While I spend most of my StackExchange time on MathematicaSE, I'm in the business and follow the questions and answers on this site with great interest.
Recently questions like the following (and some others):
- Time-series similarity measures
- How are correlation and cointegration related?
- Can the concept of entropy be applied to financial time series?
- What is the intuition behind cointegration?
And even something like this from the Wolfram blog:
have me thinking about the different approaches available to analyze time-series similarity.
These approaches could include:
- factor analysis,
- graph theory,
- cluster analysis, and
- probably others
They certainly have different uses and apply to different aspects of time-series. They all seem to have something to contribute to understanding time-series similarity.
So this leads me to a number of questions (or maybe just restatements of the same question):
- Can anyone provide a good intuition describing how they all relate to one another?
I hope here for something beyond things like some of the excellent discussion of the differences between correlation and co-integration and the kinds of series to which they apply that have appeared elsewhere on this site. I hope for something more along the line of...
Does some point of view or insight exist that provides a better idea of all of these measures?
From some meta-perspective could one view these approaches as aspects of some broader idea?
Could some meta-perspective recast all of these approaches so one could view them in similar units of measure?
Not even certain this is possible. Even as I write out this post it seems like asking someone to deliver something like a unified field theory of time-series similarity.
Still, if possible it might prove useful so, maybe the questions will spark an interesting answer or two.
Of course, any recommendations of papers or other resources that explore any of this appreciated.
Updated 15 Aug 2012 2:00 PM EDT
The following paper gives an example of the kind of thinking that moves in the direction of a broader answer to this question: Empirical Entropy Manipulation and Analysis.
The section on casting (at least some) PCA problems as entropy maximization problems follows:
And yes PCA can be done by eigenvalue decomposition of a data covariance (or correlation) matrix.
So, clearly some of these kinds of analysis have some kind of relationship.
Perhaps some view can include more of them. Maybe an entropy or information description of correlation or cointegration. I'm not sure, but to me it seems a interesting question.