pyerrors/pyerrors/fits.py

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import gc
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from collections.abc import Sequence
import warnings
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import numpy as np
import autograd.numpy as anp
import scipy.optimize
import scipy.stats
import matplotlib.pyplot as plt
from matplotlib import gridspec
from scipy.odr import ODR, Model, RealData
import iminuit
from autograd import jacobian as auto_jacobian
from autograd import hessian as auto_hessian
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from autograd import elementwise_grad as egrad
from numdifftools import Jacobian as num_jacobian
from numdifftools import Hessian as num_hessian
from .obs import Obs, derived_observable, covariance, cov_Obs
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class Fit_result(Sequence):
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"""Represents fit results.
Attributes
----------
fit_parameters : list
results for the individual fit parameters,
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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.
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"""
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def __init__(self):
self.fit_parameters = None
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def __getitem__(self, idx):
return self.fit_parameters[idx]
def __len__(self):
return len(self.fit_parameters)
def gamma_method(self, **kwargs):
"""Apply the gamma method to all fit parameters"""
[o.gamma_method(**kwargs) for o in self.fit_parameters]
gm = gamma_method
def __str__(self):
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my_str = 'Goodness of fit:\n'
if hasattr(self, 'chisquare_by_dof'):
my_str += '\u03C7\u00b2/d.o.f. = ' + f'{self.chisquare_by_dof:2.6f}' + '\n'
elif hasattr(self, 'residual_variance'):
my_str += 'residual variance = ' + f'{self.residual_variance:2.6f}' + '\n'
if hasattr(self, 'chisquare_by_expected_chisquare'):
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my_str += '\u03C7\u00b2/\u03C7\u00b2exp = ' + f'{self.chisquare_by_expected_chisquare:2.6f}' + '\n'
if hasattr(self, 'p_value'):
my_str += 'p-value = ' + f'{self.p_value:2.4f}' + '\n'
if hasattr(self, 't2_p_value'):
my_str += 't\u00B2p-value = ' + f'{self.t2_p_value:2.4f}' + '\n'
my_str += 'Fit parameters:\n'
for i_par, par in enumerate(self.fit_parameters):
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my_str += str(i_par) + '\t' + ' ' * int(par >= 0) + str(par).rjust(int(par < 0.0)) + '\n'
return my_str
def __repr__(self):
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m = max(map(len, list(self.__dict__.keys()))) + 1
return '\n'.join([key.rjust(m) + ': ' + repr(value) for key, value in sorted(self.__dict__.items())])
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def least_squares(x, y, func, priors=None, silent=False, **kwargs):
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r'''Performs a non-linear fit to y = func(x).
```
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Parameters
----------
For an uncombined fit:
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x : list
list of floats.
y : list
list of Obs.
func : object
fit function, has to be of the form
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```python
import autograd.numpy as anp
def func(a, x):
return a[0] + a[1] * x + a[2] * anp.sinh(x)
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```
For multiple x values func can be of the form
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```python
def func(a, x):
(x1, x2) = x
return a[0] * x1 ** 2 + a[1] * x2
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```
It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
will not work.
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OR For a combined fit:
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x : dict
dict of lists.
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y : dict
dict of lists of Obs.
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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}
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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.
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priors : list, optional
priors has to be a list with an entry for every parameter in the fit. The entries can either be
Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like
0.548(23), 500(40) or 0.5(0.4)
silent : bool, optional
If true all output to the console is omitted (default False).
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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
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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).
At the moment this option only works for `prior==None` and when no `method` is given.
expected_chisquare : bool
If True estimates the expected chisquare which is
corrected by effects caused by correlated input data (default False).
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resplot : bool
If True, a plot which displays fit, data and residuals is generated (default False).
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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.
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'''
if priors is not None:
return _prior_fit(x, y, func, priors, silent=silent, **kwargs)
else:
return _combined_fit(x, y, func, silent=silent, **kwargs)
def total_least_squares(x, y, func, silent=False, **kwargs):
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r'''Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.
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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
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```python
import autograd.numpy as anp
def func(a, x):
return a[0] + a[1] * x + a[2] * anp.sinh(x)
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```
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For multiple x values func can be of the form
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```python
def func(a, x):
(x1, x2) = x
return a[0] * x1 ** 2 + a[1] * x2
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```
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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).
