This paper examines the close interplay between cooperation and adaptation for distributed detection schemes over fully decentralized networks. The combined attributes of cooperation and adaptation are necessary to enable networks of detectors to continually learn from streaming data and to continually track drifts in the state of nature when deciding in favor of one hypothesis or another. The results in this paper establish a fundamental scaling law for the steady-state probabilities of miss detection and false alarm in the slow adaptation regime, when the agents interact with each other according to distributed strategies that employ small constant step-sizes. The latter are critical to enable continuous adaptation and learning. This paper establishes three key results. First, it is shown that the output of the collaborative process at each agent has a steady-state distribution. Second, it is shown that this distribution is asymptotically Gaussian in the slow adaptation regime of small step-sizes. Third, by carrying out a detailed large deviations analysis, closed-form expressions are derived for the decaying rates of the false-alarm and miss-detection probabilities. Interesting insights are gained from these expressions. In particular, it is verified that as the step-size @math decreases, the error probabilities are driven to zero exponentially fast as functions of @math , and that the exponents governing the decay increase linearly in the number of agents. It is also verified that the scaling laws governing the errors of detection and the errors of estimation over the network behave very differently, with the former having exponential decay proportional to @math , while the latter scales linearly with decay proportional to @math . Moreover, and interestingly, it is shown that the cooperative strategy allows each agent to reach the same detection performance, in terms of detection error exponents, of a centralized stochastic-gradient solution. The results of this paper are illustrated by applying them to canonical distributed detection problems.
In a decentralized hypothesis testing network, several peripheral nodes observe an environment and communicate their observations to a central node for the final decision. The presence of capacity constraints introduces theoretical and practical problems. The following problem is addressed: given that the peripheral encoders that satisfy these constraints are scalar quantizers, how should they be designed in order that the central test to be performed on their output indices is most powerful? The scheme is called cooperative design-separate encoding since the quantizers process separate observations but have a common goal; they seek to maximize a system-wide performance measure. The Bhattacharyya distance of the joint index space as such a criterion is suggested, and a design algorithm to optimize arbitrarily many quantizers cyclically is proposed. A simplified version of the algorithm, namely an independent design-separate encoding scheme, where the correlation is either absent or neglected for the sake of simplicity, is outlined. Performances are compared through worked examples. > In this paper, we investigate a binary decentralized detection problem in which a network of wireless sensors provides relevant information about the state of nature to a fusion center. Each sensor transmits its data over a multiple access channel. Upon reception of the information, the fusion center attempts to accurately reconstruct the state of nature. We consider the scenario where the sensor network is constrained by the capacity of the wireless channel over which the sensors are transmitting, and we study the structure of an optimal sensor configuration. For the problem of detecting deterministic signals in additive Gaussian noise, we show that having a set of identical binary sensors is asymptotically optimal, as the number of observations per sensor goes to infinity. Thus, the gain offered by having more sensors exceeds the benefits of getting detailed information from each sensor. A thorough analysis of the Gaussian case is presented along with some extensions to other observation distributions. This paper reviews the classical decentralized decision theory in the light of new constraints and requirements. The central theme that transcends various aspects of signal processing design is that an integrated channel-aware approach needs to be taken for optimal detection performance given the available resources. Detection problems provide a productive starting point for the study of more general statistical inference problems in sensor networks. In this article, the classical framework for decentralized detection is reviewed and argued that, while this framework provides a useful basis for developing a theory for detection in sensor networks, it has serious limitations. The classical framework does not adequately take into account important features of sensor technology and of the communication link between the sensors and the fusion center. An alternative framework for detection in sensor networks that has emerged over the last few years is discussed. Several design and optimization strategies may be gleaned from this new framework 1 Introduction.- 1.1 Distributed Detection Systems.- 1.2 Outline of the Book.- 2 Elements of Detection Theory.- 2.1 Introduction.- 2.2 Bayesian Detection Theory.- 2.3 Minimax Detection.- 2.4 Neyman-Pearson Test.- 2.5 Sequential Detection.- 2.6 Constant False Alarm Rate (CFAR) Detection.- 2.7 Locally Optimum Detection.- 3 Distributed Bayesian Detection: Parallel Fusion Network.- 3.1 Introduction.- 3.2 Distributed Detection Without Fusion.- 3.3 Design of Fusion Rules.- 3.4 Detection with Parallel Fusion Network.- 4 Distributed Bayesian Detection: Other Network Topologies.- 4.1 Introduction.- 4.2 The Serial Network.- 4.3 Tree Networks.- 4.4 Detection Networks with Feedback.- 4.5 Generalized Formulation for Detection Networks.- 5 Distributed Detection with False Alarm Rate Constraints.- 5.1 Introduction.- 5.2 Distributed Neyman-Pearson Detection.- 5.3 Distributed CFAR Detection.- 5.4 Distributed Detection of Weak Signals.- 6 Distributed Sequential Detection.- 6.1 Introduction.- 6.2 Sequential Test Performed at the Sensors.- 6.3 Sequential Test Performed at the Fusion Center.- 7 Information Theory and Distributed Hypothesis Testing.- 7.1 Introduction.- 7.2 Distributed Detection Based on Information Theoretic Criterion.- 7.3 Multiterminal Detection with Data Compression.- Selected Bibliography. In this paper basic results on distributed detection are reviewed. In particular we consider the parallel and the serial architectures in some detail and discuss the decision rules obtained from their optimization based an the Neyman-Pearson (NP) criterion and the Bayes formulation. For conditionally independent sensor observations, the optimality of the likelihood ratio test (LRT) at the sensors is established. General comments on several important issues are made including the computational complexity of obtaining the optimal solutions the design of detection networks with more general topologies, and applications to different areas. We consider the problem of classifying among a set of M hypotheses via distributed noisy sensors. The sensors can collaborate over a communication network and the task is to arrive at a consensus about the event after exchanging messages. We apply a variant of belief propagation as a strategy for collaboration to arrive at a solution to the distributed classification problem. We show that the message evolution can be reformulated as the evolution of a linear dynamical system, which is primarily characterized by network connectivity. We show that a consensus to the centralized maximum a posteriori (MAP) estimate can almost always reached by the sensors for any arbitrary network. We then extend these results in several directions. First, we demonstrate that these results continue to hold with quantization of the messages, which is appealing from the point of view of finite bit rates supportable between links. We then demonstrate robustness against packet losses, which implies that optimal decisions can be achieved with asynchronous transmissions as well. Next, we present an account of energy requirements for distributed detection and demonstrate significant improvement over conventional decentralized detection. Finally, extensions to distributed estimation are described Following the foundational work that established basic ideas for optimum distributed defection schemes using multiple sensors (as reviewed in Part I of this two-part review), further work on distributed detection has developed many useful and interesting extensions of the basic concepts. These more recent developments parallel those that arose from the early work on centralized, classical signal detection, resulting in new ideas of asymptotically optimum nonparametric, robust, and sequential centralized detection. Recent developments on these topics in the setting of distributed signal detection are reviewed in the present paper. Results in these directions are important in practice because they allow cases of modeling uncertainty to be addressed, and they provide more efficient detection schemes by optimizing more general performance criteria.