envelope.Rd
\name{envelope}
\alias{envelope}
\alias{envelope.ppp}
\alias{envelope.ppm}
\alias{envelope.kppm}
\title{Simulation Envelopes of Summary Function}
\description{
Computes simulation envelopes of a summary function.
}
\usage{
envelope(Y, fun, ...)
\method{envelope}{ppp}(Y, fun=Kest, nsim=99, nrank=1, \dots,
simulate=NULL, verbose=TRUE, clipdata=TRUE,
transform=NULL,global=FALSE,ginterval=NULL,
savefuns=FALSE, savepatterns=FALSE,
nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim)
\method{envelope}{ppm}(Y, fun=Kest, nsim=99, nrank=1, \dots,
simulate=NULL, verbose=TRUE, clipdata=TRUE,
start=NULL,control=list(nrep=1e5, expand=default.expand(Y)),
transform=NULL,global=FALSE,ginterval=NULL,
savefuns=FALSE, savepatterns=FALSE,
nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim)
\method{envelope}{kppm}(Y, fun=Kest, nsim=99, nrank=1, \dots,
simulate=NULL, verbose=TRUE, clipdata=TRUE,
transform=NULL,global=FALSE,ginterval=NULL,
savefuns=FALSE, savepatterns=FALSE,
nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim)
}
\arguments{
\item{Y}{
Object containing point pattern data.
A point pattern (object of class
\code{"ppp"}) or a fitted point process model
(object of class \code{"ppm"} or \code{"kppm"}).
}
\item{fun}{
Function that computes the desired summary statistic
for a point pattern.
}
\item{nsim}{
Number of simulated point patterns to be generated
when computing the envelopes.
}
\item{nrank}{
Integer. Rank of the envelope value amongst the \code{nsim} simulated
values. A rank of 1 means that the minimum and maximum
simulated values will be used.
}
\item{\dots}{
Extra arguments passed to \code{fun}.
}
\item{simulate}{
Optional. Specifies how to generate the simulated point patterns.
If \code{simulate} is an expression in the R language, then this
expression will be evaluated \code{nsim} times,
to obtain \code{nsim} point patterns which are taken as the
simulated patterns from which the envelopes are computed.
If \code{simulate} is a list of point patterns, then the entries
in this list will be treated as the simulated patterns from which
the envelopes are computed.
Alternatively \code{simulate} may be an object produced by the
\code{envelope} command: see Details.
}
\item{verbose}{
Logical flag indicating whether to print progress reports
during the simulations.
}
\item{clipdata}{
Logical flag indicating whether the data point pattern should be
clipped to the same window as the simulated patterns,
before the summary function for the data is computed.
This should usually be \code{TRUE} to ensure that the
data and simulations are properly comparable.
}
\item{start,control}{
Optional. These specify the arguments \code{start} and \code{control}
of \code{rmh}, giving complete control over the simulation
algorithm. Applicable only when \code{Y} is a fitted model
of class \code{"ppm"}.
}
\item{transform}{
Optional. A transformation to be applied to the
function values, before the envelopes are computed.
An expression object (see Details).
}
\item{global}{
Logical flag indicating whether envelopes should be pointwise
(\code{global=FALSE}) or simultaneous (\code{global=TRUE}).
}
\item{ginterval}{
Optional.
A vector of length 2 specifying
the interval of \eqn{r} values for the simultaneous critical
envelopes. Only relevant if \code{global=TRUE}.
}
\item{savefuns}{
Logical flag indicating whether to save all the simulated
function values.
}
\item{savepatterns}{
Logical flag indicating whether to save all the simulated
point patterns.
}
\item{nsim2}{
Number of extra simulated point patterns to be generated
if it is necessary to use simulation to estimate the theoretical
mean of the summary function. Only relevant when \code{global=TRUE}
and the simulations are not based on CSR.
