  Large-scale and multidimensional spatiotemporal data sets are becoming
ubiquitous in many real-world applications such as monitoring urban traffic and
air quality. Making predictions on these time series has become a critical
challenge due to not only the large-scale and high-dimensional nature but also
the considerable amount of missing data. In this paper, we propose a Bayesian
temporal factorization (BTF) framework for modeling multidimensional time
series -- in particular spatiotemporal data -- in the presence of missing
values. By integrating low-rank matrix/tensor factorization and vector
autoregressive (VAR) process into a single probabilistic graphical model, this
framework can characterize both global and local consistencies in large-scale
time series data. The graphical model allows us to effectively perform
probabilistic predictions and produce uncertainty estimates without imputing
those missing values. We develop efficient Gibbs sampling algorithms for model
inference and model updating for real-time prediction and test the proposed BTF
framework on several real-world spatiotemporal data sets for both missing data
imputation and multi-step rolling prediction tasks. The numerical experiments
demonstrate the superiority of the proposed BTF approaches over existing
state-of-the-art methods.
