Spatiotemporal Hawkes Models

• Spatiotemporal Hawkes models are point processes that model self-exciting events by using a conditional intensity function combining a baseline rate with a triggering kernel.
• They leverage flexible parametric and nonparametric kernels, efficient discretization on a uniform grid, and precomputed tensors to achieve scalable and accurate inference.
• Real-world applications include earthquake aftershock analysis, disease spread tracking, and social network dynamics, often outperforming traditional ETAS models.

A spatiotemporal Hawkes model is a point process on space and time, designed to capture self-exciting phenomena where the occurrence of events increases the rate of future events in their spatiotemporal neighborhood. These models are essential in domains such as seismology, epidemiology, finance, criminology, and social sciences, where events exhibit complex clustering and triggering behaviors both temporally and spatially. Mathematically, such a process is characterized by a conditional intensity function which depends on a baseline background rate and a triggering kernel encoding the influence of past events in space and time.

1. Mathematical Formulation

Let $\mathcal S \subset \mathbb R^2$ be a compact spatial domain and $[0,T]$ a finite time interval. For a multivariate process with $D$ event types, the conditional intensity for events of type $i$ at location $(x,y)$ and time $t$—given history $\mathcal H_t$—takes the general form: $$\lambda_i(x, y, t \mid \mathcal{H}_t) = \mu_i + \sum_{j=1}^D \sum_{u_n^j \in \mathcal{H}_t^j} \alpha_{ij}\, g_{ij}(x-x_n^j, y-y_n^j, t-t_n^j)$$ where:

• $\mu_i > 0$ is the baseline (background) rate for type $i$,
• $[0,T]$0 quantifies the strength by which type $[0,T]$1 events excite type $[0,T]$2,
• $[0,T]$3 is the normalized spatiotemporal triggering kernel, often constructed as a product of spatial and temporal components, with finite support enforced for computational tractability (Siviero et al., 2024).

For compactly supported kernels,

$[0,T]$4

with $[0,T]$5 as spatial and temporal support widths. Kernels can be parametric (e.g., Gaussian, truncated power law, exponential, Kumaraswamy) or even nonparametric (Siviero et al., 2024, Kwon et al., 2022).

2. Discretization and Efficient Inference

To enable tractable inference, the space-time domain is discretized onto a uniform grid,

$[0,T]$6

with event data projected to their nearest grid cell. The discrete intensity is then expressed via convolution with the sampled kernel: $[0,T]$7 where $[0,T]$8 is the kernel sampled at grid steps, and $[0,T]$9 is the count of past events of type $D$0 in each cell.

The inference is based on minimizing an $D$1 loss (least squares) over the discretized intensity: $D$2 which provides unbiased parameter estimation as the grid size $D$3 tends to zero, with discretization error decaying linearly in $D$4 (Siviero et al., 2024).

For computational gains, all terms in the loss that do not depend on unknown parameters are precomputed (major efficiency advantages for large-scale data). This enables rapid, closed-form gradient evaluations and efficient optimization via gradient descent or quasi-Newton algorithms.

3. Model Flexibility and Kernel Choices

Parametric kernels used in spatiotemporal Hawkes processes are chosen for both expressiveness and computational properties. Common specifications include:

• Spatial kernels: truncated Gaussian ($D$5), truncated power-law ($D$6)
• Temporal kernels: exponential ($D$7), Kumaraswamy, truncated Gaussian
• Separable product: $D$8 (Siviero et al., 2024)

These flexible kernels outperform fixed-form separable kernels (e.g., Gaussian-exponential) typical of classical ETAS (Epidemic-Type Aftershock Sequence) models, especially in real data such as earthquake aftershocks.

4. Precomputation and Scalability

Through the use of precomputed tensors (such as the summary statistics $D$9 defined on the discrete grid), highly efficient evaluation of the loss and its gradients is possible. In cases where the double-sum structure of the quadratic terms would otherwise be prohibitive ($i$0), an approximation

$i$1

reduces the cost to $i$2, with negligible bias for large enough $i$3 (Siviero et al., 2024).

This framework enables fitting spatiotemporal Hawkes models to large-scale datasets with high statistical and computational efficiency, supporting rapid convergence even in nonconvex settings.

5. Empirical Performance and Real-World Applications

Synthetic Experiments

• As grid steps $i$4, estimation error in kernel parameters decays linearly with $i$5
• For fixed discretization, statistical error decreases at rate $i$6 as the number of events increases
• Approximating precomputed tensors yields order-of-magnitude computational speedups with vanishing error as data size grows (Siviero et al., 2024).

Real Data: Earthquake Catalogs

• Applying the method to earthquake aftershocks (Northern California dataset), best held-out log-likelihoods are obtained by tailoring spatial (e.g., truncated Gaussian vs power-law) and temporal (exponential, Gaussian, Kumaraswamy) kernels to seismic catalogs
• Outperforms standard ETAS and classical Hawkes fits with rigid kernel choices

Summary Table: Computational Advantages

Aspect	Standard Approaches	Flexible Discretized Inference (Siviero et al., 2024)
Kernel Form	Fixed (Gaussian/Exponential)	Any finite-support, differentiable kernel
Loss/Objective	Likelihood (costly)	$i$7 loss (fast, closed-form gradients)
Grid Requirement	Not explicit	Uniform grid, steps $i$8
Precomputation	Rare	$i$9, $(x,y)$0 tensors, approximable
Scaling	$(x,y)$1 (likelihood sum)	$(x,y)$2 (linear in grid and kernel size)

6. Extensions and Broader Implications

The described methodology supports a wide class of spatiotemporal point-process models, including multivariate settings, arbitrary compactly-supported differentiable kernels, and nonstationary or cross-type excitation structures. Its modularity in kernel specification and ability to leverage extensive precomputation lends itself to broad application in domains such as seismology, epidemiology, criminology, and finance (Siviero et al., 2024). This framework substantially extends prior inference schemes and represents a state-of-the-art template for statistically rigorous, flexible, and computationally tractable inference in spatiotemporal Hawkes modeling.