Dirichlet depths for point process

Abstract

Statistical depths have been well studied for multivariate and functional data over the past few decades, but remain under-explored for point processes. A first attempt on the notion of point process depth was conducted recently where the depth was defined as a weighted product of two terms: (1) the probability of the number of events in each process and (2) the depth of the event times conditioned on the number of events by using a Mahalanobis depth. We point out that multivariate depths such as the Mahalanobis depth cannot be directly used because they often neglect the important ordering property in the point process events. To deal with this problem, we propose a model-based approach for point process systematically. In particular, we develop a Dirichlet-distribution-based framework on the conditional depth term, where the new methods are referred to as Dirichlet depths. We examine mathematical properties of the new depths and conduct asymptotic analysis. In addition, we illustrate the new methods using various simulated and real experiment data. It is found that the proposed framework provides a reasonable center-outward rank and the new methods have accurate decoding in one neural spike train dataset.