Optimal Sensor Placement for Intruder Detection

TL;DR

This paper investigates the optimal placement of sensors for intruder detection, demonstrating that uniform placement is not always optimal and analyzing how detection and false alarm probabilities influence placement strategies.

Contribution

It provides theoretical proofs on the structure of optimal sensor placements, including conditions where uniform placement is suboptimal and the impact of detection parameters.

Findings

Uniform placement is never optimal when sensors equal points.

Optimal placement structure remains consistent across different numbers of points for certain conditions.

Increasing false alarm probability shifts optimal placements higher in the majorization scale.

Abstract

We consider the centralized detection of an intruder, whose location is modeled as uniform across a specified set of points, using an optimally placed team of sensors. These sensors make conditionally independent observations. The local detectors at the sensors are also assumed to be identical, with detection probability $(P_{_{D}})$ and false alarm probability $(P_{_{F}})$. We formulate the problem as an N-ary hypothesis testing problem, jointly optimizing the sensor placement and detection policies at the fusion center. We prove that uniform sensor placement is never strictly optimal when the number of sensors $(M)$ equals the number of placement points $(N)$. We prove that for $N_{2} > N_{1} > M$, where $N_{1},N_{2}$ are number of placement points, the framework utilizing $M$ sensors and $N_{1}$ placement points has the same optimal placement structure as the one utilizing $M$ sensors and $N_{2}$ placement points. For $M\leq 5$ and for fixed $P_{_{D}}$, increasing $P_{_{F}}$ leads to optimal placements that are higher in the majorization-based placement scale. Similarly for $M\leq 5$ and for fixed $P_{_{F}}$, increasing $P_{_{D}}$ leads to optimal placements that are higher in the majorization-based placement scale. For $M>5$, this result does not necessarily hold and we provide a simple counterexample. It is conjectured that the set of optimal placements for a given $(M,N)$ can always be placed on a majorization-based placement scale.