Rotated EOFs^*

Sometimes, the interpretation of the EOF patterns may be difficult because the adjacent modes are degenerate (not well-resolved in terms of their eigenvalues): any combination of degenerate patterns is equally valid. Furthermore, the order of degenerate modes are arbitrary. In order to resolve the modes, it is possible to rotate the EOFs. The rotation transforms the EOFs into a non-orthogonal linear basis.

The Varimax rotation (Kaiser, 1958) is one of the most commonly used type of rotation that minimises the 'simplicity' functional:

$${\bf V}_k^* = \frac{L \sum_{i=1}^L {\bf E}_{i,k}^4 - (\sum_{i=1}^L {\bf E}_{i,k}^2)^2}{L^2}$$

Maximises if ${\bf E}_{\vec{r},k}$ are all 0 or 1. ${\bf E}^{(R)} = {\bf E} {\bf T}^{-1}$.

If two patterns are degenerate and located in different regions, rotated EOFs should resolve them. Of course, there is a catch: i) if two waves are degenerate; ii) if you retain too many EOFs.

But, what physical meaning do the EOFs actually have? Coherent spatial patterns with maximum variance. Modes of energy? Just convenient mathematical abstractions? Depends on the nature of the problem.