}
\item{VARIANCE}{
Logical. If \code{TRUE}, critical envelopes will be calculated
as sample mean plus or minus \code{nSD} times sample standard
deviation.
}
\item{nSD}{
Number of estimated standard deviations used to determine
the critical envelopes, if \code{VARIANCE=TRUE}.
}
\item{Yname}{
Character string that should be used as the name of the
data point pattern \code{Y} when printing or plotting the results.
}
\item{maxnerr}{
Maximum number of rejected patterns.
If \code{fun} yields an error when applied to a simulated point
pattern (for example, because the pattern is empty and \code{fun}
requires at least one point), the pattern will be rejected
and a new random point pattern will be generated. If this happens
more than \code{maxnerr} times, the algorithm will give up.
}
}
\value{
An object of class \code{"fv"}, see \code{\link{fv.object}},
which can be printed and plotted directly.
Essentially a data frame containing columns
\item{r}{the vector of values of the argument \eqn{r}
at which the summary function \code{fun} has been estimated
}
\item{obs}{
values of the summary function for the data point pattern
}
\item{lo}{
lower envelope of simulations
}
\item{hi}{
upper envelope of simulations
}
and \emph{either}
\item{theo}{
theoretical value of the summary function under CSR
(Complete Spatial Randomness, a uniform Poisson point process)
if the simulations were generated according to CSR
}
\item{mmean}{
estimated theoretical value of the summary function,
computed by averaging simulated values,
if the simulations were not generated according to CSR.
}
Additionally, if \code{savepatterns=TRUE}, the return value has an attribute
\code{"simpatterns"} which is a list containing the \code{nsim}
simulated patterns. If \code{savefuns=TRUE}, the return value
has an attribute \code{"simfuns"} which is an object of class
\code{"fv"} containing the summary functions
computed for each of the \code{nsim} simulated patterns.
}
\details{
The \code{envelope} command performs simulations and
computes envelopes of a summary statistic based on the simulations.
The result is an object that can be plotted to display the envelopes.
The envelopes can be used to assess the goodness-of-fit of
a point process model to point pattern data.
For the most basic use, if you have a point pattern \code{X} and
you want to test Complete Spatial Randomness (CSR), type
\code{plot(envelope(X, Kest,nsim=39))} to see the \eqn{K} function
for \code{X} plotted together with the envelopes of the
\eqn{K} function for 39 simulations of CSR.
The \code{envelope} function is generic, with methods for
the classes \code{"ppp"}, \code{"ppm"} and \code{"kppm"}
described here. There is also a method for the class \code{"pp3"}
which is described separately as \code{\link{envelope.pp3}}.
To create simulation envelopes, the command \code{envelope(Y, ...)}
first generates \code{nsim} random point patterns
in one of the following ways.
\itemize{
\item
If \code{Y} is a point pattern (an object of class \code{"ppp"})
and \code{simulate=NULL},
then we generate \code{nsim} simulations of
Complete Spatial Randomness (i.e. \code{nsim} simulated point patterns
each being a realisation of the uniform Poisson point process)
with the same intensity as the pattern \code{Y}.
(If \code{Y} is a multitype point pattern, then the simulated patterns
are also given independent random marks; the probability
distribution of the random marks is determined by the
relative frequencies of marks in \code{Y}.)
\item
If \code{Y} is a fitted point process model (an object of class
\code{"ppm"} or \code{"kppm"}) and \code{simulate=NULL},
then this routine generates \code{nsim} simulated
realisations of that model.
\item
If \code{simulate} is supplied, then it determines how the
simulated point patterns are generated. It may be either
\itemize{
\item
an expression in the R language, typically containing a call
to a random generator. This expression will be evaluated
\code{nsim} times to yield \code{nsim} point patterns. For example
if \code{simulate=expression(runifpoint(100))} then each simulated
pattern consists of exactly 100 independent uniform random points.
\item
a list of point patterns.
The entries in this list will be taken as the simulated patterns.
