cs.LGMar 24, 2026

Kronecker-Structured Nonparametric Spatiotemporal Point Processes

Authors: Zhitong Xu, Qiwei Yuan, Yinghao Chen, Yan Sun, Bin Shen, Shandian Zhe

Organizations: Kahlert School of Computing, The University of Utah · College of Arts & Sciences, Utah State University · Celonis AI

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Abstract

Events in spatiotemporal domains arise in numerous real-world applications, where uncovering event relationships and enabling accurate prediction are central challenges. Classical Poisson and Hawkes processes rely on restrictive parametric assumptions that limit their ability to capture complex interaction patterns, while recent neural point process models increase representational capacity but integrate event information in a black-box manner, hindering interpretable relationship discovery. To address these limitations, we propose a Kronecker-Structured Nonparametric Spatiotemporal Point Process (KSTPP) that enables transparent event-wise relationship discovery while retaining high modeling flexibility. We model the background intensity with a spatial Gaussian process (GP) and the influence kernel as a spatiotemporal GP, allowing rich interaction patterns including excitation, inhibition, neutrality, and time-varying effects. To enable scalable training and prediction, we adopt separable product kernels and represent the GPs on structured grids, inducing Kronecker-structured covariance matrices. Exploiting Kronecker algebra substantially reduces computational cost and allows the model to scale to large event collections. In addition, we develop a tensor-product Gauss-Legendre quadrature scheme to efficiently evaluate intractable likelihood integrals. Extensive experiments demonstrate the effectiveness of our framework.

