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. In case of correlated fits the guess is used to perform
 99        an uncorrelated fit which then serves as guess for the correlated fit.
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)
123
124
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
309
310
311def _prior_fit(x, y, func, priors, silent=False, **kwargs):
312    output = Fit_result()
313
314    output.fit_function = func
315
316    x = np.asarray(x)
317
318    if not callable(func):
319        raise TypeError('func has to be a function.')
320
321    for i in range(100):
322        try:
323            func(np.arange(i), 0)
324        except Exception:
325            pass
326        else:
327            break
328
329    n_parms = i
330
331    if n_parms != len(priors):
332        raise Exception('Priors does not have the correct length.')
333
334    def extract_val_and_dval(string):
335        split_string = string.split('(')
336        if '.' in split_string[0] and '.' not in split_string[1][:-1]:
337            factor = 10 ** -len(split_string[0].partition('.')[2])
338        else:
339            factor = 1
340        return float(split_string[0]), float(split_string[1][:-1]) * factor
341
342    loc_priors = []
343    for i_n, i_prior in enumerate(priors):
344        if isinstance(i_prior, Obs):
345            loc_priors.append(i_prior)
346        else:
347            loc_val, loc_dval = extract_val_and_dval(i_prior)
348            loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}"))
349
350    output.priors = loc_priors
351
352    if not silent:
353        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
354
355    y_f = [o.value for o in y]
356    dy_f = [o.dvalue for o in y]
357
358    if np.any(np.asarray(dy_f) <= 0.0):
359        raise Exception('No y errors available, run the gamma method first.')
360
361    p_f = [o.value for o in loc_priors]
362    dp_f = [o.dvalue for o in loc_priors]
363
364    if np.any(np.asarray(dp_f) <= 0.0):
365        raise Exception('No prior errors available, run the gamma method first.')
366
367    if 'initial_guess' in kwargs:
368        x0 = kwargs.get('initial_guess')
369        if len(x0) != n_parms:
370            raise Exception('Initial guess does not have the correct length.')
371    else:
372        x0 = p_f
373
374    def chisqfunc(p):
375        model = func(p, x)
376        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((p_f - p) / dp_f) ** 2)
377        return chisq
378
379    if not silent:
380        print('Method: migrad')
381
382    m = iminuit.Minuit(chisqfunc, x0)
383    m.errordef = 1
384    m.print_level = 0
385    if 'tol' in kwargs:
386        m.tol = kwargs.get('tol')
387    else:
388        m.tol = 1e-4
389    m.migrad()
390    params = np.asarray(m.values)
391
392    output.chisquare_by_dof = m.fval / len(x)
393
394    output.method = 'migrad'
395
396    if not silent:
397        print('chisquare/d.o.f.:', output.chisquare_by_dof)
398
399    if not m.fmin.is_valid:
400        raise Exception('The minimization procedure did not converge.')
401
402    hess_inv = np.linalg.pinv(jacobian(jacobian(chisqfunc))(params))
403
404    def chisqfunc_compact(d):
405        model = func(d[:n_parms], x)
406        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)
407        return chisq
408
409    jac_jac = jacobian(jacobian(chisqfunc_compact))(np.concatenate((params, y_f, p_f)))
410
411    deriv = -hess_inv @ jac_jac[:n_parms, n_parms:]
412
413    result = []
414    for i in range(n_parms):
415        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])))
416
417    output.fit_parameters = result
418    output.chisquare = chisqfunc(np.asarray(params))
419
420    if kwargs.get('resplot') is True:
421        residual_plot(x, y, func, result)
422
423    if kwargs.get('qqplot') is True:
424        qqplot(x, y, func, result)
425
426    return output
427
428
429def _standard_fit(x, y, func, silent=False, **kwargs):
430
431    output = Fit_result()
432
433    output.fit_function = func
434
435    x = np.asarray(x)
436
437    if x.shape[-1] != len(y):
438        raise Exception('x and y input have to have the same length')
439
440    if len(x.shape) > 2:
441        raise Exception('Unknown format for x values')
442
443    if not callable(func):
444        raise TypeError('func has to be a function.')
