Distributed Detection/Isolation Procedures for Quickest Event Detection in Large Extent Wireless Sensor Networks

We study a problem of distributed detection of a stationary point event in a large extent wireless sensor network ($\wsn$), where the event influences the observations of the sensors only in the vicinity of where it occurs. An event occurs at an unknown time and at a random location in the coverage region (or region of interest ($\ROI$)) of the $\wsn$. We consider a general sensing model in which the effect of the event at a sensor node depends on the distance between the event and the sensor node; in particular, in the Boolean sensing model, all sensors in a disk of a given radius around the event are equally affected. Following the prior work reported in \cite{nikiforov95change_isolation}, \cite{nikiforov03lower-bound-for-det-isolation}, \cite{tartakovsky08multi-decision}, {\em the problem is formulated as that of detecting the event and locating it to a subregion of the $\ROI$ as early as possible under the constraints that the average run length to false alarm ($\tfa$) is bounded below by $\gamma$, and the probability of false isolation ($\pfi$) is bounded above by $\alpha$}, where $\gamma$ and $\alpha$ are target performance requirements. In this setting, we propose distributed procedures for event detection and isolation (namely $\mx$, $\all$, and $\hall$), based on the local fusion of $\CUSUM$s at the sensors. For these procedures, we obtain bounds on the maximum mean detection/isolation delay ($\add$), and on $\tfa$ and $\pfi$, and thus provide an upper bound on $\add$ as $\min{\gamma,1/\alpha} \to \infty$. For the Boolean sensing model, we show that an asymptotic upper bound on the maximum mean detection/isolation delay of our distributed procedure scales with $\gamma$ and $\alpha$ in the same way as the asymptotically optimal centralised procedure \cite{nikiforov03lower-bound-for-det-isolation}.

💡 Research Summary

The paper addresses the problem of quickly detecting and isolating a stationary point event in a large‑extent wireless sensor network (WSN) where the event influences only those sensors that lie in its vicinity. Unlike classical change‑detection settings that assume a single, globally observable change, here the event’s location is unknown and its effect on each sensor depends on the distance between the sensor and the event. The authors formulate the task as a quickest change‑detection‑and‑isolation problem under two performance constraints: the average run length to false alarm (ARL₂FA) must be at least a prescribed value γ, and the probability of false isolation (PFI) must not exceed a prescribed value α. The goal is to minimize the worst‑case mean detection/isolation delay (SADD) while respecting these constraints.

System and Sensing Model
The region of interest (ROI) A ⊂ ℝ² is covered by n identical sensors, each synchronized in time and sampling once per slot k = 0,1,… . An event occurs at an unknown time T and unknown location ℓₑ ∈ A and remains permanently active. The observation at sensor s before the event is pure noise Wₖ^{(s)} (zero‑mean Gaussian with variance σ²). After the event, the observation becomes Xₖ^{(s)} = hₑ·ρ(dₑ,s) + Wₖ^{(s)}, where dₑ,s = ‖ℓ(s) – ℓₑ‖ is the Euclidean distance, hₑ is the signal amplitude, and ρ(d) is a non‑increasing distance‑dependent attenuation function with ρ(0)=1. Two concrete models are considered: (i) the Boolean model, where ρ(d)=1 for d ≤ r_d and 0 otherwise, and (ii) a power‑law path‑loss model, ρ(d)=d^{‑η} (η>0). The Boolean model yields a clear binary distinction between affected and unaffected sensors, while the power‑law model leads to a gradual decay of signal strength.

Detection Partition
Because each sensor can only “see” within a finite detection radius r_d (defined by a design threshold µ₁ such that hₑ·ρ(r_d)=µ₁), the ROI is partitioned into a minimal set of sub‑regions A₁,…,A_N. Each sub‑region A_i is uniquely covered by a set of sensors N_i = {s : ℓ(s) is within r_d of any point in A_i}. By construction, the N_i are distinct and the A_i are disjoint, providing a natural “detection partition”. This structure enables the design of distributed algorithms that operate locally on each sensor group N_i.

Local CUSUM Statistics
Since the exact distance dₑ,s is unknown, the authors adopt a worst‑case approach: each sensor computes a CUSUM statistic using the log‑likelihood ratio (LLR) evaluated at the detection radius r_d, i.e., Zₖ^{(s)}(r_d) = log