Classification and clustering for samples of event time data using non-homogeneous Poisson process models

Data of the form of event times arise in various applications. A simple model for such data is a non-homogeneous Poisson process (NHPP) which is specified by a rate function that depends on time. We consider the problem of having access to multiple independent samples of event time data, observed on a common interval, from which we wish to classify or cluster the samples according to their rate functions. Each rate function is unknown but assumed to belong to a small set of rate functions defining distinct classes. We model the rate functions using a spline basis expansion, the coefficients of which need to be estimated from data. The classification approach consists of using training data for which the class membership is known and to calculate maximum likelihood estimates of the coefficients for each group, then assigning test samples to a class by a maximum likelihood criterion. For clustering, by analogy to the Gaussian mixture model approach for Euclidean data, we consider a mixture of NHPP models and use the expectation maximisation algorithm to estimate the coefficients of the rate functions for the component models and probability of membership for each sample to each model. The classification and clustering approaches perform well on both synthetic and real-world data sets considered. Code associated with this paper is available at https://github.com/duncan-barrack/NHPP.