As food for thought, I think what's missing in various El Nino analyses is that the phenomenon itself exists on a continuum. So certainly there are cases of strong El Nino that get the top billing, but the rest of the peaks and valleys are still there and that's what this style of SOI analysis is all about.
The fundamental concept of the SOI dipole is that a sloshing of the ocean and of the thermocline is taking place and due to the enormous inertial mass of the ocean's waters, this sloshing can't stop on a dime. But what it can do is respond to forcing functions that are at approximately at the same frequency as the characteristic (i.e. resonant) frequency of the underlying wave equation.
That means that frequencies close to the Clarke characteristic period of 4.25 years are very effective at controlling the dynamics of ENSO. So what frequencies are close to 4 years? The QBO at 2.33 years is an obvious candidate to consider. As is the Chandler wobble beat frequency of 6.4 years. The TSI signal has a period of about 10.6 years, which makes it less of a factor, but then again it does likely have higher harmonics due to the fast rise on a cycle. That is part of the reason that they are included in the analysis. It doesn't hurt that the literature is full of assertions that the QBO clearly interacts with ENSO, that the Chandler wobble is associated with oceanic motion, and suggestions that the Solar Hale cycle of sunspot activity has interactions with the QBO and certainly with the ocean absorbing periodic incoming radiation.
(The other frequencies such as tides, seasonal, and daily operate too quickly and are not as close to creating resonances with the 4.25 year characteristic period. Although one of the tidal beat frequencies is 1/2 of the 8.85 year [lunar precession](http://en.wikipedia.org/wiki/Lunar_month#Anomalistic_month))
In essence, the DiffEq solution provides the fundamental behavior that any strong El Nino behavior will likely spring from. It appears to be a low probability occurence that an El Nino year will not coincide with a negative excursion corresponding to the SOI model. See comments #1 and #6.
As a diversion to John's suggestion of applying statistical measures, I have been experimenting with the machine learning Eureqa tool, and with its AIC-based error measure. Sure enough, the tool isolates the 2.33 year QBO, the 6.5 year CW, and a 10-11 year period on certain intervals. And the 4 year characteristic period is found when cast as a wave equation. The AIC error metric penalizes too complicated a model and thus it appears that it is providing evidence that this (DiffEq in comment #33) could be the simplest / canonical formulation for the SOI dipole. So there is at least some Aikike Information Criteria statistical support that the model is on the right track. Criteria are often useless unless other candidate models are supplied to evaluate the target model. So by creating a constantly evolving set of alternative symbolic formulations, Eureqa can gauge the most efficient information-weighted model on a Pareto curve.
I was able to mimic what Eureqa is doing to some extent on an Excel spreadsheet and the results are very interesting. This basically parameterizes the symbolic expression in comment #33 and the Excel evolutionary solver does a nice job of finding the optimal set in the context of an error-minimizing regression analysis. It really excels (!) at evaluating a training interval and then showing whether the correlation extends outside of the training interval. I will show these graphs when I finish polishing them up.