 Range restriction in singular-covariance spectral fitting

Range restriction in singular-covariance spectral fitting requires a residual to lie in the subspace where the measurement model allows random error. Let $\mathbf{y}=\mathbf{m}(\theta )+\mathbf{ϵ}$ with zero-mean error covariance $C$. Every vector $\mathbf{v}$ in the kernel of $C$ satisfiesso ${\mathbf{v}}^T\mathbf{ϵ}=0$ almost surely. Thus a permissible residual obeyswhere $C^{+}$ is the Moore-Penrose inverse. The matrix $C{C}^{+}$ is the orthogonal projector onto the range of the covariance.

Only after enforcing this condition should a generalized least squares objective use $(\mathbf{y}-\mathbf{m}{)}^T{C}^{+}(\mathbf{y}-\mathbf{m})$. The quadratic form alone assigns zero penalty to the forbidden directions and therefore cannot impose the condition. Rank deficiency from deterministic spectral upsampling gives its physical spectral-grid origin through $C=A{C}_{\mathrm{in}}{A}^T$. Covariance propagation preserves those exact dependencies.^[ref]

For an original idealized two-channel example, take $C=\mathrm{diag}(1,0)$, mean $\mathbf{m}(\theta )=(\theta ,2\theta {)}^T$ and observation $(3,4{)}^T$. The second channel has no random error in this stipulated model, so the range condition forces $\theta =2$. The remaining squared residual is one. Minimizing only the pseudoinverse quadratic would instead choose $\theta =3$ and objective zero, while predicting the impossible second-channel value six.

Real measurements need not contain a perfect channel: deterministic rebinning produces exact noise-free combinations of otherwise noisy channels. Additional physical noise in such a combination must be modelled if present. If covariance ranges change with the parameters, pseudoinverse fitting with a parameter-dependent covariance range requires enforcing the appropriate support for each candidate; a single fixed projection need not apply.

Raman detector-array spectra are recorded before numerical redistribution onto another spectral grid.^[ref] Fitting the original independent measurement coordinates often makes these support restrictions explicit and avoids treating interpolation points as new observations.