gcd.coef(mat, indices, pcindices = seq(1:length(indices)))indices.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 (co)variance 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.
Ramsay, J.O., ten Berge, J. and Styan, G.P.H. (1984), "Matrix Correlation", Psychometrika, 49, 403-423.
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] 0.6035127Run the code above in your browser using DataLab