Wow, I determined that the first 1-D time-series chart of QBO is at a higher altitude than the one one I plotted at #6.
I was intrigued by nad's suggestion that it had some interesting structure. It also seemed to have a greater signal-to-noise ratio as the periods appear stronger and more distinct than the lower altitude results.
So I ran the higher-altitude QBO on Eureqa to find the Fourier components:
I looked for the lowest-error/minimum complexity representation on the Pareto curve, highlighted in blue on the left columns and red in the right columns.
There is a main frequency of 2.665 rads/yr with symmetric sidelobes at 2.487 rads/yr and 2.841 rads/year. The 2.665 corresponds to the mean period of the QBO = 28 months. The sidelobes are weaker.
Also a pair of high-frequency components are generated at 153 rads/yr and 154 yrs/yr. These are approximately equal in amplitude and correspond to about 1/2-month period each (a 1/2 month tidal factor ?). But since they are close in frequency, we should be able to take the difference (154-153)=1 rads/yr and use that as an envelope. Look at the Eureqa results in the following and you can see how the machine learning actually started with 1 rad/yr and then switches over to the higher frequency representation, since that must reduce the error in some incremental fashion.
This is where it gets neat, IMO.
I decided to apply the 2.665, 2.487, 2.841, and 1 rads/yr components as forcing factors in my SOM Mathieu differential equation evaluation, most recently evaluated [here](http://forum.azimuthproject.org/discussion/1451/enso-proxy-records) and specifically for the SOI set, which overlaps the QBOM time span.
Recall further up in this thread where I observed that the QBO is quite periodic in its waveform, while the ENSO is highly erratic. In the case of SOI specifically, the measure shows the same erratic waveform.
In the solution below, I left the Mathieu modulation as before and chose a restricted time interval for the SOI, yet I still backcasted 20 years prior to when the actual QBO data was collected. Note that I did modify the *amplitudes* of the factors to improve the fit, so that the lower sidelobe is stronger than the main. The correlation coefficient is 0.63, which isn't extremely high, but the general agreement seems quite good to me.
The weak fits occur at the start of the 1990's and 1980's and around 1964. (Incidentally, these do correspond to significant volcanic events, Pinatubo 1991, El Chicon 1982, and Agung 1963)
The idea here is that the non-linear Mathieu modulation (LHS of DiffEq) is transforming the regular QBO forcing (RHS of DiffEq) into something much more erratic. The Mathieu modulation is rationalized as a low-order effect in the sloshing dynamics of the equatorial Pacific Ocean.
The process now is to hammer on this formulation to determine the likelihood of this solution being statistically significant. So the questions to ask are (1) is inadvertent bias being introduced to guide the solution? (2) how much can the coefficients be tweaked without being accused of over-fitting? (3) is this a case of over-fitting as it is? and (4) the big question, justifying the math as plausible physics. In other words, am I fooling myself by going down this path?
As far as I know there is only one way to improve the fit, and that is to use a brute force differential evolution search as suggested by Dara.