Testing the Structure of a Gaussian Graphical Model With Reduced Transmissions in a Distributed Setting

Abstract

Testing a covariance matrix following a Gaussian graphical model (GGM) is considered in this paper based on observations made at a set of distributed sensors grouped into clusters. Ordered transmissions are proposed to achieve the same Bayes risk as the optimum centralized energy unconstrained approach but with fewer transmissions and a completely distributed approach. In this approach, we represent the Bayes optimum test statistic as a sum of local test statistics which can be calculated by only utilizing the observations available at one cluster. We select one sensor to be the cluster head (CH) to collect and summarize the observed data in each cluster and intercluster communications are assumed to be inexpensive. The CHs with more informative observations transmit their data to the fusion center (FC) first. By halting before all transmissions have taken place, transmissions can be saved without performance loss. It is shown that this ordering approach can guarantee a lower bound on the average number of transmissions saved for any given GGM and the lower bound can approach approximately half the number of clusters when the minimum eigenvalue of the covariance matrix under the alternative hypothesis in each cluster becomes sufficiently large.

Ordering for Communication-Efficient Quickest Change Detection in a Decomposable Graphical Model

“…The QCD problem can be modeled as a hypothesis testing problem, given by H 0 : no change occurs H 1 : change occurs at a finite unknown time slot τ. (16) Note that when the change occurs, all sensors are assumed to be affected simultaneously as mentioned in Assumption 1.…”

“…Assumption 5: For the hypothesis testing problem considered in (16) with L k (X n,C k ) as per ( 24)-( 26), we assume that the probability Pr(L k (X n,C k ) < 0) → 1 as s → ∞ for all k = 1, ..., K and n < τ .…”

“…Proof: The proof of this theorem is omitted, since it follows from the proof in [15] and [16]. Specifically, for the problem in (49), as s → ∞ with s = min k µ C k , the result in (54) can be obtained by following Theorem 2 in [15].…”

“…Specifically, for the problem in (49), as s → ∞ with s = min k µ C k , the result in (54) can be obtained by following Theorem 2 in [15]. For the problem in (50), as s → ∞ with s = min k λ min,k , then (54) can be obtained by following Theorem 3 in [16].…”

“…In [14], it was shown that this ordered transmission approach can reduce the number of transmissions without losing any detection performance for cases with statistically independent observations. The detection of a mean shift or covariance matrix change in statistically dependent Gaussian observations following a decomposable Gaussian graphical model (GGM) is considered in [15] and [16], respectively. In this paper, we provide the first communication efficient QCD algorithm for rapidly detecting a change with statistically dependent observations across the sensors.…”