import warnings
from . import _minpack
import numpy as np
from numpy import (atleast_1d, triu, shape, transpose, zeros, prod, greater,
asarray, inf,
finfo, inexact, issubdtype, dtype)
from scipy import linalg
from scipy.linalg import svd, cholesky, solve_triangular, LinAlgError
from scipy._lib._util import _asarray_validated, _lazywhere, _contains_nan
from scipy._lib._util import getfullargspec_no_self as _getfullargspec
from ._optimize import OptimizeResult, _check_unknown_options, OptimizeWarning
from ._lsq import least_squares
# from ._lsq.common import make_strictly_feasible
from ._lsq.least_squares import prepare_bounds
from scipy.optimize._minimize import Bounds
error = _minpack.error
__all__ = ['fsolve', 'leastsq', 'fixed_point', 'curve_fit']
def _check_func(checker, argname, thefunc, x0, args, numinputs,
output_shape=None):
res = atleast_1d(thefunc(*((x0[:numinputs],) + args)))
if (output_shape is not None) and (shape(res) != output_shape):
if (output_shape[0] != 1):
if len(output_shape) > 1:
if output_shape[1] == 1:
return shape(res)
msg = "{}: there is a mismatch between the input and output " \
"shape of the '{}' argument".format(checker, argname)
func_name = getattr(thefunc, '__name__', None)
if func_name:
msg += " '%s'." % func_name
else:
msg += "."
msg += f'Shape should be {output_shape} but it is {shape(res)}.'
raise TypeError(msg)
if issubdtype(res.dtype, inexact):
dt = res.dtype
else:
dt = dtype(float)
return shape(res), dt
def fsolve(func, x0, args=(), fprime=None, full_output=0,
col_deriv=0, xtol=1.49012e-8, maxfev=0, band=None,
epsfcn=None, factor=100, diag=None):
"""
Find the roots of a function.
Return the roots of the (non-linear) equations defined by
``func(x) = 0`` given a starting estimate.
Parameters
----------
func : callable ``f(x, *args)``
A function that takes at least one (possibly vector) argument,
and returns a value of the same length.
x0 : ndarray
The starting estimate for the roots of ``func(x) = 0``.
args : tuple, optional
Any extra arguments to `func`.
fprime : callable ``f(x, *args)``, optional
A function to compute the Jacobian of `func` with derivatives
across the rows. By default, the Jacobian will be estimated.
full_output : bool, optional
If True, return optional outputs.
col_deriv : bool, optional
Specify whether the Jacobian function computes derivatives down
the columns (faster, because there is no transpose operation).
xtol : float, optional
The calculation will terminate if the relative error between two
consecutive iterates is at most `xtol`.
maxfev : int, optional
The maximum number of calls to the function. If zero, then
``100*(N+1)`` is the maximum where N is the number of elements
in `x0`.
band : tuple, optional
If set to a two-sequence containing the number of sub- and
super-diagonals within the band of the Jacobi matrix, the
Jacobi matrix is considered banded (only for ``fprime=None``).
epsfcn : float, optional
A suitable step length for the forward-difference
approximation of the Jacobian (for ``fprime=None``). If
`epsfcn` is less than the machine precision, it is assumed
that the relative errors in the functions are of the order of
the machine precision.
factor : float, optional
A parameter determining the initial step bound
(``factor * || diag * x||``). Should be in the interval
``(0.1, 100)``.
diag : sequence, optional
N positive entries that serve as a scale factors for the
variables.
Returns
-------
x : ndarray
The solution (or the result of the last iteration for
an unsuccessful call).
infodict : dict
A dictionary of optional outputs with the keys:
``nfev``
number of function calls
``njev``
number of Jacobian calls
``fvec``
function evaluated at the output
``fjac``
the orthogonal matrix, q, produced by the QR
factorization of the final approximate Jacobian
matrix, stored column wise
``r``
upper triangular matrix produced by QR factorization
of the same matrix
``qtf``
the vector ``(transpose(q) * fvec)``
ier : int
An integer flag. Set to 1 if a solution was found, otherwise refer
to `mesg` for more information.
mesg : str
If no solution is found, `mesg` details the cause of failure.
See Also
--------
root : Interface to root finding algorithms for multivariate
functions. See the ``method='hybr'`` in particular.
Notes
-----
``fsolve`` is a wrapper around MINPACK's hybrd and hybrj algorithms.
Examples
--------
Find a solution to the system of equations:
``x0*cos(x1) = 4, x1*x0 - x1 = 5``.
