Modelagem de processos pontuais de cox log-gaussianos: uma abordagem local e global com covariáveis para análise espaço-temporal

Abstract

Spatio-temporal point processes (STPP) are stochastic models used to describe the distribution of discrete events in space and time. The simplest model is the Poisson process, which assumes a constant intensity over space and time; however, this assumption is often unrealistic for real data, which frequently exhibit varying intensities. Spatio-temporal Cox processes, such as Log-Gaussian Cox Processes (LGCP), allow for the modeling of varying intensities through Gaussian random fields, incorporating spatio-temporal dependence. However, the intractability of the log-likelihood function for these models poses significant computational challenges. Methods such as Markov Chain Monte Carlo (MCMC) and Integrated Nested Laplace Approximation (INLA) are commonly used, but they can be computationally expensive. Alternatively, the minimum contrast method provides a more efficient and less computationally demanding solution. Some extensions of the LGCP model consider local variations in intensity, allowing for the capture of more complex spatial and temporal dependence patterns. The local approach can be particularly useful in scenarios where parameters such as the spatial range vary across regions, as in wildfire modeling. Nevertheless, there are still limitations in these approaches that prevent the inclusion of covariates. This thesis proposes two novel approaches for LGCP model estimation that allow for the inclusion of covariates. In the first approach, generalized additive models (GAMs) are employed, and the tractability of the Gaussian random field is achieved through the estimation of second-order parameters using the minimum contrast method, approximated via the pair correlation function. In the second approach, GAMs are also used, and the Gaussian random field (GRF) is approximated through linear combinations of spatio-temporal basis functions. Both approaches were applied to spatio-temporal location data of wildfire occurrences in the Brazilian Pantanal and in the state of New Mexico, USA. The results show that the proposed STPP models are computationally feasible and provide improved estimation accuracy, making them more suitable for analyzing data with high spatio-temporal variability. Keywords: Spatio-Temporal Point Processes; Log-Gaussian Cox Models; Generalized Additive Models; Spatio-Temporal Dependence; Minimum Contrast; Gaussian Random Field.