Closed-Form Error Exponent for the Neyman–Pearson Fusion of Dependent Local Decisions in a One-Dimensional Sensor Network

We consider a distributed detection system formed by a large number of local detectors and a data fusion center that performs a Neyman-Pearson fusion of the binary quantizations of the sensor observations. In the analyzed two-stage detection system the local decisions are taken with no kind of cooperation among the devices and they are transmitted to the fusion center over an error free parallel access channel. In addition, the sensors are randomly deployed along a straight line, and the corresponding sensor spacings are drawn independently from a common probability density function (pdf). For both hypothesis, H₀ and H₁, depending on the correlation structure of the observed phenomenon the local decisions might be dependent. In the case of being dependent, their correlation structure is modelled with a one-dimensional Markov random field with nearest neighbor dependency and binary state space. Under this scenario, we first derive a closed-form error exponent for the Neyman-Pearson fusion of the local decisions when the involved data fusion center only knows the distribution of the sensor spacings. Second, based on a single parameter that captures the mean correlation strength among the local decisions, some analytical properties of the error exponent are investigated. Finally, we develop a physical model for the conditional probabilities of the Markov random fields that might be present under each hypothesis. Using this model we characterize the error exponent for two well-known models of the sensor spacing: i) equispaced sensors with failures, and ii) exponentially spaced sensors with failures.