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 full_output : boolean, optional If True, this function returns additional information: `infodict`, `mesg`, and `ier`. .. versionadded:: 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 NaNs are present is implementation-dependent and may change. Note that if this value is specified explicitly (not None), `check_finite` will be set as False. .. versionadded:: 1.11 **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 : 2-D array The estimated approximate 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))``. Note that the relationship between `cov` and parameter error estimates is derived based on a linear approximation to the model function around the optimum [1]_. When this approximation becomes inaccurate, `cov` may not provide an accurate measure of uncertainty. 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. Covariance matrices with large condition numbers (e.g. computed with `numpy.linalg.cond`) may indicate that results are unreliable. infodict : dict (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 not count function calls for numerical Jacobian approximation, as opposed to 'lm' method. ``fvec`` The residual values evaluated at the solution, for a 1-D `sigma` this is ``(f(x, *popt) - ydata)/sigma``. ``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. Method 'lm' only provides this information. ``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. Method 'lm' only provides this information. ``qtf`` The vector (transpose(q) * fvec). Method 'lm' only provides this information. .. versionadded:: 1.9 mesg : str (returned only if `full_output` is True) A string message giving information about the solution. .. versionadded:: 1.9 ier : int (returned only if `full_output` is True) 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. .. versionadded:: 1.9 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 ----- Users should ensure that inputs `xdata`, `ydata`, and the output of `f` are ``float64``, or else the optimization may return incorrect results. 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. Parameters to be fitted must have similar scale. Differences of multiple orders of magnitude can lead to incorrect results. For the 'trf' and 'dogbox' methods, the `x_scale` keyword argument can be used to scale the parameters. References ---------- .. [1] K. Vugrin et al. Confidence region estimation techniques for nonlinear regression in groundwater flow: Three case studies. Water Resources Research, 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) >>> popt array([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])) >>> popt array([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 some cases, poorer quality fits. As a quick check of whether the model may be overparameterized, 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 were to add a fourth parameter ``d`` to `func` with the same effect as ``a``: >>> def func2(x, a, b, c, d): ... return a * d * np.exp(-b * x) + c # a and d are redundant >>> popt, pcov = curve_fit(func2, xdata, ydata) >>> np.linalg.cond(pcov) 1.13250718925596e+32 # may vary Such a large value is cause for concern. The diagonal elements of the covariance matrix, which is related to uncertainty of the fit, gives more information: >>> 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 and that only one of these parameters is needed in the model. If the optimal parameters of `f` differ by multiple orders of magnitude, the resulting fit can be inaccurate. Sometimes, `curve_fit` can fail to find any results: >>> ydata = func(xdata, 500000, 0.01, 15) >>> try: ... popt, pcov = curve_fit(func, xdata, ydata, method = 'trf') ... except RuntimeError as e: ... print(e) Optimal parameters not found: The maximum number of function evaluations is exceeded. If parameter scale is roughly known beforehand, it can be defined in `x_scale` argument: >>> popt, pcov = curve_fit(func, xdata, ydata, method = 'trf', ... x_scale = [1000, 1, 1]) >>> popt array([5.00000000e+05, 1.00000000e-02, 1.49999999e+01]) NrR