The information given by a correlation coefficient

The information given by a correlation coefficient is not enough to define the dependence structure between random variables. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is a multivariate normal distribution. (See diagram above.) In the case of elliptical distributions it characterizes the (hyper-)ellipses of equal density, however, it does not completely characterize the dependence structure (for example, a multivariate t-distribution's degrees of freedom determine the level of tail dependence).
Distance correlation and Brownian covariance / Brownian correlation [9][10] were introduced to address the deficiency of Pearson's correlation that it can be zero for dependent random variables; zero distance correlation and zero Brownian correlation imply independence.
The correlation ratio is able to detect almost any functional dependency[citation needed][clarification needed], and the entropy-based mutual information, total correlation and dual total correlation are capable of detecting even more general dependencies. These are sometimes referred to as multi-moment correlation measures[citation needed], in comparison to those that consider only second moment (pairwise or quadratic) dependence.
The polychoric correlation is another correlation applied to ordinal data that aims to estimate the correlation between theorised latent variables.
One way to capture a more complete view of dependence structure is to consider a copula between them.


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