motivation: sensor selection adaptive sensor selection in
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Adaptive Sensor Selection in Sequential Decision Making
Vaibhav Srivastava, Kurt Plarre and Francesco Bullo
Center for Control, Dynamical Systems & Computation
University of California at Santa Barbara
http://motion.me.ucsb.edu/∼vaibhav
14 December 2011
Conference on Decision & Control and European Control Conference
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 1 / 11
Motivation: Sensor selection
Attention in Camera Sensor Network Sensors for UAV Surveillance
Which camera to choose?
Cloud Rain WindEO X X XSAR XFPR X XIR X XMTIR X
Which sensor to choose?
1 how to avoid operator overload?
2 how to select most informative sensors and focus attention?
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 2 / 11
Motivation: Sensor selection
Attention in Camera Sensor Network Sensors for UAV Surveillance
Which camera to choose?
Cloud Rain WindEO X X XSAR XFPR X XIR X XMTIR X
Which sensor to choose?
1 how to avoid operator overload?
2 how to select most informative sensors and focus attention?
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 2 / 11
Relevant Literature
Human Decision Making
R. Bogacz, E. Brown, J. Moehlis, P. Holmes, and J. D. Cohen. The physics of optimal decisionmaking: A formal analysis of performance in two-alternative forced choice tasks. Psychological
Review, 113(4):700–765, 2006
Sequential Test of Hypothesis
A. Wald. Sequential tests of statistical hypotheses. 16(2):117–186, 1945
C. W. Baum and V. V. Veeravalli. A sequential procedure for multihypothesis testing. IEEE Trans In-
formation Theory, 40(6):1994–2007, 1994
Sensor Selection
D. Bajovic, B. Sinopoli, and J. Xavier. Sensor selection for hypothesis testing in wireless sensornetworks: a Kullback-Leibler based approach. In Proc CDC, pages 1659–1664, Shanghai, China,December 2009S. Joshi and S. Boyd. Sensor selection via convex optimization. IEEE Trans Signal Processing,57(2):451–462, 2009
Search
J. P. Hespanha, H. J. Kim, and S. S. Sastry. Multiple-agent probabilistic pursuit-evasion games. InProc CDC, pages 2432–2437, Phoenix, AZ, USA, December 1999T. H. Chung and J. W. Burdick. A decision-making framework for control strategies in probabilisticsearch. In Proc ICRA, pages 4386–4393, Roma, Italy, April 2007
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 3 / 11
Problem Setup
1 Binary decision making tasks
2 N information sources
3 operator focuses attentionto only one source at a time
4 collection+transmission+processing timeof sensor s is a random variable Ts > 0
Problem: how to select sensors to minimize decision time?
Issues: Non-i.i.d. data, sensor selection problem is NP-hard, in general
Two phase approach
1 determine optimal stationary sensor selection scheme
2 adapt the stationary scheme at each step
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 4 / 11
Problem Setup
1 Binary decision making tasks
2 N information sources
3 operator focuses attentionto only one source at a time
4 collection+transmission+processing timeof sensor s is a random variable Ts > 0
Problem: how to select sensors to minimize decision time?
Issues: Non-i.i.d. data, sensor selection problem is NP-hard, in general
Two phase approach
1 determine optimal stationary sensor selection scheme
2 adapt the stationary scheme at each step
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 4 / 11
Stationary Scheme
two alternative hypotheses: H0,H1
given pdfs fs(y |Hk) = fks (y), k ∈ {0, 1}
sensor selection probability N−tuple q
decision thresholds η0 < 0 < η1
SPRT with stationary sensor selection
At each time t
(a) Sample a sensor st from q
(b) Compute likelihood: lt ≡ log(f 1st (yt)/f0st (yt))
(c) Lt :=�t
τ=1 lτ
(d)η1 < Lt =⇒ sayH1
Lt < η0 =⇒ sayH0
η0 < Lt < η1 =⇒ continue sampling
Stationary Scheme
SPRT Evolutions
Stationary scheme makes sequence {(st , yt)}t∈N i.i.d.
