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This index has been mentioned elsewhere: http://forum.azimuthproject.org/discussion/169/el-nino-southern-oscillation-enso/?Focus=11994#Comment_11994

I ran a Eureqa machine learning dispersion analysis on the MEI numbers and found some rather predictable periods.

The numbers that pop up in the solution space are frequencies of 6.3 rads/year and 12.6 rads/year, which are the annual and biannual signals

$ f''(t) = -12.6 f+ 6.3 f \sin(2 \pi t) - 6.3 f \sin(4 \pi t) + 13.22 \sin(4 \pi t) - 9.922 \sin(12.84 t) $

Because of the Mathieu modulation in the 2nd-order DiffEq and a forcing signal that is slightly off the biannual frequency, the result is a very erratic waveform with no clearly discernible period. The biannual forcing is interacting with a modulation that is annual and biannual, which makes it very sensitive to phase changes.

According to the documentation, MEI is a heavily processed index that goes through a data reduction technique known as principal components analysis (PCA).

```
"Here we attempt to monitor ENSO by basing the Multivariate ENSO Index (MEI) on the six main observed variables over the tropical Pacific. These six variables are: sea-level pressure (P), zonal (U) and meridional (V) components of the surface wind, sea surface temperature (S), surface air temperature (A), and total cloudiness fraction of the sky (C). These observations have been collected and published in ICOADS for many years. The MEI is computed separately for each of twelve sliding bi-monthly seasons (Dec/Jan, Jan/Feb,..., Nov/Dec). After spatially filtering the individual fields into clusters (Wolter, 1987), the MEI is calculated as the first unrotated Principal Component (PC) of all six observed fields combined. This is accomplished by normalizing the total variance of each field first, and then performing the extraction of the first PC on the co-variance matrix of the combined fields (Wolter and Timlin, 1993). In order to keep the MEI comparable, all seasonal values are standardized with respect to each season and to the 1950-93 reference period. "
```

PCA may be removing other frequencies, as only one principal component is retained.

## Comments

Paul

I do not see any multi-variate variables in this formulation? D

`Paul I do not see any multi-variate variables in this formulation? D`

6

`6`

As Jim pointed out with his very succinct response, the multivariate formulation is baked in to the results. Perhaps it is better to keep the variates isolated and see if one can compose these in more interesting ways.

For example, in the tidal records thread, It appears that the SOI may compose as a sea-level height factor plus some other factor, where the sea-level height is measured from the tidal gauge.

This is where the machine learning may become invaluable.

And the fact that PCA is a data reduction technique, which may be losing information, is obviously not a good thing. There is currently an interesting discussion going on at Nick Stokes blog (and eleswhere for that matter) where Nick has pointed out how PCA can cause "issues". The point is that not doing the data reduction may have been a better approach. Hopefully Nick is reading this and may want to chime in.

`As Jim pointed out with his very succinct response, the multivariate formulation is baked in to the results. Perhaps it is better to keep the variates isolated and see if one can compose these in more interesting ways. For example, in the [tidal records thread](http://forum.azimuthproject.org/discussion/1480/tidal-records-and-enso/), It appears that the SOI may compose as a sea-level height factor plus some other factor, where the sea-level height is measured from the tidal gauge. This is where the machine learning may become invaluable. And the fact that PCA is a data reduction technique, which may be losing information, is obviously not a good thing. There is currently an interesting discussion going on at [Nick Stokes blog](moyhu.blogspot.com/2014/10/analysis-of-short-centered-pca.html) (and eleswhere for that matter) where Nick has pointed out how PCA can cause "issues". The point is that not doing the data reduction may have been a better approach. Hopefully Nick is reading this and may want to chime in.`

Does Jim mean "6" as in 6 coefficient for the diff EQ?

I added them as sliders to our CDF but that does not make it multi-variate, since the coefficients do not participate in the differentiations, perhaps FREE VARIABLES a better term?