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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).
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Notes
-----
Based on the orthogonal distance regression module of scipy.
Returns
-------
output : Fit_result
Parameters and information on the fitted result.
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'''
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output = Fit_result()
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output.fit_function = func
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x = np.array(x)
x_shape = x.shape
if kwargs.get('num_grad') is True:
jacobian = num_jacobian
hessian = num_hessian
else:
jacobian = auto_jacobian
hessian = auto_hessian
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if not callable(func):
raise TypeError('func has to be a function.')
for i in range(42):
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try:
func(np.arange(i), x.T[0])
except TypeError:
continue
except IndexError:
continue
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else:
break
else:
raise RuntimeError("Fit function is not valid.")
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n_parms = i
if not silent:
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print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
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x_f = np.vectorize(lambda o: o.value)(x)
dx_f = np.vectorize(lambda o: o.dvalue)(x)
y_f = np.array([o.value for o in y])
dy_f = np.array([o.dvalue for o in y])
if np.any(np.asarray(dx_f) <= 0.0):
raise Exception('No x errors available, run the gamma method first.')
if np.any(np.asarray(dy_f) <= 0.0):
raise Exception('No y errors available, run the gamma method first.')
if 'initial_guess' in kwargs:
x0 = kwargs.get('initial_guess')
if len(x0) != n_parms:
raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
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else:
x0 = [1] * n_parms
data = RealData(x_f, y_f, sx=dx_f, sy=dy_f)
model = Model(func)
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odr = ODR(data, model, x0, partol=np.finfo(np.float64).eps)
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odr.set_job(fit_type=0, deriv=1)
out = odr.run()
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output.residual_variance = out.res_var
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output.method = 'ODR'
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output.message = out.stopreason
output.xplus = out.xplus
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if not silent:
print('Method: ODR')
print(*out.stopreason)
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print('Residual variance:', output.residual_variance)
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if out.info > 3:
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raise Exception('The minimization procedure did not converge.')
m = x_f.size
def odr_chisquare(p):
model = func(p[:n_parms], p[n_parms:].reshape(x_shape))
chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((x_f - p[n_parms:].reshape(x_shape)) / dx_f) ** 2)
return chisq
if kwargs.get('expected_chisquare') is True:
W = np.diag(1 / np.asarray(np.concatenate((dy_f.ravel(), dx_f.ravel()))))
if kwargs.get('covariance') is not None:
cov = kwargs.get('covariance')
else:
cov = covariance(np.concatenate((y, x.ravel())))
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number_of_x_parameters = int(m / x_f.shape[-1])
old_jac = jacobian(func)(out.beta, out.xplus)
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fused_row1 = np.concatenate((old_jac, np.concatenate((number_of_x_parameters * [np.zeros(old_jac.shape)]), axis=0)))
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])))
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new_jac = np.concatenate((fused_row1, fused_row2), axis=1)
A = W @ new_jac
P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
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expected_chisquare = np.trace((np.identity(P_phi.shape[0]) - P_phi) @ W @ cov @ W)
if expected_chisquare <= 0.0:
warnings.warn("Negative expected_chisquare.", RuntimeWarning)
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expected_chisquare = np.abs(expected_chisquare)
output.chisquare_by_expected_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel()))) / expected_chisquare
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if not silent:
print('chisquare/expected_chisquare:',
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output.chisquare_by_expected_chisquare)
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fitp = out.beta
try:
hess = hessian(odr_chisquare)(np.concatenate((fitp, out.xplus.ravel())))
except TypeError:
raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
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def odr_chisquare_compact_x(d):
model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
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)
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return chisq
jac_jac_x = hessian(odr_chisquare_compact_x)(np.concatenate((fitp, out.xplus.ravel(), x_f.ravel())))
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# Compute hess^{-1} @ jac_jac_x[:n_parms + m, n_parms + m:] using LAPACK dgesv
try:
deriv_x = -scipy.linalg.solve(hess, jac_jac_x[:n_parms + m, n_parms + m:])
except np.linalg.LinAlgError:
raise Exception("Cannot invert hessian matrix.")
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def odr_chisquare_compact_y(d):
model = func(d[:n_parms], d[n_parms:n_parms + m].reshape(x_shape))
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)
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return chisq
jac_jac_y = hessian(odr_chisquare_compact_y)(np.concatenate((fitp, out.xplus.ravel(), y_f)))
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# Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
try:
deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms + m, n_parms + m:])
except np.linalg.LinAlgError:
raise Exception("Cannot invert hessian matrix.")
