204 x0 = kwargs.get('initial_guess') 205 if len(x0) != n_parms: 206 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) 207 else: 208 x0 = [1] * n_parms 209 210 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) 211 model = Model(func) 212 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) 213 odr.set_job(fit_type=0, deriv=1) 214 out = odr.run() 215 216 output.residual_variance = out.res_var 217 218 output.method = 'ODR' 219 220 output.message = out.stopreason 221 222 output.xplus = out.xplus 223 224 if not silent: 225 print('Method: ODR') 226 print(*out.stopreason) 227 print('Residual variance:', output.residual_variance) 228 229 if out.info > 3: 230 raise Exception('The minimization procedure did not converge.') 231 232 m = x_f.size 233 234 def odr_chisquare(p): 235 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) 236 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) 237 return chisq 238 239 if kwargs.get('expected_chisquare') is True: 240 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) 241 242 if kwargs.get('covariance') is not None: 243 cov = kwargs.get('covariance') 244 else: 245 cov = covariance(np.concatenate((y, x.ravel()))) 246 247 number_of_x_parameters = int(m / x_f.shape[-1]) 248 249 old_jac = jacobian(func)(out.beta, out.xplus) 250 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) 251 fused_row2 = np.concatenate((jacobian(lambda x, y: func(y, x))(out.xplus, out.beta).reshape(x_f.shape[-1], x_f.shape[-1] * number_of_x_parameters), np.identity(number_of_x_parameters * old_jac.shape[0]))) 252 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) 253 254 A = W @ new_jac 255 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T 256 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) 257 if expected_chisquare <= 0.0: 258 warnings.warn("Negative expected_chisquare.", RuntimeWarning) 259 expected_chisquare = np.abs(expected_chisquare) 260 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare 261 if not silent: 262 print('chisquare/expected_chisquare:', 263 output.chisquare_by_expected_chisquare) 264 265 fitp = out.beta 266 try: 267 hess = jacobian(jacobian(odr_chisquare))(np.concatenate((fitp, out.xplus.ravel()))) 268 except TypeError: 269 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None 270 271 def odr_chisquare_compact_x(d): 272 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) 273 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((d[n_parms + m:].reshape(x_shape) - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) 274 return chisq 275 276 jac_jac_x = jacobian(jacobian(odr_chisquare_compact_x))(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) 277 278 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv 279 try: 280 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) 281 except np.linalg.LinAlgError: 282 raise Exception("Cannot invert hessian matrix.") 283 284 def odr_chisquare_compact_y(d): 285 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) 286 chisq = anp.sum(((d[n_parms + m:] - model) / dy_f) ** 2) + anp.sum(((x_f - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) 287 return chisq 288 289 jac_jac_y = jacobian(jacobian(odr_chisquare_compact_y))(np.concatenate((fitp, out.xplus.ravel(), y_f))) 290 291 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv 292 try: 293 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) 294 except np.linalg.LinAlgError: 295 raise Exception("Cannot invert hessian matrix.") 296 297 result = [] 298 for i in range(n_parms): 299 result.append(derived_observable(lambda my_var, **kwargs: (my_var[0] + np.finfo(np.float64).eps) / (x.ravel()[0].value + np.finfo(np.float64).eps) * out.beta[i], list(x.ravel()) + list(y), man_grad=list(deriv_x[i]) + list(deriv_y[i]))) 300 301 output.fit_parameters = result 302 303 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) 304 output.dof = x.shape[-1] - n_parms 305 output.p_value = 1 - chi2.cdf(output.odr_chisquare, output.dof) 306 307 return output 308 309 310def _prior_fit(x, y, func, priors, silent=False, **kwargs): 311 output = Fit_result() 312 313 output.fit_function = func 314 315 x = np.asarray(x) 316 317 if not callable(func): 318 raise TypeError('func has to be a function.') 319 320 for i in range(100): 321 try: 322 func(np.arange(i), 0) 323 except Exception: 324 pass 325 else: 326 break 327 328 n_parms = i 329 330 if n_parms != len(priors): 331 raise Exception('Priors does not have the correct length.') 