= Noise-whitened angle between spectral signal and nuisance shapes
{title2=$\cos\phi=(\mathbf s^TC^{-1}\mathbf x)/\sqrt{(\mathbf s^TC^{-1}\mathbf s)(\mathbf x^TC^{-1}\mathbf x)}$}
The \b[noise-whitened angle between spectral signal and nuisance shapes] measures their separation in units of measurement noise rather than raw plotting units. For nonzero shapes $\mathbf s,\mathbf x$ and a positive-definite <covariance> $C$, choose a square <whitening transformation> $W$ with $WCW^T=I$. The angle is
$$
\cos\phi=\frac{(W\mathbf s)^T(W\mathbf x)}{\|W\mathbf s\|\|W\mathbf x\|}
=\frac{\mathbf s^TC^{-1}\mathbf x}
{\sqrt{(\mathbf s^TC^{-1}\mathbf s)(\mathbf x^TC^{-1}\mathbf x)}}.
$$
It is the ordinary Euclidean angle after whitening. Both parallel and antiparallel shapes are indistinguishable up to an unknown nuisance amplitude; the precision penalty depends on $\sin^2\phi$, as in <noise penalty from nearly parallel signal and reference shapes>.
For an original example, take raw shapes $\mathbf s=(1,1)^T$, $\mathbf x=(1,-1)^T$ and $C=\operatorname{diag}(1,100)$. The plotted shapes are orthogonal, but whitening gives $(1,0.1)^T$ and $(1,-0.1)^T$. Thus $\cos\phi=0.99/1.01\approx0.98020$ and $\phi\approx11.42^{\circ}$. The second channel, which separates their signs, is noisy. Rejecting the nuisance multiplies the optimum <variance> by $1/\sin^2\phi=25.5025$, or standard uncertainty by $5.05$.
The <invariance of noise-whitened spectral angles to the whitening factor> follows from $W^TW=C^{-1}$. Different valid square whitening factors can rotate the whitened coordinates without changing the angle. Likewise an invertible change of measurement units transforms both shapes and <covariance>, leaving the inverse-<covariance> expression unchanged. Rescaling only the displayed curves while leaving their noise model fixed is a different operation.
The <covariance> must describe the actual measured channels and shared errors, using the full propagation rule.<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} If $C$ is singular, first restrict to its physically allowed range; an unrestricted inverse whitening factor does not exist.
The primary stimulated-Raman experiment described systematic spectral noise and subtraction of a pure-solvent reference.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Results and Discussion, observed noise versus shot-noise limit and pure-solvent subtraction]{ref} The angle above assesses which desired patterns remain distinguishable after such nuisance rejection under a stated <covariance>. It does not establish that an estimated reference shape or noise model is itself accurate.
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