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result = []
for i in range(n_parms):
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])))
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output.fit_parameters = result
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output.odr_chisquare = odr_chisquare(np.concatenate((out.beta, out.xplus.ravel())))
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output.dof = x.shape[-1] - n_parms
output.p_value = 1 - scipy.stats.chi2.cdf(output.odr_chisquare, output.dof)
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return output
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def _prior_fit(x, y, func, priors, silent=False, **kwargs):
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output = Fit_result()
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output.fit_function = func
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x = np.asarray(x)
if kwargs.get('num_grad') is True:
hessian = num_hessian
else:
hessian = auto_hessian
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if not callable(func):
raise TypeError('func has to be a function.')
for i in range(100):
try:
func(np.arange(i), 0)
except TypeError:
continue
except IndexError:
continue
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else:
break
else:
raise RuntimeError("Fit function is not valid.")
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n_parms = i
if n_parms != len(priors):
raise Exception('Priors does not have the correct length.')
def extract_val_and_dval(string):
split_string = string.split('(')
if '.' in split_string[0] and '.' not in split_string[1][:-1]:
factor = 10 ** -len(split_string[0].partition('.')[2])
else:
factor = 1
return float(split_string[0]), float(split_string[1][:-1]) * factor
loc_priors = []
for i_n, i_prior in enumerate(priors):
if isinstance(i_prior, Obs):
loc_priors.append(i_prior)
else:
loc_val, loc_dval = extract_val_and_dval(i_prior)
loc_priors.append(cov_Obs(loc_val, loc_dval ** 2, '#prior' + str(i_n) + f"_{np.random.randint(2147483647):010d}"))
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output.priors = loc_priors
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if not silent:
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print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
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y_f = [o.value for o in y]
dy_f = [o.dvalue for o in y]
if np.any(np.asarray(dy_f) <= 0.0):
raise Exception('No y errors available, run the gamma method first.')
p_f = [o.value for o in loc_priors]
dp_f = [o.dvalue for o in loc_priors]
if np.any(np.asarray(dp_f) <= 0.0):
raise Exception('No prior errors available, run the gamma method first.')
if 'initial_guess' in kwargs:
x0 = kwargs.get('initial_guess')
if len(x0) != n_parms:
raise Exception('Initial guess does not have the correct length.')
else:
x0 = p_f
def chisqfunc(p):
model = func(p, x)
chisq = anp.sum(((y_f - model) / dy_f) ** 2) + anp.sum(((p_f - p) / dp_f) ** 2)
return chisq
if not silent:
print('Method: migrad')
m = iminuit.Minuit(chisqfunc, x0)
m.errordef = 1
m.print_level = 0
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if 'tol' in kwargs:
m.tol = kwargs.get('tol')
else:
m.tol = 1e-4
m.migrad()
params = np.asarray(m.values)
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output.chisquare_by_dof = m.fval / len(x)
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output.method = 'migrad'
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if not silent:
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print('chisquare/d.o.f.:', output.chisquare_by_dof)
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if not m.fmin.is_valid:
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raise Exception('The minimization procedure did not converge.')
hess = hessian(chisqfunc)(params)
hess_inv = np.linalg.pinv(hess)
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def chisqfunc_compact(d):
model = func(d[:n_parms], x)
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)
return chisq
jac_jac = hessian(chisqfunc_compact)(np.concatenate((params, y_f, p_f)))
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deriv = -hess_inv @ jac_jac[:n_parms, n_parms:]
result = []
for i in range(n_parms):
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])))
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output.fit_parameters = result
output.chisquare = chisqfunc(np.asarray(params))
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if kwargs.get('resplot') is True:
residual_plot(x, y, func, result)
if kwargs.get('qqplot') is True:
qqplot(x, y, func, result)
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return output
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def _combined_fit(x, y, func, silent=False, **kwargs):
output = Fit_result()
if (type(x) == dict and type(y) == dict and type(func) == dict):
xd = x
yd = y
funcd = func
output.fit_function = func
elif (type(x) == dict or type(y) == dict or type(func) == dict):
raise TypeError("All arguments have to be dictionaries in order to perform a combined fit.")