\item
an object of class \code{"envelope"}. This should have been
produced by calling \code{envelope} with the
argument \code{savepatterns=TRUE}.
The simulated point patterns that were saved in this object
will be extracted and used as the simulated patterns for the
new envelope computation. This makes it possible to plot envelopes
for two different summary functions based on exactly the same set of
simulated point patterns.
}
}
The summary statistic \code{fun} is applied to each of these simulated
patterns. Typically \code{fun} is one of the functions
\code{Kest}, \code{Gest}, \code{Fest}, \code{Jest}, \code{pcf},
\code{Kcross}, \code{Kdot}, \code{Gcross}, \code{Gdot},
\code{Jcross}, \code{Jdot}, \code{Kmulti}, \code{Gmulti},
\code{Jmulti} or \code{Kinhom}. It may also be a character string
containing the name of one of these functions.
The statistic \code{fun} can also be a user-supplied function;
if so, then it must have arguments \code{X} and \code{r}
like those in the functions listed above, and it must return an object
of class \code{"fv"}.
Upper and lower critical envelopes are computed in one of the following ways:
\describe{
\item{pointwise:}{by default, envelopes are calculated pointwise
(i.e. for each value of the distance argument \eqn{r}), by sorting the
\code{nsim} simulated values, and taking the \code{m}-th lowest
and \code{m}-th highest values, where \code{m = nrank}.
For example if \code{nrank=1}, the upper and lower envelopes
are the pointwise maximum and minimum of the simulated values.
The pointwise envelopes are \bold{not} \dQuote{confidence bands}
for the true value of the function! Rather,
they specify the critical points for a Monte Carlo test
(Ripley, 1981). The test is constructed by choosing a
\emph{fixed} value of \eqn{r}, and rejecting the null hypothesis if the
observed function value
lies outside the envelope \emph{at this value of} \eqn{r}.
This test has exact significance level
\code{alpha = 2 * nrank/(1 + nsim)}.
}
\item{simultaneous:}{if \code{global=TRUE}, then the envelopes are
determined as follows. First we calculate the theoretical mean value of
the summary statistic (if we are testing CSR, the theoretical
value is supplied by \code{fun}; otherwise we perform a separate
set of \code{nsim2} simulations, compute the
average of all these simulated values, and take this average
as an estimate of the theoretical mean value). Then, for each simulation,
we compare the simulated curve to the theoretical curve, and compute the
maximum absolute difference between them (over the interval
of \eqn{r} values specified by \code{ginterval}). This gives a
deviation value \eqn{d_i}{d[i]} for each of the \code{nsim}
simulations. Finally we take the \code{m}-th largest of the
deviation values, where \code{m=nrank}, and call this
\code{dcrit}. Then the simultaneous envelopes are of the form
\code{lo = expected - dcrit} and \code{hi = expected + dcrit} where
\code{expected} is either the theoretical mean value \code{theo}
(if we are testing CSR) or the estimated theoretical value
\code{mmean} (if we are testing another model). The simultaneous critical
envelopes have constant width \code{2 * dcrit}.
The simultaneous critical envelopes allow us to perform a different
Monte Carlo test (Ripley, 1981). The test rejects the null
hypothesis if the graph of the observed function
lies outside the envelope \bold{at any value of} \eqn{r}.
This test has exact significance level
\code{alpha = nrank/(1 + nsim)}.
}
\item{based on sample moments:}{if \code{VARIANCE=TRUE},
the algorithm calculates the
(pointwise) sample mean and sample variance of
the simulated functions. Then the envelopes are computed
as mean plus or minus \code{nSD} standard deviations.
These envelopes do not have an exact significance interpretation.
They are a naive approximation to
the critical points of the Neyman-Pearson test
assuming the summary statistic is approximately Normally
distributed.
}
}
The return value is an object of class \code{"fv"} containing
the summary function for the data point pattern,
the upper and lower simulation envelopes, and
the theoretical expected value (exact or estimated) of the summary function
for the model being tested. It can be plotted
using \code{\link{plot.envelope}}.
