Regularized Principal Component Analysis for Spatial Data

Abstract: In many atmospheric and earth sciences, it is of interest to identify dominant spatial patterns of variation based on data observed at p locations with n repeated measurements. While principal component analysis (PCA) is commonly applied to find the patterns, the eigenimages produced from PCA may be noisy or exhibit patterns that are not physically meaningful when p is large relative to n. To obtain more precise estimates of eigenimages (eigenfunctions), we propose a regularization approach incorporating smoothness and sparseness of eigenfunctions, while accounting for their orthogonality. Our method allows data taken at irregularly spaced or sparse locations. In addition, the resulting optimization problem can be solved using the alternating direction method of multipliers, which is computationally fast, easy to implement, and applicable to a large spatial dataset. Furthermore, the estimated eigenfunctions provide a natural basis for representing the underlying spatial process in a spatial randomeffects model, from which spatial covariance function estimation and spatial prediction can be efficiently performed using a regularized fixed-rank kriging method. Finally, the effectiveness of the proposed method is demonstrated by several numerical examples.