https://github.com/lmfit/lmfit-py
Tip revision: 555c29e63c18262a0b3ab9648c9d2e0d56f7a023 authored by Matthew Newville on 08 July 2023, 15:51:20 UTC
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Tip revision: 555c29e
minimizer.py
"""Implementation of the Minimizer class and fitting algorithms.
Simple minimizer is a wrapper around scipy.leastsq, allowing a user to
build a fitting model as a function of general purpose fit Parameters that
can be fixed or varied, bounded, and written as a simple expression of
other fit Parameters.
The user sets up a model in terms of instance of Parameters and writes a
function-to-be-minimized (residual function) in terms of these Parameters.
Original copyright:
Copyright (c) 2011 Matthew Newville, The University of Chicago
See LICENSE for more complete authorship information and license.
"""
from collections import namedtuple
from copy import deepcopy
import inspect
import multiprocessing
import numbers
import warnings
import numpy as np
from scipy import __version__ as scipy_version
from scipy.linalg import LinAlgError, inv
from scipy.optimize import basinhopping as scipy_basinhopping
from scipy.optimize import brute as scipy_brute
from scipy.optimize import differential_evolution
from scipy.optimize import dual_annealing as scipy_dual_annealing
from scipy.optimize import least_squares
from scipy.optimize import leastsq as scipy_leastsq
from scipy.optimize import minimize as scipy_minimize
from scipy.optimize import shgo as scipy_shgo
from scipy.sparse import issparse
from scipy.sparse.linalg import LinearOperator
from scipy.stats import cauchy as cauchy_dist
from scipy.stats import norm as norm_dist
from ._ampgo import ampgo
from .parameter import Parameter, Parameters
from .printfuncs import fitreport_html_table
# check for EMCEE
try:
import emcee
from emcee.autocorr import AutocorrError
HAS_EMCEE = int(emcee.__version__[0]) >= 3
except ImportError:
HAS_EMCEE = False
# check for pandas
try:
import pandas as pd
from pandas import isnull
HAS_PANDAS = True
except ImportError:
HAS_PANDAS = False
isnull = np.isnan
# check for numdifftools
try:
import numdifftools as ndt
HAS_NUMDIFFTOOLS = True
except ImportError:
HAS_NUMDIFFTOOLS = False
# check for dill
try:
import dill # noqa: F401
HAS_DILL = True
except ImportError:
HAS_DILL = False
# define the namedtuple here so pickle will work with the MinimizerResult
Candidate = namedtuple('Candidate', ['params', 'score'])
maxeval_warning = "ignoring `{}` argument to `{}()`. Use `max_nfev` instead."
def thisfuncname():
"""Return the name of calling function."""
try:
return inspect.stack()[1].function
except AttributeError:
return inspect.stack()[1][3]
class MinimizerException(Exception):
"""General Purpose Exception."""
def __init__(self, msg):
Exception.__init__(self)
self.msg = msg
def __str__(self):
"""string"""
return f"{self.msg}"
class AbortFitException(MinimizerException):
"""Raised when a fit is aborted by the user."""
SCALAR_METHODS = {'nelder': 'Nelder-Mead',
'powell': 'Powell',
'cg': 'CG',
'bfgs': 'BFGS',
'newton': 'Newton-CG',
'lbfgsb': 'L-BFGS-B',
'l-bfgsb': 'L-BFGS-B',
'tnc': 'TNC',
'cobyla': 'COBYLA',
'slsqp': 'SLSQP',
'dogleg': 'dogleg',
'trust-ncg': 'trust-ncg',
'differential_evolution': 'differential_evolution',
'trust-constr': 'trust-constr',
'trust-exact': 'trust-exact',
'trust-krylov': 'trust-krylov'}
def reduce_chisquare(r):
"""Reduce residual array to scalar (chi-square).
Calculate the chi-square value from residual array `r` as
``(r*r).sum()``.
Parameters
----------
r : numpy.ndarray
Residual array.
Returns
-------
float
Chi-square calculated from the residual array.
"""
return (r*r).sum()
def reduce_negentropy(r):
"""Reduce residual array to scalar (negentropy).
Reduce residual array `r` to scalar using negative entropy and the
normal (Gaussian) probability distribution of `r` as pdf:
``(norm.pdf(r)*norm.logpdf(r)).sum()``
since ``pdf(r) = exp(-r*r/2)/sqrt(2*pi)``, this is
``((r*r/2 - log(sqrt(2*pi))) * exp(-r*r/2)).sum()``.
Parameters
----------
r : numpy.ndarray
Residual array.
Returns
-------
float
Negative entropy value calculated from the residual array.
"""
return (norm_dist.pdf(r)*norm_dist.logpdf(r)).sum()
def reduce_cauchylogpdf(r):
"""Reduce residual array to scalar (cauchylogpdf).
Reduce residual array `r` to scalar using negative log-likelihood and
a Cauchy (Lorentzian) distribution of `r`:
``-scipy.stats.cauchy.logpdf(r)``
where the Cauchy pdf = ``1/(pi*(1+r*r))``.
This gives better suppression of outliers compared to the default
sum-of-squares.
Parameters
----------
r : numpy.ndarray
Residual array.
Returns
-------
float
Negative entropy value calculated from the residual array.
"""
return -cauchy_dist.logpdf(r).sum()
class MinimizerResult:
r"""The results of a minimization.
Minimization results include data such as status and error messages,
fit statistics, and the updated (i.e., best-fit) parameters themselves
in the :attr:`params` attribute.
The list of (possible) `MinimizerResult` attributes is given below:
Attributes
----------
residual : numpy.ndarray
Residual array :math:`{\rm Resid_i}`. Return value of the objective
function when using the best-fit values of the parameters.
params : Parameters
The best-fit Parameters resulting from the fit.
uvars : dict
Dictionary of uncertainties ufloats from Parameters
var_names : list
list of variable Parameter names used in optimization in the
same order as the values in :attr:`init_vals` and :attr:`covar`.
covar : numpy.ndarray or None
Covariance matrix from minimization, with rows and columns
corresponding to :attr:`var_names`. If uncertainties cannot
be determined, this value will be ``None``.
init_vals : list
List of initial values for variable parameters using
:attr:`var_names`.
init_values : dict
Dictionary of initial values for variable parameters.
aborted : bool
Whether the fit was aborted.
status : int
Termination status of the optimizer. Its value depends on the
underlying solver. Refer to `message` for details.
success : bool
True if the fit succeeded, otherwise False. This is an optimistic
view of success, meaning that the method finished without error.
errorbars : bool
whether uncertainties were estimated for variable Parameters.
message : str
Message about fit success.
ier : int
Integer error value from :scipydoc:`optimize.leastsq` (`'leastsq'`
method only).
lmdif_message : str
Message from :scipydoc:`optimize.leastsq` (`'leastsq'` method only).
call_kws : dict
Keyword arguments sent to underlying solver.
flatchain : pandas.DataFrame
A flatchain view of the sampling chain from the `emcee` method.
nfev : int
Number of function evaluations.
nvarys : int
Number of variables in fit: :math:`N_{\rm varys}`.
ndata : int
Number of data points: :math:`N`.
nfree : int
Degrees of freedom in fit: :math:`N - N_{\rm varys}`.
chisqr : float
Chi-square: :math:`\chi^2 = \sum_i^N [{\rm Resid}_i]^2`.
redchi : float
Reduced chi-square:
:math:`\chi^2_{\nu}= {\chi^2} / {(N - N_{\rm varys})}`.
aic : float
Akaike Information Criterion statistic:
:math:`N \ln(\chi^2/N) + 2 N_{\rm varys}`.
bic : float
Bayesian Information Criterion statistic:
:math:`N \ln(\chi^2/N) + \ln(N) N_{\rm varys}`.
Methods
-------
show_candidates
:meth:`pretty_print` representation of candidates from the `brute`
fitting method.
"""
def __init__(self, **kws):
for key, val in kws.items():
setattr(self, key, val)
@property
def flatchain(self):
"""Show flatchain view of the sampling chain from `emcee` method."""
if not hasattr(self, 'chain'):
return None
if not HAS_PANDAS:
raise NotImplementedError('Please install Pandas to see the '
'flattened chain')
if len(self.chain.shape) == 4:
return pd.DataFrame(self.chain[0, ...].reshape((-1, self.nvarys)),
columns=self.var_names)
elif len(self.chain.shape) == 3:
return pd.DataFrame(self.chain.reshape((-1, self.nvarys)),
columns=self.var_names)
def show_candidates(self, candidate_nmb='all'):
"""Show pretty_print() representation of candidates.
Showing all stored candidates (default) or the specified
candidate-# obtained from the `brute` fitting method.
Parameters
----------
candidate_nmb : int or 'all', optional
The candidate-number to show using the :meth:`pretty_print`
method (default is 'all').
"""
if hasattr(self, 'candidates'):
if candidate_nmb == 'all':
for i, candidate in enumerate(self.candidates):
print(f"\nCandidate #{i + 1}, chisqr = {candidate.score:.3f}")
candidate.params.pretty_print()
elif (candidate_nmb < 1 or candidate_nmb > len(self.candidates)):
raise ValueError(f"'candidate_nmb' should be between 1 and {len(self.candidates)}.")
else:
candidate = self.candidates[candidate_nmb-1]
print(f"\nCandidate #{candidate_nmb}, chisqr = {candidate.score:.3f}")
candidate.params.pretty_print()
def _calculate_statistics(self):
"""Calculate the fitting statistics."""
self.nvarys = len(self.init_vals)
if not hasattr(self, 'residual'):
self.residual = -np.inf
if isinstance(self.residual, np.ndarray):
self.chisqr = (self.residual**2).sum()
self.ndata = len(self.residual)
self.nfree = self.ndata - self.nvarys
else:
self.chisqr = self.residual
self.ndata = 1
self.nfree = 1
self.redchi = self.chisqr / max(1, self.nfree)
# this is -2*loglikelihood
self.chisqr = max(self.chisqr, 1.e-250*self.ndata)
_neg2_log_likel = self.ndata * np.log(self.chisqr / self.ndata)
self.aic = _neg2_log_likel + 2 * self.nvarys
self.bic = _neg2_log_likel + np.log(self.ndata) * self.nvarys
def _repr_html_(self, show_correl=True, min_correl=0.1):
"""Return a HTML representation of parameters data."""
report = fitreport_html_table(self, show_correl=show_correl,
min_correl=min_correl)
return f"<h2>Fit Result</h2> {report}"
class Minimizer:
"""A general minimizer for curve fitting and optimization."""
