Diffusion-Based Adaptive Distributed Detection: Steady-State Performance in the Slow Adaptation Regime

This work 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 the 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. The work 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. And 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. In particular, it is verified that as the step-size $\mu$ decreases, the error probabilities are driven to zero exponentially fast as functions of $1/\mu$, and that the error exponents increase linearly in the number of agents. It is also verified that the scaling laws governing errors of detection and errors of estimation over networks behave very differently, with the former having an exponential decay proportional to $1/\mu$, while the latter scales linearly with decay proportional to $\mu$. 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.Comment: The paper will appear in IEEE Trans. Inf. Theor

Distributed Detection With Multiple Sensors: Part I—Fundamentals

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 on 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

Decision Fusion in Non-stationary Environments

A parallel distributed detection system consists of multiple local sensors/detectors that observe a phenomenon and process the gathered observations using inbuilt processing capabilities. The end product of the local processing is transmitted from each sensor/detector to a centrally located data fusion center for integration and decision making. The data fusion center uses a specific optimization criterion to obtain global decisions about the environment seen by the sensors/detectors. In this study, the overall objective is to make a globally-optimal binary (target/non-target) decision with respect to a Bayesian cost, or to satisfy the Neyman-Pearson criterion. We also note that in some cases a globally-optimal Bayesian decision is either undesirable or impractical, in which case other criteria or localized decisions are used. In this thesis, we investigate development of several fusion algorithms under different constraints including sequential availability of data and dearth of statistical information. The main contribution of this study are: (1) an algorithm that provides a globally optimal solution for local detector design that satisfies a Neyman-Pearson criterion for systems with identical local sensors; (2) an adaptive fusion algorithm that fuses local decisions without a prior knowledge of the local sensor performance; and (3) a fusion rule that applies a genetic In addition, we develop a parallel decision fusion system where each local sensor is a sequential decision maker that implements the modified Wald's sequential probability test (SPRT) as proposed by Lee and Thomas (1984).Ph.D., Electrical Engineering -- Drexel University, 201

Optimal and Suboptimal Detection of Gaussian Signals in Noise: Asymptotic Relative Efficiency

The performance of Bayesian detection of Gaussian signals using noisy observations is investigated via the error exponent for the average error probability. Under unknown signal correlation structure or limited processing capability it is reasonable to use the simple quadratic detector that is optimal in the case of an independent and identically distributed (i.i.d.) signal. Using the large deviations principle, the performance of this detector (which is suboptimal for non-i.i.d. signals) is compared with that of the optimal detector for correlated signals via the asymptotic relative efficiency defined as the ratio between sample sizes of two detectors required for the same performance in the large-sample-size regime. The effects of SNR on the ARE are investigated. It is shown that the asymptotic efficiency of the simple quadratic detector relative to the optimal detector converges to one as the SNR increases without bound for any bounded spectrum, and that the simple quadratic detector performs as well as the optimal detector for a wide range of the correlation values at high SNR.Comment: To appear in the Proceedings of the SPIE Conference on Advanced Signal Processing Algorithms, Architectures and Implementations XV, San Diego, CA, Jul. 1 - Aug. 4, 200

Hypothesis Testing Using Spatially Dependent Heavy-Tailed Multisensor Data

The detection of spatially dependent heavy-tailed signals is considered in this dissertation. While the central limit theorem, and its implication of asymptotic normality of interacting random processes, is generally useful for the theoretical characterization of a wide variety of natural and man-made signals, sensor data from many different applications, in fact, are characterized by non-Gaussian distributions. A common characteristic observed in non-Gaussian data is the presence of heavy-tails or fat tails. For such data, the probability density function (p.d.f.) of extreme values decay at a slower-than-exponential rate, implying that extreme events occur with greater probability. When these events are observed simultaneously by several sensors, their observations are also spatially dependent. In this dissertation, we develop the theory of detection for such data, obtained through heterogeneous sensors. In order to validate our theoretical results and proposed algorithms, we collect and analyze the behavior of indoor footstep data using a linear array of seismic sensors. We characterize the inter-sensor dependence using copula theory. Copulas are parametric functions which bind univariate p.d.f. s, to generate a valid joint p.d.f. We model the heavy-tailed data using the class of alpha-stable distributions. We consider a two-sided test in the Neyman-Pearson framework and present an asymptotic analysis of the generalized likelihood test (GLRT). Both, nested and non-nested models are considered in the analysis. We also use a likelihood maximization-based copula selection scheme as an integral part of the detection process. Since many types of copula functions are available in the literature, selecting the appropriate copula becomes an important component of the detection problem. The performance of the proposed scheme is evaluated numerically on simulated data, as well as using indoor seismic data. With appropriately selected models, our results demonstrate that a high probability of detection can be achieved for false alarm probabilities of the order of 10^-4. These results, using dependent alpha-stable signals, are presented for a two-sensor case. We identify the computational challenges associated with dependent alpha-stable modeling and propose alternative schemes to extend the detector design to a multisensor (multivariate) setting. We use a hierarchical tree based approach, called vines, to model the multivariate copulas, i.e., model the spatial dependence between multiple sensors. The performance of the proposed detectors under the vine-based scheme are evaluated on the indoor footstep data, and significant improvement is observed when compared against the case when only two sensors are deployed. Some open research issues are identified and discussed

