To finish off that sentence, "If the glove don't fit, you must acquit". One of the most common complaints that I've received whenever I describe fitting models to data, is that "correlation does not equal causation". I especially get this at Daily Kos with respect to any comment I make to science posts. One knee jerk follows me around and reminds me of this bit of wisdom like clockwork. I can't really argue the assertion out of hand because it's indeed true that there are plenty of coincidental correlations that don't imply anything significant in terms of causation.
But one aspect that they forget with respect to the correlation≠causation cliche is that having a correlation to a theoretical model provides strong support for not immediately rejecting the hypothesis. "If the glove fits .. um .. uh oh". Recall what QBO theorist Richard Lindzen said about finding a correlation in an atmospheric measure:
"it is unlikely that lunar periods could be produced by anything other than the lunar tidal potential."
That assertion is actually much stronger than what I am implying.The much more solid complement to that statement is that if a period is measured that is not related to a lunar period, then it is a strong reason to outright reject a lunar hypothesis as a cause. "If the glove don't fit, you must acquit"
As a rationale for why Lindzen never found a correlation in the QBO to lunar periods, I think it has to do with his never grasping the possibility that aliasing of the lunar period with a seasonal modulation could have forced the oscillating behavior. So he obviously considered only the conventional non-aliased lunar tidal periods, found no match, and then moved on to an ill-advised overly complex model to explain QBO. After 40+ years, Lindzen's model is still considered the official understanding for the origin of the QBO -- whereby the oscillations are related to a resonant condition of the atmosphere.
I constructed the original QBO model by calculating the expected aliased frequencies and then running a multiple linear regression against the data to calculate a fit. This is no different than what is being done for tidal analysis  -- with of course the one difference.
Another way to do this is to run a machine-learning experiment on a QBO time-series interval. Using Eureqa, we get this after letting it loose on a short 16 year interval of the QBO 30 hPa time series, looking at the 2nd derivative -- so after 10 trillion formula calculations:
Without any prompting, Eureqa found the exact same set of aliased harmonics that should occur if Draconic month aliasing was occurring, which is the top row in Figure 2.
2.363 and 0.6939 -- 2902.193*sin(3.19778396*t)*cos(5.8565998*t) 0.413966 -- 1032.57681*cos(15.1785856*t)
These are the comparisons:
- 2.363 vs. 2.366 expected
- 0.6939 vs. 0.7029 expected
- 0.413966 vs. 0.41278 expected
which is well within any phase propagation error in a 16 year interval.
For completeness, another factor Eureqa selected, 1598.2936cos(0.07492t^2), was a spread around 0.7 years when unaliased. That's a bit complicated as it looks as if the frequency changes slightly over the interval. Yet, this is acceptable as other periods around 0.7 occur according to the table. Since this is a second derivative, the sensitivity was higher on these shorter periods as well. The 2.363 factor was the final piece according to the Pareto front.
The other was the known 0.53 year tidal factor 857.8cos(11.9297t)
So that is what Eureqa found after trillions of trials, a nice validation of the lunar aliasing mechanism.
And how sensitive are the individual aliased signals? The following chart shows how the correlation changes with slight tweaking of the unaliased long period.
Recently I have discovered that the Chandler Wobble also nicely fits to an aliased Draconic lunar forcing mechanism. Consider the fit below, this time using an Excel evolutionary solver, which picked the solution in red.
Again, is this a case of correlation resulting in a causation?
The Chandler wobble has a period of 1.185 years while the QBO is 2.37, which is half as fast per cycle. This can be reconciled if one considers a symmetric (North vs South) declination forcing versus an asymmetric forcing.
Once again, of course correlation does not equal causation, but this is one of those cases in which an agreement with a plausible theory does not allow one to rule out that theory. For example, if those numbers in Figure 5 were just a little off, one could rule out the lunar cycle as a forcing mechanism to the Chandler wobble.
Yet the numbers lining up still could be just a coincidence. Could it be that all these little pieces of evidence when put together are not much stronger than the evidence for a moon landing hoax? As I recall, that was largely a sequence of coincidences.
Ultimately, what motivates this analysis is the mystery behind the moon's control over something as significant an effect as oceanic tides, but not much else, including ENSO. Have scientists simply not looked hard enough? Or are they starting to finally pursue this path ?
Should we acquit the moon for the behavior of the QBO and the Chandler wobble? Or is it guilty as charged?
Lots more evidence to come, stay tuned ...
 R. Mousavian and M. M. Hossainali, “Detection of main tidal frequencies using least squares harmonic estimation method,” Journal of Geodetic Science, vol. 2, no. 3, pp. 224–233, 2012.
 Tidal Analysis and Prediction - NOAA Tides and Currents -- http://tidesandcurrents.noaa.gov/publications/Tidal_Analysis_and_Predictions.pdf
 There is also this recent paper on finding a strong correlation (but fractionally weak as a perturbation) between upper atmospheric rainfall and the Lunar cycle. "Rainfall variations induced by the lunar gravitational atmospheric tide and their implications for the relationship between tropical rainfall and humidity", Kohyama & Wallace, Geophysical Research Letters, 2016