$$(\alpha , \tau )$$ ( α , τ ) -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 $$(\alpha ,\tau )$$(ź,ź)-monitoring in WSNs. For a given monitored threshold $$\alpha $$ź, we prove that (i) the tight upper bound of $$\Pr [{S(t)} \ge \alpha ]$$Pr[S(t)źź] is $$O\left( {\exp \left\{ { - n\ell \left( {\frac{\alpha }{{nsup}},\frac{{\mu (t)}}{{nsup}}} \right) } \right\} } \right) $$Oexp-nlźnsup,μ(t)nsup, if $$\mu (t) \alpha $$μ(t)>ź, where $$\Pr [X]$$Pr[X] is the probability of random event $$X,\, S(t)$$X,S(t) is the sum of the monitored area at time $$t,\, n$$t,n is the number of the sensor nodes, $$sup$$sup is the upper bound of sensed data, $$ \mu (t)$$μ(t) is the expectation of $$S(t)$$S(t), and $$\ell ({x_1},{x_2}) = {x_1}\ln \left( {\frac{{{x_1}}}{{{x_2}}}} \right) + (1 - {x_1})\ln \left( {\frac{{1 - {x_1}}}{{1 - {x_2}}}} \right) $$l(x1,x2)=x1lnx1x2+(1-x1)ln1-x11-x2. An instant $$(\alpha ,\tau )$$(ź,ź)-monitoring scheme is then developed based on the upper bound. Moreover, approximate continuous $$(\alpha , \tau )$$(ź,ź)-monitoring is investigated. We prove that the probability of false negative alarm is $$\delta $$ź, if the sample size is [InlineEquation not available: see fulltext.] for a given precision requirement, where [InlineEquation not available: see fulltext.] is the [InlineEquation not available: see fulltext.] fractile of a standard normal distribution. Finally, the performance of the proposed algorithms is validated through experiments.