Quickest Detection of a Change Process Across a Sensor Array Vasanthan Raghavan and Venugopal V. Veeravalli Presented by: Kuntal Ray

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

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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.

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

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.

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.

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

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.

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PROBLEM FORMULATION Consider L sensors that observes L dimensional Discrete time Stochastic process Disruption in the sensing environment occurs at random time constant Γ 1 Hence the density of the observations at each sensor undergoes a change from the null density f 0 to the alternate density f 1.

Problem Formulation… Previous work considers change to be instantaneous to all the sensors at time Γ 1 In this paper they consider change process which evolves across the sensor array and the change seen by the l th sensor is given by Γ l Also assume the evolution of the change process is Markovian Process across the sensor

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

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. f 0 if k < Γ, i.i.d. f 1 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}

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)

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).

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

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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.

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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

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

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Structure for Optimal Stopping Rule

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References [1] M. Basseville and I. V. Nikiforov, Detection of Abrupt Changes: Theory and Applications. Prentice Hall, Englewood Cliffs, [2] T. L. Lai, “Sequential changepoint detection in quality control and dynamical systems,” J. R. Statist. Soc. B, Vol. 57, No. 4, pp. 613–658, [3] G. Lorden, “Procedures for reacting to a change in distribution,” Ann. Math. Statist., Vol. 42, pp. 1987– 1908, [4] G. V. Moustakides, “Optimal stopping times for detecting changes in distributions,” Ann. Statist., Vol. 14, pp. 1379–1387, [5] M. Pollak, “Optimal detection of a change in distribution,” Ann. Statist., Vol. 13, pp. 206–227, [6] A. N. Shiryaev, “On optimum methods in quickest detection problems,” Theory Probab. Appl., Vol. 8, pp. 22–46, [7] A. N. Shiryaev, Optimal Stopping Rules. Springer- Verlag, NY, [8] A. G. Tartakovsky, Sequential Methods in the Theory of Information Systems. Radio i Svyaz’, Moscow, 1991 (In Russian).