Fitting Statistics Reference

A practical guide to statistics used in spectroscopic peak fitting. The examples assume result is a MultiPeakFitResult obtained from fit_peaks:

result = fit_peaks(x, y, (region_min, region_max))

Goodness-of-fit statistics are plain fields on the result (result.r_squared, result.rss, result.mse); per-parameter estimates come from the per-peak PeakFitResult reached by indexing — result[i] is peak i, and result[i][:center] returns a (value, err, ci) named tuple. The only CurveFit accessor extended for this type is residuals. (To use the solver-level accessors coef/stderror/confint, fit a NonlinearCurveFitProblem directly and call them on the returned CurveFitSolution.)

Parameter Estimates

Coefficients

The fitted parameter values (amplitude, peak position, FWHM, offset) obtained by minimizing the residual sum of squares. Access them per peak, by name:

peak = result[1]               # PeakFitResult for the first peak
peak[:center].value            # fitted peak position
peak.params                    # available parameter names, e.g. [:amplitude, :center, :fwhm, :offset]
[peak[:center].value for peak in result]   # centers of every peak

Uncertainty Quantification

Standard Error

The standard error estimates uncertainty in each fitted parameter. Computed from the diagonal of the variance-covariance matrix:

\[\text{SE}(\hat{p}_i) = \sqrt{\text{Var}(\hat{p}_i)} = \sqrt{C_{ii}}\]

where $C$ is the covariance matrix.

se = result[1][:center].err   # standard error of the first peak's center

When to use: Report as parameter ± SE for quick uncertainty estimates. Assumes errors are symmetric and normally distributed.

Confidence Intervals

A range likely to contain the true parameter value. The 95% CI means: if we repeated the experiment many times, 95% of computed intervals would contain the true value.

\[\text{CI} = \hat{p} \pm t_{\alpha/2, \nu} \cdot \text{SE}(\hat{p})\]

where $t_{\alpha/2, \nu}$ is the t-distribution critical value and $\nu$ is degrees of freedom.

ci = result[1][:center].ci  # 95% CI tuple (low, high) for the first peak's center

The intervals are computed at the 95% level. For a different level, fit a NonlinearCurveFitProblem directly and call confint(sol; level=0.99) on the solver result.

When to use: Preferred for publications. More informative than SE alone, especially for small sample sizes where t-distribution differs significantly from normal.

Variance-Covariance Matrix

The full covariance matrix captures correlations between parameters. Off-diagonal elements indicate how uncertainties in different parameters are related.

\[C = \sigma^2 (J^T J)^{-1}\]

where $J$ is the Jacobian matrix and $\sigma^2$ is estimated from MSE.

Per-parameter standard errors (result[i][name].err) and confidence intervals (result[i][name].ci) are exposed directly; the full covariance matrix is not currently surfaced as a convenience. For error propagation across derived quantities, use the per-parameter .err under the independent-parameter approximation, or access the underlying solver result.

When to use: Unusually large stderror values flag potential parameter correlations — a good diagnostic for overparameterized models (e.g. two overlapping peaks when one suffices).

Goodness of Fit

Residual Sum of Squares (RSS)

The sum of squared differences between observed and fitted values:

\[\text{RSS} = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2\]

ss_res = result.rss

When to use: Comparing fits of the same model to the same data. Lower is better. Not useful for comparing across different datasets or region sizes.

Mean Squared Error (MSE)

RSS normalized by degrees of freedom:

\[\text{MSE} = \frac{\text{RSS}}{n - p}\]

where $n$ is number of points and $p$ is number of parameters.

mse_val = result.mse

When to use: Comparing fits across different region sizes. Estimates the variance of the residuals. Also used internally to compute parameter uncertainties.

Coefficient of Determination (R²)

Proportion of variance explained by the model:

\[R^2 = 1 - \frac{\text{RSS}}{\text{TSS}} = 1 - \frac{\sum(y_i - \hat{y}_i)^2}{\sum(y_i - \bar{y})^2}\]

r_squared = result.r_squared          # computed by the fit
# equivalently, from the residual and total sums of squares:
ss_tot = sum((y .- mean(y)).^2)
r_squared == 1 - result.rss / ss_tot

When to use: Quick assessment of fit quality. R² > 0.99 is typical for good spectroscopic fits. However, R² can be misleading:

• Always high for peaked data (even bad fits)
• Doesn't detect systematic deviations
• Can't compare models with different numbers of parameters

Caution: Always check residuals visually. A high R² does not guarantee a good fit.

