Asymptotic locally optimal detector for large-scale sensor networks under the Poisson regime

We consider distributed detection with a large number of identical sensors deployed over a region where the phenomenon of interest (POI) has unknown spatially varying strength. Each sensor makes a decision based on its own measurement of the signal at its location and the local decision of each sensor is sent to a fusion center through a multiple access channel. The fusion center decides whether the POI has occurred in the region, under a global size constraint in the Neyman-Pearson formulation. Assuming that the initial distribution of sensors is a homogeneous spatial Poisson process, we show that the Poisson process of ´alarmed´ sensors satisfies the locally asymptotic normality (LAN) condition as the number of sensor goes to infinity. We derive a new asymptotically locally most powerful (ALMP) detector jointly over the fusion scheme and the sensor threshold. We also derive the conditions on the spatial signal shape to guarantee the existence of the ALMP detector. We show that the optimal test statistic is a weighted sum of local decisions, the optimal weight function being the shape of the spatial signal, but the exact value of the spatial signal is not required. The optimal threshold for a single sensor is also derived. For the case of independent, identically-distributed (i.i.d.) sensor observation, we show that the counting-based detector is also asymptotic locally optimal.