Pg 1 of 10 AGI Sherman’s Theorem Fundamental Technology for ODTK Jim Wright

Pg 2 of 10 AGI Why? Satisfaction of Sherman's Theorem guarantees that the mean-squared state estimate error on each state estimate component is minimized

Pg 3 of 10 AGI Sherman Probability Density

Pg 4 of 10 AGI Sherman Probability Distribution

Pg 5 of 10 AGI Notational Convention Here Bold symbols denote known quantities (e.g., denote the optimal state estimate by ΔX k+1|k+1, after processing measurement residual Δy k+1|k ) Non-bold symbols denote true unknown quantities (e.g., the error ΔX k+1|k in propagated state estimate X k+1|k )

Pg 6 of 10 AGI Admissible Loss Function L L = L(ΔX k+1|k ) a scalar-valued function of state L(ΔX k+1|k ) ≥ 0; L(0) = 0 L(ΔX k+1|k ) is a non-decreasing function of distance from the origin: lim ΔX → 0 L(ΔX) = 0 L(-ΔX k+1|k ) = L(ΔX k+1|k ) Example of interest (squared state error): L(ΔX k+1|k ) = (ΔX k+1|k ) T (ΔX k+1|k )

Pg 7 of 10 AGI Performance Function J(ΔX k+1|k ) J(ΔX k+1|k ) = E{L(ΔX k+1|k )} Goal: Minimize J(ΔX k+1|k ), the mean value of loss on the unknown state error ΔX k+1|k in the propagated state estimate X k+1|k. Example (mean-squared state error): J(ΔX k+1|k ) = E{(ΔX k+1|k ) T (ΔX k+1|k )}

Pg 8 of 10 AGI Aurora Response to CME

Pg 9 of 10 AGI Minimize Mean-Squared State Error

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

Pg 11 of 10 AGI Doob’s First Theorem Mean-Square State Error If L(ΔX k+1|k ) = (ΔX k+1|k ) T (ΔX k+1|k ) Then the optimal estimate ΔX k+1|k+1 of ΔX k+1|k is the conditional mean: ΔX k+1|k+1 = E{ΔX k+1|k | Δy k+1|k } The conditional distribution function need not be Sherman; i.e., not symmetric nor convex

Pg 12 of 10 AGI Doob’s Second Theorem Gaussian ΔX k+1|k and Δy k+1|k If: ΔX k+1|k and Δy k+1|k have Gaussian probability distribution functions Then the optimal estimate ΔX k+1|k+1 of ΔX k+1|k is the conditional mean: ΔX k+1|k+1 = E{ΔX k+1|k | Δy k+1|k }

Pg 13 of 10 AGI 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

Pg 14 of 10 AGI 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

Pg 15 of 10 AGI 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)

Pg 16 of 10 AGI Measurement System Calibration Definition from Gaussian probability density function Radar range spacecraft tracking system example

Pg 17 of 10 AGI Gaussian Probability Density N(μ,R 2 ) = N(0,1/4)

Pg 18 of 10 AGI Gaussian Probability Distribution N(μ,R 2 ) = N(0,1/4)

Pg 19 of 10 AGI Calibration (1) N(μ,R 2 ) = N(0,[σ/σ input ] 2 ) N(μ,R 2 ) = N(0,1) ↔ σ input = σ σ input > σ Histogram peaked relative to N(0,1) Filter gain too large Estimate correction too large Mean-squared state error not minimized

Pg 20 of 10 AGI Calibration (2) σ input < 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

Pg 21 of 10 AGI Before Calibration

Pg 22 of 10 AGI After Calibration

Pg 23 of 10 AGI Nonlinear Real-Time Multidimensional Estimation Requirements - Validation Conclusions - Operations

Pg 24 of 10 AGI 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

Pg 25 of 10 AGI 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

Pg 26 of 10 AGI Necessary & Sufficient Validation Requirements Satisfy rigorous necessary conditions for real data validation Satisfy rigorous sufficient conditions for realistic simulated data validation

Pg 27 of 10 AGI 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

Pg 28 of 10 AGI 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

Pg 29 of 10 AGI 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)