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
 11import iminuit
 12from autograd import jacobian as auto_jacobian
 13from autograd import hessian as auto_hessian
 14from autograd import elementwise_grad as egrad
 15from numdifftools import Jacobian as num_jacobian
 16from numdifftools import Hessian as num_hessian
 17from .obs import Obs, derived_observable, covariance, cov_Obs, invert_corr_cov_cholesky
 18
 19
 20class Fit_result(Sequence):
 21    """Represents fit results.
 22
 23    Attributes
 24    ----------
 25    fit_parameters : list
 26        results for the individual fit parameters,
 27        also accessible via indices.
 28    chisquare_by_dof : float
 29        reduced chisquare.
 30    p_value : float
 31        p-value of the fit
 32    t2_p_value : float
 33        Hotelling t-squared p-value for correlated fits.
 34    """
 35
 36    def __init__(self):
 37        self.fit_parameters = None
 38
 39    def __getitem__(self, idx):
 40        return self.fit_parameters[idx]
 41
 42    def __len__(self):
 43        return len(self.fit_parameters)
 44
 45    def gamma_method(self, **kwargs):
 46        """Apply the gamma method to all fit parameters"""
 47        [o.gamma_method(**kwargs) for o in self.fit_parameters]
 48
 49    gm = gamma_method
 50
 51    def __str__(self):
 52        my_str = 'Goodness of fit:\n'
 53        if hasattr(self, 'chisquare_by_dof'):
 54            my_str += '\u03C7\u00b2/d.o.f. = ' + f'{self.chisquare_by_dof:2.6f}' + '\n'
 55        elif hasattr(self, 'residual_variance'):
 56            my_str += 'residual variance = ' + f'{self.residual_variance:2.6f}' + '\n'
 57        if hasattr(self, 'chisquare_by_expected_chisquare'):
 58            my_str += '\u03C7\u00b2/\u03C7\u00b2exp  = ' + f'{self.chisquare_by_expected_chisquare:2.6f}' + '\n'
 59        if hasattr(self, 'p_value'):
 60            my_str += 'p-value   = ' + f'{self.p_value:2.4f}' + '\n'
 61        if hasattr(self, 't2_p_value'):
 62            my_str += 't\u00B2p-value = ' + f'{self.t2_p_value:2.4f}' + '\n'
 63        my_str += 'Fit parameters:\n'
 64        for i_par, par in enumerate(self.fit_parameters):
 65            my_str += str(i_par) + '\t' + ' ' * int(par >= 0) + str(par).rjust(int(par < 0.0)) + '\n'
 66        return my_str
 67
 68    def __repr__(self):
 69        m = max(map(len, list(self.__dict__.keys()))) + 1
 70        return '\n'.join([key.rjust(m) + ': ' + repr(value) for key, value in sorted(self.__dict__.items())])
 71
 72
 73def least_squares(x, y, func, priors=None, silent=False, **kwargs):
 74    r'''Performs a non-linear fit to y = func(x).
 75        ```
 76
 77    Parameters
 78    ----------
 79    For an uncombined fit:
 80
 81    x : list
 82        list of floats.
 83    y : list
 84        list of Obs.
 85    func : object
 86        fit function, has to be of the form
 87
 88        ```python
 89        import autograd.numpy as anp
 90
 91        def func(a, x):
 92            return a[0] + a[1] * x + a[2] * anp.sinh(x)
 93        ```
 94
 95        For multiple x values func can be of the form
 96
 97        ```python
 98        def func(a, x):
 99            (x1, x2) = x
100            return a[0] * x1 ** 2 + a[1] * x2
101        ```
102        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
103        will not work.
104
105    OR For a combined fit:
106
107    x : dict
108        dict of lists.
109    y : dict
110        dict of lists of Obs.
111    funcs : dict
112        dict of objects
113        fit functions have to be of the form (here a[0] is the common fit parameter)
114        ```python
115        import autograd.numpy as anp
116        funcs = {"a": func_a,
117                "b": func_b}
118
119        def func_a(a, x):
120            return a[1] * anp.exp(-a[0] * x)
121
122        def func_b(a, x):
123            return a[2] * anp.exp(-a[0] * x)
124
125        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
126        will not work.
127
128    priors : dict or list, optional
129        priors can either be a dictionary with integer keys and the corresponding priors as values or
130        a list with an entry for every parameter in the fit. The entries can either be
131        Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like
132        0.548(23), 500(40) or 0.5(0.4)
133    silent : bool, optional
134        If true all output to the console is omitted (default False).
135    initial_guess : list
136        can provide an initial guess for the input parameters. Relevant for
137        non-linear fits with many parameters. In case of correlated fits the guess is used to perform
138        an uncorrelated fit which then serves as guess for the correlated fit.
139    method : str, optional
140        can be used to choose an alternative method for the minimization of chisquare.
141        The possible methods are the ones which can be used for scipy.optimize.minimize and
142        migrad of iminuit. If no method is specified, Levenberg-Marquard is used.
143        Reliable alternatives are migrad, Powell and Nelder-Mead.
144    tol: float, optional
145        can be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence
146        to a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly
147        invalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values
148        The stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
149    correlated_fit : bool
150        If True, use the full inverse covariance matrix in the definition of the chisquare cost function.
151        For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`.
152        In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).
153        This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
154    inv_chol_cov_matrix [array,list], optional
155        array: shape = (no of y values) X (no of y values)
156        list:   for an uncombined fit: [""]
157                for a combined fit: list of keys belonging to the corr_matrix saved in the array, must be the same as the keys of the y dict in alphabetical order
158        If correlated_fit=True is set as well, can provide an inverse covariance matrix (y errors, dy_f included!) of your own choosing for a correlated fit.
159        The matrix must be a lower triangular matrix constructed from a Cholesky decomposition: The function invert_corr_cov_cholesky(corr, inverrdiag) can be
160        used to construct it from a correlation matrix (corr) and the errors dy_f of the data points (inverrdiag = np.diag(1 / np.asarray(dy_f))). For the correct
161        ordering the correlation matrix (corr) can be sorted via the function sort_corr(corr, kl, yd) where kl is the list of keys and yd the y dict.
162    expected_chisquare : bool
163        If True estimates the expected chisquare which is
164        corrected by effects caused by correlated input data (default False).
165    resplot : bool
166        If True, a plot which displays fit, data and residuals is generated (default False).
167    qqplot : bool
168        If True, a quantile-quantile plot of the fit result is generated (default False).
169    num_grad : bool
170        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
171
172    Returns
173    -------
174    output : Fit_result
175        Parameters and information on the fitted result.
176    Examples
177    ------
178    >>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set
179    >>> import numpy as np
180    >>> from scipy.stats import norm
181    >>> from scipy.linalg import cholesky
182    >>> import pyerrors as pe
183    >>> # generating the random data set
184    >>> num_samples = 400
185    >>> N = 3
186    >>> x = np.arange(N)
187    >>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers
188    >>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers
189    >>> r = r1 = r2 = np.zeros((N, N))
190    >>> y = {}
191    >>> for i in range(N):
192    >>>    for j in range(N):
193    >>>        r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix
194    >>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors
195    >>> for i in range(N):
196    >>>    for j in range(N):
197    >>>        r[i, j] *= errl[i] * errl[j] # element in covariance matrix
198    >>> c = cholesky(r, lower=True)
199    >>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined
200    >>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built
201    >>> x_dict = {}
202    >>> y_dict = {}
203    >>> chol_inv_dict = {}
204    >>> data = []
205    >>> for key in y.keys():
206    >>>    x_dict[key] = x
207    >>>    for i in range(N):
208    >>>        data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data
209    >>>    [o.gamma_method() for o in data]
210    >>>    corr = pe.covariance(data, correlation=True)
211    >>>    inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data]))
212    >>>    chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below
213    >>> y_dict = {'a': data[:3], 'b': data[3:]}
214    >>> # common fit parameter p[0] in combined fit
215    >>> def fit1(p, x):
216    >>>    return p[0] + p[1] * x
217    >>> def fit2(p, x):
218    >>>    return p[0] + p[2] * x
219    >>> fitf_dict = {'a': fit1, 'b':fit2}
220    >>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']])