>>> import numpy as np
>>> from scipy.optimize import fsolve
>>> def func(x):
... return [x[0] * np.cos(x[1]) - 4,
... x[1] * x[0] - x[1] - 5]
>>> root = fsolve(func, [1, 1])
>>> root
array([6.50409711, 0.90841421])
>>> np.isclose(func(root), [0.0, 0.0]) # func(root) should be almost 0.0.
array([ True, True])
"""
options = {'col_deriv': col_deriv,
'xtol': xtol,
'maxfev': maxfev,
'band': band,
'eps': epsfcn,
'factor': factor,
'diag': diag}
res = _root_hybr(func, x0, args, jac=fprime, **options)
if full_output:
x = res['x']
info = {k: res.get(k)
for k in ('nfev', 'njev', 'fjac', 'r', 'qtf') if k in res}
info['fvec'] = res['fun']
return x, info, res['status'], res['message']
else:
status = res['status']
msg = res['message']
if status == 0:
raise TypeError(msg)
elif status == 1:
pass
elif status in [2, 3, 4, 5]:
warnings.warn(msg, RuntimeWarning)
else:
raise TypeError(msg)
return res['x']
def _root_hybr(func, x0, args=(), jac=None,
col_deriv=0, xtol=1.49012e-08, maxfev=0, band=None, eps=None,
factor=100, diag=None, **unknown_options):
"""
Find the roots of a multivariate function using MINPACK's hybrd and
hybrj routines (modified Powell method).
Options
-------
col_deriv : bool
Specify whether the Jacobian function computes derivatives down
the columns (faster, because there is no transpose operation).
xtol : float
The calculation will terminate if the relative error between two
consecutive iterates is at most `xtol`.
maxfev : int
The maximum number of calls to the function. If zero, then
``100*(N+1)`` is the maximum where N is the number of elements
in `x0`.
band : tuple
If set to a two-sequence containing the number of sub- and
super-diagonals within the band of the Jacobi matrix, the
Jacobi matrix is considered banded (only for ``fprime=None``).
eps : float
A suitable step length for the forward-difference
approximation of the Jacobian (for ``fprime=None``). If
`eps` is less than the machine precision, it is assumed
that the relative errors in the functions are of the order of
the machine precision.
factor : float
A parameter determining the initial step bound
(``factor * || diag * x||``). Should be in the interval
``(0.1, 100)``.
diag : sequence
N positive entries that serve as a scale factors for the
variables.
"""
_check_unknown_options(unknown_options)
epsfcn = eps
x0 = asarray(x0).flatten()
n = len(x0)
if not isinstance(args, tuple):
args = (args,)
shape, dtype = _check_func('fsolve', 'func', func, x0, args, n, (n,))
if epsfcn is None:
epsfcn = finfo(dtype).eps
Dfun = jac
if Dfun is None:
if band is None:
ml, mu = -10, -10
else:
ml, mu = band[:2]
if maxfev == 0:
maxfev = 200 * (n + 1)
retval = _minpack._hybrd(func, x0, args, 1, xtol, maxfev,
ml, mu, epsfcn, factor, diag)
else:
_check_func('fsolve', 'fprime', Dfun, x0, args, n, (n, n))
if (maxfev == 0):
maxfev = 100 * (n + 1)
retval = _minpack._hybrj(func, Dfun, x0, args, 1,
col_deriv, xtol, maxfev, factor, diag)
x, status = retval[0], retval[-1]
errors = {0: "Improper input parameters were entered.",
1: "The solution converged.",
2: "The number of calls to function has "
"reached maxfev = %d." % maxfev,
3: "xtol=%f is too small, no further improvement "
"in the approximate\n solution "
"is possible." % xtol,
4: "The iteration is not making good progress, as measured "
"by the \n improvement from the last five "
"Jacobian evaluations.",
5: "The iteration is not making good progress, "
"as measured by the \n improvement from the last "
"ten iterations.",
'unknown': "An error occurred."}
info = retval[1]
info['fun'] = info.pop('fvec')
sol = OptimizeResult(x=x, success=(status == 1), status=status)
sol.update(info)
try:
sol['message'] = errors[status]
except KeyError:
sol['message'] = errors['unknown']
return sol
LEASTSQ_SUCCESS = [1, 2, 3, 4]
LEASTSQ_FAILURE = [5, 6, 7, 8]
def leastsq(func, x0, args=(), Dfun=None, full_output=False,
col_deriv=False, ftol=1.49012e-8, xtol=1.49012e-8,
gtol=0.0, maxfev=0, epsfcn=None, factor=100, diag=None):
"""
Minimize the sum of squares of a set of equations.
::
x = arg min(sum(func(y)**2,axis=0))
y
Parameters
----------
func : callable
Should take at least one (possibly length ``N`` vector) argument and
returns ``M`` floating point numbers. It must not return NaNs or
fitting might fail. ``M`` must be greater than or equal to ``N``.
x0 : ndarray
The starting estimate for the minimization.
args : tuple, optional
Any extra arguments to func are placed in this tuple.
