Spatial matrix completion for spatially misaligned and high‐dimensional air pollution data

In health‐pollution cohort studies, accurate predictions of pollutant concentrations at new locations are needed, since the locations of fixed monitoring sites and study participants are often spatially misaligned. For multi‐pollution data, principal component analysis (PCA) is often incorporated to obtain low‐rank (LR) structure of the data prior to spatial prediction. Recently developed predictive PCA modifies the traditional algorithm to improve the overall predictive performance by leveraging both LR and spatial structures within the data. However, predictive PCA requires complete data or an initial imputation step. Nonparametric imputation techniques without accounting for spatial information may distort the underlying structure of the data, and thus further reduce the predictive performance. We propose a convex optimization problem inspired by the LR matrix completion framework and develop a proximal algorithm to solve it. Missing data are imputed and handled concurrently within the algorithm, which eliminates the necessity of a separate imputation step. We review the connections among those existing methods developed for spatially misaligned multivariate data, and show that our algorithm has lower computational burden and leads to reliable predictive performance as the severity of missing data increases.