Figures & tables

 SYN1SYN2
STHP1.45e-01 ±2.41e−02\pm_{2.41e-02}8.42e-02 ±1.87e−02\pm_{1.87e-02}
NSTPP5.57e-02 ±6.50e−03\pm_{6.50e-03}2.99e-02 ±4.78e−03\pm_{4.78e-03}
DeepSTPP1.25e-01 ±1.89e−02\pm_{1.89e-02}7.77e-02 ±1.52e−02\pm_{1.52e-02}
NHP1.35e-01 ±3.08e−02\pm_{3.08e-02}2.34e-02 ±4.37e−03\pm_{4.37e-03}
THP8.14e-01 ±7.10e−02\pm_{7.10e-02}1.91e-01 ±2.87e−02\pm_{2.87e-02}
KSTPP4.44e-02 ±3.96e−03\pm_{3.96e-03}2.00e-02 ±3.50e−03\pm_{3.50e-03}
Table 1: Relative L2L_{2} error of the learned marginal intensity λ(t∣Ht)\lambda(t|{\mathcal{H}}_{t}) on synthetic datasets. The smallest error is shown in bold.
Figure 1: Temporal conditional intensity on example test sequence from SYN1 .
 SYN1SYN2
STHP1.77e-01 ±7.44e−02\pm_{7.44e-02}1.23e-01 ±4.73e−02\pm_{4.73e-02}
NSTPP8.68e-01 ±9.41e−03\pm_{9.41e-03}8.64e-01 ±3.26e−03\pm_{3.26e-03}
DeepSTPP3.78e-01 ±3.65e−02\pm_{3.65e-02}3.87e-01 ±2.47e−02\pm_{2.47e-02}
KSTPP6.52e-02 ±4.80e−03\pm_{4.80e-03}3.67e-02 ±1.43e−02\pm_{1.43e-02}
Table 2: Relative L2L_{2} error of the learned spatiotemporal intensity λ(t,x,y∣Ht)\lambda(t,x,y|{\mathcal{H}}_{t}) on synthetic datasets.
Figure 2: Spatial conditional intensity p(x,y∣t,Ht)=λ(t,x,y∣Ht)/λ(t∣Ht)p(x,y\mid t,{\mathcal{H}}_{t})=\lambda(t,x,y\mid{\mathcal{H}}_{t})/\lambda(t\mid{\mathcal{H}}_{t}) on an example test sequence from SYN1 . GT denotes the ground-truth.
Figure 3: Learned influence kernel from SYN1 .
 EarthquakeCOVID-19Citibike
ModelSpatial ↓\downarrowTemporal ↓\downarrowSpatial ↓\downarrowTemporal ↓\downarrowSpatial ↓\downarrowTemporal ↓\downarrow
RMTPP-0.424 ±\pm 0.009-1.32 ±\pm 0.024-2.07 ±\pm 0.015
NHP-1.86 ±\pm 0.023-2.13 ±\pm 0.100-2.36 ±\pm 0.056
THP-2.44 ±\pm 0.021-0.611 ±\pm 0.008-1.46 ±\pm 0.009
Poisson9.45 ±\pm 0.0000.412 ±\pm 0.0000.818 ±\pm 0.0000.113 ±\pm 0.0000.452 ±\pm 0.0000.239 ±\pm 0.000
STHP8.35 ±\pm 0.2520.424 ±\pm 0.0180.422 ±\pm 0.0000.100 ±\pm 0.0010.032 ±\pm 0.0000.633 ±\pm 0.126
Table 3: Root-mean-square error (RMSE) and Euclidean distance for next-event time and location prediction. The best two results are highlighted in bold.
Figure 4: Learned influence kernel by KSTPP on Earthquake .
Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Temporal conditional intensity on example test sequences from SYN2 .
Figure 6: Spatial conditional intensities on an example test sequence from SYN2 at different time points.
Figure 7: Learned influence kernel from SYN2 dataset.
DatasetKernelInducing Grid SizeQuadrature Grid SizeOptimizer / LR
EarthquakeSEF: 32×32×32F{:}\,32{\times}32{\times}32 ; G: 12×12G{:}\,12{\times}128×12×128{\times}12{\times}12Adam / 10−310^{-3}
COVID-19Matérn- 5/25/2F: 32×32×32F{:}\,32{\times}32{\times}32 ; G: 16×16G{:}\,16{\times}168×16×168{\times}16{\times}16Adam / 10−310^{-3}
CitibikeMatérn- 5/25/2F: 32×32×32F{:}\,32{\times}32{\times}32 ; G: 12×12G{:}\,12{\times}128×12×128{\times}12{\times}12Adam / 10−310^{-3}
Appendix
Table 4: Hyperparameters used by KSTPP on each real-world dataset.
Figure 8: Overall KSTPP model pipeline: (1) observed spatiotemporal event sequences; (2) conditional intensity with a background rate, aggregated influence from past events, and a softplus link function; (3a) background spatial GP and (3b) influence-kernel spatiotemporal GP, both represented on Kronecker-structured inducing grids; (4) intensity integration via a tensor-product Gauss–Legendre quadrature grid for the log-likelihood; (5) GP priors for regularization; (6) optimization of the posterior to obtain the trained KSTPP model; and (7) outputs, including predictive intensity, next-event prediction, and interpretable influence patterns.
Quadrature ResolutionDistanceTime RMSE
4×4×44{\times}4{\times}40.4450.103
8×8×88{\times}8{\times}80.4000.101
12×12×1212{\times}12{\times}120.3970.102
16×16×1616{\times}16{\times}160.3930.102
Appendix
Table 5: Effect of quadrature resolution on prediction error (COVID-19).
Inducing Grid ResolutionDistanceTime RMSE
4×4×44{\times}4{\times}40.3930.0986
8×8×88{\times}8{\times}80.3910.0985
12×12×1212{\times}12{\times}120.3910.0999
16×16×1616{\times}16{\times}160.3940.101
Appendix
Table 6: Effect of inducing-grid resolution on prediction error (COVID-19).
KernelDistanceTime RMSE
Matérn- 5/25/20.3920.100
SE0.3930.101
Appendix
Table 7: Effect of kernel choice on prediction error (COVID-19).
Seq. LengthTraining TimePrediction TimeTraining MemoryTest Memory
501.390.401.030.96
1002.431.181.721.44
1504.123.913.072.39
2006.826.814.323.20
25010.3710.616.374.61
Appendix
Table 8: Runtime (s) and memory (GB) under different event sequence lengths.
Quadrature SizeTraining TimePrediction TimeTraining MemoryTest Memory
41.541.051.791.72
82.141.032.001.78
123.201.202.271.74
163.611.102.701.93
245.391.203.912.72
Appendix
Table 9: Effect of quadrature resolution on runtime (s) and memory (GB).
Inducing Grid SizeTraining TimePrediction TimeTraining MemoryTest Memory
40.860.101.491.28
120.960.202.301.58
160.950.202.501.84
240.990.203.311.96
Appendix
Table 10: Effect of inducing-grid resolution on runtime (s) and memory (GB).
MethodCOVID-19Earthquake
NSTPP21.325.5
DeepSTPP1.30.8
DSTPP43.929.3
KSTPP24.519.7
Appendix
Table 11: Per-epoch training time (seconds).
MethodΔt=0.00\Delta t=0.00Δt=0.26\Delta t=0.26Δt=1.77\Delta t=1.77
STHP0.64090.6409−0.6409-0.6409
KSTPP0.99320.99320.8706
Appendix
Table 12: Correlation between the learned and ground-truth influence kernel on SYN1.
MethodΔt=0.00\Delta t=0.00Δt=0.26\Delta t=0.26Δt=0.52\Delta t=0.52
STHP−0.4379-0.4379−0.4379-0.4379−0.4379-0.4379
KSTPP0.75020.75030.7507
Appendix
Table 13: Correlation between the learned and ground-truth influence kernel on SYN2.
MetricΔt=0.00\Delta t=0.00Δt=0.26\Delta t=0.26Δt=0.52\Delta t=0.52Δt=10.35\Delta t=10.35
Sparsity Ratio88.6%96.7%99.1%100%
Appendix
Table 14: Sparsity ratio of the learned influence function on the Earthquake dataset.
Figure 9: Recovery of a nonseparable influence kernel. Top row: the ground-truth (GT) influence kernel f(Δt,Δx,Δy)f(\Delta t,\Delta x,\Delta y) shown as spatial surface slices at increasing temporal lags Δt∈{0.00,0.06,0.13,0.19,0.26,0.32,0.48,0.65}\Delta t\in\{0.00,0.06,0.13,0.19,0.26,0.32,0.48,0.65\} . Bottom row: the corresponding influence kernel recovered by KSTPP at the same Δt\Delta t slices. KSTPP closely matches the ground-truth shape and its temporal decay despite using a product-kernel covariance structure.
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