I am having a terminology issue here I think...

Dara

`Does Jim mean "6" as in 6 coefficient for the diff EQ? I added them as sliders to our CDF but that does not make it multi-variate, since the coefficients do not participate in the differentiations, perhaps FREE VARIABLES a better term? I am having a terminology issue here I think... Dara`

I think the problem is that there is not a target "index" that we are shooting for. If we knew what the true ENSO index was, we could create a multi-variate combination of factors in which to optimally construct that index. Yet, who is to say that the combination that went into MEI is the correct one?

I can't tell for certain but it almost appears that the combination in the MEI case was constructed to accentuate the strongest El Nino events. That may be good if one wants to isolate the conditions for what constitutes a strong event, but it may not be useful if one wants to represent the underling physical phenomena.

As an example of what works, consider how one can search for strong dipoles in climate measurements. Any strong dipole (such as SOI) is likely represented as an underlying standing wave phenomenon. So that constructing the SOI as a difference makes sense both physically and as a good indicator of El Nino.

So the search is on for a better index of variates, factors, variables, or whatever we should call them.

`I think the problem is that there is not a target "index" that we are shooting for. If we knew what the true ENSO index was, we could create a multi-variate combination of factors in which to optimally construct that index. Yet, who is to say that the combination that went into MEI is the correct one? I can't tell for certain but it almost appears that the combination in the MEI case was constructed to accentuate the strongest El Nino events. That may be good if one wants to isolate the conditions for what constitutes a strong event, but it may not be useful if one wants to represent the underling physical phenomena. As an example of what works, consider how one can search for strong dipoles in climate measurements. Any strong dipole (such as SOI) is likely represented as an underlying standing wave phenomenon. So that constructing the SOI as a difference makes sense both physically and as a good indicator of El Nino. So the search is on for a better index of variates, factors, variables, or whatever we should call them.`

This is an interesting Mathieu DiffEq fit to MEI

The correlation coefficient is only 0.39 but the detail is more evident. What I did was place a Mathieu modulation of close to one-year (6.2566=0.996*$2\pi$) instead of the 6 to 10 year period that has provided a better fit for the SOI data.

The issue I am finding with this fit is that the fluctuations are so sharp that any search will too readily get stuck in local minima, and so this fit may not be representative of a global optimal across the search space.

`This is an interesting Mathieu DiffEq fit to MEI ![MEI](http://imageshack.com/a/img673/8920/rY02jY.gif) The correlation coefficient is only 0.39 but the detail is more evident. What I did was place a Mathieu modulation of close to one-year (6.2566=0.996*$2\pi$) instead of the 6 to 10 year period that has provided a better fit for the SOI data. The issue I am finding with this fit is that the fluctuations are so sharp that any search will too readily get stuck in local minima, and so this fit may not be representative of a global optimal across the search space.`

I am ready for some experimentation to mix your diff EQ with Neural Networks, perhaps as early as tonight.

Somewhat slower these days since I am traveling around, Bee Keepers convention at Chase Manhattan building

D

`I am ready for some experimentation to mix your diff EQ with Neural Networks, perhaps as early as tonight. Somewhat slower these days since I am traveling around, Bee Keepers convention at Chase Manhattan building D`

Dara, Good luck with the NN approach.

I have been learning some search rules that are effective as heuristics, and can offer these up if either NN or Differential Evolution can incorporate such rules.

`Dara, Good luck with the NN approach. I have been learning some search rules that are effective as heuristics, and can offer these up if either NN or Differential Evolution can incorporate such rules.`

Paul I am going to code some NN algorithms and try on data and diff EQ, let's say OCT/NOV time frame so to see if we could use them

D

`Paul I am going to code some NN algorithms and try on data and diff EQ, let's say OCT/NOV time frame so to see if we could use them D`

Thanks, looking forward to what you come up with!

`Thanks, looking forward to what you come up with!`