Deep spatio-temporal point processes: Advances and new directions (2504.06364v1)

Abstract: Spatio-temporal point processes (STPPs) model discrete events distributed in time and space, with important applications in areas such as criminology, seismology, epidemiology, and social networks. Traditional models often rely on parametric kernels, limiting their ability to capture heterogeneous, nonstationary dynamics. Recent innovations integrate deep neural architectures -- either by modeling the conditional intensity function directly or by learning flexible, data-driven influence kernels, substantially broadening their expressive power. This article reviews the development of the deep influence kernel approach, which enjoys statistical explainability, since the influence kernel remains in the model to capture the spatiotemporal propagation of event influence and its impact on future events, while also possessing strong expressive power, thereby benefiting from both worlds. We explain the main components in developing deep kernel point processes, leveraging tools such as functional basis decomposition and graph neural networks to encode complex spatial or network structures, as well as estimation using both likelihood-based and likelihood-free methods, and address computational scalability for large-scale data. We also discuss the theoretical foundation of kernel identifiability. Simulated and real-data examples highlight applications to crime analysis, earthquake aftershock prediction, and sepsis prediction modeling, and we conclude by discussing promising directions for the field.

• The paper shows how deep learning, particularly deep kernels, enhances spatio-temporal point process models to capture complex, nonstationary event dynamics.
• Empirical studies demonstrate these models' effectiveness in real-world applications like earthquake prediction and crime analysis, achieving enhanced predictive accuracy and interpretable kernel estimates.
• Theoretical insights include kernel identifiability and computational efficiency techniques, while future directions involve integrating uncertainty quantification and causal inference.

Deep Spatio-temporal Point Processes: Advances and New Directions

The reviewed paper, "Deep Spatio-temporal Point Processes: Advances and New Directions," by Xiuyuan Cheng, Zheng Dong, and Yao Xie, explores the development of deep influence kernel approaches in spatio-temporal point processes (STPPs). These models are pivotal for handling discrete events occurring in temporal and spatial contexts, with applications spanning criminology, seismology, epidemiology, and social networks. Traditional STPP models rely on parametric kernels, which often fall short of capturing the complexity inherent in nonstationary and heterogeneous real-world data. This paper articulates how incorporating deep learning architectures enhances the expressive power of STPPs through a non-parametric and interpretable modeling framework.

Overview and Contributions

The paper begins by reviewing classical STPP models, which typically utilize a self-exciting structure derived from the Hawkes process. These models traditionally employ simple parametric forms, such as exponential decay kernels, to ensure tractability and interpretability. Despite their convenience, these kernels assume stationarity and monotonicity, which limit their applicability to complex datasets with nonstationary influences.

In addressing these limitations, the paper introduces deep learning–based methodologies that utilize the representational capabilities of neural networks. These approaches fall into two main categories:

1. Direct Intensity Modeling: This approach leverages autoregressive neural architectures, including recurrent neural networks (RNNs) and self-attention mechanisms, to model the conditional intensity function directly. These models can capture intricate temporal dependencies within event sequences but often do so at the cost of interpretability.
2. Kernel-based Modeling: Maintaining the lineage of the Hawkes process, this methodology focuses on generalizing and learning the kernel function itself through flexible, non-parametric representations. By leveraging neural architectures, these deep kernels provide a more transparent model of how past events influence future occurrences and offer robust modeling capacity for high-dimensional and nonstationary dynamics.

The authors propose a low-rank kernel decomposition schema, exploiting the theoretical groundwork provided by Mercer's theorem. This approach allows for efficient approximation of functional data, capturing nonstationary processes through a combination of basis functions parameterized by neural networks. This formulation facilitates the modeling of spatio-temporal processes and can be generalized for graphs using graph neural network (GNN) paradigms.

Key Results and Theoretical Insights

The paper provides empirical validation through multiple case studies, demonstrating the applicability and effectiveness of deep kernel approaches in real-world applications. For instance, in the context of earthquake prediction and crime dynamics analysis, these models deliver enhanced predictive accuracy and interpretable kernel estimates, exhibiting superior performance over traditional methods and simpler neural architectures.

Theoretical contributions include:

• Kernel Identifiability: The paper discusses theoretical aspects of kernel identifiability, emphasizing conditions where unique maximum likelihood solutions exist. This is vital for ensuring model robustness and reliability in practical applications.
• Computational Efficiency: Emphasis is placed on techniques to enhance computation, such as the adoption of log-barrier methods in maximum likelihood estimation to facilitate scalable learning from large datasets.

Practical and Theoretical Implications

The capability to model complex, non-linear interactions via deep kernels promises substantial practical benefits across domains. For urban security, these models can reveal insights into crime contagion dynamics, allowing for more informed resource allocation in patrolling. In seismology or epidemiology, understanding nonstationary patterns in aftershock or epidemic spreads can significantly improve forecasting and intervention strategies.

Looking forward, the integration of uncertainty quantification, causal inference, and hybrid models combining multiplicative and additive effects are outlined as promising research directions. Such endeavors could further enhance the robustness and interpretability of these models, opening avenues for broader application and theoretical exploration in AI and beyond.

In conclusion, this paper provides a comprehensive overview of the advances in deep-kernel STPPs, effectively articulating the transition towards more expressive and interpretable spatio-temporal models. While challenges remain, the methodologies discussed present exciting steps towards richer modeling paradigms capable of capturing complex event dynamics in diverse applications.