 Transverse-noise limit of a rotating rank-one covariance model

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.