compositions (version 1.01-1)

princomp.acomp: Principal component analysis for Aitchison compositions

Description

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

Usage

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

Arguments

x

a acomp-dataset (in princomp) or a result from princomp.acomp

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

scale.sdev

the multiple of sigma to use plotting the loadings

main

title of the plot

object

a fitted princomp.acomp object

newdata

another compositional dataset of class acomp

...

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 robustnessInCompositions for details.

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". The last eigenvector is removed since it should contain the irrelevant scaling.
• center
  the clr-transformed vector of means used to center the dataset
• Center
  the acomp 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
  compositions that represent a perturbation with the vectors represented by the loadings of each of the factors
• DownLoadings
  compositions that represent a perturbation with the inverse 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.

Details

As a metric euclidean space the Aitchison simplex has its own principal component analysis, that should be performed in terms of the covariance matrix and not in terms of the meaningless correlation matrix. To aid the interpretation we added some extra functionality to a normal princomp(clr(x)). First of all the result contains as additional information the compositional representation of the returned vectors in the space of the data: the center as a composition Center, and the loadings in terms of a composition to perturbe with, either positively (Loadings) or negatively (DownLoadings). The Up- and DownLoadings are normalized to the number of parts in the simplex and not to one to simplify the interpretation. A value of about one means no change in the specific component. To avoid confusion the meaningless last principal component is removed. 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 biplot strongly differs from a classical one. The relevant variables are not the arrows drawn (one for each component), but rather the links (i.e., the differences) between two arrow heads, which represents the log-ratio between the two components represented by the arrows. The compositional loading plot is introduced with this package. The loadings of all component can be seen as an orthogonal basis in the space of clr-transformed data. These vectors are displayed by a barplot with their corresponding composition. For a better interpretation the total of these compositons is set to the number of parts in the composition, such that a portion of one means no effect. This is similar to (but not exactly the same as) 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 composition beeing represented by the unit vector of loadings in the clr-transformed space. If scale.sdev is numeric we use this composition 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.

References

Aitchison, J, C. Barcel'o-Vidal, J.J. Egozcue, V. Pawlowsky-Glahn (2002) A consise guide to the algebraic geometric structure of the simplex, the sample space for compositional data analysis, Terra Nostra, Schriften der Alfred Wegener-Stiftung, 03/2003 Aitchison, J. and M. Greenacre (2002) Biplots for Compositional Data Journal of the Royal Statistical Society, Series C (Applied Statistics) 51 (4) 375-392 http://ima.udg.es/Activitats/CoDaWork03 http://ima.udg.es/Activitats/CoDaWork05

See Also

clr,acomp, relativeLoadings princomp.aplus, princomp.rcomp, barplot.acomp, mean.acomp, var.acomp

Examples

data(SimulatedAmounts)
pc <- princomp(acomp(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))