905 lines
33 KiB
Python
905 lines
33 KiB
Python
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from __future__ import division, print_function, absolute_import
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import threading
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import warnings
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from . import _minpack
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import numpy as np
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from numpy import (atleast_1d, dot, take, triu, shape, eye,
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transpose, zeros, product, greater, array,
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all, where, isscalar, asarray, inf, abs,
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finfo, inexact, issubdtype, dtype)
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from scipy.linalg import svd, cholesky, solve_triangular, LinAlgError
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from scipy._lib._util import _asarray_validated, _lazywhere
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from .optimize import OptimizeResult, _check_unknown_options, OptimizeWarning
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from ._lsq import least_squares
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from ._lsq.common import make_strictly_feasible
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from ._lsq.least_squares import prepare_bounds
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_MINPACK_LOCK = threading.RLock()
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error = _minpack.error
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__all__ = ['fsolve', 'leastsq', 'fixed_point', 'curve_fit']
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def _check_func(checker, argname, thefunc, x0, args, numinputs,
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output_shape=None):
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res = atleast_1d(thefunc(*((x0[:numinputs],) + args)))
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if (output_shape is not None) and (shape(res) != output_shape):
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if (output_shape[0] != 1):
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if len(output_shape) > 1:
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if output_shape[1] == 1:
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return shape(res)
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msg = "%s: there is a mismatch between the input and output " \
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"shape of the '%s' argument" % (checker, argname)
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func_name = getattr(thefunc, '__name__', None)
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if func_name:
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msg += " '%s'." % func_name
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else:
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msg += "."
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msg += 'Shape should be %s but it is %s.' % (output_shape, shape(res))
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raise TypeError(msg)
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if issubdtype(res.dtype, inexact):
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dt = res.dtype
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else:
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dt = dtype(float)
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return shape(res), dt
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def fsolve(func, x0, args=(), fprime=None, full_output=0,
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col_deriv=0, xtol=1.49012e-8, maxfev=0, band=None,
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epsfcn=None, factor=100, diag=None):
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"""
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Find the roots of a function.
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Return the roots of the (non-linear) equations defined by
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``func(x) = 0`` given a starting estimate.
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Parameters
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----------
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func : callable ``f(x, *args)``
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A function that takes at least one (possibly vector) argument,
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and returns a value of the same length.
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x0 : ndarray
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The starting estimate for the roots of ``func(x) = 0``.
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args : tuple, optional
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Any extra arguments to `func`.
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fprime : callable ``f(x, *args)``, optional
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A function to compute the Jacobian of `func` with derivatives
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across the rows. By default, the Jacobian will be estimated.
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full_output : bool, optional
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If True, return optional outputs.
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col_deriv : bool, optional
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Specify whether the Jacobian function computes derivatives down
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the columns (faster, because there is no transpose operation).
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xtol : float, optional
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The calculation will terminate if the relative error between two
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consecutive iterates is at most `xtol`.
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maxfev : int, optional
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The maximum number of calls to the function. If zero, then
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``100*(N+1)`` is the maximum where N is the number of elements
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in `x0`.
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band : tuple, optional
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If set to a two-sequence containing the number of sub- and
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super-diagonals within the band of the Jacobi matrix, the
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Jacobi matrix is considered banded (only for ``fprime=None``).
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epsfcn : float, optional
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A suitable step length for the forward-difference
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approximation of the Jacobian (for ``fprime=None``). If
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`epsfcn` is less than the machine precision, it is assumed
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that the relative errors in the functions are of the order of
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the machine precision.
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factor : float, optional
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A parameter determining the initial step bound
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(``factor * || diag * x||``). Should be in the interval
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``(0.1, 100)``.
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diag : sequence, optional
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N positive entries that serve as a scale factors for the
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variables.
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Returns
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-------
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x : ndarray
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The solution (or the result of the last iteration for
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an unsuccessful call).
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infodict : dict
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A dictionary of optional outputs with the keys:
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``nfev``
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number of function calls
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``njev``
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number of Jacobian calls
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``fvec``
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function evaluated at the output
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``fjac``
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the orthogonal matrix, q, produced by the QR
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factorization of the final approximate Jacobian
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matrix, stored column wise
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``r``
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upper triangular matrix produced by QR factorization
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of the same matrix
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``qtf``
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the vector ``(transpose(q) * fvec)``
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ier : int
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An integer flag. Set to 1 if a solution was found, otherwise refer
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to `mesg` for more information.
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mesg : str
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If no solution is found, `mesg` details the cause of failure.
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See Also
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--------
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root : Interface to root finding algorithms for multivariate
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functions. See the 'hybr' `method` in particular.
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Notes
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-----
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``fsolve`` is a wrapper around MINPACK's hybrd and hybrj algorithms.