332 333 def extract_val_and_dval(string): 334 split_string = string.split('(') 335 if '.' in split_string[0] and '.' not in split_string[1][:-1]: 336 factor = 10 ** -len(split_string[0].partition('.')[2]) 337 else: 338 factor = 1 339 return float(split_string[0]), float(split_string[1][:-1]) * factor 340 341 loc_priors = [] 342 for i_n, i_prior in enumerate(priors): 343 if isinstance(i_prior, Obs): 344 loc_priors.append(i_prior) 345 else: 346 loc_val, loc_dval = extract_val_and_dval(i_prior) 347 loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}")) 348 349 output.priors = loc_priors 350 351 if not silent: 352 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) 353 354 y_f = [o.value for o in y] 355 dy_f = [o.dvalue for o in y] 356 357 if np.any(np.asarray(dy_f) <= 0.0): 358 raise Exception('No y errors available, run the gamma method first.') 359 360 p_f = [o.value for o in loc_priors] 361 dp_f = [o.dvalue for o in loc_priors] 362 363 if np.any(np.asarray(dp_f) <= 0.0): 364 raise Exception('No prior errors available, run the gamma method first.') 365 366 if 'initial_guess' in kwargs: 367 x0 = kwargs.get('initial_guess') 368 if len(x0) != n_parms: 369 raise Exception('Initial guess does not have the correct length.') 370 else: 371 x0 = p_f 372 373 def chisqfunc(p): 374 model = func(p, x) 375 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((p_f - p) / dp_f) ** 2) 376 return chisq 377 378 if not silent: 379 print('Method: migrad') 380 381 m = iminuit.Minuit(chisqfunc, x0) 382 m.errordef = 1 383 m.print_level = 0 384 if 'tol' in kwargs: 385 m.tol = kwargs.get('tol') 386 else: 387 m.tol = 1e-4 388 m.migrad() 389 params = np.asarray(m.values) 390 391 output.chisquare_by_dof = m.fval / len(x) 392 393 output.method = 'migrad' 394 395 if not silent: 396 print('chisquare/d.o.f.:', output.chisquare_by_dof) 397 398 if not m.fmin.is_valid: 399 raise Exception('The minimization procedure did not converge.') 400 401 hess_inv = np.linalg.pinv(jacobian(jacobian(chisqfunc))(params)) 402 403 def chisqfunc_compact(d): 404 model = func(d[:n_parms], x) 405 chisq = anp.sum(((d[n_parms: n_parms + len(x)] - model) / dy_f) ** 2) + anp.sum(((d[n_parms + len(x):] - d[:n_parms]) / dp_f) ** 2) 406 return chisq 407 408 jac_jac = jacobian(jacobian(chisqfunc_compact))(np.concatenate((params, y_f, p_f))) 409 410 deriv = -hess_inv @ jac_jac[:n_parms, n_parms:] 411 412 result = [] 413 for i in range(n_parms): 414 result.append(derived_observable(lambda x, **kwargs: (x[0] + np.finfo(np.float64).eps) / (y[0].value + np.finfo(np.float64).eps) * params[i], list(y) + list(loc_priors), man_grad=list(deriv[i]))) 415 416 output.fit_parameters = result 417 output.chisquare = chisqfunc(np.asarray(params)) 418 419 if kwargs.get('resplot') is True: 420 residual_plot(x, y, func, result) 421 422 if kwargs.get('qqplot') is True: 423 qqplot(x, y, func, result) 424 425 return output 426 427 428def _standard_fit(x, y, func, silent=False, **kwargs): 429 430 output = Fit_result() 431 432 output.fit_function = func 433 434 x = np.asarray(x) 435 436 if x.shape[-1] != len(y): 437 raise Exception('x and y input have to have the same length') 438 439 if len(x.shape) > 2: 440 raise Exception('Unknown format for x values') 441 442 if not callable(func): 443 raise TypeError('func has to be a function.') 444 445 for i in range(25): 446 try: 447 func(np.arange(i), x.T[0]) 448 except Exception: 449 pass 450 else: 451 break 452 453 n_parms = i 454 455 if not silent: 456 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) 457 458 y_f = [o.value for o in y] 459 dy_f = [o.dvalue for o in y] 460 461 if np.any(np.asarray(dy_f) <= 0.0): 462 raise Exception('No y errors available, run the gamma method first.') 