RV coefficient

In statistics, the RV coefficient^[1] is a multivariate generalization of the squared Pearson correlation coefficient (because the RV coefficient takes values between 0 and 1).^[2] It measures the closeness of two set of points that may each be represented in a matrix.

The major approaches within statistical multivariate data analysis can all be brought into a common framework in which the RV coefficient is maximised subject to relevant constraints. Specifically, these statistical methodologies include:^[1]

• principal component analysis
• canonical correlation analysis
• multivariate regression
• statistical classification (linear discrimination).

One application of the RV coefficient is in functional neuroimaging where it can measure the similarity between two subjects' series of brain scans^[3] or between different scans of a same subject.^[4]

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

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

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

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

The scalar-valued variance is defined correspondingly:

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

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]

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

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

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