Perform Canonical Variate Analysis (CVA)

This function appends the biplot object with elements resulting from performing CVA.

CVA(bp, classes=bp$classes, dim.biplot = c(2, 1, 3), e.vects = 1:ncol(bp$X),
           weightedCVA = "weighted", show.class.means = TRUE,
           low.dim = "sample.opt")

Arguments

bp: an object of class biplot obtained from preceding function biplot().
classes: a vector of the same length as the number of rows in the data matrix with the class indicator for the samples.
dim.biplot: the dimension of the biplot. Only values 1, 2 and 3 are accepted, with default 2.
e.vects: the vector indicating which eigenvectors (canonical variates) should be plotted in the biplot, with default 1:dim.biplot.
weightedCVA: a character string indicating which type of CVA to perform. One of "weighted" (default) for a weighted CVA to be performed (The centring matrix will be a diagonal matrix with the class sizes (\(\mathbf{C} = \mathbf{N}\)), "unweightedCent" for unweighted CVA to be performed (The centring matrix is the usual centring matrix (\(\mathbf{C} = \mathbf{I}_{G} - G^{-1}\mathbf{1}_{G}\mathbf{1}_{G}'\))) or "unweightedI" for unweighted CVA to be performed while retaining the weighted centroid (The centring matrix is an indicator matrix (\(\mathbf{C} = \mathbf{I}_{G}\))).
show.class.means: a logical value indicating whether to plot the class means on the biplot.
low.dim: a character string indicating which method to use to construct additional dimension(s) if the dimension of the canonical space is smaller than dim.biplot. One of "sample.opt" (default) for maximising the sample predictivity of the individual samples in the biplot or "Bhattacharyya.dist" which is based on the decomposition of the Bhattacharyya distance into a component for the sample means and a component for the dissimilarity between the sample covariance matrices.

Value

Object of class CVA with the following elements:

X: the matrix of the centered and scaled numeric variables.
Xcat: the data frame of the categorical variables.
raw.X: the original data.
classes: the vector of category levels for the class variable. This is to be used for colour, pch and cex specifications.
na.action: the vector of observations that have been removed.
center: a logical value indicating whether \(\mathbf{X}\) is centered.
scaled: a logical value indicating whether \(\mathbf{X}\) is scaled.
means: the vector of means for each numerical variable.
sd: the vector of standard deviations for each numerical variable.
n: the number of observations.
p: the number of variables.
group.aes: the vector of category levels for the grouping variable. This is to be used for colour, pch and cex specifications.
g.names: the descriptive names to be used for group labels.
g: the number of groups.
Title: the title of the biplot rendered.
Lmat: the matrix for transformation to the canonical space.
Linv: the inverse of \(\mathbf{L}\).
eigenvalues: the vector of eigenvalues of the two-sided eigenvalue problem.
Z: the matrix with each row containing the details of the points to be plotted (i.e. coordinates).
ax.one.unit: one unit in the positive direction of each biplot axis.
Gmat: the indicator matrix defining membership of the classes.
Xmeans: the matrix of the class means.
Zmeans: the matrix of the class mean coordinates that are plotted in the biplot.
e.vects: the vector indicating which canonical variates are plotted in the biplot.
Cmat: the centring matrix based on different choices of weighting described in arguments.
Bmat: the between class sums of squares and cross products matrix.
Wmat: the within class sums of squares and cross products matrix.
Mrr: the matrix used for prediction from the canonical space (the inverse of \(\mathbf{M}=\mathbf{LV})\).
Mr: the first r dimensions of the solution to be plotted.
Nmat: the matrix with the class sizes on the diagonal.
lambda.mat: the matrix with the eigenvalues of \(\mathbf{W}^{-1/2}\mathbf{BW}^{-1/2}\) on the diagonal.
class.means: a logical value indicating whether the class means should be plotted in the biplot.
dim.biplot: the dimension of the biplot.
low.dim: the method used to construct additional dimension(s).

Examples

biplot(iris[,1:4]) |> CVA(classes=iris[,5])
#> Object of class biplot, based on 150 samples and 4 variables.
#> 4 numeric variables.
#> 3 classes: setosa versicolor virginica 
# create a CVA biplot
biplot(iris[,1:4]) |> CVA(classes=iris[,5]) |> plot()

Perform Canonical Variate Analysis (CVA)

Arguments

Value

See also

Examples