463 464 if 'initial_guess' in kwargs: 465 x0 = kwargs.get('initial_guess') 466 if len(x0) != n_parms: 467 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) 468 else: 469 x0 = [0.1] * n_parms 470 471 if kwargs.get('correlated_fit') is True: 472 corr = covariance(y, correlation=True, **kwargs) 473 covdiag = np.diag(1 / np.asarray(dy_f)) 474 condn = np.linalg.cond(corr) 475 if condn > 0.1 / np.finfo(float).eps: 476 raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})") 477 if condn > 1 / np.sqrt(np.finfo(float).eps): 478 warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning) 479 chol = np.linalg.cholesky(corr) 480 chol_inv = np.linalg.inv(chol) 481 chol_inv = np.dot(chol_inv, covdiag) 482 483 def chisqfunc_corr(p): 484 model = func(p, x) 485 chisq = anp.sum(anp.dot(chol_inv, (y_f - model)) ** 2) 486 return chisq 487 488 def chisqfunc(p): 489 model = func(p, x) 490 chisq = anp.sum(((y_f - model) / dy_f) ** 2) 491 return chisq 492 493 output.method = kwargs.get('method', 'Levenberg-Marquardt') 494 if not silent: 495 print('Method:', output.method) 496 497 if output.method != 'Levenberg-Marquardt': 498 if output.method == 'migrad': 499 fit_result = iminuit.minimize(chisqfunc, x0, tol=1e-4) # Stopping criterion 0.002 * tol * errordef 500 if kwargs.get('correlated_fit') is True: 501 fit_result = iminuit.minimize(chisqfunc_corr, fit_result.x, tol=1e-4) # Stopping criterion 0.002 * tol * errordef 502 output.iterations = fit_result.nfev 503 else: 504 fit_result = scipy.optimize.minimize(chisqfunc, x0, method=kwargs.get('method'), tol=1e-12) 505 if kwargs.get('correlated_fit') is True: 506 fit_result = scipy.optimize.minimize(chisqfunc_corr, fit_result.x, method=kwargs.get('method'), tol=1e-12) 507 output.iterations = fit_result.nit 508 509 chisquare = fit_result.fun 510 511 else: 512 if kwargs.get('correlated_fit') is True: 513 def chisqfunc_residuals_corr(p): 514 model = func(p, x) 515 chisq = anp.dot(chol_inv, (y_f - model)) 516 return chisq 517 518 def chisqfunc_residuals(p): 519 model = func(p, x) 520 chisq = ((y_f - model) / dy_f) 521 return chisq 522 523 fit_result = scipy.optimize.least_squares(chisqfunc_residuals, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) 524 if kwargs.get('correlated_fit') is True: 525 fit_result = scipy.optimize.least_squares(chisqfunc_residuals_corr, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15) 526 527 chisquare = np.sum(fit_result.fun ** 2) 528 529 output.iterations = fit_result.nfev 530 531 if not fit_result.success: 532 raise Exception('The minimization procedure did not converge.') 533 534 if x.shape[-1] - n_parms > 0: 535 output.chisquare_by_dof = chisquare / (x.shape[-1] - n_parms) 536 else: 537 output.chisquare_by_dof = float('nan') 538 539 output.message = fit_result.message 540 if not silent: 541 print(fit_result.message) 542 print('chisquare/d.o.f.:', output.chisquare_by_dof) 543 544 if kwargs.get('expected_chisquare') is True: 545 if kwargs.get('correlated_fit') is not True: 546 W = np.diag(1 / np.asarray(dy_f)) 547 cov = covariance(y) 548 A = W @ jacobian(func)(fit_result.x, x) 549 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T 550 expected_chisquare = np.trace((np.identity(x.shape[-1]) - P_phi) @ W @ cov @ W) 551 output.chisquare_by_expected_chisquare = chisquare / expected_chisquare 552 if not silent: 553 print('chisquare/expected_chisquare:', 554 output.chisquare_by_expected_chisquare) 555 556 fitp = fit_result.x 557 try: 558 hess = jacobian(jacobian(chisqfunc))(fitp) 559 except TypeError: 560 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None 561 562 if kwargs.get('correlated_fit') is True: 563 def chisqfunc_compact(d): 564 model = func(d[:n_parms], x) 565 chisq = anp.sum(anp.dot(chol_inv, (d[n_parms:] - model)) ** 2) 566 return chisq 567 568 else: 569 def chisqfunc_compact(d): 570 model = func(d[:n_parms], x) 571 chisq = anp.sum(((d[n_parms:] - model) / dy_f) ** 2) 572 return chisq 573 574 jac_jac = jacobian(jacobian(chisqfunc_compact))(np.concatenate((fitp, y_f))) 575 576 # Compute hess^{-1} @ jac_jac[:n_parms, n_parms:] using LAPACK dgesv 577 try: 578 deriv = -scipy.linalg.solve(hess, jac_jac[:n_parms, n_parms:]) 579 except np.linalg.LinAlgError: 580 raise Exception("Cannot invert hessian matrix.") 581 582 result = [] 583 for i in range(n_parms): 584 result.append(derived_observable(lambda x, **kwargs: (x[0] + np.finfo(np.float64).eps) / (y[0].value + np.finfo(np.float64).eps) * fit_result.x[i], list(y), man_grad=list(deriv[i]))) 585 586 output.fit_parameters = result 587 588 output.chisquare = chisqfunc(fit_result.x) 589 output.dof = x.shape[-1] - n_parms 590 output.p_value = 1 - chi2.cdf(output.chisquare, output.dof) 591 592 if kwargs.get('resplot') is True: 593 residual_plot(x, y, func, result) 594 595 if kwargs.get('qqplot') is True: 596 qqplot(x, y, func, result) 597 598 return output 599 600 601def fit_lin(x, y, **kwargs): 602 """Performs a linear fit to y = n + m * x and returns two Obs n, m. 603 604 Parameters 605 ---------- 606 x : list 607 Can either be a list of floats in which case no xerror is assumed, or 608 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. 