_err_nonparam = ("params must be a minimizer.Parameters() instance or"
" list of Parameters()")
_err_max_evals = ("Too many function calls (max set to {:i})! Use:"
" minimize(func, params, ..., max_nfev=NNN)"
" to increase this maximum.")
def __init__(self, userfcn, params, fcn_args=None, fcn_kws=None,
iter_cb=None, scale_covar=True, nan_policy='raise',
reduce_fcn=None, calc_covar=True, max_nfev=None, **kws):
"""
Parameters
----------
userfcn : callable
Objective function that returns the residual (difference
between model and data) to be minimized in a least-squares
sense. This function must have the signature::
userfcn(params, *fcn_args, **fcn_kws)
params : Parameters
Contains the Parameters for the model.
fcn_args : tuple, optional
Positional arguments to pass to `userfcn`.
fcn_kws : dict, optional
Keyword arguments to pass to `userfcn`.
iter_cb : callable, optional
Function to be called at each fit iteration. This function
should have the signature::
iter_cb(params, iter, resid, *fcn_args, **fcn_kws)
where `params` will have the current parameter values, `iter`
the iteration number, `resid` the current residual array, and
`*fcn_args` and `**fcn_kws` are passed to the objective
function.
scale_covar : bool, optional
Whether to automatically scale the covariance matrix (default
is True).
nan_policy : {'raise', 'propagate', 'omit}, optional
Specifies action if `userfcn` (or a Jacobian) returns NaN
values. One of:
- `'raise'` : a `ValueError` is raised (default)
- `'propagate'` : the values returned from `userfcn` are un-altered
- `'omit'` : non-finite values are filtered
reduce_fcn : str or callable, optional
Function to convert a residual array to a scalar value for the
scalar minimizers. Optional values are (where `r` is the
residual array):
- None : sum-of-squares of residual (default)
= (r*r).sum()
- `'negentropy'` : neg entropy, using normal distribution
= rho*log(rho).sum()`, where rho = exp(-r*r/2)/(sqrt(2*pi))
- `'neglogcauchy'` : neg log likelihood, using Cauchy distribution
= -log(1/(pi*(1+r*r))).sum()
- callable : must take one argument (`r`) and return a float.
calc_covar : bool, optional
Whether to calculate the covariance matrix (default is True)
for solvers other than ``'leastsq'`` and ``'least_squares'``.
Requires the ``numdifftools`` package to be installed.
max_nfev : int or None, optional
Maximum number of function evaluations (default is None). The
default value depends on the fitting method.
**kws : dict, optional
Options to pass to the minimizer being used.
Notes
-----
The objective function should return the value to be minimized.
For the Levenberg-Marquardt algorithm from :meth:`leastsq` or
:meth:`least_squares`, this returned value must be an array, with
a length greater than or equal to the number of fitting variables
in the model. For the other methods, the return value can either be
a scalar or an array. If an array is returned, the sum-of-squares
of the array will be sent to the underlying fitting method,
effectively doing a least-squares optimization of the return
values. If the objective function returns non-finite values then a
`ValueError` will be raised because the underlying solvers cannot
deal with them.
A common use for the `fcn_args` and `fcn_kws` would be to pass in
other data needed to calculate the residual, including such things
as the data array, dependent variable, uncertainties in the data,
and other data structures for the model calculation.
"""
self.userfcn = userfcn
self.userargs = fcn_args
if self.userargs is None:
self.userargs = []
self.userkws = fcn_kws
if self.userkws is None:
self.userkws = {}
for maxnfev_alias in ('maxfev', 'maxiter'):
if maxnfev_alias in kws:
warnings.warn(maxeval_warning.format(maxnfev_alias, 'Minimizer'),
RuntimeWarning)
kws.pop(maxnfev_alias)
self.kws = kws
self.iter_cb = iter_cb
self.calc_covar = calc_covar
self.scale_covar = scale_covar
self.max_nfev = max_nfev
self.nfev = 0
self.nfree = 0
self.ndata = 0
self.ier = 0
self._abort = False
self.success = True
self.errorbars = False
self.message = None
self.lmdif_message = None
self.chisqr = None
self.redchi = None
self.covar = None
self.residual = None
self.reduce_fcn = reduce_fcn
self.params = params
self.col_deriv = False
self.jacfcn = None
self.nan_policy = nan_policy
def set_max_nfev(self, max_nfev=None, default_value=100000):
"""Set maximum number of function evaluations.
If `max_nfev` is None, use the provided `default_value`.
>>> self.set_max_nfev(max_nfev, 2000*(result.nvarys+1))
"""
if max_nfev is not None:
self.max_nfev = max_nfev
if self.max_nfev in (None, np.inf):
self.max_nfev = default_value
@property
def values(self):
"""Return Parameter values in a simple dictionary."""
return {name: p.value for name, p in self.result.params.items()}
def __residual(self, fvars, apply_bounds_transformation=True):
"""Residual function used for least-squares fit.
With the new, candidate values of `fvars` (the fitting variables),
this evaluates all parameters, including setting bounds and
evaluating constraints, and then passes those to the user-supplied
function to calculate the residual.
Parameters
----------
fvars : numpy.ndarray
Array of new parameter values suggested by the minimizer.
apply_bounds_transformation : bool, optional
Whether to apply lmfits parameter transformation to constrain
parameters (default is True). This is needed for solvers
without built-in support for bounds.
Returns
-------
numpy.ndarray
The evaluated function values for given `fvars`.
"""
params = self.result.params
if fvars.shape == ():
fvars = fvars.reshape((1,))
for name, val in zip(self.result.var_names, fvars):
if apply_bounds_transformation:
params[name].value = params[name].from_internal(val)
else:
params[name].value = val
params.update_constraints()
if self.max_nfev is None:
self.max_nfev = 200000*(len(fvars)+1)
self.result.nfev += 1
self.result.last_internal_values = fvars
if self.result.nfev > self.max_nfev:
self.result.aborted = True
m = f"number of function evaluations > {self.max_nfev}"
self.result.message = f"Fit aborted: {m}"
self.result.success = False
raise AbortFitException(f"fit aborted: too many function evaluations {self.max_nfev}")
out = self.userfcn(params, *self.userargs, **self.userkws)
if callable(self.iter_cb):
abort = self.iter_cb(params, self.result.nfev, out,
*self.userargs, **self.userkws)
self._abort = self._abort or abort
if self._abort:
self.result.residual = out
self.result.aborted = True
self.result.message = "Fit aborted by user callback. Could not estimate error-bars."
self.result.success = False
raise AbortFitException("fit aborted by user.")
else:
return coerce_float64(out, nan_policy=self.nan_policy)
def __jacobian(self, fvars):
"""Return analytical jacobian to be used with Levenberg-Marquardt.
modified 02-01-2012 by Glenn Jones, Aberystwyth University
modified 06-29-2015 by M Newville to apply gradient scaling for
bounded variables (thanks to JJ Helmus, N Mayorov)
"""
pars = self.result.params
grad_scale = np.ones_like(fvars)
for ivar, name in enumerate(self.result.var_names):
val = fvars[ivar]
pars[name].value = pars[name].from_internal(val)
grad_scale[ivar] = pars[name].scale_gradient(val)
pars.update_constraints()
# compute the jacobian for "internal" unbounded variables,
# then rescale for bounded "external" variables.
jac = self.jacfcn(pars, *self.userargs, **self.userkws)
jac = coerce_float64(jac, nan_policy=self.nan_policy, ravel=False)
if self.col_deriv:
jac = (jac.transpose()*grad_scale).transpose()
else:
jac *= grad_scale
return jac
def penalty(self, fvars):
"""Penalty function for scalar minimizers.
Parameters
----------
fvars : numpy.ndarray
Array of values for the variable parameters.
Returns
-------
r : float
The evaluated user-supplied objective function.
If the objective function is an array of size greater than 1,
use the scalar returned by `self.reduce_fcn`. This defaults to
sum-of-squares, but can be replaced by other options.
"""
if self.result.method in ['brute', 'shgo', 'dual_annealing']:
apply_bounds_transformation = False
else:
apply_bounds_transformation = True
r = self.__residual(fvars, apply_bounds_transformation)
if isinstance(r, np.ndarray) and r.size > 1:
r = self.reduce_fcn(r)
if isinstance(r, np.ndarray) and r.size > 1:
r = r.sum()
return r
def prepare_fit(self, params=None):
"""Prepare parameters for fitting.
Prepares and initializes model and Parameters for subsequent
fitting. This routine prepares the conversion of
:class:`Parameters` into fit variables, organizes parameter bounds,
and parses, "compiles" and checks constrain expressions. The method
also creates and returns a new instance of a
:class:`MinimizerResult` object that contains the copy of the
Parameters that will actually be varied in the fit.
Parameters
----------
params : Parameters, optional
Contains the Parameters for the model; if None, then the
Parameters used to initialize the Minimizer object are used.
Returns
-------
MinimizerResult
Notes
-----
This method is called directly by the fitting methods, and it is
generally not necessary to call this function explicitly.
.. versionchanged:: 0.9.0
Return value changed to :class:`MinimizerResult`.
"""
self._abort = False
self.result = MinimizerResult()
result = self.result
if params is not None:
self.params = params
if isinstance(self.params, Parameters):
result.params = deepcopy(self.params)
elif isinstance(self.params, (list, tuple)):
result.params = Parameters()
for par in self.params:
if not isinstance(par, Parameter):
raise MinimizerException(self._err_nonparam)
else:
result.params[par.name] = par
elif self.params is None:
raise MinimizerException(self._err_nonparam)
# determine which parameters are actually variables
# and which are defined expressions.
result.var_names = [] # note that this *does* belong to self...
result.init_vals = []
result._init_vals_internal = []
result.params.update_constraints()
result.nfev = 0
result.call_kws = {}
result.errorbars = False
result.aborted = False
result.success = True
result.covar = None
for name, par in self.result.params.items():
par.stderr = None
par.correl = None
if par.expr is not None:
par.vary = False
if par.vary:
result.var_names.append(name)
result._init_vals_internal.append(par.setup_bounds())
result.init_vals.append(par.value)
par.init_value = par.value
if par.name is None:
par.name = name
result.nvarys = len(result.var_names)
result.init_values = {n: v for n, v in zip(result.var_names,
result.init_vals)}
# set up reduce function for scalar minimizers
# 1. user supplied callable
# 2. string starting with 'neglogc' or 'negent'
# 3. sum-of-squares
if not callable(self.reduce_fcn):
if isinstance(self.reduce_fcn, str):
if self.reduce_fcn.lower().startswith('neglogc'):
self.reduce_fcn = reduce_cauchylogpdf
elif self.reduce_fcn.lower().startswith('negent'):
self.reduce_fcn = reduce_negentropy
if self.reduce_fcn is None:
self.reduce_fcn = reduce_chisquare
return result
def unprepare_fit(self):
"""Clean fit state.
This methods Removes AST compilations of constraint expressions.
Subsequent fits will need to call prepare_fit().
