DOUBLY STOCHASTIC MODEL WITH COVARIATES FOR REPLICATED POISSON POINT PROCESSES

Abstract

Poisson point processes (PPPs) are powerful tools for modeling random point occurrencesin multidimensional spaces, with applications across various fields. Although the traditional literature has focused on single realizations, replicated point processes are becoming increasingly common due to the growing availability of complex data. This dissertation develops a doubly stochastic model for replicated PPPs that incorporates covariates, extending latent component models to capture external effects. The proposed model expresses the log-intensity function as the sum of a mean function and latent component scores that vary with covariates. To ensure identifiability, component scores are constrained to be zero-mean and uncorrelated via centering and orthogonality. Parameter estimation is performed using penalized maximum likelihood, employing Newton–Raphson updates and the Laplace approximation for conditional distributions. Simulation studies assess the model’s stability across various covariate structures (linear and nonlinear), baseline rates, and sample sizes. The results demonstrate decreasing error with increasing sample size, confirming the estimators’ consistency. The model is applied to real data from the Divvy bicycle-sharing system in Chicago, analyzing daily usage at a representative station. The results reveal a nonlinear relationship between temperature and ridership, with peak usage occurring at moderate temperatures and declines observed under extreme heat or cold. This modeling framework improves the interpretability and predictive accuracy of PPPs with covariates, offering practical insights for applications such as fleet allocation in bicycle-sharing systems.