# Universal Correlation measure — ranking correlations

I have time series data of experimental observations for two related processes. I want to measure correlation for use in further analysis.

Correlation of the series changes over time and across different length sliding windows on the data. To clarify, I might want to look at correlation over 10, 20, 30, 40, ..., n periods, each of these essentially sliding windows across the data. Kind of analogous to looking at a bunch of simple moving average windows.

Historically and over future observations, one of these correlation windows will prove a better representation of the data than the rest. But the random nature of the underlying processes (whose distribution one may not know) makes settling on one window by evaluating the data historically an unsound approach.

Universal - A possible approach?

Information theory applied to the areas of data compression and portfolio allocation has produced what often gets referred to as a “Universal” approach to attacking a similar problem.

Thomas Cover a principle advocate for the idea saw the universal approach as a general method for multi-variate optimization of random processes even where one had no idea of the underlying distribution.

Cover's book, Elements of Information Theory, looks at a couple of examples of this idea. Cover seeing this as an optimization technique led me to ask this question here on the Mathematics SE site rather that say the statistics site.

Example - Universal data compression

In Universal Data Compression, one can see that an internet router can’t know the optimal data compression algorithm to use prior to its actually reading the packet of information or stream of data. Universal Data Compression addresses the problem with the following steps:

• Ranks each of its available compression algorithms as to their effectiveness after each stream of data has passed through the router.

• Calculates the cumulative ranking for each of the compression algorithms

• Identifies the mean cumulative rank weighted compression algorithm.

• Uses this mean cumulative rank weighted algorithm on the next stream of data it receives.

Note that the approach does not use a follow the leader strategy.

Over time this method will asymptopically approach the effectiveness of the best single algorithm that one could have picked at the beginning. A Darwin quote captures the key idea behind a Universal approach:

"It is not the strongest of the species that survive, nor the most intelligent, but the one most responsive to change."

Universal approaches keep one in a pretty good place most of the time, which does very well over time.

Universal Correlation

So it occurred to me that I might develop a related method, Universal Correlation

with the following steps:

• Calculate the cumulative ranking for each of the correlation windows

• Identify the mean cumulative rank weighted correlation window (or the one nearest the mean).

I'll then use this in another stage of analysis.

Now my question

What measure can I use to rank each correlation window’s effectiveness at each time step?

...

The following earlier question may have some bearing on an answer:

What is a common way to measure the “goodness of fit” of an individual data point to a correlation?

This may help clarify: At a given point in time, I have calculated correlation for some number of windows on the data up until that point in time (going back, 10 periods, 20 periods, 30, periods, ...).

Now I need a measure to rank the effectiveness or accuracy of each of the time periods correlation relative to each other.

Put yet another way: What do I rank the measures of correlation against?

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Don't believe the hype about universality. The concept of correlation is open-ended. You can define your own correlation. First of all even the definition of entropy evolves (Hartley-->Shannon-->Renyi). There's a new correlation method just published in Science 2011: Reshef et al "Detecting novel associations in large data sets." Also don't forget that pairwise correlation does not imply mutual correlation. Similarly, pairwise mutual information doesn't say much about mutual information. – alancalvitti Aug 28 '12 at 2:36
@alancalvitti - Can you elaborate re: "...hype abut universiality."? I started my search for something more robust than Pearson Correlation because its general use seems so arbitrary and unsatisfactory. Not having an obvious measure of it effectiveness underscores the problem. Since posting the question someone suggested that I look at Solomonoff algorithmic probability, suggesting that the most natural definition for universal correlation would be the mutual information in the algorithmic probability sense. 'K(X) - K(X|Y) ~ K(Y) - K(Y|X)'. Any thoughts appreciated. Thx for the comment. – Jagra Aug 28 '12 at 13:35
Sure I can expand on the above as soon as I have some time, but first are you aware of existing rank correlation methods? Also, re Kolmogorov-Solomonoff-Chaitin algorithmic information: Murray Gell-Mann pointed out that a random string has high KSC information content because it's incompressible, despite the fact that it carries no interpretable structure (ie, hierarchical organization). So every method has blind spots. – alancalvitti Aug 28 '12 at 14:33
PS I can post online a moving-quantile analysis of ~2000 year record of reconstructed precipitation record that I implemented in Mathematica. Will send you the link. Maybe you can tell me to what degree it fits your needs, since quantiles and quantile regression is an optimal L1-norm (ie, robust) approach. – alancalvitti Aug 28 '12 at 14:38
@alancalvitti - I've looked at the rank correlation methods available in Mathematica, but I don't have an immediate idea of how they might apply in this context. I'll do some research. I downloaded Chaitin's book Meta Math! The Quest of Omega, just last week. A friend had suggested it might have some useful ideas for this question. Indeed, every method has blind spots, just hoping for something not deaf, dumb, and blind ;-) re: your moving-quantile analysis, you can email me directly. You can find my address in my user profile or jagra24891atMypacks.net. – Jagra Aug 28 '12 at 15:43