Fusing of binary correlated data with unknown statistics

We address the problem of distributed target detection with correlated observations, i.e., where local detectors transmit their binary decisions to a fusion center but these decisions are correlated in an unknown manner. We propose three Separating Function Estimation Tests (SFETs) and a Generalized Likelihood Ratio Test (GLRT) to fuse the binary data. SFETs convert the detection problem into a problem of estimating a separating function that is positive under the alternative (to the null) hypothesis. Detection decisions are achieved by comparing the estimate of the Separating Function (SF) with a threshold, where the threshold is set to satisfy a probability of false alarm constraint. The SFETs are derived based on the asymptotically optimal SF (AOSF) theorem (SFET₁), the Euclidean distance (SFET₂) and Kullback-Leibler (K-L) divergence (SFET₃) of the probability mass function (pmf) of the observations under each hypothesis. Since the correlations are unknown, we formulate a linear optimization program to estimate the pmf. The simulation results show that the probability of detection of the SFETs using the AOSF and the Euclidean distance is greater than the GLRT and the SFET using K-L divergence. Interestingly, when the observations are independent SFET₁ and SFET₂ provide optimal performance for the problem.