Asymmetric weighted discriminant coordinates as defined in Hennig (2003). Asymmetric discriminant projection means that there are two classes, one of which is treated as the homogeneous class (i.e., it should appear homogeneous and separated in the resulting projection) while the other may be heterogeneous. The principle is to maximize the ratio between the projection of a between classes separation matrix and the projection of the covariance matrix within the homogeneous class. Points are weighted according to their (robust) Mahalanobis distance to the homogeneous class.
awcoord(xd, clvecd, clnum=1, mahal="square", method="classical",
clweight=switch(method,classical=FALSE,TRUE),
alpha=0.99, subsample=0, countmode=1000, ...)List with the following components
eigenvalues in descending order.
columns are coordinates of projection basis vectors.
New points x can be projected onto the projection basis vectors
by x %*% units
projections of xd onto units.
the data matrix; a numerical object which can be coerced to a matrix.
integer vector of class numbers; length must equal
nrow(xd).
integer. Number of the homogeneous class.
"md" or "square". If "md", the points are weighted by the
square root of the alpha-quantile of the
corresponding chi squared distribution
over the roots of their Mahalanobis distance to the
homogeneous class, unless
this is smaller than 1. If "square" (which is recommended), the
(originally squared) Mahalanobis distance and the
unrooted quantile is used.
one of
"mve", "mcd" or "classical". Covariance matrix used within the
homogeneous class and for the computation of the Mahalanobis distances.
"mcd" and "mve" are robust covariance matrices as implemented
in cov.rob. "classical" refers to the classical
covariance matrix.
logical. If FALSE, only the points of the
heterogeneous class are weighted. This, together with
method="classical", computes AWC as defined in Hennig (2003). If
TRUE, all points are weighted. This, together with
method="mcd", computes ARC as defined in Hennig (2003).
numeric between 0 and 1. The corresponding quantile of the chi squared distribution is used for the downweighting of points. Points with a smaller Mahalanobis distance to the homogeneous class get full weight.
integer. If 0, all points are used. Else, only a
subsample of subsample of the points is used.
optional positive integer. Every countmode
algorithm runs awcoord shows a message.
no effect
Christian Hennig christian.hennig@unibo.it https://www.unibo.it/sitoweb/christian.hennig/en/
The square root of the homogeneous classes covariance matrix
is inverted by use of
tdecomp, which can be expected to give
reasonable results for singular within-class covariance matrices.
Hennig, C. (2004) Asymmetric linear dimension reduction for classification. Journal of Computational and Graphical Statistics 13, 930-945 .
Hennig, C. (2005) A method for visual cluster validation. In: Weihs, C. and Gaul, W. (eds.): Classification - The Ubiquitous Challenge. Springer, Heidelberg 2005, 153-160.
plotcluster for straight forward discriminant plots.
discrproj for alternatives.
rFace for generation of the example data used below.
set.seed(4634)
face <- rFace(600,dMoNo=2,dNoEy=0)
grface <- as.integer(attr(face,"grouping"))
awcf <- awcoord(face,grface==1)
# awcf2 <- ancoord(face,grface==1, method="mcd")
plot(awcf$proj,col=1+(grface==1))
# plot(awcf2$proj,col=1+(grface==1))
# ...done in one step by function plotcluster.
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