"""
def _calculate_covariance_matrix(self, fvars):
"""Calculate the covariance matrix.
The ``numdiftoools`` package is used to estimate the Hessian
matrix, and the covariance matrix is calculated as:
.. math::
cov_x = inverse(Hessian) * 2.0
Parameters
----------
fvars : numpy.ndarray
Array of the optimal internal, freely variable parameters.
Returns
-------
cov_x : numpy.ndarray or None
Covariance matrix if successful, otherwise None.
"""
warnings.filterwarnings(action="ignore", module="scipy",
message="^internal gelsd")
nfev = deepcopy(self.result.nfev)
best_vals = self.result.params.valuesdict()
try:
Hfun = ndt.Hessian(self.penalty, step=1.e-4)
hessian_ndt = Hfun(fvars)
cov_x = inv(hessian_ndt) * 2.0
if cov_x.diagonal().min() < 0:
# we know the calculated covariance is incorrect, so we set the covariance to None
cov_x = None
except (LinAlgError, ValueError):
cov_x = None
finally:
self.result.nfev = nfev
# restore original values
for name in self.result.var_names:
self.result.params[name].value = best_vals[name]
return cov_x
def _int2ext_cov_x(self, cov_int, fvars):
"""Transform covariance matrix to external parameter space.
It makes use of the gradient scaling according to the MINUIT
recipe:
cov_ext = np.dot(grad.T, grad) * cov_int
Parameters
----------
cov_int : numpy.ndarray
Covariance matrix in the internal parameter space.
fvars : numpy.ndarray
Array of the optimal internal, freely variable, parameter
values.
Returns
-------
cov_ext : numpy.ndarray
Covariance matrix, transformed to external parameter space.
"""
g = [self.result.params[name].scale_gradient(fvars[i]) for i, name in
enumerate(self.result.var_names)]
grad2d = np.atleast_2d(g)
grad = np.dot(grad2d.T, grad2d)
cov_ext = cov_int * grad
return cov_ext
def _calculate_uncertainties_correlations(self):
"""Calculate parameter uncertainties and correlations."""
self.result.errorbars = True
if self.scale_covar:
self.result.covar *= self.result.redchi
for par in self.result.params.values():
par.stderr, par.correl = 0, None
for ivar, name in enumerate(self.result.var_names):
par = self.result.params[name]
par.stderr = np.sqrt(self.result.covar[ivar, ivar])
par.correl = {}
try:
self.result.errorbars = self.result.errorbars and (par.stderr > 0.0)
for jvar, varn2 in enumerate(self.result.var_names):
if jvar != ivar:
par.correl[varn2] = (self.result.covar[ivar, jvar] /
(par.stderr * np.sqrt(self.result.covar[jvar, jvar])))
except ZeroDivisionError:
self.result.errorbars = False
if self.result.errorbars:
self.result.uvars = self.result.params.create_uvars(covar=self.result.covar)
def scalar_minimize(self, method='Nelder-Mead', params=None, max_nfev=None,
**kws):
"""Scalar minimization using :scipydoc:`optimize.minimize`.
Perform fit with any of the scalar minimization algorithms
supported by :scipydoc:`optimize.minimize`. Default argument
values are:
+-------------------------+---------------+-----------------------+
| :meth:`scalar_minimize` | Default Value | Description |
| arg | | |
+=========================+===============+=======================+
| `method` | 'Nelder-Mead' | fitting method |
+-------------------------+---------------+-----------------------+
| `tol` | 1.e-7 | fitting and parameter |
| | | tolerance |
+-------------------------+---------------+-----------------------+
| `hess` | None | Hessian of objective |
| | | function |
+-------------------------+---------------+-----------------------+
Parameters
----------
method : str, optional
Name of the fitting method to use. One of:
- `'Nelder-Mead'` (default)
- `'L-BFGS-B'`
- `'Powell'`
- `'CG'`
- `'Newton-CG'`
- `'COBYLA'`
- `'BFGS'`
- `'TNC'`
- `'trust-ncg'`
- `'trust-exact'`
- `'trust-krylov'`
- `'trust-constr'`
- `'dogleg'`
- `'SLSQP'`
- `'differential_evolution'`
params : Parameters, optional
Parameters to use as starting point.
max_nfev : int or None, optional
Maximum number of function evaluations. Defaults to
``2000*(nvars+1)``, where ``nvars`` is the number of variable
parameters.
**kws : dict, optional
Minimizer options pass to :scipydoc:`optimize.minimize`.
Returns
-------
MinimizerResult
Object containing the optimized parameters and several
goodness-of-fit statistics.
.. versionchanged:: 0.9.0
Return value changed to :class:`MinimizerResult`.
Notes
-----
If the objective function returns a NumPy array instead of the
expected scalar, the sum-of-squares of the array will be used.
Note that bounds and constraints can be set on Parameters for any
of these methods, so are not supported separately for those
designed to use bounds. However, if you use the
``differential_evolution`` method you must specify finite
``(min, max)`` for each varying Parameter.
"""
result = self.prepare_fit(params=params)
result.method = method
variables = result._init_vals_internal
params = result.params
self.set_max_nfev(max_nfev, 2000*(result.nvarys+1))
fmin_kws = dict(method=method, options={'maxiter': 2*self.max_nfev})
if method == 'L-BFGS-B':
fmin_kws['options']['maxfun'] = 2*self.max_nfev
elif method == 'COBYLA':
# for this method, we explicitly let the solver reach
# the users max nfev, and do not abort in _residual.
fmin_kws['options']['maxiter'] = self.max_nfev
self.max_nfev = 5*self.max_nfev
# fmin_kws = dict(method=method, options={'maxfun': 2*self.max_nfev})
fmin_kws.update(self.kws)
if 'maxiter' in kws:
warnings.warn(maxeval_warning.format('maxiter', thisfuncname()),
RuntimeWarning)
kws.pop('maxiter')
fmin_kws.update(kws)
# hess supported only in some methods
if 'hess' in fmin_kws and method not in ('Newton-CG', 'dogleg',
'trust-constr', 'trust-ncg',
'trust-krylov', 'trust-exact'):
fmin_kws.pop('hess')
# Accept Jacobians given as Dfun argument
if 'jac' not in fmin_kws and fmin_kws.get('Dfun', None) is not None:
fmin_kws['jac'] = fmin_kws.pop('Dfun')
# Wrap Jacobian function to deal with bounds
if 'jac' in fmin_kws:
self.jacfcn = fmin_kws.pop('jac')
fmin_kws['jac'] = self.__jacobian
# Ignore jac argument for methods that do not support it
if 'jac' in fmin_kws and method not in ('CG', 'BFGS', 'Newton-CG',
'L-BFGS-B', 'TNC', 'SLSQP',
'dogleg', 'trust-ncg',
'trust-krylov', 'trust-exact'):
self.jacfcn = None
fmin_kws.pop('jac')
# workers / updating keywords only supported in differential_evolution
for kwd in ('workers', 'updating'):
if kwd in fmin_kws and method != 'differential_evolution':
fmin_kws.pop(kwd)
if method == 'differential_evolution':
for par in params.values():
if (par.vary and
not (np.isfinite(par.min) and np.isfinite(par.max))):
raise ValueError('differential_evolution requires finite '
'bound for all varying parameters')
_bounds = [(-np.pi / 2., np.pi / 2.)] * len(variables)
kwargs = dict(args=(), strategy='best1bin', maxiter=self.max_nfev,
popsize=15, tol=0.01, mutation=(0.5, 1),
recombination=0.7, seed=None, callback=None,
disp=False, polish=True, init='latinhypercube',
atol=0, updating='immediate', workers=1)
for k, v in fmin_kws.items():
if k in kwargs:
kwargs[k] = v
fmin_kws = kwargs
result.call_kws = fmin_kws
try:
ret = differential_evolution(self.penalty, _bounds, **fmin_kws)
except AbortFitException:
pass
else:
result.call_kws = fmin_kws
try:
ret = scipy_minimize(self.penalty, variables, **fmin_kws)
except AbortFitException:
pass
if not result.aborted:
if isinstance(ret, dict):
for attr, value in ret.items():
setattr(result, attr, value)
else:
for attr in dir(ret):
if not attr.startswith('_'):
setattr(result, attr, getattr(ret, attr))
result.x = np.atleast_1d(result.x)
result.residual = self.__residual(result.x)
result.nfev -= 1
else:
result.x = result.last_internal_values
self.result.nfev -= 2
self._abort = False
result.residual = self.__residual(result.x)
result.nfev += 1
result._calculate_statistics()
# calculate the cov_x and estimate uncertainties/correlations
self.result.uvars = None
if (not result.aborted and self.calc_covar and HAS_NUMDIFFTOOLS and
len(result.residual) > len(result.var_names)):
_covar_ndt = self._calculate_covariance_matrix(result.x)
if _covar_ndt is not None:
result.covar = self._int2ext_cov_x(_covar_ndt, result.x)
self._calculate_uncertainties_correlations()
return result
def _lnprob(self, theta, userfcn, params, var_names, bounds, userargs=(),
userkws=None, float_behavior='posterior', is_weighted=True,
nan_policy='raise'):
"""Calculate the log-posterior probability.
See the `Minimizer.emcee` method for more details.
Parameters
----------
theta : sequence
Float parameter values (only those being varied).
userfcn : callable
User objective function.
params : Parameters
The entire set of Parameters.
var_names : list
The names of the parameters that are varying.
bounds : numpy.ndarray
Lower and upper bounds of parameters, with shape
``(nvarys, 2)``.
userargs : tuple, optional
Extra positional arguments required for user objective function.
userkws : dict, optional
Extra keyword arguments required for user objective function.
float_behavior : {'posterior', 'chi2'}, optional
Specifies meaning of objective when it returns a float. Use
`'posterior'` if objective function returns a log-posterior
probability (default) or `'chi2'` if it returns a chi2 value.
is_weighted : bool, optional
If `userfcn` returns a vector of residuals then `is_weighted`
(default is True) specifies if the residuals have been weighted
by data uncertainties.
nan_policy : {'raise', 'propagate', 'omit'}, optional
Specifies action if `userfcn` returns NaN values. Use `'raise'`
(default) to raise a `ValueError`, `'propagate'` to use values
as-is, or `'omit'` to filter out the non-finite values.
Returns
-------
lnprob : float
Log posterior probability.
