|
transpose, zeros, product, greater, array, all, where, isscalar, asarray, inf, abs, finfo, inexact, issubdtype, dtype)
output_shape=None): if (output_shape[0] != 1): if len(output_shape) > 1: if output_shape[1] == 1: return shape(res) msg = "%s: there is a mismatch between the input and output " \ "shape of the '%s' argument" % (checker, argname) func_name = getattr(thefunc, '__name__', None) if func_name: msg += " '%s'." % func_name else: msg += "." msg += 'Shape should be %s but it is %s.' % (output_shape, shape(res)) raise TypeError(msg) else: dt = dtype(float)
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 'hybr' `method` in particular.
Notes ----- ``fsolve`` is a wrapper around MINPACK's hybrd and hybrj algorithms.
""" 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 = dict((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']
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) with _MINPACK_LOCK: 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) with _MINPACK_LOCK: 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
col_deriv=0, 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. 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 non-zero to return all optional outputs. col_deriv : bool, optional non-zero to 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 Uses the fjac and ipvt optional outputs to construct an estimate of the jacobian around the solution. None if a singular matrix encountered (indicates very flat curvature in some direction). This matrix must be multiplied by the residual variance to get the covariance of the parameter estimates -- see curve_fit. infodict : dict a dictionary of optional outputs with the key s:
``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).
mesg : str A string message giving information about the cause of failure. 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.
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 1D array, regardless of the shape of `x0`, or whether `x0` is a scalar. """ args = (args,) raise TypeError('Improper input: N=%s must not exceed M=%s' % (n, m)) 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)
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]}
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])
except (LinAlgError, ValueError): pass else: return (retval[0], info)
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)
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
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
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()
""" # 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 = _initialize_feasible(lb, ub)
method = 'trf' else:
raise ValueError("Method 'lm' only works for unconstrained problems. " "Use 'trf' or 'dogbox' instead.")
# NaNs can not be handled else: ydata = np.asarray(ydata)
# `xdata` is passed straight to the user-defined `f`, so allow # non-array_like `xdata`. else: xdata = np.asarray(xdata)
# Determine type of sigma 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:
jac = _wrap_jac(jac, xdata, transform) jac = '2-point'
# Remove full_output from kwargs, otherwise we're passing it in twice. 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
# indeterminate covariance pcov = zeros((len(popt), len(popt)), dtype=float) pcov.fill(inf) warn_cov = True else: pcov.fill(inf) warn_cov = True
warnings.warn('Covariance of the parameters could not be estimated', category=OptimizeWarning)
return popt, pcov, infodict, errmsg, ier else:
"""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)
return p0 - np.square(p1 - p0) / d
return (actual - desired) / desired
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)
""" 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) |