Quickest Detection of a Change Process Across a Sensor Array

1. Quickest Detection of a Change Process Across aSensor Array VasanthanRaghavan and Venugopal V. Veeravalli Presented by: Kuntal Ray

2. Outline • Introduction • Problem Formulation • DP Frame Work • Recursion for Sufficient Statistics • Structure for Optimal Stopping Rule • References

3. Outline • Introduction • Problem Formulation • DP Frame Work • Recursion for Sufficient Statistics • Structure for Optimal Stopping Rule • References

4. INTRODUCTION • Sensors take observations ,responds to disruptive change • The goal is to detect this change point, subjected to false alarm constraints • Sequence of observations ,densities changes at unknown time has to be detected.

5. Two Approaches to change detection: - • Bayesian Approach • Change point is assumed to be a random variable with a prior density known as a priori • Goal is to minimize the expected detection delay subject to a bound on the false alarm Probability • Minimax Approach • Goal is to minimize the worst case delay subject to lower bound on the mean time between false alarm

6. Introduction… • Significant advances in theory of change detection has been made using single sensor • Also extension of those framework to the multi-sensor case has been studied, where information available for decision making is decentralized. • The above work assume that the statistical properties of the sensors’ observations change at the same time. • However, in many scenarios, it is more suitable to consider the case where each sensor’s observations may change at different points in time.

7. Introduction… • An application to such a model is detection of pollutants where the change process is governed by the movement of the agent through the medium under consideration. • This paper considers Bayesian version of this problem and assumes point of disruption is a random variable with Geometric Distribution.

8. INTRODUCTION… • Assume L sensors placed in an array • Fusion center has complete information about the observations. • This is applicable when ample bandwidth is available for communication between the sensors and the fusion center

9. INTRODUCTION… • The goal of the fusion center is to come up with a strategy to declare change, subject to false alarm constraints • Towards this goal, pose the problem in a dynamic programming (DP) framework and first obtain sufficient statistics for the DP under consideration • We then establish a recursion for the sufficient statistics which generalizes the recursion established in previous paper • Following along the logic of previous work they establish the optimality of a more general stopping rule for change detection.

10. Outline • Introduction • Problem Formulation • DP Frame Work • Recursion for Sufficient Statistics • Structure for Optimal Stopping Rule • References

11. PROBLEM FORMULATION • Consider L sensors that observes L dimensional Discrete time Stochastic process • Disruption in the sensing environment occurs at random time constant Γ1 • Hence the density of the observations at each sensor undergoes a change from the null density f0to the alternate density f1.

12. Problem Formulation… • Previous work considers change to be instantaneous to all the sensors at time Γ1 • In this paper they consider change process which evolves across the sensor array and the change seen by the lth sensor is given by Γl • Also assume the evolution of the change process is Markovian Process across the sensor

13. Problem Formulation… • Under this model, the change point evolves as a geometric random variable with parameter ρ. • P({Γ1 = m}) = ρ (1 − ρ)m, m ≥ 0. • As ρ 1 corresponds to case where instantaneous disruption has high probability of occurrence • As ρ 0 uniformizes the change point in the sense that the disruption is equally likely to happen at any point at any time

14. Problem Formulation… • Observations at every sensor are independent and identically distributed (i.i.d.) conditioned on the change hypothesis corresponding to that sensor. • Zk, ∼ i.i.d. f0 if k < Γ, i.i.d. f1 if k ≥ Γ. • Consider a centralized, Bayesian setup where a fusion center has complete knowledge of the observations from all the sensors • Ik {Z1, . . . ,Zk}

15. Problem Formulation… • The fusion center decides whether a change has happened or not based on the information, Ik, available to it at time instant k (equivalently, it provides a stopping time τ)

16. Problem Formulation… • Two conflicting performance measures on change detection are: - • Probability of false Alarm • PFA = P({τ < Γ1}) • The average detection delay, • EDD = E [(τ − Γ1)+] where x+ = max(x, 0).

17. Problem Formulation • The previous two conflicts are captured by Bayes Risk which is defined as: • R(c) = PFA + cEDD • For an appropriate choice of per-unit delay cost ‘c’ • The goal of the fusion center is to come up with a strategy (a stopping time τ) to minimize the Bayes risk

18. Outline • Introduction • Problem Formulation • DP Frame Work • Recursion for Sufficient Statistics • Structure for Optimal Stopping Rule • References

19. DP Framework • In their previous paper they had rewritten Bayes Risk as: • The state of the system at time k is the vector • Sk =[Sk,1, . . . , Sk,L] • With Sk, denoting the state at sensor . • The state Sk, can take the value 1 (post-change), 0 (prechange), or t (terminal). • The system goes to the terminal state t, once a change-point decision τ has been declared.

20. Outline • Introduction • Problem Formulation • DP Frame Work • Recursion for Sufficient Statistics • Structure for Optimal Stopping Rule • Asymptotically Optimal Test • Numerical Results • References

21. Recursion for Sufficient Statistics • Consider case where changes to all sensors happen at same instant. • In this setting, it can be shown that Random Variable P({Γ1 ≤ k}|Ik) serves as the sufficient statistics for the dynamic program and affords a recursion • But, we consider general case

22. Recursion for Sufficient Statistics • But as we consider general case, i.e. slow propagation of change

23. Outline • Introduction • Problem Formulation • DP Frame Work • Recursion for Sufficient Statistics • Structure for Optimal Stopping Rule • References

24. Structure for Optimal Stopping Rule

25. Outline • Introduction • Problem Formulation • DP Frame Work • Recursion for Sufficient Statistics • Structure for Optimal Stopping Rule • References

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