Wednesday, February 4, 2015

Taken's Theorem and Dynamic Correlation

As much as I dislike regurgitating content instead of producing it, this Sugihara et al. paper is possibly the coolest thing I have ever seen, and has been for a couple years now.

Since MathJax (what we are using to format everything in $\LaTeX$) probably doesn't have BibTeX support, I'm going to go ahead and do an academic no-no by just providing the link to the paper, and no citation[1].

The goal of this paper is to introduce a new method, convergent cross mapping (CCM), a test meant to help determine whether one event in a nonlinear system causes another. As the introduction notes, two populations interacting nonlinearly can go through phases where the behavior is similar, they behave oppositely, or there appears to be no relation. This makes applying traditional measures of correlation or causation useless in such situations.

Enter Taken's theorem: a theorem stating (in extreme layman's terms) that it is possible to 'reconstruct' a chaotic attractor using one of its components. WHICH IS SO COOL. Sugihara et al. concluded that if two components were members of the same system, they would not only be able to reconstruct the original system, but one component would predict the behavior of the other. Hence the 'nearest neighbors' to a data point in the first component should be associated timewise with the nearest neighbors to the corresponding data point in the second component, providing the systems are related, and this predictive ability should get better as more data are taken into account. This is the gist of CCM, which is explained more clearly below and in the paper.

What happens when this method is applied to real data on sardine and anchovy population? Read the paper to find out! If you're a member of the general public, it's less intense and more explanatory than math papers generally tend to go, and a great read if you're even a tiny bit into population ecology. (Not a whole lot is said on the exact implementation, but the numbers they're getting look like correlation coefficients between a variable and its nearest-neighbor estimate as more data is added. I should try to do this in MATLAB and post code.)

Did I mention the videos? George Sugihara's son made two brilliant videos to illustrate where the idea came from. (I want to be his friend.) Here's one on Taken's theorem:

There's also one demonstrating the manifold reconstruction:

Lastly, a brief description of how CCM works:

Yeah. I'm not kidding about this being the coolest thing ever!

I learned of this paper during a talk George Sugihara gave in 2013. Of course, the videos are very pretty, but the topic also illustrates something bigger: how applied mathematicians can make breakthroughs by studying "useless" theoretical topics. Some of my old professors were fond of claiming all pure math eventually becomes applied math. This is a great recent example of such creativity; who would have expected Taken's theorem to relate to causality in ecosystems?

Theory: it's what separates us from the engineers! Or just another excuse for the applied folks to read analysis textbooks.

[1]... Nope, my heart won't let me do it. Here's the citation:
Detecting Causality in Complex Ecosystems. George Sugihara, Robert May, Hao Ye, Chih-hao Hsieh, Ethan Deyle, Michael Fogarty, and Stephan Munch. Science. 26 October 2012: 338 (6106), 496-500. Published online 20 September 2012 [DOI:10.1126/science.1227079]

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