quickest detection of a change process across a sensor array vasanthan raghavan and venugopal v....
TRANSCRIPT
Quickest Detection of a Change Process Across aSensor 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
Outline
•Introduction•Problem Formulation•DP Frame Work•Recursion for Sufficient Statistics•Structure for Optimal Stopping Rule•References
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.
Outline
•Introduction•Problem Formulation•DP Frame Work•Recursion for Sufficient Statistics•Structure for Optimal Stopping Rule•References
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 f0 to the alternate density f1.
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
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. 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}
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
Outline
•Introduction•Problem Formulation•DP Frame Work•Recursion for Sufficient Statistics•Structure for Optimal Stopping Rule•References
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.
Outline
•Introduction•Problem Formulation•DP Frame Work•Recursion for Sufficient Statistics•Structure for Optimal Stopping Rule•Asymptotically Optimal Test•Numerical Results•References
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
Outline
•Introduction•Problem Formulation•DP Frame Work•Recursion for Sufficient Statistics•Structure for Optimal Stopping Rule•References
Structure for Optimal Stopping Rule
Outline
•Introduction•Problem Formulation•DP Frame Work•Recursion for Sufficient Statistics•Structure for Optimal Stopping Rule•References
References[1] M. Basseville and I. V. Nikiforov, Detection of AbruptChanges: Theory and Applications. Prentice Hall, EnglewoodCliffs, 1993.[2] T. L. Lai, “Sequential changepoint detection in qualitycontrol and dynamical systems,” J. R. Statist. Soc. B,Vol. 57, No. 4, pp. 613–658, 1995.[3] G. Lorden, “Procedures for reacting to a change indistribution,” Ann. Math. Statist., Vol. 42, pp. 1987–1908, 1971.[4] G. V. Moustakides, “Optimal stopping times for detectingchanges in distributions,” Ann. Statist., Vol. 14,pp. 1379–1387, 1986.[5] M. Pollak, “Optimal detection of a change in distribution,”Ann. Statist., Vol. 13, pp. 206–227, 1985.[6] A. N. Shiryaev, “On optimum methods in quickest detectionproblems,” Theory Probab. Appl., Vol. 8, pp.22–46, 1963.[7] A. N. Shiryaev, Optimal Stopping Rules. Springer-Verlag, NY, 1978.[8] A. G. Tartakovsky, Sequential Methods in the Theoryof Information Systems. Radio i Svyaz’, Moscow,1991 (In Russian).