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.

翻译：作为一种空间统计学中的回归技术，时空变系数模型（STVC）是发现响应-协变量在空间和时间上非平稳且可解释关联关系的重要工具。然而，由于计算成本高昂，STVC难以应用于大规模时空分析。针对这一挑战，我们将时空变系数归纳为三阶张量结构，并提出将时空变系数模型重新表述为特殊的低秩张量回归问题。低秩分解能够以显著减少的参数数量有效建模大规模数据集的全局模式。为进一步融入局部时空依赖性，我们在空间和时间因子矩阵上采用高斯过程（GP）先验。我们将整体框架称为贝叶斯核化张量回归（BKTR），核化张量分解可视为一种新型可扩展方法，用于对具有低秩协方差结构的多元时空过程进行建模。在模型推理方面，我们开发了高效的马尔可夫链蒙特卡洛（MCMC）算法，该算法使用吉布斯采样更新因子矩阵，并使用切片采样更新核超参数。我们在合成数据集和真实数据集上进行了大量实验，结果证实了BKTR在模型估计和参数推断方面的卓越性能与效率。