---
title: Predictive Validity Analysis
url: https://www.emergentmind.com/topics/predictive-validity-analysis
type: topic
---

# Predictive Validity Analysis

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 \(\xi\) the true state then  
     \[
       e \;:=\; \phi_{\hat X^{-1}}(\xi)\;\in\;\calM.
     \]  
     –  At \(e=\bar\xi\) one recovers \(\xi=\hat\xi:=\phi_{\hat X}(\bar\xi)\).  
     –  Because \(\phi\) is a group action, \(e\) is intrinsically and globally well‐defined.  

3. Equivariant lift of the plant dynamics  
   •  By equivariance of the original system  
     \[
       f(\xi,u)\;=\;T_\id\phi_{X}\,\Lambda(\xi,u)
       \quad\text{with}\quad
       \Lambda:\calM\times\vecL\to\gothg,
     \]  
     where \(\gothg=T_\id G\).  
   •  The **lifted** (internal‐model) dynamics on \(G\) are  
     \[
       \dot X 
       \;=\; X\,\Lambda\!\bigl(\phi_{\bar\xi}(X),\,u\bigr)\,, 
       \quad X(0)=\Id.
     \]  
   •  **Equivariance of the lift**: for all \(X\in G\),  
     \[
       \Ad_{X^{-1}}\!\bigl[\Lambda(\xi,u)\bigr]
       \;=\;\Lambda\bigl(\phi_X(\xi),\,\psi_X(u)\bigr),
     \]  
     where \(\psi\) is the induced input action on \(\vecL\).  

4. Equivariant error‐dynamics  
   •  Differentiating \(e=\phi_{\hat X^{-1}}(\xi)\) under  
     \(\dot{\hat X}= \hat X\,\Lambda(\hat\xi,u)-\Delta\hat X\) and \(\dot\xi=f(\xi,u)\) yields  
     \[
       \dot e
       \;=\;
       T_e\phi\,\Bigl[
         \Lambda\bigl(e,\psi_{\hat X^{-1}}(u)\bigr)
         -\;\Lambda\bigl(\bar\xi,\psi_{\hat X^{-1}}(u)\bigr)
       \Bigr]
       \;-\;
       T_e\phi\,\bigl[\Delta\bigr].
     \]  
   •  Denote \(\bar u=\psi_{\hat X^{-1}}(u)\).  Then one has the compact form  
     \[
       \dot e
       = T_e\phi\,\bigl[\Lambda(e,\bar u)-\Lambda(\bar\xi,\bar u)\bigr]
       \;-\;
       T_e\phi\,[\Delta].
     \]  
   •  **Key simplification**: the dependence on \(\hat X\) enters only via the single “origin‐input” \(\bar u\).  

5. Equivariant Filter (EqF) via EKF on the group error  
   (a) **Reference trajectory** \(\check X(t)\in G\): unforced lift  
     \(\dot{\check X}=\check X\,\Lambda(\phi_{\bar\xi}(\check X),u)\).  
   (b) **Reference error** \(\check e=\phi_{\check X^{-1}}(\xi)\).  
   (c) **Linearisation** about \((\bar\xi,\bar y)\) in local coordinates  
     \(\varepsilon:\calM\!\supset\!U\to\R^m,\;\delta:\calN\!\supset\!V\to\R^n\) yields  
     \[
       \begin{cases}
         \dot{\check\varepsilon}
           = A_t\,\check\varepsilon + B_t\,w,\\[3pt]
         \check\delta
           = C_t\,\check\varepsilon + D_t\,v,
       \end{cases}
     \]  
     with process‐noise \(w\sim\mathcal N(0,Q_t)\), measurement‐noise \(v\sim\mathcal N(0,R_t)\).  
   (d) **Kalman‐gain** \(K_t = \Sigma_t\,C_t^\top\,(R_t)^{-1}\).  
   (e) **Riccati with curvature**  
     \[
       \dot \Sigma
       = A_t\,\Sigma + \Sigma\,A_t^\top + Q_t
         \;-\;\Sigma\,C_t^\top\,R_t^{-1}\,C_t\,\Sigma
         \;+\;\mathcal C(\Sigma),
     \]  
     where \(\mathcal C(\Sigma)\) is the **curvature correction** arising from infinitesimal parallel‐transport of \(\Sigma\) under the “steering” input.  
   (f) **Observer update** on \(G\):  
     \[
       \Delta_t 
       = T_{\bar\xi}\phi^\dagger\,
         \bigl[
           \tfrac{\pd\varepsilon}{\pd e}^{-1}\,
           K_t\;\bigl(\delta - \hat\delta\bigr)
         \bigr],
     \]
     \[
       \dot{\hat X}
       = \hat X\,\Lambda\bigl(\phi_{\bar\xi}(\hat X),u\bigr)
         - \Delta_t\,\hat X,
       \quad \hat X(0)=\Id.
     \]  
   (g) **State estimate** \(\hat\xi(t)=\phi_{\hat X(t)}(\bar\xi)\), with covariance \(\Sigma_t\).  

6. Theoretical results: robustness & performance  
   •  Local second‐order optimality in the equivariant error coordinates.  
   •  **Uniform regularity** of \((A_t,B_t,C_t)\) along any trajectory, since linearisation point is fixed at \(\bar\xi\).  
   •  Curvature term \(\mathcal C(\Sigma)\) guarantees correct propagation of uncertainty on non‐flat \(\calM\).  
   •  Provable improvements over standard EKF:  
     –  Larger region of attraction (no coordinate‐switching).  
     –  Lower linearisation error ⇒ faster transient, less bias.  
     –  Better consistency (filter‐energy \(\varepsilon^\top\Sigma^{-1}\varepsilon\) stays small).  

7. Illustrative examples  
   (i) **Direction‐kinematics on \(\Sph^2\) (attitude‐like)**  
   \[
     \dot\eta = -\Omega\times\eta,\quad \Omega\in\R^3,
     \quad \phi_Q(\eta)=Q^\top\eta,\;Q\in\SO(3).
   \]  
   –  Lift: \(\Lambda(\eta,\Omega)=\Omega^\times\in\so(3)\).  
   –  Error: \(e = \hat Q\,\eta\).  
   –  Innovation: \(d=\hat Q\,y\).  
   –  EqF Riccati w/o curvature in *normal* coordinates ⇒ very fast bearing convergence.  
   (ii) **Second‐order linear kinematics with range/bearing**  
   \[
     \dot p=v,\;\dot v=a;\quad
     y_1=\tfrac p{\|p\|},\;y_2=\|p\|.
   \]  
   –  Polar symmetry \(G=(\SO(3)\!\times\!\R^+)\ltimes\R^3\) with action  
     \(\phi_{(R,r,\beta)}(p,v)\!=\!(r^{-1}R^\top p,\;r^{-1}R^\top(v-\beta))\).  
   –  Lift \(\Lambda((p,v),a)\) state‐dependent; equivariant innovation linear in *one* local chart.  
   –  EqF with curvature term \(\mathcal C(\Sigma)\) ⇒ robust range/bearing‐aided velocity‐estimation, outperforming both naive EKF and linear KF with algebraic position reconstruction.  

— End of exposition.

Source: https://www.emergentmind.com/topics/predictive-validity-analysis