Performance Bounds for Parameter Estimation under Misspecified Models: Fundamental findings and applications

Inferring information from a set of acquired data is the main objective of any signal processing (SP) method. In particular, the common problem of estimating the value of a vector of parameters from a set of noisy measurements is at the core of a plethora of scientific and technological advances in the last decades; for example, wireless communications, radar and sonar, biomedicine, image processing, and seismology, just to name a few. Developing an estimation algorithm often begins by assuming a statistical model for the measured data, i.e. a probability density function (pdf) which if correct, fully characterizes the behaviour of the collected data/measurements. Experience with real data, however, often exposes the limitations of any assumed data model since modelling errors at some level are always present. Consequently, the true data model and the model assumed to derive the estimation algorithm could differ. When this happens, the model is said to be mismatched or misspecified. Therefore, understanding the possible performance loss or regret that an estimation algorithm could experience under model misspecification is of crucial importance for any SP practitioner. Further, understanding the limits on the performance of any estimator subject to model misspecification is of practical interest. Motivated by the widespread and practical need to assess the performance of a mismatched estimator, the goal of this paper is to help to bring attention to the main theoretical findings on estimation theory, and in particular on lower bounds under model misspecification, that have been published in the statistical and econometrical literature in the last fifty years. Secondly, some applications are discussed to illustrate the broad range of areas and problems to which this framework extends, and consequently the numerous opportunities available for SP researchers.Comment: To appear in the IEEE Signal Processing Magazin

Borné de Cramér-Rao sous contraintes pour l’estimation simultanée de paramètres aléatoires et non aléatoires

In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both non-random and random parameters. On the other hand, numerous works have shown the versatility of deterministic constrained Cramér-Rao bound for estimation performance analysis and design of a system of measurement. In this communication, we propose a constrained hybrid lower bound which takes into account equality constraints on deterministic parameters. The proposed bound is then compared to previous bounds of the literature. Finally, the usefulness of the proposed bound is illustrated with an application to radar Doppler estimation

Recursive joint Cramér‐Rao lower bound for parametric systems with two‐adjacent‐states dependent measurements

Joint Cramér-Rao lower bound (JCRLB) is very useful for the performance evaluation of joint state and parameter estimation (JSPE) of non-linear systems, in which the current measurement only depends on the current state. However, in reality, the non-linear systems with two-adjacent-states dependent (TASD) measurements, that is, the current measurement is dependent on the current state as well as the most recent previous state, are also common. First, the recursive JCRLB for the general form of such non-linear systems with unknown deterministic parameters is developed. Its relationships with the posterior CRLB for systems with TASD measurements and the hybrid CRLB for regular parametric systems are also provided. Then, the recursive JCRLBs for two special forms of parametric systems with TASD measurements, in which the measurement noises are autocorrelated or cross-correlated with the process noises at one time step apart, are presented, respectively. Illustrative examples in radar target tracking show the effectiveness of the JCRLB for the performance evaluation of parametric TASD systems

Recursive Hybrid Cramér–Rao Bound for Discrete-Time Markovian Dynamic Systems

Abstract—In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both non-random and random parameters. As a contribution to the hybrid estimation framework, we introduce a recursive hybrid Cramér–Rao lower bound for discrete-time Markovian dynamic systems depending on unknown deterministic parameters. Additionally, the regularity conditions required for its existence and its use are clarified

A Constrained Hybrid Cramér-Rao Bound for Parameter Estimation

In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both non-random and random parameters. Numerous works have shown the versatility of deterministic constrained Cramér-Rao bound for estimation performance analysis and design of a system of measurement. However in many systems both random and non-random parameters may occur simultaneously. In this communication, we propose a constrained hybrid lower bound which take into account of equality constraint on deterministic parameters. The usefulness of the proposed bound is illustrated with an application to radar Doppler estimation

Caractérisation des performances minimales d'estimation pour des modèles d'observations non-standards

