Curve Fit

curve_fit() fits a custom function to data by adjusting its parameters to minimize the difference between the function’s predictions and the actual data points, ensuring the best possible match.

Syntax

popt, pcov = 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)

• f: This is the function to fit to the data.
• xdata: An array-like input representing the independent variable values (e.g., time, x-coordinates, etc.).
• ydata: An array-like input representing the dependent variable values (e.g., measured data corresponding to xdata).
• p0: Initial guess of the parameters.
• sigma: Defines the uncertainty in ydata.
• absolute_sigma: If True, sigma is interpreted absolutely, and the parameter covariance pcov reflects absolute values. If False which is the default, sigma is scaled to normalize residual variance. Here, pcov(absolute_sigma=False) = pcov(absolute_sigma=True) * chisq(popt)/(M-N).
• check_finite: Ensures input arrays do not contain NaN or inf. If True, a ValueError is raised when such values are found. Defaults to True unless nan_policy is explicitly specified.
• bounds: Specifies parameter bounds. Defaults to no bounds. Options include:
  • An instance of the Bounds class.
  • A 2-tuple of array-like objects or scalars: Scalars apply bounds uniformly, and np.inf can disable bounds partially.
• method - Optimization method. Choices are:
  • 'lm' (default for unconstrained problems): Levenberg-Marquardt.
  • 'trf' (default if bounds are set): Trust Region Reflective.
  • 'dogbox': Dogleg. 'lm' cannot handle cases where observations < variables. Use 'trf' or 'dogbox' instead.
• jac: Jacobian matrix computation for jac(x, ...). Defaults to numerical estimation if None. Supports finite difference schemes for 'trf' and 'dogbox' methods.
• full_output: If True, returns additional information such as infodict, mesg, ier.
• nan_policy: Decides behavior when NaN values exist in input data:
  • None (default): No special handling; behavior depends on implementation.
  • 'raise': Throws an error.
  • 'omit': Ignores NaN values during computation.
• **kwargs: Additional keyword arguments passed to leastsq (if method = ‘lm’) or least_squares otherwise.

It returns:

• popt: A 1D array containing the optimal values of the parameters (a, b, c, etc.) that minimize the difference between the function and the data (ydata).
• pcov: A 2D array representing the covariance matrix of the estimated parameters, which provides an estimate of the uncertainties (or standard errors) associated with the optimized parameters.

Example

This code uses curve_fit to fit a custom exponential decay function to noisy data and estimates the parameters a, b, and c:

import numpy as np
from scipy.optimize import curve_fit

# Define your custom function
def func(x, a, b, c):
  return a * np.exp(-b * x) + c

# Define your data
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

# Fit for the parameters a, b, c of the function func:
popt, pcov = curve_fit(func, xdata, ydata)
print(popt)

The above will give the following output:

[2.47681145 1.32375703 0.53063146]

Note: Since we are using np.random() for rng variable the output is expected to change for each run.