a tractable approach to probabilistically accurate mode estimation
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
September 7, 2005
Massachusetts Institute of Technology
A Tractable Approach to Probabilistically Accurate Mode
Estimation
Oliver B. MartinSeung H. ChungBrian C. Williams
i-SAIRAS 2005Munich, Germany
2
ClosedClosed
ValveValve
OpenOpenStuckStuckopenopen
StuckStuckclosedclosed
OpenOpen CloseClose
0. 010. 01
0. 010. 01
0.010.01
0.010.01
inflow = outflow = 0
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
closed open stuck-open
cmd=openobs=flow
cmd=closedobs=no-flow
0.7
0.2
0.1
4
Hidden Markov ModelBelief State Update Equations
• Propagation step computes aprior probability
• Update step computes posterior probability
0, 0, 0, 0, 11 1, , ,t ti
t t t tt t t t tj j i is Ss o s s s o
P P P
1 1
0, 0,1 1 1
0, 1 0,1
0, 0,1 1 1
,,
,t ti
t tt t tj jt tt
jt tt t t
i is S
s o o ss o
s o o s
P PP
P P
t t+1
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+11.0 or 0.0
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
0.50
0.15
0.35
Approximations to Estimation
Deep Space One Earth Observing One
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
Approximations to Estimation
0.65
0.05
0.30
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
Approximations to Estimation
0.65
0.30
0.05
1 1
1 1 1 1
1 if M ,
0 if M ,
1/ otherwise.
t tj
t t t tj j
s o
o s s o
m
P
‘
‘
9
BFBSE vs. BFBSU: The Consequence ofIncorrect Observation Probability
Sensor = High
Working
Broken
Observe (Sensor = High) at every time step
Assume: P(Sensor = High | Broken) = 1 P(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 Estimation Best-First Belief State Update
Working State Estimate Probability
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200 250 300 350 400 450 500
Estimation TimestepSta
te e
stim
ate
d p
roba
bili
ty
BFBSE
BFBSU
10
BSBSU ComputesHMM 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 consistentobservations for k states
– Model counting for kproblems
0, 0, 0, 0, 11 1, , ,t ti
t t t tt t t t tj j i is Ss o s s s o
P P P
1 1
0, 0,1 1 1
0, 1 0,1
0, 0,1 1 1
,,
,t ti
t tt t tj jt tt
jt tt t t
i is S
s o o ss o
s o o s
P PP
P P
1 1
1 1 1 1
1 if M ,
0 if M ,
1/ otherwise.
t tj
t t t tj j
s o
o s s o
m
P
‘
‘
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
x P o x
M Q is inconsistent o x o x
number of inconsistent observationsi
iom o
o
D
1
mP o x
Model Max # of OPRs # OPRs Required Online
EO-1 1.77×108 64
MarsEDL 1.46×106 307
ST7-A 1.44×104 8
12
Solve Mode Estimation as anOptimal 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)
1
1
1
1 1
0, 0, 1
, ,
max , ,
,
tg g
t t ti h h h h
t t t tg g g g ix v n
t t t t th h h h is S x v n v x
t tti
x v x v s
f n x v x v s o n
s o
D
P
P P
P
13
ST7A Model Results:State Estimate Accuracy
• Single-point Failure state estimate probability
• Nominal state estimate probability
14
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)
0 5 10 15 20 25 3010
20
30
40
50
60
70
Ma
x n
um
ber
of
nod
es
in q
ue
ue
Estimate Enumeration Maximum Queue Size for ST7A Model
BFBSE, single estimateBFBSE, k estimatesBFBSU, single estimateBFBSU, k estimates
0 5 10 15 20 25 300
10
20
30
40
50
60
70
80
Tim
e (
ms)
Estimate Enumeration Time for ST7A Model
BFBSE, single estimateBFBSE, k estimatesBFBSU, single estimateBFBSU, k estimates
Number of initial states, k Number of initial states, k
b Average number of modesn Number of componentsk Number of estimates being trackedC Constant factor for data manipulationR OPR lookup time
15
EO1 Model Results:State Estimate Performance
• Space Performance • Time Performance
0 5 10 15 20 25 300
50
100
150
200
250
300
350
400
450
Number of initial states,k
Max
num
ber
of
nodes
in q
ueue
Estimate Enumeration Maximum Queue Size for EO1 Model
BFBSE, single estimateBFBSE, k estimatesBFBSU, single estimateBFBSU, k estimates
0 5 10 15 20 25 300
50
100
150
200
250
300
350
400
450
Number of initial states,k
Tim
e (
ms)
Estimate Enumeration Time for EO1 Model
BFBSE, single estimateBFBSE, k estimatesBFBSU, single estimateBFBSU, k estimates
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.