221    Fit with 3 parameters
222    Method: Levenberg-Marquardt
223    `ftol` termination condition is satisfied.
224    chisquare/d.o.f.: 0.5388013574561786 # random
225    fit parameters [1.11897846 0.96361162 0.92325319] # random
226
227    '''
228    output = Fit_result()
229
230    if (isinstance(x, dict) and isinstance(y, dict) and isinstance(func, dict)):
231        xd = {key: anp.asarray(x[key]) for key in x}
232        yd = y
233        funcd = func
234        output.fit_function = func
235    elif (isinstance(x, dict) or isinstance(y, dict) or isinstance(func, dict)):
236        raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.")
237    else:
238        x = np.asarray(x)
239        xd = {"": x}
240        yd = {"": y}
241        funcd = {"": func}
242        output.fit_function = func
243
244    if kwargs.get('num_grad') is True:
245        jacobian = num_jacobian
246        hessian = num_hessian
247    else:
248        jacobian = auto_jacobian
249        hessian = auto_hessian
250
251    key_ls = sorted(list(xd.keys()))
252
253    if sorted(list(yd.keys())) != key_ls:
254        raise ValueError('x and y dictionaries do not contain the same keys.')
255
256    if sorted(list(funcd.keys())) != key_ls:
257        raise ValueError('x and func dictionaries do not contain the same keys.')
258
259    x_all = np.concatenate([np.array(xd[key]).transpose() for key in key_ls]).transpose()
260    y_all = np.concatenate([np.array(yd[key]) for key in key_ls])
261
262    y_f = [o.value for o in y_all]
263    dy_f = [o.dvalue for o in y_all]
264
265    if len(x_all.shape) > 2:
266        raise ValueError("Unknown format for x values")
267
268    if np.any(np.asarray(dy_f) <= 0.0):
269        raise Exception("No y errors available, run the gamma method first.")
270
271    # number of fit parameters
272    n_parms_ls = []
273    for key in key_ls:
274        if not callable(funcd[key]):
275            raise TypeError('func (key=' + key + ') is not a function.')
276        if np.asarray(xd[key]).shape[-1] != len(yd[key]):
277            raise ValueError('x and y input (key=' + key + ') do not have the same length')
278        for n_loc in range(100):
279            try:
280                funcd[key](np.arange(n_loc), x_all.T[0])
281            except TypeError:
282                continue
283            except IndexError:
284                continue
285            else:
286                break
287        else:
288            raise RuntimeError("Fit function (key=" + key + ") is not valid.")
289        n_parms_ls.append(n_loc)
290
291    n_parms = max(n_parms_ls)
292
293    if len(key_ls) > 1:
294        for key in key_ls:
295            if np.asarray(yd[key]).shape != funcd[key](np.arange(n_parms), xd[key]).shape:
296                raise ValueError(f"Fit function {key} returns the wrong shape ({funcd[key](np.arange(n_parms), xd[key]).shape} instead of {np.asarray(yd[key]).shape})\nIf the fit function is just a constant you could try adding x*0 to get the correct shape.")
297
298    if not silent:
299        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
300
301    if priors is not None:
302        if isinstance(priors, (list, np.ndarray)):
303            if n_parms != len(priors):
304                raise ValueError("'priors' does not have the correct length.")
305
306            loc_priors = []
307            for i_n, i_prior in enumerate(priors):
308                loc_priors.append(_construct_prior_obs(i_prior, i_n))
309
310            prior_mask = np.arange(len(priors))
311            output.priors = loc_priors
312
313        elif isinstance(priors, dict):
314            loc_priors = []
315            prior_mask = []
316            output.priors = {}
317            for pos, prior in priors.items():
318                if isinstance(pos, int):
319                    prior_mask.append(pos)
320                else:
321                    raise TypeError("Prior position needs to be an integer.")
322                loc_priors.append(_construct_prior_obs(prior, pos))
323
324                output.priors[pos] = loc_priors[-1]
325            if max(prior_mask) >= n_parms:
326                raise ValueError("Prior position out of range.")
327        else:
328            raise TypeError("Unkown type for `priors`.")
329
330        p_f = [o.value for o in loc_priors]
331        dp_f = [o.dvalue for o in loc_priors]
332        if np.any(np.asarray(dp_f) <= 0.0):
333            raise Exception("No prior errors available, run the gamma method first.")
334    else:
335        p_f = dp_f = np.array([])
336        prior_mask = []
337        loc_priors = []
338
339    if 'initial_guess' in kwargs:
340        x0 = kwargs.get('initial_guess')
341        if len(x0) != n_parms:
342            raise ValueError('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
343    else:
344        x0 = [0.1] * n_parms
345
346    if priors is None:
347        def general_chisqfunc_uncorr(p, ivars, pr):
348            model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls])
349            return (ivars - model) / dy_f
350    else:
351        def general_chisqfunc_uncorr(p, ivars, pr):
352            model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls])
353            return anp.concatenate(((ivars - model) / dy_f, (p[prior_mask] - pr) / dp_f))
354
355    def chisqfunc_uncorr(p):
356        return anp.sum(general_chisqfunc_uncorr(p, y_f, p_f) ** 2)
357
358    if kwargs.get('correlated_fit') is True:
359        if 'inv_chol_cov_matrix' in kwargs:
360            chol_inv = kwargs.get('inv_chol_cov_matrix')
361            if (chol_inv[0].shape[0] != len(dy_f)):
362                raise TypeError('The number of columns of the inverse covariance matrix handed over needs to be equal to the number of y errors.')
363            if (chol_inv[0].shape[0] != chol_inv[0].shape[1]):
364                raise TypeError('The inverse covariance matrix handed over needs to have the same number of rows as columns.')
365            if (chol_inv[1] != key_ls):
366                raise ValueError('The keys of inverse covariance matrix are not the same or do not appear in the same order as the x and y values.')
367            chol_inv = chol_inv[0]
368            if np.any(np.diag(chol_inv) <= 0) or (not np.all(chol_inv == np.tril(chol_inv))):
369                raise ValueError('The inverse covariance matrix inv_chol_cov_matrix[0] has to be a lower triangular matrix constructed from a Cholesky decomposition.')
370        else:
371            corr = covariance(y_all, correlation=True, **kwargs)
372            inverrdiag = np.diag(1 / np.asarray(dy_f))
373            chol_inv = invert_corr_cov_cholesky(corr, inverrdiag)
374
375        def general_chisqfunc(p, ivars, pr):
376            model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls])
377            return anp.concatenate((anp.dot(chol_inv, (ivars - model)), (p[prior_mask] - pr) / dp_f))
378
379        def chisqfunc(p):
380            return anp.sum(general_chisqfunc(p, y_f, p_f) ** 2)
381    else:
382        general_chisqfunc = general_chisqfunc_uncorr
383        chisqfunc = chisqfunc_uncorr
384
385    output.method = kwargs.get('method', 'Levenberg-Marquardt')
386    if not silent:
387        print('Method:', output.method)
388
389    if output.method != 'Levenberg-Marquardt':
390        if output.method == 'migrad':
391            tolerance = 1e-4  # default value of 1e-1 set by iminuit can be problematic
392            if 'tol' in kwargs:
393                tolerance = kwargs.get('tol')
394            fit_result = iminuit.minimize(chisqfunc_uncorr, x0, tol=tolerance)  # Stopping criterion 0.002 * tol * errordef
395            if kwargs.get('correlated_fit') is True:
396                fit_result = iminuit.minimize(chisqfunc, fit_result.x, tol=tolerance)
397            output.iterations = fit_result.nfev
398        else:
399            tolerance = 1e-12
400            if 'tol' in kwargs:
401                tolerance = kwargs.get('tol')
402            fit_result = scipy.optimize.minimize(chisqfunc_uncorr, x0, method=kwargs.get('method'), tol=tolerance)
403            if kwargs.get('correlated_fit') is True:
404                fit_result = scipy.optimize.minimize(chisqfunc, fit_result.x, method=kwargs.get('method'), tol=tolerance)
405            output.iterations = fit_result.nit
406
407        chisquare = fit_result.fun
408
409    else:
410        if 'tol' in kwargs:
411            print('tol cannot be set for Levenberg-Marquardt')
412
413        def chisqfunc_residuals_uncorr(p):
414            return general_chisqfunc_uncorr(p, y_f, p_f)
415
416        fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
417        if kwargs.get('correlated_fit') is True:
418            def chisqfunc_residuals(p):
419                return general_chisqfunc(p, y_f, p_f)
420
421            fit_result = scipy.optimize.least_squares(chisqfunc_residuals, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
422
423        chisquare = np.sum(fit_result.fun ** 2)
424        assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14)
425
426        output.iterations = fit_result.nfev
427
428    if not fit_result.success:
