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

Detecting abnormal events is one of the fundamental issues in wireless sensor networks (WSNs). In this paper, we investigate $$(\alpha ,\tau )$$ ( α , τ ) -monitoring in WSNs. For a given monitored threshold $$\alpha $$ α , we prove that (i) the tight upper bound of $$\Pr [{S(t)} \ge \alpha ]$$ Pr [ S ( t ) ≥ α ] is $$O\left( {\exp \left\{ { - n\ell \left( {\frac{\alpha }{{nsup}},\frac{{\mu (t)}}{{nsup}}} \right) } \right\} } \right) $$ O exp - n ℓ α n s u p , μ ( t ) n s u p , if $$\mu (t) \alpha $$ μ ( t ) > α , 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, $$sup$$ s u p 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 ( x 1 , x 2 ) = x 1 ln x 1 x 2 + ( 1 - x 1 ) ln 1 - x 1 1 - x 2 . An instant $$(\alpha ,\tau )$$ ( α , τ ) -monitoring scheme is then developed based on the upper bound. Moreover, approximate continuous $$(\alpha , \tau )$$ ( α , τ ) -monitoring is investigated. We prove that the probability of false negative alarm is $$\delta $$ δ , 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.