Submodularity and Optimality of Fusion Rules in Balanced Binary Relay Trees

We study the distributed detection problem in a balanced binary relay tree, where the leaves of the tree are sensors generating binary messages. The root of the tree is a fusion center that makes the overall decision. Every other node in the tree is a fusion node that fuses two binary messages from its child nodes into a new binary message and sends it to the parent node at the next level. We assume that the fusion nodes at the same level use the same fusion rule. We call a string of fusion rules used at different levels a fusion strategy. We consider the problem of finding a fusion strategy that maximizes the reduction in the total error probability between the sensors and the fusion center. We formulate this problem as a deterministic dynamic program and express the solution in terms of Bellman's equations. We introduce the notion of stringsubmodularity and show that the reduction in the total error probability is a stringsubmodular function. Consequentially, we show that the greedy strategy, which only maximizes the level-wise reduction in the total error probability, is within a factor of the optimal strategy in terms of reduction in the total error probability

Detection Performance in Balanced Binary Relay Trees with Node and Link Failures

We study the distributed detection problem in the context of a balanced binary relay tree, where the leaves of the tree correspond to $N$ identical and independent sensors generating binary messages. The root of the tree is a fusion center making an overall decision. Every other node is a relay node that aggregates the messages received from its child nodes into a new message and sends it up toward the fusion center. We derive upper and lower bounds for the total error probability $P_N$ as explicit functions of $N$ in the case where nodes and links fail with certain probabilities. These characterize the asymptotic decay rate of the total error probability as $N$ goes to infinity. Naturally, this decay rate is not larger than that in the non-failure case, which is $\sqrt N$. However, we derive an explicit necessary and sufficient condition on the decay rate of the local failure probabilities $p_k$ (combination of node and link failure probabilities at each level) such that the decay rate of the total error probability in the failure case is the same as that of the non-failure case. More precisely, we show that $\log P_N^{-1}=\Theta(\sqrt N)$ if and only if $\log p_k^{-1}=\Omega(2^{k/2})$

Impact of Channel Errors on Decentralized Detection Performance of Wireless Sensor Networks: A Study of Binary Modulations, Rayleigh-Fading and Nonfading Channels, and Fusion-Combiners

We provide new results on the performance of wireless sensor networks in which a number of identical sensor nodes transmit their binary decisions, regarding a binary hypothesis, to a fusion center (FC) by means of a modulation scheme. Each link between a sensor and the fusion center is modeled independent and identically distributed (i.i.d.) either as slow Rayleigh-fading or as nonfading. The FC employs a counting rule (CR) or another combining scheme to make a final decision. Main results obtained are the following: 1) in slow fading, a) the correctness of using an average bit error rate of a link, averaged with respect to the fading distribution, for assessing the performance of a CR and b) with proper choice of threshold, ON/OFF keying (OOK), in addition to energy saving, exhibits asymptotic (large number of sensors) performance comparable to that of FSK; and 2) for a large number of sensors, a) for slow fading and a counting rule, given a minimum sensor-to-fusion link SNR, we determine a minimum sensor decision quality, in order to achieve zero asymptotic errors and b) for Rayleigh-fading and nonfading channels and PSK (FSK) modulation, using a large deviation theory, we derive asymptotic error exponents of counting rule, maximal ratio (square law), and equal gain combiners

Optimal Inference for Distributed Detection

In distributed detection, there does not exist an automatic way of generating optimal decision strategies for non-affine decision functions. Consequently, in a detection problem based on a non-affine decision function, establishing optimality of a given decision strategy, such as a generalized likelihood ratio test, is often difficult or even impossible. In this thesis we develop a novel detection network optimization technique that can be used to determine necessary and sufficient conditions for optimality in distributed detection for which the underlying objective function is monotonic and convex in probabilistic decision strategies. Our developed approach leverages on basic concepts of optimization and statistical inference which are provided in appendices in sufficient detail. These basic concepts are combined to form the basis of an optimal inference technique for signal detection. We prove a central theorem that characterizes optimality in a variety of distributed detection architectures. We discuss three applications of this result in distributed signal detection. These applications include interactive distributed detection, optimal tandem fusion architecture, and distributed detection by acyclic graph networks. In the conclusion we indicate several future research directions, which include possible generalizations of our optimization method and new research problems arising from each of the three applications considered

Decision Fusion for Large-Scale Sensor Networks with Nonideal Channels

Since there has been an increasing interest in the areas of Internet of Things (IoT) and artificial intelligence that often deals with a large number of sensors, this chapter investigates the decision fusion problem for large-scale sensor networks. Due to unavoidable transmission channel interference, we consider sensor networks with nonideal channels that are prone to errors. When the fusion rule is fixed, we present the necessary condition for the optimal sensor rules that minimize the Monte Carlo cost function. For the K-out-of-L fusion rule chosen very often in practice, we analytically derive the optimal sensor rules. For general fusion rules, a Monte Carlo Gauss-Seidel optimization algorithm is developed to search for the optimal sensor rules. The complexity of the new algorithm is of the order of OLN compared with OLNL of the previous algorithm that was based on Riemann sum approximation, where L is the number of sensors and N is the number of samples. Thus, the proposed method allows us to design the decision fusion rule for large-scale sensor networks. Moreover, the algorithm is generalized to simultaneously search for the optimal sensor rules and the optimal fusion rule. Finally, numerical examples show the effectiveness of the new algorithms for large-scale sensor networks with nonideal channels