[mathjax]Given the fact that very short training intervals will reveal the underlying fundamental frequencies of QBO, could we do the same for ENSO? Not nearly as short, but about 40 years is the interval required to uncover the fundamental frequencies. Again none of this is possible unless we make the assumption of a phase reversal for ENSO between the years 1980 and 1996.

Using the Excel Solver again instead of the multiple linear regression approach I started with, the ENSO data set was split into three non-overlapping intervals, with 1/6 of the 1/3 split equally in the lower and higher range.

Fig. 1: Lower 1/3 fit, training intervals indicated by blue horizontal bars. The fit is noticeably better within these intervals, with some overfitting apparent outside.

Fig. 2: Middle 1/3 fit

Fig. 3: Upper 1/3 fit

The fits were very aggressive for each of the intervals, with correlation coefficients near or above 0.9. Yet, even with such a demanding fit, the coherence within the validation interval is readily apparent.  What’s more, the strong El Nino of 2016 clearly emerges, with never having applied training data past the year 2013.  Figure 4 is the entire interval fit with a correlation coefficient almost at 0.8

Fig. 4:  Entire interval

What seals the analysis is the robustness of the known wobble and lunar tidal periods with respect to the orthogonal intervals.  Figure 5 below shows a sinusoidal amplitude mapping with the phase retained.  In this case, due north is the cosine component and due east is the sine component. For each triangle shown, the vertices represent the sinusoidal factor amplitude+phase values — one vertex for each of the 1/3-interval training segments.  The round symbol represents the value for the entire fitting interval.  In general, the location for the entire fitting interval is enclosed by the triangle or is nearby.

Fig. 5: Amplitude with phase indicated on this quasi-radial plot.

What will completely destroy the fitting process is to include a sinusoidal factor not related to the wobble or lunar tidal periods.  That will cause a significant amount of over-fitting in one interval, which is not seen in another interval.  ENSO is essentially a stationary ergodic process, much like ocean tides, and once these periods are set in one interval, they will obviously occur in other intervals. What a “phantom” period accomplishes is either fitting to the noise or taking away amplitude from a real factor in favor of a spurious signal that doesn’t actually exist — thus referring to it as phantom.

The convenience aspect of using the Excel Solver is not to be understated. Anyone can create a column of sinusoidal factors with unknown amplitude and phase and then execute the plugin. The GRG Nonlinear algorithm accepts an input of, e.g. I1:I10, J1:J10, for the amplitudes and phases and it will find a good fit in short order .. as long as the biennial modulated wobble and lunisolar terms are set. If these periods are not set and left to vary, it will take longer depending on the seed used.

Next is a set of ENSO split up into two intervals, with the halfway point set at the time that the Cleveland Indians last won the World Series.

Fig. 6 : Interval 1/2+1/2 training

As a summary, the key to choosing the periods is to apply only the known wobble and lunisolar periods (seasonally aliased and non-aliased long-period terms). Then add in all the $$npi$$-shifted harmonics introduced by the biennial nonlinear modulation of the Mathieu formulation. None of these harmonics cause overfitting as they will align with the primary factors, much like what is observed with the QBO model.   (From a comment in the prior thread, as is commonly observed the QBO can be tantalizingly close in aligning withe ENSO, but the much stronger globally symmetric nodal periods comprising the QBO forcing has to compete with the other wobble and lunisolar — anomalistic and tropical — periods driving ENSO. Also the 1980 phase inversion did not impact QBO. These will all work together to prevent coherence over any longer interval).