$(\alpha , \tau )$-Monitoring for event detection in wireless sensor networks

Abstract

Detecting abnormal events is one of the fundamental issues in wireless sensor networks (WSNs). In this paper, we investigate $(\alpha ,\tau )$-monitoring in WSNs. For a given monitored threshold $\alpha$, we prove that (i) the tight upper bound of $\Pr [{S(t)} \ge \alpha ]$ is $O\left( {\exp \left\{ { - n\ell \left( {\frac{\alpha }{{nsup}},\frac{{\mu (t)}}{{nsup}}} \right) } \right\} } \right)$, if $\mu (t) < \alpha$; and (ii) the tight upper bound of $\Pr [{S(t)} \le \alpha ]$ is $O\left( {\exp \left\{ { - n\ell \left( {\frac{\alpha }{{nsup}},\frac{{\mu (t)}}{{nsup}}} \right) } \right\} } \right)$, if $\mu (t) > \alpha$, where $\Pr [X]$ is the probability of random event $X,\, S(t)$ is the sum of the monitored area at time $t,\, n$ is the number of the sensor nodes, $sup$ is the upper bound of sensed data, $\mu (t)$ is the expectation of $S(t)$, and $\ell ({x_1},{x_2}) = {x_1}\ln \left( {\frac{{{x_1}}}{{{x_2}}}} \right) + (1 - {x_1})\ln \left( {\frac{{1 - {x_1}}}{{1 - {x_2}}}} \right)$. An instant $(\alpha ,\tau )$-monitoring scheme is then developed based on the upper bound. Moreover, approximate continuous $(\alpha , \tau )$-monitoring is investigated. We prove that the probability of false negative alarm is $\delta$, if the sample size is for a given precision requirement, where is the fractile of a standard normal distribution. Finally, the performance of the proposed algorithms is validated through experiments.