\name{kstest.ppm}
\alias{kstest}
\alias{kstest.ppm}
\alias{kstest.ppp}
\title{Kolmogorov-Smirnov Test for Point Process Model} 
\description{
  Performs a Kolmogorov-Smirnov test of goodness-of-fit
  of a Poisson point process model. The test compares the observed
  and predicted distributions of the values of a spatial covariate.
}
\usage{
kstest(...)
\method{kstest}{ppp}(X, covariate, ..., jitter=TRUE)
\method{kstest}{ppm}(model, covariate, ..., jitter=TRUE)
}
\arguments{
  \item{X}{
    A point pattern (object of class \code{"ppp"}).
  }
  \item{model}{
    A fitted point process model (object of class \code{"ppm"}).
  }
  \item{covariate}{
    The spatial covariate on which the test will be based.
    An image (object of class \code{"im"})
    or a function.
  }
  \item{\dots}{
    Arguments passed to \code{\link{ks.test}}
    to control the test.
  }
  \item{jitter}{
    Logical flag. If \code{jitter=TRUE}, values of the covariate
    will be slightly perturbed at random, to avoid tied values in the test.
  }
}
\details{
  These functions perform a goodness-of-fit test of a Poisson point
  process model fitted to point pattern data. The observed distribution
  of the values of a spatial covariate at the data points,
  and the predicted distribution of the same values under the model,
  are compared using the Kolmogorov-Smirnov test. 

  The function \code{kstest} is generic, with methods for
  point patterns (\code{"ppp"}) and point process models (\code{"ppm"}).
  \itemize{
    \item 
    If \code{X} is a point pattern dataset (object of class \code{"ppp"}), then
    \code{kstest(X, ...)} performs a goodness-of-fit test of the
    uniform Poisson point process (Complete Spatial Randomness, CSR)
    for this dataset.
    \item
    If \code{model} is a fitted point process model
    (object of class \code{"ppm"}) then \code{kstest(model, ...)} performs
    a test of goodness-of-fit for this fitted model. In this case,
    \code{model} should be a Poisson point process.
  }
  
  The test is performed by comparing the observed distribution
  of the values of a spatial covariate at the data points,
  and the predicted distribution of the same covariate under the model,
  using the classical Kolmogorov-Smirnov test. Thus, you must nominate
  a spatial covariate for this test.
  
  The argument \code{covariate} should be either a \code{function(x,y)}
  or a pixel image (object of class \code{"im"} containing the values
  of a spatial function.
  If \code{covariate} is an image, it should have numeric values,
  and its domain should cover the observation window of the
  \code{model}. If \code{covariate} is a function, it should expect
  two arguments \code{x} and \code{y} which are vectors of coordinates,
  and it should return a numeric vector of the same length
  as \code{x} and \code{y}.  

  First the original data point pattern is extracted from \code{model}.
  The values of the \code{covariate} at these data points are
  collected. 

  The predicted distribution of the values of the \code{covariate}
  under the fitted \code{model} is computed as follows.
  The values of the \code{covariate} at all locations in the
  observation window are evaluated,
  weighted according to the point process intensity of the fitted model,
  and compiled into a cumulative distribution function \eqn{F} using
  \code{\link{ewcdf}}.

  The probability integral transformation is then applied:
  the values of the \code{covariate} at the original data points
  are transformed by the predicted cumulative distribution function
  \eqn{F} into numbers between 0 and 1. If the model is correct,
  these numbers are i.i.d. uniform random numbers. The
  Kolmogorov-Smirnov test of uniformity is applied using
  \code{\link{ks.test}}.

  This test was apparently first described (in the context of
  spatial data) by Berman (1986). See also Baddeley et al (2005).

  The return value is an object of class \code{"htest"} containing the
  results of the hypothesis test. The print method for this class
  gives an informative summary of the test outcome.

  The return value also belongs to the class \code{"kstest"}
  for which there is an (undocumented) plot method.
  The plot method displays the empirical cumulative distribution
  function of the covariate at the data points, and the predicted
  cumulative distribution function of the covariate under the model,
  plotted against the value of the covariate.

  The argument \code{jitter} controls whether covariate values are
  randomly perturbed, in order to avoid ties.
  If the original data contains any ties in the covariate (i.e. points
  with equal values of the covariate), and if \code{jitter=FALSE}, then 
  the Kolmogorov-Smirnov test implemented in \code{\link{ks.test}}
  will issue a warning that it cannot calculate the exact \eqn{p}-value.
  To avoid this, if \code{jitter=TRUE} each value of the covariate will
  be perturbed by adding a small random value. The perturbations are
  normally distributed with standard deviation equal to one hundredth of
  the range of values of the covariate. This prevents ties, 
  and the \eqn{p}-value is still correct. There is
  a very slight loss of power.
}
\value{
  An object of class \code{"htest"} containing the results of the
  test. See \code{\link{ks.test}} for details. The return value can be
  printed to give an informative summary of the test.

  The value also belongs to the class \code{"kstest"} for which there is
  a plot method.
}
\section{Warning}{
  The outcome of the test involves a small amount of random variability,
  because (by default) the coordinates are randomly perturbed to
  avoid tied values. Hence, if \code{kstest} is executed twice, the
  \eqn{p}-values will not be exactly the same. To avoid this behaviour,
  set \code{jitter=FALSE}.
}
\author{Adrian Baddeley
  \email{adrian@maths.uwa.edu.au}
  \url{http://www.maths.uwa.edu.au/~adrian/}
  and Rolf Turner
  \email{r.turner@auckland.ac.nz}
}
\seealso{
  \code{\link{ks.test}},
  \code{\link{quadrat.test}},
  \code{\link{ppm}}
}
\references{
  Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005)
  Residual analysis for spatial point processes.
  \emph{Journal of the Royal Statistical Society, Series B}
  \bold{67}, 617--666.

  Berman, M. (1986)
  Testing for spatial association between a point process
  and another stochastic process.
  \emph{Applied Statistics} \bold{35}, 54--62.
}
\examples{
   # real data: Swedish Pines
   data(swedishpines)

   # test covariate = x coordinate
   xcoord <- function(x,y) { x }

   # test of CSR
   kstest(swedishpines, xcoord)

   # fit inhomogeneous Poisson model and test
   model <- ppm(swedishpines, ~x + y)
   kstest(model, xcoord)

   # synthetic data: nonuniform Poisson process
   X <- rpoispp(function(x,y) { 100 * exp(x) }, win=square(1))

   # fit uniform Poisson process
   fit0 <- ppm(X, ~1)
   # fit correct nonuniform Poisson process
   fit1 <- ppm(X, ~x)

   # test wrong model
   kstest(fit0, xcoord)
   # test right model
   kstest(fit1, xcoord)
}
\keyword{htest}
\keyword{spatial}