Reduced Chi-Squared (χ²/ν)

If measurement uncertainties ($\sigma_i$) are known:

\[\chi^2_\nu = \frac{1}{\nu} \sum_{i=1}^{n} \frac{(y_i - \hat{y}_i)^2}{\sigma_i^2}\]

where $\nu = n - p$ is degrees of freedom.

When to use: When you have reliable error estimates for each data point (e.g., from detector noise specifications or repeated measurements).

Residual Analysis

Residuals

The differences between observed and fitted values:

\[r_i = y_i - \hat{y}_i\]

res = residuals(result)

Visual inspection is essential. Look for:

• Random scatter around zero — good fit
• Systematic patterns — wrong model or missing physics
• Trends — baseline problems
• Outliers — bad data points or model breakdown

Autocorrelation in Residuals

Consecutive residuals should be independent. Correlated residuals indicate model inadequacy.

Durbin-Watson statistic:

\[d = \frac{\sum_{i=2}^{n}(r_i - r_{i-1})^2}{\sum_{i=1}^{n} r_i^2}\]

Model Comparison

Akaike Information Criterion (AIC)

Balances goodness of fit against model complexity:

\[\text{AIC} = n \ln(\text{RSS}/n) + 2p\]

Lower AIC is better. Penalizes extra parameters.

Bayesian Information Criterion (BIC)

Similar to AIC but with stronger penalty for parameters:

\[\text{BIC} = n \ln(\text{RSS}/n) + p \ln(n)\]

When to use AIC vs BIC:

• AIC: prediction-focused, may favor more complex models
• BIC: tends toward simpler models, better for identifying "true" model

F-test for Nested Models

Compare a simpler model (fewer parameters) to a more complex one:

\[F = \frac{(\text{RSS}_1 - \text{RSS}_2)/(p_2 - p_1)}{\text{RSS}_2/(n - p_2)}\]

Example: Testing whether a second Lorentzian peak is statistically justified.

Practical Guidelines

Minimum Reporting for Publications

1. Parameter values with uncertainties: peak = 2055.3 ± 0.2 cm⁻¹
2. Confidence intervals (preferred): 95% CI (2054.9, 2055.7)
3. Fit quality metric: R² = 0.9998 or χ²/ν = 1.02
4. Residual plot in supplementary material

Quick Quality Check

result = fit_peaks(spec, (1900, 2200))

# Good fit indicators:
# - R² > 0.999 for clean spectroscopic data
# - Parameter errors << parameter values
# - Residuals show no systematic pattern
# - MSE comparable to expected noise level

Red Flags

• R² high but residuals show clear patterns
• Parameter uncertainty larger than parameter value
• Confidence interval includes physically impossible values (e.g., negative FWHM)
• Fit result changes significantly with small changes to region bounds

References

1. Bevington, P.R. & Robinson, D.K. (2003). Data Reduction and Error Analysis for the Physical Sciences, 3rd ed. McGraw-Hill.

   • Classic reference for error analysis in experimental physics

2. Press, W.H. et al. (2007). Numerical Recipes: The Art of Scientific Computing, 3rd ed. Cambridge University Press.

   • Chapter 15: Modeling of Data (least squares, confidence limits)

3. Motulsky, H. & Christopoulos, A. (2004). Fitting Models to Biological Data Using Linear and Nonlinear Regression. Oxford University Press.

   • Practical guide with emphasis on proper statistical interpretation

4. IUPAC (1997). Compendium of Analytical Nomenclature (Orange Book), 3rd ed.

   • Standard definitions for analytical chemistry

5. Hug, W. & Fedorovsky, M. (2018). A comprehensive approach to statistical analysis of vibrational spectra. J. Raman Spectrosc. 49, 3-16.

   • Modern treatment specific to vibrational spectroscopy