\name{decorana}
\alias{decorana}
\alias{summary.decorana}
\alias{print.summary.decorana}
\alias{plot.decorana}
\alias{downweight}
\alias{scores.decorana}
\alias{points.decorana}
\alias{text.decorana}

\title{Detrended Correspondence Analysis and Basic Reciprocal Averaging }
\description{
  Performs detrended correspondence analysis and basic reciprocal
  averaging or orthogonal correspondence analysis.
}
\usage{
decorana(veg, iweigh=0, iresc=4, ira=0, mk=26, short=0,
         before=NULL, after=NULL)

\method{plot}{decorana}(x, choices=c(1,2), origin=TRUE,
     display=c("both","sites","species","none"),
     cex = 0.8, cols = c(1,2), type, xlim, ylim, ...)

\method{text}{decorana}(x, display = c("sites", "species"), labels,
     choices = 1:2, origin = TRUE, select,  ...)

\method{points}{decorana}(x, display = c("sites", "species"),
       choices=1:2, origin = TRUE, select, ...)

\method{summary}{decorana}(object, digits=3, origin=TRUE,
        display=c("both", "species","sites","none"), ...)

\method{print}{summary.decorana}(x, head = NA, tail = head, ...)

downweight(veg, fraction = 5)

\method{scores}{decorana}(x, display=c("sites","species"), choices=1:4,
       origin=TRUE, ...)
}

\arguments{
  \item{veg}{Community data, a matrix-like object. }
  \item{iweigh}{Downweighting of rare species (0: no). }
  \item{iresc}{Number of rescaling cycles (0: no rescaling). }
  \item{ira}{Type of analysis (0: detrended, 1: basic reciprocal averaging). }
  \item{mk}{Number of segments in rescaling. }
  \item{short}{Shortest gradient to be rescaled. }
  \item{before}{Hill's piecewise transformation: values before transformation. }
  \item{after}{Hill's piecewise transformation: values after
  transformation -- these must correspond to values in \code{before}.}
  \item{x, object}{A \code{decorana} result object.}
  \item{choices}{Axes shown.}
  \item{origin}{Use true origin even in detrended correspondence analysis.}
  \item{display}{Display only sites, only species, both or neither.}
  \item{cex}{Plot character size.}
  \item{cols}{Colours used for sites and species.}
  \item{type}{Type of plots, partial match to \code{"text"},
    \code{"points"} or \code{"none"}.}
  \item{labels}{Optional text to be used instead of row names.}
  \item{select}{Items to be displayed.  This can either be a logical
    vector which is \code{TRUE} for displayed items or a vector of indices
    of displayed items.}
  \item{xlim, ylim}{the x and y limits (min,max) of the plot.}
  \item{digits}{Number of digits in summary output.}
  \item{head, tail}{Number of rows printed from the head and tail of
    species and site scores. Default \code{NA} prints all.}
  \item{fraction}{Abundance fraction where downweighting begins.}
  \item{...}{Other arguments for \code{plot} function.}
  }

\details{
  In late 1970s, correspondence analysis became the method of choice for
  ordination in vegetation science, since it seemed better able to cope 
  with non-linear species responses than principal components
  analysis. However, even correspondence analysis can produce an arc-shaped
  configuration of a single gradient. Mark Hill developed detrended
  correspondence analysis to correct two assumed `faults' in 
  correspondence analysis: curvature of straight gradients and packing
  of sites at the ends of the gradient.  

  The curvature is removed by replacing the orthogonalization of axes
  with detrending.  In orthogonalization successive axes are made
  non-correlated, but detrending should remove all systematic dependence
  between axes.  Detrending is performed using a five-segment smoothing
  window with weights (1,2,3,2,1) on \code{mk} segments --- which indeed
  is more robust than the suggested alternative of detrending by
  polynomials. The packing of sites at the ends of the gradient is
  undone by rescaling the axes after extraction.  After rescaling, the
  axis is supposed to be scaled by `SD' units, so that the average width
  of Gaussian species responses is supposed to be one over whole axis.
  Other innovations were the piecewise linear transformation of species
  abundances and downweighting of rare species which were regarded to
  have an unduly high influence on ordination axes.

  It seems that detrending actually works by twisting the ordination
  space, so that the results look non-curved in two-dimensional projections
  (`lolly paper effect').  As a result, the points usually have an
  easily recognized triangular or diamond shaped pattern, obviously an
  artefact of detrending.  Rescaling works differently than commonly
  presented, too. \code{decorana} does not use, or even evaluate, the
  widths of species responses.  Instead, it tries to equalize the
  weighted variance of species scores on axis segments (parameter
  \code{mk} has only a small effect, since \code{decorana} finds the
  segment number from the current estimate of axis length).  This 
  equalizes response widths only for the idealized species packing
  model, where all species initially have unit width responses and
  equally spaced modes.

