RV coefficient

The definition of the RV-coefficient makes use of ideas[5] 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

${\displaystyle \Sigma _{XY}=\operatorname {E} (XY^{\top })\,,}$

then the scalar-valued covariance (denoted by COVV) is defined by[5]

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

The scalar-valued variance is defined correspondingly:

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

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.[5]

Then the RV-coefficient is defined by[5]

${\displaystyle \mathrm {RV} (X,Y)={\frac {\operatorname {COVV} (X,Y)}{\sqrt {\operatorname {VAV} (X)\operatorname {VAV} (Y)}}}\,.}$

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.[6]

Given known diagonal blocks ${\displaystyle \Sigma _{XX}}$ and ${\displaystyle \Sigma _{YY}}$ of dimensions ${\displaystyle p\times p}$ and ${\displaystyle q\times q}$ respectively, assuming that ${\displaystyle p\leq q}$ without loss of generality, it has been proved[7] that the maximal attainable numerator is ${\displaystyle \operatorname {Tr} (\Lambda _{X}\Pi \Lambda _{Y}),}$ where ${\displaystyle \Lambda _{X}}$ (resp. ${\displaystyle \Lambda _{Y}}$) denotes the diagonal matrix of the eigenvalues of ${\displaystyle \Sigma _{XX}}$(resp. ${\displaystyle \Sigma _{YY}}$) sorted decreasingly from the upper leftmost corner to the lower rightmost corner and ${\displaystyle \Pi }$ is the ${\displaystyle p\times q}$ matrix ${\displaystyle (I_{p}\ 0_{p\times (q-p)})}$.

In light of this, Mordant and Segers[7] proposed an adjusted version of the RV coefficient in which the denominator is the maximal value attainable by the numerator. It reads

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

The impact of this adjustment is clearly visible in practice.[7]

• Congruence coefficient
• Distance correlation