Ordering for Reduced Transmission Energy Detection in Sensor Networks Testing a Shift in the Mean of a Gaussian Graphical Model

Abstract

Detection of a shift in the mean of a vector following a decomposable Gaussian graphical model (DGGM) is considered, where each component of the vector is measured at a different sensor in a network. We provide a new method eliminating transmissions normally needed in an optimum clustering approach by using ordered transmissions while achieving the same Bayes risk as the optimum clustering approach. In the new approach, the Bayes optimum test statistic is represented as a sum of local test statistics, where each local test statistic depends only on the observations made at one clique in a generalization of the ordered transmission approach previously suggested for statistically independent observations. Hence, we propose to organize the sensors into clusters based on the clique of the DGGM they belong to, and each cluster selects one sensor to be its cluster head (CH). After collecting and summarizing the observed data at each cluster, the ordered transmission approach is employed over the CHs in an attempt to reduce the number of communications from the CHs to the fusion center where the decision is made. It is shown that the developed approach can guarantee a lower bound on the average number of transmissions saved from the ordered transmission approach for any given DGGM which approaches approximately half the number of cliques when the norm of the mean-shift vector in each clique becomes sufficiently large. In all the cases considered, numerical results imply that a significant portion of the transmissions can be saved, and the developed lower bound is obeyed. The development of the appropriate local processing and the proof of savings are highly nontrivial generalizations of ordering for statistically independent observations and this paper represents the first justification of an ordering approach for cases with statistically dependent observations.