Regularized Principal Component Analysis for Spatial Data

TL;DR

This paper introduces a regularized PCA method tailored for spatial data that enhances the interpretability of eigenimages by incorporating smoothness and sparsity, suitable for large and irregular datasets.

Contribution

It proposes a novel regularization approach for PCA in spatial settings, enabling more accurate eigenimage estimation and efficient spatial modeling with large, irregular datasets.

Findings

Regularized PCA produces clearer, more meaningful spatial patterns.

The method is computationally efficient and scalable to large datasets.

Numerical examples demonstrate improved spatial covariance estimation.

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