How the Chi-Squared Is Built

Everything AZURE2 fits or samples is driven by one objective function. It is worth knowing exactly what goes into it, because the number reported in chiSquared.out is not the whole objective, and the difference matters as soon as normalizations are free.

The data term

For every point in every active segment, AZURE2 compares the calculated cross section against the measured one, with the segment’s normalization \(n\) applied to the data rather than to the theory:

\[\chi^2_{\text{data}} = \sum_{s}\ \sum_{i \in s} \frac{\bigl(\sigma^{\text{fit}}_i - n_s\,\sigma^{\text{data}}_i\bigr)^2} {\bigl(n_s\,\delta\sigma^{\text{data}}_i\bigr)^2}\]

Points with zero uncertainty are skipped rather than producing an infinity. Because both the value and its error are scaled by \(n_s\), a normalization that is free to move cannot reduce \(\chi^2\) simply by shrinking the uncertainties.

The normalization penalty

A free normalization is not free of consequence. Each segment that has Vary Norm? enabled and a non-zero Norm Error contributes

\[\chi^2_{\text{norm},s} = \left(\frac{n_s - n_s^{\text{nom}}}{n_s^{\text{nom}} \cdot \epsilon_s / 100}\right)^2\]

where \(n_s^{\text{nom}}\) is the nominal normalization and \(\epsilon_s\) is the quoted systematic uncertainty in percent — the value typed into the Norm Error column of the Segments tab. This is what keeps a dataset’s normalization near its experimental value instead of letting it absorb every discrepancy in the model.

A segment with Vary Norm? enabled but no quoted error has no penalty, and its normalization is genuinely unconstrained.

The energy-shift penalty

Segments with Vary Energy Shift? enabled contribute the analogous term, this time with an absolute rather than a percentage error:

\[\chi^2_{\text{shift},s} = \left(\frac{\Delta E_s - \Delta E_s^{\text{nom}}}{\delta(\Delta E_s)}\right)^2\]

Nuisance parameters

Level energies and widths flagged Use as Nuisance in the Fitting tab add a Gaussian term of the same shape, using the Error column as the width. This is how prior knowledge of a resonance energy enters a least-squares fit.

What `chiSquared.out` actually reports

The file separates the two contributions:

Segment#, Chi-Squared,  N,  Norm,  Norm-Chi-Squared
1,823.88,17,1,0
2,24.335,20,1,0
...
Total-Chi-Squared: 107456 Total-Norm-Chi-Squared: 0 Total-N: 415

Chi-Squared (per segment) and Total-Chi-Squared are the data term only.
Norm-Chi-Squared and Total-Norm-Chi-Squared are the normalization penalty.
N is the number of points in the segment, and Total-N their sum.

The quantity the minimizer actually descends is the sum of all of them. Quoting Total-Chi-Squared as “the χ²” of a fit with free normalizations understates the objective.

Note

Total-N counts data points, not degrees of freedom. Subtract the number of free parameters yourself before forming a reduced χ².

Consistency with the MCMC

The Bayesian sampler reaches the same objective by a different route, which is worth understanding if you compare a fit against a posterior.

Its log-likelihood is built from the data term alone:

\[\ln \mathcal{L} = -\tfrac{1}{2}\,\chi^2_{\text{data}}\]

The normalization and energy-shift penalties reappear as priors instead: AZURE2 derives a Gaussian prior for every varying normalization and energy shift straight from the quoted experimental errors, with exactly the mean and width implied by the penalties above. The resulting log-posterior therefore matches the fit objective term for term, and a posterior mode coincides with a χ² minimum. See MCMC Tab.

The same distinction applies to pyazr — the Python Interface: calculate_chi2_rwa and residual_jacobian return the data χ² only. A least-squares fit driven from Python that minimises those residuals alone will let the normalizations drift to absorb every discrepancy, reaching a “better” χ² that AZURE2 itself would never find. Append the penalty rows explicitly — the recipe is in the pyazr chapter.