445
446    for i in range(25):
447        try:
448            func(np.arange(i), x.T[0])
449        except Exception:
450            pass
451        else:
452            break
453
454    n_parms = i
455
456    if not silent:
457        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
458
459    y_f = [o.value for o in y]
460    dy_f = [o.dvalue for o in y]
461
462    if np.any(np.asarray(dy_f) <= 0.0):
463        raise Exception('No y errors available, run the gamma method first.')
464
465    if 'initial_guess' in kwargs:
466        x0 = kwargs.get('initial_guess')
467        if len(x0) != n_parms:
468            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
469    else:
470        x0 = [0.1] * n_parms
471
472    if kwargs.get('correlated_fit') is True:
473        corr = covariance(y, correlation=True, **kwargs)
474        covdiag = np.diag(1 / np.asarray(dy_f))
475        condn = np.linalg.cond(corr)
476        if condn > 0.1 / np.finfo(float).eps:
477            raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})")
478        if condn > 1 / np.sqrt(np.finfo(float).eps):
479            warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning)
480        chol = np.linalg.cholesky(corr)
481        chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True)
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        if kwargs.get('correlated_fit') is True:
529            assert np.isclose(chisquare, chisqfunc_corr(fit_result.x), atol=1e-14)
530        else:
531            assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14)
532
533        output.iterations = fit_result.nfev
534
535    if not fit_result.success:
536        raise Exception('The minimization procedure did not converge.')
537
538    if x.shape[-1] - n_parms > 0:
539        output.chisquare_by_dof = chisquare / (x.shape[-1] - n_parms)
540    else:
541        output.chisquare_by_dof = float('nan')
542
543    output.message = fit_result.message
544    if not silent:
545        print(fit_result.message)
546        print('chisquare/d.o.f.:', output.chisquare_by_dof)
547
548    if kwargs.get('expected_chisquare') is True:
549        if kwargs.get('correlated_fit') is not True:
550            W = np.diag(1 / np.asarray(dy_f))
551            cov = covariance(y)
552            A = W @ jacobian(func)(fit_result.x, x)
553            P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
554            expected_chisquare = np.trace((np.identity(x.shape[-1]) - P_phi) @ W @ cov @ W)
555            output.chisquare_by_expected_chisquare = chisquare / expected_chisquare
556            if not silent:
557                print('chisquare/expected_chisquare:',
558                      output.chisquare_by_expected_chisquare)
559
560    fitp = fit_result.x
561    try:
562        hess = jacobian(jacobian(chisqfunc))(fitp)
563    except TypeError:
564        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
565
566    if kwargs.get('correlated_fit') is True:
567        def chisqfunc_compact(d):
568            model = func(d[:n_parms], x)
569            chisq = anp.sum(anp.dot(chol_inv, (d[n_parms:] - model)) ** 2)
570            return chisq
571
572    else:
573        def chisqfunc_compact(d):
574            model = func(d[:n_parms], x)
575            chisq = anp.sum(((d[n_parms:] - model) / dy_f) ** 2)
576            return chisq
577
578    jac_jac = jacobian(jacobian(chisqfunc_compact))(np.concatenate((fitp, y_f)))
579
580    # Compute hess^{-1} @ jac_jac[:n_parms, n_parms:] using LAPACK dgesv
581    try:
582        deriv = -scipy.linalg.solve(hess, jac_jac[:n_parms, n_parms:])
583    except np.linalg.LinAlgError:
584        raise Exception("Cannot invert hessian matrix.")
585
586    result = []
587    for i in range(n_parms):
588        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])))
589
590    output.fit_parameters = result
591
592    output.chisquare = chisquare
593    output.dof = x.shape[-1] - n_parms
594    output.p_value = 1 - chi2.cdf(output.chisquare, output.dof)
595
596    if kwargs.get('resplot') is True:
597        residual_plot(x, y, func, result)
598
599    if kwargs.get('qqplot') is True:
600        qqplot(x, y, func, result)
601
602    return output
603
604
605def fit_lin(x, y, **kwargs):
606    """Performs a linear fit to y = n + m * x and returns two Obs n, m.
