Non-homogeneous Poisson process intensity modeling and estimation using measure transport

Abstract

Non-homogeneous Poisson processes are used in a wide range of scientific disciplines, ranging from the environmental sciences to the health sciences. Often, the central object of interest in a point process is the underlying intensity function. Here, we present a general model for the intensity function of a non-homogeneous Poisson process using measure transport. The model is built from a flexible bijective mapping that maps from the underlying intensity function of interest to a simpler reference intensity function. We enforce bijectivity by modeling the map as a composition of multiple bijective maps that have increasing triangular structure, and show that the model exhibits an important approximation property. Estimation of the flexible mapping is accomplished within an optimization framework, wherein computations are efficiently done using tools originally designed to facilitate deep learning, and a graphics processing unit. Point process simulation and uncertainty quantification are straightforward to do with the proposed model. We demonstrate the potential benefits of our proposed method over conventional approaches to intensity modeling through various simulation studies. We also illustrate the use of our model on a real data set containing the locations of seismic events near Fiji since 1964.