envelope {spatstat} | R Documentation |
Computes simulation envelopes of a summary function.
envelope(Y, fun, ...) ## S3 method for class 'ppp' envelope(Y, fun=Kest, nsim=99, nrank=1, ..., 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) ## S3 method for class 'ppm' envelope(Y, fun=Kest, nsim=99, nrank=1, ..., simulate=NULL, verbose=TRUE, clipdata=TRUE, start=NULL, control=default.rmhcontrol(Y, nrep=nrep), nrep=1e5, transform=NULL, global=FALSE, ginterval=NULL, savefuns=FALSE, savepatterns=FALSE, nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim) ## S3 method for class 'kppm' envelope(Y, fun=Kest, nsim=99, nrank=1, ..., 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)
Y |
Object containing point pattern data.
A point pattern (object of class
|
fun |
Function that computes the desired summary statistic for a point pattern. |
nsim |
Number of simulated point patterns to be generated when computing the envelopes. |
nrank |
Integer. Rank of the envelope value amongst the |
... |
Extra arguments passed to |
simulate |
Optional. Specifies how to generate the simulated point patterns.
If |
verbose |
Logical flag indicating whether to print progress reports during the simulations. |
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 |
start,control |
Optional. These specify the arguments |
nrep |
Number of iterations in the Metropolis-Hastings simulation
algorithm. Applicable only when |
transform |
Optional. A transformation to be applied to the function values, before the envelopes are computed. An expression object (see Details). |
global |
Logical flag indicating whether envelopes should be pointwise
( |
ginterval |
Optional.
A vector of length 2 specifying
the interval of r values for the simultaneous critical
envelopes. Only relevant if |
savefuns |
Logical flag indicating whether to save all the simulated function values. |
savepatterns |
Logical flag indicating whether to save all the simulated point patterns. |
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 |
VARIANCE |
Logical. If |
nSD |
Number of estimated standard deviations used to determine
the critical envelopes, if |
Yname |
Character string that should be used as the name of the
data point pattern |
maxnerr |
Maximum number of rejected patterns.
If |
The 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 X
and
you want to test Complete Spatial Randomness (CSR), type
plot(envelope(X, Kest,nsim=39))
to see the K function
for X
plotted together with the envelopes of the
K function for 39 simulations of CSR.
The envelope
function is generic, with methods for
the classes "ppp"
, "ppm"
and "kppm"
described here. There is also a method for the class "pp3"
which is described separately as envelope.pp3
.
To create simulation envelopes, the command envelope(Y, ...)
first generates nsim
random point patterns
in one of the following ways.
If Y
is a point pattern (an object of class "ppp"
)
and simulate=NULL
,
then we generate nsim
simulations of
Complete Spatial Randomness (i.e. nsim
simulated point patterns
each being a realisation of the uniform Poisson point process)
with the same intensity as the pattern Y
.
(If 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 Y
.)
If Y
is a fitted point process model (an object of class
"ppm"
or "kppm"
) and simulate=NULL
,
then this routine generates nsim
simulated
realisations of that model.
If simulate
is supplied, then it determines how the
simulated point patterns are generated. It may be either
an expression in the R language, typically containing a call
to a random generator. This expression will be evaluated
nsim
times to yield nsim
point patterns. For example
if simulate=expression(runifpoint(100))
then each simulated
pattern consists of exactly 100 independent uniform random points.
a list of point patterns. The entries in this list will be taken as the simulated patterns.
an object of class "envelope"
. This should have been
produced by calling envelope
with the
argument 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 fun
is applied to each of these simulated
patterns. Typically fun
is one of the functions
Kest
, Gest
, Fest
, Jest
, pcf
,
Kcross
, Kdot
, Gcross
, Gdot
,
Jcross
, Jdot
, Kmulti
, Gmulti
,
Jmulti
or Kinhom
. It may also be a character string
containing the name of one of these functions.
The statistic fun
can also be a user-supplied function;
if so, then it must have arguments X
and r
like those in the functions listed above, and it must return an object
of class "fv"
.
Upper and lower critical envelopes are computed in one of the following ways:
by default, envelopes are calculated pointwise
(i.e. for each value of the distance argument r), by sorting the
nsim
simulated values, and taking the m
-th lowest
and m
-th highest values, where m = nrank
.
For example if nrank=1
, the upper and lower envelopes
are the pointwise maximum and minimum of the simulated values.
The pointwise envelopes are not “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
fixed value of r, and rejecting the null hypothesis if the
observed function value
lies outside the envelope at this value of r.
This test has exact significance level
alpha = 2 * nrank/(1 + nsim)
.
if 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 fun
; otherwise we perform a separate
set of 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 r values specified by ginterval
). This gives a
deviation value d[i] for each of the nsim
simulations. Finally we take the m
-th largest of the
deviation values, where m=nrank
, and call this
dcrit
. Then the simultaneous envelopes are of the form
lo = expected - dcrit
and hi = expected + dcrit
where
expected
is either the theoretical mean value theo
(if we are testing CSR) or the estimated theoretical value
mmean
(if we are testing another model). The simultaneous critical
envelopes have constant width 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 at any value of r.