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"""
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options = {'col_deriv': col_deriv,
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'xtol': xtol,
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'maxfev': maxfev,
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'band': band,
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'eps': epsfcn,
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'factor': factor,
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'diag': diag}
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res = _root_hybr(func, x0, args, jac=fprime, **options)
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if full_output:
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x = res['x']
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info = dict((k, res.get(k))
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for k in ('nfev', 'njev', 'fjac', 'r', 'qtf') if k in res)
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info['fvec'] = res['fun']
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return x, info, res['status'], res['message']
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else:
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status = res['status']
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msg = res['message']
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if status == 0:
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raise TypeError(msg)
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elif status == 1:
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pass
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elif status in [2, 3, 4, 5]:
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warnings.warn(msg, RuntimeWarning)
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else:
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raise TypeError(msg)
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return res['x']
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def _root_hybr(func, x0, args=(), jac=None,
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col_deriv=0, xtol=1.49012e-08, maxfev=0, band=None, eps=None,
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factor=100, diag=None, **unknown_options):
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"""
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Find the roots of a multivariate function using MINPACK's hybrd and
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hybrj routines (modified Powell method).
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Options
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-------
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col_deriv : bool
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Specify whether the Jacobian function computes derivatives down
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the columns (faster, because there is no transpose operation).
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xtol : float
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The calculation will terminate if the relative error between two
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consecutive iterates is at most `xtol`.
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maxfev : int
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The maximum number of calls to the function. If zero, then
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``100*(N+1)`` is the maximum where N is the number of elements
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in `x0`.
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band : tuple
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If set to a two-sequence containing the number of sub- and
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super-diagonals within the band of the Jacobi matrix, the
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Jacobi matrix is considered banded (only for ``fprime=None``).
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eps : float
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A suitable step length for the forward-difference
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approximation of the Jacobian (for ``fprime=None``). If
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`eps` is less than the machine precision, it is assumed
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that the relative errors in the functions are of the order of
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the machine precision.
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factor : float
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A parameter determining the initial step bound
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(``factor * || diag * x||``). Should be in the interval
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``(0.1, 100)``.
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diag : sequence
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N positive entries that serve as a scale factors for the
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variables.
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"""
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_check_unknown_options(unknown_options)
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epsfcn = eps
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x0 = asarray(x0).flatten()
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n = len(x0)
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if not isinstance(args, tuple):
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args = (args,)
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shape, dtype = _check_func('fsolve', 'func', func, x0, args, n, (n,))
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if epsfcn is None:
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epsfcn = finfo(dtype).eps
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Dfun = jac
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if Dfun is None:
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if band is None:
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ml, mu = -10, -10
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else:
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ml, mu = band[:2]
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if maxfev == 0:
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maxfev = 200 * (n + 1)
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with _MINPACK_LOCK:
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retval = _minpack._hybrd(func, x0, args, 1, xtol, maxfev,
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ml, mu, epsfcn, factor, diag)
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else:
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_check_func('fsolve', 'fprime', Dfun, x0, args, n, (n, n))
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if (maxfev == 0):
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maxfev = 100 * (n + 1)
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with _MINPACK_LOCK:
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retval = _minpack._hybrj(func, Dfun, x0, args, 1,
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col_deriv, xtol, maxfev, factor, diag)
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x, status = retval[0], retval[-1]
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errors = {0: "Improper input parameters were entered.",
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1: "The solution converged.",
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2: "The number of calls to function has "
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"reached maxfev = %d." % maxfev,
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3: "xtol=%f is too small, no further improvement "
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"in the approximate\n solution "
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"is possible." % xtol,
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4: "The iteration is not making good progress, as measured "
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"by the \n improvement from the last five "
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"Jacobian evaluations.",
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5: "The iteration is not making good progress, "
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"as measured by the \n improvement from the last "
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"ten iterations.",
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'unknown': "An error occurred."}
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info = retval[1]
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info['fun'] = info.pop('fvec')
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sol = OptimizeResult(x=x, success=(status == 1), status=status)
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sol.update(info)
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try:
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sol['message'] = errors[status]
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except KeyError:
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sol['message'] = errors['unknown']
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return sol
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def leastsq(func, x0, args=(), Dfun=None, full_output=0,
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col_deriv=0, ftol=1.49012e-8, xtol=1.49012e-8,
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gtol=0.0, maxfev=0, epsfcn=None, factor=100, diag=None):
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"""
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Minimize the sum of squares of a set of equations.
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::
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x = arg min(sum(func(y)**2,axis=0))
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y
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Parameters
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----------
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func : callable
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should take at least one (possibly length N vector) argument and
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returns M floating point numbers. It must not return NaNs or
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fitting might fail.
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x0 : ndarray
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The starting estimate for the minimization.
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args : tuple, optional
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Any extra arguments to func are placed in this tuple.