Decentralized detection

Cover title. "To appear in Advances in Statistical Signal Processing, Vol. 2: Signal Detection, H.V. Poor and J.B. Thomas, Editors."--Cover.Includes bibliographical references (p. 40-43).Research supported by the ONR. N00014-84-K-0519 (NR 649-003) Research supported by the ARO. DAAL03-86-K-0171John N. Tsitsiklis

Neyman-Pearson detection in sensor networks with dependent observations

In this thesis, within the context of sensor networks, we are interested in the distributed detection problem under the Neyman-Pearson formulation and conditionally dependent sensor observations. In order to exploit all the detection potential of the network, the literature on this issue has faced optimal distributed detection problems, where optimality usually consists in properly designing the parameters of the network with the aim of minimizing some cost function related to the overall detection performance of the network. However, this problem of optimization has usually constraints regarding the possible physical and design parameters that we can choose when maximizing the detection performance of the network. In many applications, some physical and design parameters, for instance the network architecture or the local processing scheme of the sensor observations, are either strongly constrained to a set of possible design alternatives or either cannot be design variables in our problem of optimization. Despite the fact that those parameters can be related to the overall performance of the network, the previous constraints might be imposed by factors such as the environment where the network has to be deployed, the energy budget of the system or the processing capabilities of the available sensors. Consequently, it is necessary to characterize optimal decentralized detection systems with various architectures, different observation processes and different local processing schemes. The mayor part of the works addressing the characterization of distributed detection systems have assumed settings where, under each one of the possible states of our phenomenon of interest, the observations are independent across the sensors. However, there are many practical scenarios where the conditional independence assumption is violated because of the presence of different spatial correlation sources. In spite of this, very few works have faced the aforementioned characterizations under the same variety of settings as under the conditional independence assumption. Actually, when the strategy of the network is not an optimization parameter, under the assumption of conditionally dependent observations the existing literature has only obtained asymptotic characterizations of the detection performance associated with parallel networks whose local processing rules are based on amplify-and-relay schemes. Motivated by this last fact, in this thesis, under the Neyman-Pearson formulation, we undertake the characterization of distributed detection systems with dependent observations, various network architectures and binary quantization rules at the sensors. In particular, considering a parallel network randomly deployed along a straight line, we derive a closed-form error exponent for the Neyman-Pearson fusion of Markov local decisions when the involved fusion center only knows the distribution of the sensor spacings. After studying some analytical properties of the derived error exponent, we carry out evaluations of the closed-form expression in order to assess which kind of trends of detection performance can appear with increasing dependency and under two well-known models of the sensor spacing. These models are equispaced sensors with failures and exponentially spaced sensors with failures. Later, the previous results are extended to a two-dimensional parallel network that, formed by a set of local detectors equally spaced on a rectangular lattice, performs a Neyman-Pearson test discriminating between two di erent two-dimensional Markov causal fields defined on a binary state space. Next, under conditionally dependent observations and under the Neyman-Pearson set up, this thesis dissertation focuses on the characterization of the detection performance of optimal tandem networks with binary communications between the fusion units. We do so by deriving conditions under which, in an optimal tandem network with an arbitrary constraint on the overall probability of false alarm, the probability of misdetection of the system, i.e. at the last fusion node of the network, converges to zero as the number of fusion stages approaches infinity. Finally, after extending this result under the Bayesian set up, we provide two examples where these conditions are applied in order to characterize the detection performance of the network. From these examples we illustrate different dependence scenarios where an optimal tandem network can or cannot achieve asymptotic perfect detection under either the Bayesian set up or the Neyman-Pearson formulation. