quadrat.test.Rd
\name{quadrat.test}
\alias{quadrat.test}
\alias{quadrat.test.ppp}
\alias{quadrat.test.ppm}
\alias{quadrat.test.quadratcount}
\title{Dispersion Test for Spatial Point Pattern Based on
Quadrat Counts}
\description{
Performs a test of Complete Spatial Randomness
for a given point pattern, based on quadrat counts.
Alternatively performs a goodness-of-fit test of a fitted
inhomogeneous Poisson model.
By default performs chi-squared tests; can also perform
Monte Carlo based tests.
}
\usage{
quadrat.test(X, ...)
\method{quadrat.test}{ppp}(X, nx=5, ny=nx,
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE,
...,
xbreaks=NULL, ybreaks=NULL, tess=NULL,
nsim=1999)
\method{quadrat.test}{ppm}(X, nx=5, ny=nx,
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE,
...,
xbreaks=NULL, ybreaks=NULL, tess=NULL,
nsim=1999)
\method{quadrat.test}{quadratcount}(X,
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE,
...,
nsim=1999)
}
\arguments{
\item{X}{
A point pattern (object of class \code{"ppp"})
to be subjected to the goodness-of-fit test.
Alternatively a fitted point process model (object of class
\code{"ppm"}) to be tested.
Alternatively \code{X} can be the result of applying
\code{\link{quadratcount}} to a point pattern.
}
\item{nx,ny}{
Numbers of quadrats in the \eqn{x} and \eqn{y} directions.
Incompatible with \code{xbreaks} and \code{ybreaks}.
}
\item{alternative}{
Character string (partially matched) specifying the alternative
hypothesis.
}
\item{method}{
Character string (partially matched) specifying the test to use:
either \code{method="Chisq"} for the chi-squared test (the default),
or \code{method="MonteCarlo"} for a Monte Carlo test.
}
\item{conditional}{
Logical. Should the Monte Carlo test be conducted
conditionally upon the observed number of points of the pattern?
Ignored if \code{method="Chisq"}.
}
\item{\dots}{Ignored.}
\item{xbreaks}{
Optional. Numeric vector giving the \eqn{x} coordinates of the
boundaries of the quadrats. Incompatible with \code{nx}.
}
\item{ybreaks}{
Optional. Numeric vector giving the \eqn{y} coordinates of the
boundaries of the quadrats. Incompatible with \code{ny}.
}
\item{tess}{
Tessellation (object of class \code{"tess"}) determining the
quadrats. Incompatible with \code{nx, ny, xbreaks, ybreaks}.
}
\item{nsim}{
The number of simulated samples to generate when
\code{method="MonteCarlo"}.
}
}
\details{
These functions perform \eqn{\chi^2}{chi^2} tests or Monte Carlo tests
of goodness-of-fit for a point process model, based on quadrat counts.
The function \code{quadrat.test} is generic, with methods for
point patterns (class \code{"ppp"}), split point patterns
(class \code{"splitppp"}), point process models
(class \code{"ppm"}) and quadrat count tables (class \code{"quadratcount"}).
\itemize{
\item
if \code{X} is a point pattern, we test the null hypothesis
that the data pattern is a realisation of Complete Spatial
Randomness (the uniform Poisson point process). Marks in the point
pattern are ignored.
\item
if \code{X} is a split point pattern, then for each of the
component point patterns (taken separately) we test
the null hypotheses of Complete Spatial Randomness.
See \code{\link{quadrat.test.splitppp}} for documentation.
\item
If \code{X} is a fitted point process model, then it should be
a Poisson point process model. The
data to which this model was fitted are extracted from the model
object, and are treated as the data point pattern for the test.
We test the null hypothesis
that the data pattern is a realisation of the (inhomogeneous) Poisson point
process specified by \code{X}.
}
In all cases, the window of observation is divided
into tiles, and the number of data points in each tile is
counted, as described in \code{\link{quadratcount}}.
The quadrats are rectangular by default, or may be regions of arbitrary shape
specified by the argument \code{tess}.
The expected number of points in each quadrat is also calculated,
as determined by CSR (in the first case) or by the fitted model
(in the second case). Then we perform either the
\eqn{\chi^2}{chi^2} test of goodness-of-fit to the quadrat counts
(if \code{method="Chisq"})
or a Monte Carlo test (if \code{method="MonteCarlo"}).