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Dfun : callable, optional
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A function or method to compute the Jacobian of func with derivatives
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across the rows. If this is None, the Jacobian will be estimated.
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full_output : bool, optional
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non-zero to return all optional outputs.
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col_deriv : bool, optional
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non-zero to specify that the Jacobian function computes derivatives
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down the columns (faster, because there is no transpose operation).
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ftol : float, optional
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Relative error desired in the sum of squares.
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xtol : float, optional
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Relative error desired in the approximate solution.
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gtol : float, optional
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Orthogonality desired between the function vector and the columns of
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the Jacobian.
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maxfev : int, optional
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The maximum number of calls to the function. If `Dfun` is provided
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then the default `maxfev` is 100*(N+1) where N is the number of elements
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in x0, otherwise the default `maxfev` is 200*(N+1).
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epsfcn : float, optional
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A variable used in determining a suitable step length for the forward-
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difference approximation of the Jacobian (for Dfun=None).
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Normally the actual step length will be sqrt(epsfcn)*x
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If epsfcn is less than the machine precision, it is assumed that the
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relative errors are of the order of the machine precision.
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factor : float, optional
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A parameter determining the initial step bound
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||
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(``factor * || diag * x||``). Should be in interval ``(0.1, 100)``.
|
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diag : sequence, optional
|
||
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N positive entries that serve as a scale factors for the variables.
|
||
|
|
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Returns
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-------
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||
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x : ndarray
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The solution (or the result of the last iteration for an unsuccessful
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call).
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cov_x : ndarray
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Uses the fjac and ipvt optional outputs to construct an
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estimate of the jacobian around the solution. None if a
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singular matrix encountered (indicates very flat curvature in
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some direction). This matrix must be multiplied by the
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residual variance to get the covariance of the
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parameter estimates -- see curve_fit.
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infodict : dict
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||
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a dictionary of optional outputs with the key s:
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``nfev``
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The number of function calls
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``fvec``
|
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The function evaluated at the output
|
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``fjac``
|
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A permutation of the R matrix of a QR
|
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factorization of the final approximate
|
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Jacobian matrix, stored column wise.
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Together with ipvt, the covariance of the
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estimate can be approximated.
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``ipvt``
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An integer array of length N which defines
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a permutation matrix, p, such that
|
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fjac*p = q*r, where r is upper triangular
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with diagonal elements of nonincreasing
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magnitude. Column j of p is column ipvt(j)
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of the identity matrix.
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``qtf``
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The vector (transpose(q) * fvec).
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mesg : str
|
||
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A string message giving information about the cause of failure.
|
||
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ier : int
|
||
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An integer flag. If it is equal to 1, 2, 3 or 4, the solution was
|
||
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found. Otherwise, the solution was not found. In either case, the
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||
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optional output variable 'mesg' gives more information.
|
||
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|
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Notes
|
||
|
|
-----
|
||
|
|
"leastsq" is a wrapper around MINPACK's lmdif and lmder algorithms.
|
||
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|
||
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cov_x is a Jacobian approximation to the Hessian of the least squares
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||
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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 1D array, regardless of the shape of `x0`,
|
||
|
|
or whether `x0` is a scalar.
|
||
|
|
"""
|
||
|
|
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('Improper input: N=%s must not exceed M=%s' % (n, m))
|
||
|
|
if epsfcn is None:
|
||
|
|
epsfcn = finfo(dtype).eps
|
||
|
|
if Dfun is None:
|
||
|
|
if maxfev == 0:
|
||
|
|
maxfev = 200*(n + 1)
|
||
|
|
with _MINPACK_LOCK:
|
||
|
|
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)
|
||
|
|
with _MINPACK_LOCK:
|
||
|
|
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" % (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],
|
||
|
|
'unknown': ["Unknown error.", TypeError]}
|
||
|
|
|
||
|
|
info = retval[-1] # The FORTRAN return value
|
||
|
|
|
||
|
|
if info not in [1, 2, 3, 4] and not full_output:
|
||
|
|
if info in [5, 6, 7, 8]:
|
||
|
|
warnings.warn(errors[info][0], RuntimeWarning)
|
||
|
|
else:
|
||
|
|
try:
|
||
|
|
raise errors[info][1](errors[info][0])
|
||
|
|
except KeyError:
|
||
|
|
raise errors['unknown'][1](errors['unknown'][0])
|
||
|
|
|
||
|
|
mesg = errors[info][0]
|
||
|
|
if full_output:
|
||
|
|
cov_x = None
|
||
|
|
if info in [1, 2, 3, 4]:
|
||
|
|
from numpy.dual import inv
|
||
|
|
perm = take(eye(n), retval[1]['ipvt'] - 1, 0)
|
||
|
|
r = triu(transpose(retval[1]['fjac'])[:n, :])
|
||
|
|
R = dot(r, perm)
|
||
|
|
try:
|
||
|
|
cov_x = inv(dot(transpose(R), R))
|
||
|
|
except (LinAlgError, ValueError):
|
||
|
|
pass
|
||
|
|
return (retval[0], cov_x) + retval[1:-1] + (mesg, info)
|
||
|
|
else:
|
||
|
|
return (retval[0], info)
|
||
|
|
|
||
|
|
|
||
|
|
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
|
||
|
|
|
||
|
|
|
||
|
|
def curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False,
|
||
|
|
check_finite=True, bounds=(-np.inf, np.inf), method=None,
|
||
|
|
jac=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 : An M-length sequence or an (k,M)-shaped array for functions with k predictors
|
||
|
|
The independent variable where the data is measured.
