Computes the RM coefficient for variable subset selection
Computes the RM coefficient, measuring the similarity of the spectral decompositions of a p-variable data matrix, and of the matrix which results from regressing all the variables on a subset of only k variables.
rm.coef(mat, indices)
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. |
Computes the RM coefficient that measures the similarity of the spectral decompositions of a p-variable data matrix, and of the matrix which results from regressing those variables on a subset (given by "indices") of the variables. 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 definition of the RM coefficient is as follows:
RM = sqrt(tr(X' Pv X)/tr(X'X))
where X is the full (column-centered) data matrix and Pv is the matrix of orthogonal projections on the subspace spanned by a k-variable subset.
This definition is equivalent to:
RM = sqrt(sum_i(lambda_i r_i^2)/sum_j(lambda_j))
where lambda_i stands for the i-th largest
eigenvalue of the covariance matrix defined by X and
r stands for the multiple correlation between the
i-th Principal Component and the k-variable subset.
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 value of the RM coefficient.
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.
McCabe, G.P. (1986) "Prediction of Principal Components by Variable Subsets", Technical Report 86-19, Department of Statistics, Purdue University.
Ramsay, J.O., ten Berge, J. and Styan, G.P.H. (1984), "Matrix Correlation", Psychometrika, 49, 403-423.
## An example with a very small data set. data(iris3) x<-iris3[,,1] rm.coef(var(x),c(1,3)) ## [1] 0.8724422 ## An example computing the RMs 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") rm.coef(cor(swiss),rvresults$subsets) ## Card.2 ##Solution 1 0.7982296 ##Solution 2 0.7945390 ##Solution 3 0.7649296 ##Solution 4 0.7623326
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