A unified framework on defining depth for point process using function smoothing

TL;DR

This paper introduces a unified framework for defining depth in point processes by transforming them into functions through smoothing, enabling better ranking and analysis of such data.

Contribution

It proposes a novel smoothing-based approach to unify the randomness in point processes and applies functional depth concepts for improved analysis.

Findings

The proposed depth effectively ranks point process observations.

The method demonstrates good performance in classification tasks.

Mathematical properties and asymptotics of the depth are established.

Abstract

The notion of statistical depth has been extensively studied in multivariate and functional data over the past few decades. In contrast, the depth on temporal point process is still under-explored. The problem is challenging because a point process has two types of randomness: 1) the number of events in a process, and 2) the distribution of these events. Recent studies proposed depths in a weighted product of two terms, describing the above two types of randomness, respectively. In this paper, we propose to unify these two randomnesses under one framework by a smoothing procedure. Basically, we transform the point process observations into functions using conventional kernel smoothing methods, and then adopt the well-known functional $h$-depth and its modified, center-based, version to describe the center-outward rank in the original data. To do so, we define a proper metric on the point processes with smoothed functions. We then propose an efficient algorithm to estimated the defined "center". We further explore the mathematical properties of the newly defined depths and study asymptotics. Simulation results show that the proposed depths can properly rank the point process observations. Finally, we demonstrate the new method in a classification task using a real neuronal spike train dataset.