Dfun : callable, optional
A function or method to compute the Jacobian of func with derivatives
across the rows. If this is None, the Jacobian will be estimated.
full_output : bool, optional
If ``True``, return all optional outputs (not just `x` and `ier`).
col_deriv : bool, optional
If ``True``, specify that the Jacobian function computes derivatives
down the columns (faster, because there is no transpose operation).
ftol : float, optional
Relative error desired in the sum of squares.
xtol : float, optional
Relative error desired in the approximate solution.
gtol : float, optional
Orthogonality desired between the function vector and the columns of
the Jacobian.
maxfev : int, optional
The maximum number of calls to the function. If `Dfun` is provided,
then the default `maxfev` is 100*(N+1) where N is the number of elements
in x0, otherwise the default `maxfev` is 200*(N+1).
epsfcn : float, optional
A variable used in determining a suitable step length for the forward-
difference approximation of the Jacobian (for Dfun=None).
Normally the actual step length will be sqrt(epsfcn)*x
If epsfcn is less than the machine precision, it is assumed that the
relative errors are of the order of the machine precision.
factor : float, optional
A parameter determining the initial step bound
(``factor * || diag * x||``). Should be in interval ``(0.1, 100)``.
diag : sequence, optional
N positive entries that serve as a scale factors for the variables.
Returns
-------
x : ndarray
The solution (or the result of the last iteration for an unsuccessful
call).
cov_x : ndarray
The inverse of the Hessian. `fjac` and `ipvt` are used to construct an
estimate of the Hessian. A value of None indicates a singular matrix,
which means the curvature in parameters `x` is numerically flat. To
obtain the covariance matrix of the parameters `x`, `cov_x` must be
multiplied by the variance of the residuals -- see curve_fit. Only
returned if `full_output` is ``True``.
infodict : dict
a dictionary of optional outputs with the keys:
``nfev``
The number of function calls
``fvec``
The function evaluated at the output
``fjac``
A permutation of the R matrix of a QR
factorization of the final approximate
Jacobian matrix, stored column wise.
Together with ipvt, the covariance of the
estimate can be approximated.
``ipvt``
An integer array of length N which defines
a permutation matrix, p, such that
fjac*p = q*r, where r is upper triangular
with diagonal elements of nonincreasing
magnitude. Column j of p is column ipvt(j)
of the identity matrix.
``qtf``
The vector (transpose(q) * fvec).
Only returned if `full_output` is ``True``.
mesg : str
A string message giving information about the cause of failure.
Only returned if `full_output` is ``True``.
ier : int
An integer flag. If it is equal to 1, 2, 3 or 4, the solution was
found. Otherwise, the solution was not found. In either case, the
optional output variable 'mesg' gives more information.
See Also
--------
least_squares : Newer interface to solve nonlinear least-squares problems
with bounds on the variables. See ``method='lm'`` in particular.
Notes
-----
"leastsq" is a wrapper around MINPACK's lmdif and lmder algorithms.
cov_x is a Jacobian approximation to the Hessian of the least squares
objective function.
This approximation assumes that the objective function is based on the
difference between some observed target data (ydata) and a (non-linear)
function of the parameters `f(xdata, params)` ::
func(params) = ydata - f(xdata, params)
so that the objective function is ::
min sum((ydata - f(xdata, params))**2, axis=0)
params
The solution, `x`, is always a 1-D array, regardless of the shape of `x0`,
or whether `x0` is a scalar.
