 Raman peak-model bias

Raman peak-model bias is a systematic displacement of an estimated Raman shift or another fitted band parameter caused by an inadequate spectral model. The discrepancy can persist in repeated low-noise measurements. It differs from the random scatter contributing to Raman peak-fit uncertainty. Manufacturer documentation explains that fitted band parameters depend on theoretical curves and warns that a broad spectral background can contaminate simpler intensity measurements.^[ref]

An original three-channel example isolates a Raman baseline-induced center shift. Write $x$ for displacement from the true band center and let the true sampled signal be a Gaussian function plus a small residual baseline slope:Suppose an equal-weight least squares fit omits that slope and estimates only the center $\mu$, keeping $H$ and $s$ fixed:Near $\mu =0$, its derivative with respect to the center is $j_i=(H{x}_i/s^2)e^{-x_i^2/(2s^2)}$, so linearizing gives $m(x_i;\mu )\approx m(x_i;0)+\mu j_i$. Minimizing the resulting squared residuals givesThe odd slope in the unmodeled background therefore projects onto the odd center-shift response of the symmetric band.

For $H=10$ intensity units, $s=2\mathrm{c}{\mathrm{m}}^{-1}$ and $b=0.05$ intensity units per $\mathrm{c}{\mathrm{m}}^{-1}$, the first-order center bias is about $0.0330\mathrm{c}{\mathrm{m}}^{-1}$. This is a deliberately restricted toy fit, not a universal bias formula: fitting the background, width and height too, or using other sample positions and statistical weights, changes the result. The calculation nevertheless demonstrates why reducing random noise alone cannot remove a wrong-model displacement.

In Raman mapping, a spatially varying background or unresolved neighboring band can consequently create an apparent Raman peak-position map contrast. Comparing physically defensible background/line-shape models and examining their residuals tests that interpretation. A good residual plot is useful evidence but does not prove uniqueness of a spectral decomposition. The GUM requires significant corrections and inadequately represented influences to be included in the measurement model; its discussion of fit-parameter uncertainties does not make an incomplete model adequate.^[ref]