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\name{pcoa}
\alias{pcoa}
\alias{biplot.pcoa}
\title{Principal Coordinate Analysis}
\description{
Function \code{\link{pcoa}} computes principal coordinate decomposition
(also called classical scaling) of a distance matrix D (Gower 1966). It
implements two correction methods for negative eigenvalues.
}
\usage{
pcoa(D, correction="none", rn=NULL)
\method{biplot}{pcoa}(x, Y=NULL, plot.axes = c(1,2), dir.axis1=1,
dir.axis2=1, rn=NULL, main=NULL, ...)
}
\arguments{
\item{D}{A distance matrix of class \code{dist} or \code{matrix}.}
\item{correction}{Correction methods for negative eigenvalues (details
below): \code{"lingoes"} and \code{"cailliez"}. Default value:
\code{"none"}.}
\item{rn}{An optional vector of row names, of length n, for the n
objects.}
\item{x}{Output object from \code{\link{pcoa}}.}
\item{Y}{Any rectangular data table containing explanatory variables
to be projected onto the ordination plot. That table may contain,
for example, the community composition data used to compute D, or
any transformation of these data; see examples.}
\item{plot.axes}{The two PCoA axes to plot.}
\item{dir.axis1}{= -1 to revert axis 1 for the projection of points
and variables. Default value: +1.}
\item{dir.axis2}{= -1 to revert axis 2 for the projection of points
and variables. Default value: +1.}
\item{main}{An optional title.}
\item{...}{Other graphical arguments passed to function.}
}
\details{
This function implements two methods for correcting for negative
values in principal coordinate analysis (PCoA). Negative eigenvalues
can be produced in PCoA when decomposing distance matrices produced by
coefficients that are not Euclidean (Gower and Legendre 1986,Legendre
and Legendre 1998).
In \code{pcoa}, when negative eigenvalues are present in the
decomposition results, the distance matrix D can be modified using
either the Lingoes or the Cailliez procedure to produce results
without negative eigenvalues.
In the Lingoes (1971) procedure, a constant c1, equal to twice
absolute value of the largest negative value of the original principal
coordinate analysis, is added to each original squared distance in the
distance matrix, except the diagonal values. A newe principal
coordinate analysis, performed on the modified distances, has at most
(n-2) positive eigenvalues, at least 2 null eigenvalues, and no
negative eigenvalue.
In the Cailliez (1983) procedure, a constant c2 is added to the
original distances in the distance matrix, except the diagonal
values. The calculation of c2 is described in Legendre and Legendre
(1998). A new principal coordinate analysis, performed on the modified
distances, has at most (n-2) positive eigenvalues, at least 2 null
eigenvalues, and no negative eigenvalue.
In all cases, only the eigenvectors corresponding to positive
eigenvalues are shown in the output list. The eigenvectors are scaled
to the square root of the corresponding eigenvalues. Gower (1966) has
shown that eigenvectors scaled in that way preserve the original
distance (in the D matrix) among the objects. These eigenvectors can
be used to plot ordination graphs of the objects.
We recommend not to use PCoA to produce ordinations from the chord,
chi-square, abundance profile, or Hellinger distances. It is easier to
first transform the community composition data using the following
transformations, available in the \code{decostand} function of the
\code{vegan} package, and then carry out a principal component
analysis (PCA) on the transformed data:
\itemize{
\item Chord transformation: decostand(spiders,"normalize")
\item Transformation to relative abundance profiles:
decostand(spiders,"total")
\item Hellinger transformation: decostand(spiders,"hellinger")
\item Chi-square transformation: decostand(spiders,"chi.square")
}
The ordination results will be identical and the calculations
shorter. This two-step ordination method, called transformation-based
PCA (tb-PCA), was described by Legendre and Gallagher (2001).
The \code{biplot.pcoa} function produces plots for any pair of
principal coordinates. The original variables can be projected onto
the ordination plot.
}
\value{
\item{correction}{The values of parameter \code{correction} and
variable 'correct' in the function.}
\item{note}{A note describing the type of correction done, if any.}
\item{values}{The eigenvalues and related information:}
\item{Eigenvalues}{All eigenvalues (positive, null, negative).}
\item{Relative_eig}{Relative eigenvalues.}
\item{Corr_eig}{Corrected eigenvalues (Lingoes correction); Legendre
and Legendre (1998, p. 438, eq. 9.27).}
\item{Rel_corr_eig}{Relative eigenvalues after Lingoes or Cailliez
correction.}
\item{Broken_stick}{Expected fractions of variance under the broken
stick model.}
\item{Cumul_eig}{Cumulative relative eigenvalues.}
\item{Cum_corr_eig}{Cumulative corrected relative eigenvalues.}
\item{Cumul_br_stick}{Cumulative broken stick fractions.}
\item{vectors}{The principal coordinates with positive eigenvalues.}
\item{trace}{The trace of the distance matrix. This is also the sum of
all eigenvalues, positive and negative.}
\item{vectors.cor}{The principal coordinates with positive
eigenvalues from the distance matrix corrected using the method
specified by parameter \code{correction}.}
\item{trace.cor}{The trace of the corrected distance matrix. This is
also the sum of its eigenvalues.}
}
\references{
Cailliez, F. (1983) The analytical solution of the additive constant
problem. \emph{Psychometrika}, \bold{48}, 305--308.
Gower, J. C. (1966) Some distance properties of latent root and vector
methods used in multivariate analysis. \emph{Biometrika}, \bold{53},
325--338.
Gower, J. C. and Legendre, P. (1986) Metric and Euclidean properties
of dissimilarity coefficients. \emph{Journal of Classification},
\bold{3}, 5--48.
Legendre, P. and Gallagher, E. D. (2001) Ecologically meaningful
transformations for ordination of species data. \emph{Oecologia},
\bold{129}, 271--280.
Legendre, P. and Legendre, L. (1998) \emph{Numerical Ecology, 2nd
English edition.} Amsterdam: Elsevier Science BV.
Lingoes, J. C. (1971) Some boundary conditions for a monotone analysis
of symmetric matrices. \emph{Psychometrika}, \bold{36}, 195--203.
}
\author{Pierre Legendre, Universite de Montreal}
\examples{
## Oribatid mite data from Borcard and Legendre (1994)
\dontrun{
if (require(vegan)) {
data(mite) # Community composition data, 70 peat cores, 35 species
## Select rows 1:30. Species 35 is absent from these rows. Transform to log
mite.log <- log(mite[1:30, -35] + 1) # Equivalent: log1p(mite[1:30, -35])
## Principal coordinate analysis and simple ordination plot
mite.D <- vegdist(mite.log, "bray")
res <- pcoa(mite.D)
res$values
biplot(res)
## Project unstandardized and standardized species on the PCoA ordination plot
mite.log.st = apply(mite.log, 2, scale, center=TRUE, scale=TRUE)
par(mfrow=c(1,2))
biplot(res, mite.log)
biplot(res, mite.log.st)
# Reverse the ordination axes in the plot
par(mfrow=c(1,2))
biplot(res, mite.log, dir.axis1=-1, dir.axis2=-1)
biplot(res, mite.log.st, dir.axis1=-1, dir.axis2=-1)
}}}
\keyword{multivariate}
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