Graham, After trying to digest your response some more, I don't think you understand exactly what I am trying to do. I am only using FindFormula to find an empirical and analytical fit to QBO, which then can be used to extrapolate to out-of-band numbers for the past where the measurements were never done (because radiosonde equipment had yet to be invented).

You ask whether I am checking the QBO to 2013. Yes, of course, this the fit to 2013

I use all available QBO data from 1953-2013 to estimate what it will be from 1880-1953. I don't throw away any useful results that will help with a fit. We will never know what the actual QBO values are before 1953, because that is the stratosphere and no traces are left in any records.

> "Your formula looks odd to me because all the cos terms are equal to 1 when Y=0."

That's not quite my formula as I have more terms. It is well known that machine learning techniques such as Eureqa and Mathematica's FindFormula balance complexity against accuracy in their results. In this case, a result such as Cos(t) is generated because it is less complex than Cos(0.97*t + 0.45) , even though the latter is more accurate. The same holds true for the other terms -- I refer to this as a "lever arm" effect as small adjustments in the frequency can model phase adjustments. You really have to spend time working with these tools to understand their intricacies. Machine learning techniques are not a panacea, they are just a tool.

Yet what I show is just the tip of the iceberg as far as results are concerned. Get a license for Eureqa and you can let it automatically find solutions. Here is an experiment where it deduces the DiffEq result along with a Mathieu modulation.


The reality is that we are way beyond the climatologists in terms of how we are looking at the data. They are stuck in the stone age of analysis is all I can figure. No one is looking at machine learning and signal processing apart from what were are doing here and what some geophysicists such as Astudillo are finding. Like I said, there has to be some explanation why no one has found this relationship before -- curious because it has always been hidden in plain sight.

[1] H. Astudillo, R. Abarca-del-Rio, and F. Borotto, “Long-term non-linear predictability of ENSO events over the 20th century,” arXiv preprint arXiv:1506.04066, 2015.