= Raman peak-model bias
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<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.<What Raman images can reveal by Renishaw>[“How can I analyse Raman images?”, curve-fit parameters and background contribution]{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:
$$
y_i=H e^{-x_i^2/(2s^2)}+b x_i,\qquad x_i\in\{-s,0,s\}.
$$
Suppose an equal-weight <least squares> fit omits that slope and estimates only the center $\mu$, keeping $H$ and $s$ fixed:
$$
m(x;\mu)=H e^{-(x-\mu)^2/(2s^2)}.
$$
Near $\mu=0$, its <derivative> with respect to the center is $j_i=(Hx_i/s^2)e^{-x_i^2/(2s^2)}$, so linearizing gives $m(x_i;\mu)\approx m(x_i;0)+\mu j_i$. Minimizing the resulting squared residuals gives
$$
\widehat\mu\approx\frac{\sum_i j_i b x_i}{\sum_i j_i^2}
=\frac{b s^2}{H}e^{1/2}.
$$
The odd slope in the unmodeled background therefore projects onto the odd center-shift response of the symmetric band.
For $H=10$ intensity units, $s=2\,\mathrm{cm^{-1}}$ and $b=0.05$ intensity units per $\mathrm{cm^{-1}}$, the first-order center bias is about $0.0330\,\mathrm{cm^{-1}}$. 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.<Guide to the expression of uncertainty in measurement by JCGM>[§4.1.2, incomplete measurement models; §4.2.5, least-squares fit parameters]{ref}
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