"""
# the comparison has to be done on theta and bounds. DO NOT inject theta
# values into Parameters, then compare Parameters values to the bounds.
# Parameters values are clipped to stay within bounds.
if np.any(theta > bounds[:, 1]) or np.any(theta < bounds[:, 0]):
return -np.inf
for name, val in zip(var_names, theta):
params[name].value = val
userkwargs = {}
if userkws is not None:
userkwargs = userkws
# update the constraints
params.update_constraints()
# now calculate the log-likelihood
out = userfcn(params, *userargs, **userkwargs)
self.result.nfev += 1
if callable(self.iter_cb):
abort = self.iter_cb(params, self.result.nfev, out,
*userargs, **userkwargs)
self._abort = self._abort or abort
if self._abort:
self.result.residual = out
self._lastpos = theta
raise AbortFitException("fit aborted by user.")
else:
out = coerce_float64(out, nan_policy=self.nan_policy)
lnprob = coerce_float64(out, nan_policy=self.nan_policy)
if len(lnprob) == 0:
lnprob = np.array([-1.e100])
if lnprob.size > 1:
# objective function returns a vector of residuals
if '__lnsigma' in params and not is_weighted:
# marginalise over a constant data uncertainty
__lnsigma = params['__lnsigma'].value
c = np.log(2 * np.pi) + 2 * __lnsigma
lnprob = -0.5 * np.sum((lnprob / np.exp(__lnsigma)) ** 2 + c)
else:
lnprob = -0.5 * (lnprob * lnprob).sum()
else:
# objective function returns a single value.
# use float_behaviour to figure out if the value is posterior or chi2
if float_behavior == 'posterior':
pass
elif float_behavior == 'chi2':
lnprob *= -0.5
else:
raise ValueError("float_behaviour must be either 'posterior' "
"or 'chi2' " + float_behavior)
return lnprob
def emcee(self, params=None, steps=1000, nwalkers=100, burn=0, thin=1,
ntemps=1, pos=None, reuse_sampler=False, workers=1,
float_behavior='posterior', is_weighted=True, seed=None,
progress=True, run_mcmc_kwargs={}):
r"""Bayesian sampling of the posterior distribution.
The method uses the ``emcee`` Markov Chain Monte Carlo package and
assumes that the prior is Uniform. You need to have ``emcee``
version 3 or newer installed to use this method.
Parameters
----------
params : Parameters, optional
Parameters to use as starting point. If this is not specified
then the Parameters used to initialize the Minimizer object
are used.
steps : int, optional
How many samples you would like to draw from the posterior
distribution for each of the walkers?
nwalkers : int, optional
Should be set so :math:`nwalkers >> nvarys`, where ``nvarys``
are the number of parameters being varied during the fit.
'Walkers are the members of the ensemble. They are almost like
separate Metropolis-Hastings chains but, of course, the proposal
distribution for a given walker depends on the positions of all
the other walkers in the ensemble.' - from the `emcee` webpage.
burn : int, optional
Discard this many samples from the start of the sampling regime.
thin : int, optional
Only accept 1 in every `thin` samples.
ntemps : int, deprecated
ntemps has no effect.
pos : numpy.ndarray, optional
Specify the initial positions for the sampler, an ndarray of
shape ``(nwalkers, nvarys)``. You can also initialise using a
previous chain of the same `nwalkers` and ``nvarys``. Note that
``nvarys`` may be one larger than you expect it to be if your
``userfcn`` returns an array and ``is_weighted=False``.
reuse_sampler : bool, optional
Set to True if you have already run `emcee` with the
`Minimizer` instance and want to continue to draw from its
``sampler`` (and so retain the chain history). If False, a
new sampler is created. The keywords `nwalkers`, `pos`, and
`params` will be ignored when this is set, as they will be set
by the existing sampler.
**Important**: the Parameters used to create the sampler must
not change in-between calls to `emcee`. Alteration of Parameters
would include changed ``min``, ``max``, ``vary`` and ``expr``
attributes. This may happen, for example, if you use an altered
Parameters object and call the `minimize` method in-between
calls to `emcee`.
workers : Pool-like or int, optional
For parallelization of sampling. It can be any Pool-like object
with a map method that follows the same calling sequence as the
built-in `map` function. If int is given as the argument, then
a multiprocessing-based pool is spawned internally with the
corresponding number of parallel processes. 'mpi4py'-based
parallelization and 'joblib'-based parallelization pools can
also be used here. **Note**: because of multiprocessing
overhead it may only be worth parallelising if the objective
function is expensive to calculate, or if there are a large
number of objective evaluations per step
(``nwalkers * nvarys``).
float_behavior : str, optional
Meaning of float (scalar) output of objective function. Use
`'posterior'` if it returns a log-posterior probability or
`'chi2'` if it returns :math:`\chi^2`. See Notes for further
details.
is_weighted : bool, optional
Has your objective function been weighted by measurement
uncertainties? If ``is_weighted=True`` then your objective
function is assumed to return residuals that have been divided
by the true measurement uncertainty ``(data - model) / sigma``.
If ``is_weighted=False`` then the objective function is
assumed to return unweighted residuals, ``data - model``. In
this case `emcee` will employ a positive measurement
uncertainty during the sampling. This measurement uncertainty
will be present in the output params and output chain with the
name ``__lnsigma``. A side effect of this is that you cannot
use this parameter name yourself.
**Important**: this parameter only has any effect if your
objective function returns an array. If your objective function
returns a float, then this parameter is ignored. See Notes for
more details.
seed : int or numpy.random.RandomState, optional
If `seed` is an ``int``, a new `numpy.random.RandomState`
instance is used, seeded with `seed`.
If `seed` is already a `numpy.random.RandomState` instance,
then that `numpy.random.RandomState` instance is used. Specify
`seed` for repeatable minimizations.
progress : bool, optional
Print a progress bar to the console while running.
run_mcmc_kwargs : dict, optional
Additional (optional) keyword arguments that are passed to
``emcee.EnsembleSampler.run_mcmc``.
Returns
-------
MinimizerResult
MinimizerResult object containing updated params, statistics,
etc. The updated params represent the median of the samples,
while the uncertainties are half the difference of the 15.87
and 84.13 percentiles. The `MinimizerResult` contains a few
additional attributes: `chain` contain the samples and has
shape ``((steps - burn) // thin, nwalkers, nvarys)``.
`flatchain` is a `pandas.DataFrame` of the flattened chain,
that can be accessed with `result.flatchain[parname]`.
`lnprob` contains the log probability for each sample in
`chain`. The sample with the highest probability corresponds
to the maximum likelihood estimate. `acor` is an array
containing the auto-correlation time for each parameter if the
auto-correlation time can be computed from the chain. Finally,
`acceptance_fraction` (an array of the fraction of steps
accepted for each walker).
Notes
-----
This method samples the posterior distribution of the parameters
using Markov Chain Monte Carlo. It calculates the log-posterior
probability of the model parameters, `F`, given the data, `D`,
:math:`\ln p(F_{true} | D)`. This 'posterior probability' is
given by:
.. math::
\ln p(F_{true} | D) \propto \ln p(D | F_{true}) + \ln p(F_{true})
where :math:`\ln p(D | F_{true})` is the 'log-likelihood' and
:math:`\ln p(F_{true})` is the 'log-prior'. The default log-prior
encodes prior information known about the model that the log-prior
probability is ``-numpy.inf`` (impossible) if any of the parameters
is outside its limits, and is zero if all the parameters are inside
their bounds (uniform prior). The log-likelihood function is [1]_:
.. math::
\ln p(D|F_{true}) = -\frac{1}{2}\sum_n \left[\frac{(g_n(F_{true}) - D_n)^2}{s_n^2}+\ln (2\pi s_n^2)\right]
The first term represents the residual (:math:`g` being the
generative model, :math:`D_n` the data and :math:`s_n` the
measurement uncertainty). This gives :math:`\chi^2` when summed
over all data points. The objective function may also return the
log-posterior probability, :math:`\ln p(F_{true} | D)`. Since the
default log-prior term is zero, the objective function can also
just return the log-likelihood, unless you wish to create a
non-uniform prior.
If the objective function returns a float value, this is assumed
by default to be the log-posterior probability, (`float_behavior`
default is 'posterior'). If your objective function returns
:math:`\chi^2`, then you should use ``float_behavior='chi2'``
instead.
By default objective functions may return an ndarray of (possibly
weighted) residuals. In this case, use `is_weighted` to select
whether these are correctly weighted by measurement uncertainty.
Note that this ignores the second term above, so that to calculate
a correct log-posterior probability value your objective function
should return a float value. With ``is_weighted=False`` the data
uncertainty, `s_n`, will be treated as a nuisance parameter to be
marginalized out. This uses strictly positive uncertainty
(homoscedasticity) for each data point,
:math:`s_n = \exp(\rm{\_\_lnsigma})`. ``__lnsigma`` will be
present in `MinimizerResult.params`, as well as `Minimizer.chain`
and ``nvarys`` will be increased by one.
References
----------
.. [1] https://emcee.readthedocs.io
"""
if not HAS_EMCEE:
raise NotImplementedError('emcee version 3 is required.')
if ntemps > 1:
msg = ("'ntemps' has no effect anymore, since the PTSampler was "
"removed from emcee version 3.")
raise DeprecationWarning(msg)
tparams = params
# if you're reusing the sampler then nwalkers have to be
# determined from the previous sampling
if reuse_sampler:
if not hasattr(self, 'sampler') or not hasattr(self, '_lastpos'):
raise ValueError("You wanted to use an existing sampler, but "
"it hasn't been created yet")
if len(self._lastpos.shape) == 2:
nwalkers = self._lastpos.shape[0]
elif len(self._lastpos.shape) == 3:
nwalkers = self._lastpos.shape[1]
tparams = None
result = self.prepare_fit(params=tparams)
params = result.params
# check if the userfcn returns a vector of residuals
out = self.userfcn(params, *self.userargs, **self.userkws)
out = np.asarray(out).ravel()
if out.size > 1 and is_weighted is False and '__lnsigma' not in params:
# __lnsigma should already be in params if is_weighted was
# previously set to True.
params.add('__lnsigma', value=0.01, min=-np.inf, max=np.inf,
vary=True)
# have to re-prepare the fit
result = self.prepare_fit(params)
params = result.params
result.method = 'emcee'
# Removing internal parameter scaling. We could possibly keep it,
# but I don't know how this affects the emcee sampling.
bounds = []
var_arr = np.zeros(len(result.var_names))
i = 0
for par in params:
param = params[par]
if param.expr is not None:
param.vary = False
if param.vary:
var_arr[i] = param.value
i += 1
else:
# don't want to append bounds if they're not being varied.
continue
param.from_internal = lambda val: val
lb, ub = param.min, param.max
if lb is None or lb is np.nan:
lb = -np.inf
if ub is None or ub is np.nan:
ub = np.inf
bounds.append((lb, ub))
bounds = np.array(bounds)
self.nvarys = len(result.var_names)
# set up multiprocessing options for the samplers
auto_pool = None
sampler_kwargs = {}
if isinstance(workers, int) and workers > 1 and HAS_DILL:
auto_pool = multiprocessing.Pool(workers)
sampler_kwargs['pool'] = auto_pool
elif hasattr(workers, 'map'):