609 y : list 610 List of Obs, the dvalues of the Obs are used as yerror for the fit. 611 """ 612 613 def f(a, x): 614 y = a[0] + a[1] * x 615 return y 616 617 if all(isinstance(n, Obs) for n in x): 618 out = total_least_squares(x, y, f, **kwargs) 619 return out.fit_parameters 620 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): 621 out = least_squares(x, y, f, **kwargs) 622 return out.fit_parameters 623 else: 624 raise Exception('Unsupported types for x') 625 626 627def qqplot(x, o_y, func, p): 628 """Generates a quantile-quantile plot of the fit result which can be used to 629 check if the residuals of the fit are gaussian distributed. 630 """ 631 632 residuals = [] 633 for i_x, i_y in zip(x, o_y): 634 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) 635 residuals = sorted(residuals) 636 my_y = [o.value for o in residuals] 637 probplot = scipy.stats.probplot(my_y) 638 my_x = probplot[0][0] 639 plt.figure(figsize=(8, 8 / 1.618)) 640 plt.errorbar(my_x, my_y, fmt='o') 641 fit_start = my_x[0] 642 fit_stop = my_x[-1] 643 samples = np.arange(fit_start, fit_stop, 0.01) 644 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') 645 plt.plot(samples, probplot[1][0] * samples + probplot[1][1], zorder=10, label='Least squares fit, r=' + str(np.around(probplot[1][2], 3)), marker='', ls='-') 646 647 plt.xlabel('Theoretical quantiles') 648 plt.ylabel('Ordered Values') 649 plt.legend() 650 plt.draw() 651 652 653def residual_plot(x, y, func, fit_res): 654 """ Generates a plot which compares the fit to the data and displays the corresponding residuals""" 655 sorted_x = sorted(x) 656 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) 657 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) 658 x_samples = np.arange(xstart, xstop + 0.01, 0.01) 659 660 plt.figure(figsize=(8, 8 / 1.618)) 661 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) 662 ax0 = plt.subplot(gs[0]) 663 ax0.errorbar(x, [o.value for o in y], yerr=[o.dvalue for o in y], ls='none', fmt='o', capsize=3, markersize=5, label='Data') 664 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) 665 ax0.set_xticklabels([]) 666 ax0.set_xlim([xstart, xstop]) 667 ax0.set_xticklabels([]) 668 ax0.legend() 669 670 residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], x)) / np.asarray([o.dvalue for o in y]) 671 ax1 = plt.subplot(gs[1]) 672 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) 673 ax1.tick_params(direction='out') 674 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) 675 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") 676 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') 677 ax1.set_xlim([xstart, xstop]) 678 ax1.set_ylabel('Residuals') 679 plt.subplots_adjust(wspace=None, hspace=None) 680 plt.draw() 681 682 683def error_band(x, func, beta): 684 """Returns the error band for an array of sample values x, for given fit function func with optimized parameters beta.""" 685 cov = covariance(beta) 686 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): 687 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) 688 689 deriv = [] 690 for i, item in enumerate(x): 691 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) 692 693 err = [] 694 for i, item in enumerate(x): 695 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) 696 err = np.array(err) 697 698 return err 699 700 701def ks_test(objects=None): 702 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. 703 704 Parameters 705 ---------- 706 objects : list 707 List of fit results to include in the analysis (optional). 708 """ 709 710 if objects is None: 711 obs_list = [] 712 for obj in gc.get_objects(): 713 if isinstance(obj, Fit_result): 714 obs_list.append(obj) 715 else: 716 obs_list = objects 717 718 p_values = [o.p_value for o in obs_list] 719 720 bins = len(p_values) 721 x = np.arange(0, 1.001, 0.001) 722 plt.plot(x, x, 'k', zorder=1) 723 plt.xlim(0, 1) 724 plt.ylim(0, 1) 725 plt.xlabel('p-value') 726 plt.ylabel('Cumulative probability') 727 plt.title(str(bins) + ' p-values') 728 729 n = np.arange(1, bins + 1) / np.float64(bins) 730 Xs = np.sort(p_values) 731 plt.step(Xs, n) 732 diffs = n - Xs 733 loc_max_diff = np.argmax(np.abs(diffs)) 734 loc = Xs[loc_max_diff] 735 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) 736 plt.draw() 737 738 print(scipy.stats.kstest(p_values, 'uniform'))
