# 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]

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]

## Definitions

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

$\Sigma_{XY}=E( X^TY) \,,$

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

$\mathrm{COVV}(X,Y)= Tr(\Sigma_{XY}\Sigma_{YX}) \, .$

The scalar-valued variance is defined correspondingly:

$\mathrm{VAV}(X)= Tr(\Sigma_{XX}^2) \, .$

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]

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