Pertaining to this [Roundy article]( about applying EOF ([empirical orthogonal functions]( and PCA ([principal component analysis]( to ENSO and MJO,
the figure below shows a set of two principle components that Roundy found. On the right is the phase relationship pattern between the two. Note that it doesn't fill up space very well -- and also that the orthogonality is somewhat in question for the two factors, as many points fall along a proportional x ~ y line.


It is then instructive to look at how orthogonality applies to solutions of Laplace's Tidal Equations. The fact that the solution factors are orthogonal is trivially true for the Mach-Zehnder-like solutions to LTE. Each solution -- sin( k1⋅f(t)), sin( k2⋅f(t)), etc -- is automatically orthogonal for different values of k, which are essentially different standing-wave patterns showing an average cross-correlation of zero over all time.

Let's look at one canonical fit to the ENSO time series, which features two standing-wave M-Z LTE patterns where the ratio between the two values of k is approximately 8.3.

Below is the fit, with the tidal forcing input in the top panel, and the decomposition of the two superposed M-Z LTE solutions in the lower panel. For the strong El Nino events in 1982 and 1998, the superposition of the peaks is constructive (explaining their large amplitude).

The charts below illustrate the phase relationship between the two components, revealing apparent fragments of a [Lissajous pattern](, which comes about from graphing a pair of parametric equations. Note the similarity to a pure non-fragmentary Lissajous curve created from two sine waves with the same relative amplifying factor of 8.3 in the lower graph. The unusual aspect of this comparison is that the time is not the parameter of the upper curve, as you can see from the apparently random sizing of the points, thus explaining it's fragmentary character.


In contrast to my results which clearly show orthogonality with an ergodic space-filling character, Roundy's results appear at most a rough initial heuristic. In other words, one approach reduces complexity in an elegant manner and the other one doesn't.