Examples
--------
>>> from scipy.optimize import leastsq
>>> def func(x):
... return 2*(x-3)**2+1
>>> leastsq(func, 0)
(array([2.99999999]), 1)
"""
x0 = asarray(x0).flatten()
n = len(x0)
if not isinstance(args, tuple):
args = (args,)
shape, dtype = _check_func('leastsq', 'func', func, x0, args, n)
m = shape[0]
if n > m:
raise TypeError(f"Improper input: func input vector length N={n} must"
f" not exceed func output vector length M={m}")
if epsfcn is None:
epsfcn = finfo(dtype).eps
if Dfun is None:
if maxfev == 0:
maxfev = 200*(n + 1)
retval = _minpack._lmdif(func, x0, args, full_output, ftol, xtol,
gtol, maxfev, epsfcn, factor, diag)
else:
if col_deriv:
_check_func('leastsq', 'Dfun', Dfun, x0, args, n, (n, m))
else:
_check_func('leastsq', 'Dfun', Dfun, x0, args, n, (m, n))
if maxfev == 0:
maxfev = 100 * (n + 1)
retval = _minpack._lmder(func, Dfun, x0, args, full_output,
col_deriv, ftol, xtol, gtol, maxfev,
factor, diag)
errors = {0: ["Improper input parameters.", TypeError],
1: ["Both actual and predicted relative reductions "
"in the sum of squares\n are at most %f" % ftol, None],
2: ["The relative error between two consecutive "
"iterates is at most %f" % xtol, None],
3: ["Both actual and predicted relative reductions in "
"the sum of squares\n are at most {:f} and the "
"relative error between two consecutive "
"iterates is at \n most {:f}".format(ftol, xtol), None],
4: ["The cosine of the angle between func(x) and any "
"column of the\n Jacobian is at most %f in "
"absolute value" % gtol, None],
5: ["Number of calls to function has reached "
"maxfev = %d." % maxfev, ValueError],
6: ["ftol=%f is too small, no further reduction "
"in the sum of squares\n is possible." % ftol,
ValueError],
7: ["xtol=%f is too small, no further improvement in "
"the approximate\n solution is possible." % xtol,
ValueError],
8: ["gtol=%f is too small, func(x) is orthogonal to the "
"columns of\n the Jacobian to machine "
"precision." % gtol, ValueError]}
# The FORTRAN return value (possible return values are >= 0 and <= 8)
info = retval[-1]
if full_output:
cov_x = None
if info in LEASTSQ_SUCCESS:
# This was
# perm = take(eye(n), retval[1]['ipvt'] - 1, 0)
# r = triu(transpose(retval[1]['fjac'])[:n, :])
# R = dot(r, perm)
# cov_x = inv(dot(transpose(R), R))
# but the explicit dot product was not necessary and sometimes
# the result was not symmetric positive definite. See gh-4555.
perm = retval[1]['ipvt'] - 1
n = len(perm)
r = triu(transpose(retval[1]['fjac'])[:n, :])
inv_triu = linalg.get_lapack_funcs('trtri', (r,))
try:
# inverse of permuted matrix is a permuation of matrix inverse
invR, trtri_info = inv_triu(r) # default: upper, non-unit diag
if trtri_info != 0: # explicit comparison for readability
raise LinAlgError(f'trtri returned info {trtri_info}')
invR[perm] = invR.copy()
cov_x = invR @ invR.T
except (LinAlgError, ValueError):
pass
return (retval[0], cov_x) + retval[1:-1] + (errors[info][0], info)
else:
if info in LEASTSQ_FAILURE:
warnings.warn(errors[info][0], RuntimeWarning)
elif info == 0:
raise errors[info][1](errors[info][0])
return retval[0], info
def _lightweight_memoizer(f):
# very shallow memoization - only remember the first set of parameters
# and corresponding function value to address gh-13670
def _memoized_func(params):
if np.all(_memoized_func.last_params == params):
return _memoized_func.last_val
val = f(params)
if _memoized_func.last_params is None:
_memoized_func.last_params = np.copy(params)
_memoized_func.last_val = val
return val
_memoized_func.last_params = None
_memoized_func.last_val = None
return _memoized_func
def _wrap_func(func, xdata, ydata, transform):
if transform is None:
def func_wrapped(params):
return func(xdata, *params) - ydata
elif transform.ndim == 1:
def func_wrapped(params):
return transform * (func(xdata, *params) - ydata)
else:
# Chisq = (y - yd)^T C^{-1} (y-yd)
# transform = L such that C = L L^T
# C^{-1} = L^{-T} L^{-1}
# Chisq = (y - yd)^T L^{-T} L^{-1} (y-yd)
# Define (y-yd)' = L^{-1} (y-yd)
# by solving
# L (y-yd)' = (y-yd)
# and minimize (y-yd)'^T (y-yd)'
def func_wrapped(params):
return solve_triangular(transform, func(xdata, *params) - ydata, lower=True)
return func_wrapped
def _wrap_jac(jac, xdata, transform):
if transform is None:
def jac_wrapped(params):
return jac(xdata, *params)
elif transform.ndim == 1:
def jac_wrapped(params):
return transform[:, np.newaxis] * np.asarray(jac(xdata, *params))
else:
def jac_wrapped(params):
return solve_triangular(transform, np.asarray(jac(xdata, *params)), lower=True)
return jac_wrapped
def _initialize_feasible(lb, ub):
p0 = np.ones_like(lb)
lb_finite = np.isfinite(lb)
ub_finite = np.isfinite(ub)
mask = lb_finite & ub_finite
p0[mask] = 0.5 * (lb[mask] + ub[mask])
mask = lb_finite & ~ub_finite
p0[mask] = lb[mask] + 1
mask = ~lb_finite & ub_finite
p0[mask] = ub[mask] - 1
return p0
[docs]def curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False,
check_finite=None, bounds=(-np.inf, np.inf), method=None,
jac=None, *, full_output=False, nan_policy=None,
**kwargs):
"""
Use non-linear least squares to fit a function, f, to data.
