Day: February 13, 2009

Correlation and Directionality

This is the second in a series of postings about correlation modelling.  In the first posting we discussed the idea of correlation as representing a proxy model of dependency between random variables. In this posting, we discuss the idea the often overlooked concept that relationships of correlation do not necessarily imply any directionality.

In modelling time series of financial asset prices, it is common practice to correlate the changes (returns) in prices (within each period).  One way to see that the variables will not necessarily trend in the same direction is by simple reference to the formula for correlation (see for example Excel’s Help and search for the CORREL function). The formula is based on the differences of each point from the average. If for example one variable has a positive average change and the other a negative average change, then they will drift in different directions, even if the changes are positively correlated.  Correlation refers to the idea that each variable moves relative to its own average in a common way.

The screenshot shows an example of two times series with this property; one has a positive average drift and the other a negative average drift, and the series have a positive correlation coefficient of 34%.

Capturing the directionality between variables is something that requires a different modelling approach, such as establishing the long-run equilibrium of the spread, ratio, or differences between the prices and modelling these.  This is the topic of co-integrated time series, which are becoming more and more important in financial econometrics, and will be the subject of a future posting.

Dr. Michael Rees
Director of Training and Consulting

Okay, Charlie, Where do we go from here?

Two hundred years ago yesterday Charles Darwin was born.  It was the midst of the Industrial Revolution, and machines had just begun to replace human labor.  Darwin had some formal medical education and a good bit of informal scientific education.  He also had an imagination powerful enough to envision the links between geological time and biological variation. 

Darwin hypothesized about  biological vehicles for introducing variation in living organisms–he called these "gemmules"–but neither he nor anyone else in his generation had knowledge of genes.   He would have found remarkable the mathematical processes that emulate biological processes.  I am thinking, of course, of genetic algorithms and neural networks.  And I think he would have found it fascinating that our use of these mathematical stand-ins has progressed to the point where robotics–the latest, most sophisticated example of machines taking over for human labor–is an everyday occurrence. 

But I bet Darwin would have been blown away by by last week’s announcement of a robot that "evolves."  Engineers at Robert Gordon University (UK) have combined neural network technology, like those used to analyze CRM data, with evolutionary algorithms, like the ones used for genetic algorithm optimization, to create a robot that has a "brain" that gradually "evolves" an optimal system to control the robots movements.  The result is a ever more smoothly running robot.  In other words, this machine is now taking over the human work of improving itself.

Okay, Charlie, where do we go from here?