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This question and its answer might highlight my naivete regarding Brownian/distance correlation.

I'm using the difference between a matrix of distance correlations, as calculated by energy::dcor(), and absolute value Pearson correlations, as calculated by cor(), to highlight potential nonlinear dependencies introduced with a certain estimation technique.

In my resulting difference matrix, I have a handful of negative values indicating that the Pearson correlation is larger in magnitude than the distance correlation (range from -.07 to -.01).

First, is my approach adequate? If so, how do I explain why the Pearson correlations might be larger in magnitude than distance correlation?

MarianD
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topherwhatever
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  • For a bit of clarification: My understanding of distance correlation is that dCor = 1 (and r = 1) when two variables are perfectly co-linear. However, as you can see via this demonstration [link](https://en.wikipedia.org/wiki/Distance_correlation#/media/File:Distance_Correlation_Examples.svg) versus the identical for Pearson [link] (https://en.wikipedia.org/wiki/Correlation_and_dependence#/media/File:Correlation_examples2.svg), the correlation index for the very strongly dependent linear data is lower for distance correlations. Why? – topherwhatever Mar 11 '19 at 13:22

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Distance correlation or distance covariance is a measure of dependence between two paired random vectors of arbitrary, not necessarily equal dimension.

The population distance correlation coefficient is zero if and only if the random vectors are independent. Thus, distance correlation measures both linear and nonlinear association between two random variables or random vectors.

This is in contrast to Pearson's correlation, which can only detect linear association between two random variables.

Bane
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