In the parametric estimation context, estimators performances can be characterized, inter alia, by the mean square error and the resolution limit. The first quantities the accuracy of estimated values and the second defines the ability of the estimator to allow a correct resolvability. This thesis deals first with the prediction the "optimal" MSE by using lower bounds in the hybrid estimation context (i.e. when the parameter vector contains both random and non-random parameters), second with the extension of Cramér-Rao bounds for non-standard estimation problems and finally to the characterization of estimators resolution. This manuscript is then divided into three parts :First, we fill some lacks of hybrid lower bound on the MSE by using two existing Bayesian lower bounds: the Weiss-Weinstein bound and a particular form of Ziv-Zakai family lower bounds. We show that these extended lower bounds are tighter than the existing hybrid lower bounds in order to predict the optimal MSE.Second, we extend Cramer-Rao lower bounds for uncommon estimation contexts. Precisely: (i) Where the non-random parameters are subject to equality constraints (linear or nonlinear). (ii) For discrete-time filtering problems when the evolution of states are defined by a Markov chain. (iii) When the observation model differs to the real data distribution.Finally, we study the resolution of the estimators when their probability distributions are known. This approach is an extension of the work of Oh and Kashyap and the work of Clark to multi-dimensional parameters estimation problems.Dans le contexte de l'estimation paramétrique, les performances d'un estimateur peuvent être caractérisées, entre autre, par son erreur quadratique moyenne (EQM) et sa résolution limite. La première quantifie la précision des valeurs estimées et la seconde définit la capacité de l'estimateur à séparer plusieurs paramètres. Cette thèse s'intéresse d'abord à la prédiction de l'EQM "optimale" à l'aide des bornes inférieures pour des problèmes d'estimation simultanée de paramètres aléatoires et non-aléatoires (estimation hybride), puis à l'extension des bornes de Cramér-Rao pour des modèles d'observation moins standards. Enfin, la caractérisation des estimateurs en termes de résolution limite est également étudiée. Ce manuscrit est donc divisé en trois parties :Premièrement, nous complétons les résultats de littérature sur les bornes hybrides en utilisant deux bornes bayésiennes : la borne de Weiss-Weinstein et une forme particulière de la famille de bornes de Ziv-Zakaï. Nous montrons que ces bornes "étendues" sont plus précises pour la prédiction de l'EQM optimale par rapport à celles existantes dans la littérature.Deuxièmement, nous proposons des bornes de type Cramér-Rao pour des contextes d'estimation moins usuels, c'est-à-dire : (i) Lorsque les paramètres non-aléatoires sont soumis à des contraintes d'égalité linéaires ou non-linéaires (estimation sous contraintes). (ii) Pour des problèmes de filtrage à temps discret où l'évolution des états (paramètres) est régit par une chaîne de Markov. (iii) Lorsque la loi des observations est différente de la distribution réelle des données.Enfin, nous étudions la résolution et la précision des estimateurs en proposant un critère basé directement sur la distribution des estimées. Cette approche est une extension des travaux de Oh et Kashyap et de Clark pour des problèmes d'estimation de paramètres multidimensionnels

Exploiting Sparse Structures in Source Localization and Tracking

This thesis deals with the modeling of structured signals under different sparsity constraints. Many phenomena exhibit an inherent structure that may be exploited when setting up models, examples include audio waves, radar, sonar, and image objects. These structures allow us to model, identify, and classify the processes, enabling parameter estimation for, e.g., identification, localisation, and tracking.In this work, such structures are exploited, with the goal to achieve efficient localisation and tracking of a structured source signal. Specifically, two scenarios are considered. In papers A and B, the aim is to find a sparse subset of a structured signal such that the signal parameters and source locations maybe estimated in an optimal way. For the sparse subset selection, a combinatorial optimization problem is approximately solved by means of convex relaxation, with the results of allowing for different types of a priori information to be incorporated in the optimization. In paper C, a sparse subset of data is provided, and a generative model is used to find the location of an unknown number of jammers in a wireless network, with the jammers’ movement in the network being tracked as additional observations become available

