scipy.optimize.curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False, check_finite=True, **kw)[source]

Use non-linear least squares to fit a function, f, to data.

Assumes ydata = f(xdata, *params) + eps


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.


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 covariance of the parameters can not be estimated.


if ydata and xdata contain NaNs.

See also



The algorithm uses the Levenberg-Marquardt algorithm through leastsq. Additional keyword arguments are passed directly to that algorithm.


>>> 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)