Optimal Sensor Placement for Intruder Detection

The fields of detection, resource allocation, and security have seen tremendous research activity in the past few decades. Different frameworks have been designed to tackle various problems using a host of techniques from statistics, engineering, and economics. The fields of centralized and decentralized detection, see [1] for a review, have been a major research focus in the communications, controls, and statistics communities and are now considered mature fields of research. Our main focus in this paper is to develop and pursue research ideas in the intersection of these exciting fields. In this paper, we operate within the centralized detection framework under which complete observations are available to the decision makers. This preference for a relatively simple and more developed framework enables us to fully extract the benefits offered by the diversity of these research ideas. We consider the centralized detection of an intruder, whose location is modeled as uniform across a specified set of points. A team of sensors is tasked to make observations which are completely reported to a centralized decision making authority. The challenge is to optimally place the sensors, before measurements are made, and to compute an optimal decision rule for detecting the location of the intruder. Simplifying assumptions like a uniform prior distribution and conditional independence of observations are made. It should be noted that sensors observations are noisy as a result of their limited accuracy. The limited accuracy of the sensors makes the conditional independence assumption practically reasonable [2]. We exploit majorization theory, see [3] and [4] for a review, within the framework of this paper to establish general laws governing the jointly optimal detection and placement policies. Majorization is an important mathematical technique for partial ordering of vectors of real