subselect (version 0.15.5)

gcd.coef: Computes Yanai's GCD in the context of the variable-subset selection problem

Description

Computes Yanai's Generalized Coefficient of Determination for the similarity of the subspaces spanned by a subset of variables and a subset of the full data set's Principal Components.

Usage

gcd.coef(mat, indices, pcindices =  NULL)

Value

The value of the GCD 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.

pcindices

a numerical vector of indices of Principal Components. By default, the first k PCs are chosen, where k is the cardinality of the subset of variables whose criterion value is being computed. If a vector of PCs is specified by the user, those PCs will be used for all cardinalities that were requested.

Details

Computes Yanai's Generalized Coefficient of Determination for the similarity of the subspaces spanned by a subset of variables (specified by indices) and a subset of the full-data set's Principal Components (specified by pcindices). Input data is expected in the form of a (co)variance or correlation matrix. If a non-square matrix is given, it is assumed to be a data matrix, and its correlation matrix is used as input. The number of variables (k) and of PCs (q) does not have to be the same.

Yanai's GCD is defined as: $$GCD = \frac{\mathrm{tr}(P_v\cdot P_c)}{\sqrt{k\cdot q}}$$ where \(P_v\) and \(P_c\) are the matrices of orthogonal projections on the subspaces spanned by the k-variable subset and by the q-Principal Component subset, respectively.

This definition is equivalent to: $$GCD = \frac{1}{\sqrt{k q}} \sum\limits_{i}(r_m)_i^2$$ where \((r_m)_i\) stands for the multiple correlation between the i-th Principal Component and the k-variable subset, and the sum is carried out over the q PCs (i=1,...,q) selected.

These definitions are also 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 GCD 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

Cadima, J. and Jolliffe, I.T. (2001), "Variable Selection and the Interpretation of Principal Subspaces", Journal of Agricultural, Biological and Environmental Statistics, Vol. 6, 62-79.

Ramsay, J.O., ten Berge, J. and Styan, G.P.H. (1984), "Matrix Correlation", Psychometrika, 49, 403-423.

Examples

## An example with a very small data set.

data(iris3) 
x<-iris3[,,1]
gcd.coef(cor(x),c(1,3))
## [1] 0.7666286
gcd.coef(cor(x),c(1,3),pcindices=c(1,3))
## [1] 0.584452
gcd.coef(cor(x),c(1,3),pcindices=1)
## [1] 0.6035127

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

data(swiss)
rvresults<-anneal(cor(swiss),2,nsol=4,niter=5,criterion="Rv")
gcd.coef(cor(swiss),rvresults$subsets)

##              Card.2
##Solution 1 0.4962297
##Solution 2 0.7092591
##Solution 3 0.4748525
##Solution 4 0.4649259