scipy.optimize.curve_fit

scipy.optimize.curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False, check_finite=True, bounds=(-inf, 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, optional

If not None, the uncertainties in the ydata array. These are used as weights in the least-squares problem i.e. minimising np.sum( ((f(xdata, *popt) - ydata) / sigma)**2 ) If None, the uncertainties are assumed to be 1.

absolute_sigma : bool, optional

If False, sigma denotes relative weights of the data points. The returned covariance matrix pcov is based on estimated errors in the data, and is not affected by the overall magnitude of the values in sigma. Only the relative magnitudes of the sigma values matter.

If True, sigma describes one standard deviation errors of the input data points. The estimated covariance in pcov is based on these values.

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 independent variables. 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.

New in version 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.

New in version 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.

New in version 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 error 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.

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
>>> from scipy.optimize import curve_fit
>>> def func(x, a, b, c):
...     return a * np.exp(-b * x) + c

>>> xdata = np.linspace(0, 4, 50)
>>> y = func(xdata, 2.5, 1.3, 0.5)
>>> ydata = y + 0.2 * np.random.normal(size=len(xdata))

>>> popt, pcov = curve_fit(func, xdata, ydata)

Constrain the optimization to the region of 0 < a < 3, 0 < b < 2 and 0 < c < 1:

>>> popt, pcov = curve_fit(func, xdata, ydata, bounds=(0, [3., 2., 1.]))