429        raise Exception('The minimization procedure did not converge.')
430
431    output.chisquare = chisquare
432    output.dof = y_all.shape[-1] - n_parms + len(loc_priors)
433    output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof)
434    if output.dof > 0:
435        output.chisquare_by_dof = output.chisquare / output.dof
436    else:
437        output.chisquare_by_dof = float('nan')
438
439    output.message = fit_result.message
440    if not silent:
441        print(fit_result.message)
442        print('chisquare/d.o.f.:', output.chisquare_by_dof)
443        print('fit parameters', fit_result.x)
444
445    def prepare_hat_matrix():
446        hat_vector = []
447        for key in key_ls:
448            if (len(xd[key]) != 0):
449                hat_vector.append(jacobian(funcd[key])(fit_result.x, xd[key]))
450        hat_vector = [item for sublist in hat_vector for item in sublist]
451        return hat_vector
452
453    if kwargs.get('expected_chisquare') is True:
454        if kwargs.get('correlated_fit') is not True:
455            W = np.diag(1 / np.asarray(dy_f))
456            cov = covariance(y_all)
457            hat_vector = prepare_hat_matrix()
458            A = W @ hat_vector
459            P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
460            expected_chisquare = np.trace((np.identity(y_all.shape[-1]) - P_phi) @ W @ cov @ W)
461            output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare
462            if not silent:
463                print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare)
464
465    fitp = fit_result.x
466
467    try:
468        hess = hessian(chisqfunc)(fitp)
469    except TypeError:
470        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
471
472    len_y = len(y_f)
473
474    def chisqfunc_compact(d):
475        return anp.sum(general_chisqfunc(d[:n_parms], d[n_parms: n_parms + len_y], d[n_parms + len_y:]) ** 2)
476
477    jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f, p_f)))
478
479    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
480    try:
481        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:])
482    except np.linalg.LinAlgError:
483        raise Exception("Cannot invert hessian matrix.")
484
485    result = []
486    for i in range(n_parms):
487        result.append(derived_observable(lambda x_all, **kwargs: (x_all[0] + np.finfo(np.float64).eps) / (y_all[0].value + np.finfo(np.float64).eps) * fitp[i], list(y_all) + loc_priors, man_grad=list(deriv_y[i])))
488
489    output.fit_parameters = result
490
491    # Hotelling t-squared p-value for correlated fits.
492    if kwargs.get('correlated_fit') is True:
493        n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all))
494        output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare,
495                                                  output.dof, n_cov - output.dof)
496
497    if kwargs.get('resplot') is True:
498        for key in key_ls:
499            residual_plot(xd[key], yd[key], funcd[key], result, title=key)
500
501    if kwargs.get('qqplot') is True:
502        for key in key_ls:
503            qqplot(xd[key], yd[key], funcd[key], result, title=key)
504
505    return output
506
507
508def total_least_squares(x, y, func, silent=False, **kwargs):
509    r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
510
511    Parameters
512    ----------
513    x : list
514        list of Obs, or a tuple of lists of Obs
515    y : list
516        list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
517    func : object
518        func has to be of the form
519
520        ```python
521        import autograd.numpy as anp
522
523        def func(a, x):
524            return a[0] + a[1] * x + a[2] * anp.sinh(x)
525        ```
526
527        For multiple x values func can be of the form
528
529        ```python
530        def func(a, x):
531            (x1, x2) = x
532            return a[0] * x1 ** 2 + a[1] * x2
533        ```
534
535        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
536        will not work.
537    silent : bool, optional
538        If true all output to the console is omitted (default False).
539    initial_guess : list
540        can provide an initial guess for the input parameters. Relevant for non-linear
541        fits with many parameters.
542    expected_chisquare : bool
543        If true prints the expected chisquare which is
544        corrected by effects caused by correlated input data.
545        This can take a while as the full correlation matrix
546        has to be calculated (default False).
547    num_grad : bool
548        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
549
550    Notes
551    -----
552    Based on the orthogonal distance regression module of scipy.
553
554    Returns
555    -------
556    output : Fit_result
557        Parameters and information on the fitted result.
558    '''
559
560    output = Fit_result()
561
562    output.fit_function = func
563
564    x = np.array(x)
565
566    x_shape = x.shape
567
568    if kwargs.get('num_grad') is True:
569        jacobian = num_jacobian
570        hessian = num_hessian
571    else:
572        jacobian = auto_jacobian
573        hessian = auto_hessian
574
575    if not callable(func):
576        raise TypeError('func has to be a function.')
577
578    for i in range(42):
579        try:
580            func(np.arange(i), x.T[0])
581        except TypeError:
582            continue
583        except IndexError:
584            continue
585        else:
586            break
587    else:
588        raise RuntimeError("Fit function is not valid.")
589
590    n_parms = i
591    if not silent:
592        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
593
594    x_f = np.vectorize(lambda o: o.value)(x)
595    dx_f = np.vectorize(lambda o: o.dvalue)(x)
596    y_f = np.array([o.value for o in y])
597    dy_f = np.array([o.dvalue for o in y])
598
599    if np.any(np.asarray(dx_f) <= 0.0):
600        raise Exception('No x errors available, run the gamma method first.')
601
602    if np.any(np.asarray(dy_f) <= 0.0):
603        raise Exception('No y errors available, run the gamma method first.')
604
605    if 'initial_guess' in kwargs:
606        x0 = kwargs.get('initial_guess')
607        if len(x0) != n_parms:
608            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
609    else:
610        x0 = [1] * n_parms
611
612    data = RealData(x_f, y_f, sx=dx_f, sy=dy_f)
613    model = Model(func)
614    odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps)
615    odr.set_job(fit_type=0, deriv=1)
616    out = odr.run()
617
618    output.residual_variance = out.res_var
619
620    output.method = 'ODR'
621
622    output.message = out.stopreason
623
624    output.xplus = out.xplus
625
626    if not silent:
627        print('Method: ODR')
628        print(*out.stopreason)
629        print('Residual variance:', output.residual_variance)
630
631    if out.info > 3:
632        raise Exception('The minimization procedure did not converge.')
633
634    m = x_f.size
635
636    def odr_chisquare(p):
637        model = func(p[:n_parms], p[n_parms:].reshape(x_shape))
638        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2)
639        return chisq
640
641    if kwargs.get('expected_chisquare') is True:
642        W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel()))))
643
644        if kwargs.get('covariance') is not None:
645            cov = kwargs.get('covariance')
646        else:
647            cov = covariance(np.concatenate((y, x.ravel())))
648
649        number_of_x_parameters = int(m / x_f.shape[-1])
650
651        old_jac = jacobian(func)(out.beta, out.xplus)
652        fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0)))
653        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])))
654        new_jac = np.concatenate((fused_row1, fused_row2), axis=1)
655
656        A = W @ new_jac
657        P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
658        expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W)
659        if expected_chisquare <= 0.0:
660            warnings.warn("Negative expected_chisquare.", RuntimeWarning)
661            expected_chisquare = np.abs(expected_chisquare)
662        output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare
663        if not silent:
664            print('chisquare/expected_chisquare:',
665                  output.chisquare_by_expected_chisquare)
666
667    fitp = out.beta
668    try:
669        hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel())))
670    except TypeError:
671        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
672
673    def odr_chisquare_compact_x(d):
674        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
675        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)
676        return chisq
677
678    jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel())))
679
680    # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv
681    try:
682        deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:])
683    except np.linalg.LinAlgError:
684        raise Exception("Cannot invert hessian matrix.")
685
686    def odr_chisquare_compact_y(d):
