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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.

The transverse-noise limit of a rotating rank-one covariance model describes how an ordinary two-dimensional noise distribution concentrates onto a line as one physical noise scale tends to zero. For orthonormal vectors ${\mathbf{u}}_{\theta },{\mathbf{v}}_{\theta }$, setA zero-mean Gaussian observation has densityThe pseudoinverse fitting with a parameter-dependent covariance range article explains the range restriction at the exactly singular endpoint.

For a fixed point off the line, ${\mathbf{v}}_{\theta }^T\mathbf{y}\ne 0$, the exponential drives this density to zero as $ϵ\downarrow 0$. On the line the two-dimensional density grows like $1/ϵ$. There is no finite pointwise two-dimensional density limit: probability concentrates onto a one-dimensional Gaussian measure. A density on that line uses a different reference measure. Simply substituting zero for $ϵ$ in the full-rank density formula is invalid.

For an original angular example, let an observed vector have length $R=3$ along orientation ${\theta }_0$, with $\sigma =1$ and $ϵ=0.06$. At angular displacement $\delta =\theta -{\theta }_0$,At $\delta =0.02$ radians the density ratio is about $0.608$. The angular resolution set by transverse spectral noise has characteristic small-angle scale $ϵ/R\approx 0.02$ radians, or $1.15^{\circ }$, when the transverse noise is much smaller than the longitudinal noise.

SciPy's pinned implementation treats singular covariance using its rank and a separate numerical support check, rather than only discarding zero eigenvalues in a quadratic.^[ref] Numerical support tolerance and actual transverse measurement noise are different quantities. Choose the latter from the detector and processing model; a tiny arbitrary regularization can otherwise imply unjustified angular precision.