else:
x = np.asarray(x)
xd = {"": x}
yd = {"": y}
funcd = {"": func}
output.fit_function = func
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if kwargs.get('num_grad') is True:
jacobian = num_jacobian
hessian = num_hessian
else:
jacobian = auto_jacobian
hessian = auto_hessian
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key_ls = sorted(list(xd.keys()))
if sorted(list(yd.keys())) != key_ls:
raise Exception('x and y dictionaries do not contain the same keys.')
if sorted(list(funcd.keys())) != key_ls:
raise Exception('x and func dictionaries do not contain the same keys.')
x_all = np.concatenate([np.array(xd[key]) for key in key_ls])
y_all = np.concatenate([np.array(yd[key]) for key in key_ls])
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y_f = [o.value for o in y_all]
dy_f = [o.dvalue for o in y_all]
if len(x_all.shape) > 2:
raise Exception('Unknown format for x values')
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if np.any(np.asarray(dy_f) <= 0.0):
raise Exception('No y errors available, run the gamma method first.')
# number of fit parameters
n_parms_ls = []
for key in key_ls:
if not callable(funcd[key]):
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raise TypeError('func (key=' + key + ') is not a function.')
if np.asarray(xd[key]).shape[-1] != len(yd[key]):
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raise Exception('x and y input (key=' + key + ') do not have the same length')
for i in range(100):
try:
funcd[key](np.arange(i), x_all.T[0])
except TypeError:
continue
except IndexError:
continue
else:
break
else:
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raise RuntimeError("Fit function (key=" + key + ") is not valid.")
n_parms = i
n_parms_ls.append(n_parms)
n_parms = max(n_parms_ls)
if not silent:
print('Fit with', n_parms, 'parameter' + 's' * (n_parms > 1))
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if 'initial_guess' in kwargs:
x0 = kwargs.get('initial_guess')
if len(x0) != n_parms:
raise Exception('Initial guess does not have the correct length: %d vs. %d' % (len(x0), n_parms))
else:
x0 = [0.1] * n_parms
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def general_chisqfunc_uncorr(p, ivars):
model = anp.concatenate([anp.array(funcd[key](p, anp.asarray(xd[key]))).reshape(-1) for key in key_ls])
return ((ivars - model) / dy_f)
def chisqfunc_uncorr(p):
return anp.sum(general_chisqfunc_uncorr(p, y_f) ** 2)
if kwargs.get('correlated_fit') is True:
corr = covariance(y_all, correlation=True, **kwargs)
covdiag = np.diag(1 / np.asarray(dy_f))
condn = np.linalg.cond(corr)
if condn > 0.1 / np.finfo(float).eps:
raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})")
if condn > 1e13:
warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning)
chol = np.linalg.cholesky(corr)
chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True)
def general_chisqfunc(p, ivars):
model = anp.concatenate([anp.array(funcd[key](p, anp.asarray(xd[key]))).reshape(-1) for key in key_ls])
return anp.dot(chol_inv, (ivars - model))
def chisqfunc(p):
return anp.sum(general_chisqfunc(p, y_f) ** 2)
else:
general_chisqfunc = general_chisqfunc_uncorr
chisqfunc = chisqfunc_uncorr
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output.method = kwargs.get('method', 'Levenberg-Marquardt')
if not silent:
print('Method:', output.method)
if output.method != 'Levenberg-Marquardt':
if output.method == 'migrad':
tolerance = 1e-4 # default value of 1e-1 set by iminuit can be problematic
if 'tol' in kwargs:
tolerance = kwargs.get('tol')
fit_result = iminuit.minimize(chisqfunc_uncorr, x0, tol=tolerance) # Stopping criterion 0.002 * tol * errordef
if kwargs.get('correlated_fit') is True:
fit_result = iminuit.minimize(chisqfunc, fit_result.x, tol=tolerance)
output.iterations = fit_result.nfev
else:
tolerance = 1e-12
if 'tol' in kwargs:
tolerance = kwargs.get('tol')
fit_result = scipy.optimize.minimize(chisqfunc_uncorr, x0, method=kwargs.get('method'), tol=tolerance)
if kwargs.get('correlated_fit') is True:
fit_result = scipy.optimize.minimize(chisqfunc, fit_result.x, method=kwargs.get('method'), tol=tolerance)
output.iterations = fit_result.nit
chisquare = fit_result.fun
else:
if 'tol' in kwargs:
print('tol cannot be set for Levenberg-Marquardt')
def chisqfunc_residuals_uncorr(p):
return general_chisqfunc_uncorr(p, y_f)
fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
if kwargs.get('correlated_fit') is True:
def chisqfunc_residuals(p):
return general_chisqfunc(p, y_f)
fit_result = scipy.optimize.least_squares(chisqfunc_residuals, fit_result.x, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
chisquare = np.sum(fit_result.fun ** 2)
assert np.isclose(chisquare, chisqfunc(fit_result.x), atol=1e-14)
output.iterations = fit_result.nfev
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if not fit_result.success:
raise Exception('The minimization procedure did not converge.')
