  Accurate predictions of pollutant concentrations at new locations are often
of interest in air pollution studies on fine particulate matters (PM$_{2.5}$),
in which data is usually not measured at all study locations. PM$_{2.5}$ is
also a mixture of many different chemical components. Principal component
analysis (PCA) can be incorporated to obtain lower-dimensional representative
scores of such multi-pollutant data. Spatial prediction can then be used to
estimate these scores at new locations. Recently developed predictive PCA
modifies the traditional PCA algorithm to obtain scores with spatial structures
that can be well predicted at unmeasured locations. However, these approaches
require complete data, whereas multi-pollutant data tends to have complex
missing patterns in practice. We propose probabilistic versions of predictive
PCA which allow for flexible model-based imputation that can account for
spatial information and subsequently improve the overall predictive
performance.
