subselect (version 0.16.0)

rv.coef: Computes the RV-coefficient applied to the variable subset selection problem

Description

Computes the RV coefficient, measuring the similarity (after rotations, translations and global re-sizing) of two configurations of n points given by: (i) observations on each of p variables, and (ii) the regression of those p observed variables on a subset of the variables.

Usage

rv.coef(mat, indices)

Value

The value of the RV-coefficient.

Arguments

mat

the full data set's covariance (or correlation) matrix

indices

a numerical vector, matrix or 3-d array of integers giving the indices of the variables in the subset. If a matrix is specified, each row is taken to represent a different k-variable subset. If a 3-d array is given, it is assumed that the third dimension corresponds to different cardinalities.

Details

Input data is expected in the form of a (co)variance or correlation matrix of the full data set. If a non-square matrix is given, it is assumed to be a data matrix, and its correlation matrix is used as input. The subset of variables on which the full data set will be regressed is given by indices.

The RV-coefficient, for a (coumn-centered) data matrix (with p variables/columns) X, and for the regression of these columns on a k-variable subset, is given by: $$RV=\frac{\mathrm{tr}(X{X}^t\cdot (P_{v}X)(P_{v}X{)}^t)}{\sqrt{\mathrm{tr}((X{X}^t{)}^2)\cdot \mathrm{tr}(((P_{v}X)(P_{v}X{)}^t{)}^2)}}$$ where $P_v$ is the matrix of orthogonal projections on the subspace defined by the k-variable subset.

This definition is equivalent to the expression used in the code, which only requires the covariance (or correlation) matrix of the data under consideration.

The fact that indices can be a matrix or 3-d array allows for the computation of the RV values of subsets produced by the search functions anneal, genetic and improve (whose output option $subsets are matrices or 3-d arrays), using a different criterion (see the example below).

References

Robert, P. and Escoufier, Y. (1976), "A Unifying tool for linear multivariate statistical methods: the RV-coefficient", Applied Statistics, Vol.25, No.3, p. 257-265.

Examples

# A simple example with a trivially small data set

data(iris3) 
x<-iris3[,,1]
rv.coef(var(x),c(1,3))
## [1] 0.8659685


## An example computing the RVs of three subsets produced when the
## anneal function attempted to optimize the RM criterion (using an
## absurdly small number of iterations).

data(swiss)
rmresults<-anneal(cor(swiss),2,nsol=4,niter=5,criterion="Rm")
rv.coef(cor(swiss),rmresults$subsets)

##              Card.2
##Solution 1 0.8389669
##Solution 2 0.8663006
##Solution 3 0.8093862
##Solution 4 0.7529066