= Noninvertible spectral compression versus a coordinate change
{title2=$J_B\leq J$}
\b[Noninvertible spectral compression versus a coordinate change] distinguishes discarding spectral coordinates from expressing the same measurements in different units. The <covariance-consistent rescaling of whitened spectral coordinates> article assumes an invertible transformation. A compression $\mathbf z=B\mathbf y$ with fewer independent rows than input coordinates instead has <covariance> $C_z=BCB^T$ and may lose information even when this <covariance> is propagated correctly.<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}
For positive-definite $C$ and a known signal shape $\mathbf s$, compare the squared noise-weighted signal sizes
$$
J=\mathbf s^TC^{-1}\mathbf s,\qquad
J_B=(B\mathbf s)^T(BCB^T)^{-1}(B\mathbf s),
$$
assuming $B$ has independent rows. Put $\mathbf t=C^{-1/2}\mathbf s$ and $M=BC^{1/2}$. Then
$$
J_B=\mathbf t^TP\mathbf t\leq\mathbf t^T\mathbf t=J,\qquad
P=M^T(MM^T)^{-1}M.
$$
The matrix $P$ is an <orthogonal projector>. This proves that compression cannot increase this signal measure under the stated <covariance> model. The <condition for preserving a spectral signal under linear compression> is that its whitened shape lie entirely in the retained subspace.
For an original example, let $C=I_2$, desired shape $\mathbf s=(1,1)^T$ and nuisance shape $\mathbf x=(1,-1)^T$. Keeping only the first channel, $B=(1,0)$, reduces $J$ from two to one and makes both shapes the same scalar. An unknown nuisance amplitude can then imitate the desired signal exactly.
Summing instead, $B=(1,1)$, gives desired response two, nuisance response zero and noise <variance> two. It retains $J_B=2$ for this particular signal while rejecting that nuisance. Thus a noninvertible map can preserve a chosen estimation task, but cannot in general be reversed to recover every original spectral component.
The primary Raman apparatus records array spectra before subsequent spectral processing.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Instrumentation, array detection; Data Collection, equation 4]{ref} Choose compression against the scientific signal and nuisance model, and propagate the full <covariance>. Correct uncertainty accounting does not restore a spectral distinction removed by the map.
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