 Pseudoinverse fitting with a parameter-dependent covariance range

Pseudoinverse fitting with a parameter-dependent covariance range requires checking the allowed residual subspace separately for each fitted parameter. If the covariance is $C_{\theta }$ and the residual is ${\mathbf{r}}_{\theta }=\mathbf{y}-\mathbf{m}(\theta )$, the range restriction in singular-covariance spectral fitting becomesThe Moore-Penrose inverse quadratic ${\mathbf{r}}_{\theta }^T{C}_{\theta }^{+}{\mathbf{r}}_{\theta }$ alone does not enforce this condition. A projector computed at one trial parameter need not describe another trial's allowed noise.

For an original example, let ${\mathbf{u}}_{\theta }=(\cos \theta ,\sin \theta {)}^T$, zero mean and $C_{\theta }={\sigma }^2{\mathbf{u}}_{\theta }{\mathbf{u}}_{\theta }^T$. Noise lies on a line that rotates with $\theta$. With observation $\mathbf{y}=(2,0{)}^T$ and $\sigma =1$, the model at $\theta =0$ permits the observation and gives quadratic value four. At $\theta =\pi /2$, the pseudoinverse quadratic is zero, yet the observation is outside the allowed vertical line. Minimizing that quadratic without its range constraint would reward an impossible candidate.

The pinned SciPy documentation describes singular-covariance evaluation using pseudo-determinants, pseudoinverses and covariance rank; its implementation additionally applies a numerical support mask and returns zero density, or negative-infinite log density, off support.^[ref] Parameter-dependent supports also require care about which measure defines a density. Values defined on different subspaces are not automatically a common likelihood merely because each uses the same-looking formula.

A physically justified transverse noise resolves the idealized singularity. With perpendicular unit vector ${\mathbf{v}}_{\theta }$, use $C_{\theta ,ϵ}={\sigma }^2{\mathbf{u}}_{\theta }{\mathbf{u}}_{\theta }^T+{ϵ}^2{\mathbf{v}}_{\theta }{\mathbf{v}}_{\theta }^T$. For $ϵ=0.1$, the two quadratic values become four and four hundred. The determinant is the same at these orientations, so this full-rank Gaussian model correctly disfavors the transverse observation.

The transverse-noise limit of a rotating rank-one covariance model is an inference about a stated physical noise model, not permission to add arbitrary diagonal noise for numerical convenience. In spectral work, distinguish a parameter-dependent physical noise direction from a fixed deterministic rebinning matrix; the latter has a fixed output range when its input covariance is positive definite.