687        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
688        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)
689        return chisq
690
691    jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f)))
692
693    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
694    try:
695        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:])
696    except np.linalg.LinAlgError:
697        raise Exception("Cannot invert hessian matrix.")
698
699    result = []
700    for i in range(n_parms):
701        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])))
702
703    output.fit_parameters = result
704
705    output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel())))
706    output.dof = x.shape[-1] - n_parms
707    output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof)
708
709    return output
710
711
712def fit_lin(x, y, **kwargs):
713    """Performs a linear fit to y = n + m * x and returns two Obs n, m.
714
715    Parameters
716    ----------
717    x : list
718        Can either be a list of floats in which case no xerror is assumed, or
719        a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
720    y : list
721        List of Obs, the dvalues of the Obs are used as yerror for the fit.
722
723    Returns
724    -------
725    fit_parameters : list[Obs]
726        LIist of fitted observables.
727    """
728
729    def f(a, x):
730        y = a[0] + a[1] * x
731        return y
732
733    if all(isinstance(n, Obs) for n in x):
734        out = total_least_squares(x, y, f, **kwargs)
735        return out.fit_parameters
736    elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
737        out = least_squares(x, y, f, **kwargs)
738        return out.fit_parameters
739    else:
740        raise TypeError('Unsupported types for x')
741
742
743def qqplot(x, o_y, func, p, title=""):
744    """Generates a quantile-quantile plot of the fit result which can be used to
745       check if the residuals of the fit are gaussian distributed.
746
747    Returns
748    -------
749    None
750    """
751
752    residuals = []
753    for i_x, i_y in zip(x, o_y):
754        residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
755    residuals = sorted(residuals)
756    my_y = [o.value for o in residuals]
757    probplot = scipy.stats.probplot(my_y)
758    my_x = probplot[0][0]
759    plt.figure(figsize=(8, 8 / 1.618))
760    plt.errorbar(my_x, my_y, fmt='o')
761    fit_start = my_x[0]
762    fit_stop = my_x[-1]
763    samples = np.arange(fit_start, fit_stop, 0.01)
764    plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
765    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='-')
766
767    plt.xlabel('Theoretical quantiles')
768    plt.ylabel('Ordered Values')
769    plt.legend(title=title)
770    plt.draw()
771
772
773def residual_plot(x, y, func, fit_res, title=""):
774    """Generates a plot which compares the fit to the data and displays the corresponding residuals
775
776    For uncorrelated data the residuals are expected to be distributed ~N(0,1).
777
778    Returns
779    -------
780    None
781    """
782    sorted_x = sorted(x)
783    xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
784    xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
785    x_samples = np.arange(xstart, xstop + 0.01, 0.01)
786
787    plt.figure(figsize=(8, 8 / 1.618))
788    gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
789    ax0 = plt.subplot(gs[0])
790    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')
791    ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
792    ax0.set_xticklabels([])
793    ax0.set_xlim([xstart, xstop])
794    ax0.set_xticklabels([])
795    ax0.legend(title=title)
796
797    residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], np.asarray(x))) / np.asarray([o.dvalue for o in y])
798    ax1 = plt.subplot(gs[1])
799    ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
800    ax1.tick_params(direction='out')
801    ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
802    ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
803    ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
804    ax1.set_xlim([xstart, xstop])
805    ax1.set_ylabel('Residuals')
806    plt.subplots_adjust(wspace=None, hspace=None)
807    plt.draw()
808
809
810def error_band(x, func, beta):
811    """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
812
813    Returns
814    -------
815    err : np.array(Obs)
816        Error band for an array of sample values x
817    """
818    cov = covariance(beta)
819    if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
820        warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
821
822    deriv = []
823    for i, item in enumerate(x):
824        deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
825
826    err = []
827    for i, item in enumerate(x):
828        err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
829    err = np.array(err)
830
831    return err
832
833
834def ks_test(objects=None):
835    """Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
836
837    Parameters
838    ----------
839    objects : list
840        List of fit results to include in the analysis (optional).
841
842    Returns
843    -------
844    None
845    """
846
847    if objects is None:
848        obs_list = []
849        for obj in gc.get_objects():
850            if isinstance(obj, Fit_result):
851                obs_list.append(obj)
852    else:
853        obs_list = objects
854
855    p_values = [o.p_value for o in obs_list]
856
857    bins = len(p_values)
858    x = np.arange(0, 1.001, 0.001)
859    plt.plot(x, x, 'k', zorder=1)
860    plt.xlim(0, 1)
861    plt.ylim(0, 1)
862    plt.xlabel('p-value')
863    plt.ylabel('Cumulative probability')
864    plt.title(str(bins) + ' p-values')
865
866    n = np.arange(1, bins + 1) / np.float64(bins)
867    Xs = np.sort(p_values)
868    plt.step(Xs, n)
869    diffs = n - Xs
870    loc_max_diff = np.argmax(np.abs(diffs))
871    loc = Xs[loc_max_diff]
872    plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
873    plt.draw()
874
875    print(scipy.stats.kstest(p_values, 'uniform'))
876
877
878def _extract_val_and_dval(string):
879    split_string = string.split('(')
880    if '.' in split_string[0] and '.' not in split_string[1][:-1]:
881        factor = 10 ** -len(split_string[0].partition('.')[2])
882    else:
883        factor = 1
884    return float(split_string[0]), float(split_string[1][:-1]) * factor
885
886
887def _construct_prior_obs(i_prior, i_n):
888    if isinstance(i_prior, Obs):
889        return i_prior
890    elif isinstance(i_prior, str):
891        loc_val, loc_dval = _extract_val_and_dval(i_prior)
892        return cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}")
893    else:
894        raise TypeError("Prior entries need to be 'Obs' or 'str'.")
class Fit_result(collections.abc.Sequence):
21class Fit_result(Sequence):
22    """Represents fit results.
23
24    Attributes
25    ----------
26    fit_parameters : list
27        results for the individual fit parameters,
28        also accessible via indices.
29    chisquare_by_dof : float
30        reduced chisquare.
31    p_value : float
32        p-value of the fit
33    t2_p_value : float
34        Hotelling t-squared p-value for correlated fits.
35    """
36
37    def __init__(self):
38        self.fit_parameters = None
39
40    def __getitem__(self, idx):
41        return self.fit_parameters[idx]
42
43    def __len__(self):
44        return len(self.fit_parameters)
45
46    def gamma_method(self, **kwargs):
47        """Apply the gamma method to all fit parameters"""
48        [o.gamma_method(**kwargs) for o in self.fit_parameters]
49
50    gm = gamma_method
51
52    def __str__(self):
53        my_str = 'Goodness of fit:\n'
54        if hasattr(self, 'chisquare_by_dof'):
55            my_str += '\u03C7\u00b2/d.o.f. = ' + f'{self.chisquare_by_dof:2.6f}' + '\n'
56        elif hasattr(self, 'residual_variance'):
57            my_str += 'residual variance = ' + f'{self.residual_variance:2.6f}' + '\n'
58        if hasattr(self, 'chisquare_by_expected_chisquare'):
59            my_str += '\u03C7\u00b2/\u03C7\u00b2exp  = ' + f'{self.chisquare_by_expected_chisquare:2.6f}' + '\n'
60        if hasattr(self, 'p_value'):
61            my_str += 'p-value   = ' + f'{self.p_value:2.4f}' + '\n'
62        if hasattr(self, 't2_p_value'):
63            my_str += 't\u00B2p-value = ' + f'{self.t2_p_value:2.4f}' + '\n'
64        my_str += 'Fit parameters:\n'
65        for i_par, par in enumerate(self.fit_parameters):
66            my_str += str(i_par) + '\t' + ' ' * int(par >= 0) + str(par).rjust(int(par < 0.0)) + '\n'
67        return my_str
68
69    def __repr__(self):
70        m = max(map(len, list(self.__dict__.keys()))) + 1
71        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.
  • chisquare_by_dof (float): reduced chisquare.
  • p_value (float): p-value of the fit
  • t2_p_value (float): Hotelling t-squared p-value for correlated fits.
fit_parameters
def gamma_method(self, **kwargs):
46    def gamma_method(self, **kwargs):
47        """Apply the gamma method to all fit parameters"""
48        [o.gamma_method(**kwargs) for o in self.fit_parameters]

