# Empirical Orthogonal Function (EOF) Analysis and Rotated EOF Analysis

The following is by Dennis Shea (NCAR)

## EmpiricalOrthogonalFunction (EOF) analysis

In climate studies, EOF analysis is often used to study possible spatial modes (ie, patterns) of variability and how they change with time (e.g., the North Atlantic Oscilliation). In statistics, EOF analysis is known as Principal Component Analysis (PCA). As such, EOF analysis is sometimes classified as a multivariate statistical technique. However, there is no a priori hypothesis based on some probability distribution and, hence, no statistical test. Further, EOF analysis is not based on physical principles. Rather, a field is partitioned into mathematically orthogonal (independent) modes which sometimes may be interpreted as atmospheric and oceanographic modes ('structures'). Typically, the EOFs are found by computing the eigenvalues and eigenvectors of a spatially weighted anomaly covariance matrix of a field. Most commonly, the spatial weights are the cos(latitude) or, better for EOF analysis, the sqrt(cos(latitude)). The derived eigenvalues provide a measure of the percent variance explained by each mode. Unfortunately, the eigenvalues are not necessarily distinct due to sampling issues. North et al (Mon. Wea. Rev., 1982, eqns 24-26) provide a 'rule of thumb' for determining if a particular eigenvalue (mode) is distinct from its nearest neighbor. Atmospheric and oceanographic processes are typically 'red' which means that most of the variance (power) is contained within the first few modes. The time series of each mode (aka, principle components) are determined by projecting the derived eigenvectors onto the spatially weighted anomalies. This will result in the amplitude of each mode over the period of record.

By construction, the EOF patterns and the principal components are independent. Two factors inhibit physical interpretation of EOFs: (1) the orthogonality constraint and (2) the derived patterns may be domain dependent.  Physical systems are not necessarily orthogonal and if the patterns depend on the region used they may not exist if the the domain changes. Still, even with these short comings, classical EOF (PCA) analysis has proved to be useful.

## Rotated EOF analysis (REOF):

To address some of the limitations of classical EOF analysis, researchers 'rotate' the classical EOFs. Some rotational methods retain the orthogonality of the modes but not the principal components or vice versa. Oblique rotations preserved neither. Most commonly, the varimax rotation has been used. The objective is to minimize the mode complexity by making the large loadings larger and the small loadings smaller. Initially, a standard EOF analysis is performed and an EOF subset (eg, 10) is retained and subjected to varimax rotation. Some feel that the resulting patterns are more physically interpretable. Still, REOF methods have issues. The patterns are may still be domain dependent and the initial number of EOFs retained is arbitrary