Curve Fitting with Python

1.1 Look at the Fit

curve_fit returns numbers whether or not the model describes your data. Plot it before you believe it:

import matplotlib.pyplot as plt

x_model = np.linspace(x_data.min(), x_data.max(), 200)

plt.plot(x_data, y_data, "o", label="Data")
plt.plot(x_model, linear(x_model, *popt), "-", label="Fit")
plt.xlabel("x")
plt.ylabel("y")
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

Two details worth copying. The model is evaluated on a dense x_model rather than the five data points, so the curve is smooth rather than a join-the-dots polyline — which matters as soon as the model isn’t a straight line. And linear(x_model, *popt) uses * to unpack the fitted parameters into the function’s arguments, so the same line works whatever the parameter count.

An eyeball check catches the failures that no summary number will: a model that’s the wrong shape, a fit that latched onto the wrong branch, one bad point dragging the line. Residuals make the same check quantitative once the plot looks broadly right.

4.1 Residuals

Residuals are the differences between data and fit, \(r_i = y_i - y_\text{fit}(x_i)\). Plotting them is the fastest way to see trouble: a good fit leaves residuals scattered randomly around zero, while a systematic curve or fanning pattern signals a wrong model or a missing effect.

import numpy as np
import matplotlib.pyplot as plt

y_fit = linear(x_data, *popt)
residuals = y_data - y_fit

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 8),
                               gridspec_kw={'height_ratios': [3, 1]})
ax1.scatter(x_data, y_data, label='Data')
ax1.plot(x_data, y_fit, 'r-', label='Fit')
ax1.set_ylabel('y'); ax1.legend(); ax1.grid(True, alpha=0.3)

ax2.scatter(x_data, residuals)
ax2.axhline(0, color='r', linestyle='--')
ax2.set_xlabel('x'); ax2.set_ylabel('Residuals'); ax2.grid(True, alpha=0.3)

plt.tight_layout(); plt.show()

Ask: are the residuals centered on zero? Random, or is there structure? Are they larger in some region of x than others (which would argue for a weighted fit or point to a systematic effect there)?

4.2 Chi-Squared and Reduced Chi-Squared

When you have uncertainties on your points, χ² quantifies goodness of fit:

\[\chi^2 = \sum_i \frac{\big(y_i - y_\text{fit}(x_i)\big)^2}{\sigma_i^2}, \qquad \chi^2_\text{red} = \frac{\chi^2}{N - n}\]

where \(N\) is the number of data points and \(n\) the number of fit parameters (\(N-n\) is the degrees of freedom).

import numpy as np

def reduced_chi_squared(y_data, y_fit, y_err, num_params):
    residuals = y_data - y_fit
    chi2 = np.sum((residuals / y_err) ** 2)
    dof = len(y_data) - num_params
    return chi2, chi2 / dof

chi2, chi2_red = reduced_chi_squared(y_data, linear(x_data, *popt), y_err, num_params=2)
print(f"chi-squared = {chi2:.2f},  reduced = {chi2_red:.2f}")

Interpreting reduced χ²:

• ≈ 1 — the fit is consistent with the data given your uncertainties. Good.
• ≫ 1 — the model fits poorly, or you underestimated the uncertainties, or there’s a systematic effect the model doesn’t capture.
• ≪ 1 — usually a sign the uncertainties are overestimated (an honest, smaller error bar would push χ² up toward 1).

A χ² test only means something if your sigma values are honest. Overstated uncertainties make almost any fit “look good.”

4.3 When curve_fit Underestimates the Uncertainty

The uncertainties from pcov assume the only error is independent, random scatter in your y-values and that the model is exactly correct. Real experiments break those assumptions, and when they do the reported uncertainty is too small:

• Systematic offsets (e.g., a miscalibrated position axis) shift all points the same way; the fit can’t see this, so it isn’t in the parameter uncertainty.
• Model limitations — if the true behavior departs slightly from your model, the fit uncertainty (which assumes the model is exact) understates reality.
• Correlated noise (e.g., 60 Hz pickup affecting neighboring points together) violates the independence assumption, again shrinking the reported error.

The takeaway: curve_fit’s uncertainty is a lower bound that captures statistical scatter. A complete uncertainty also accounts for systematics — see Error Propagation.