Decision theory and sampling algorithms for spatial and spatio-temporal point processes
In this work, we ﬁrst present a ﬂexible hierarchical Bayesian model and develop a comprehensive Bayesian decision theoretic framework for point process theory. Then, we provide a comparative study of the approximate Bayesian computation (ABC) for Gibbs point processes based on various summary statistics and diﬀerent approaches of constructing the discrepancy measure. Finally, we propose a ﬂexible spatio-temporal area-interaction point (STAI) process for ﬁtting spatial point patterns with discrete time stamps. Under a Bayesian decision theoretic framework, we closely investigate the Pois son process, using a ﬁnite mixture of exponential family components to model the intensity function. We demonstrate the eﬀectiveness of the Bayes rule under the Kullback-Leibler and Hellinger loss functions and compare them with the usual estimator, the posterior mean under squared error loss. By applying the ABC for Gibbs point processes, we demonstrate the issue of identiﬁability of the parameter values for Gibbs point processes and provide a solution for parameter estimation. We further propose robust choices for the discrepancy measures for diﬀerent point processes through an intensive simulation study. For the STAI process, a hierarchical model is implemented in order to incorporate the underlying evolution process of the model parameters. For parameter estimation, a double Metropolis-Hastings within Gibbs sampler is used. We exempliﬁed the proposed estimators, sampling algorithm and point process model through simulations and applications to the Chicago crime data, the Swedish pines data and the United States natural caused wildﬁre data from 2002 to 2019.
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