Theoretical Aspects of Spatial-Temporal Modeling

In the last decade, the area of multiple target tracking has witnessed the introduction of important concepts and methods, aiming at establishing principled approaches for dealing with the estimation of multiple objects in an efficient way. One of the most successful classes of multi-object filters that have been derived out of these new grounds includes all the variants of the Probability Hypothesis Density (phd) filter. In spite of the attention that these methods have attracted, their theoretical performances are still not fully understood. In this chapter, we first focus on the different ways of establishing the equations of the phd filter, using a consistent set of notations. The objective is then to introduce the idea of observation path, upon which association measures are defined. We will see how these concepts highlight the structure of the first moment of the multi-object distributions in time, and how they allow for devising solutions to practical estimation problems.

Nonlinear non-Gaussian state-space models arise in numerous applications in statistics and signal processing. In this context, one of the most successful and popular approximation techniques is the sequential Monte-Carlo (SMC) algorithm, also known as the particle filter. Nevertheless, this method tends to be inefficient when applied to high-dimensional problems. In this chapter, we present, an overview of recent contributions related to Monte-Carlo methods for sequential simulation from ultra high-dimensional distributions, often arising for instance in Bayesian applications.

Currently, we are witnessing the proliferation of wireless sensor networks and the superposition of several communicating objects which have a heterogeneous nature. Those are merely the beginnings of an evolution toward the so-called Internet of Things. The advent of these networks as well as the increasing demand for improved quality and services will increase the complexity of communications and put a strain on current techniques and models. Indeed, they must first adapt to the temporal and spatial evolutions and second, they must take into account the rare and unpredictable events that can have disastrous consequences for decision-making. This chapter provides an overview of the various spectral techniques used in signal processing and statistics literature to describe a communication channel having an impulsive behavior. This project is mainly motivated by the historical success of the interaction between probability, statistics and the world of communications, information theory and signal processing. The second motivation is the scarcity of references and literature summarizing mathematical developments on the application of alpha-stable process for channel modeling. This chapter will be divided into two parts: the first is devoted to the synthesis of various developments on alpha-stable variables and processes in a purely mathematical mind. The second part will be devoted to applications in the context of communications. The two sides will combine two fundamentally linked aspects: first, a theoretical approach, necessary for a good formalization of problems and identifying the best solutions. Second, the use of these models in real work of channel modeling.

The theory of random graphs goes back to the late 1950s when Paul Erdős and Alfréd Rényi introduced the Erdős-Rényi random graph. Since then many models have been developed, and the study of random graph models has become popular for real-life network modelling such as social networks and financial networks. The aim of this overview is to review relevant random graph models for real-life network modelling. Therefore, we analyse their properties in terms of stylised facts of real-life networks.