= Covariance of count-conserving rebinned Raman channels
{title2=$C'=ACA^T$}
The \b[<covariance> of count-conserving rebinned Raman channels] records the shared noise created when one measured channel contributes to several output channels. For a fixed deterministic rebinning matrix $A$ and an input count vector $\mathbf N$ with <covariance matrix> $C$, the output is $\mathbf N'=A\mathbf N$ and
$$
C'=ACA^T.
$$
This identity needs finite second moments, not a Gaussian approximation. <Raman spectral rebinning with conserved integrated counts> describes the count-preserving map; retaining its full <covariance matrix> is necessary for uncertainties of sums and fitted spectral features.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3, covariance terms in propagated uncertainty]{ref}
Consider independent input counts with means $100$ and $300$ and <Poisson distribution> <variances> equal to those means. Split them deterministically using
$$
A=\begin{pmatrix}1/2&0\\1/2&1/2\\0&1/2\end{pmatrix},\qquad
C'=\begin{pmatrix}25&25&0\\25&100&75\\0&75&75\end{pmatrix}.
$$
The three output means are $50,200,150$. Their total has <variance> $\mathbf1^TC'\mathbf1=400$, exactly the original total <variance>, and <standard deviation> $20$. Keeping only diagonal entries would give <variance> $200$ and <standard deviation> about $14.14$, understating the uncertainty of the conserved total.
The <rank deficiency from deterministic spectral upsampling> is visible without matrix inversion: the vector $(1,-1,1)^T$ has zero output contrast for every possible input. Three displayed channels therefore contain at most two independent degrees of freedom in this model. Fractional redistribution creates correlated numerical channels; it is different from physically assigning each detected <photon> randomly to one new channel.
The primary stimulated-Raman apparatus records spectra with array detectors and calibrates their spectral coordinates.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Instrumentation, array detection and frequency-shift calibration; Data Collection, equation 4]{ref} When those data are placed on a new spectral grid, the original detector counts remain the source of statistical information. Additional interpolation points do not supply independent <photon> measurements.
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