On Optimal Quantized Non-Bayesian Quickest Change Detection With Energy Harvesting

Abstract

In this paper, we consider a problem of decentralized non-Bayesian quickest change detection using a wireless sensor network where the sensor nodes are powered by harvested energy from the environment. The underlying random process being monitored by the sensors is subject to change in its distribution at an unknown but deterministic time point, and the sensors take samples (sensing) periodically, compute the likelihood ratio based on the distributions before and after the change, quantize it and send it to a remote fusion centre (FC) over fading channels for performing a sequential test to detect the change. Due to the unpredictable and intermittent nature of harvested energy arrivals, the sensors need to decide whether they want to sense, and at what rate they want to quantize their information before sending them to the FC, since higher quantization rates result in higher accuracy and better detection performance, at the cost of higher energy consumption. We formulate an optimal sensing and quantization rate allocation problem (in order to minimize the expected detection delay subject to false alarm rate constraint) based on the availability (at the FC) of non-causal and causal information of sensors’ energy state information, and channel state information between the sensors and the FC. Motivated by the asymptotically inverse relationship between the expected detection delay (under a vanishingly small probability of false alarm) and the Kullback-Leibler (KL) divergence measure at the FC, we maximize an expected sum of the KL divergence measure over a finite horizon to obtain the optimal sensing and quantization rate allocation policy, subject to energy causality constraints at each sensor. The optimal solution is obtained using a typical dynamic programming based technique, and based on the optimal quantization rate, the optimal quantization thresholds are found by maximizing the KL information measure per slot. We also provide suboptimal threshold design policies using uniform quantization and an asymptotically optimal quantization policy for higher number of quantization bits. We provide an asymptotic approximation for the loss due to quantization of the KL measure, and also consider an alternative optimization problem with minimizing the expected sum of the inverse the KL divergence measure as the cost per time slot. Numerical results are provided comparing the various optimal and suboptimal quantization strategies for both optimization problem formulations, illustrating the comparative performance of these strategies at different regimes of quantization rates.