19class Fit_result(Sequence): 20 """Represents fit results. 21 22 Attributes 23 ---------- 24 fit_parameters : list 25 results for the individual fit parameters, 26 also accessible via indices. 27 """ 28 29 def __init__(self): 30 self.fit_parameters = None 31 32 def __getitem__(self, idx): 33 return self.fit_parameters[idx] 34 35 def __len__(self): 36 return len(self.fit_parameters) 37 38 def gamma_method(self): 39 """Apply the gamma method to all fit parameters""" 40 [o.gamma_method() for o in self.fit_parameters] 41 42 def __str__(self): 43 my_str = 'Goodness of fit:\n' 44 if hasattr(self, 'chisquare_by_dof'): 45 my_str += '\u03C7\u00b2/d.o.f. = ' + f'{self.chisquare_by_dof:2.6f}' + '\n' 46 elif hasattr(self, 'residual_variance'): 47 my_str += 'residual variance = ' + f'{self.residual_variance:2.6f}' + '\n' 48 if hasattr(self, 'chisquare_by_expected_chisquare'): 49 my_str += '\u03C7\u00b2/\u03C7\u00b2exp = ' + f'{self.chisquare_by_expected_chisquare:2.6f}' + '\n' 50 if hasattr(self, 'p_value'): 51 my_str += 'p-value = ' + f'{self.p_value:2.4f}' + '\n' 52 my_str += 'Fit parameters:\n' 53 for i_par, par in enumerate(self.fit_parameters): 54 my_str += str(i_par) + '\t' + ' ' * int(par >= 0) + str(par).rjust(int(par < 0.0)) + '\n' 55 return my_str 56 57 def __repr__(self): 58 m = max(map(len, list(self.__dict__.keys()))) + 1 59 return '\n'.join([key.rjust(m) + ': ' + repr(value) for key, value in sorted(self.__dict__.items())])
Represents fit results.
Attributes
- fit_parameters (list): results for the individual fit parameters, also accessible via indices.
38 def gamma_method(self): 39 """Apply the gamma method to all fit parameters""" 40 [o.gamma_method() for o in self.fit_parameters]
Apply the gamma method to all fit parameters
Inherited Members
- collections.abc.Sequence
- index
- count
62def least_squares(x, y, func, priors=None, silent=False, **kwargs): 63 r'''Performs a non-linear fit to y = func(x). 64 65 Parameters 66 ---------- 67 x : list 68 list of floats. 69 y : list 70 list of Obs. 71 func : object 72 fit function, has to be of the form 73 74 ```python 75 import autograd.numpy as anp 76 77 def func(a, x): 78 return a[0] + a[1] * x + a[2] * anp.sinh(x) 79 ``` 80 81 For multiple x values func can be of the form 82 83 ```python 84 def func(a, x): 85 (x1, x2) = x 86 return a[0] * x1 ** 2 + a[1] * x2 87 ``` 88 89 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation 90 will not work. 91 priors : list, optional 92 priors has to be a list with an entry for every parameter in the fit. The entries can either be 93 Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like 94 0.548(23), 500(40) or 0.5(0.4) 95 silent : bool, optional 96 If true all output to the console is omitted (default False). 97 initial_guess : list 98 can provide an initial guess for the input parameters. Relevant for 99 non-linear fits with many parameters. 100 method : str, optional 101 can be used to choose an alternative method for the minimization of chisquare. 102 The possible methods are the ones which can be used for scipy.optimize.minimize and 103 migrad of iminuit. If no method is specified, Levenberg-Marquard is used. 104 Reliable alternatives are migrad, Powell and Nelder-Mead. 105 correlated_fit : bool 106 If True, use the full inverse covariance matrix in the definition of the chisquare cost function. 107 For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`. 108 In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix). 109 This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning). 110 At the moment this option only works for `prior==None` and when no `method` is given. 111 expected_chisquare : bool 112 If True estimates the expected chisquare which is 113 corrected by effects caused by correlated input data (default False). 114 resplot : bool 115 If True, a plot which displays fit, data and residuals is generated (default False). 116 qqplot : bool 117 If True, a quantile-quantile plot of the fit result is generated (default False). 118 ''' 119 if priors is not None: 120 return _prior_fit(x, y, func, priors, silent=silent, **kwargs) 121 else: 122 return _standard_fit(x, y, func, silent=silent, **kwargs)
Performs a non-linear fit to y = func(x).
Parameters
- x (list): list of floats.
