Multivariate Spatio-Temporal Neural Hawkes Processes

Christopher Chukwuemeka; Hojun You; Mikyoung Jun

stat.MLFeb 27, 2026

Multivariate Spatio-Temporal Neural Hawkes Processes

Authors: Christopher Chukwuemeka, Hojun You, Mikyoung Jun

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.

Figures & tables

Panel of Figure 1 — Fig. 1: (a): Location of terrorist attacks by four major groups in Pakistan during 2008-2020. (b): Daily counts of attacks for the same data.

Fig. 2: True intensity for a simulated data example used to compare six different neural network approaches with ( 1 ) and ( 2 ).

Panel of Figure 3 — Fig. 3: Fitted conditional intensity for the simulation data with six methods in EasyTPP. Each model is trained until convergence based on validation log-likelihood.

Panel of Figure 4 — Fig. 4: Architecture of the Proposed MSTNHP. The model processes spatio-temporal events $(t_{i},\mathbf{s}_{i})$ through a continuous-time LSTM, where intensities $\lambda_{k}$ and baselines jump at event times and decay smoothly between them.

Parameters	Biv 1	Biv 2	Biv 3
$\alpha_{11},\alpha_{12},$	0.25, 0.1,	0.25, -0.1,	0.25, 0.1,
$\alpha_{21},\alpha_{22}$	0.1, 0.25	-0.1, 0.25	0.1, 0.25
$\beta_{11},\beta_{12},$	0.3, 0.3,	0.3, 0.3,	0.1, 0.1,
$\beta_{21},\beta_{22}$	0.3, 0.3	0.3, 0.3	0.1, 0.1
$\sigma_{11}^{2},\sigma_{12}^{2},$	0.5, 0.5,	0.5, 0.5,	0.5, 0.25,
$\sigma_{21}^{2},\sigma_{22}^{2}$	0.5, 0.5	0.5, 0.5	0.25, 0.5

TABLE I: Parameter values for the separable triggering structure in ( 7 )

Dataset	# Param.	# Event Tokens			Sequence Length
		Train	Valid	Test	Min	Mean	Max
Biv 1	20,960	86,520	4,692	4,900	60	97	134
Biv 2	20,960	77,106	4,181	4,304	59	87	119
Biv 3	20,960	87,313	4,726	4,944	60	97	135
Biv 4	82,880	133,615	7,131	7,693	88	149	242

TABLE II: Statistics of simulated datasets for biv 1–4

Panel of Figure 7 — Fig. 5: True vs. Fitted temporal intensities for simulated datasets using parameter configurations from Table I .

Panel of Figure 8 — Fig. 6: Cumulative time-averaged spatial intensity maps at $\tau=100$ for the four bivariate simulation settings (Row 1–4 represents Biv 1–4 respectively). For each setting, columns 1 and 3 display the true spatial intensities, while columns 2 and 4 show the fitted counterparts for the two event types. Intensities are averaged over the entire observation window [0,100]. Cumulative maps at horizons $\tau\in$ {20,40,60,80,100 $\}$ are reported in section S.2 of the supplementary material.

Panel of Figure 9 — Fig. 7: Fitted temporal intensities for the simulated datasets as used in Figure 5 , from MTNHP method.

Panel of Figure 10 — Fig. 8: Fitted temporal conditional intensity functions for Pakistan terrorism data. (a) is for MSTNHP and (b) for MTNHP. For (b), some colors are displayed with dotted lines for legibility.

Panel of Figure 11 — Fig. 9: Fitted spatial conditional intensity maps for selected dates in Pakistan in 2014. Left: BRA/BLA. Right: TTP/BLF. Markers indicate observed attack locations on the day: BRA ( $\times$ , red), BLA ( $\diamond$ , yellow), TTP ( $\circ$ , orange), BLF ( $\triangle$ , magenta). Per-frame color scaling is applied.

Model	Hyperparameters	Value
RMTPP	num_layers	2
	hidden_size	32
	time_emb_size	16
NHP	num_layers	2
	hidden_size	64
	time_emb_size	16

TABLE S1: Hyperparameter settings for trainining temporal point process (TPP) models.

MModel	Trainable Parameters	Runtime (min)
RMTPP	2,342	0.651
NHP	58,112	47.173
ODETPP	1,218	645.758
SAHP	6,514	0.996
THP	25,282	0.956
AttNHP	11,202	22.321

TABLE S2: Comparison of model complexity and training runtime for the temporal point process models. Number of trainable parameters and the total wall clock time required to complete 2,000 training epochs using the hyperparameter configurations in Table S1 are reported.

Panel of Figure 17 — Fig. S1: Log-likelihood values of training, testing, validation data sets for six temporal point process models implemented in EasyTPP.

Panel of Figure 18 — Fig. S2: SAHP runs under different training budgets. Left: Right: the evolution of the log-likelihood as a function of epoch. Right: fitted temporal conditional intensity for a representative sequence.

Panel of Figure 19 — Fig. S3: Convergence of the log-likelihood for the MSTNHP approach.

Panel of Figure 20 — Fig. S4: Cumulative time-averaged spatial intensity maps for Biv 1. Columns 1 and 3 give true spatial intensities, columns 2 and 4 corresponding MSTNHP-fitted intensities for each event type. Each panel displays the average spatial intensity accumulated from $t=0$ up to the indicated time horizon (i.e., a cumulative moving average over $[0,t]$ ).

Panel of Figure 21 — Fig. S5: Same as Figure S4 except for Biv 2.

Panel of Figure 22 — Fig. S6: Same as Figure S4 except for Biv 3.

Panel of Figure 23 — Fig. S7: Same as Figure S4 except for Biv 4.

Panel of Figure 24 — Fig. S8: MTNHP results for Biv 1–4 datasets. All models were trained for 2k epochs with 58,112 trainable parameters (64 hidden units), whereas the corresponding MSTNHP models used 32 hidden units for Biv 1–3 and 64 hidden units for Biv 4.

Panel of Figure 25 — Fig. S9: True and fitted temporal intensity from MTNHP with data simulated under Biv 1 model, except for shorter spatial triggering distance (i.e., $\sigma^{2}=10^{-4}$ ).

Fig. S10: Log-likelihoods against epoch for the Biv 1 simulation with

\sigma^{2}=10^{-4}

. Validation likelihood has its maximum at epoch=85.

Panel of Figure 27 — Fig. S11: Log-likelihood values for MSTNHP and MTNHP for the Pakistan terrorism data.

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