Apply the gamma method to all fit parameters

def gm(self, **kwargs):
46    def gamma_method(self, **kwargs):
47        """Apply the gamma method to all fit parameters"""
48        [o.gamma_method(**kwargs) for o in self.fit_parameters]

Apply the gamma method to all fit parameters

def least_squares(x, y, func, priors=None, silent=False, **kwargs):
 74def least_squares(x, y, func, priors=None, silent=False, **kwargs):
 75    r'''Performs a non-linear fit to y = func(x).
 76        ```
 77
 78    Parameters
 79    ----------
 80    For an uncombined fit:
 81
 82    x : list
 83        list of floats.
 84    y : list
 85        list of Obs.
 86    func : object
 87        fit function, has to be of the form
 88
 89        ```python
 90        import autograd.numpy as anp
 91
 92        def func(a, x):
 93            return a[0] + a[1] * x + a[2] * anp.sinh(x)
 94        ```
 95
 96        For multiple x values func can be of the form
 97
 98        ```python
 99        def func(a, x):
100            (x1, x2) = x
101            return a[0] * x1 ** 2 + a[1] * x2
102        ```
103        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
104        will not work.
105
106    OR For a combined fit:
107
108    x : dict
109        dict of lists.
110    y : dict
111        dict of lists of Obs.
112    funcs : dict
113        dict of objects
114        fit functions have to be of the form (here a[0] is the common fit parameter)
115        ```python
116        import autograd.numpy as anp
117        funcs = {"a": func_a,
118                "b": func_b}
119
120        def func_a(a, x):
121            return a[1] * anp.exp(-a[0] * x)
122
123        def func_b(a, x):
124            return a[2] * anp.exp(-a[0] * x)
125
126        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
127        will not work.
128
129    priors : dict or list, optional
130        priors can either be a dictionary with integer keys and the corresponding priors as values or
131        a list with an entry for every parameter in the fit. The entries can either be
132        Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like
133        0.548(23), 500(40) or 0.5(0.4)
134    silent : bool, optional
135        If true all output to the console is omitted (default False).
136    initial_guess : list
137        can provide an initial guess for the input parameters. Relevant for
138        non-linear fits with many parameters. In case of correlated fits the guess is used to perform
139        an uncorrelated fit which then serves as guess for the correlated fit.
140    method : str, optional
141        can be used to choose an alternative method for the minimization of chisquare.
142        The possible methods are the ones which can be used for scipy.optimize.minimize and
143        migrad of iminuit. If no method is specified, Levenberg-Marquard is used.
144        Reliable alternatives are migrad, Powell and Nelder-Mead.
145    tol: float, optional
146        can be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence
147        to a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly
148        invalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values
149        The stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
150    correlated_fit : bool
151        If True, use the full inverse covariance matrix in the definition of the chisquare cost function.
152        For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`.
153        In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).
154        This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
155    inv_chol_cov_matrix [array,list], optional
156        array: shape = (no of y values) X (no of y values)
157        list:   for an uncombined fit: [""]
158                for a combined fit: list of keys belonging to the corr_matrix saved in the array, must be the same as the keys of the y dict in alphabetical order
159        If correlated_fit=True is set as well, can provide an inverse covariance matrix (y errors, dy_f included!) of your own choosing for a correlated fit.
160        The matrix must be a lower triangular matrix constructed from a Cholesky decomposition: The function invert_corr_cov_cholesky(corr, inverrdiag) can be
161        used to construct it from a correlation matrix (corr) and the errors dy_f of the data points (inverrdiag = np.diag(1 / np.asarray(dy_f))). For the correct
162        ordering the correlation matrix (corr) can be sorted via the function sort_corr(corr, kl, yd) where kl is the list of keys and yd the y dict.
163    expected_chisquare : bool
164        If True estimates the expected chisquare which is
165        corrected by effects caused by correlated input data (default False).
166    resplot : bool
167        If True, a plot which displays fit, data and residuals is generated (default False).
168    qqplot : bool
169        If True, a quantile-quantile plot of the fit result is generated (default False).
170    num_grad : bool
171        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
172
173    Returns
174    -------
175    output : Fit_result
176        Parameters and information on the fitted result.
177    Examples
178    ------
179    >>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set
180    >>> import numpy as np
181    >>> from scipy.stats import norm
182    >>> from scipy.linalg import cholesky
183    >>> import pyerrors as pe
184    >>> # generating the random data set
185    >>> num_samples = 400
186    >>> N = 3
187    >>> x = np.arange(N)
188    >>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers
189    >>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers
190    >>> r = r1 = r2 = np.zeros((N, N))
191    >>> y = {}
192    >>> for i in range(N):
193    >>>    for j in range(N):
194    >>>        r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix
195    >>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors
196    >>> for i in range(N):
197    >>>    for j in range(N):
198    >>>        r[i, j] *= errl[i] * errl[j] # element in covariance matrix
199    >>> c = cholesky(r, lower=True)
200    >>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined
201    >>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built
202    >>> x_dict = {}
203    >>> y_dict = {}
204    >>> chol_inv_dict = {}
205    >>> data = []
206    >>> for key in y.keys():
207    >>>    x_dict[key] = x
208    >>>    for i in range(N):
209    >>>        data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data
210    >>>    [o.gamma_method() for o in data]
211    >>>    corr = pe.covariance(data, correlation=True)
212    >>>    inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data]))
213    >>>    chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below
214    >>> y_dict = {'a': data[:3], 'b': data[3:]}
215    >>> # common fit parameter p[0] in combined fit
216    >>> def fit1(p, x):
217    >>>    return p[0] + p[1] * x
218    >>> def fit2(p, x):
219    >>>    return p[0] + p[2] * x
220    >>> fitf_dict = {'a': fit1, 'b':fit2}
221    >>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']])
222    Fit with 3 parameters
223    Method: Levenberg-Marquardt
224    `ftol` termination condition is satisfied.
225    chisquare/d.o.f.: 0.5388013574561786 # random
226    fit parameters [1.11897846 0.96361162 0.92325319] # random
227
228    '''
229    output = Fit_result()
230
231    if (isinstance(x, dict) and isinstance(y, dict) and isinstance(func, dict)):
232        xd = {key: anp.asarray(x[key]) for key in x}
233        yd = y
234        funcd = func
235        output.fit_function = func
236    elif (isinstance(x, dict) or isinstance(y, dict) or isinstance(func, dict)):
237        raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.")
238    else:
239        x = np.asarray(x)
240        xd = {"": x}
241        yd = {"": y}
242        funcd = {"": func}
243        output.fit_function = func
244
245    if kwargs.get('num_grad') is True:
246        jacobian = num_jacobian
247        hessian = num_hessian
248    else:
249        jacobian = auto_jacobian
250        hessian = auto_hessian
251
252    key_ls = sorted(list(xd.keys()))
253
254    if sorted(list(yd.keys())) != key_ls:
255        raise ValueError('x and y dictionaries do not contain the same keys.')
256
257    if sorted(list(funcd.keys())) != key_ls:
258        raise ValueError('x and func dictionaries do not contain the same keys.')
259
260    x_all = np.concatenate([np.array(xd[key]).transpose() for key in key_ls]).transpose()
261    y_all = np.concatenate([np.array(yd[key]) for key in key_ls])
262
263    y_f = [o.value for o in y_all]
264    dy_f = [o.dvalue for o in y_all]
265
266    if len(x_all.shape) > 2:
267        raise ValueError("Unknown format for x values")
268
269    if np.any(np.asarray(dy_f) <= 0.0):
270        raise Exception("No y errors available, run the gamma method first.")
271
272    # number of fit parameters
273    n_parms_ls = []
274    for key in key_ls:
275        if not callable(funcd[key]):
276            raise TypeError('func (key=' + key + ') is not a function.')
277        if np.asarray(xd[key]).shape[-1] != len(yd[key]):
278            raise ValueError('x and y input (key=' + key + ') do not have the same length')
279        for n_loc in range(100):
280            try:
281                funcd[key](np.arange(n_loc), x_all.T[0])
282            except TypeError:
283                continue
284            except IndexError:
285                continue
286            else:
287                break
288        else:
289            raise RuntimeError("Fit function (key=" + key + ") is not valid.")
290        n_parms_ls.append(n_loc)
291
292    n_parms = max(n_parms_ls)
293
294    if len(key_ls) > 1:
