Estimation, Decision and Applications to Target Tracking

This dissertation mainly consists of three parts. The first part proposes generalized linear minimum mean-square error (GLMMSE) estimation for nonlinear point estimation. The second part proposes a recursive joint decision and estimation (RJDE) algorithm for joint decision and estimation (JDE). The third part analyzes the performance of sequential probability ratio test (SPRT) when the log-likelihood ratios (LLR) are independent but not identically distributed. The linear minimum mean-square error (LMMSE) estimation plays an important role in nonlinear estimation. It searches for the best estimator in the set of all estimators that are linear in the measurement. A GLMMSE estimation framework is proposed in this disser- tation. It employs a vector-valued measurement transform function (MTF) and finds the best estimator among all estimators that are linear in MTF. Several design guidelines for the MTF based on a numerical example were provided. A RJDE algorithm based on a generalized Bayes risk is proposed in this dissertation for dynamic JDE problems. It is computationally efficient for dynamic problems where data are made available sequentially. Further, since existing performance measures for estimation or decision are effective to evaluate JDE algorithms, a joint performance measure is proposed for JDE algorithms for dynamic problems. The RJDE algorithm is demonstrated by applications to joint tracking and classification as well as joint tracking and detection in target tracking. The characteristics and performance of SPRT are characterized by two important functions—operating characteristic (OC) and average sample number (ASN). These two functions have been studied extensively under the assumption of independent and identically distributed (i.i.d.) LLR, which is too stringent for many applications. This dissertation relaxes the requirement of identical distribution. Two inductive equations governing the OC and ASN are developed. Unfortunately, they have non-unique solutions in the general case. They do have unique solutions in two special cases: (a) the LLR sequence converges in distributions and (b) the LLR sequence has periodic distributions. Further, the analysis can be readily extended to evaluate the performance of the truncated SPRT and the cumulative sum test

Some topics in sequential analysis

Sequential analysis refers to the statistical theory and methods that can be applied to situations where the sample size is not fixed in advance. Instead, the data are collected sequentially over time, and the sampling is stopped according to a pre-specified stopping rule as soon as the accumulated information is deemed sufficient. The goal of this adaptive approach is to reach a reliable decision as soon as possible. This dissertation investigates two problems in sequential analysis. In the first problem, assuming that data are collected sequentially from independent streams, we consider the simultaneous testing of multiple hypotheses. We start with the class of procedures that control the classical familywise error probabilities of both type I and type II under two general setups: when the number of signals (correct alternatives) is known in advance, and when we only have a lower and an upper bound for it. Then we continue to study two generalized error metrics: under the first one, the probability of at least k mistakes, of any kind, is controlled; under the second, the probabilities of at least k1 false positives and at least k2 false negatives are simultaneously controlled. For each formulation, the optimal expected sample size is characterized, to a first-order asymptotic approximation as the error probabilities vanish, and a novel multiple testing procedure is proposed and shown to be asymptotically efficient under every signal configuration. In the second problem, we propose a generalization of the Bayesian sequential change detection problem, where the change is a latent event that should be not only detected but also accelerated. It is assumed that the sequentially collected observations are responses to treatments selected in real time. The assigned treatments not only determine the distribution of responses before and after the change, but also influence when the change happens. The problem is to find a treatment assignment rule and a stopping rule to minimize the average total number of observations subject to a bound on the false-detection probability. We propose an intuitive solution, which is easy to implement and achieves for a large class of change-point models the optimal performance up to a first-order asymptotic approximation. A simulation study suggests the almost exact optimality of the proposed scheme under a Markovian change-point model

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