= Projection of shared-scale uncertainty onto a spectral contrast
{title2=$u_{\rm scale}^2=\sigma_k^2(\mathbf a^T\mathbf x)^2$}
\b[Projection of shared-scale uncertainty onto a spectral contrast] determines which measured spectral features are sensitive to an uncertain common subtraction coefficient. In <correlated spectral residuals from a shared subtraction scale>, a fixed reference shape $\mathbf x$ and scale error $\delta k$ contribute $-\mathbf x\delta k$ to the spectrum. For a linear contrast $c=\mathbf a^T\mathbf z$,
$$
\delta c_{\rm scale}=-(\mathbf a^T\mathbf x)\delta k,\qquad
u_{\rm scale}^2(c)=\sigma_k^2(\mathbf a^T\mathbf x)^2.
$$
The contrast is insensitive to this particular error when $\mathbf a^T\mathbf x=0$. This is an exact cancellation for a fixed reference shape; it does not require the scale error to have a Gaussian distribution.
Suppose the desired spectral signal has known shape $\mathbf s$ and unknown amplitude $\theta$, so that $\mathbf z=\theta\mathbf s-\delta k\mathbf x+\boldsymbol\epsilon$. To retain its amplitude while rejecting the scale error, require
$$
\mathbf a^T\mathbf s=1,\qquad \mathbf a^T\mathbf x=0.
$$
For an original two-channel example, take $\mathbf x=(1,1)^T$ and $\mathbf s=(1,0)^T$. The difference $c=z_1-z_2$ has unit signal response and cancels any common scale error. If the two independent reading noises each have <standard deviation> $3$ in the chosen units, the difference has <standard deviation> $\sqrt{18}\approx4.24$. Using channel 1 alone would have reading-noise <standard deviation> $3$ but would retain the unknown common contribution. Rejection has a noise cost even when the nuisance cancellation is exact.
If $\mathbf s$ is proportional to $\mathbf x$, the two requirements are incompatible: cancelling that reference direction also cancels the signal. In more channels, a <minimum-variance spectral contrast orthogonal to a reference shape> can balance detector noise among the remaining directions. The relevant <variance> is $\mathbf a^TC\mathbf a$, including correlations.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref} Uncertain reference shape or scale estimates correlated with the measured spectra require their own <covariance> contributions.
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