How to statistically describe chain-like patterns in bivariate data?

Question

I am reading an article that explores correlation between two variables, X and Y. Usually, if the scatter plot show something like this, we can claim that there is a strong correlation of X and Y.

Y
|
|              o
|           o o
|          o
|        o o    
|       o
|    o
|  o
+--------------------->X

What about the following case?

Y
|    o          o
|    o          o
|     o        oo
|     o         o
|    oo         o
|    o          o 
|    o          o
| o oo o oo oo o o o oo
+--------------------->X
    x1          x2

Basically, the scatter plot show strong clustering and spikes around a few data point along X axis, e.g. x1 and x2.

What kind of statistical property does this imply?

Answer 1

The correlation between X and Y measures the presence or absence of a linear relationship between them. If y=aX+B -- a pure line with no scatter and no error - the correlation will be 1 (if a>0) and -1 (if a < 0). It is not a general measure of causality or relatedness.

Your second scatter plot shows no linearity at all. A correlation is not an appropriate measure in this case.

Answer 2

The pattern you describe is completely uncorrelated but can be picked up with some information theoretic measure such as mutual information. There are a number of packages implementing it, e.g. entropy and infotheo.

There is also a quite recent dependency measure called MIC, implemented in java with an R wrapper that you can get at www.exploredata.net. It has a bunch of nice properties such as being on the [0, 1] interval and not favoring any particular type of relationship (e.g. linear) over others. It is not entirely uncontroversial though, so I recommend reading the orgignal article by Reshef et al (Science 2012).