Principal component analysis for amounts in log geometry
Description
A principal component analysis is done in the Aitchison geometry (i.e. ilttransform). Some gimics simplify the interpretation of the computed components as perturbations of amounts.
Usage
1 2 3 4 5 6 7 8 9 10 11 12  ## S3 method for class 'aplus'
princomp(x,...,scores=TRUE,center=attr(covmat,"center"),
covmat=var(x,robust=robust,giveCenter=TRUE),
robust=getOption("robust"))
## S3 method for class 'princomp.aplus'
print(x,...)
## S3 method for class 'princomp.aplus'
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 class 'princomp.aplus'
predict(object,newdata,...)

Arguments
x 
an aplus dataset (for princomp) or a result from princomp.aplus 
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: 
scale.sdev 
the multiple of sigma to use when plotting the loadings 
main 
title of the plot 
object 
a fitted princomp.aplus object 
newdata 
another amount dataset of class aplus 
... 
further arguments to pass to internallycalled 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 
Details
As a metric euclidean space, the positive real space described in
aplus
has its own
principal component analysis, that can be performed either in terms of the
covariance matrix or the correlation matrix. However, since all parts in a composition
or in an amount vector share a natural scaling, they do not need the
standardization (which in fact would produce a loss of important information).
For this reason, princomp.aplus
works on the covariance matrix.
To aid the interpretation we added some extra functionality to a
normal princomp(ilt(x))
. First of all the result contains as
additional information the amount representation of
returned vectors in the space of the data: the center as an amount
Center
, and the loadings in terms of amounts to perturbe
with, either positively
(Loadings
) or negatively (DownLoadings
). The Up and
DownLoadings are normalized to the number of parts
and not to one to simplify the interpretation. A value of about one
means no change in the specific component.
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 (log)ratios (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 the biplot uses, additionally to the
classical one, a compositional concept: The
differences between two arrowheads can be interpreted as logratios
between the two components represented by the arrows.
The amount loading plot is introduced with this
package. The loadings of all component can be seen as an orthogonal basis
in the space of ilt
transformed data. These vectors are displayed by a barplot with
their corresponding amounts. A portion of one means no change of this
part. This is equivalent to a zero loading in a real principal component analysis.
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 ilttransformed 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 ratio. The
interpretation of the ratios plot may only be done in an Aitchisoncompositional framework
(see princomp.acomp
).
Value
princomp
gives an object of type
c("princomp.acomp","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

center 
the ilttransformed vector of means used to center the dataset 
Center 
the 
scale 
the scaling applied to each variable 
n.obs 
number of observations 
scores 
if 
call 
the matched call 
na.action 
not clearly understood 
Loadings 
vectors of amounts that represent a perturbation with the vectors represented by the loadings of each of the factors 
DownLoadings 
vectors of amounts that represent a perturbation with the inverses of the vectors represented by the loadings of each of the factors 
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
.
Author(s)
K.Gerald v.d. Boogaart http://www.stat.boogaart.de
See Also
ilt
,aplus
, relativeLoadings
princomp.acomp
, princomp.rplus
,
barplot.aplus
, mean.aplus
,
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  data(SimulatedAmounts)
pc < princomp(aplus(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$Loadings
pc$DownLoadings
barplot(pc$Loadings)
pc$sdev^2
cov(predict(pc,sa.lognormals5))
