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

TL;DR

This paper introduces a Product Depth for temporal point processes observed only up to the first $k$ events in an unbounded time domain. It defines depth as the product of a marginal depth on the last event time $S_k$ and a conditional depth on the preceding times, with explicit components $D(\boldsymbol{s}_k;P_{\mathbf{S}_k}) = \omega(s_k;P_{S_k})^{\frac{|s_k-\eta|}{M-s_0}} D_c(\boldsymbol{s}_k;P_{\boldsymbol{S}_k|S_k})$. The paper derives four desirable properties (continuity, vanishing at boundary and infinity, maximality at a center $\Theta_k$, scale/shift invariance, and monotonicity) and provides sample-based implementations, including a Poisson process specialization where the conditional part uses a Dirichlet-like structure. Through simulation studies on homogeneous Poisson and state-dependent processes, the authors show improved centrality ranking and boundary behavior over the general Mahalanobis depth, and demonstrate the method on real data from cell division and 40m sprint tests, yielding interpretable rankings that reflect both global timing and internal spacing. The work highlights potential extensions to classification, clustering, and outlier detection in temporally ordered data.

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.

Figures (12)

• Figure 1: The contour curves of the Product Depth and the general Mahalanobis Depth in HPP, where $\lambda=2$ and $k=2$; The blue number indicates the rank of the point based on the corresponding depth measure; The ranks of the points are 76, 83, 74, 77, 86, 80, 70, 84, 85, 89, 82, 81, 93 (left plot), and 34, 38, 49, 56, 58, 61, 64, 65, 67, 69, 72, 75, 76 (right plot).
• Figure 2: The contour curves of the Product Depth and the general Mahalanobis Depth, where $(\lambda_1,\lambda_2) =(2.5,10)$ and $k=2$; The blue number indicates the rank of the point based on the corresponding depth measure; The ranks of the points are 60, 61, 63, 64, 66, 72, 77, 79 (left plot), and 90, 92, 75, 73, 77, 80, 84, 93 (right plot).
• Figure 3: Left: Scatter plot of the HPP, where $\lambda=2$ and $k=2$; Middle: Scatter plot after ranking based on Product Depth; Scatter plot after ranking based on general Mahalanobis Depth. In all plots, black points correspond to the first event time $S_1$, and red points correspond to the second event time $S_2$. The black and red vertical lines represent the means of $S_1$ and $S_2$, respectively.
• Figure 4: Left: Scatter plot of the data, where $(\lambda_1,\lambda_2) =(2.5,10)$ and $k=2$; Middle: Scatter plot after ranking based on Product Depth; Scatter plot after ranking based on general Mahalanobis Depth. In all plots, black points correspond to the first event time $S_1$, and red points correspond to the second event time $S_2$. The black and red vertical lines represent the means of $S_1$ and $S_2$, respectively.
• Figure 5: The sample images, each of size $321 \times 321$, are taken from time points 1, 23, 195, and 259 in the folder E00 from celltracking.bio.nyu.edu. The first, second, and third cell division events occur at time points 22, 194, and 258, respectively.

Theorems & Definitions (6)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Lemma 1
• Lemma 2