Assumes ``ydata = f(xdata, *params) + eps``.
Parameters
----------
f : callable
The model function, f(x, ...). It must take the independent
variable as the first argument and the parameters to fit as
separate remaining arguments.
xdata : array_like
The independent variable where the data is measured.
Should usually be an M-length sequence or an (k,M)-shaped array for
functions with k predictors, and each element should be float
convertible if it is an array like object.
ydata : array_like
The dependent data, a length M array - nominally ``f(xdata, ...)``.
p0 : array_like, optional
Initial guess for the parameters (length N). If None, then the
initial values will all be 1 (if the number of parameters for the
function can be determined using introspection, otherwise a
ValueError is raised).
sigma : None or M-length sequence or MxM array, optional
Determines the uncertainty in `ydata`. If we define residuals as
``r = ydata - f(xdata, *popt)``, then the interpretation of `sigma`
depends on its number of dimensions:
- A 1-D `sigma` should contain values of standard deviations of
errors in `ydata`. In this case, the optimized function is
``chisq = sum((r / sigma) ** 2)``.
- A 2-D `sigma` should contain the covariance matrix of
errors in `ydata`. In this case, the optimized function is
``chisq = r.T @ inv(sigma) @ r``.
.. versionadded:: 0.19
None (default) is equivalent of 1-D `sigma` filled with ones.
absolute_sigma : bool, optional
If True, `sigma` is used in an absolute sense and the estimated parameter
covariance `pcov` reflects these absolute values.
If False (default), only the relative magnitudes of the `sigma` values matter.
The returned parameter covariance matrix `pcov` is based on scaling
`sigma` by a constant factor. This constant is set by demanding that the
reduced `chisq` for the optimal parameters `popt` when using the
*scaled* `sigma` equals unity. In other words, `sigma` is scaled to
match the sample variance of the residuals after the fit. Default is False.
Mathematically,
``pcov(absolute_sigma=False) = pcov(absolute_sigma=True) * chisq(popt)/(M-N)``
check_finite : bool, optional
If True, check that the input arrays do not contain nans of infs,
and raise a ValueError if they do. Setting this parameter to
False may silently produce nonsensical results if the input arrays
do contain nans. Default is True if `nan_policy` is not specified
explicitly and False otherwise.
bounds : 2-tuple of array_like or `Bounds`, optional
Lower and upper bounds on parameters. Defaults to no bounds.
There are two ways to specify the bounds:
- Instance of `Bounds` class.
- 2-tuple of array_like: Each element of the tuple must be either
an array with the length equal to the number of parameters, or a
scalar (in which case the bound is taken to be the same for all
parameters). Use ``np.inf`` with an appropriate sign to disable
bounds on all or some parameters.
method : {'lm', 'trf', 'dogbox'}, optional
Method to use for optimization. See `least_squares` for more details.
Default is 'lm' for unconstrained problems and 'trf' if `bounds` are
provided. The method 'lm' won't work when the number of observations
is less than the number of variables, use 'trf' or 'dogbox' in this
case.
.. versionadded:: 0.17
jac : callable, string or None, optional
Function with signature ``jac(x, ...)`` which computes the Jacobian
matrix of the model function with respect to parameters as a dense
array_like structure. It will be scaled according to provided `sigma`.
If None (default), the Jacobian will be estimated numerically.
String keywords for 'trf' and 'dogbox' methods can be used to select
a finite difference scheme, see `least_squares`.
.. versionadded:: 0.18
full_output : boolean, optional
If True, this function returns additioal information: `infodict`,
`mesg`, and `ier`.
.. versionadded:: 1.9
nan_policy : {'raise', 'omit', None}, optional
Defines how to handle when input contains nan.
The following options are available (default is None):
* 'raise': throws an error
* 'omit': performs the calculations ignoring nan values
* None: no special handling of NaNs is performed
(except what is done by check_finite); the behavior when NaNs
are present is implementation-dependent and may change.
Note that if this value is specified explicitly (not None),
`check_finite` will be set as False.
.. versionadded:: 1.11
**kwargs
Keyword arguments passed to `leastsq` for ``method='lm'`` or
`least_squares` otherwise.
Returns
-------
popt : array
Optimal values for the parameters so that the sum of the squared
residuals of ``f(xdata, *popt) - ydata`` is minimized.
pcov : 2-D array
The estimated approximate covariance of popt. The diagonals provide
the variance of the parameter estimate. To compute one standard
deviation errors on the parameters, use
``perr = np.sqrt(np.diag(pcov))``. Note that the relationship between
`cov` and parameter error estimates is derived based on a linear
approximation to the model function around the optimum [1].
When this approximation becomes inaccurate, `cov` may not provide an
accurate measure of uncertainty.
