RV coefficient

The definition of the RV-coefficient makes use of ideas{{Cite journal

| author1 = Escoufier, Y.

| title = Le Traitement des Variables Vectorielles

| journal = Biometrics

| year = 1973

| volume = 29

| issue = 4

| pages = 751–760

| doi = 10.2307/2529140

| publisher = International Biometric Society

| jstor = 2529140

}}

concerning the definition of scalar-valued quantities which are called the "variance" and "covariance" of vector-valued random variables. Note that standard usage is to have matrices for the variances and covariances of vector random variables.

Given these innovative definitions, the RV-coefficient is then just the correlation coefficient defined in the usual way.

Suppose that X and Y are matrices of centered random vectors (column vectors) with covariance matrix given by

:$\backslash Sigma\_\{XY\}=\backslash operatorname\{E\}(\; XY^\backslash top)\; \backslash ,,$

then the scalar-valued covariance (denoted by COVV) is defined by

:$\backslash operatorname\{COVV\}(X,Y)=\; \backslash operatorname\{Tr\}(\backslash Sigma\_\{XY\}\backslash Sigma\_\{YX\})\; \backslash ,\; .$

The scalar-valued variance is defined correspondingly:

:$\backslash operatorname\{VAV\}(X)=\; \backslash operatorname\{Tr\}(\backslash Sigma\_\{XX\}^2)\; \backslash ,\; .$

With these definitions, the variance and covariance have certain additive properties in relation to the formation of new vector quantities by extending an existing vector with the elements of another.

Then the RV-coefficient is defined by

:$\backslash mathrm\{RV\}(X,Y)\; =\; \backslash frac$

{ \operatorname{COVV}(X,Y) }

{ \sqrt{ \operatorname{VAV}(X) \operatorname{VAV}(Y) } } \, .

Even though the coefficient takes values between 0 and 1 by construction, it seldom attains values close to 1 as the denominator is often too large with respect to the maximal attainable value of the denominator.{{Cite journal

| author1 = Pucetti, G.

| title = Measuring Linear Correlation Between Random Vectors

| journal = SSRN

| year = 2019

| url = http://dx.doi.org/10.2139/ssrn.3116066s

}}

Given known diagonal blocks $\backslash Sigma\_\{XX\}$ and $\backslash Sigma\_\{YY\}$ of dimensions $p\backslash times\; p$ and $q\backslash times\; q$ respectively, assuming that $p\; \backslash le\; q$ without loss of generality, it has been proved{{Cite journal

| author1 = Mordant Gilles

| author2 = Segers Johan

| title = Measuring dependence between random vectors via optimal transport,

| journal = Journal of Multivariate Analysis

| volume = 189

| year = 2022

}}

that the maximal attainable numerator is $\backslash operatorname\{Tr\}(\backslash Lambda\_X\; \backslash Pi\; \backslash Lambda\_Y),$

where $\backslash Lambda\_X$ (resp. $\backslash Lambda\_Y$) denotes the diagonal matrix of the eigenvalues of $\backslash Sigma\_\{XX\}$(resp. $\backslash Sigma\_\{YY\}$) sorted decreasingly from the upper leftmost corner to the lower rightmost corner and $\backslash Pi$ is the $p\; \backslash times\; q$ matrix $(I\_p\; \backslash \; 0\_\{p\backslash times\; (q-p)\}\; )$.

In light of this, Mordant and Segers proposed an adjusted version of the RV coefficient in which the denominator is the maximal value attainable by the numerator. It reads

:$\backslash bar\{\backslash operatorname\{RV\}\}(X,Y)\; =\; \backslash frac\{\backslash operatorname\{Tr\}(\backslash Sigma\_\{XY\}\backslash Sigma\_\{YX\})\}\{\backslash operatorname\{Tr\}(\backslash Lambda\_X\; \backslash Pi\; \backslash Lambda\_Y)\}\; =\; \backslash frac\{\backslash operatorname\{Tr\}(\backslash Sigma\_\{XY\}\backslash Sigma\_\{YX\})\}\{\backslash sum\_\{j=1\}^\{min(p,q)\}\; (\backslash Lambda\_X)\_\{j,j\}\; (\backslash Lambda\_Y)\_\{j,j\}\}.$

The impact of this adjustment is clearly visible in practice.

• Congruence coefficient
• Distance correlation

{{Reflist}}

Category:Covariance and correlation