sampler_kwargs['pool'] = workers
# function arguments for the log-probability functions
# these values are sent to the log-probability functions by the sampler.
lnprob_args = (self.userfcn, params, result.var_names, bounds)
lnprob_kwargs = {'is_weighted': is_weighted,
'float_behavior': float_behavior,
'userargs': self.userargs,
'userkws': self.userkws,
'nan_policy': self.nan_policy}
sampler_kwargs['args'] = lnprob_args
sampler_kwargs['kwargs'] = lnprob_kwargs
# set up the random number generator
rng = _make_random_gen(seed)
# now initialise the samplers
if reuse_sampler:
if auto_pool is not None:
self.sampler.pool = auto_pool
p0 = self._lastpos
if p0.shape[-1] != self.nvarys:
raise ValueError("You cannot reuse the sampler if the number "
"of varying parameters has changed")
else:
p0 = 1 + rng.randn(nwalkers, self.nvarys) * 1.e-4
p0 *= var_arr
sampler_kwargs.setdefault('pool', auto_pool)
self.sampler = emcee.EnsembleSampler(nwalkers, self.nvarys,
self._lnprob, **sampler_kwargs)
# user supplies an initialisation position for the chain
# If you try to run the sampler with p0 of a wrong size then you'll get
# a ValueError. Note, you can't initialise with a position if you are
# reusing the sampler.
if pos is not None and not reuse_sampler:
tpos = np.asfarray(pos)
if p0.shape == tpos.shape:
pass
# trying to initialise with a previous chain
elif tpos.shape[-1] == self.nvarys:
tpos = tpos[-1]
else:
raise ValueError('pos should have shape (nwalkers, nvarys)')
p0 = tpos
# if you specified a seed then you also need to seed the sampler
if seed is not None:
self.sampler.random_state = rng.get_state()
if not isinstance(run_mcmc_kwargs, dict):
raise ValueError('run_mcmc_kwargs should be a dict of keyword arguments')
# now do a production run, sampling all the time
try:
output = self.sampler.run_mcmc(p0, steps, progress=progress, **run_mcmc_kwargs)
self._lastpos = output.coords
except AbortFitException:
result.aborted = True
result.message = "Fit aborted by user callback. Could not estimate error-bars."
result.success = False
result.nfev = self.result.nfev
# discard the burn samples and thin
chain = self.sampler.get_chain(thin=thin, discard=burn)[..., :, :]
lnprobability = self.sampler.get_log_prob(thin=thin, discard=burn)[..., :]
flatchain = chain.reshape((-1, self.nvarys))
if not result.aborted:
quantiles = np.percentile(flatchain, [15.87, 50, 84.13], axis=0)
for i, var_name in enumerate(result.var_names):
std_l, median, std_u = quantiles[:, i]
params[var_name].value = median
params[var_name].stderr = 0.5 * (std_u - std_l)
params[var_name].correl = {}
params.update_constraints()
# work out correlation coefficients
corrcoefs = np.corrcoef(flatchain.T)
for i, var_name in enumerate(result.var_names):
for j, var_name2 in enumerate(result.var_names):
if i != j:
result.params[var_name].correl[var_name2] = corrcoefs[i, j]
result.chain = np.copy(chain)
result.lnprob = np.copy(lnprobability)
result.errorbars = True
result.nvarys = len(result.var_names)
result.nfev = nwalkers*steps
try:
result.acor = self.sampler.get_autocorr_time()
except AutocorrError as e:
print(str(e))
result.acceptance_fraction = self.sampler.acceptance_fraction
# Calculate the residual with the "best fit" parameters
out = self.userfcn(params, *self.userargs, **self.userkws)
result.residual = coerce_float64(out, nan_policy=self.nan_policy,
handle_inf=False)
# If uncertainty was automatically estimated, weight the residual properly
if not is_weighted and result.residual.size > 1 and '__lnsigma' in params:
result.residual /= np.exp(params['__lnsigma'].value)
# Calculate statistics for the two standard cases:
if isinstance(result.residual, np.ndarray) or (float_behavior == 'chi2'):
result._calculate_statistics()
# Handle special case unique to emcee:
# This should eventually be moved into result._calculate_statistics.
elif float_behavior == 'posterior':
result.ndata = 1
result.nfree = 1
# assuming prior prob = 1, this is true
_neg2_log_likel = -2*result.residual
# assumes that residual is properly weighted, avoid overflowing np.exp()
result.chisqr = np.exp(min(650, _neg2_log_likel))
result.redchi = result.chisqr / result.nfree
result.aic = _neg2_log_likel + 2 * result.nvarys
result.bic = _neg2_log_likel + np.log(result.ndata) * result.nvarys
if auto_pool is not None:
auto_pool.terminate()
return result
def least_squares(self, params=None, max_nfev=None, **kws):
"""Least-squares minimization using :scipydoc:`optimize.least_squares`.
This method wraps :scipydoc:`optimize.least_squares`, which has
built-in support for bounds and robust loss functions. By default
it uses the Trust Region Reflective algorithm with a linear loss
function (i.e., the standard least-squares problem).
Parameters
----------
params : Parameters, optional
Parameters to use as starting point.
max_nfev : int or None, optional
Maximum number of function evaluations. Defaults to
``2000*(nvars+1)``, where ``nvars`` is the number of variable
parameters.
**kws : dict, optional
Minimizer options to pass to :scipydoc:`optimize.least_squares`.
Returns
-------
MinimizerResult
Object containing the optimized parameters and several
goodness-of-fit statistics.
.. versionchanged:: 0.9.0
Return value changed to :class:`MinimizerResult`.
"""
result = self.prepare_fit(params)
result.method = 'least_squares'
replace_none = lambda x, sign: sign*np.inf if x is None else x
self.set_max_nfev(max_nfev, 2000*(result.nvarys+1))
start_vals, lower_bounds, upper_bounds = [], [], []
for vname in result.var_names:
par = self.params[vname]
start_vals.append(par.value)
lower_bounds.append(replace_none(par.min, -1))
upper_bounds.append(replace_none(par.max, 1))
least_squares_kws = dict(jac='2-point', method='trf', ftol=1e-08,
xtol=1e-08, gtol=1e-08, x_scale=1.0,
loss='linear', f_scale=1.0, diff_step=None,
tr_solver=None, tr_options={},
jac_sparsity=None, max_nfev=2*self.max_nfev,
verbose=0, kwargs={})
least_squares_kws.update(self.kws)
least_squares_kws.update(kws)
least_squares_kws['kwargs'].update({'apply_bounds_transformation': False})
result.call_kws = least_squares_kws
try:
ret = least_squares(self.__residual, start_vals,
bounds=(lower_bounds, upper_bounds),
**least_squares_kws)
result.residual = ret.fun
except AbortFitException:
ret = None
result.aborted = True
# Note: scipy.optimize.least_squares is actually returning the
# "last evaluation", which is not necessarily the "best result"; so we
# do that here for consistency
if not result.aborted:
result.nfev -= 1
result.residual = self.__residual(ret.x, False)
elif result.nfev > self.max_nfev-5:
result.nfev -= 2
_best = result.last_internal_values
result.residual = self.__residual(_best, False)
result._calculate_statistics()
if not result.aborted:
for attr in ret:
outattr = attr
if attr == 'nfev':
outattr = 'least_squares_nfev'
setattr(result, outattr, ret[attr])
result.x = np.atleast_1d(result.x)
# calculate the cov_x and estimate uncertainties/correlations
try:
if issparse(ret.jac):
hess = (ret.jac.T * ret.jac).toarray()
elif isinstance(ret.jac, LinearOperator):
identity = np.eye(ret.jac.shape[1], dtype=ret.jac.dtype)
hess = (ret.jac.T * ret.jac) * identity
else:
hess = np.matmul(ret.jac.T, ret.jac)
result.covar = np.linalg.inv(hess)
self._calculate_uncertainties_correlations()
except LinAlgError:
pass
return result
def leastsq(self, params=None, max_nfev=None, **kws):
"""Use Levenberg-Marquardt minimization to perform a fit.
It assumes that the input Parameters have been initialized, and a
function to minimize has been properly set up. When possible, this
calculates the estimated uncertainties and variable correlations
from the covariance matrix.
This method calls :scipydoc:`optimize.leastsq` and, by default,
numerical derivatives are used.
Parameters
----------
params : Parameters, optional
Parameters to use as starting point.
max_nfev : int or None, optional
Maximum number of function evaluations. Defaults to
``2000*(nvars+1)``, where ``nvars`` is the number of variable
parameters.
**kws : dict, optional
Minimizer options to pass to :scipydoc:`optimize.leastsq`.
Returns
-------
MinimizerResult
Object containing the optimized parameters and several
goodness-of-fit statistics.
.. versionchanged:: 0.9.0
Return value changed to :class:`MinimizerResult`.
"""
result = self.prepare_fit(params=params)
result.method = 'leastsq'
result.nfev -= 2 # correct for "pre-fit" initialization/checks
variables = result._init_vals_internal
# Note: we set max number of function evaluations here, and send twice
# that value to the solver so it essentially never stops on its own
self.set_max_nfev(max_nfev, 2000*(result.nvarys+1))
lskws = dict(Dfun=None, full_output=1, col_deriv=0, ftol=1.5e-8,
xtol=1.5e-8, gtol=0.0, maxfev=2*self.max_nfev,
epsfcn=1.e-10, factor=100, diag=None)
if 'maxfev' in kws:
warnings.warn(maxeval_warning.format('maxfev', thisfuncname()),
RuntimeWarning)
kws.pop('maxfev')
lskws.update(self.kws)
lskws.update(kws)
self.col_deriv = False
if lskws['Dfun'] is not None:
self.jacfcn = lskws['Dfun']
self.col_deriv = lskws['col_deriv']
lskws['Dfun'] = self.__jacobian
# suppress runtime warnings during fit and error analysis
orig_warn_settings = np.geterr()
np.seterr(all='ignore')
result.call_kws = lskws
try:
lsout = scipy_leastsq(self.__residual, variables, **lskws)
except AbortFitException:
pass
if not result.aborted:
_best, _cov, _infodict, errmsg, ier = lsout
else:
_best = result.last_internal_values
_cov = None
ier = -1
errmsg = 'Fit aborted.'
result.nfev -= 1
if result.nfev >= self.max_nfev:
result.nfev = self.max_nfev - 1
self.result.nfev = result.nfev
try:
result.residual = self.__residual(_best)
result._calculate_statistics()
except AbortFitException:
pass
result.ier = ier
result.lmdif_message = errmsg
result.success = ier in [1, 2, 3, 4]
if ier in {1, 2, 3}:
result.message = 'Fit succeeded.'
elif ier == 0:
result.message = ('Invalid Input Parameters. I.e. more variables '
'than data points given, tolerance < 0.0, or '
'no data provided.')
elif ier == 4:
result.message = 'One or more variable did not affect the fit.'
elif ier == 5:
result.message = self._err_max_evals.format(lskws['maxfev'])
else:
result.message = 'Tolerance seems to be too small.'
