\name{predict.cca}
\alias{fitted.cca}
\alias{fitted.rda}
\alias{fitted.capscale}
\alias{residuals.cca}
\alias{predict.cca}
\alias{predict.rda}
\alias{predict.decorana}
\alias{coef.cca}
\alias{coef.rda}
\alias{calibrate.cca}
\alias{calibrate}

\title{Prediction Tools for [Constrained] Ordination (CCA,
  RDA, DCA, CA, PCA) }
\description{
  Function \code{predict} can be used to find site and species scores or
  estimates of the response data with new data sets, Function
  \code{calibrate} estimates values of constraints with new data set.
  Functions \code{fitted} and \code{residuals} return estimates of
  response data. 
}
\usage{
\method{fitted}{cca}(object, model = c("CCA", "CA", "pCCA"),
    type =  c("response", "working"), ...)
\method{fitted}{capscale}(object, model = c("CCA", "CA", "pCCA", "Imaginary"),
    type = c("response", "working"), ...)
\method{residuals}{cca}(object, ...)
\method{predict}{cca}(object, newdata, type = c("response", "wa", "sp", "lc", "working"),
        rank = "full", model = c("CCA", "CA"), scaling = "none",
        hill = FALSE, ...)
\method{predict}{rda}(object, newdata, type = c("response", "wa", "sp", "lc", "working"),
        rank = "full", model = c("CCA", "CA"), scaling = "none",
        correlation = FALSE, ...)
\method{calibrate}{cca}(object, newdata, rank = "full", ...)
\method{coef}{cca}(object, ...)
\method{predict}{decorana}(object, newdata, type = c("response", "sites", "species"),
    rank = 4, ...)
}

\arguments{
  \item{object}{A result object from \code{\link{cca}},
    \code{\link{rda}}, \code{\link{capscale}} or \code{\link{decorana}}. }
  \item{model}{Show constrained (\code{"CCA"}), unconstrained
    (\code{"CA"}) or conditioned \dQuote{partial} (\code{"pCCA"})
    results. For \code{fitted} method of \code{\link{capscale}} this
    can also be \code{"Imaginary"} for imaginary components with
    negative eigenvalues. }
  \item{newdata}{New data frame to be used in prediction or in
    calibration.  Usually this a new community data frame, but with
    \code{type = "lc"} and for constrained component with \code{type =
    "response"} and \code{type = "working"} it must be a data frame of
    constraints.  The \code{newdata} must have the same number of rows
    as the original community data for a \code{\link{cca}} result with
    \code{type = "response"} or \code{type = "working"}.  If the
    original model had row or column names, then new data must contain
    rows or columns with the same names (row names for species scores,
    column names for \code{"wa"} scores and constraint names of
    \code{"lc"} scores). In other cases the rows or columns must match
    directly. }
  \item{type}{The type of prediction, fitted values or residuals:
    \code{"response"} scales results so that the same ordination gives
    the same results, and \code{"working"} gives the values used
    internally, that is after Chi-square standardization in
    \code{\link{cca}} and scaling and centring in
    \code{\link{rda}}. In \code{\link{capscale}} the \code{"response"}
    gives the dissimilarities, and \code{"working"} the scaled scores
    that produce the dissimilarities as Euclidean
    distances. Alternative \code{"wa"} gives the site scores as
    weighted averages of the community data, \code{"lc"} the site
    scores as linear combinations of environmental data, and
    \code{"sp"} the species scores. In \code{predict.decorana} the
    alternatives are scores for \code{"sites"} or \code{"species"}.}
  \item{rank}{The rank or the number of axes used in the approximation.
    The default is to use all axes (full rank) of the \code{"model"} or
    all available four axes in \code{predict.decorana}.}
  \item{scaling}{logical, character, or numeric; Scaling or predicted
    scores with the same meaning as in \code{\link{cca}},
    \code{\link{rda}} and \code{\link{capscale}}. See \code{scores.cca}
    for further details on acceptable values.}
  \item{correlation, hill}{logical; correlation-like scores or Hill's
    scaling as appropriate for RDA/\code{\link{capscale}} and CCA
    respectively. See \code{\link{scores.cca}} for additional details.}
  \item{\dots}{Other parameters to the functions.}
}
\details{

  Function \code{fitted} gives the approximation of the original data
  matrix or dissimilarities from the ordination result either in the
  scale of the response or as scaled internally by the function.
  Function \code{residuals} gives the approximation of the original
  data from the unconstrained ordination.  With argument \code{type =
  "response"} the \code{fitted.cca} and \code{residuals.cca} function
  both give the same marginal totals as the original data matrix, and
  fitted and residuals do not add up to the original data.  Functions
  \code{fitted.capscale} and \code{residuals.capscale} give the
  dissimilarities with \code{type = "response"}, but these are not
  additive, but the \code{"working"} scores are additive.  All
  variants of \code{fitted} and \code{residuals} are defined so that
  for model \code{mod <- cca(y ~ x)}, \code{cca(fitted(mod))} is equal
  to constrained ordination, and \code{cca(residuals(mod))} is equal
  to unconstrained part of the ordination.  

