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def 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``.
.. versionadded:: 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.
.. 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])
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