ccr12.coef {subselect} | R Documentation |
First Squared Canonical Correlation for a multivariate linear hypothesis
Description
Computes the first squared canonical correlation. The maximization of this criterion is equivalent to the maximization of the Roy first root.
Usage
ccr12.coef(mat, H, r, indices,
tolval=10*.Machine$double.eps, tolsym=1000*.Machine$double.eps)
Arguments
mat |
the Variance or Total sums of squares and products matrix for the full data set. |
H |
the Effect description sums of squares and products matrix (defined in the same way as the |
r |
the Expected rank of the H matrix. See the |
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. |
tolval |
the tolerance level to be used in checks for
ill-conditioning and positive-definiteness of the 'total' and
'effects' (H) matrices. Values smaller than |
tolsym |
the tolerance level for symmetry of the
covariance/correlation/total matrix and for the effects ( |
Details
Different kinds of statistical methodologies are considered within the framework, of a multivariate linear model:
X = A \Psi + U
where X
is the (nxp) data matrix of
original variables, A
is a known (nxp) design matrix,
\Psi
an (qxp) matrix of unknown parameters and U
an (nxp)
matrix of residual vectors.
The ccr_1^2
index is related to the traditional test statistic
(the Roy first root) and measures the contribution of each subset to
an Effect characterized by the violation of a linear hypothesis of the form
C \Psi = 0
, where C
is a known cofficient matrix
of rank r. The Roy first root is the first eigen value of HE^{-1}
, where
H
is the Effect matrix and E
is the Error matrix.
The index ccr_1^2
is related to the Roy first root
(\lambda_1
) by:
ccr_1^2 =\frac{\lambda_1}{1+\lambda_1}
The fact that indices
can be a matrix or 3-d array allows for
the computation of the ccr_1^2
values of subsets produced
by the search
functions anneal
, genetic
,
improve
and
anneal
(whose output option $subsets
are
matrices or 3-d arrays), using a different criterion (see the example
below).
Value
The value of the ccr_1^2
coefficient.
Examples
## 1) A Linear Discriminant Analysis example with a very small data set.
## We considered the Iris data and three groups,
## defined by species (setosa, versicolor and virginica).
data(iris)
irisHmat <- ldaHmat(iris[1:4],iris$Species)
ccr12.coef(irisHmat$mat,H=irisHmat$H,r=2,c(1,3))
## [1] 0.9589055
## ---------------------------------------------------------------
## 2) An example computing the value of the ccr1_2 criteria for two
## subsets produced when the anneal function attempted to optimize
## the zeta_2 criterion (using an absurdly small number of iterations).
zetaresults<-anneal(irisHmat$mat,2,nsol=2,niter=2,criterion="zeta2",
H=irisHmat$H,r=2)
ccr12.coef(irisHmat$mat,H=irisHmat$H,r=2,zetaresults$subsets)
## Card.2
##Solution 1 0.9526304
##Solution 2 0.9558787
## ---------------------------------------------------------------