|
||
|
|
ydata : M-length sequence
|
||
|
|
The dependent data --- nominally f(xdata, ...)
|
||
|
|
p0 : None, scalar, or N-length sequence, optional
|
||
|
|
Initial guess for the parameters. 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, 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.
|
||
|
|
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.
|
||
|
|
bounds : 2-tuple of array_like, optional
|
||
|
|
Lower and upper bounds on parameters. Defaults to no bounds.
|
||
|
|
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.
|
||
|
|
|
||
|
|
.. versionadded:: 0.17
|
||
|
|
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
|
||
|
|
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 : 2d array
|
||
|
|
The estimated 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))``.
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
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
|
||
|
|
-----
|
||
|
|
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.
|
||
|
|
|
||
|
|
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)
|
||
|
|
>>> np.random.seed(1729)
|
||
|
|
>>> y_noise = 0.2 * np.random.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([ 2.55423706, 1.35190947, 0.47450618])
|
||
|
|
>>> 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([ 2.43708906, 1. , 0.35015434])
|
||
|
|
>>> 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()
|
||
|
|
|
||
|
|
"""
|
||
|
|
if p0 is None:
|
||
|
|
# determine number of parameters by inspecting the function
|
||
|
|
from scipy._lib._util import getargspec_no_self as _getargspec
|
||
|
|
args, varargs, varkw, defaults = _getargspec(f)
|
||
|
|
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
|
||
|
|
|
||
|
|
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.")
|
||
|
|
|
||
|
|
# NaNs can not be handled
|
||
|
|
if check_finite:
|
||
|
|
ydata = np.asarray_chkfinite(ydata)
|
||
|
|
else:
|
||
|
|
ydata = np.asarray(ydata)
|
||
|
|
|
||
|
|
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)
|
||
|
|
else:
|
||
|
|
xdata = np.asarray(xdata)
|
||
|
|
|
||
|
|
# 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:
|
||
|
|
raise ValueError("`sigma` must be positive definite.")
|
||
|
|
else:
|
||
|
|
raise ValueError("`sigma` has incorrect shape.")
|
||
|
|
else:
|
||
|
|
transform = None
|
||
|
|
|
||
|
|
func = _wrap_func(f, xdata, ydata, transform)
|
||
|
|
if callable(jac):
|
||
|
|
jac = _wrap_jac(jac, xdata, transform)
|
||
|
|
elif jac is None and method != 'lm':
|
||
|
|
jac = '2-point'
|
||
|
|
|
||
|
|
if method == 'lm':
|
||
|
|
# Remove full_output from kwargs, otherwise we're passing it in twice.
|
||
|
|
return_full = kwargs.pop('full_output', False)
|
||
|
|
res = leastsq(func, p0, Dfun=jac, full_output=1, **kwargs)
|
||
|
|
popt, pcov, infodict, errmsg, ier = res
|
||
|
|
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)
|
||
|
|
|
||
|
|
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)
|
||
|
|
return_full = False
|
||
|
|
|
||
|
|
warn_cov = False
|
||
|
|
if pcov is None:
|
||
|
|
# indeterminate covariance
|
||
|
|
pcov = zeros((len(popt), len(popt)), dtype=float)
|
||
|
|
pcov.fill(inf)
|
||
|
|
warn_cov = True
|
||
|
|
elif not absolute_sigma:
|
||
|
|
if ydata.size > p0.size:
|
||
|
|
s_sq = cost / (ydata.size - 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 return_full:
|
||
|
|
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
|
||
|
|
with _MINPACK_LOCK:
|
||
|
|
_minpack._chkder(m, n, x, fvec, fjac, ldfjac, xp, fvecp, 1, err)
|
||
|
|
|
||
|
|
fvecp = atleast_1d(fcn(xp, *args))
|
||
|
|
fvecp = fvecp.reshape((m,))
|
||
|
|
with _MINPACK_LOCK:
|
||
|
|
_minpack._chkder(m, n, x, fvec, fjac, ldfjac, xp, fvecp, 2, err)
|
||
|
|
|
||
|
|
good = (product(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
|
||
|
|
--------
|
||
|
|
>>> 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)
|