If \code{VARIANCE=TRUE} then the return value also includes the
sample mean, sample variance and other quantities.
Arguments can be passed to the function \code{fun} through
\code{...}. This makes it possible to select the edge correction
used to calculate the summary statistic. See the Examples.
Selecting only a single edge
correction will make the code run much faster.
If \code{Y} is a fitted cluster point process model (object of
class \code{"kppm"}), and \code{simulate=NULL},
then the model is simulated directly
using \code{\link{simulate.kppm}}.
If \code{Y} is a fitted Gibbs point process model (object of
class \code{"ppm"}), and \code{simulate=NULL},
then the model is simulated
by running the Metropolis-Hastings algorithm \code{\link{rmh}}.
Complete control over this algorithm is provided by the
arguments \code{start} and \code{control} which are passed
to \code{\link{rmh}}.
For simultaneous critical envelopes (\code{global=TRUE})
the following options are also useful:
\describe{
\item{\code{ginterval}}{determines the interval of \eqn{r} values
over which the deviation between curves is calculated.
It should be a numeric vector of length 2.
There is a sensible default (namely, the recommended plotting
interval for \code{fun(X)}, or the range of \code{r} values if
\code{r} is explicitly specified).
}
\item{\code{transform}}{specifies a transformation of the
summary function \code{fun} that will be carried out before the
deviations are computed. It must be an expression object
using the symbol \code{.} to represent the function value.
For example,
the conventional way to normalise the \eqn{K} function
(Ripley, 1981) is to transform it to the \eqn{L} function
\eqn{L(r) = \sqrt{K(r)/\pi}}{L(r) = sqrt(K(r)/pi)}
and this is implemented by setting
\code{transform=expression(sqrt(./pi))}.
Such transforms are only useful if \code{global=TRUE}.
}
}
It is also possible to extract the summary functions for each of the
individual simulated point patterns, by setting \code{savefuns=TRUE}.
Then the return value also
has an attribute \code{"simfuns"} containing all the
summary functions for the individual simulated patterns.
It is an \code{"fv"} object containing
functions named \code{sim1, sim2, ...} representing the \code{nsim}
summary functions.
It is also possible to save the simulated point patterns themselves,
by setting \code{savepatterns=TRUE}. Then the return value also has
an attribute \code{"simpatterns"} which is a list of length
\code{nsim} containing all the simulated point patterns.
See \code{\link{plot.envelope}} and \code{link{plot.fv}}
for information about how to plot the envelopes.
}
\section{Warning}{
An error may be generated if one of the simulations produces a
point pattern that is empty, or is otherwise unacceptable to the
function \code{fun}.
}
\references{
Cressie, N.A.C. \emph{Statistics for spatial data}.
John Wiley and Sons, 1991.
Diggle, P.J. \emph{Statistical analysis of spatial point patterns}.
Arnold, 2003.
Ripley, B.D. (1981)
\emph{Spatial statistics}.
John Wiley and Sons.
Ripley, B.D. \emph{Statistical inference for spatial processes}.
Cambridge University Press, 1988.
Stoyan, D. and Stoyan, H. (1994)
Fractals, random shapes and point fields:
methods of geometrical statistics.
John Wiley and Sons.