if x_all.shape[-1] - n_parms > 0:
output.chisquare = chisquare
output.dof = x_all.shape[-1] - n_parms
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output.chisquare_by_dof = output.chisquare / output.dof
output.p_value = 1 - scipy.stats.chi2.cdf(output.chisquare, output.dof)
else:
output.chisquare_by_dof = float('nan')
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output.message = fit_result.message
if not silent:
print(fit_result.message)
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print('chisquare/d.o.f.:', output.chisquare_by_dof)
print('fit parameters', fit_result.x)
def prepare_hat_matrix():
hat_vector = []
for key in key_ls:
x_array = np.asarray(xd[key])
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if (len(x_array) != 0):
hat_vector.append(jacobian(funcd[key])(fit_result.x, x_array))
hat_vector = [item for sublist in hat_vector for item in sublist]
return hat_vector
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if kwargs.get('expected_chisquare') is True:
if kwargs.get('correlated_fit') is not True:
W = np.diag(1 / np.asarray(dy_f))
cov = covariance(y_all)
hat_vector = prepare_hat_matrix()
A = W @ hat_vector
P_phi = A @ np.linalg.pinv(A.T @ A) @ A.T
expected_chisquare = np.trace((np.identity(x_all.shape[-1]) - P_phi) @ W @ cov @ W)
output.chisquare_by_expected_chisquare = output.chisquare / expected_chisquare
if not silent:
print('chisquare/expected_chisquare:', output.chisquare_by_expected_chisquare)
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fitp = fit_result.x
if np.any(np.asarray(dy_f) <= 0.0):
raise Exception('No y errors available, run the gamma method first.')
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try:
hess = hessian(chisqfunc)(fitp)
except TypeError:
raise Exception("It is required to use autograd.numpy instead of numpy within fit functions, see the documentation for details.") from None
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def chisqfunc_compact(d):
return anp.sum(general_chisqfunc(d[:n_parms], d[n_parms:]) ** 2)
jac_jac_y = hessian(chisqfunc_compact)(np.concatenate((fitp, y_f)))
# Compute hess^{-1} @ jac_jac_y[:n_parms + m, n_parms + m:] using LAPACK dgesv
try:
deriv_y = -scipy.linalg.solve(hess, jac_jac_y[:n_parms, n_parms:])
except np.linalg.LinAlgError:
raise Exception("Cannot invert hessian matrix.")
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result = []
for i in range(n_parms):
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), man_grad=list(deriv_y[i])))
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output.fit_parameters = result
# Hotelling t-squared p-value for correlated fits.
if kwargs.get('correlated_fit') is True:
n_cov = np.min(np.vectorize(lambda x_all: x_all.N)(y_all))
output.t2_p_value = 1 - scipy.stats.f.cdf((n_cov - output.dof) / (output.dof * (n_cov - 1)) * output.chisquare,
output.dof, n_cov - output.dof)
if kwargs.get('resplot') is True:
for key in key_ls:
residual_plot(xd[key], yd[key], funcd[key], result, title=key)
if kwargs.get('qqplot') is True:
for key in key_ls:
qqplot(xd[key], yd[key], funcd[key], result, title=key)
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return output
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def fit_lin(x, y, **kwargs):
"""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.
