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

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


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?

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Incidentally, can you link to the article in question? (The figure you're making could be caused by a couple of explanations, but it depends on the specific data). – David Robinson Sep 18 '12 at 19:34
For example, depending on what the figure specifically looks like, x and y could be entirely independent. Try running plot(rnorm(20000, rep(1:2, each=10000), .2), rlnorm(20000, 0, .8)) in R. It'll look roughly similar to your figure, even though x and y are perfectly independent. – David Robinson Sep 18 '12 at 19:38
@DavidRobinson, sorry the article is still under review and not publicly available. – Oliver Sep 19 '12 at 13:23

## migrated from stackoverflow.comSep 18 '12 at 23:59

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.

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 Your clarification on the term "correlation" <--> linear relationship is correct. It shows a relationship, just not the correlation. I am still trying to find the right way to describe the relationship though. – Oliver Sep 19 '12 at 13:28 It looks like a mixture model: 2 tight distributions centered around X1 and X2 respectively. That's all I can say, since I don't know where the numbers come from. It doesn't look as if knowing a Y value helps you predict X - or vice versa. – Placidia Sep 19 '12 at 14:12

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).

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 thanks for the references. Now I think about it, X Y does have a correlation, it is just not conventional "linear" type. I don't know the statistic term for this, maybe the article you referred will shed some light to it. thx – Oliver Sep 19 '12 at 1:57 Yes, regardless of if it's called "correlation" or something else, X and Y are certainly not independent. – Backlin Sep 21 '12 at 10:45