ADDITIVE POISSON PROCESS: LEARNING INTENSITY OF HIGHER-ORDER INTERACTION IN POISSON PRO-CESSES

Abstract

We present the Additive Poisson Process (APP), a novel framework that can model the higher-order interaction effects of the intensity functions in Poisson processes using projections into lower-dimensional space. Our model combines the techniques from information geometry to model higher-order interactions on a statistical manifold and in generalized additive models to use lower-dimensional projections to overcome the effects from the curse of dimensionality. Our approach solves a convex optimization problem by minimizing the KL divergence from a sample distribution in lower-dimensional projections to the distribution modeled by an intensity function in the Poisson process. Our empirical results show that our model effectively uses samples observed in lower dimensional space to estimate a higher-order intensity function with sparse observations.

1. INTRODUCTION

The Poisson process is a counting process used in a wide range of disciplines such as spatialtemporal sequential data in transportation (Zhou et al., 2021) , finance (Ilalan, 2016) and ecology (Thompson, 1955) to model the arrival rate by learning an intensity function. For a given time interval, the integral of the intensity function represents the average number of events occurring in that interval. The intensity function can be generalized to multiple dimensions. However, for most practical applications, learning the multi-dimensional intensity function is a challenge due to the sparsity of observations. Despite the recent advances of Poisson processes, current Poisson process models are unable to learn the intensity function of a multi-dimensional Poisson process. Our research question is, "Are there any good ways of approximating the high dimensional intensity function?" Our proposal, the Additive Poisson Process (APP), provides a novel solution to this problem. Throughout this paper, we use a running example in a spatial-temporal setting. Say we want to learn the intensity function for a taxi to pick up customers at a given time and location. For this setting, each event is multi-dimensional; that is, (x, y, W ), where a pair of x and y represents two spatial coordinates and W represents the day of the week. In addition, observation time t is associated with this event. For any given location or time, we can expect at most a few pick-up events, which makes it difficult for any model to learn the low-valued intensity function. Figure 1b visualizes this problem. In this problem setup, if we would like to learn the intensity function at a given location (x, y) and day of the week W , the naïve approach would be to learn the intensity at (x, y, W ) directly from observations. This is extremely difficult because there could be only few events for a given location and day. However, there is useful information in lower-dimensional space; for example, the marginalized observations at the location (x, y) across all days of the week, or on the day W at all locations. This information can be included into the model to improve the estimation of the joint intensity function. Using the information in lower-dimensional space provides a structured approach to include prior information based on the location or day of the week to improve the estimation of the joint intensity function. For example, a given location could be a shopping center or a hotel, where it is common for taxis to pick up passengers, and therefore we expect more passengers at this location. There could also be additional patterns that could be uncovered based on the day of the week. We can then use the observations of events to update our knowledge of the intensity function.