# self.errorbars = error bars were successfully estimated
result.errorbars = (_cov is not None)
if result.errorbars:
# transform the covariance matrix to "external" parameter space
result.covar = self._int2ext_cov_x(_cov, _best)
# calculate parameter uncertainties and correlations
self._calculate_uncertainties_correlations()
else:
result.message = f'{result.message} Could not estimate error-bars.'
np.seterr(**orig_warn_settings)
return result
def basinhopping(self, params=None, max_nfev=None, **kws):
"""Use the `basinhopping` algorithm to find the global minimum.
This method calls :scipydoc:`optimize.basinhopping` using the
default arguments. The default minimizer is ``BFGS``, but since
lmfit supports parameter bounds for all minimizers, the user can
choose any of the solvers present in :scipydoc:`optimize.minimize`.
Parameters
----------
params : Parameters, optional
Contains the Parameters for the model. If None, then the
Parameters used to initialize the Minimizer object are used.
max_nfev : int or None, optional
Maximum number of function evaluations (default is None). Defaults
to ``200000*(nvarys+1)``.
**kws : dict, optional
Minimizer options to pass to :scipydoc:`optimize.basinhopping`.
Returns
-------
MinimizerResult
Object containing the optimization results from the
basinhopping algorithm.
.. versionadded:: 0.9.10
"""
result = self.prepare_fit(params=params)
result.method = 'basinhopping'
self.set_max_nfev(max_nfev, 200000*(result.nvarys+1))
basinhopping_kws = dict(niter=100, T=1.0, stepsize=0.5,
minimizer_kwargs=None, take_step=None,
accept_test=None, callback=None, interval=50,
disp=False, niter_success=None, seed=None)
# FIXME: update when SciPy requirement is >= 1.8
if int(scipy_version.split('.')[1]) >= 8:
basinhopping_kws.update({'target_accept_rate': 0.5,
'stepwise_factor': 0.9})
basinhopping_kws.update(self.kws)
basinhopping_kws.update(kws)
x0 = result._init_vals_internal
result.call_kws = basinhopping_kws
try:
ret = scipy_basinhopping(self.penalty, x0, **basinhopping_kws)
except AbortFitException:
pass
if not result.aborted:
result.message = ret.message
result.residual = self.__residual(ret.x)
result.nfev -= 1
elif result.nfev > self.max_nfev-5:
result.nfev -= 2
_best = result.last_internal_values
result.residual = self.__residual(_best, False)
result._calculate_statistics()
# calculate the cov_x and estimate uncertainties/correlations
if (not result.aborted and self.calc_covar and HAS_NUMDIFFTOOLS and
len(result.residual) > len(result.var_names)):
_covar_ndt = self._calculate_covariance_matrix(ret.x)
if _covar_ndt is not None:
result.covar = self._int2ext_cov_x(_covar_ndt, ret.x)
self._calculate_uncertainties_correlations()
return result
def brute(self, params=None, Ns=20, keep=50, workers=1, max_nfev=None):
"""Use the `brute` method to find the global minimum of a function.
The following parameters are passed to :scipydoc:`optimize.brute`
and cannot be changed:
+-------------------+-------+------------------------------------+
| :meth:`brute` arg | Value | Description |
+===================+=======+====================================+
| `full_output` | 1 | Return the evaluation grid and the |
| | | objective function's values on it. |
+-------------------+-------+------------------------------------+
| `finish` | None | No "polishing" function is to be |
| | | used after the grid search. |
+-------------------+-------+------------------------------------+
| `disp` | False | Do not print convergence messages |
| | | (when finish is not None). |
+-------------------+-------+------------------------------------+
It assumes that the input Parameters have been initialized, and a
function to minimize has been properly set up.
Parameters
----------
params : Parameters, optional
Contains the Parameters for the model. If None, then the
Parameters used to initialize the Minimizer object are used.
Ns : int, optional
Number of grid points along the axes, if not otherwise
specified (see Notes).
keep : int, optional
Number of best candidates from the brute force method that are
stored in the :attr:`candidates` attribute. If `'all'`, then
all grid points from :scipydoc:`optimize.brute` are stored as
candidates.
workers : int or map-like callable, optional
For parallel evaluation of the grid (see :scipydoc:`optimize.brute`
for more details).
max_nfev : int or None, optional
Maximum number of function evaluations (default is None). Defaults
to ``200000*(nvarys+1)``.
Returns
-------
MinimizerResult
Object containing the parameters from the brute force method.
The return values (``x0``, ``fval``, ``grid``, ``Jout``) from
:scipydoc:`optimize.brute` are stored as ``brute_<parname>``
attributes. The `MinimizerResult` also contains the
:attr:``candidates`` attribute and :meth:`show_candidates`
method. The :attr:`candidates` attribute contains the
parameters and chisqr from the brute force method as a
namedtuple, ``('Candidate', ['params', 'score'])`` sorted on
the (lowest) chisqr value. To access the values for a
particular candidate one can use ``result.candidate[#].params``
or ``result.candidate[#].score``, where a lower # represents a
better candidate. The :meth:`show_candidates` method uses the
:meth:`pretty_print` method to show a specific candidate-# or
all candidates when no number is specified.
.. versionadded:: 0.9.6
Notes
-----
The :meth:`brute` method evaluates the function at each point of a
multidimensional grid of points. The grid points are generated from
the parameter ranges using `Ns` and (optional) `brute_step`.
The implementation in :scipydoc:`optimize.brute` requires finite
bounds and the ``range`` is specified as a two-tuple ``(min, max)``
or slice-object ``(min, max, brute_step)``. A slice-object is used
directly, whereas a two-tuple is converted to a slice object that
interpolates `Ns` points from ``min`` to ``max``, inclusive.
In addition, the :meth:`brute` method in lmfit, handles three other
scenarios given below with their respective slice-object:
- lower bound (:attr:`min`) and :attr:`brute_step` are specified:
``range = (min, min + Ns * brute_step, brute_step)``.
- upper bound (:attr:`max`) and :attr:`brute_step` are specified:
``range = (max - Ns * brute_step, max, brute_step)``.
- numerical value (:attr:`value`) and :attr:`brute_step` are specified:
``range = (value - (Ns//2) * brute_step`, value +
(Ns//2) * brute_step, brute_step)``.
"""
result = self.prepare_fit(params=params)
result.method = 'brute'
self.set_max_nfev(max_nfev, 200000*(result.nvarys+1))
brute_kws = dict(full_output=1, finish=None, disp=False, Ns=Ns,
workers=workers)
varying = np.asarray([par.vary for par in self.params.values()])
replace_none = lambda x, sign: sign*np.inf if x is None else x
lower_bounds = np.asarray([replace_none(i.min, -1) for i in
self.params.values()])[varying]
upper_bounds = np.asarray([replace_none(i.max, 1) for i in
self.params.values()])[varying]
value = np.asarray([i.value for i in self.params.values()])[varying]
stepsize = np.asarray([i.brute_step for i in self.params.values()])[varying]
ranges = []
for i, step in enumerate(stepsize):
if np.all(np.isfinite([lower_bounds[i], upper_bounds[i]])):
# lower AND upper bounds are specified (brute_step optional)
par_range = ((lower_bounds[i], upper_bounds[i], step)
if step else (lower_bounds[i], upper_bounds[i]))
elif np.isfinite(lower_bounds[i]) and step:
# lower bound AND brute_step are specified
par_range = (lower_bounds[i], lower_bounds[i] + Ns*step, step)
elif np.isfinite(upper_bounds[i]) and step:
# upper bound AND brute_step are specified
par_range = (upper_bounds[i] - Ns*step, upper_bounds[i], step)
elif np.isfinite(value[i]) and step:
# no bounds, but an initial value is specified
par_range = (value[i] - (Ns//2)*step, value[i] + (Ns//2)*step,
step)
else:
raise ValueError('Not enough information provided for the brute '
'force method. Please specify bounds or at '
'least an initial value and brute_step for '
'parameter "{}".'.format(result.var_names[i]))
ranges.append(par_range)
result.call_kws = brute_kws
try:
ret = scipy_brute(self.penalty, tuple(ranges), **brute_kws)
except AbortFitException:
pass
if not result.aborted:
result.brute_x0 = ret[0]
result.brute_fval = ret[1]
result.brute_grid = ret[2]
result.brute_Jout = ret[3]
# sort the results of brute and populate .candidates attribute
grid_score = ret[3].ravel() # chisqr
grid_points = [par.ravel() for par in ret[2]]
if len(result.var_names) == 1:
grid_result = np.array([res for res in zip(zip(grid_points), grid_score)],
dtype=[('par', 'O'), ('score', 'float')])
else:
grid_result = np.array([res for res in zip(zip(*grid_points), grid_score)],
dtype=[('par', 'O'), ('score', 'float')])
grid_result_sorted = grid_result[grid_result.argsort(order='score')]
result.candidates = []
if keep == 'all':
keep_candidates = len(grid_result_sorted)
else:
keep_candidates = min(len(grid_result_sorted), keep)
for data in grid_result_sorted[:keep_candidates]:
pars = deepcopy(self.params)
for i, par in enumerate(result.var_names):
pars[par].value = data[0][i]
result.candidates.append(Candidate(params=pars, score=data[1]))
result.params = result.candidates[0].params
result.residual = self.__residual(result.brute_x0,
apply_bounds_transformation=False)
result.nfev = len(result.brute_Jout.ravel())
elif result.nfev > self.max_nfev-5:
result.nfev -= 2
_best = result.last_internal_values
result.residual = self.__residual(_best, False)
result._calculate_statistics()
return result
def ampgo(self, params=None, max_nfev=None, **kws):
"""Find the global minimum of a multivariate function using AMPGO.
AMPGO stands for 'Adaptive Memory Programming for Global
Optimization' and is an efficient algorithm to find the global
minimum.
Parameters
----------
params : Parameters, optional
Contains the Parameters for the model. If None, then the
Parameters used to initialize the Minimizer object are used.
max_nfev : int, optional
Maximum number of total function evaluations. If None
(default), the optimization will stop after `totaliter` number
of iterations (see below)..
