Sensor selection for hypothesis testing in wireless sensor networks: a Kullback-Leibler based approach

Abstract — We consider the problem of selecting a subset of p out of n sensors for the purpose of event detection, in a wireless sensor network (WSN). Occurrence or not of the event of interest is modeled as a binary Gaussian hypothesis test. In this case sensor selection consists of finding, among all ( n) p combinations, the one maximizing the Kullback-Leibler (KL) distance between the induced p-dimensional distributions under the two hypotheses. An exhaustive search is impractical if n and p are large, as the resulting optimization problem is combinatorial. We propose a suboptimal approach with computational complexity of order O(n3 p). This consists of relaxing the 0/1 constraint on the entries of the selection matrices to let the optimization problem search over the set of Stiefel matrices. Although finding the Stiefel matrix is a nonconvex problem, we provide an algorithm that guarantees to produce a global optimum for p = 1, through a series of judicious problem reformulations. The case p> 1 is tackled by an incremental, greedy approach. The obtained Stiefel matrix is then used to determine the sensor selection matrix which best approximates its range space. Extensive simulations are used to assess near optimality of the proposed approach. They also show how the proposed approach performs better than exhaustive searches once an upper bound on the computation time is set. I

Consensus and Products of Random Stochastic Matrices: Exact Rate for Convergence in Probability

Distributed consensus and other linear systems with system stochastic matrices $W_k$ emerge in various settings, like opinion formation in social networks, rendezvous of robots, and distributed inference in sensor networks. The matrices $W_k$ are often random, due to, e.g., random packet dropouts in wireless sensor networks. Key in analyzing the performance of such systems is studying convergence of matrix products $W_kW_{k-1}... W_1$. In this paper, we find the exact exponential rate $I$ for the convergence in probability of the product of such matrices when time $k$ grows large, under the assumption that the $W_k$'s are symmetric and independent identically distributed in time. Further, for commonly used random models like with gossip and link failure, we show that the rate $I$ is found by solving a min-cut problem and, hence, easily computable. Finally, we apply our results to optimally allocate the sensors' transmission power in consensus+innovations distributed detection

Optimal detection and error exponents for hidden semi-Markov models

We study detection of random signals corrupted by noise that over time switch their values (states) between a finite set of possible values, where the switchings occur at unknown points in time. We model such signals as hidden semi-Markov signals (HSMS), which generalize classical Markov chains by introducing explicit (possibly non-geometric) distribution for the time spent in each state. Assuming two possible signal states and Gaussian noise, we derive optimal likelihood ratio test and show that it has a computationally tractable form of a matrix product, with the number of matrices involved in the product being the number of process observations. The product matrices are independent and identically distributed, constructed by a simple measurement modulation of the sparse semi-Markov model transition matrix that we define in the paper. Using this result, we show that the Neyman-Pearson error exponent is equal to the top Lyapunov exponent for the corresponding random matrices. Using theory of large deviations, we derive a lower bound on the error exponent. Finally, we show that this bound is tight by means of numerical simulations