How the `sigma` parameter affects the estimated covariance
depends on `absolute_sigma` argument, as described above.
If the Jacobian matrix at the solution doesn't have a full rank, then
'lm' method returns a matrix filled with ``np.inf``, on the other hand
'trf' and 'dogbox' methods use Moore-Penrose pseudoinverse to compute
the covariance matrix. Covariance matrices with large condition numbers
(e.g. computed with `numpy.linalg.cond`) may indicate that results are
unreliable.
infodict : dict (returned only if `full_output` is True)
a dictionary of optional outputs with the keys:
``nfev``
The number of function calls. Methods 'trf' and 'dogbox' do not
count function calls for numerical Jacobian approximation,
as opposed to 'lm' method.
``fvec``
The residual values evaluated at the solution, for a 1-D `sigma`
this is ``(f(x, *popt) - ydata)/sigma``.
``fjac``
A permutation of the R matrix of a QR
factorization of the final approximate
Jacobian matrix, stored column wise.
Together with ipvt, the covariance of the
estimate can be approximated.
Method 'lm' only provides this information.
``ipvt``
An integer array of length N which defines
a permutation matrix, p, such that
fjac*p = q*r, where r is upper triangular
with diagonal elements of nonincreasing
magnitude. Column j of p is column ipvt(j)
of the identity matrix.
Method 'lm' only provides this information.
``qtf``
The vector (transpose(q) * fvec).
Method 'lm' only provides this information.
.. versionadded:: 1.9
mesg : str (returned only if `full_output` is True)
A string message giving information about the solution.
.. versionadded:: 1.9
ier : int (returnned only if `full_output` is True)
An integer flag. If it is equal to 1, 2, 3 or 4, the solution was
found. Otherwise, the solution was not found. In either case, the
optional output variable `mesg` gives more information.
.. versionadded:: 1.9
Raises
------
ValueError
if either `ydata` or `xdata` contain NaNs, or if incompatible options
are used.
RuntimeError
if the least-squares minimization fails.
OptimizeWarning
if covariance of the parameters can not be estimated.
See Also
--------
least_squares : Minimize the sum of squares of nonlinear functions.
scipy.stats.linregress : Calculate a linear least squares regression for
two sets of measurements.
Notes
-----
Users should ensure that inputs `xdata`, `ydata`, and the output of `f`
are ``float64``, or else the optimization may return incorrect results.
With ``method='lm'``, the algorithm uses the Levenberg-Marquardt algorithm
through `leastsq`. Note that this algorithm can only deal with
unconstrained problems.
Box constraints can be handled by methods 'trf' and 'dogbox'. Refer to
the docstring of `least_squares` for more information.
References
----------
[1] K. Vugrin et al. Confidence region estimation techniques for nonlinear
regression in groundwater flow: Three case studies. Water Resources
Research, Vol. 43, W03423, :doi:`10.1029/2005WR004804`
Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.optimize import curve_fit
>>> def func(x, a, b, c):
... return a * np.exp(-b * x) + c
Define the data to be fit with some noise:
>>> xdata = np.linspace(0, 4, 50)
>>> y = func(xdata, 2.5, 1.3, 0.5)
>>> rng = np.random.default_rng()
>>> y_noise = 0.2 * rng.normal(size=xdata.size)
>>> ydata = y + y_noise
>>> plt.plot(xdata, ydata, 'b-', label='data')
Fit for the parameters a, b, c of the function `func`:
>>> popt, pcov = curve_fit(func, xdata, ydata)
>>> popt
array([2.56274217, 1.37268521, 0.47427475])
>>> plt.plot(xdata, func(xdata, *popt), 'r-',
... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))
Constrain the optimization to the region of ``0 <= a <= 3``,
``0 <= b <= 1`` and ``0 <= c <= 0.5``:
>>> popt, pcov = curve_fit(func, xdata, ydata, bounds=(0, [3., 1., 0.5]))
>>> popt
array([2.43736712, 1. , 0.34463856])
>>> plt.plot(xdata, func(xdata, *popt), 'g--',
... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))
>>> plt.xlabel('x')
>>> plt.ylabel('y')
>>> plt.legend()
>>> plt.show()
For reliable results, the model `func` should not be overparametrized;
redundant parameters can cause unreliable covariance matrices and, in some
cases, poorer quality fits. As a quick check of whether the model may be
overparameterized, calculate the condition number of the covariance matrix:
>>> np.linalg.cond(pcov)
34.571092161547405 # may vary
The value is small, so it does not raise much concern. If, however, we were
to add a fourth parameter ``d`` to `func` with the same effect as ``a``:
>>> def func(x, a, b, c, d):
... return a * d * np.exp(-b * x) + c # a and d are redundant
>>> popt, pcov = curve_fit(func, xdata, ydata)
>>> np.linalg.cond(pcov)
1.13250718925596e+32 # may vary
Such a large value is cause for concern. The diagonal elements of the
covariance matrix, which is related to uncertainty of the fit, gives more
information:
>>> np.diag(pcov)
array([1.48814742e+29, 3.78596560e-02, 5.39253738e-03, 2.76417220e+28]) # may vary
Note that the first and last terms are much larger than the other elements,
suggesting that the optimal values of these parameters are ambiguous and
that only one of these parameters is needed in the model.
