Distribution-Free Conformal Joint Prediction Regions for Neural Marked Temporal Point Processes

TL;DR

This work tackles uncertainty quantification for neural marked temporal point processes by applying distribution-free conformal prediction to construct joint prediction regions for the next event's arrival time $\tau$ and its mark $k$ with finite-sample marginal coverage. It introduces two principal approaches: a naive method that combines univariate CP regions and a sharper Conformal HDR (C-HDR) strategy that leverages the joint predictive density $\hat{f}(\tau,k|\boldsymbol{h})$ to exclude unlikely combinations and improve sharpness, with an asymptotic guarantee on conditional coverage if estimators are consistent. The study also develops univariate CP methods for time (C-QR/C-QRL/C-HDR-left) and for marks (RAPS/APS with conformal calibration and C-PROB), and evaluates them on real-world datasets plus synthetic Hawkes data using neural TPP models like CLNM, FNN, RMTPP, and SAHP. Empirical results show that HDR-based joint regions outperform naive combinations by accounting for interdependencies, achieving reliable marginal coverage and competitive conditional coverage, thereby enabling sharper and more informative forecasts for sequential events. The findings advance uncertainty quantification for mixed-type targets in neural TPPs and have practical implications for forecasting in domains such as healthcare, finance, and social data.

Abstract

Sequences of labeled events observed at irregular intervals in continuous time are ubiquitous across various fields. Temporal Point Processes (TPPs) provide a mathematical framework for modeling these sequences, enabling inferences such as predicting the arrival time of future events and their associated label, called mark. However, due to model misspecification or lack of training data, these probabilistic models may provide a poor approximation of the true, unknown underlying process, with prediction regions extracted from them being unreliable estimates of the underlying uncertainty. This paper develops more reliable methods for uncertainty quantification in neural TPP models via the framework of conformal prediction. A primary objective is to generate a distribution-free joint prediction region for an event's arrival time and mark, with a finite-sample marginal coverage guarantee. A key challenge is to handle both a strictly positive, continuous response and a categorical response, without distributional assumptions. We first consider a simple but conservative approach that combines individual prediction regions for the event's arrival time and mark. Then, we introduce a more effective method based on bivariate highest density regions derived from the joint predictive density of arrival times and marks. By leveraging the dependencies between these two variables, this method excludes unlikely combinations of the two, resulting in sharper prediction regions while still attaining the pre-specified coverage level. We also explore the generation of individual univariate prediction regions for events' arrival times and marks through conformal regression and classification techniques. Moreover, we evaluate the stronger notion of conditional coverage. Finally, through extensive experimentation on both simulated and real-world datasets, we assess the validity and efficiency of these methods.

Figures (40)

• Figure 1: Examples of marked events sequences. The vertical lines represent the arrival times with the colors representing the marks.
• Figure 2: Toy illustration of a joint prediction region constructed from the joint density of arrival times and marks. The three colored curves represent predictive density functions while the horizontal bars represent prediction intervals.
• Figure 3: Illustration of a MCIF for a Hawkes process with two marks. This example highlights how events can influence the occurrence of future events. For the sake of simplicity, this illustration assumes independence between events of different marks.
• Figure 4: Illustration of the input-output pairs $\mathcal{D} = \Set{(\boldsymbol{h}_{i}, \bm{e}_{i})}_{i=1}^n$ and the joint prediction region we aim to construct for $\boldsymbol{e}_{n+1}$ given a new input $\boldsymbol{h}_{n+1}$.
• Figure 5: Example of joint bivariate prediction regions with $\alpha = 0.4$ on a synthetic example with $\tau \in \mathbb{R}^+$ and marks $\mathbb{K} = \{k_1, k_2, k_3\}$.