1. Massachusetts Institute of Technology September 7, 2005 A Tractable Approach to Probabilistically Accurate Mode Estimation Oliver B. Martin Seung H. Chung Brian C. Williams i-SAIRAS 2005 Munich, Germany

2. 2 Closed Valve Open Stuckopen Stuckclosed OpenClose 0. 01 0.01 0.01 inflow = outflow = 0

3. 3 Estimation of PCCA (Probabilistic Concurrent Constraint Automata) Belief State Evolution visualized with a Trellis Diagram Complete history knowledge is captured in a single belief state by exploiting the Markov property Belief states are computed using the HMM belief state update equations Compute the belief state for each estimation cycle closedopen stuck-open cmd=open obs=flow cmd=closed obs=no-flow 0.7 0.2 0.1

4. 4 Hidden Markov Model Belief State Update Equations Propagation step computes aprior probability Update step computes posterior probability tt+1

5. 5 Approximations to PCCA Estimation k most likely estimates – The belief state can be accurately approximated by maintain the k most likely estimates Most likely trajectory – The probability of each state can be accurately approximated by the most likely trajectory to that state 1 or 0 observation probability – The observation probability can be reduced to 1.0 for observations consistent with the state, and 0.0 for observations inconsistent with the state t+1 1.0 or 0.0

6. 6 Previous Estimation Approach: Best-First Trajectory Estimation (BFTE) Livingstone –Williams and Nayak, AAAI-96 Livingstone 2 –Kurien and Nayak, AAAI-00 Titan –Williams et al., IEEE ’03 The belief state can be accurately approximated by maintain the k most likely estimates The probability of each state can be accurately approximated by the most likely trajectory to that state The observation probability can be reduced to –1.0 for observations consistent with the state, –0.0 for observations inconsistent with the state 0.4 0.2 0.50 0.15 0.35 Approximations to Estimation Deep Space One Earth Observing One

7. 7 Previous Estimation Approach: Best-First Belief State Enumeration (BFBSE) Diagnosis as Approximate Belief State Enumeration for Probabilistic Concurrent Constraint Automata –Martin et al., AAAI-05 Increased estimator accuracy by maintaining a compact belief state encoding Minimal overhead over BFTE The belief state can be accurately approximated by maintain the k most likely estimates The probability of each state can be accurately approximated by the most likely trajectory to that state The observation probability can be reduced to –1.0 for observations consistent with the state, –0.0 for observations inconsistent with the state 0.4 0.2 Approximations to Estimation 0.65 0.05 0.30

8. 8 New Approach: Best-First Belief State Update (BFBSU) Increased estimator accuracy by properly computing the observation probability Minimal overhead over BFBSE The belief state can be accurately approximated by maintain the k most likely estimates The probability of each state can be accurately approximated by the most likely trajectory to that state The observation probability can be reduced to –1.0 for observations consistent with the state, –0.0 for observations inconsistent with the state 0.4 0.2 Approximations to Estimation 0.65 0.30 0.05

9. 9 BFBSE vs. BFBSU: The Consequence of Incorrect Observation Probability Sensor = High Working Broken Observe (Sensor = High) at every time step Assume: P(Sensor = High | Broken) = 1P(Sensor = High | Broken) = 0.5 0.990 0.010 1 0 0 1 0.001 0.999 0.995 0.005 1 0 0.990 0.010 0.991 0.009 0.99 0.01 1 Best-First Belief State EstimationBest-First Belief State Update

10. 10 BSBSU Computes HMM Belief State Update Equations Propagation Step Update Step Problem: Computing the observation probability for the “otherwise” case can be expensive. –Must compute total number of consistent observations for k states –Model counting for k problems

11. 11 Solution: Pre-Compute Observation Probability Rules Observation Probability Rule (OPR): –Compute OPR for a set of partial states –Compute OPRs using the dissents –Ignore all OPRs with probability of 0 or 1 ModelMax # of OPRs# OPRs Required Online EO-11.77×10 8 64 MarsEDL1.46×10 6 307 ST7-A1.44×10 4 8

12. 12 Solve Mode Estimation as an Optimal Constraint Satisfaction Problem Use Conflict-directed A* with a tight admissible heuristic –Use greedy approximation for the cost to go (BFBSE) –Include observation probability within the heuristic (BFBSU)

13. 13 ST7A Model Results: State Estimate Accuracy Single-point Failure state estimate probability Nominal state estimate probability

14. 14 ST7A Model Results: State Estimate Performance Space Performance –Best case: n·b –Worst case: b n Time Performance –Best case: n 2 ·b·k + n·b·C –Worst case: BFBSE: b n (n·k + C) BFBSU: b n (n·k + b n ·R + C) Number of initial states, k bAverage number of modes nNumber of components kNumber of estimates being tracked CConstant factor for data manipulation ROPR lookup time

15. 15 EO1 Model Results: State Estimate Performance Space PerformanceTime Performance

16. 16 Conclusion Best-First Belief State Update –Compute k most likely estimates –Remove most likely trajectory approximation by computing the most likely belief state estimates –Remove 1 or 0 observation probability approximation by computing the proper observation probabilities Increased PCCA estimator accuracy by computing the Optimal Constraint Satisfaction Problem (OCSP) utility function directly from the HMM propagation and update equations Maintaining the computational efficiency.

17. 17 051015202530 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 State estimate probability Estimation timestep SinglePoint Failure State Estimate Probability for ST7A Model BFBSE, k=10 BFBSU, k=10 Exact State estimate probability 051015202530 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 State estimate probability Estimation timestep Nominal State Estimate Probability over Time for ST7A Model BFBSE, k=10 BFBSU, k=10 Exact State estimate probability