**kws : dict, optional
Minimizer options to pass to the ampgo algorithm, the options
are listed below::
local: str, optional
Name of the local minimization method. Valid options
are:
- `'L-BFGS-B'` (default)
- `'Nelder-Mead'`
- `'Powell'`
- `'TNC'`
- `'SLSQP'`
local_opts: dict, optional
Options to pass to the local minimizer (default is
None).
maxfunevals: int, optional
Maximum number of function evaluations. If None
(default), the optimization will stop after
`totaliter` number of iterations (deprecated: use
`max_nfev` instead).
totaliter: int, optional
Maximum number of global iterations (default is 20).
maxiter: int, optional
Maximum number of `Tabu Tunneling` iterations during
each global iteration (default is 5).
glbtol: float, optional
Tolerance whether or not to accept a solution after a
tunneling phase (default is 1e-5).
eps1: float, optional
Constant used to define an aspiration value for the
objective function during the Tunneling phase (default
is 0.02).
eps2: float, optional
Perturbation factor used to move away from the latest
local minimum at the start of a Tunneling phase
(default is 0.1).
tabulistsize: int, optional
Size of the (circular) tabu search list (default is 5).
tabustrategy: {'farthest', 'oldest'}, optional
Strategy to use when the size of the tabu list exceeds
`tabulistsize`. It can be `'oldest'` to drop the oldest
point from the tabu list or `'farthest'` (defauilt) to
drop the element farthest from the last local minimum
found.
disp: bool, optional
Set to True to print convergence messages (default is
False).
Returns
-------
MinimizerResult
Object containing the parameters from the ampgo method, with
fit parameters, statistics and such. The return values
(``x0``, ``fval``, ``eval``, ``msg``, ``tunnel``) are stored
as ``ampgo_<parname>`` attributes.
.. versionadded:: 0.9.10
Notes
-----
The Python implementation was written by Andrea Gavana in 2014
(http://infinity77.net/global_optimization/index.html).
The details of the AMPGO algorithm are described in the paper
"Adaptive Memory Programming for Constrained Global Optimization"
located here:
http://leeds-faculty.colorado.edu/glover/fred%20pubs/416%20-%20AMP%20(TS)%20for%20Constrained%20Global%20Opt%20w%20Lasdon%20et%20al%20.pdf
"""
result = self.prepare_fit(params=params)
self.set_max_nfev(max_nfev, 200000*(result.nvarys+1))
ampgo_kws = dict(local='L-BFGS-B', local_opts=None, maxfunevals=None,
totaliter=20, maxiter=5, glbtol=1e-5, eps1=0.02,
eps2=0.1, tabulistsize=5, tabustrategy='farthest',
disp=False)
ampgo_kws.update(self.kws)
ampgo_kws.update(kws)
values = result._init_vals_internal
result.method = f"ampgo, with {ampgo_kws['local']} as local solver"
result.call_kws = ampgo_kws
try:
ret = ampgo(self.penalty, values, **ampgo_kws)
except AbortFitException:
pass
if not result.aborted:
result.ampgo_x0 = ret[0]
result.ampgo_fval = ret[1]
result.ampgo_eval = ret[2]
result.ampgo_msg = ret[3]
result.ampgo_tunnel = ret[4]
for i, par in enumerate(result.var_names):
result.params[par].value = result.ampgo_x0[i]
result.residual = self.__residual(result.ampgo_x0)
result.nfev -= 1
elif result.nfev > self.max_nfev-5:
result.nfev -= 2
_best = result.last_internal_values
result.residual = self.__residual(_best, False)
result._calculate_statistics()
# calculate the cov_x and estimate uncertainties/correlations
if (not result.aborted and self.calc_covar and HAS_NUMDIFFTOOLS and
len(result.residual) > len(result.var_names)):
_covar_ndt = self._calculate_covariance_matrix(result.ampgo_x0)
if _covar_ndt is not None:
result.covar = self._int2ext_cov_x(_covar_ndt, result.ampgo_x0)
self._calculate_uncertainties_correlations()
return result
def shgo(self, params=None, max_nfev=None, **kws):
"""Use the `SHGO` algorithm to find the global minimum.
SHGO stands for "simplicial homology global optimization" and
calls :scipydoc:`optimize.shgo` using its default arguments.
Parameters
----------
params : Parameters, optional
Contains the Parameters for the model. If None, then the
Parameters used to initialize the Minimizer object are used.
max_nfev : int or None, optional
Maximum number of function evaluations. Defaults to
``200000*(nvars+1)``, where ``nvars`` is the number of variable
parameters.
**kws : dict, optional
Minimizer options to pass to the SHGO algorithm.
Returns
-------
MinimizerResult
Object containing the parameters from the SHGO method.
The return values specific to :scipydoc:`optimize.shgo`
(``x``, ``xl``, ``fun``, ``funl``, ``nfev``, ``nit``,
``nlfev``, ``nlhev``, and ``nljev``) are stored as
``shgo_<parname>`` attributes.
.. versionadded:: 0.9.14
"""
result = self.prepare_fit(params=params)
result.method = 'shgo'
self.set_max_nfev(max_nfev, 200000*(result.nvarys+1))
shgo_kws = dict(constraints=None, n=100, iters=1, callback=None,
minimizer_kwargs=None, options=None,
sampling_method='simplicial')
# FIXME: update when SciPy requirement is >= 1.7
if int(scipy_version.split('.')[1]) >= 7:
shgo_kws['n'] = None
shgo_kws.update(self.kws)
shgo_kws.update(kws)
varying = np.asarray([par.vary for par in self.params.values()])
bounds = np.asarray([(par.min, par.max) for par in
self.params.values()])[varying]
result.call_kws = shgo_kws
try:
ret = scipy_shgo(self.penalty, bounds, **shgo_kws)
except AbortFitException:
pass
if not result.aborted:
for attr, value in ret.items():
if attr in ['success', 'message']:
setattr(result, attr, value)
else:
setattr(result, f'shgo_{attr}', value)
result.residual = self.__residual(result.shgo_x, False)
result.nfev -= 1
elif result.nfev > self.max_nfev-5:
result.nfev -= 2
_best = result.last_internal_values
result.residual = self.__residual(_best, False)
result._calculate_statistics()
# calculate the cov_x and estimate uncertainties/correlations
if (not result.aborted and self.calc_covar and HAS_NUMDIFFTOOLS and
len(result.residual) > len(result.var_names)):
result.covar = self._calculate_covariance_matrix(result.shgo_x)
if result.covar is not None:
self._calculate_uncertainties_correlations()
return result
def dual_annealing(self, params=None, max_nfev=None, **kws):
"""Use the `dual_annealing` algorithm to find the global minimum.
This method calls :scipydoc:`optimize.dual_annealing` using its
default arguments.
Parameters
----------
params : Parameters, optional
Contains the Parameters for the model. If None, then the
Parameters used to initialize the Minimizer object are used.
max_nfev : int or None, optional
Maximum number of function evaluations. Defaults to
``200000*(nvars+1)``, where ``nvars`` is the number of variables.
**kws : dict, optional
Minimizer options to pass to the dual_annealing algorithm.
Returns
-------
MinimizerResult
Object containing the parameters from the dual_annealing
method. The return values specific to
:scipydoc:`optimize.dual_annealing` (``x``, ``fun``, ``nfev``,
``nhev``, ``njev``, and ``nit``) are stored as
``da_<parname>`` attributes.
.. versionadded:: 0.9.14
"""
result = self.prepare_fit(params=params)
result.method = 'dual_annealing'
self.set_max_nfev(max_nfev, 200000*(result.nvarys+1))
da_kws = dict(maxiter=1000, local_search_options={},
initial_temp=5230.0, restart_temp_ratio=2e-05,
visit=2.62, accept=-5.0, maxfun=2*self.max_nfev,
seed=None, no_local_search=False, callback=None, x0=None)
da_kws.update(self.kws)
da_kws.update(kws)
# FIXME: update when SciPy requirement is >= 1.8
# ``local_search_options`` deprecated in favor of ``minimizer_kwargs``
if int(scipy_version.split('.')[1]) >= 8:
da_kws.update({'minimizer_kwargs': da_kws.pop('local_search_options')})
varying = np.asarray([par.vary for par in self.params.values()])
bounds = np.asarray([(par.min, par.max) for par in
self.params.values()])[varying]
if not np.all(np.isfinite(bounds)):
raise ValueError('dual_annealing requires finite bounds for all'
' varying parameters')
result.call_kws = da_kws
try:
ret = scipy_dual_annealing(self.penalty, bounds, **da_kws)
except AbortFitException:
pass
if not result.aborted:
for attr, value in ret.items():
if attr in ['success', 'message']:
setattr(result, attr, value)
else:
setattr(result, f'da_{attr}', value)
result.residual = self.__residual(result.da_x, False)
result.nfev -= 1
elif result.nfev > self.max_nfev-5:
result.nfev -= 2
_best = result.last_internal_values
result.residual = self.__residual(_best, False)
result._calculate_statistics()
# calculate the cov_x and estimate uncertainties/correlations
if (not result.aborted and self.calc_covar and HAS_NUMDIFFTOOLS and
len(result.residual) > len(result.var_names)):
result.covar = self._calculate_covariance_matrix(result.da_x)
if result.covar is not None:
self._calculate_uncertainties_correlations()
return result
def minimize(self, method='leastsq', params=None, **kws):
"""Perform the minimization.
Parameters
----------
method : str, optional
Name of the fitting method to use. Valid values are:
- `'leastsq'`: Levenberg-Marquardt (default)
- `'least_squares'`: Least-Squares minimization, using Trust
Region Reflective method
- `'differential_evolution'`: differential evolution
- `'brute'`: brute force method
- `'basinhopping'`: basinhopping
- `'ampgo'`: Adaptive Memory Programming for Global
Optimization
- '`nelder`': Nelder-Mead
- `'lbfgsb'`: L-BFGS-B
- `'powell'`: Powell
- `'cg'`: Conjugate-Gradient
- `'newton'`: Newton-CG
- `'cobyla'`: Cobyla
- `'bfgs'`: BFGS
- `'tnc'`: Truncated Newton
- `'trust-ncg'`: Newton-CG trust-region
- `'trust-exact'`: nearly exact trust-region
- `'trust-krylov'`: Newton GLTR trust-region
- `'trust-constr'`: trust-region for constrained optimization
- `'dogleg'`: Dog-leg trust-region
- `'slsqp'`: Sequential Linear Squares Programming
- `'emcee'`: Maximum likelihood via Monte-Carlo Markov Chain
- `'shgo'`: Simplicial Homology Global Optimization
- `'dual_annealing'`: Dual Annealing optimization
In most cases, these methods wrap and use the method with the
same name from `scipy.optimize`, or use
`scipy.optimize.minimize` with the same `method` argument.
Thus `'leastsq'` will use `scipy.optimize.leastsq`, while
`'powell'` will use `scipy.optimize.minimizer(...,
method='powell')`.
