Recent attention in quickest change detection in the multi-sensor setting has been on the case where the densities of the observations change at the same instant at all the sensors due to the disruption. In this work, a more general scenario is considered where the change propagates across the sensors, and its propagation can be modeled as a Markov process. A centralized, Bayesian version of this problem, with a fusion center that has perfect information about the observations and a priori knowledge of the statistics of the change process, is considered. The problem of minimizing the average detection delay subject to false alarm constraints is formulated as a partially observable Markov decision process (POMDP). Insights into the structure of the optimal stopping rule are presented. In the limiting case of rare disruptions, we show that the structure of the optimal test reduces to thresholding the a posteriori probability of the hypothesis that no change has happened. We establish the asymptotic optimality (in the vanishing false alarm probability regime) of this threshold test under a certain condition on the Kullback-Leibler (K-L) divergence between the post- and the pre-change densities. In the special case of near-instantaneous change propagation across the sensors, this condition reduces to the mild condition that the K-L divergence be positive. Numerical studies show that this low complexity threshold test results in a substantial improvement in performance over naive tests such as a single-sensor test or a test that wrongly assumes that the change propagates instantaneously.
Deep Dive into Quickest Change Detection of a Markov Process Across a Sensor Array.
Recent attention in quickest change detection in the multi-sensor setting has been on the case where the densities of the observations change at the same instant at all the sensors due to the disruption. In this work, a more general scenario is considered where the change propagates across the sensors, and its propagation can be modeled as a Markov process. A centralized, Bayesian version of this problem, with a fusion center that has perfect information about the observations and a priori knowledge of the statistics of the change process, is considered. The problem of minimizing the average detection delay subject to false alarm constraints is formulated as a partially observable Markov decision process (POMDP). Insights into the structure of the optimal stopping rule are presented. In the limiting case of rare disruptions, we show that the structure of the optimal test reduces to thresholding the a posteriori probability of the hypothesis that no change has happened. We establish the
An important application area for distributed decision-making systems is in environment surveillance and monitoring. Specific applications include: i) Intrusion detection in computer networks and security systems [2], [3], ii) monitoring cracks and damages to vital bridges and highway networks [4], iii) monitoring catastrophic faults to critical infrastructures such as water and gas pipelines, electricity connections, supply chains, etc. [5], iv) biological problems characterized by an event-driven potential including monitoring human subjects for epileptic fits, seizures, dramatic changes in physiological behavior, etc. [6], [7], v) dynamic spectrum access and allocation problems [8], vi) chemical or biological warfare agent detection systems to protect against terrorist attacks, vii) detection of the onset of an epidemic, and viii) failure detection in manufacturing systems and large machines. In all of these applications, the sensors monitoring the environment take observations that undergo a change in statistical properties in response to a disruption (change) in the environment. The goal is to detect the point of disruption (change-point) as quickly as possible, subject to false alarm constraints.
In the standard formulation of the change detection problem, studied over the last fifty years, there is a sequence of observations whose density changes at some unknown point in time and the goal is to detect the change-point as soon as possible. Two classical approaches to quickest change detection are: i) The minimax approach [9], [10], where the goal is to minimize the worst-case delay subject to a lower bound on the mean time between false alarms, and ii) The Bayesian approach [11]- [13], where the change-point is assumed to be a random variable with a density that is known a priori and the goal is to minimize the expected (average) detection delay subject to a bound on the probability of false alarm. Significant advances in both the minimax and the Bayesian theories of change detection have been made, and the reader is referred to [9]- [22] for a representative sample of the body of work in this area. The reader is also referred to [9], [16], [18], [22]- [27] for performance analyses of the standard change detection approaches in the minimax context, and [28], [29] in the Bayesian context.
Extensions of the above framework to the multi-sensor case where the information available for decision-making is distributed has also been explored [29]- [32]. In this setting, the observations are taken at a set of L distributed sensors, as shown in Fig. 1. The sensors may send either quantized/unquantized versions of their observations or local decisions to a fusion center, subject to communication delay, power and bandwidth constraints, where a final decision is made, based on all the sensor messages. In particular, in recent work [29]- [32], it is assumed that the statistical properties of all the sensors’ observations change at the same time. However, in many scenarios, it is more suitable to consider the case where the statistics of each sensor’s observations may change at different points in time. An application of this model is in the detection of pollutants and We consider a Bayesian version of this problem and assume that the point of disruption (that needs to be detected) is a random variable with a geometric distribution. We assume that the L sensors are placed in an array or a line and they observe the change as it propagates through them. We model the inter-sensor delay with a Markov model and in particular, the focus is on the case where the inter-sensor delay is also geometric. More general inter-sensor delay models can be considered, but the case of a geometric prior has an intuitive and appealing interpretation due to the memorylessness property of the geometric random variable.
We study the centralized case, where the fusion center has complete information about the observations at all the L sensors, the change process statistics, and the pre-and the post-change densities. This is applicable in scenarios where: i) the fusion center is geographically collocated with the sensors so that ample bandwidth is available for reliable communication between the sensors and the fusion center; and ii) the impact of the disruption-causing agent on the statistical dynamics of the change process and the statistical nature of the change so induced can be modeled accurately.
The goal of the fusion center is to come up with a strategy (or a stopping rule) to declare change, subject to false alarm constraints. Towards this goal, we first show that the problem fits the standard partially observable Markov decision process (POMDP) framework [33] with the sufficient statistics given by the a posteriori probabilities of the state of the system conditioned on the observation process. We then establish a recursion for the sufficient statistics, which generalizes the recursion established in [32] for the case when all t
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