295        for key in key_ls:
296            if np.asarray(yd[key]).shape != funcd[key](np.arange(n_parms), xd[key]).shape:
297                raise ValueError(f"Fit function {key} returns the wrong shape ({funcd[key](np.arange(n_parms), xd[key]).shape} instead of {np.asarray(yd[key]).shape})\nIf the fit function is just a constant you could try adding x*0 to get the correct shape.")
298
299    if not silent:
300        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
301
302    if priors is not None:
303        if isinstance(priors, (list, np.ndarray)):
304            if n_parms != len(priors):
305                raise ValueError("'priors' does not have the correct length.")
306
307            loc_priors = []
308            for i_n, i_prior in enumerate(priors):
309                loc_priors.append(_construct_prior_obs(i_prior, i_n))
310
311            prior_mask = np.arange(len(priors))
312            output.priors = loc_priors
313
314        elif isinstance(priors, dict):
315            loc_priors = []
316            prior_mask = []
317            output.priors = {}
318            for pos, prior in priors.items():
319                if isinstance(pos, int):
320                    prior_mask.append(pos)
321                else:
322                    raise TypeError("Prior position needs to be an integer.")
323                loc_priors.append(_construct_prior_obs(prior, pos))
324
325                output.priors[pos] = loc_priors[-1]
326            if max(prior_mask) >= n_parms:
327                raise ValueError("Prior position out of range.")
328        else:
329            raise TypeError("Unkown type for `priors`.")
330
331        p_f = [o.value for o in loc_priors]
332        dp_f = [o.dvalue for o in loc_priors]
333        if np.any(np.asarray(dp_f) <= 0.0):
334            raise Exception("No prior errors available, run the gamma method first.")
335    else:
336        p_f = dp_f = np.array([])
337        prior_mask = []
338        loc_priors = []
339
340    if 'initial_guess' in kwargs:
341        x0 = kwargs.get('initial_guess')
342        if len(x0) != n_parms:
343            raise ValueError('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
344    else:
345        x0 = [0.1] * n_parms
346
347    if priors is None:
348        def general_chisqfunc_uncorr(p, ivars, pr):
349            model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls])
350            return (ivars - model) / dy_f
351    else:
352        def general_chisqfunc_uncorr(p, ivars, pr):
353            model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls])
354            return anp.concatenate(((ivars - model) / dy_f, (p[prior_mask] - pr) / dp_f))
355
356    def chisqfunc_uncorr(p):
357        return anp.sum(general_chisqfunc_uncorr(p, y_f, p_f) ** 2)
358
359    if kwargs.get('correlated_fit') is True:
360        if 'inv_chol_cov_matrix' in kwargs:
361            chol_inv = kwargs.get('inv_chol_cov_matrix')
362            if (chol_inv[0].shape[0] != len(dy_f)):
363                raise TypeError('The number of columns of the inverse covariance matrix handed over needs to be equal to the number of y errors.')
364            if (chol_inv[0].shape[0] != chol_inv[0].shape[1]):
365                raise TypeError('The inverse covariance matrix handed over needs to have the same number of rows as columns.')
366            if (chol_inv[1] != key_ls):
367                raise ValueError('The keys of inverse covariance matrix are not the same or do not appear in the same order as the x and y values.')
368            chol_inv = chol_inv[0]
369            if np.any(np.diag(chol_inv) <= 0) or (not np.all(chol_inv == np.tril(chol_inv))):
370                raise ValueError('The inverse covariance matrix inv_chol_cov_matrix[0] has to be a lower triangular matrix constructed from a Cholesky decomposition.')
371        else:
372            corr = covariance(y_all, correlation=True, **kwargs)
373            inverrdiag = np.diag(1 / np.asarray(dy_f))
374            chol_inv = invert_corr_cov_cholesky(corr, inverrdiag)
375
376        def general_chisqfunc(p, ivars, pr):
377            model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls])
378            return anp.concatenate((anp.dot(chol_inv, (ivars - model)), (p[prior_mask] - pr) / dp_f))
379
380        def chisqfunc(p):
381            return anp.sum(general_chisqfunc(p, y_f, p_f) ** 2)
382    else:
383        general_chisqfunc = general_chisqfunc_uncorr
384        chisqfunc = chisqfunc_uncorr
385
386    output.method = kwargs.get('method', 'Levenberg-Marquardt')
387    if not silent:
388        print('Method:', output.method)
389
390    if output.method != 'Levenberg-Marquardt':
391        if output.method == 'migrad':
392            tolerance = 1e-4  # default value of 1e-1 set by iminuit can be problematic
393            if 'tol' in kwargs:
394                tolerance = kwargs.get('tol')
395            fit_result = iminuit.minimize(chisqfunc_uncorr, x0, tol=tolerance)  # Stopping criterion 0.002 * tol * errordef
396            if kwargs.get('correlated_fit') is True:
397                fit_result = iminuit.minimize(chisqfunc, fit_result.x, tol=tolerance)
398            output.iterations = fit_result.nfev
399        else:
400            tolerance = 1e-12
401            if 'tol' in kwargs:
402                tolerance = kwargs.get('tol')
403            fit_result = scipy.optimize.minimize(chisqfunc_uncorr, x0, method=kwargs.get('method'), tol=tolerance)
404            if kwargs.get('correlated_fit') is True:
405                fit_result = scipy.optimize.minimize(chisqfunc, fit_result.x, method=kwargs.get('method'), tol=tolerance)
406            output.iterations = fit_result.nit
407
408        chisquare = fit_result.fun
409
410    else:
411        if 'tol' in kwargs:
412            print('tol cannot be set for Levenberg-Marquardt')
413
414        def chisqfunc_residuals_uncorr(p):
415            return general_chisqfunc_uncorr(p, y_f, p_f)
416
417        fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
418        if kwargs.get('correlated_fit') is True:
419            def chisqfunc_residuals(p):
420                return general_chisqfunc(p, y_f, p_f)
421
422            fit_result = scipy.optimize.least_squares(chisqfunc_residuals, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
423
424        chisquare = np.sum(fit_result.fun ** 2)
425        assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14)
426
427        output.iterations = fit_result.nfev
428
429    if not fit_result.success:
430        raise Exception('The minimization procedure did not converge.')
431
432    output.chisquare = chisquare
433    output.dof = y_all.shape[-1] - n_parms + len(loc_priors)
434    output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof)
435    if output.dof > 0:
436        output.chisquare_by_dof = output.chisquare / output.dof
437    else:
438        output.chisquare_by_dof = float('nan')
439
440    output.message = fit_result.message
441    if not silent:
442        print(fit_result.message)
443        print('chisquare/d.o.f.:', output.chisquare_by_dof)
444        print('fit parameters', fit_result.x)
445
446    def prepare_hat_matrix():
447        hat_vector = []
448        for key in key_ls:
449            if (len(xd[key]) != 0):
450                hat_vector.append(jacobian(funcd[key])(fit_result.x, xd[key]))
451        hat_vector = [item for sublist in hat_vector for item in sublist]
452        return hat_vector
453
454    if kwargs.get('expected_chisquare') is True:
455        if kwargs.get('correlated_fit') is not True:
456            W = np.diag(1 / np.asarray(dy_f))
457            cov = covariance(y_all)
458            hat_vector = prepare_hat_matrix()
459            A = W @ hat_vector
460            P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
461            expected_chisquare = np.trace((np.identity(y_all.shape[-1]) - P_phi) @ W @ cov @ W)
462            output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare
463            if not silent:
464                print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare)
465
466    fitp = fit_result.x
467
468    try:
469        hess = hessian(chisqfunc)(fitp)
470    except TypeError:
471        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
472
473    len_y = len(y_f)
474
475    def chisqfunc_compact(d):
476        return anp.sum(general_chisqfunc(d[:n_parms], d[n_parms: n_parms + len_y], d[n_parms + len_y:]) ** 2)
477
478    jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f, p_f)))
479
480    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
481    try:
482        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:])
483    except np.linalg.LinAlgError:
484        raise Exception("Cannot invert hessian matrix.")
485
486    result = []
487    for i in range(n_parms):
488        result.append(derived_observable(lambda x_all, **kwargs: (x_all[0] + np.finfo(np.float64).eps) / (y_all[0].value + np.finfo(np.float64).eps) * fitp[i], list(y_all) + loc_priors, man_grad=list(deriv_y[i])))
489
490    output.fit_parameters = result
491
492    # Hotelling t-squared p-value for correlated fits.
493    if kwargs.get('correlated_fit') is True:
494        n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all))
495        output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare,
496                                                  output.dof, n_cov - output.dof)
497
498    if kwargs.get('resplot') is True:
499        for key in key_ls:
500            residual_plot(xd[key], yd[key], funcd[key], result, title=key)
501
502    if kwargs.get('qqplot') is True:
503        for key in key_ls:
504            qqplot(xd[key], yd[key], funcd[key], result, title=key)
505
506    return output

