Resumen de Modelling, Estimation and Applications of Second-Order Spatio-Temporal Characteristics of Point Processes

Francico Javier Rodríguez Cortés

• español

  Esta tesis se centra principalmente en el desarrollo de propiedades y características de los estimadores de segundo orden de procesos puntuales y espacio-temporales. En primer lugar, se presenta un marco teórico acerca de procesos puntuales espaciales y espacio-temporales. El resto de la tesis se organiza como sigue. En el capítulo 2, se presenta una nueva familia de kernel positivos y óptimos, además se propone un estimador insensgado alternativo para la función de la densidad del producto. Su rendimiento se compara para varios kernel mediante MISE. En el capítulo 3, se dada un nuevo estimador kernel de la función de la densidad producto espacio-temporal y también se desarrollan expresiones cerradas para la varianza en el caso de Poisson. En el Capítulo 4, nos centramos en los métodos de orientación de segundo orden los cuales proporcionan una herramienta para el análisis natural para los datos espaciales anisótropicos. Finalmente, se proporciona una descripción general de los proyectos de investigación actualmente en curso que han surgido motivadas por la estrecha relación con las propiedades de segundo orden de los procesos puntuales espaciales y espacio-temporales.

• English

  There is an extensive literature on the analysis of point process data in both time and space, separately. However, methods for the analysis of spatio-temporal point processes are less well established. Many spatial processes of scientific interest also have a temporal component that may need to be considered when modelling the underlying phenomenon. The spatio-temporal behaviour analysis is fundamental in areas such as environmental sciences, climate prediction and meteorology, epidemiology, image analysis, agriculture, seismology and astronomy, and so spatio-temporal point processes, rather than purely spatial point processes, must then be considered as potential models. A natural starting point for the analysis of spatio-temporal point process data is to investigate the nature of any stochastic interactions among the points of the process. For these processes, second-order properties play an important role for exploratory and inferential analysis. Second-order methods provide indeed a natural starting point for such analysis. This thesis is mainly focused on developing properties and estimators for second-order characteristics of spatio-temporal point processes, and every chapter adds some valuable information over the previous ones. In Chapter 1 we present a theoretical framework of spatial and spatio-temporal point processes as a mathematical tool for dealing with the concepts shown along the next chapters of this thesis. The final part of this chapter consists of a first compilation of the most recent developments in the literature of spatio-temporal point processes. In Chapter 2 we consider kernel-based non-parametric estimation of second-order product densities of spatial point patterns. We present a new family of optimal and positive kernels showing less variance than that for optimal kernels. This family generalises most of the classical and widely used kernel functions, such as Box or Epanechnikov kernels. We propose an alternative unbiased estimator for the product density function, and compare the performance of the estimator for several members of the family of optimal and positive kernels through MISE and relative efficiency. We present a simulation study to analyse the behaviour of such kernel functions, for three different spatial structures, for which we know the exact analytical form of the product density, and under small sample sizes. Some known datasets are revisited. In Chapter 3 a new kernel estimator of the second-order product density function of a spatio-temporal point process with and without considering first- and second-order spatio-temporal separability is given. The spatio-temporal second-order product density function is of interest as can be used to discriminate amongst several spatio-temporal point structures. Further, the expectation and variance of this estimator are obtained. In addition, as we have eveloped close expressions for the variance under the Poisson case, we use them to generate the corresponding confidence surfaces. A simulation study is presented. We have used functions of the R library stpp in connection with Fortran subroutines. Finally, we apply the resulting estimator to data on the spatio-temporal distribution of invasive meningococcal disease in Germany. In Chapter 4 we focus on second-order orientation methods which provide a natural tool for the analysis of anisotropic spatial point process data. Here we extend to the spatio-temporal setting the spatial point pair orientation distribution function. The new spatio-temporal orientation distribution function is used to detect spatio-temporal anisotropic configurations. An edge-corrected estimator is defined and illustrated through a simulation study. We apply the resulting estimator to data on the spatio-temporal distribution of fire ignition events caused by humans in a square area of 30 x 30 square kilometers during four years. Our results confirm that our approach is able to detect directional components at distinct spatio-temporal scales. Finally, we provide a general description of the currently ongoing research projects which have emerged motivated by the close relationship with the second-order properties of spatial and spatio-temporal point processes. In particular, we have adapted our methodology to spatio-temporal local clustering analysis, and to modelling orbital debris using a new and innovative adaptation over the sphere of the classical theory.