= Minimum-variance spectral contrast orthogonal to a reference shape
{title2=$\mathbf a=C^{-1}\mathbf s_\perp/(\mathbf s_\perp^TC^{-1}\mathbf s_\perp)$}
A \b[minimum-<variance> spectral contrast orthogonal to a reference shape] estimates a desired signal amplitude while exactly rejecting one fixed nuisance shape. Let $\mathbf z=\theta\mathbf s+\beta\mathbf x+\boldsymbol\epsilon$, with known shapes $\mathbf s,\mathbf x$ and a positive-definite noise <covariance matrix> $C$. Choose $\widehat\theta=\mathbf a^T\mathbf z$ to minimize $\mathbf a^TC\mathbf a$, subject to
$$
\mathbf a^T\mathbf s=1,\qquad\mathbf a^T\mathbf x=0.
$$
The constraints retain unit signal response and reject the reference amplitude, as in <projection of shared-scale uncertainty onto a spectral contrast>. The quadratic <variance> includes the full noise <covariance>.<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}
Remove the reference component in the inverse-<covariance> geometry:
$$
\mathbf s_\perp=\mathbf s-\mathbf x\frac{\mathbf x^TC^{-1}\mathbf s}{\mathbf x^TC^{-1}\mathbf x},\qquad
D=\mathbf s_\perp^TC^{-1}\mathbf s_\perp.
$$
For nonzero $\mathbf x$ and $D>0$, the minimizing contrast and its <variance> are
$$
\mathbf a=\frac{C^{-1}\mathbf s_\perp}{D},\qquad
\operatorname{Var}(\widehat\theta)=\frac1D.
$$
For a short proof, every other feasible contrast is $\mathbf a+\mathbf v$ with $\mathbf v^T\mathbf s=\mathbf v^T\mathbf x=0$. Since $C\mathbf a$ is a linear combination of those shapes, the cross term vanishes, leaving the added nonnegative <variance> $\mathbf v^TC\mathbf v$. This is a constrained <best linear unbiased estimator>.
In an original three-channel example, take $\mathbf s=(1,0,0)^T$, $\mathbf x=(1,1,1)^T$ and $C=\operatorname{diag}(1,4,1)$. The optimum is $\mathbf a=(1,-1/5,-4/5)^T$, with <variance> $1.8$. Equal subtraction of the two reference channels, $(1,-1/2,-1/2)^T$, has <variance> $2.25$. The quieter third channel receives more of the subtraction.
If the desired and reference shapes are proportional, $D=0$ and no unit-response rejecting contrast exists. The <noise penalty from nearly parallel signal and reference shapes> already becomes large as $D$ approaches zero. This construction assumes fixed known shapes and <covariance>; estimating them from the same noisy spectra requires additional uncertainty terms.
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