Asymptotically Optimal Truncated Hypothesis Test for a Large Sensor Network Described by a Multivariate Gaussian Distribution

While recent advances have provided extremely efficient distributed methods for computing optimal test statistics for many hypothesis testing problems occurring in large sensor networks, the popular multivariate Gaussian hypothesis testing problem involving a change in both the mean vector and covariance matrix is not one of these. The difficultly is that these test statistics generally require long range communications. A truncated test is studied which only requires that each sensor shares information with 2k neighboring sensors out of a set of L total sensors. Sufficient conditions are given on the k as a function of L for a given sequence of hypothesis testing problems to ensure no loss in deflection performance as L approaches infinity when compared to the optimal untruncated detector. For several popular classes of system and process models, including observations from some wide-sense stationary limiting processes as L→ ∞ (after the mean is subtracted), the sufficient conditions are shown to be satisfied for k increasing very slowly compared to L even when the difficulty of the hypothesis testing problem scales in the least favorable manner. Numerical results imply the fixed-false-alarm-rate detection probability of the truncated detector converges rapidly to the detection probability of the optimal untruncated detector