For more details on the fitting methods please refer to the
`SciPy documentation
<https://docs.scipy.org/doc/scipy/reference/optimize.html>`__.
params : Parameters, optional
Parameters of the model to use as starting values.
**kws : optional
Additional arguments are passed to the underlying minimization
method.
Returns
-------
MinimizerResult
Object containing the optimized parameters and several
goodness-of-fit statistics.
.. versionchanged:: 0.9.0
Return value changed to :class:`MinimizerResult`.
"""
kwargs = {'params': params}
kwargs.update(self.kws)
for maxnfev_alias in ('maxfev', 'maxiter'):
if maxnfev_alias in kws:
warnings.warn(maxeval_warning.format(maxnfev_alias, thisfuncname()),
RuntimeWarning)
kws.pop(maxnfev_alias)
kwargs.update(kws)
user_method = method.lower()
if user_method.startswith('leasts'):
function = self.leastsq
elif user_method.startswith('least_s'):
function = self.least_squares
elif user_method == 'brute':
function = self.brute
elif user_method == 'basinhopping':
function = self.basinhopping
elif user_method == 'ampgo':
function = self.ampgo
elif user_method == 'emcee':
function = self.emcee
elif user_method == 'shgo':
function = self.shgo
elif user_method == 'dual_annealing':
function = self.dual_annealing
else:
function = self.scalar_minimize
for key, val in SCALAR_METHODS.items():
if (key.lower().startswith(user_method) or
val.lower().startswith(user_method)):
kwargs['method'] = val
return function(**kwargs)
def _make_random_gen(seed):
"""Turn seed into a numpy.random.RandomState instance.
If `seed` is None, return the RandomState singleton used by numpy.random.
If `seed` is an int, return a new RandomState instance seeded with seed.
If `seed` is already a RandomState instance, return it.
Otherwise raise a `ValueError`.
"""
if seed is None or seed is np.random:
return np.random.mtrand._rand
if isinstance(seed, (numbers.Integral, np.integer)):
return np.random.RandomState(seed)
if isinstance(seed, np.random.RandomState):
return seed
raise ValueError(f'{seed:r} cannot be used to seed a numpy.random.RandomState'
' instance')
def coerce_float64(arr, nan_policy='raise', handle_inf=True,
ravel=True, ravel_order='C'):
"""coerce array-like objects to be a float64 ndarrays, usually forcing to 1D arrays.
also handles behaviour when array contains ``numpy.nan`` or ``numpy.inf``.
Parameters
----------
arr : array_like
Input array to consider.
nan_policy : {'raise', 'propagate', 'omit'}, optional
policy for handling NaN values. One of:
`'raise'` - raise a `ValueError` if `arr` contains NaN (default)
`'propagate'` - propagate NaN
`'omit'` - filter NaN from input array
handle_inf : bool, optional
Whether to apply the `nan_policy` to +/-Inf (default is True).
ravel : bool, optional
Whether to force to be 1D array (default is True).
ravel_order : str, optional
array ordering to assume when unravelling array (default is 'C')
Returns
-------
array
ndarray of type np.float64, possibly after applying the `nan_policy`,
and usually raveling to 1-D array
Notes
-----
Parts of this function are based on scipy/stats/stats.py/_contains_nan
support for 'array-like` objects is from numpy `asfarray`, which includes
lists of numbers, pandas.Series, h5py.Datasets, and many other array-like
Python objects
"""
if np.iscomplexobj(arr):
arr = np.asfarray(arr, dtype=np.complex128).view(np.float64)
else:
arr = np.asfarray(arr, dtype=np.float64)
if ravel:
arr = arr.ravel(order=ravel_order)
if nan_policy not in ('propagate', 'omit', 'raise'):
raise ValueError("nan_policy must be 'propagate', 'omit', or 'raise'.")
if handle_inf:
handler_func = lambda x: ~np.isfinite(x)
else:
handler_func = isnull
if nan_policy == 'omit':
# mask locates any values to remove
mask = ~handler_func(arr)
if not np.all(mask): # there are some NaNs/infs/missing values
return arr[mask]
if nan_policy == 'raise':
try:
# Calling np.sum to avoid creating a huge array into memory
# e.g. np.isnan(a).any()
with np.errstate(invalid='ignore'):
contains_nan = handler_func(np.sum(arr))
except TypeError:
# If the check cannot be properly performed we fallback to omitting
# nan values and raising a warning. This can happen when attempting to
# sum things that are not numbers (e.g. as in the function `mode`).
contains_nan = False
warnings.warn("The input array could not be checked for NaNs. "
"NaNs will be ignored.", RuntimeWarning)
if contains_nan:
msg = ('NaN values detected in your input data or the output of '
'your objective/model function - fitting algorithms cannot '
'handle this! Please read https://lmfit.github.io/lmfit-py/faq.html#i-get-errors-from-nan-in-my-fit-what-can-i-do '
'for more information.')
raise ValueError(msg)
return arr
# coerce_float64 replaces _nan_policy. That was never part of the public API,
# but we'll have it raise a DeprecationWarning for a while.
# This change happened in June, 2023, v 1.2.1, so this function can removed
# sometime in 2024, or after v 1.3.
def _nan_policy(arr, nan_policy='raise', handle_inf=True, **kws):
warnings.warn('`_nan_policy` has been replaced with coerce_float64`', DeprecationWarning)
return coerce_float64(arr, nan_policy=nan_policy, handle_inf=handle_inf, **kws)
def minimize(fcn, params, method='leastsq', args=None, kws=None, iter_cb=None,
scale_covar=True, nan_policy='raise', reduce_fcn=None,
calc_covar=True, max_nfev=None, **fit_kws):
"""Perform the minimization of the objective function.
The minimize function takes an objective function to be minimized,
a dictionary (:class:`~lmfit.parameter.Parameters` ; Parameters) containing
the model parameters, and several optional arguments including the fitting
method.
Parameters
----------
fcn : callable
Objective function to be minimized. When method is `'leastsq'` or
'`least_squares`', the objective function should return an array
of residuals (difference between model and data) to be minimized
in a least-squares sense. With the scalar methods the objective
function can either return the residuals array or a single scalar
value. The function must have the signature::
fcn(params, *args, **kws)
params : Parameters
Contains the Parameters for the model.
method : str, optional
Name of the fitting method to use. Valid values are:
- `'leastsq'`: Levenberg-Marquardt (default)
- `'least_squares'`: Least-Squares minimization, using Trust Region Reflective method
- `'differential_evolution'`: differential evolution
- `'brute'`: brute force method
- `'basinhopping'`: basinhopping
- `'ampgo'`: Adaptive Memory Programming for Global Optimization
- '`nelder`': Nelder-Mead
- `'lbfgsb'`: L-BFGS-B
- `'powell'`: Powell
- `'cg'`: Conjugate-Gradient
- `'newton'`: Newton-CG
- `'cobyla'`: Cobyla
- `'bfgs'`: BFGS
- `'tnc'`: Truncated Newton
- `'trust-ncg'`: Newton-CG trust-region
- `'trust-exact'`: nearly exact trust-region
- `'trust-krylov'`: Newton GLTR trust-region
- `'trust-constr'`: trust-region for constrained optimization
- `'dogleg'`: Dog-leg trust-region
- `'slsqp'`: Sequential Linear Squares Programming
- `'emcee'`: Maximum likelihood via Monte-Carlo Markov Chain
- `'shgo'`: Simplicial Homology Global Optimization
- `'dual_annealing'`: Dual Annealing optimization
In most cases, these methods wrap and use the method of the same
name from `scipy.optimize`, or use `scipy.optimize.minimize` with
the same `method` argument. Thus `'leastsq'` will use
`scipy.optimize.leastsq`, while `'powell'` will use
`scipy.optimize.minimizer(..., method='powell')`
For more details on the fitting methods please refer to the
`SciPy docs <https://docs.scipy.org/doc/scipy/reference/optimize.html>`__.
args : tuple, optional
Positional arguments to pass to `fcn`.
kws : dict, optional
Keyword arguments to pass to `fcn`.
iter_cb : callable, optional
Function to be called at each fit iteration. This function should
have the signature::
iter_cb(params, iter, resid, *args, **kws),
where `params` will have the current parameter values, `iter` the
iteration number, `resid` the current residual array, and `*args`
and `**kws` as passed to the objective function.
scale_covar : bool, optional
Whether to automatically scale the covariance matrix (default is
True).
nan_policy : {'raise', 'propagate', 'omit'}, optional
Specifies action if `fcn` (or a Jacobian) returns NaN values. One
of:
- `'raise'` : a `ValueError` is raised
- `'propagate'` : the values returned from `userfcn` are un-altered
- `'omit'` : non-finite values are filtered
reduce_fcn : str or callable, optional
Function to convert a residual array to a scalar value for the
scalar minimizers. See Notes in `Minimizer`.
calc_covar : bool, optional
Whether to calculate the covariance matrix (default is True) for
solvers other than `'leastsq'` and `'least_squares'`. Requires the
`numdifftools` package to be installed.
max_nfev : int or None, optional
Maximum number of function evaluations (default is None). The
default value depends on the fitting method.
**fit_kws : dict, optional
Options to pass to the minimizer being used.
Returns
-------
MinimizerResult
Object containing the optimized parameters and several
goodness-of-fit statistics.
.. versionchanged:: 0.9.0
Return value changed to :class:`MinimizerResult`.
Notes
-----
The objective function should return the value to be minimized. For
the Levenberg-Marquardt algorithm from leastsq(), this returned value
must be an array, with a length greater than or equal to the number of
fitting variables in the model. For the other methods, the return
value can either be a scalar or an array. If an array is returned, the
sum-of- squares of the array will be sent to the underlying fitting
method, effectively doing a least-squares optimization of the return
values.
A common use for `args` and `kws` would be to pass in other data needed
to calculate the residual, including such things as the data array,
dependent variable, uncertainties in the data, and other data structures
for the model calculation.
On output, `params` will be unchanged. The best-fit values and, where
appropriate, estimated uncertainties and correlations, will all be
contained in the returned :class:`MinimizerResult`. See
:ref:`fit-results-label` for further details.
This function is simply a wrapper around :class:`Minimizer` and is
equivalent to::
fitter = Minimizer(fcn, params, fcn_args=args, fcn_kws=kws,
iter_cb=iter_cb, scale_covar=scale_covar,
nan_policy=nan_policy, reduce_fcn=reduce_fcn,
calc_covar=calc_covar, **fit_kws)
fitter.minimize(method=method)
"""
fitter = Minimizer(fcn, params, fcn_args=args, fcn_kws=kws,
iter_cb=iter_cb, scale_covar=scale_covar,
nan_policy=nan_policy, reduce_fcn=reduce_fcn,
calc_covar=calc_covar, max_nfev=max_nfev, **fit_kws)
return fitter.minimize(method=method)