- y (list): list of Obs.
func (object): fit function, has to be of the form
import autograd.numpy as anp def func(a, x): return a[0] + a[1] * x + a[2] * anp.sinh(x)
For multiple x values func can be of the form
def func(a, x): (x1, x2) = x return a[0] * x1 ** 2 + a[1] * x2
It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation will not work.
- priors (list, optional): priors has to be a list with an entry for every parameter in the fit. The entries can either be Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like 0.548(23), 500(40) or 0.5(0.4)
- silent (bool, optional): If true all output to the console is omitted (default False).
- initial_guess (list): can provide an initial guess for the input parameters. Relevant for non-linear fits with many parameters.
- method (str, optional): can be used to choose an alternative method for the minimization of chisquare. The possible methods are the ones which can be used for scipy.optimize.minimize and migrad of iminuit. If no method is specified, Levenberg-Marquard is used. Reliable alternatives are migrad, Powell and Nelder-Mead.
- correlated_fit (bool):
If True, use the full inverse covariance matrix in the definition of the chisquare cost function.
For details about how the covariance matrix is estimated see
pyerrors.obs.covariance
. In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix). This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning). At the moment this option only works forprior==None
and when nomethod
is given. - expected_chisquare (bool): If True estimates the expected chisquare which is corrected by effects caused by correlated input data (default False).
- resplot (bool): If True, a plot which displays fit, data and residuals is generated (default False).
- qqplot (bool): If True, a quantile-quantile plot of the fit result is generated (default False).
125def total_least_squares(x, y, func, silent=False, **kwargs): 126 r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters. 127 128 Parameters 129 ---------- 130 x : list 131 list of Obs, or a tuple of lists of Obs 132 y : list 133 list of Obs. The dvalues of the Obs are used as x- and yerror for the fit. 134 func : object 135 func has to be of the form 136 137 ```python 138 import autograd.numpy as anp 139 140 def func(a, x): 141 return a[0] + a[1] * x + a[2] * anp.sinh(x) 142 ``` 143 144 For multiple x values func can be of the form 145 146 ```python 147 def func(a, x): 148 (x1, x2) = x 149 return a[0] * x1 ** 2 + a[1] * x2 150 ``` 151 152 It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation 153 will not work. 154 silent : bool, optional 155 If true all output to the console is omitted (default False). 156 initial_guess : list 157 can provide an initial guess for the input parameters. Relevant for non-linear 158 fits with many parameters. 159 expected_chisquare : bool 160 If true prints the expected chisquare which is 161 corrected by effects caused by correlated input data. 162 This can take a while as the full correlation matrix 163 has to be calculated (default False). 164 165 Notes 166 ----- 167 Based on the orthogonal distance regression module of scipy 168 ''' 169 170 output = Fit_result() 171 172 output.fit_function = func 173 174 x = np.array(x) 175 176 x_shape = x.shape 177 178 if not callable(func): 179 raise TypeError('func has to be a function.') 180 181 for i in range(25): 182 try: 183 func(np.arange(i), x.T[0]) 184 except Exception: 185 pass 186 else: 187 break 188 189 n_parms = i 190 if not silent: 191 print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1)) 192 193 x_f = np.vectorize(lambda o: o.value)(x) 194 dx_f = np.vectorize(lambda o: o.dvalue)(x) 195 y_f = np.array([o.value for o in y]) 196 dy_f = np.array([o.dvalue for o in y]) 197 198 if np.any(np.asarray(dx_f) <= 0.0): 199 raise Exception('No x errors available, run the gamma method first.') 200 201 if np.any(np.asarray(dy_f) <= 0.0): 202 raise Exception('No y errors available, run the gamma method first.') 203 204 if 'initial_guess' in kwargs: 205 x0 = kwargs.get('initial_guess') 206 if len(x0) != n_parms: 207 raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms)) 208 else: 209 x0 = [1] * n_parms 210 211 data = RealData(x_f, y_f, sx=dx_f, sy=dy_f) 212 model = Model(func) 213 odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps) 214 odr.set_job(fit_type=0, deriv=1) 215 out = odr.run() 216 217 output.residual_variance = out.res_var 218 219 output.method = 'ODR' 220 221 output.message = out.stopreason 222 223 output.xplus = out.xplus 224 225 if not silent: 226 print('Method: ODR') 227 print(*out.stopreason) 228 print('Residual variance:', output.residual_variance) 229 230 if out.info > 3: 231 raise Exception('The minimization procedure did not converge.') 