High-dimensional data with hundreds of thousands of observations are becoming commonplace in many disciplines. The analysis of such data poses many computational challenges, especially when the observations are correlated over time and/or across space. In this paper we propose flexible hierarchical regression models for analyzing such data that accommodate serial and/or spatial correlation. We address the computational challenges involved in fitting these models by adopting an approximate inference framework. We develop an online variational Bayes algorithm that works by incrementally reading the data into memory one portion at a time. The performance of the method is assessed through simulation studies. We applied the methodology to analyze signal intensity in MRI images of subjects with knee osteoarthritis, using data from the Osteoarthritis Initiative.
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