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John had asked me to use Darwin anomaly data and when I decomposed it with wavelets, multi-Trend appeared and the trends were spread out in Energy Fractions:

{{1} -> 0.226711, {0, 1} -> 0.120706, {0, 0, 1} -> 0.105509, {0, 0, 0, 1} -> 0.152306, {0, 0, 0, 0} -> 0.394767}

For the forecast algorithm, often I do a delta forecast to avoid the order of magnitude of the numbers to increase accuracy i.e.

Signal(t+1) - Signal(t)

used in place of Signal (t) i.e. time-shift differenced.

While I did that for Dariwn anomaly data, I got a lop-sided Energy Fractions for the same wavelet decomposition reported above:

{{1} -> 0.724091, {0, 1} -> 0.17549, {0, 0, 1} ->0.0364491, {0, 0, 0, 1} -> 0.0120768, {0, 0, 0, 0} -> 0.0518935}

As you can see {1} comprises most of the signal i.e. 70%.

Premature conclusion: The Darwin anomaly data is actually an unraveling of a time-shift data or an Integral (as in calculus integral or anti-derivative) of some other data.

Dara

## Comments

That's interesting. Maybe someone can guess that the time derivative of the Darwin surface air pressure anomaly is (approximately) equal to something interesting....

Hello John

I did the scalogram for Darwin Delta Anomaly i.e. Darwin Anomaly[t+1] - Darwin Anomaly[t] and 70% of the signal was in the index {1}, which reports period of 3 months. Might I infer that the derivative relates to something seasonal as in 4 seasons of the year. But where Darwin is, are there 4 seasons?

Dara

One useful characterization technique is the parametric phase plot, where one plots the variate against the derivative of the variate

This is the phase plot for Darwin, plotted along with a Mathieu function phase plot

The general pattern imbedded in the plot is characteristic of the differential equation evolution or attractor.

This is what I would call a "peering through a binoculars" pattern. If some erratic underlying forcing occurs, then the pattern starts to shift from its ideal.

What makes this challenging is that a "red noise" signal can also approximate this pattern:

red noise signal

red noise phase plot

Paul Pukite

Paul I will hack your code later on tonight.

Meanwhile, I noted that the wavelet trend {1} of Darwin Delta Anomaly i.e. Darwin[t+1]-Darwin[t] is very similar to a form f(x)Sin(x) where f(x) is either slow-monotonic or bounded between two values and changes slowly, but the entire signal has period of 3 months.

That being said, if the signal has period of 3 months and the data collection is monthly then we are in a compromised situation.

I got a FUDGED SVR forecast working for 30 months of Darwin Anomaly for predicting the UP vs. DOWN i.e. forecasting the sign Darwin[t+1]-Darwin[t], Fudged:

Had to multiply the original signal by 100 in order to get SVR not giving me a almost 0 constant function due to low sampling

Phase-shift (time axis translation) required to get the forecast match since the learning algorithms suffer from a slight drift like a cat chasing a mouse

I try to run more forecasts and publish the results here and investigate what to do next. Even if the results are negative.

D

I am thinking of generating interim numeric and symbolic computations for forecast algorithms for John and others devise new variations or custom-tailor algorithms for a particular data sets. This is to reduce the need for John to program in person, it really needs to show the history of computation to understand how things work or not.

Dara, keep us informed and keep posting graphics as you proceed.

Here is a very fascinating preprint read from arXiv : Krauskopf, Bernd, and Jan Sieber. "Bifurcation analysis of delay-induced resonances of the El-Nino Southern Oscillation." http://arxiv.org/abs/1109.2818 (2011).

In the conclusions they state

They go on to say that the more sensitive the model is to initial conditions, the less robust the outcomes become and as they get "smeared out" (in their words), the harder it becomes to make predictions. The first step on the road to success may be to simply establish that ENSO is not random. The characterizations would then need to be designed to distinguish randomness from determinism. The phase plot I showed above demonstrates just this difficulty.

I am sorry will post more images tonight and some pdf files.

ONE. As I mentioned earlier f might be random or unstable or chaotic but f inverse could be well-behaved. This is the guts of control system assumption for the navigation systems for UAVs and drones. So it might be that instead of working with f we need to work with inverse f for the climate! I mean that is my intuition as a computer scientist.

TWO. I loved your phase plot, my machine is running another Mathematica application I do not want to corrupt the variables so I am delayed recreating your example. Actually let's code some systems in phase space as you did and show it to John and others here perhaps they might be able to pursue a new line of thought. I coded the delayed diff eq when I first arrived here and it worked ;) so there is hope, also Mathematica 10 has a huge set of new support functions for such solutions.

THREE. Let's work on volumetric data, whatever is the system of the climate cannot possibly be a 1-D time series!

BTW I am reading the paper you posted

Dara

"whatever is the system of the climate cannot possibly be a 1-D time series!"

I assume you are correct on this, yet if the principal terms factor out so that the reduced system appears as a 1-D time series I wouldn't get duly upset :)

What keeps me motivated is that seemingly complex phenomena such as tides can still be represented as 1-D time series. That happens because a first-order factor swamps out many of the other factors and then we can use that as a predictor.

Here is an interesting example of how a climate characteristic can be reduced into a simpler forcing.

Li, Guoqing, Haifeng Zong, and Qingyun Zhang. "27.3-day and average 13.6-day periodic oscillations in the Earth’s rotation rate and atmospheric pressure fields due to celestial gravitation forcing." Advances in Atmospheric Sciences 28 (2011): 45-58. http://www.iapjournals.ac.cn/aas/CN/article/downloadArticleFile.do?attachType=PDF&id=2065

This is not simply a tidal effect but impacts the atmospheric pressure as well. I may be missing something but the analysis here gives very good agreement with the data. This could be useful to consider in the El-Nino-related-data comment thread http://azimuth.mathforge.org/discussion/1415/some-el-nino-related-data/

of course there are many simple patterns: atmosphere gets colder when sun sets and hotter when sun rises. WINTER caused by some tilting of the earth's rotation axis and so on.

But I looked at the patterns slicer produced, specially the ones Mahler found, and they tell tales of patterns uncovered by 2D derivatives on a 4D grid/volume. These patterns could not be discerned by any 1D model. And these patterns are the suspect culprits in explaining El Nino.

I hope I am utterly wrong for all of our sakes.

Dara