How do we determine confidence that we are not fitting to noise for the ENSO model ? One way to do this is to compare the data against another model; in this case, a model that provides an instrumentally independent measure. One can judge data quality by comparing an index such as NINO34 against SOI, which are instrumentally independent measures (one based on temperature and one on atmospheric pressure).

If you look at a sliding correlation coefficient of these two indices along the complete interval, you will see certain years that are poorly correlated (see **RED **line below). Impressively, these are the same years that give poor agreement against the ENSO model (see **BLUE **line below). What this tells us is the poorly correlated years are ones with poor signal-to-noise ratio. But more importantly, it also indicates that the model is primarily fitting to the real ENSO signal (especially the peak values) and the noisy parts (closer to zero crossings or neutral ENSO conditions) are likely not contributing to the fit. And this is not a situation where the model will fit SOI better than NINO34 -- because it doesn't.

The tracking of SOI correlating to NINO34 matches that of Model to NINO34 across the range with the exception of some excursions during the 1950's, where SOI fit NINO34 better that the model fit NINO34. The average correlation coefficient of SOI to NINO34 across the entire range is 0.75 while the model against NINO34 is less but depending on the parameterization always above 0.6.

As a result of this finding, I started to use a modification of a correlation coefficient called a weighted correlation coefficient, whereby the third parameter set is a density function that remains near 1 when the signal-to-noise (SNR) ratio is high and closer to zero where the SNR is closer to zero. This allows the fit to concentrate on the intervals of strong SNR, thus reducing the possibility of over-fitting against noise.

## Or is it really all noise? (Added: 5/17/2017)

As I derived earlier, the solution to Laplace's tidal equations at the equator for a behavior such as QBO leads to a *sin(k sin(f(t)))* modulated time-series, where the inner sinusoid is essentially the forcing. This particular formulation (referred to as the *sin-sin* envelope) has interesting properties. For one, it has an amplitude limiting property due to the fact that a sinuosoid can't exceed an amplitude of unity. Besides this excursion-limiting behavior, this formulation can also show **amplitude folding** at the positive and negative extremes. In other words, if the amplitude is too large, the outer sin modulation starts to *shrink* the excursion, instead of just limiting it. So if there is a massive amplitude, what happens is that the folding will occur multiple times within the peak interval, thus resulting in a rapid up and down oscillation. This potentially can have the appearance of noise as the oscillations are so rapid that (1) they may blur the data record or (2) may be unsustainable and lead to some form of wave-breaking. I am not sure if the latter is related to folding of geological strata.

So the question is: can this happen for ENSO? I have been feeding the solution to the delayed differential Mathieu equation as a forcing to the sin-sin envelope and find that it works effectively to match the "noisy" regions identified above. In the figure below, the diamonds represent intervals with the poorest correlation between NINO34 and SOI and perhaps the noisiest in terms of SOI. In particular, the regions labelled 1 and 6 indicate rapid cyclic excursions.

By comparison, the model fit to ENSO shows the rapid oscillations near many of the same regions. In particular look at intervals indicated by diamonds 1 and 6 below, as well as the interval just before 1950.

Now, consider that these just happen to be the same regions that the ENSO model shows excessive amplitude folding. The pattern isn't 100% but also doesn't appear to be coincidental, nor is it biased or forced (as the fitting procedure has no idea that these are considered the noisy intervals). So the suggestion is that these are points in time that could have developed into massive El Nino or La Nina, but didn't because the forcing amplitude became folded. Thus they could not grow and instead the strong lunar gravitational forcing went into rapid oscillations which dissipated that energy. In fact, it's really the rate of change in the kinetic energy that scales with forcing, and the rapid oscillations identify that change. Connecting back to the theory, that's what the sin-sin envelope describes — its essentially a solution to a Hamiltonian that conserves the energy of the system. From the Sturm-Liouville equation that Laplace's tidal equations reduce to, this answer is analytically precise and provided in closed-form.

The caveat to this idea of course is that no one else in climate science is even close to considering such a sin-sin formulation. Consider this:

An alternative model that matches ENSO does not exist, so there is nothing at the moment to refute. And see above how it fits in balance with known physics.

This is again really interesting, and I again suggest that you publish a *predictive* version of your best-available model. When you go out on a limb and it holds, then people start to understand that there's something real on the other side.

Keith, I'm thinking that's a dead end. From what I have been able to understand, they use that to make you go away. In other words,

"make a prediction, wait 10 years and I will get back to you ... but until that time run along"This guy has now blocked me because he thinks that the entire theory is wrong. In other words, it won't matter if it is able to predict as long as it doesn't agree with the consensus theory. So the only approach is to keep hammering away on the evidence that already exists.

Paul, can't you do a 10 year forecast from 2007 -2017 by only using data prior to 2007? A hindcast/forecast?

I can do that, but they think any forecast of that sort is tainted by prior knowledge.

None of the data is under lock and key so even if you claim to not peak at out-of-band data, they will think you cheated.

The lack of access to controlled experiments causes lots of grief.

The most impressive forecast is probably the 1900 to 1920 training interval which captures the pattern outside.

Hi Paul,

Nice work.

It seems the current theory advocated by Roundy is not very good. So your alternative should be published in a peer reviewed journal (if possible) and the experts can refute if there are errors.

Dennis, The wind-driven theory still sounds like a heuristic backed by a correlation. However, there are lots of correlations in climate variables, largely due to the fact that these variables are related by laws such as the ideal gas law.

Pingback: Limits to Goodness of Fit | context/Earth