[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] 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 then the scalar-valued covariance (denoted by COVV) is defined by[5] 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.
without loss of generality, it has been proved[7] that the maximal attainable numerator is
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 The impact of this adjustment is clearly visible in practice.