I am juggling three primary data sets, the Southern Oscillation Index (SOI) time-series from 1880-present, the QBO time-series from 1952-present, and the Universal ENSO Proxy (UEP) records from 1650-1976.
The connection so far is:
1. The QBO provides the RHS forcing function to the LHS sloshing dynamics Mathieu differential equation.
2. One DiffEq solution fits the UEP time series
3. One DiffEq solution fits the SOI time series
This is an experiment to see how well a fit to the UEP time-series works with the SOI time-series, using essentially the same RHS forcing function, apart from any initial conditions required to align the two series in amplitude and phase. They also share the same LHS DiffEq.
I used the UEP model fit from [comment #41 above ](http://forum.azimuthproject.org/discussion/1451/enso-proxy-records/?Focus=12376#Comment_12376), repeated below, as the baseline. This is optimized over a 300+ year time span, with no favored interval.
## UEP results ##
Then using the same RHS forcing and LHS DiffEq in the UEP fit, we can apply it to the SOI fit. The parameters are kept constant except for the initial conditions for y', y, and the constant and linear forcing terms, which are necessary to account for offset and drift.
## SOI results ##
The SOI fit is deceptively good considering that the UEP fit ends at 1975, while the SOI data continues to 2013. Both cases have a correlation coefficient slightly above 0.42
This indicates that the UEP data is providing an extended training interval, where we can establish a periodic pattern for the RHS forcing and a stable LHS featuring a slightly modulated Mathieu factor. That solution, when applied to SOI generates a very good fit -- considering how difficult it is to map to such an erratic waveform. (it is very difficult to get to a CC much above 0.6 using an arbitrary Fourier series of a handful of terms )
*We are getting so close, I can taste it*
The qualifier to this is that the UEP data is calibrated to the SOI for recent years. However, this calibration is not perfect and the fit puts more emphasis in the 200 years prior to 1880.
The statistics question is how much of the correlation is due to a real deterministic behavior and how much could be due to a random alignment of signals? I have a feeling that the statistical significance of it being a real correlation is overwhelming, and that this behavior is describing the erratic dynamics of ENSO effectively.
Feedback appreciated, because as usual, I don't want to be fooling myself.