Partially Linear Single-Index Models and Functional Principal Component Analysis of Spatially and Temporally Indexed Point Processes

We model spatially and temporally indexed point process data as a multi-level log-Gaussian Cox process where the log intensity function depends on a partially linear single-index structure of spatio-temporal covariates and three latent functional random effects representing the spatial and temporal random effects as well as their interactions. We assume that the latent functional effects are Gaussian processes with Karhunen-Loève representations, and model the unknown link function of the single-index as well as the covariance functions of the latent functional effects as splines. We propose to estimate the partially linear coefficients and the single-index link function using a Poisson maximum likelihood method, and the covariance functions of the latent processes using maximum composite likelihood methods. We also propose approaches to predict the functional principal component scores. Under the multi-level dependence structure and allowing the spatio-temporal covariates to be non-stationary, the proposed estimators follow rather unconventional convergence rates which depend on both the number of locations and the number of repeated measures in time. We illustrate the proposed methods through simulation studies and a real-data application in modeling bike-sharing events.