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I ran across another ENSO index, called BEST for Bivariate EnSo Timeseries
This is a good one for doing machine learning on because it is relatively free from noise and shows little by way of a trend. It is all oscillations.
The machine learning finds the usual QBO forcing period of around 28 months, and a Mathieu-like modulation of 9 to 12 year periods. It also finds a characteristic period of a little over 4 years, spanning an interval running back to 1880.
The top chart has double the complexity fit as the second chart. Both correlation coefficients are above 0.85.