}
\seealso{
\code{\link{fv.object}},
\code{\link{plot.envelope}},
\code{\link{plot.fv}},
\code{\link{Kest}},
\code{\link{Gest}},
\code{\link{Fest}},
\code{\link{Jest}},
\code{\link{pcf}},
\code{\link{ppp}},
\code{\link{ppm}},
\code{\link{default.expand}}
}
\examples{
data(simdat)
X <- simdat
# Envelope of K function under CSR
\dontrun{
plot(envelope(X))
}
\testonly{
plot(envelope(X, nsim=4))
}
# Translation edge correction (this is also FASTER):
\dontrun{
plot(envelope(X, correction="translate"))
}
\testonly{
plot(envelope(X, nsim=4, correction="translate"))
}
# Envelope of K function for simulations from Gibbs model
data(cells)
fit <- ppm(cells, ~1, Strauss(0.05))
\dontrun{
plot(envelope(fit))
plot(envelope(fit), global=TRUE)
}
\testonly{
plot(envelope(fit, nsim=4))
plot(envelope(fit, nsim=4, global=TRUE))
}
# Envelope of K function for simulations from cluster model
data(redwood)
fit <- kppm(redwood, ~1, "Thomas")
\dontrun{
plot(envelope(fit, Gest))
plot(envelope(fit, Gest, global=TRUE))
}
\testonly{
plot(envelope(fit, Gest, nsim=4))
plot(envelope(fit, Gest, nsim=4, global=TRUE))
}
# Envelope of G function under CSR
\dontrun{
plot(envelope(X, Gest))
}
\testonly{
plot(envelope(X, Gest, nsim=4))
}
# Envelope of L function under CSR
# L(r) = sqrt(K(r)/pi)
\dontrun{
E <- envelope(X, Kest)
plot(E, sqrt(./pi) ~ r)
}
\testonly{
E <- envelope(X, Kest, nsim=4)
plot(E, sqrt(./pi) ~ r)
}
# Simultaneous critical envelope for L function
# (alternatively, use Lest)
\dontrun{
plot(envelope(X, Kest, transform=expression(sqrt(./pi)), global=TRUE))
}
\testonly{
plot(envelope(X, Kest, nsim=4,transform=expression(sqrt(./pi)), global=TRUE))
}
# How to pass arguments needed to compute the summary functions:
# We want envelopes for Jcross(X, "A", "B")
# where "A" and "B" are types of points in the dataset 'demopat'
data(demopat)
\dontrun{
plot(envelope(demopat, Jcross, i="A", j="B"))
}
\testonly{
plot(envelope(demopat, Jcross, i="A", j="B", nsim=4))
}
# Use of `simulate'
\dontrun{
plot(envelope(cells, Gest, simulate=expression(runifpoint(42))))
plot(envelope(cells, Gest, simulate=expression(rMaternI(100,0.02))))
}
\testonly{
plot(envelope(cells, Gest, simulate=expression(runifpoint(42)), nsim=4))
plot(envelope(cells, Gest, simulate=expression(rMaternI(100, 0.02)), nsim=4))
plot(envelope(cells, Gest, simulate=expression(runifpoint(42)),
nsim=4, global=TRUE))
plot(envelope(cells, Gest, simulate=expression(rMaternI(100, 0.02)),
nsim=4, global=TRUE))
}
# Envelope under random toroidal shifts
data(amacrine)
\dontrun{
plot(envelope(amacrine, Kcross, i="on", j="off",
simulate=expression(rshift(amacrine, radius=0.25))))
}
# Envelope under random shifts with erosion
\dontrun{
plot(envelope(amacrine, Kcross, i="on", j="off",
simulate=expression(rshift(amacrine, radius=0.1, edge="erode"))))
}
# Envelope of INHOMOGENEOUS K-function with fitted trend
\dontrun{
trend <- density.ppp(X, 1.5)
plot(envelope(X, Kinhom, lambda=trend,
simulate=expression(rpoispp(trend))))
}
# Precomputed list of point patterns
X <- rpoispp(50)
PatList <- list()
for(i in 1:20) PatList[[i]] <- runifpoint(X$n)
plot(envelope(X, Kest, nsim=20, simulate=PatList))
# re-using the same point patterns
EK <- envelope(X, Kest, nsim=10, savepatterns=TRUE)
EG <- envelope(X, Kest, nsim=10, simulate=EK)
}
\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}
}
\keyword{spatial}
\keyword{htest}
\keyword{hplot}
\keyword{iteration}