607
608    Parameters
609    ----------
610    x : list
611        Can either be a list of floats in which case no xerror is assumed, or
612        a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
613    y : list
614        List of Obs, the dvalues of the Obs are used as yerror for the fit.
615    """
616
617    def f(a, x):
618        y = a[0] + a[1] * x
619        return y
620
621    if all(isinstance(n, Obs) for n in x):
622        out = total_least_squares(x, y, f, **kwargs)
623        return out.fit_parameters
624    elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
625        out = least_squares(x, y, f, **kwargs)
626        return out.fit_parameters
627    else:
628        raise Exception('Unsupported types for x')
629
630
631def qqplot(x, o_y, func, p):
632    """Generates a quantile-quantile plot of the fit result which can be used to
633       check if the residuals of the fit are gaussian distributed.
634    """
635
636    residuals = []
637    for i_x, i_y in zip(x, o_y):
638        residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
639    residuals = sorted(residuals)
640    my_y = [o.value for o in residuals]
641    probplot = scipy.stats.probplot(my_y)
642    my_x = probplot[0][0]
643    plt.figure(figsize=(8, 8 / 1.618))
644    plt.errorbar(my_x, my_y, fmt='o')
645    fit_start = my_x[0]
646    fit_stop = my_x[-1]
647    samples = np.arange(fit_start, fit_stop, 0.01)
648    plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
649    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='-')
650
651    plt.xlabel('Theoretical quantiles')
652    plt.ylabel('Ordered Values')
653    plt.legend()
654    plt.draw()
655
656
657def residual_plot(x, y, func, fit_res):
658    """ Generates a plot which compares the fit to the data and displays the corresponding residuals"""
659    sorted_x = sorted(x)
660    xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
661    xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
662    x_samples = np.arange(xstart, xstop + 0.01, 0.01)
663
664    plt.figure(figsize=(8, 8 / 1.618))
665    gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
666    ax0 = plt.subplot(gs[0])
667    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')
668    ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
669    ax0.set_xticklabels([])
670    ax0.set_xlim([xstart, xstop])
671    ax0.set_xticklabels([])
672    ax0.legend()
673
674    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])
675    ax1 = plt.subplot(gs[1])
676    ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
677    ax1.tick_params(direction='out')
678    ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
679    ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
680    ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
681    ax1.set_xlim([xstart, xstop])
682    ax1.set_ylabel('Residuals')
683    plt.subplots_adjust(wspace=None, hspace=None)
684    plt.draw()
685
686
687def error_band(x, func, beta):
688    """Returns the error band for an array of sample values x, for given fit function func with optimized parameters beta."""
689    cov = covariance(beta)
690    if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
691        warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
692
693    deriv = []
694    for i, item in enumerate(x):
695        deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
696
697    err = []
698    for i, item in enumerate(x):
699        err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
700    err = np.array(err)
701
702    return err
703
704
705def ks_test(objects=None):
706    """Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
707
708    Parameters
709    ----------
710    objects : list
711        List of fit results to include in the analysis (optional).
712    """
713
714    if objects is None:
715        obs_list = []
716        for obj in gc.get_objects():
717            if isinstance(obj, Fit_result):
718                obs_list.append(obj)
719    else:
720        obs_list = objects
721
722    p_values = [o.p_value for o in obs_list]
723
724    bins = len(p_values)
725    x = np.arange(0, 1.001, 0.001)
726    plt.plot(x, x, 'k', zorder=1)
727    plt.xlim(0, 1)
728    plt.ylim(0, 1)
729    plt.xlabel('p-value')
730    plt.ylabel('Cumulative probability')
731    plt.title(str(bins) + ' p-values')
732
733    n = np.arange(1, bins + 1) / np.float64(bins)
734    Xs = np.sort(p_values)
735    plt.step(Xs, n)
736    diffs = n - Xs
737    loc_max_diff = np.argmax(np.abs(diffs))
738    loc = Xs[loc_max_diff]
739    plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
740    plt.draw()
741
742    print(scipy.stats.kstest(p_values, 'uniform'))