Chapter 6 - Empirical Orthogonal Functions

Abstract

Empirical orthogonal function (EOF) analyses are often used to study possible spatial patterns of climate variability and how they change with time. One of the important results from EOF analysis is the discovery of several oscillations in the climate system, including the Pacific Decadal Oscillation and the Arctic Oscillation. Similarly to Fourier analysis and wavelet analysis, in EOF analysis, one also projects the original climate data on an orthogonal basis. However, this orthogonal basis is derived by computing the eigenvectors of a spatially weighted anomaly covariance matrix, and the corresponding eigenvalues provide a measure of the percent variance explained by each pattern. Therefore, EOFs of a space-time physical process can represent mutually orthogonal space patterns where the data variance is concentrated, with the first pattern being responsible for the largest part of the variance, the second for the largest part of the remaining variance, and so on. Later, in order to overcome some limitations of classical EOF analysis and make the resulting patterns more physically interpretable, rotated EOFs, Hilbert EOFs, and complex EOFs are developed. In this chapter, we focus on the theory and algorithms of EOF analyses and their generalization. At the same time, we introduce singular spectrum analysis, canonical correlation analysis, and principal oscillation pattern analysis.