Performs a non-linear fit to y = func(x). ```

Parameters
  • For an uncombined fit:
  • 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.

  • OR For a combined fit:
  • x (dict): dict of lists.
  • y (dict): dict of lists of Obs.
  • funcs (dict): dict of objects fit functions have to be of the form (here a[0] is the common fit parameter) ```python import autograd.numpy as anp funcs = {"a": func_a, "b": func_b}

    def func_a(a, x): return a[1] * anp.exp(-a[0] * x)

    def func_b(a, x): return a[2] * anp.exp(-a[0] * x)

    It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation will not work.

  • priors (dict or list, optional): priors can either be a dictionary with integer keys and the corresponding priors as values or 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. In case of correlated fits the guess is used to perform an uncorrelated fit which then serves as guess for the correlated fit.
  • 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.
  • tol (float, optional): can be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence to a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly invalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values The stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
  • 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).
  • inv_chol_cov_matrix [array,list], optional: array: shape = (no of y values) X (no of y values) list: for an uncombined fit: [""] for a combined fit: list of keys belonging to the corr_matrix saved in the array, must be the same as the keys of the y dict in alphabetical order If correlated_fit=True is set as well, can provide an inverse covariance matrix (y errors, dy_f included!) of your own choosing for a correlated fit. The matrix must be a lower triangular matrix constructed from a Cholesky decomposition: The function invert_corr_cov_cholesky(corr, inverrdiag) can be used to construct it from a correlation matrix (corr) and the errors dy_f of the data points (inverrdiag = np.diag(1 / np.asarray(dy_f))). For the correct ordering the correlation matrix (corr) can be sorted via the function sort_corr(corr, kl, yd) where kl is the list of keys and yd the y dict.
  • 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).
  • num_grad (bool): Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
Returns
  • output (Fit_result): Parameters and information on the fitted result.
Examples
>>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set
>>> import numpy as np
>>> from scipy.stats import norm
>>> from scipy.linalg import cholesky
>>> import pyerrors as pe
>>> # generating the random data set
>>> num_samples = 400
>>> N = 3
>>> x = np.arange(N)
>>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers
>>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers
>>> r = r1 = r2 = np.zeros((N, N))
>>> y = {}
>>> for i in range(N):
>>>    for j in range(N):
>>>        r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix
>>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors
>>> for i in range(N):
>>>    for j in range(N):
>>>        r[i, j] *= errl[i] * errl[j] # element in covariance matrix
>>> c = cholesky(r, lower=True)
>>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined
>>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built
>>> x_dict = {}
>>> y_dict = {}
>>> chol_inv_dict = {}
>>> data = []
>>> for key in y.keys():
>>>    x_dict[key] = x
>>>    for i in range(N):
>>>        data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data
>>>    [o.gamma_method() for o in data]
>>>    corr = pe.covariance(data, correlation=True)
>>>    inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data]))
>>>    chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below
>>> y_dict = {'a': data[:3], 'b': data[3:]}
>>> # common fit parameter p[0] in combined fit
>>> def fit1(p, x):
>>>    return p[0] + p[1] * x
>>> def fit2(p, x):
>>>    return p[0] + p[2] * x
>>> fitf_dict = {'a': fit1, 'b':fit2}
>>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']])
Fit with 3 parameters
Method: Levenberg-Marquardt
`ftol` termination condition is satisfied.
chisquare/d.o.f.: 0.5388013574561786 # random
fit parameters [1.11897846 0.96361162 0.92325319] # random
def total_least_squares(x, y, func, silent=False, **kwargs):
509def total_least_squares(x, y, func, silent=False, **kwargs):
510    r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
511
512    Parameters
513    ----------
514    x : list
515        list of Obs, or a tuple of lists of Obs
516    y : list
517        list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
518    func : object
519        func has to be of the form
520
521        ```python
522        import autograd.numpy as anp
523
524        def func(a, x):
525            return a[0] + a[1] * x + a[2] * anp.sinh(x)
526        ```
527
528        For multiple x values func can be of the form
529
530        ```python
531        def func(a, x):
532            (x1, x2) = x
533            return a[0] * x1 ** 2 + a[1] * x2
534        ```
535
536        It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
537        will not work.
538    silent : bool, optional
539        If true all output to the console is omitted (default False).
540    initial_guess : list
541        can provide an initial guess for the input parameters. Relevant for non-linear
542        fits with many parameters.
543    expected_chisquare : bool
544        If true prints the expected chisquare which is
545        corrected by effects caused by correlated input data.
546        This can take a while as the full correlation matrix
547        has to be calculated (default False).
548    num_grad : bool
549        Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
550
551    Notes
552    -----
553    Based on the orthogonal distance regression module of scipy.
554
555    Returns
556    -------
557    output : Fit_result
558        Parameters and information on the fitted result.
559    '''
560
561    output = Fit_result()
562
563    output.fit_function = func
564
565    x = np.array(x)
566
567    x_shape = x.shape
568
569    if kwargs.get('num_grad') is True:
570        jacobian = num_jacobian
571        hessian = num_hessian
572    else:
573        jacobian = auto_jacobian
574        hessian = auto_hessian
575
576    if not callable(func):
577        raise TypeError('func has to be a function.')
578
579    for i in range(42):
580        try:
581            func(np.arange(i), x.T[0])
582        except TypeError:
583            continue
584        except IndexError:
585            continue
586        else:
587            break
588    else:
589        raise RuntimeError("Fit function is not valid.")
590
591    n_parms = i
592    if not silent:
593        print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
594
595    x_f = np.vectorize(lambda o: o.value)(x)
596    dx_f = np.vectorize(lambda o: o.dvalue)(x)
597    y_f = np.array([o.value for o in y])
598    dy_f = np.array([o.dvalue for o in y])
599
600    if np.any(np.asarray(dx_f) <= 0.0):
601        raise Exception('No x errors available, run the gamma method first.')
602
603    if np.any(np.asarray(dy_f) <= 0.0):
604        raise Exception('No y errors available, run the gamma method first.')
605
606    if 'initial_guess' in kwargs:
607        x0 = kwargs.get('initial_guess')
608        if len(x0) != n_parms:
609            raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
610    else:
611        x0 = [1] * n_parms
612
613    data = RealData(x_f, y_f, sx=dx_f, sy=dy_f)
614    model = Model(func)
615    odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps)
616    odr.set_job(fit_type=0, deriv=1)
617    out = odr.run()
618
619    output.residual_variance = out.res_var
620
621    output.method = 'ODR'
622
623    output.message = out.stopreason
624
625    output.xplus = out.xplus
626
627    if not silent:
628        print('Method: ODR')
629        print(*out.stopreason)
630        print('Residual variance:', output.residual_variance)
631
632    if out.info > 3:
633        raise Exception('The minimization procedure did not converge.')
634
635    m = x_f.size
636
637    def odr_chisquare(p):
638        model = func(p[:n_parms], p[n_parms:].reshape(x_shape))
639        chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2)
640        return chisq
641
642    if kwargs.get('expected_chisquare') is True:
643        W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel()))))
644
645        if kwargs.get('covariance') is not None:
646            cov = kwargs.get('covariance')
647        else:
648            cov = covariance(np.concatenate((y, x.ravel())))
649
650        number_of_x_parameters = int(m / x_f.shape[-1])
651
652        old_jac = jacobian(func)(out.beta, out.xplus)
653        fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0)))
654        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])))
655        new_jac = np.concatenate((fused_row1, fused_row2), axis=1)
656
657        A = W @ new_jac
658        P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
659        expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W)
660        if expected_chisquare <= 0.0:
661            warnings.warn("Negative expected_chisquare.", RuntimeWarning)
662            expected_chisquare = np.abs(expected_chisquare)
663        output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare
664        if not silent:
665            print('chisquare/expected_chisquare:',
666                  output.chisquare_by_expected_chisquare)
667
668    fitp = out.beta
669    try:
670        hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel())))
671    except TypeError:
672        raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
673
674    def odr_chisquare_compact_x(d):
675        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
676        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)
677        return chisq
678
679    jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel())))
680
681    # Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv
682    try:
683        deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:])
684    except np.linalg.LinAlgError:
685        raise Exception("Cannot invert hessian matrix.")
686
687    def odr_chisquare_compact_y(d):
688        model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
689        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)
690        return chisq
691
692    jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f)))
693
694    # Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
695    try:
696        deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:])
697    except np.linalg.LinAlgError:
698        raise Exception("Cannot invert hessian matrix.")
699
700    result = []
701    for i in range(n_parms):
702        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])))
703
704    output.fit_parameters = result
705
706    output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel())))
707    output.dof = x.shape[-1] - n_parms
708    output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof)
709
710    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).
  • num_grad (bool): Use numerical differentation instead of automatic differentiation to perform the error propagation (default False).
Notes

