princomp.aplus {compositions}

Principal component analysis for amounts in log geometry

Description

A principal component analysis is done in the Aitchison geometry (i.e. ilt-transform). Some gimics simplify the interpretation of the computed components as perturbations of amounts.

Usage

## 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: "screeplot" is a lined screeplot, "variance" is a boxplot-like screeplot, "biplot" is a biplot, "loadings" displays the loadings as a barplot.acomp

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 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

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 log-ratios 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 ilt-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 ratio. The interpretation of the ratios plot may only be done in an Aitchison-compositional 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 "loadings".

center

the ilt-transformed vector of means used to center the dataset

Center

the aplus vector of means used to center the dataset

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

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

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))