Predictive Validity Analysis

• Predictive Validity Analysis is the systematic assessment of a model’s ability to forecast future states by comparing predictions with actual outcomes.
• It leverages methods like EKF-style linearization and Riccati equations with curvature corrections to enhance robustness and reduce bias in state estimation.
• The approach ensures uniform regularity and improved performance, demonstrated in applications such as attitude estimation and range/bearing-aided velocity determination.

1. Definition of a Natural Filter Equivariant (NFE)
   • In the setting of a control‐affine system $\dot\xi \;=\; f(\xi,u),\quad y = h(\xi)\,, \quad \xi\in\calM,\;u\in\vecL,\;y\in\calN,$ a Natural Filter Equivariant (NFE) is an observer‐filter that (a) exploits a transitive Lie‐group symmetry $\phi:\,G\times\calM\to\calM$, (b) preserves equivariance at every step (i.e.\ is itself a group action on its information state), (c) uses an intrinsic “equivariant error” on $\calM$ and a lifted state on $G$, (d) applies an EKF‐style linearisation around the group identity to yield a Riccati equation modified by group‐curvature terms.
2. Lifting to the symmetry group and the globally defined error
   • Transitive action $\phi_X:\calM\to\calM,\;X\in G$ with fixed reference $\bar\xi\in\calM$.
   • Lift: any state $\xi\in\calM$ can be written $\xi=\phi_{X}(\bar\xi)$ for some $X\in G$.
   • Equivariant error: if $\hat X\in G$ is the observer’s group‐state and $\phi:\,G\times\calM\to\calM$0 the true state then $\phi:\,G\times\calM\to\calM$1
     • At $\phi:\,G\times\calM\to\calM$2 one recovers $\phi:\,G\times\calM\to\calM$3.
     • Because $\phi:\,G\times\calM\to\calM$4 is a group action, $\phi:\,G\times\calM\to\calM$5 is intrinsically and globally well‐defined.
3. Equivariant lift of the plant dynamics
   • By equivariance of the original system $\phi:\,G\times\calM\to\calM$6 where $\phi:\,G\times\calM\to\calM$7.
   • The lifted (internal‐model) dynamics on $\phi:\,G\times\calM\to\calM$8 are $\phi:\,G\times\calM\to\calM$9
   • Equivariance of the lift: for all $\calM$0, $\calM$1 where $\calM$2 is the induced input action on $\calM$3.
4. Equivariant error‐dynamics
   • Differentiating $\calM$4 under $\calM$5 and $\calM$6 yields $\calM$7
   • Denote $\calM$8. Then one has the compact form $\calM$9
   • Key simplification: the dependence on $G$0 enters only via the single “origin‐input” $G$1.
5. Equivariant Filter (EqF) via EKF on the group error (a) Reference trajectory $G$2: unforced lift $G$3. (b) Reference error $G$4. (c) Linearisation about $G$5 in local coordinates $G$6 yields $G$7 with process‐noise $G$8, measurement‐noise $G$9. (d) Kalman‐gain $\phi_X:\calM\to\calM,\;X\in G$0. (e) Riccati with curvature $\phi_X:\calM\to\calM,\;X\in G$1 where $\phi_X:\calM\to\calM,\;X\in G$2 is the curvature correction arising from infinitesimal parallel‐transport of $\phi_X:\calM\to\calM,\;X\in G$3 under the “steering” input. (f) Observer update on $\phi_X:\calM\to\calM,\;X\in G$4:

   $\phi_X:\calM\to\calM,\;X\in G$5

   $\phi_X:\calM\to\calM,\;X\in G$6 (g) State estimate $\phi_X:\calM\to\calM,\;X\in G$7, with covariance $\phi_X:\calM\to\calM,\;X\in G$8.

6. Theoretical results: robustness & performance
   • Local second‐order optimality in the equivariant error coordinates.
   • Uniform regularity of $\phi_X:\calM\to\calM,\;X\in G$9 along any trajectory, since linearisation point is fixed at $\bar\xi\in\calM$0.
   • Curvature term $\bar\xi\in\calM$1 guarantees correct propagation of uncertainty on non‐flat $\bar\xi\in\calM$2.
   • Provable improvements over standard EKF:
     • Larger region of attraction (no coordinate‐switching).
     • Lower linearisation error ⇒ faster transient, less bias.
     • Better consistency (filter‐energy $\bar\xi\in\calM$3 stays small).
7. Illustrative examples (i) Direction‐kinematics on $\bar\xi\in\calM$4 (attitude‐like) $\bar\xi\in\calM$5
   • Lift: $\bar\xi\in\calM$6.
   • Error: $\bar\xi\in\calM$7.
   • Innovation: $\bar\xi\in\calM$8.
   • EqF Riccati w/o curvature in normal coordinates ⇒ very fast bearing convergence. (ii) Second‐order linear kinematics with range/bearing $\bar\xi\in\calM$9
   • Polar symmetry $\xi\in\calM$0 with action $\xi\in\calM$1.
   • Lift $\xi\in\calM$2 state‐dependent; equivariant innovation linear in one local chart.
   • EqF with curvature term $\xi\in\calM$3 ⇒ robust range/bearing‐aided velocity‐estimation, outperforming both naive EKF and linear KF with algebraic position reconstruction.

— End of exposition.