Concepts: reduced χ² and error propagation

What is reduced χ²?

For a data set with \(N\) points and a fit with \(k\) free parameters, the reduced chi-square is

\[\chi^2_\nu = \frac{1}{N - k} \sum_{i=1}^{N} \left( \frac{y_i - M(x_i)}{\sigma_i} \right)^2\]

where \(M(x)\) is the model prediction.

Interpretation:

\(\chi^2_\nu \approx 1\) → consistent fit.
\(\chi^2_\nu > 1\) → scatter larger than expected → overestimate errors, weak model, neglected structure, or outliers.
\(\chi^2_\nu < 1\) → scatter smaller than expected → underestimated errors, overfitting, or data selection bias.

LabFit reports reduced_chi2 on every FitResult. Plotting the standardized residuals helps diagnose which assumption failed.

Correlated errors

If your uncertainties share a common systematic, include a correlation matrix sigma_cov in Series.

series = Series(x=x, y=y, sigma=sigma, sigma_cov=C)

LabFit uses Generalized Least Squares when sigma_cov is provided, so the fit accounts for correlated noise.

Asymmetric errors

When sigma_low and sigma_high differ, store them as a sigma tuple or use labfit.types.AsymmetricError.

labsig = (lower, upper)
fitter.fit(x, y, sigma=labsig)

Propagation to the fitted parameters is done via the covariance matrix returned by the optimizer. The reported errors are the square root of the diagonal of the covariance matrix.

Errors on derived quantities

Use labfit.utils.propagate_errors() to transform parameter uncertainties into function uncertainty.

from labfit.utils import propagate_errors

def half_life_from_decay(decay):
    return decay / np.log(2.0)

half_life, half_life_error = propagate_errors(
    half_life_from_decay,
    decay=result.params["decay"],
    jacobian=lambda decay: 1.0 / np.log(2.0),
)

Initial guesses and bounds

Always provide an initial guess when possible. If the model is highly nonlinear, use parameter bounds to prevent the optimizer from jumping to unphysical minima.

fitter = Fitter(
    model=gaussian,
    p0={"amplitude": 1.0, "mean": 0.0, "sigma": 1.0},
    bounds={
        "amplitude": (0.0, None),
        "mean": (-5.0, 5.0),
        "sigma": (1e-6, None),
    },
)

Choosing weights

By default LabFit interprets sigma as 1-σ uncertainties. If you are supplying a design matrix regression, set sigma=None and pass weights instead.

result = fitter.fit(x, y, weights=1.0 / sigma**2)

Weighted regression is mathematically equivalent to correlated errors with a diagonal covariance matrix.