princomp.rplus: Principal component analysis for real amounts

Description

A principal component analysis is done in real geometry (i.e. using iit-transform).

Usage

# S3 method for rplus
princomp(x,...,scores=TRUE,center=attr(covmat,"center"),
                           covmat=var(x,robust=robust,giveCenter=TRUE),
                           robust=getOption("robust"))
 # S3 method for princomp.rplus
print(x,...)
 # S3 method for princomp.rplus
plot(x,y=NULL,...,npcs=min(10,length(x$sdev)),
          type=c("screeplot","variance","biplot","loadings","relative"),
          main=NULL,scale.sdev=1)
 # S3 method for princomp.rplus
predict(object,newdata,...)

Value

princomp gives an object of type

c("princomp.rcomp","princomp") with the following content:

sdev: the standard deviation of the principal components
loadings: the matrix of variable loadings (i.e., a matrix which columns contain the eigenvectors). This is of class "loadings"
Loadings: the loadings as an "rmult"-object
center: the iit-transformed vector of means used to center the dataset
Center: the rplus vector of means used to center the dataset (center and Center have no difference, except that the second has a class)
scale: the scaling applied to each variable
n.obs: number of observations
scores: if scores = TRUE, the scores of the supplied data on the principal components. Scores are coordinates in a basis given by the principal components and thus not compositions
call: the matched call
na.action: not clearly understood

predict returns a matrix of scores of the observations in the

newdata dataset.

The other routines are mainly called for their side effect of plotting or printing and return the object x.

Arguments

x: an rplus-dataset (for princomp) or a result from princomp.rplus
y: not used
scores: a logical indicating whether scores should be computed or not
npcs: the number of components to be drawn in the scree plot
type: type of the plot: "screeplot" is a lined screeplot, "variance" is a boxplot-like screeplot, "biplot" is a biplot, "loadings" displays the loadings as a barplot
scale.sdev: the multiple of sigma to use when plotting the loadings
main: title of the plot
object: a fitted princomp.rplus object
newdata: another amount dataset of class rcomp
...: further arguments to pass to internally-called functions
covmat: provides the covariance matrix to be used for the principle component analysis
center: provides the be used for the computation of scores
robust: Gives the robustness type for the calculation of the covariance matrix. See var.rmult for details.

Details

Mainly a princomp(iit(x)) is performed. Note all parts in a composition or in an amount vector share a natural scaling. Therefore, they do not need any preliminary standardization (which in fact would produce a loss of important information). For this reason, princomp.rplus works on the covariance matrix.
The plot routine provides screeplots (type = "s",type= "v"), biplots (type = "b"), plots of the effect of loadings (type = "b") in scale.sdev*sdev-spread, and loadings of pairwise differences (type = "r").
The interpretation of a screeplot does not differ from ordinary screeplots. It shows the eigenvalues of the covariance matrix, which represent the portions of variance explained by the principal components.
The interpretation of the biplot uses, additionally to the classical interperation, a compositional concept: the differences between two arrowheads can be interpreted as the shift of mass between the two components represented by the arrows.
The amount loading plot is more or less a standard loadings plot. The loadings are displayed by a barplot as positive and negative changes of amounts.
The loadings plot can work in two different modes: If scale.sdev is set to NA it displays the amount vector being represented by the unit vector of loadings in the iit-transformed space. If scale.sdev is numeric we use this amount vector scaled by the standard deviation of the respective component.
The relative plot displays the relativeLoadings as a barplot. The deviation from a unit bar shows the effect of each principal component on the respective differences.

Examples

Run this code

data(SimulatedAmounts)
pc <- princomp(rplus(sa.lognormals5))
pc
summary(pc)
plot(pc)           #plot(pc,type="screeplot")
plot(pc,type="v")
plot(pc,type="biplot")
plot(pc,choice=c(1,3),type="biplot")
plot(pc,type="loadings")
plot(pc,type="loadings",scale.sdev=-1) # Downward
plot(pc,type="relative",scale.sdev=NA) # The directions
plot(pc,type="relative",scale.sdev=1) # one sigma Upward 
plot(pc,type="relative",scale.sdev=-1) # one sigma Downward
biplot(pc)
screeplot(pc)
loadings(pc)
relativeLoadings(pc,mult=FALSE)
relativeLoadings(pc)
relativeLoadings(pc,scale.sdev=1)
relativeLoadings(pc,scale.sdev=2)

pc$sdev^2
cov(predict(pc,sa.lognormals5))