In statistics, the RV coefficient is a multivariate generalization of the squared Pearson correlation coefficient (because the RV coefficient takes values between 0 and 1). 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:
The definition of the RV-coefficient makes use of ideas 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
then the scalar-valued covariance (denoted by COVV) is defined by
The scalar-valued variance is defined correspondingly:
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
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