Sherman’s Theorem

1. Sherman’s Theorem Fundamental Technology for ODTK Jim Wright

2. Why? Satisfaction of Sherman's Theorem guarantees that the mean-squared state estimate error on each state estimate component is minimized

3. Sherman Probability Density

4. Sherman Probability Distribution

5. Notational Convention Here • Bold symbols denote known quantities (e.g., denote the optimal state estimate by ΔXk+1|k+1, after processing measurement residual Δyk+1|k) • Non-bold symbols denote true unknown quantities (e.g., the error ΔXk+1|k in propagated state estimate Xk+1|k)

6. Admissible Loss Function L • L = L(ΔXk+1|k) a scalar-valued function of state • L(ΔXk+1|k) ≥ 0; L(0) = 0 • L(ΔXk+1|k) is a non-decreasing function of distance from the origin: limΔX→ 0L(ΔX) = 0 • L(-ΔXk+1|k) = L(ΔXk+1|k) Example of interest (squared state error): L(ΔXk+1|k) = (ΔXk+1|k)T (ΔXk+1|k)

7. Performance Function J(ΔXk+1|k) J(ΔXk+1|k) = E{L(ΔXk+1|k)} Goal: Minimize J(ΔXk+1|k), the mean value of loss on the unknown state error ΔXk+1|k in the propagated state estimate Xk+1|k. Example (mean-squared state error): J(ΔXk+1|k) = E{(ΔXk+1|k)T (ΔXk+1|k)}

8. Aurora Response to CME

9. Minimize Mean-Squared State Error

10. Sherman’s Theorem Given any admissible loss function L(ΔXk+1|k), and any Sherman conditional probability distribution function F(ξ|Δyk+1|k), then the optimal estimate ΔXk+1|k+1 of ΔXk+1|k is the conditional mean: ΔXk+1|k+1 = E{ΔXk+1|k| Δyk+1|k}

11. Doob’s First TheoremMean-Square State Error If L(ΔXk+1|k) = (ΔXk+1|k)T (ΔXk+1|k) Then the optimal estimate ΔXk+1|k+1 of ΔXk+1|k is the conditional mean: ΔXk+1|k+1 = E{ΔXk+1|k| Δyk+1|k} The conditional distribution function need not be Sherman; i.e., not symmetric nor convex

12. Doob’s Second TheoremGaussian ΔXk+1|k and Δyk+1|k If: ΔXk+1|k and Δyk+1|k have Gaussian probability distribution functions Then the optimal estimate ΔXk+1|k+1 of ΔXk+1|k is the conditional mean: ΔXk+1|k+1 = E{ΔXk+1|k| Δyk+1|k}

13. Sherman’s Papers • Sherman proved Sherman’s Theorem in his 1955 paper. • Sherman demonstrated the equivalence in optimal performance using the conditional mean in all three cases, in his 1958 paper

14. Kalman • Kalman’s filter measurement update algorithm is derived from the Gaussian probability distribution function • Explicit filter measurement update algorithm not possible from Sherman probability distribution function

15. Gaussian Hypothesis is Correct • Don’t waste your time looking for a Sherman measurement update algorithm • Post-filtered measurement residuals are zero mean Gaussian white noise • Post-filtered state estimate errors are zero mean Gaussian white noise (due to Kalman’s linear map)

16. Measurement System Calibration • Definition from Gaussian probability density function • Radar range spacecraft tracking system example

17. Gaussian Probability DensityN(μ,R2) = N(0,1/4)

18. Gaussian Probability DistributionN(μ,R2) = N(0,1/4)

19. Calibration (1) N(μ,R2) = N(0,[σ/σinput]2) N(μ,R2) = N(0,1) ↔ σinput = σ σinput > σ • Histogram peaked relative to N(0,1) • Filter gain too large • Estimate correction too large • Mean-squared state error not minimized

20. Calibration (2) σinput < σ • Histogram flattened relative to N(0,1) • Filter gain too small • Estimate correction too small • Residual editor discards good measurements – information lost • Mean-squared state error not minimized

21. Before Calibration

22. After Calibration

23. Nonlinear Real-Time Multidimensional Estimation • Requirements - Validation • Conclusions - Operations

24. Requirements (1 of 2) • Adopt Kalman’s linear map from measurement residuals to state estimate errors • Measurement residuals must be calibrated: Identify and model constant mean biases and variances • Estimate and remove time-varying measurement residual biases in real time • Process measurements sequentially with time • Apply Sherman's Theorem anew at each measurement time

25. Requirements (2 of 2) • Specify a complete state estimate structure • Propagate the state estimate with a rigorous nonlinear propagator • Apply all known physics appropriately to state estimate propagation and to associated forcing function modeling error covariance • Apply all sensor dependent random stochastic measurement sequence components to the measurement covariance model

26. Necessary & Sufficient ValidationRequirements • Satisfy rigorous necessary conditions for real data validation • Satisfy rigorous sufficient conditions for realistic simulated data validation

27. Conclusions (1 of 2) • Measurement residuals produced by optimal estimators are Gaussian white residuals with zero mean • Gaussian white residuals with zero mean imply Gaussian white state estimate errors with zero mean (due to linear map) • Sherman's Theorem is satisfied with unbiased Gaussian white residuals and Gaussian white state estimate errors

28. Conclusions (2 of 2) • Sherman's Theorem maps measurement residuals to optimal state estimate error corrections via Kalman's linear measurement update operation • Sherman's Theorem guarantees that the mean-squared state estimate error on each state estimate component is minimized • Sherman's Theorem applies to all real-time estimation problems that have nonlinear measurement representations and nonlinear state estimate propagations

29. Operational Capabilities • Calculate realistic state estimate error covariance functions (real-time filter and all smoothers) • Calculate realistic state estimate accuracy performance assessment (real-time filter and all smoothers) • Perform autonomous data editing (real-time filter, near-real-time fixed-lag smoother)