# 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 array_like or object

      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, but can actually be any 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 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 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`.

      >   New 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.

    bounds 2-tuple of array_like, optional

      Lower and upper bounds on parameters. 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 residuals of `f(xdata, *popt) - ydata` is minimized.

    pcov 2-D 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.

  `scipy.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 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()

scipy.optimize.lsq_linear scipy.optimize.root_scalar