232 233 m = x_f.size 234 235 def odr_chisquare(p): 236 model = func(p[:n_parms], p[n_parms:].reshape(x_shape)) 237 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2) 238 return chisq 239 240 if kwargs.get('expected_chisquare') is True: 241 W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel())))) 242 243 if kwargs.get('covariance') is not None: 244 cov = kwargs.get('covariance') 245 else: 246 cov = covariance(np.concatenate((y, x.ravel()))) 247 248 number_of_x_parameters = int(m / x_f.shape[-1]) 249 250 old_jac = jacobian(func)(out.beta, out.xplus) 251 fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0))) 252 fused_row2 = np.concatenate((jacobian(lambda x, y: func(y, x))(out.xplus, out.beta).reshape(x_f.shape[-1], x_f.shape[-1] * number_of_x_parameters), np.identity(number_of_x_parameters * old_jac.shape[0]))) 253 new_jac = np.concatenate((fused_row1, fused_row2), axis=1) 254 255 A = W @ new_jac 256 P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T 257 expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W) 258 if expected_chisquare <= 0.0: 259 warnings.warn("Negative expected_chisquare.", RuntimeWarning) 260 expected_chisquare = np.abs(expected_chisquare) 261 output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare 262 if not silent: 263 print('chisquare/expected_chisquare:', 264 output.chisquare_by_expected_chisquare) 265 266 fitp = out.beta 267 try: 268 hess = jacobian(jacobian(odr_chisquare))(np.concatenate((fitp, out.xplus.ravel()))) 269 except TypeError: 270 raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None 271 272 def odr_chisquare_compact_x(d): 273 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) 274 chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((d[n_parms + m:].reshape(x_shape) - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) 275 return chisq 276 277 jac_jac_x = jacobian(jacobian(odr_chisquare_compact_x))(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel()))) 278 279 # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv 280 try: 281 deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:]) 282 except np.linalg.LinAlgError: 283 raise Exception("Cannot invert hessian matrix.") 284 285 def odr_chisquare_compact_y(d): 286 model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape)) 287 chisq = anp.sum(((d[n_parms + m:] - model) / dy_f) ** 2) + anp.sum(((x_f - d[n_parms:n_parms + m].reshape(x_shape)) / dx_f) ** 2) 288 return chisq 289 290 jac_jac_y = jacobian(jacobian(odr_chisquare_compact_y))(np.concatenate((fitp, out.xplus.ravel(), y_f))) 291 292 # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv 293 try: 294 deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:]) 295 except np.linalg.LinAlgError: 296 raise Exception("Cannot invert hessian matrix.") 297 298 result = [] 299 for i in range(n_parms): 300 result.append(derived_observable(lambda my_var, **kwargs: (my_var[0] + np.finfo(np.float64).eps) / (x.ravel()[0].value + np.finfo(np.float64).eps) * out.beta[i], list(x.ravel()) + list(y), man_grad=list(deriv_x[i]) + list(deriv_y[i]))) 301 302 output.fit_parameters = result 303 304 output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) 305 output.dof = x.shape[-1] - n_parms 306 output.p_value = 1 - chi2.cdf(output.odr_chisquare, output.dof) 307 308 return output
Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
Parameters
- x (list): list of Obs, or a tuple of lists of Obs
- y (list): list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
func (object): func has to be of the form
import autograd.numpy as anp def func(a, x): return a[0] + a[1] * x + a[2] * anp.sinh(x)
For multiple x values func can be of the form
def func(a, x): (x1, x2) = x return a[0] * x1 ** 2 + a[1] * x2
It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation will not work.
- silent (bool, optional): If true all output to the console is omitted (default False).
- initial_guess (list): can provide an initial guess for the input parameters. Relevant for non-linear fits with many parameters.
- expected_chisquare (bool): If true prints the expected chisquare which is corrected by effects caused by correlated input data. This can take a while as the full correlation matrix has to be calculated (default False).
Notes
Based on the orthogonal distance regression module of scipy
602def fit_lin(x, y, **kwargs): 603 """Performs a linear fit to y = n + m * x and returns two Obs n, m. 604 605 Parameters 606 ---------- 607 x : list 608 Can either be a list of floats in which case no xerror is assumed, or 609 a list of Obs, where the dvalues of the Obs are used as xerror for the fit. 610 y : list 611 List of Obs, the dvalues of the Obs are used as yerror for the fit. 612 """ 613 614 def f(a, x): 615 y = a[0] + a[1] * x 616 return y 617 618 if all(isinstance(n, Obs) for n in x): 619 out = total_least_squares(x, y, f, **kwargs) 620 return out.fit_parameters 621 elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray): 622 out = least_squares(x, y, f, **kwargs) 623 return out.fit_parameters 624 else: 625 raise Exception('Unsupported types for x')
Performs a linear fit to y = n + m * x and returns two Obs n, m.
Parameters
- x (list): Can either be a list of floats in which case no xerror is assumed, or a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
- y (list): List of Obs, the dvalues of the Obs are used as yerror for the fit.