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------En esta tesis, dentro del contexto de las redes de sensores, estamos interesados en el problema de detección distribuida bajo la formulación de Neyman-Pearson y observaciones condicionalmente dependientes. Con objeto de explotar el potencial de detección de la red, la literatura sobre este tema se ha enfrentado a problemas de detección distribuida óptima, donde la optimalidad normalmente hace referencia al diseño adecuado de diferentes parámetros de la red con el objeto de minimizar alguna función de coste relacionada con las prestaciones globales de detección. Sin embargo, este problema de optimización tiene normalmente restricciones asociadas con los posibles parámetros físicos y de diseño de la red que pueden ser seleccionados a la hora de maximizar las prestaciones de detección de la misma. En muchas aplicaciones algunos parámetros físicos y de diseño, como por ejemplo la arquitectura de la red o los esquemas de procesado local de las observaciones de los sensores, bien están fuertemente restringidos a un conjunto de posibles alternativas de diseño, o bien no pueden ser variables de diseño en nuestro problema de optimización. A pesar de que estos parámetros pueden estar relacionados con las prestaciones de detección de la red, las anteriores restricciones podrán estar impuestas por factores tales como el entorno en el que la red se despliega, el presupuesto de energía disponible de la red o las capacidades de procesado de los sensores. Consecuentemente, es necesario caracterizar sistemas de detección distribuidos óptimos con varias arquitecturas, diferentes procesos de observación y diferentes esquemas de procesado local. La mayor parte de los trabajos tratando la caracterización de sistemas de detección distribuida han asumido escenarios en los que, bajo cada uno de los posibles estados del fenómeno de interés, las observaciones son independientes de un sensor a otro. Sin embargo, hay muchos escenarios prácticos donde la asunción de independencia condicional es violada como consecuencia de la presencia de diferentes fuentes de correlación. A pesar de esto, muy pocos trabajos han tratado las anteriores caracterizaciones bajo la misma variedad de escenarios que bajo la asunción de independencia condicional. De hecho, cuando la estrategia de la red no es un parámetro a optimizar, bajo la asunción de observaciones condicionalmente dependientes la literatura existente sólo ha obtenido caracterizaciones asintóticas de las prestaciones de detección asociadas con redes paralelas cuyas reglas de procesado local se basan en esquemas de amplificación y retransmisión. Motivado por este útimo hecho, en esta tesis, bajo la formulación de Neyman-Pearson, llevamos a cabo la caracterización de sistemas de detección distribuida con observaciones dependientes, varias arquitecturas de red y reglas de cuantificación binaria en los sensores. En particular, considerando una red paralela desplegada aleatoriamente a lo largo de una línea recta, bajo la formulación de Neyman-Pearson derivamos una expresión cerrada del exponente de error asociado a la fusión de decisiones locales Makovianas cuando, con respecto a los espaciados entre sensores, sólo se conoce su distribución. Después de analizar algunas propiedades analíticas del derivado exponente de error, llevamos a cabo evaluaciones de su expresión cerrada con el objeto de determinar las diferentes tendencias de detección que pueden aparecer con dependencia creciente y bajo dos modelos de espaciado entre sensores muy conocidos. Estos dos modelos son sensores equiespaciados con fallos y sensores exponencialmente espaciados con fallos. Más tarde, los anteriores resultados son extendidos a una red paralela bidimensional que, formada por un conjunto de dispositivos equiespaciados sobre una rejilla rectangular, lleva a cabo un test de Neyman-Pearson para discriminar entre dos diferentes campos aleatorios causales de Markov definidos en un espacio de estados binario. Seguidamente, bajo observaciones condicionalmente dependientes y bajo la formulación de Neyman-Pearson, esta tesis se centra en la caracterización de las prestaciones de detección asociada a redes tándem óptimas con comunicación binaria entre los nodos de fusión. Para hacer eso, derivamos condiciones bajo las cuales, en una red t andem óptima con una arbitraria restricci ón en la probabilidad de falsa alarma global, la probabilidad de pérdida de la red, es decir la asociada último nodo de fusión, converge a cero seg un el número de etapas de fusión tiende a infinito. Finalmente, después de extender este resultado bajo la formulación bayesiana, proporcionamos dos ejemplos donde estas condiciones son aplicadas para caracterizar las prestaciones de detección de la red. A partir de estos ejemplos ilustramos diferentes escenarios de dependencia en los que una red t ándem óptima puede o no lograr detección asintóticamente perfecta tanto bajo la formulación bayesiana como bajo la formulación de Neyman-Pearson