Temporal Point Processes

Networks of timestamped interactions arise across social, financial, and biological domains, where forecasting future events requires modeling both evolving topology and temporal ordering. Temporal link prediction methods typically frame the task as binary classification with negative sampling, discarding the sequential and correlated nature of real-world interactions. We introduce STEP (STochastic Event Predictor), a framework that reformulates temporal link prediction as a sequential forecasting problem in continuous time. STEP models event dynamics through discrete temporal motif transitions governed by Poisson processes, maintaining a set of open motif instances that evolve as new interactions arrive. At each step, the framework decides whether to initiate a new temporal motif or extend an existing one, selecting the most probable event via Bayesian scoring of temporal likelihoods and structural priors. STEP also produces compact, temporal motif-based feature vectors that can be concatenated with existing temporal graph neural network outputs, enriching their representations without architectural modifications. Experiments on five real-world datasets demonstrate up to 21% average precision gains over state-of-the-art baselines in classification and 0.99 precision in next $k$ sequential forecasting, with consistently lower runtime than competing motif-aware methods.

Cox processes model overdispersed point process data via a latent stochastic intensity, but both nonparametric estimation of the intensity model and posterior inference over intensity paths are typically intractable, relying on expensive MCMC methods. We introduce Neural Diffusion Intensity Models, a variational framework for Cox processes driven by neural SDEs. Our key theoretical result, based on enlargement of filtrations, shows that conditioning on point process observations preserves the diffusion structure of the latent intensity with an explicit drift correction. This guarantees the variational family contains the true posterior, so that ELBO maximization coincides with maximum likelihood estimation under sufficient model capacity. We design an amortized encoder architecture that maps variable-length event sequences to posterior intensity paths by simulating the drift-corrected SDE, replacing repeated MCMC runs with a single forward pass. Experiments on synthetic and real-world data demonstrate accurate recovery of latent intensity dynamics and posterior paths, with orders-of-magnitude speedups over MCMC-based methods.

Longitudinal electronic health record (EHR) data are often left-censored, making diagnosis records incomplete and unreliable for determining disease onset. In contrast, outpatient prescriptions form renewal-based trajectories that provide a continuous signal of disease management. We propose a probabilistic framework to infer chronic treatment onset by modeling prescription dynamics as a renewal process and detecting transitions from sporadic to sustained therapy via change-point detection between a baseline Poisson (sporadic prescribing) regime and a regime-specific Weibull (sustained therapy) renewal model. Using a nationwide ePrescription dataset of 2.4 million individuals, we show that the approach yields more temporally plausible onset estimates than naive rule-based triggering, substantially reducing implausible early detections under strong left censoring. Detection performance varies across diseases and is strongly associated with prescription density, highlighting both the strengths and limits of treatment-based onset inference.

We propose a Multivariate Spatio-Temporal Neural Hawkes Process for modeling complex multivariate event data with spatio-temporal dynamics. The proposed model extends continuous-time neural Hawkes processes by integrating spatial information into latent state evolution through learned temporal and spatial decay dynamics, enabling flexible modeling of excitation and inhibition without predefined triggering kernels. By analyzing fitted intensity functions of deep learning-based temporal Hawkes process models, we identify a modeling gap in how fitted intensity behavior is captured beyond likelihood-based performance, which motivates the proposed spatio-temporal approach. Simulation studies show that the proposed method successfully recovers sensible temporal and spatial intensity structure in multivariate spatio-temporal point patterns, while existing temporal neural Hawkes process approach fails to do so. An application to terrorism data from Pakistan further demonstrates the proposed model's ability to capture complex spatio-temporal interaction across multiple event types.

Modeling asynchronous event sequences is crucial in numerous real-world applications such as healthcare monitoring, financial transaction analysis, and so on. Traditional temporal point processes, including Hawkes Processes, often fail to capture complex dependencies due to their parametric limitations. While neural approaches like RNNs and Transformers have improved flexibility, they struggle with computational inefficiency, and attention saturation. In this paper, we introduce the Mamba Hawkes Process (MHP), the first framework to integrate selective state space model (Mamba) with temporal point processes. MHP leverages time-varying state transitions and input-dependent gating to efficiently encode event history and capture long-term dependencies with linear complexity. Importantly, we provide theoretical guarantees showing that MHP generalizes both classical multi-exponential Hawkes processes and exponential-decay gated RNNs, underscoring its expressive power and theoretical soundness. To address the inherent constraints of pure state space models in handling heterogeneous event interactions, we further develop Adaptive Mamba Hawkes Process (A-MHP) that incorporates two novel mechanisms: a Time-Scaling Mechanism that adaptively weights time intervals based on event type and history, and a Dual-Channel State Transition that adaptively processes event content and temporal dynamics for more refined state updates. Extensive experiments on synthetic and realworld datasets demonstrate that MHP and A-MHP consistently outperform state-of-the-art baselines in event prediction tasks, particularly in long-sequence scenarios. Our work establishes a scalable and theoretically grounded paradigm for event sequence modeling, with practical implications for predictive maintenance, anomaly detection, and dynamic system analysis.