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Regularized Principal Component Analysis for Spatial Data
In many atmospheric and earth sciences, it is of interest to identify
dominant spatial patterns of variation based on data observed at locations
and time points with the possibility that . While principal component
analysis (PCA) is commonly applied to find the dominant patterns, the
eigenimages produced from PCA may exhibit patterns that are too noisy to be
physically meaningful when is large relative to . To obtain more precise
estimates of eigenimages, we propose a regularization approach incorporating
smoothness and sparseness of eigenimages, 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 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 random-effects 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 example
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