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 5 / 11
Optimal Stationary Scheme
KL divergence D(f 1, f 0) ≡ Ef 1�log f 1(Y )
f 0(Y )
�
Stationary Decision Time
E[Td |Hk ] =const×
�ns=1 qsTs�n
s=1 qsD(f ks , f ∗s )Linear fractional function
Optimal Sensor Selection Probability
Given prior probability πk of Hk
q∗ = argmin{π0E[Td |H0] + π1E[Td |H1]}
Sum of Linear fractional function
– A non-convex problem, but efficiently solvable
– Conditioned on a hypothesis, optimal policy is deterministic
– An optimal policy samples at most two sensors
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 6 / 11
Optimal Stationary Scheme
KL divergence D(f 1, f 0) ≡ Ef 1�log f 1(Y )
f 0(Y )
�
Stationary Decision Time
E[Td |Hk ] =const×
�ns=1 qsTs�n
s=1 qsD(f ks , f ∗s )Linear fractional function
Optimal Sensor Selection Probability
Given prior probability πk of Hk
q∗ = argmin{π0E[Td |H0] + π1E[Td |H1]}
Sum of Linear fractional function
– A non-convex problem, but efficiently solvable
– Conditioned on a hypothesis, optimal policy is deterministic
– An optimal policy samples at most two sensors
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 6 / 11
Adaptive Policy
Adaptive Sensor Selection Probability
At each time t + 1
1: Determine posterior probabilities
π0(t) = 1/(1 + exp(Lt)) & π1(t) = 1− π0(t)
2: Adapt the sensor selection probability
q∗t+1=argmin{π0(t)E[Td |H0] + π1(t)E[Td |H1]}
Posterior probabilities Sensor selection probabilities
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 7 / 11
Performance of Adaptive Policy
Global Lower Bound
E[Td |Hk ] ≥ mins∈{1,...,n}
const× Ts
D(f ks , f ∗s )=
const× Tsk
D(f ksk , f∗sk )
Upper Bound for Adaptive Policy
E[Td |Hk ] ≤ minq
max {E[Td |H0],E[Td |H1]}
Adaptive policy is asymptotically optimal
provided each sensor is informative
Performance bounds
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 8 / 11
Performance of Adaptive Policy
Global Lower Bound
E[Td |Hk ] ≥ mins∈{1,...,n}
const× Ts
D(f ks , f ∗s )=
const× Tsk
D(f ksk , f∗sk )
Upper Bound for Adaptive Policy
E[Td |Hk ] ≤ minq
max {E[Td |H0],E[Td |H1]}
Adaptive policy is asymptotically optimal
provided each sensor is informative
Performance bounds
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 8 / 11
Performance of Adaptive Policy
Global Lower Bound
E[Td |Hk ] ≥ mins∈{1,...,n}
const× Ts
D(f ks , f ∗s )=
const× Tsk
D(f ksk , f∗sk )
Upper Bound for Adaptive Policy
E[Td |Hk ] ≤ minq
max {E[Td |H0],E[Td |H1]}
Adaptive policy is asymptotically optimal
provided each sensor is informative
Performance bounds
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 8 / 11
Control of Decision Time
Assumption: Identical processing time of the sensors
Reorder sensors in decreasing order of D(f 0s , f1s )
Pick first ζ ≤ n sensors, s0� , � ∈ {1, . . . , ζ}Similarly, pick sensors s1� , � ∈ {1, . . . , ζ}
Asymptotically optimal policy
Apply adaptive policy to sets Ξk = {sk� | � ∈ {1, . . . , ζ}}, k ∈ {0, 1}Asymptotic decision time: E[Td |Hk ] = const/(
�ζ�=1D(f 0s� , f
1s�))
Control of Decision Time
Given a desired feasible expected decision time, a randomized cardinality ζof the set can be designed
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 9 / 11
Application: Search in a Camera Network
Treasure at region k with prior prob. πk
Search ≡ MSPRT
One camera at each region
(evidence | treas. at loc. k) ∼ f1k ; f
0k o.w.
Region selection probability M-tuple: q
Each sensor non-informative about other regions
Region selection probability Posterior probability of treasure
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 10 / 11
Application: Search in a Camera Network
Treasure at region k with prior prob. πk
Search ≡ MSPRT
One camera at each region
(evidence | treas. at loc. k) ∼ f1k ; f
0k o.w.
Region selection probability M-tuple: q
Each sensor non-informative about other regions
Region selection probability Posterior probability of treasure
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 10 / 11
Conclusions & Future directions
Conclusions
Identification of most pertinent information sources
Max. cardinality of optimal source set = No. of hypothesis
Adaptive source selection is asymptotically optimal
Communication-decision time trade-off
Application to decision theoretic search
Future Directions
Extension to dynamic hypothesis
Extension of GLR
Vaibhav Srivastava (UCSB) Adaptive Sensor Selection 12-14-11 CDC-ECC 11 / 11
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