scipy.optimize.curve_fit — SciPy v1.12.0 Manual (2024)

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

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

Parameters:

fcallable

The model function, f(x, …). It must take the independentvariable as the first argument and the parameters to fit asseparate remaining arguments.

xdataarray_like

The independent variable where the data is measured.Should usually be an M-length sequence or an (k,M)-shaped array forfunctions with k predictors, and each element should be floatconvertible if it is an array like object.

ydataarray_like

The dependent data, a length M array - nominally f(xdata, ...).

p0array_like, optional

Initial guess for the parameters (length N). If None, then theinitial values will all be 1 (if the number of parameters for thefunction can be determined using introspection, otherwise aValueError is raised).

sigmaNone or scalar or M-length sequence or MxM array, optional

Determines the uncertainty in ydata. If we define residuals asr = ydata - f(xdata, *popt), then the interpretation of sigmadepends on its number of dimensions:

• A scalar or 1-D sigma should contain values of standard deviations oferrors in ydata. In this case, the optimized function ischisq = sum((r / sigma) ** 2).

• A 2-D sigma should contain the covariance matrix oferrors in ydata. In this case, the optimized function ischisq = r.T @ inv(sigma) @ r.

  New in version 0.19.

None (default) is equivalent of 1-D sigma filled with ones.

absolute_sigmabool, optional

If True, sigma is used in an absolute sense and the estimated parametercovariance 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 scalingsigma by a constant factor. This constant is set by demanding that thereduced chisq for the optimal parameters popt when using thescaled sigma equals unity. In other words, sigma is scaled tomatch 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_finitebool, optional

If True, check that the input arrays do not contain nans of infs,and raise a ValueError if they do. Setting this parameter toFalse may silently produce nonsensical results if the input arraysdo contain nans. Default is True if nan_policy is not specifiedexplicitly and False otherwise.

bounds2-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 eitheran array with the length equal to the number of parameters, or ascalar (in which case the bound is taken to be the same for allparameters). Use np.inf with an appropriate sign to disablebounds 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 areprovided. The method ‘lm’ won’t work when the number of observationsis less than the number of variables, use ‘trf’ or ‘dogbox’ in thiscase.

New in version 0.17.

jaccallable, string or None, optional

Function with signature jac(x, ...) which computes the Jacobianmatrix of the model function with respect to parameters as a densearray_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 selecta finite difference scheme, see least_squares.

See Also

What it takes to rewire a CPG company to outcompete in digital and AIPart numbering systems: 7 best practices to optimize the process - Duro

New in version 0.18.

full_outputboolean, optional

If True, this function returns additioal information: infodict,mesg, and ier.

New in version 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 NaNsare present is implementation-dependent and may change.

Note that if this value is specified explicitly (not None),check_finite will be set as False.

New in version 1.11.

**kwargs

Keyword arguments passed to leastsq for method='lm' orleast_squares otherwise.

Returns:

poptarray

Optimal values for the parameters so that the sum of the squaredresiduals of f(xdata, *popt) - ydata is minimized.

pcov2-D array

The estimated approximate covariance of popt. The diagonals providethe variance of the parameter estimate. To compute one standarddeviation errors on the parameters, useperr = np.sqrt(np.diag(pcov)). Note that the relationship betweencov and parameter error estimates is derived based on a linearapproximation to the model function around the optimum [1].When this approximation becomes inaccurate, cov may not provide anaccurate measure of uncertainty.

How the sigma parameter affects the estimated covariancedepends 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 computethe covariance matrix. Covariance matrices with large condition numbers(e.g. computed with numpy.linalg.cond) may indicate that results areunreliable.

infodictdict (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 notcount function calls for numerical Jacobian approximation,as opposed to ‘lm’ method.

fvec

The residual values evaluated at the solution, for a 1-D sigmathis is (f(x, *popt) - ydata)/sigma.

fjac

A permutation of the R matrix of a QRfactorization of the final approximateJacobian matrix, stored column wise.Together with ipvt, the covariance of theestimate can be approximated.Method ‘lm’ only provides this information.

ipvt

An integer array of length N which definesa permutation matrix, p, such thatfjac*p = q*r, where r is upper triangularwith diagonal elements of nonincreasingmagnitude. 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.

New in version 1.9.

mesgstr (returned only if full_output is True)

A string message giving information about the solution.

New in version 1.9.

ierint (returned only if full_output is True)

An integer flag. If it is equal to 1, 2, 3 or 4, the solution wasfound. Otherwise, the solution was not found. In either case, theoptional output variable mesg gives more information.

New in version 1.9.

Raises:

ValueError

if either ydata or xdata contain NaNs, or if incompatible optionsare used.

RuntimeError

if the least-squares minimization fails.

OptimizeWarning

if covariance of the parameters can not be estimated.

Notes

Users should ensure that inputs xdata, ydata, and the output of fare float64, or else the optimization may return incorrect results.

With method='lm', the algorithm uses the Levenberg-Marquardt algorithmthrough leastsq. Note that this algorithm can only deal withunconstrained problems.

Box constraints can be handled by methods ‘trf’ and ‘dogbox’. Refer tothe 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 ResourcesResearch, 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)>>> poptarray([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]))>>> poptarray([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 somecases, poorer quality fits. As a quick check of whether the model may beoverparameterized, 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 wereto 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 thecovariance matrix, which is related to uncertainty of the fit, gives moreinformation:

>>> 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 andthat only one of these parameters is needed in the model.