Based on the orthogonal distance regression module of scipy.

Returns
  • output (Fit_result): Parameters and information on the fitted result.
def fit_lin(x, y, **kwargs):
713def fit_lin(x, y, **kwargs):
714    """Performs a linear fit to y = n + m * x and returns two Obs n, m.
715
716    Parameters
717    ----------
718    x : list
719        Can either be a list of floats in which case no xerror is assumed, or
720        a list of Obs, where the dvalues of the Obs are used as xerror for the fit.
721    y : list
722        List of Obs, the dvalues of the Obs are used as yerror for the fit.
723
724    Returns
725    -------
726    fit_parameters : list[Obs]
727        LIist of fitted observables.
728    """
729
730    def f(a, x):
731        y = a[0] + a[1] * x
732        return y
733
734    if all(isinstance(n, Obs) for n in x):
735        out = total_least_squares(x, y, f, **kwargs)
736        return out.fit_parameters
737    elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
738        out = least_squares(x, y, f, **kwargs)
739        return out.fit_parameters
740    else:
741        raise TypeError('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.
Returns
  • fit_parameters (list[Obs]): LIist of fitted observables.
def qqplot(x, o_y, func, p, title=''):
744def qqplot(x, o_y, func, p, title=""):
745    """Generates a quantile-quantile plot of the fit result which can be used to
746       check if the residuals of the fit are gaussian distributed.
747
748    Returns
749    -------
750    None
751    """
752
753    residuals = []
754    for i_x, i_y in zip(x, o_y):
755        residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
756    residuals = sorted(residuals)
757    my_y = [o.value for o in residuals]
758    probplot = scipy.stats.probplot(my_y)
759    my_x = probplot[0][0]
760    plt.figure(figsize=(8, 8 / 1.618))
761    plt.errorbar(my_x, my_y, fmt='o')
762    fit_start = my_x[0]
763    fit_stop = my_x[-1]
764    samples = np.arange(fit_start, fit_stop, 0.01)
765    plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
766    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='-')
767
768    plt.xlabel('Theoretical quantiles')
769    plt.ylabel('Ordered Values')
770    plt.legend(title=title)
771    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.

Returns
  • None
def residual_plot(x, y, func, fit_res, title=''):
774def residual_plot(x, y, func, fit_res, title=""):
775    """Generates a plot which compares the fit to the data and displays the corresponding residuals
776
777    For uncorrelated data the residuals are expected to be distributed ~N(0,1).
778
779    Returns
780    -------
781    None
782    """
783    sorted_x = sorted(x)
784    xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
785    xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
786    x_samples = np.arange(xstart, xstop + 0.01, 0.01)
787
788    plt.figure(figsize=(8, 8 / 1.618))
789    gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
790    ax0 = plt.subplot(gs[0])
791    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')
792    ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
793    ax0.set_xticklabels([])
794    ax0.set_xlim([xstart, xstop])
795    ax0.set_xticklabels([])
796    ax0.legend(title=title)
797
798    residuals = (np.asarray([o.value for o in y]) - func([o.value for o in fit_res], np.asarray(x))) / np.asarray([o.dvalue for o in y])
799    ax1 = plt.subplot(gs[1])
800    ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
801    ax1.tick_params(direction='out')
802    ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
803    ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
804    ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
805    ax1.set_xlim([xstart, xstop])
806    ax1.set_ylabel('Residuals')
807    plt.subplots_adjust(wspace=None, hspace=None)
808    plt.draw()

Generates a plot which compares the fit to the data and displays the corresponding residuals

For uncorrelated data the residuals are expected to be distributed ~N(0,1).

Returns
  • None
def error_band(x, func, beta):
811def error_band(x, func, beta):
812    """Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.
813
814    Returns
815    -------
816    err : np.array(Obs)
817        Error band for an array of sample values x
818    """
819    cov = covariance(beta)
820    if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
821        warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
822
823    deriv = []
824    for i, item in enumerate(x):
825        deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
826
827    err = []
828    for i, item in enumerate(x):
829        err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
830    err = np.array(err)
831
832    return err

Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.

Returns
  • err (np.array(Obs)): Error band for an array of sample values x
def ks_test(objects=None):
835def ks_test(objects=None):
836    """Performs a Kolmogorov–Smirnov test for the p-values of all fit object.
837
838    Parameters
839    ----------
840    objects : list
841        List of fit results to include in the analysis (optional).
842
843    Returns
844    -------
845    None
846    """
847
848    if objects is None:
849        obs_list = []
850        for obj in gc.get_objects():
851            if isinstance(obj, Fit_result):
852                obs_list.append(obj)
853    else:
854        obs_list = objects
855
856    p_values = [o.p_value for o in obs_list]
857
858    bins = len(p_values)
859    x = np.arange(0, 1.001, 0.001)
860    plt.plot(x, x, 'k', zorder=1)
861    plt.xlim(0, 1)
862    plt.ylim(0, 1)
863    plt.xlabel('p-value')
864    plt.ylabel('Cumulative probability')
865    plt.title(str(bins) + ' p-values')
866
867    n = np.arange(1, bins + 1) / np.float64(bins)
868    Xs = np.sort(p_values)
869    plt.step(Xs, n)
870    diffs = n - Xs
871    loc_max_diff = np.argmax(np.abs(diffs))
872    loc = Xs[loc_max_diff]
873    plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
874    plt.draw()
875
876    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).
Returns
  • None