THEORETICAL ASPECTS AND REAL ISSUES IN AN INTEGRATED MULTIRADAR SYSTEM

In the last few years Homeland Security (HS) has gained a considerable interest in the research community. From a scientific point of view, it is a difficult task to provide a definition of this research area and to exactly draw up its boundaries. In fact, when we talk about the security and the surveillance, several problems and aspects must be considered. In particular, the following factors play a crucial role and define the complexity level of the considered application field: the number of potential threats can be high and uncertain; the threat detection and identification can be made more complicated by the use of camouflaging techniques; the monitored area is typically wide and it requires a large and heterogeneous sensor network; the surveillance operation is strongly related to the operational scenario, so that it is not possible to define a unique approach to solve the problem [1]. Information Technology (IT) can provide an important support to HS in preventing, detecting and early warning of threats. Even though the link between IT and HS is relatively recent, sensor integration and collaboration is a widely applied technique aimed to aggregate data from multiple sources, to yield timely information on potential threats and to improve the accuracy in monitoring events [2]. A large number of sensors have already been developed to support surveillance operations. Parallel to this technological effort in developing new powerful and dedicated sensors, interest in integrating a set of stand-alone sensors into an integrated multi-sensor system has been increasing. In fact, rather than to develop new sensors to achieve more accurate tracking and surveillance systems, it is more useful to integrate existing stand-alone sensors into a single system in order to obtain performance improvements In this dissertation, a notional integrated multi-sensor system acting in a maritime border control scenario for HS is considered. In general, a border surveillance system is composed of multiple land based and moving platforms carrying different types of sensors [1]. In a typical scenario, described in [1], the integrated system is composed of a land based platform, located on the coast, and an airborne platform moving in front of the coast line. In this dissertation, we handle two different fundamental aspects. In Part I, we focus on a single sensor in the system, i.e. the airborne radar. We analyze the tracking performance of such a kind of sensor in the presence of two different atmospheric problems: the turbulence (in Chapter 1) and the tropospheric refraction (in Chapter 2). In particular, in Chapter 1, the losses in tracking accuracy of a turbulence-ignorant tracking filter (i.e. a filter that does not take into account the effects of the atmospheric turbulences) acting in a turbulent scenario, is quantified. In Chapter 2, we focus our attention on the tropospheric propagation effects on the radar electromagnetic (em) signals and their correction for airborne radar tracking. It is well known that the troposphere is characterized by a refractive index that varies with the altitude and with the local weather. This variability of the refractive index causes an error in the radar measurements. First, a mathematical model to describe and calculate the em radar signal ray path in the troposphere is discussed. Using this mathematical model, the errors due to the tropospheric propagation are evaluated and the corrupted radar measurements are then numerically generated. Second, a tracking algorithm, based on the Kalman Filter, that is able to mitigate the tropospheric errors during the tracking procedure, is proposed. In Part II, we consider the integrated system in its wholeness to investigate a fundamental prerequisite of any data fusion process: the sensor registration process. The problem of sensor registration (also termed, for naval system, the grid-locking problem) arises when a set of data coming from two or more sensors must be combined. This problem involves a coordinate transformation and the reciprocal alignment among the various sensors: streams of data from different sensors must be converted into a common coordinate system (or frame) and aligned before they could be used in a tracking or surveillance system. If not corrected, registration errors can seriously degrade the global system performance by increasing tracking errors and even introducing ghost tracks. A first basic distinction is usually made between relative grid-locking and absolute grid-locking. The relative grid-locking process aligns remote data to local data under the assumption that the local data are bias free and that all biases reside with the remote sensor. The problem is that, actually, also the local sensor is affected by bias. Chapter 3 of this dissertation is dedicated to the solution of the relative grid-locking problem. Two different estimation algorithms are proposed: a linear Least Squares (LS) algorithm and an Expectation-Maximization-based (EM) algorithm. The linear LS algorithm is a simple and fast algorithm, but numerical results have shown that the LS estimator is not efficient for most of the registration bias errors. Such non-efficiency could be caused by the linearization implied by the linear LS algorithm. Then, in order to obtain a more efficient estimation algorithm, an Expectation-Maximization algorithm is derived. In Chapter 4 we generalize our findings to the absolute grid-locking problem. Part III of this dissertation is devoted to a more theoretical aspect of fundamental importance in a lot of practical applications: the estimate of the disturbance covariance matrix. Due to its relevance, in literature it can be found a huge quantity of works on this topic. Recently, a new geometrical concept has been applied to this estimation problem: the Riemann (or intrinsic) geometry. In Chapter 5, we give an overview on the state of the art of the application of the Riemann geometry for the covariance matrix estimation in radar problems. Particular attention is given for the detection problem in additive clutter. Some covariance matrix estimators and a new decision rule based on the Riemann geometry are analyzed and their performance are compared with the classical ones. [1] Sofia Giompapa, “Analysis, modeling, and simulation of an integrated multi-sensor system for maritime border control”, PhD dissertation, University of Pisa, April 2008. [2] H. Chen, F. Y. Wang, and D. Zeng, “Intelligence and security informatics for Homeland Security: information, communication and transportation,” Intelligent Transportation Systems, IEEE Transactions on, vol. 5, no. 4, pp. 329-341, December 2004