\name{idw}
\alias{idw}
\title{Inverse-distance weighted smoothing of observations at irregular points}
\description{
  Performs spatial smoothing of numeric values observed
  at a set of irregular locations using inverse-distance weighting.
}
\usage{
idw(X, power=2, at="pixels", ...)
}
\arguments{
  \item{X}{A marked point pattern (object of class \code{"ppp"}).}
  \item{power}{Numeric. Power of distance used in the weighting.}
  \item{at}{
    String specifying whether to compute the intensity values
    at a grid of pixel locations (\code{at="pixels"}) or
    only at the points of \code{X} (\code{at="points"}).
  }
  \item{\dots}{Arguments passed to \code{\link{as.mask}}
    to control the pixel resolution of the result.}
}
\details{
  This function performs spatial smoothing of numeric values
  observed at a set of irregular locations.
  
  Smoothing is performed by inverse distance weighting. If the
  observed values are \eqn{v_1,\ldots,v_n}{v[1],...,v[n]}
  at locations \eqn{x_1,\ldots,x_n}{x[1],...,x[n]} respectively,
  then the smoothed value at a location \eqn{u} is
  \deqn{
    g(u) = \frac{\sum_i w_i v_i}{\sum_i w_i}
  }{
    g(u) = (sum of w[i] * v[i])/(sum of w[i])
  }
  where the weights are the inverse \eqn{p}-th powers of distance,
  \deqn{
    w_i = \frac 1 {d(u,x_i)^p}
  }{
    w[i] = 1/d(u,x[i])^p
  }
  where \eqn{d(u,x_i) = ||u - x_i||}{d(u,x[i])}
  is the Euclidean distance from \eqn{u} to \eqn{x_i}{x[i]}.
  
  The argument \code{X} must be a marked point pattern (object
  of class \code{"ppp"}, see \code{\link{ppp.object}}).
  The points of the pattern are taken to be the
  observation locations \eqn{x_i}{x[i]}, and the marks of the pattern
  are taken to be the numeric values \eqn{v_i}{v[i]} observed at these
  locations.

  The marks are allowed to be a data frame.
  Then the smoothing procedure is applied to each
  column of marks. 
  
  If \code{at="pixels"} (the default), the smoothed mark value
  is calculated at a grid of pixels, and the result is a pixel image.
  The arguments \code{\dots} control the pixel resolution.
  See \code{\link{as.mask}}.

  If \code{at="points"}, the smoothed mark values are calculated
  at the data points only, using a leave-one-out rule (the mark value
  at a data point is excluded when calculating the smoothed value
  for that point). 

  An alternative to  inverse-distance weighting is kernel smoothing,
  which is performed by \code{\link{smooth.ppp}}.
}
\value{
  \emph{If \code{X} has a single column of marks:}
  \itemize{
    \item 
    If \code{at="pixels"} (the default), the result is
    a pixel image (object of class \code{"im"}). 
    Pixel values are values of the interpolated function.
    \item
    If \code{at="points"}, the result is a numeric vector
    of length equal to the number of points in \code{X}.
    Entries are values of the interpolated function at the points of \code{X}.
  }
  \emph{If \code{X} has a data frame of marks:}
  \itemize{
    \item 
    If \code{at="pixels"} (the default), the result is a named list of 
    pixel images (object of class \code{"im"}). There is one
    image for each column of marks. This list also belongs to
    the class \code{listof}, for which there is a plot method.
    \item
    If \code{at="points"}, the result is a data frame
    with one row for each point of \code{X},
    and one column for each column of marks. 
    Entries are values of the interpolated function at the points of \code{X}.
  }
}
\seealso{
  \code{\link{density.ppp}},
  \code{\link{ppp.object}},
  \code{\link{im.object}}.

  See \code{\link{smooth.ppp}} for kernel smoothing.
  
  To perform interpolation, see also the \code{akima} package.
}
\examples{
   # data frame of marks: trees marked by diameter and height
   data(finpines)
   plot(idw(finpines))
   idw(finpines, at="points")[1:5,]
}
\author{Adrian Baddeley
  \email{Adrian.Baddeley@csiro.au}
  \url{http://www.maths.uwa.edu.au/~adrian/}
  and Rolf Turner
  \email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{methods}
\keyword{smooth}