  Function \code{predict} can find the estimate of the original data
  matrix or dissimilarities (\code{type = "response"}) with any rank.
  With \code{rank = "full"} it is identical to \code{fitted}.  In
  addition, the function can find the species scores or site scores from
  the community data matrix for \code{\link{cca}} or \code{\link{rda}}.
  The function can be used with new data, and it can be used to add new
  species or site scores to existing ordinations.  The function returns
  (weighted) orthonormal scores by default, and you must specify
  explicit \code{scaling} to add those scores to ordination
  diagrams. With \code{type = "wa"} the function finds the site scores
  from species scores. In that case, the new data can contain new sites,
  but species must match in the original and new data.  With \code{type="sp"} 
  the function finds species scores from site constraints
  (linear combination scores). In that case the new data can contain new
  species, but sites must match in the original and new data. With
  \code{type = "lc"} the function finds the linear combination scores
  for sites from environmental data. In that case the new data frame
  must contain all constraining and conditioning environmental variables
  of the model formula. With \code{type = "response"} or 
  \code{type = "working"} the new data must contain environmental variables 
  if constrained component is desired, and community data matrix if
  residual or unconstrained component is desired.  With these types, the
  function uses \code{newdata} to find new \code{"lc"} (constrained) or
  \code{"wa"} scores (unconstrained) and then finds the response or
  working data from these new row scores and species scores.  The
  original site (row) and species (column) weights are used for
  \code{type = "response"} and \code{type = "working"} in correspondence
  analysis (\code{\link{cca}}) and therefore the number of rows must
  match in the original data and \code{newdata}.

  If a completely new data frame is created, extreme care is needed
  defining variables similarly as in the original model, in particular
  with (ordered) factors. If ordination was performed with the formula
  interface, the \code{newdata} can be a data frame or matrix, but
  extreme care is needed that the columns match in the original and
  \code{newdata}.

  Function \code{calibrate.cca} finds estimates of constraints from
  community ordination or \code{"wa"} scores from \code{\link{cca}},
  \code{\link{rda}} and \code{\link{capscale}}. This is often known as
  calibration, bioindication or environmental reconstruction.
  Basically, the method is similar to projecting site scores onto biplot
  arrows, but it uses regression coefficients.  The function can be called
  with \code{newdata} so that cross-validation is possible.  The
  \code{newdata} may contain new sites, but species must match in the
  original and new data.  The function
  does not work with \sQuote{partial} models with \code{Condition} term,
  and it cannot be used with \code{newdata} for \code{\link{capscale}}
  results.  The results may only be interpretable for continuous variables.
  
  Function \code{coef} will give the regression coefficients from centred
  environmental variables (constraints and conditions) to linear
  combination scores. The coefficients are for unstandardized environmental
  variables. The coefficients will be \code{NA} for aliased effects.

  Function \code{predict.decorana} is similar to \code{predict.cca}.
  However, \code{type = "species"} is not available in detrended
  correspondence analysis  (DCA), because detrending destroys the mutual
  reciprocal averaging (except for the first axis when rescaling is not
  used). Detrended CA does not attempt to approximate the original data
  matrix, so \code{type = "response"} has no meaning in detrended
  analysis (except with \code{rank = 1}).
}

\value{
  The functions return matrices, vectors or dissimilarities as is appropriate.
}
\references{
  Greenacre, M. J. (1984). Theory and applications of correspondence
  analysis. Academic Press, London.
}
\author{Jari Oksanen.}


\seealso{\code{\link{cca}}, \code{\link{rda}}, \code{\link{capscale}},
  \code{\link{decorana}}, \code{\link[car]{vif}}, \code{\link{goodness.cca}}. }
\examples{
data(dune)
data(dune.env)
mod <- cca(dune ~ A1 + Management + Condition(Moisture), data=dune.env)
# Definition of the concepts 'fitted' and 'residuals'
mod
cca(fitted(mod))
cca(residuals(mod))
# Remove rare species (freq==1) from 'cca' and find their scores
# 'passively'.
freq <- specnumber(dune, MARGIN=2)
freq
mod <- cca(dune[, freq>1] ~ A1 + Management + Condition(Moisture), dune.env)
predict(mod, type="sp", newdata=dune[, freq==1], scaling=2)
# New sites
predict(mod, type="lc", new=data.frame(A1 = 3, Management="NM", Moisture="2"), scal=2)
# Calibration and residual plot
mod <- cca(dune ~ A1 + Moisture, dune.env)
pred <- calibrate(mod)
pred
with(dune.env, plot(A1, pred[,"A1"] - A1, ylab="Prediction Error"))
abline(h=0)
}
\keyword{ multivariate }

