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

2025-11-24

AI Summary

Existing statistical depth functions for temporal point processes (TPPs) lack theoretical and practical support for truncated observations—i.e., sequences containing only the first *k* events. Method: This paper introduces the first product-depth function tailored to truncated TPPs, jointly modeling a normalized marginal depth (characterizing the distribution of the final event time) and a conditional depth (capturing the joint dependence structure among earlier events). Contribution/Results: The proposed depth is theoretically rigorous: we formally prove its affine invariance, monotonicity with respect to the center, and continuity. It enables central sequence identification, ordering, and outlier detection—even for incomplete sequences. Extensive experiments on synthetic and real-world datasets demonstrate that our method significantly outperforms existing approaches in estimating central trends and detecting anomalous sequences, exhibiting robust performance across diverse settings.

Technology Category

• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Security and Privacy: Data transparency and provenance

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.

Problem

• Develops product depth for temporal point processes
• Analyzes centrality with limited event observations
• Combines marginal and conditional depth components

Innovation

• Product depth for temporal point processes
• Normalized marginal depth for final event
• Conditional depth for preceding events