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"""
def f(a, x):
y = a[0] + a[1] * x
return y
if all(isinstance(n, Obs) for n in x):
out = total_least_squares(x, y, f, **kwargs)
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return out.fit_parameters
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elif all(isinstance(n, float) or isinstance(n, int) for n in x) or isinstance(x, np.ndarray):
out = least_squares(x, y, f, **kwargs)
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return out.fit_parameters
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else:
raise Exception('Unsupported types for x')
def qqplot(x, o_y, func, p, title=""):
"""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
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"""
residuals = []
for i_x, i_y in zip(x, o_y):
residuals.append((i_y - func(p, i_x)) / i_y.dvalue)
residuals = sorted(residuals)
my_y = [o.value for o in residuals]
probplot = scipy.stats.probplot(my_y)
my_x = probplot[0][0]
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plt.figure(figsize=(8, 8 / 1.618))
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plt.errorbar(my_x, my_y, fmt='o')
fit_start = my_x[0]
fit_stop = my_x[-1]
samples = np.arange(fit_start, fit_stop, 0.01)
plt.plot(samples, samples, 'k--', zorder=11, label='Standard normal distribution')
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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='-')
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plt.xlabel('Theoretical quantiles')
plt.ylabel('Ordered Values')
plt.legend(title=title)
plt.draw()
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def residual_plot(x, y, func, fit_res, title=""):
"""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
"""
sorted_x = sorted(x)
xstart = sorted_x[0] - 0.5 * (sorted_x[1] - sorted_x[0])
xstop = sorted_x[-1] + 0.5 * (sorted_x[-1] - sorted_x[-2])
x_samples = np.arange(xstart, xstop + 0.01, 0.01)
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plt.figure(figsize=(8, 8 / 1.618))
gs = gridspec.GridSpec(2, 1, height_ratios=[3, 1], wspace=0.0, hspace=0.0)
ax0 = plt.subplot(gs[0])
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')
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ax0.plot(x_samples, func([o.value for o in fit_res], x_samples), label='Fit', zorder=10, ls='-', ms=0)
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ax0.set_xticklabels([])
ax0.set_xlim([xstart, xstop])
ax0.set_xticklabels([])
ax0.legend(title=title)
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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])
ax1 = plt.subplot(gs[1])
ax1.plot(x, residuals, 'ko', ls='none', markersize=5)
ax1.tick_params(direction='out')
ax1.tick_params(axis="x", bottom=True, top=True, labelbottom=True)
ax1.axhline(y=0.0, ls='--', color='k', marker=" ")
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ax1.fill_between(x_samples, -1.0, 1.0, alpha=0.1, facecolor='k')
ax1.set_xlim([xstart, xstop])
ax1.set_ylabel('Residuals')
plt.subplots_adjust(wspace=None, hspace=None)
plt.draw()
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def error_band(x, func, beta):
"""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
"""
cov = covariance(beta)
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if np.any(np.abs(cov - cov.T) > 1000 * np.finfo(np.float64).eps):
warnings.warn("Covariance matrix is not symmetric within floating point precision", RuntimeWarning)
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deriv = []
for i, item in enumerate(x):
deriv.append(np.array(egrad(func)([o.value for o in beta], item)))
err = []
for i, item in enumerate(x):
err.append(np.sqrt(deriv[i] @ cov @ deriv[i]))
err = np.array(err)
return err
def ks_test(objects=None):
"""Performs a KolmogorovSmirnov test for the p-values of all fit object.
Parameters
----------
objects : list
List of fit results to include in the analysis (optional).
Returns
-------
None
"""
if objects is None:
obs_list = []
for obj in gc.get_objects():
if isinstance(obj, Fit_result):
obs_list.append(obj)
else:
obs_list = objects
p_values = [o.p_value for o in obs_list]
bins = len(p_values)
x = np.arange(0, 1.001, 0.001)
plt.plot(x, x, 'k', zorder=1)
plt.xlim(0, 1)
plt.ylim(0, 1)
plt.xlabel('p-value')
plt.ylabel('Cumulative probability')
plt.title(str(bins) + ' p-values')
n = np.arange(1, bins + 1) / np.float64(bins)
Xs = np.sort(p_values)
plt.step(Xs, n)
diffs = n - Xs
loc_max_diff = np.argmax(np.abs(diffs))
loc = Xs[loc_max_diff]
plt.annotate('', xy=(loc, loc), xytext=(loc, loc + diffs[loc_max_diff]), arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0))
plt.draw()
print(scipy.stats.kstest(p_values, 'uniform'))