  The \code{summary} method prints the ordination scores,
  possible prior weights used in downweighting, and the marginal totals
  after applying these weights. The \code{plot} method plots
  species and site scores.  Classical \code{decorana} scaled the axes
  so that smallest site score was 0 (and smallest species score was
  negative), but \code{summary}, \code{plot} and
  \code{scores} use the true origin, unless \code{origin = FALSE}.

  In addition to proper eigenvalues, the function also reports `decorana
  values' in detrended analysis. These `decorana values' are the values
  that the legacy code of \code{decorana} returns as `eigenvalues'.
  They are estimated internally during iteration, and it seems that
  detrending interferes the estimation so that these values are
  generally too low and have unclear interpretation. Moreover, `decorana
  values' are estimated before rescaling which will change the
  eigenvalues. The proper eigenvalues are estimated after extraction of
  the axes and they are the ratio of biased weighted variances of site
  and species scores even in detrended and rescaled solutions. The
  `decorana values' are provided only for the compatibility with
  legacy software, and they should not be used.
}
\value{
  \code{decorana} returns an object of class \code{"decorana"}, which has
  \code{print}, \code{summary} and \code{plot} methods.
}
\references{
  Hill, M.O. and Gauch, H.G. (1980). Detrended correspondence analysis:
  an improved ordination technique. \emph{Vegetatio} \strong{42},
  47--58.

  Oksanen, J. and Minchin, P.R. (1997). Instability of ordination
  results under changes in input data order: explanations and
  remedies. \emph{Journal of Vegetation Science} \strong{8}, 447--454.
}
\author{Mark O. Hill wrote the original Fortran code, the \R port was by
  Jari Oksanen. }
\note{
  \code{decorana} uses the central numerical engine of the
  original Fortran code (which is in the public domain), or about 1/3 of
  the original program.  I have tried to implement the original
  behaviour, although a great part of preparatory steps were written in
  \R language, and may differ somewhat from the original code. However,
  well-known bugs are corrected and strict criteria used (Oksanen &
  Minchin 1997). 

  Please note that there really is no need for piecewise transformation
  or even downweighting within \code{decorana}, since there are more
  powerful and extensive alternatives in \R, but these options are
  included for compliance with the original software.  If a different
  fraction of abundance is needed in downweighting, function
  \code{downweight} must be applied before \code{decorana}.  Function
  \code{downweight} indeed can be applied prior to correspondence
  analysis, and so it can be used together with \code{\link{cca}}, too.

  The function finds only four axes: this is not easily changed.
}

\seealso{
  For unconstrained ordination, non-metric multidimensional scaling in
  \code{\link{monoMDS}} may be more robust (see also
  \code{\link{metaMDS}}).  Constrained (or
  \sQuote{canonical}) correspondence analysis can be made with
  \code{\link{cca}}.  Orthogonal correspondence analysis can be
  made with \code{\link[MASS]{corresp}}, or with \code{decorana} or
  \code{\link{cca}}, but the scaling of results vary (and the one in
  \code{decorana} corresponds to \code{scaling = -1} in
  \code{\link{cca}}.).
  See \code{\link{predict.decorana}} for adding new points to an
  ordination.
}

\examples{
data(varespec)
vare.dca <- decorana(varespec)
vare.dca
summary(vare.dca)
plot(vare.dca)

### the detrending rationale:
gaussresp <- function(x,u) exp(-(x-u)^2/2)
x <- seq(0,6,length=15) ## The gradient
u <- seq(-2,8,len=23)   ## The optima
pack <- outer(x,u,gaussresp)
matplot(x, pack, type="l", main="Species packing")
opar <- par(mfrow=c(2,2))
plot(scores(prcomp(pack)), asp=1, type="b", main="PCA")
plot(scores(decorana(pack, ira=1)), asp=1, type="b", main="CA")
plot(scores(decorana(pack)), asp=1, type="b", main="DCA")
plot(scores(cca(pack ~ x), dis="sites"), asp=1, type="b", main="CCA")

### Let's add some noise:
noisy <- (0.5 + runif(length(pack)))*pack
par(mfrow=c(2,1))
matplot(x, pack, type="l", main="Ideal model")
matplot(x, noisy, type="l", main="Noisy model")
par(mfrow=c(2,2))
plot(scores(prcomp(noisy)), type="b", main="PCA", asp=1)
plot(scores(decorana(noisy, ira=1)), type="b", main="CA", asp=1)
plot(scores(decorana(noisy)), type="b", main="DCA", asp=1)
plot(scores(cca(noisy ~ x), dis="sites"), asp=1, type="b", main="CCA")
par(opar)
}
\keyword{ multivariate }