""" # noqa
if p0 is None:
# determine number of parameters by inspecting the function
sig = _getfullargspec(f)
args = sig.args
if len(args) < 2:
raise ValueError("Unable to determine number of fit parameters.")
n = len(args) - 1
else:
p0 = np.atleast_1d(p0)
n = p0.size
if isinstance(bounds, Bounds):
lb, ub = bounds.lb, bounds.ub
else:
lb, ub = prepare_bounds(bounds, n)
if p0 is None:
p0 = _initialize_feasible(lb, ub)
bounded_problem = np.any((lb > -np.inf) | (ub < np.inf))
if method is None:
if bounded_problem:
method = 'trf'
else:
method = 'lm'
if method == 'lm' and bounded_problem:
raise ValueError("Method 'lm' only works for unconstrained problems. "
"Use 'trf' or 'dogbox' instead.")
if check_finite is None:
check_finite = True if nan_policy is None else False
# optimization may produce garbage for float32 inputs, cast them to float64
if check_finite:
ydata = np.asarray_chkfinite(ydata, float)
else:
ydata = np.asarray(ydata, float)
if isinstance(xdata, (list, tuple, np.ndarray)):
# `xdata` is passed straight to the user-defined `f`, so allow
# non-array_like `xdata`.
if check_finite:
xdata = np.asarray_chkfinite(xdata, float)
else:
xdata = np.asarray(xdata, float)
if ydata.size == 0:
raise ValueError("`ydata` must not be empty!")
# nan handling is needed only if check_finite is False because if True,
# the x-y data are already checked, and they don't contain nans.
if not check_finite and nan_policy is not None:
if nan_policy == "propagate":
raise ValueError("`nan_policy='propagate'` is not supported "
"by this function.")
policies = [None, 'raise', 'omit']
x_contains_nan, nan_policy = _contains_nan(xdata, nan_policy,
policies=policies)
y_contains_nan, nan_policy = _contains_nan(ydata, nan_policy,
policies=policies)
if (x_contains_nan or y_contains_nan) and nan_policy == 'omit':
# ignore NaNs for N dimensional arrays
has_nan = np.isnan(xdata)
has_nan = has_nan.any(axis=tuple(range(has_nan.ndim-1)))
has_nan |= np.isnan(ydata)
xdata = xdata[..., ~has_nan]
ydata = ydata[~has_nan]
# Determine type of sigma
if sigma is not None:
sigma = np.asarray(sigma)
# if 1-D, sigma are errors, define transform = 1/sigma
if sigma.shape == (ydata.size, ):
transform = 1.0 / sigma
# if 2-D, sigma is the covariance matrix,
# define transform = L such that L L^T = C
elif sigma.shape == (ydata.size, ydata.size):
try:
# scipy.linalg.cholesky requires lower=True to return L L^T = A
transform = cholesky(sigma, lower=True)
except LinAlgError as e:
raise ValueError("`sigma` must be positive definite.") from e
else:
raise ValueError("`sigma` has incorrect shape.")
else:
transform = None
func = _lightweight_memoizer(_wrap_func(f, xdata, ydata, transform))
if callable(jac):
jac = _lightweight_memoizer(_wrap_jac(jac, xdata, transform))
elif jac is None and method != 'lm':
jac = '2-point'
if 'args' in kwargs:
# The specification for the model function `f` does not support
# additional arguments. Refer to the `curve_fit` docstring for
# acceptable call signatures of `f`.
raise ValueError("'args' is not a supported keyword argument.")
if method == 'lm':
# if ydata.size == 1, this might be used for broadcast.
if ydata.size != 1 and n > ydata.size:
raise TypeError(f"The number of func parameters={n} must not"
f" exceed the number of data points={ydata.size}")
res = leastsq(func, p0, Dfun=jac, full_output=1, **kwargs)
popt, pcov, infodict, errmsg, ier = res
ysize = len(infodict['fvec'])
cost = np.sum(infodict['fvec'] ** 2)
if ier not in [1, 2, 3, 4]:
raise RuntimeError("Optimal parameters not found: " + errmsg)
else:
# Rename maxfev (leastsq) to max_nfev (least_squares), if specified.
if 'max_nfev' not in kwargs:
kwargs['max_nfev'] = kwargs.pop('maxfev', None)
res = least_squares(func, p0, jac=jac, bounds=bounds, method=method,
**kwargs)
if not res.success:
raise RuntimeError("Optimal parameters not found: " + res.message)
infodict = dict(nfev=res.nfev, fvec=res.fun)
ier = res.status
errmsg = res.message
ysize = len(res.fun)
cost = 2 * res.cost # res.cost is half sum of squares!
