Product Depth for Temporal Point Processes Observed Only Up to the First k Events

TL;DR

This paper introduces a new product depth measure for temporal point processes observed only up to the first k events, enabling better analysis of their centrality and distribution.

Contribution

The paper develops a novel product depth function tailored for TPPs observed up to the first k events, with theoretical properties and practical applications.

Findings

The proposed depth captures the temporal distribution of the final event.

It effectively characterizes the joint distribution of preceding events.

Simulation and real data demonstrate its utility.

Abstract

Temporal point processes (TPPs) model the timing of discrete events along a timeline and are widely used in fields such as neuroscience and fi- nance. Statistical depth functions are powerful tools for analyzing centrality and ranking in multivariate and functional data, yet existing depth notions for TPPs remain limited. In this paper, we propose a novel product depth specifically designed for TPPs observed only up to the first k events. Our depth function comprises two key components: a normalized marginal depth, which captures the temporal distribution of the final event, and a conditional depth, which characterizes the joint distribution of the preceding events. We establish its key theoretical properties and demonstrate its practical utility through simulation studies and real data applications.