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scipy.optimize.

# curve_fit

scipy.optimize.curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False, check_finite=None, bounds=(-inf, inf), method=None, jac=None, *, full_output=False, nan_policy=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 array_like

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

    ydata array_like

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

    p0 array_like, optional

      Initial guess for the parameters (length N). 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 scalar 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 scalar or 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`.

        Added in version 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 (default), 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. Default is False. 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 if nan_policy is not specified explicitly and False otherwise.

    bounds 2-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 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.

    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.

      Added 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`.

      Added in version 0.18.

    full_output boolean, optional

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

      Added 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 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.

      Added in version 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.

      Added in version 1.9.

    mesg str (returned only if full_output is True)

      A string message giving information about the solution.

      Added in version 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.

      Added in version 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.

  `curve_fit` is for local optimization of parameters to minimize the sum of squares of residuals. For global optimization, other choices of objective function, and other advanced features, consider using SciPy’s Global optimization tools or the LMFIT package.

  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])