popt = res.x
# Do Moore-Penrose inverse discarding zero singular values.
_, s, VT = svd(res.jac, full_matrices=False)
threshold = np.finfo(float).eps * max(res.jac.shape) * s[0]
s = s[s > threshold]
VT = VT[:s.size]
pcov = np.dot(VT.T / s**2, VT)
warn_cov = False
if pcov is None or np.isnan(pcov).any():
# indeterminate covariance
pcov = zeros((len(popt), len(popt)), dtype=float)
pcov.fill(inf)
warn_cov = True
elif not absolute_sigma:
if ysize > p0.size:
s_sq = cost / (ysize - p0.size)
pcov = pcov * s_sq
else:
pcov.fill(inf)
warn_cov = True
if warn_cov:
warnings.warn('Covariance of the parameters could not be estimated',
category=OptimizeWarning)
if full_output:
return popt, pcov, infodict, errmsg, ier
else:
return popt, pcov
def check_gradient(fcn, Dfcn, x0, args=(), col_deriv=0):
"""Perform a simple check on the gradient for correctness.
"""
x = atleast_1d(x0)
n = len(x)
x = x.reshape((n,))
fvec = atleast_1d(fcn(x, *args))
m = len(fvec)
fvec = fvec.reshape((m,))
ldfjac = m
fjac = atleast_1d(Dfcn(x, *args))
fjac = fjac.reshape((m, n))
if col_deriv == 0:
fjac = transpose(fjac)
xp = zeros((n,), float)
err = zeros((m,), float)
fvecp = None
_minpack._chkder(m, n, x, fvec, fjac, ldfjac, xp, fvecp, 1, err)
fvecp = atleast_1d(fcn(xp, *args))
fvecp = fvecp.reshape((m,))
_minpack._chkder(m, n, x, fvec, fjac, ldfjac, xp, fvecp, 2, err)
good = (prod(greater(err, 0.5), axis=0))
return (good, err)
def _del2(p0, p1, d):
return p0 - np.square(p1 - p0) / d
def _relerr(actual, desired):
return (actual - desired) / desired
def _fixed_point_helper(func, x0, args, xtol, maxiter, use_accel):
p0 = x0
for i in range(maxiter):
p1 = func(p0, *args)
if use_accel:
p2 = func(p1, *args)
d = p2 - 2.0 * p1 + p0
p = _lazywhere(d != 0, (p0, p1, d), f=_del2, fillvalue=p2)
else:
p = p1
relerr = _lazywhere(p0 != 0, (p, p0), f=_relerr, fillvalue=p)
if np.all(np.abs(relerr) < xtol):
return p
p0 = p
msg = "Failed to converge after %d iterations, value is %s" % (maxiter, p)
raise RuntimeError(msg)
def fixed_point(func, x0, args=(), xtol=1e-8, maxiter=500, method='del2'):
"""
Find a fixed point of the function.
Given a function of one or more variables and a starting point, find a
fixed point of the function: i.e., where ``func(x0) == x0``.
Parameters
----------
func : function
Function to evaluate.
x0 : array_like
Fixed point of function.
args : tuple, optional
Extra arguments to `func`.
xtol : float, optional
Convergence tolerance, defaults to 1e-08.
maxiter : int, optional
Maximum number of iterations, defaults to 500.
method : {"del2", "iteration"}, optional
Method of finding the fixed-point, defaults to "del2",
which uses Steffensen's Method with Aitken's ``Del^2``
convergence acceleration [1]_. The "iteration" method simply iterates
the function until convergence is detected, without attempting to
accelerate the convergence.
References
----------
.. [1] Burden, Faires, "Numerical Analysis", 5th edition, pg. 80
Examples
--------
>>> import numpy as np
>>> from scipy import optimize
>>> def func(x, c1, c2):
... return np.sqrt(c1/(x+c2))
>>> c1 = np.array([10,12.])
>>> c2 = np.array([3, 5.])
>>> optimize.fixed_point(func, [1.2, 1.3], args=(c1,c2))
array([ 1.4920333 , 1.37228132])
"""
use_accel = {'del2': True, 'iteration': False}[method]
x0 = _asarray_validated(x0, as_inexact=True)
return _fixed_point_helper(func, x0, args, xtol, maxiter, use_accel)