Scalable Spatiotemporally Varying Coefficient Modelling with Bayesian Kernelized Tensor Regression

TL;DR

This work presents Bayesian Kernelized Tensor Regression (BKTR), a scalable approach for spatiotemporally varying coefficient modeling that combines a low-rank CP decomposition of the coefficient tensor with Gaussian process priors on spatial and temporal factors. By expressing the coefficient tensor as $\boldsymbol{B}=\sum_{r=1}^R \boldsymbol{u}_r\circ\boldsymbol{v}_r\circ\boldsymbol{w}_r$ and placing GP priors on $\boldsymbol{U}$ and $\boldsymbol{V}$, BKTR achieves substantial computational savings (down to $O\left(R^3\left(M^3+N^3+P^3\right)\right)$) while preserving the ability to capture local spatiotemporal structure. Inference is fully Bayesian via MCMC, with Gibbs updates for the factor matrices and slice sampling for kernel hyperparameters, and it can handle partially observed responses. Through extensive simulations and a real bike-sharing application, BKTR demonstrates accurate coefficient recovery, uncertainty quantification, and superior scalability compared with traditional STVC and tensor regression baselines. The framework enables interpretable, nonstationary spatiotemporal modeling at large scales and offers flexible extensions for non-Gaussian data and adaptive rank.

Abstract

As a regression technique in spatial statistics, the spatiotemporally varying coefficient model (STVC) is an important tool for discovering nonstationary and interpretable response-covariate associations over both space and time. However, it is difficult to apply STVC for large-scale spatiotemporal analyses due to its high computational cost. To address this challenge, we summarize the spatiotemporally varying coefficients using a third-order tensor structure and propose to reformulate the spatiotemporally varying coefficient model as a special low-rank tensor regression problem. The low-rank decomposition can effectively model the global patterns of large data sets with a substantially reduced number of parameters. To further incorporate the local spatiotemporal dependencies, we use Gaussian process (GP) priors on the spatial and temporal factor matrices. We refer to the overall framework as Bayesian Kernelized Tensor Regression (BKTR), and kernelized tensor factorization can be considered a new and scalable approach to modeling multivariate spatiotemporal processes with a low-rank covariance structure. For model inference, we develop an efficient Markov chain Monte Carlo (MCMC) algorithm, which uses Gibbs sampling to update factor matrices and slice sampling to update kernel hyperparameters. We conduct extensive experiments on both synthetic and real-world data sets, and our results confirm the superior performance and efficiency of BKTR for model estimation and parameter inference.

Figures (11)

• Figure 1: Illustration of the proposed BKTR framework.
• Figure 2: Graphical model of BKTR.
• Figure 3: BKTR ($R=10$) estimated coefficients for Simulation 1 (50% of $\boldsymbol{Y}$ is observed).
• Figure 4: Sensitivity test of BKTR for Simulation 1: (a) Effects of the rank $R$; (b) Effects of the observation rate $\frac{|\Omega|}{MN}$. For each case, the figure shows the boxplots and mean values of the corresponding metrics calculated from 40 replications.
• Figure 5: Comparison of BKTR in different settings for Simulation 1. (a) and (b) plot the estimated $\boldsymbol{\mathcal{B}}$ (mean with 95% CI) of one simulation at location #3 ($m=3$) for the 6th and 3rd covariates (i.e., $\boldsymbol{\mathcal{B}}(3,:,6)\text{ and }\boldsymbol{\mathcal{B}}(3,:,3)$), respectively.