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A Tractable Approach to Probabilistically Accurate Mode Estimation. Oliver B. Martin Seung H. Chung Brian C. Williams i-SAIRAS 2005 Munich, Germany. Valve. Stuck open. 0.01. Open. 0. 01. Open. Close. 0. 01. Stuck closed. Closed. 0.01. inflow = outflow = 0.

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a tractable approach to probabilistically accurate mode estimation

A Tractable Approach to Probabilistically Accurate Mode Estimation

Oliver B. Martin

Seung H. Chung

Brian C. Williams

i-SAIRAS 2005

Munich, Germany

slide2

Valve

Stuck

open

0.01

Open

0. 01

Open

Close

0. 01

Stuck

closed

Closed

0.01

inflow = outflow = 0

estimation of pcca p robabilistic c oncurrent c onstraint a utomata
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

stuck-open

closed

open

0.7

0.2

0.1

cmd=openobs=flow

cmd=closedobs=no-flow

hidden markov model belief state update equations

t

t+1

Hidden Markov ModelBelief State Update Equations
  • Propagation step computes aprior probability
  • Update step computes posterior probability
approximations to pcca estimation
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

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

0.50

0.15

0.35

Previous Estimation Approach:Best-First Trajectory Estimation (BFTE)

Approximations to Estimation

Deep Space One

Earth Observing One

0.4

0.2

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

0.65

0.05

0.30

Previous Estimation Approach: Best-First Belief State Enumeration (BFBSE)

Approximations to Estimation

0.4

0.2

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

0.65

0.30

0.05

New Approach: Best-First Belief State Update (BFBSU)

Approximations to Estimation

0.4

0.2

bfbse vs bfbsu the consequence of incorrect observation probability

Working State Estimate Probability

1

BFBSE

0.8

BFBSU

0.6

State estimated probability

0.4

0.2

0

0

50

100

150

200

250

300

350

400

450

500

Estimation Timestep

0.990

0.001

0

0.995

0.991

0.990

0.010

0.999

1

0.005

0.009

0.010

BFBSE vs. BFBSU: The Consequence ofIncorrect Observation Probability

0.99

Sensor = High

Working

0.01

Broken

1

Best-First Belief State Estimation

Best-First Belief State Update

Assume: P(Sensor = High | Broken) = 1

P(Sensor = High | Broken) = 0.5

1

1

0

0

Observe (Sensor = High) at every time step

bsbsu computes hmm belief state update equations
BSBSU ComputesHMM Belief State Update Equations
  • Propagation Step
  • Update Step
  • Problem: Computing the observation probability for the “otherwise” case can be expensive.
    • Must compute totalnumber of consistentobservations for k states
    • Model counting for kproblems
solution pre compute observation probability rules
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
solve mode estimation as an optimal constraint satisfaction problem
Solve Mode Estimation as anOptimal Constraint Satisfaction Problem
  • UseConflict-directed A* with a tight admissible heuristic
    • Use greedy approximation for the cost to go (BFBSE)
    • Include observation probability within the heuristic (BFBSU)
st7a model results state estimate performance
Space Performance

Best case: n·b

Worst case: bn

Time Performance

Best case: n2·b·k + n·b·C

Worst case:

BFBSE: bn(n·k + C)

BFBSU: bn(n·k + bn·R + C)

ST7A Model Results:State Estimate Performance

Number of initial states, k

Number of initial states, k

eo1 model results state estimate performance
EO1 Model Results:State Estimate Performance
  • Space Performance
  • Time Performance
conclusion
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.