Spatio-temporal families of overdispersed and zero-inflated counts through new counting distributions - Vinueza Cajas, Johanna Gabriela (2025)
Spatial and spatial-temporal count models have become essential in several scientific research fields, such as ecology, epidemiology, geography, and urban crime. Stochastic processes or random fields are mathematical models that can deal with spatial or spatio-temporal count data. However, these random fields typically face problems such as excess of zeros and overdispersion in the count data generated as a part of a natural phenomenon, becoming a modeling challenge. In this work, we provide theoretical results of two new families of zero-inflated and overdispersed random fields called zero-inflated Poisson-Erlang mixture and Poisson-log Gaussian mixture random fields, including the zero-inflated Poisson-log Gaussian mixture random field. These models are based on the work of Morales-Navarrete, Diego, who proposed a new counting distribution based on the Poisson counting process. An application of our proposed model, a zero-inflated Poisson-Erlang mixture random field, has been developed by analyzing crime incident counts, such as smooth thefts on streets in Valencia, Spain. Since this model cannot entirely model the excess zeros and overdispersed of the random field, the marginal mean and variance are still moderately high com-pared with Poisson, zero-inflated Poisson, and Poisson-Erlang mixture models. For that reason, as an alternate model, it motivates the proposal of a new model called a Poisson-log Gaussian mixture random field. This doubly stochastic random field could better control the excess zeros and overdispersion of the count data. The weighted pairwise likelihood method has been used for the estimation procedure. In future work, we will use an optimal linear predictor to study the predictive performance.