Curve fitting

optimize.curve_fit fits nonlinear model parameters to data by least squares—common in labs, finance decay curves, and sensor calibration.

Model function signature

Define def model(x, a, b, ...): returning predictions. curve_fit returns optimal parameters and covariance estimate.

Workflow

1. Plot raw (x, y) data
2. Choose model with physical meaning
3. Provide initial guess p0
4. Inspect fitted curve and parameter uncertainties

Exponential decay fit

import numpy as np
from scipy import optimize

def exp_decay(t, a, b):
    return a * np.exp(-b * t)

t = np.linspace(0, 4, 20)
y = 2.5 * np.exp(-0.8 * t) + 0.05 * np.random.default_rng(0).normal(size=20)
popt, pcov = optimize.curve_fit(exp_decay, t, y, p0=[2, 1])
print('fitted a, b:', popt)

Important interview questions and answers

1. Q: p0 importance?
   A: Poor initial guesses can converge to wrong local minima—plot and try multiple starts.
2. Q: pcov diagonal?
   A: Variance estimates for parameters—sqrt gives approximate standard errors.

Self-check

1. What does curve_fit return?
2. Why plot data before trusting fitted parameters?

Pitfall: Bad p0 guesses send curve_fit to local minima—try multiple starts.