If \code{method="Chisq"} then the \eqn{\chi^2}{chi^2} test of
goodness-of-fit is performed. The Pearson \eqn{X^2} statistic
\deqn{
X^2 = sum((observed - expected)^2/expected)
}
is computed, and compared to the \eqn{\chi^2}{chi^2} distribution
with \eqn{m-k} degrees of freedom, where \code{m} is the number of
quadrats and \eqn{k} is the number of fitted parameters
(equal to 1 for \code{quadrat.test.ppp}). The default is to
compute the \emph{two-sided} \eqn{p}-value, so that the test will
be declared significant if \eqn{X^2} is either very large or very
small. One-sided \eqn{p}-values can be obtained by specifying the
\code{alternative}. An important requirement of the
\eqn{\chi^2}{chi^2} test is that the expected counts in each quadrat
be greater than 5.
If \code{method="MonteCarlo"} then a Monte Carlo test is performed,
obviating the need for all expected counts to be at least 5. In the
Monte Carlo test, \code{nsim} random point patterns are generated
from the null hypothesis (either CSR or the fitted point process
model). The Pearson \eqn{X^2} statistic is computed as above.
The \eqn{p}-value is determined by comparing the \eqn{X^2}
statistic for the observed point pattern, with the values obtained
from the simulations. Again the default is to
compute the \emph{two-sided} \eqn{p}-value.
If \code{conditional} is \code{TRUE} then the simulated samples are
generated from the multinomial distribution with the number of \dQuote{trials}
equal to the number of observed points and the vector of probabilities
equal to the expected counts divided by the sum of the expected counts.
Otherwise the simulated samples are independent Poisson counts, with
means equal to the expected counts.
The return value is an object of class \code{"htest"}.
Printing the object gives comprehensible output
about the outcome of the test.
The return value also belongs to
the special class \code{"quadrat.test"}. Plotting the object
will display the quadrats, annotated by their observed and expected
counts and the Pearson residuals. See the examples.
}
\seealso{
\code{\link{quadrat.test.splitppp}},
\code{\link{quadratcount}},
\code{\link{quadrats}},
\code{\link{quadratresample}},
\code{\link{chisq.test}},
\code{\link{kstest}}.
To test a Poisson point process model against a specific alternative,
use \code{\link{anova.ppm}}.
}
\value{
An object of class \code{"htest"}. See \code{\link{chisq.test}}
for explanation.
The return value is also an object of the special class
\code{"quadrattest"}, and there is a plot method for this class.
See the examples.
}
\examples{
data(simdat)
quadrat.test(simdat)
quadrat.test(simdat, 4, 3)
quadrat.test(simdat, alternative="regular")
quadrat.test(simdat, alternative="clustered")
# Using Monte Carlo p-values
quadrat.test(swedishpines) # Get warning, small expected values.
\dontrun{
quadrat.test(swedishpines, method="M", nsim=4999)
quadrat.test(swedishpines, method="M", nsim=4999, conditional=FALSE)
}
\testonly{
quadrat.test(swedishpines, method="M", nsim=19)
quadrat.test(swedishpines, method="M", nsim=19, conditional=FALSE)
}
# quadrat counts
qS <- quadratcount(simdat, 4, 3)
quadrat.test(qS)
# fitted model: inhomogeneous Poisson
fitx <- ppm(simdat, ~x, Poisson())
quadrat.test(fitx)
te <- quadrat.test(simdat, 4)
residuals(te) # Pearson residuals
plot(te)
plot(simdat, pch="+", cols="green", lwd=2)
plot(te, add=TRUE, col="red", cex=1.4, lty=2, lwd=3)
sublab <- eval(substitute(expression(p[chi^2]==z),
list(z=signif(te$p.value,3))))
title(sub=sublab, cex.sub=3)
# quadrats of irregular shape
B <- dirichlet(runifpoint(6, simdat$window))
qB <- quadrat.test(simdat, tess=B)
plot(simdat, main="quadrat.test(simdat, tess=B)", pch="+")
plot(qB, add=TRUE, col="red", lwd=2, cex=1.2)
}
\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{htest}
