Simple Bounds for the Probability of False Alarm to Design Sensor Detection Systems Using the Scan Statistic

Abstract

In a distributed sensor system to detect a target emitter in a large region of interest, only a small subset of sensors is close enough to the emitter to collect strong measurements, which motivates the use of fusion rules based on the scan statistic. However, it is difficult to design the scan statistic fusion rule because the analytical derivation of the probability of false alarm is intractable. Elaborate upper bounds for such a probability are available; however, they are difficult to apply and are applicable only in certain scenarios and conditions. Simple upper bounds are also available; however, they are usually dismissed for being too loose and leading to too conservative designs. This paper argues that, when designing sensor detection systems at typical configurations, one of these simple upper bounds (the product bound) is not as loose and the resulting design is not as conservative as generally considered. A simple lower bound is also proposed to allow a designer evaluate how conservative the resulting design is. This paper further shows that, for a large class of distributions, the design based on the product bound becomes less and less conservative as the size of the region increases. Application of the results to scenarios with irregular sensor placement is also presented.