628def qqplot(x, o_y, func, p): 629 """Generates a quantile-quantile plot of the fit result which can be used to 630 check if the residuals of the fit are gaussian distributed. 631 """ 632 633 residuals = [] 634 for i_x, i_y in zip(x, o_y): 635 residuals.append((i_y - func(p, i_x)) / i_y.dvalue) 636 residuals = sorted(residuals) 637 my_y = [o.value for o in residuals] 638 probplot = scipy.stats.probplot(my_y) 639 my_x = probplot[0][0] 640 plt.figure(figsize=(8, 8 / 1.618)) 641 plt.errorbar(my_x, my_y, fmt='o') 642 fit_start = my_x[0] 643 fit_stop = my_x[-1] 644 samples = np.arange(fit_start, fit_stop, 0.01) 645 plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution') 646 plt.plot(samples, probplot[1][0] * samples + probplot[1][1], zorder=10, label='Least squares fit, r=' + str(np.around(probplot[1][2], 3)), marker='', ls='-') 647 648 plt.xlabel('Theoretical quantiles') 649 plt.ylabel('Ordered Values') 650 plt.legend() 651 plt.draw()
Generates a quantile-quantile plot of the fit result which can be used to check if the residuals of the fit are gaussian distributed.
654def residual_plot(x, y, func, fit_res): 655 """ Generates a plot which compares the fit to the data and displays the corresponding residuals""" 656 sorted_x = sorted(x) 657 xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0]) 658 xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2]) 659 x_samples = np.arange(xstart, xstop + 0.01, 0.01) 660 661 plt.figure(figsize=(8, 8 / 1.618)) 662 gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0) 663 ax0 = plt.subplot(gs[0]) 664 ax0.errorbar(x, [o.value for o in y], yerr=[o.dvalue for o in y], ls='none', fmt='o', capsize=3, markersize=5, label='Data') 665 ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0) 666 ax0.set_xticklabels([]) 667 ax0.set_xlim([xstart, xstop]) 668 ax0.set_xticklabels([]) 669 ax0.legend() 670 671 residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], x)) / np.asarray([o.dvalue for o in y]) 672 ax1 = plt.subplot(gs[1]) 673 ax1.plot(x, residuals, 'ko', ls='none', markersize=5) 674 ax1.tick_params(direction='out') 675 ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True) 676 ax1.axhline(y=0.0, ls='--', color='k', marker=" ") 677 ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k') 678 ax1.set_xlim([xstart, xstop]) 679 ax1.set_ylabel('Residuals') 680 plt.subplots_adjust(wspace=None, hspace=None) 681 plt.draw()
Generates a plot which compares the fit to the data and displays the corresponding residuals
684def error_band(x, func, beta): 685 """Returns the error band for an array of sample values x, for given fit function func with optimized parameters beta.""" 686 cov = covariance(beta) 687 if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps): 688 warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning) 689 690 deriv = [] 691 for i, item in enumerate(x): 692 deriv.append(np.array(egrad(func)([o.value for o in beta], item))) 693 694 err = [] 695 for i, item in enumerate(x): 696 err.append(np.sqrt(deriv[i] @ cov @ deriv[i])) 697 err = np.array(err) 698 699 return err
Returns the error band for an array of sample values x, for given fit function func with optimized parameters beta.
702def ks_test(objects=None): 703 """Performs a Kolmogorov–Smirnov test for the p-values of all fit object. 704 705 Parameters 706 ---------- 707 objects : list 708 List of fit results to include in the analysis (optional). 709 """ 710 711 if objects is None: 712 obs_list = [] 713 for obj in gc.get_objects(): 714 if isinstance(obj, Fit_result): 715 obs_list.append(obj) 716 else: 717 obs_list = objects 718 719 p_values = [o.p_value for o in obs_list] 720 721 bins = len(p_values) 722 x = np.arange(0, 1.001, 0.001) 723 plt.plot(x, x, 'k', zorder=1) 724 plt.xlim(0, 1) 725 plt.ylim(0, 1) 726 plt.xlabel('p-value') 727 plt.ylabel('Cumulative probability') 728 plt.title(str(bins) + ' p-values') 729 730 n = np.arange(1, bins + 1) / np.float64(bins) 731 Xs = np.sort(p_values) 732 plt.step(Xs, n) 733 diffs = n - Xs 734 loc_max_diff = np.argmax(np.abs(diffs)) 735 loc = Xs[loc_max_diff] 736 plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0)) 737 plt.draw() 738 739 print(scipy.stats.kstest(p_values, 'uniform'))
Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
Parameters
- objects (list): List of fit results to include in the analysis (optional).