Fit quality metrics

MultiFitResult.metrics is a frozen Metrics object. The fields below are always present after a successful fit.

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Figure of Merit (FOM)

\[ \text{FOM} = 100 \cdot \frac{\displaystyle\sum_{i=1}^{n} |I_i - \hat{I}_i|} {\displaystyle\sum_{i=1}^{n} I_i} \quad [\%] \]

Defined by Balian & Eddy (1977). It measures the mean absolute residual relative to the total observed signal area.

FOM value	Interpretation
< 2.5 %	Excellent — publication quality
2.5 – 5 %	Acceptable for most dosimetry applications
5 – 10 %	Model mismatch or a missing peak component
> 10 %	Poor fit — revisit model choice or number of peaks

FOM is not sensitive to systematic under- or over-fitting because it uses absolute values; complement it with a visual residual plot.

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Coefficient of determination (R²)

\[ R^2 = 1 - \frac{\text{SSR}}{\text{SS}_\text{tot}}, \quad \text{SSR} = \sum_i (I_i - \hat{I}_i)^2, \quad \text{SS}_\text{tot} = \sum_i (I_i - \bar{I})^2 \]

• \(R^2 = 1\): perfect fit.
• \(R^2 < 0\): fit is worse than the mean.
• Typical good TL fits: \(R^2 > 0.999\).

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Reduced chi-square (RCS)

\[ \tilde{\chi}^2 = \frac{\text{SSR}}{n - k} \]

where \(n\) is the number of data points and \(k\) is the number of free parameters.

• \(\tilde{\chi}^2 \approx 1\): residuals consistent with assumed noise level.
• \(\tilde{\chi}^2 \gg 1\): underfitting or noise underestimated.
• \(\tilde{\chi}^2 \ll 1\): overfitting or noise overestimated.

RCS is useful for absolute goodness-of-fit, but requires a noise estimate. In practice, FOM and visual inspection are more diagnostic.

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AIC and BIC (information criteria)

Used for model comparison (e.g. "does adding a fifth peak improve the fit beyond the penalty for the extra parameters?"):

\[ \text{AIC} = n \ln\!\left(\frac{\text{SSR}}{n}\right) + 2k \]

\[ \text{BIC} = n \ln\!\left(\frac{\text{SSR}}{n}\right) + k \ln n \]

Lower is better. BIC penalises parameters more heavily than AIC for \(n > 7\).

print(f"AIC (4 peaks): {result_4p.metrics.AIC:.2f}")
print(f"AIC (5 peaks): {result_5p.metrics.AIC:.2f}")
# Negative ΔAIC → 5-peak model is better;
# positive ΔAIC → extra peak is not justified.
delta_aic = result_5p.metrics.AIC - result_4p.metrics.AIC
print(f"ΔAIC = {delta_aic:.2f}  ({'5p better' if delta_aic < 0 else '4p sufficient'})")

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Convergence diagnostics

Beyond metrics, always check:

Field	Meaning	Action if bad
result.converged	Optimizer exit flag	Check bounds and init values
result.hit_bounds	Params pinned at bound	Widen bounds or revise init
result.jac_cond	Jacobian condition number	> 1e10 → parameters correlated
result.n_iter	Function evaluations	Very high → slow convergence

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Uncertainty metrics

Available when FitOptions(uncertainty=UncertaintyOptions(enabled=True)):

Field	Unit	Meaning
metrics.uc_global	%	Area-weighted combined uncertainty (ISO GUM)
metrics.uc_max	%	Maximum channel-wise \(u_c(T)\) in the ROI
metrics.uc_p95	%	95th percentile of \(u_c(T)\) in the ROI
metrics.contrib_E	%	Fraction of \(u_c\) from \(E\) parameters
metrics.contrib_Tm	%	Fraction from \(T_m\) parameters
metrics.contrib_Im	%	Fraction from \(I_m\) parameters
metrics.contrib_bg	%	Fraction from background parameters

The ROI (region of interest) excludes channels where the fitted signal falls below 5 % of its maximum to avoid division-by-zero in relative uncertainty.

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References

• Balian, H. G., & Eddy, N. W. (1977). Figure-of-merit (FOM), an improved criterion over the normalized chi-squared test for assessing goodness-of-fit of gamma-ray spectral peaks. Nucl. Instr. Meth. 145, 389.