Multivariate Spatio-Temporal Neural Hawkes Processes
| Authors | Christopher Chukwuemeka et al. |
| Year | 2026 |
| Field | Statistics / ML |
| arXiv | 2602.23629 |
| Download | |
| Categories | stat.ML, cs.LG, stat.AP, stat.ME |
Abstract
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.
Engineering Breakdown
Plain English
This paper proposes a neural Hawkes process model that simultaneously captures temporal dynamics and spatial structure in multivariate event data—situations where events occur at different times and locations and can influence each other. Traditional Hawkes processes use fixed triggering kernels that define how past events influence future ones, but this approach learns both temporal decay and spatial decay patterns directly from data without predefined kernel shapes. The authors identify a gap in how previous deep learning-based Hawkes models capture fitted intensity behavior beyond just likelihood metrics, and their spatio-temporal extension addresses this by integrating spatial information into the latent state evolution. Simulation studies demonstrate the model successfully recovers both temporal and spatial intensity structure, suggesting it can capture realistic excitation and inhibition patterns in complex real-world event sequences.
Core Technical Contribution
The core novelty is extending neural Hawkes processes from purely temporal models to joint spatio-temporal models by learning adaptive spatial and temporal decay dynamics within the latent state evolution, eliminating the need for predefined triggering kernels. Rather than using fixed functional forms (like exponential or Hawkes kernels), the model learns these decay patterns end-to-end via neural networks, enabling flexible modeling of how events at different locations and times influence each other. The authors explicitly identify and address a modeling gap: previous deep learning-based Hawkes processes optimize likelihood well but fail to capture the actual fitted intensity function behavior, which this approach remedies through explicit spatial-temporal architecture. This represents a shift from kernel-based to learned-dynamics approaches in point process modeling, similar to how RNNs replaced hand-crafted features in other domains.
How It Works
The model takes a sequence of events with timestamps and spatial coordinates as input. For each event, it maintains a latent state representation that captures the current 'intensity' of the event process—essentially a learned summary of how likely future events are. When a new event arrives at time t and location s, the model updates its latent state by combining information about all previous events, weighted by learned temporal decay functions (closer events in time have more influence, but the decay pattern is learned) and learned spatial decay functions (events near the current location have more influence, again learned non-parametrically). The intensity at any point in space-time is computed from this latent state through a neural network, which outputs the conditional probability of an event occurring. During training, the model optimizes the log-likelihood of observed events while minimizing a regularization term that encourages reasonable spatial and temporal decay patterns, effectively learning both when and where events are likely to occur based on historical context.
Production Impact
This approach enables systems to model complex, spatially-distributed event sequences in domains like crime prediction, earthquake aftershock modeling, network intrusions, social media cascades, and disease spread—situations where event locations and times are both critical. Engineers adopting this would replace hand-crafted spatial-temporal features with end-to-end learned representations, reducing feature engineering overhead and potentially improving prediction accuracy through more flexible dynamics. However, this introduces significant computational cost: maintaining and updating latent states for each event in a streaming setting requires GPU acceleration for large-scale deployment, and inference latency increases with sequence length (the model must process all prior events to predict the next one). Production systems would need to implement windowing strategies (only looking back so far) and potentially train separate models per region to manage computational complexity; the trade-off is improved spatial-temporal understanding at the cost of higher infrastructure requirements than simpler methods like separate temporal Hawkes processes per region.
Limitations and When Not to Use This
The paper does not address scalability to extremely long sequences or very high-dimensional spatial domains; the mechanism for handling thousands of simultaneous locations or year-long event sequences remains unclear, and training time complexity versus sequence length is not reported. The approach assumes events can be meaningfully modeled as a single point process—it may struggle with events that have duration, spatial extent, or hierarchical structure, and the paper provides no guidance on how to discretize continuous space or how sensitive results are to spatial resolution choices. The simulation studies use synthetic data with known ground truth, which is substantially easier than real-world validation where true intensity functions are unobservable; no results on real datasets (crime, seismic, social) are presented, limiting confidence in practical effectiveness. Additionally, the paper identifies a modeling gap in previous work but doesn't quantitatively demonstrate that their approach actually closes it—no direct comparisons showing improved intensity function reconstruction versus baselines are provided.
Research Context
This work extends continuous-time neural point processes (particularly neural Hawkes processes pioneered by Mei et al. and Hawkes et al.) by adding explicit spatial structure, following recent trends in spatio-temporal deep learning that combine graph neural networks or attention mechanisms with temporal models. It builds on the recognition that Hawkes processes are powerful but rigid due to fixed kernels, motivating the shift toward learned dynamics seen in models like Neural Point Processes and Transformer Hawkes. The paper contributes to a broader research direction treating point processes as flexible intensity modeling problems rather than as kernel-based statistical models, aligning with deep learning's philosophy of learning functional forms from data. This opens opportunities for future work on combining these spatio-temporal Hawkes processes with graph structure (for networked events), multi-task learning (different regions with shared dynamics), and online learning (adapting to non-stationary patterns in production systems).
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