Spatial marked point processes: Models and inferences

Yen-Ning E Huang, Purdue University

Abstract

A spatial marked point process describes the locations of randomly distributed events in a region, with a mark attached to each observed point. Nowadays, the availability of spatiotemporal data is increasing and many spatiotemporal models are studied with applications in a wide range of disciplines. Spatial marked point processes are then extended to spatiotemporal marked point processes if time component is taken into account. In general, the marks can be quantitative or categorical variables. Independence between points and marks is a convenient assumption, but may not be true in practice. Tests for independence between points and marks are proposed previously, though only a few models have been developed to describe dependence between points and marks. In this dissertation, I focus on quantitative marks and the objective is to provide flexible models for both spatial and spatiotemporal marked point processes when points and marks are dependent. Three approaches to describe dependence between points and marks are studied in this dissertation, while the first two approaches are for spatial marked point processes and the last is for spatiotemporal marked point processes. First, we derive a covariance function of additive models for marked point processes. This covariance function carries information of dependence between points and marks, which can be used in kriging to make predictions of marks at unknown locations. We expect to obtain better prediction results by using this covariance function when the points and marks are dependent. The second approach is to consider intensity-dependent models. We study both univariate and bivariate intensity marked Log Gaussian Cox processes and apply an empirical Bayesian estimation procedure with implementation of Markov Chain Monte Carlo methodology for statistical inference. We allow dependence between marks after conditioning on the intensity which is more flexible than conditional independence assumption. The influence of adding cross covariance in modeling bivariate marks is also explored. The first two approaches are applied to model the dependence between points and marks of a white oak data. The last approach is to consider the partially stationary spatiotemporal marked point process, where the distribution of the spatiotemporal marked process is invariant under parallel shift of time, but may not be invariant under parallel shift of points or marks. It can be classified as a location-dependent model. To determine the potential usefulness of this approach, we illustrate through two typical examples in natural hazards: a forest wild fire study and an earthquake study. The results show that the distribution of marks and points is significantly different at local scale. It is expected that the proposed approach will have wide applications in the study of natural hazards.

Degree

Ph.D.

Advisors

Zhang, Purdue University.

Subject Area

Statistics

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