Regularized Principal Component Analysis for Spatial Data |
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Authors: | Wen-Ting Wang Hsin-Cheng Huang |
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Institution: | 1. Institute of Statistics, National Chiao Tung University, Hsinchu City, Taiwan;2. Institute of Statistical Science, Academia Sinica, Taipei City, Taiwan |
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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 and n time points with the possibility that p > n. 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 p is large relative to n. 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 examples. |
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Keywords: | Alternating direction method of multipliers Empirical orthogonal functions Fixed rank kriging Lasso Nonstationary spatial covariance estimation Orthogonal constraint Smoothing splines |
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