This test has exact significance level
alpha = nrank/(1 + nsim)
.
if 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 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 "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 plot.envelope
.
If VARIANCE=TRUE
then the return value also includes the
sample mean, sample variance and other quantities.
Arguments can be passed to the function fun
through
...
. 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 Y
is a fitted cluster point process model (object of
class "kppm"
), and simulate=NULL
,
then the model is simulated directly
using simulate.kppm
.
If Y
is a fitted Gibbs point process model (object of
class "ppm"
), and simulate=NULL
,
then the model is simulated
by running the Metropolis-Hastings algorithm rmh
.
Complete control over this algorithm is provided by the
arguments start
and control
which are passed
to rmh
.
For simultaneous critical envelopes (global=TRUE
)
the following options are also useful:
ginterval
determines the interval of 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 fun(X)
, or the range of r
values if
r
is explicitly specified).
transform
specifies a transformation of the
summary function fun
that will be carried out before the
deviations are computed. It must be an expression object
using the symbol .
to represent the function value.
For example,
the conventional way to normalise the K function
(Ripley, 1981) is to transform it to the L function
L(r) = sqrt(K(r)/pi)
and this is implemented by setting
transform=expression(sqrt(./pi))
.
Such transforms are only useful if global=TRUE
.
It is also possible to extract the summary functions for each of the
individual simulated point patterns, by setting savefuns=TRUE
.
Then the return value also
has an attribute "simfuns"
containing all the
summary functions for the individual simulated patterns.
It is an "fv"
object containing
functions named sim1, sim2, ...
representing the nsim
summary functions.
It is also possible to save the simulated point patterns themselves,
by setting savepatterns=TRUE
. Then the return value also has
an attribute "simpatterns"
which is a list of length
nsim
containing all the simulated point patterns.
See plot.envelope
and plot.fv
for information about how to plot the envelopes.
Different envelopes can be recomputed from the same data
using envelope.envelope
.
Envelopes can be combined using pool.envelope
.
An object of class "fv"
, see fv.object
,
which can be printed and plotted directly.
Essentially a data frame containing columns
r |
the vector of values of the argument r
at which the summary function |
obs |
values of the summary function for the data point pattern |
lo |
lower envelope of simulations |
hi |
upper envelope of simulations |
and either
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 |
mmean |
estimated theoretical value of the summary function, computed by averaging simulated values, if the simulations were not generated according to CSR. |
Additionally, if savepatterns=TRUE
, the return value has an attribute
"simpatterns"
which is a list containing the nsim
simulated patterns. If savefuns=TRUE
, the return value
has an attribute "simfuns"
which is an object of class
"fv"
containing the summary functions
computed for each of the nsim
simulated patterns.
An error may be generated if one of the simulations produces a
point pattern that is empty, or is otherwise unacceptable to the
function fun
.
The upper envelope may be NA
(plotted as plus or minus
infinity) if some of the function values
computed for the simulated point patterns are NA
.
Whether this occurs will depend on the function fun
,
but it usually happens when the simulated point pattern does not contain
enough points to compute a meaningful value.
Simulation envelopes do not compute confidence intervals;
they generate significance bands.
If you really need a confidence interval for the true mean
of the data pattern, use varblock
.
Adrian Baddeley Adrian.Baddeley@csiro.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner r.turner@auckland.ac.nz
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Arnold, 2003.
Ripley, B.D. (1981) Spatial statistics. John Wiley and Sons.
Ripley, B.D. 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.
fv.object
,
plot.envelope
,
plot.fv
,
envelope.envelope
,
pool.envelope
,
Kest
,
Gest
,
Fest
,
Jest
,
pcf
,
ppp
,
ppm
,
default.expand
data(simdat) X <- simdat # Envelope of K function under CSR ## Not run: plot(envelope(X)) ## End(Not run) # Translation edge correction (this is also FASTER): ## Not run: plot(envelope(X, correction="translate")) ## End(Not run) # Envelope of K function for simulations from Gibbs model data(cells) fit <- ppm(cells, ~1, Strauss(0.05)) ## Not run: plot(envelope(fit)) plot(envelope(fit), global=TRUE) ## End(Not run) # Envelope of K function for simulations from cluster model data(redwood) fit <- kppm(redwood, ~1, "Thomas") ## Not run: plot(envelope(fit, Gest)) plot(envelope(fit, Gest, global=TRUE)) ## End(Not run) # Envelope of G function under CSR ## Not run: plot(envelope(X, Gest)) ## End(Not run) # Envelope of L function under CSR # L(r) = sqrt(K(r)/pi) ## Not run: E <- envelope(X, Kest) plot(E, sqrt(./pi) ~ r) ## End(Not run) # Simultaneous critical envelope for L function # (alternatively, use Lest) ## Not run: plot(envelope(X, Kest, transform=expression(sqrt(./pi)), global=TRUE)) ## End(Not run) # 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) ## Not run: plot(envelope(demopat, Jcross, i="A", j="B")) ## End(Not run) # Use of `simulate' ## Not run: plot(envelope(cells, Gest, simulate=expression(runifpoint(42)))) plot(envelope(cells, Gest, simulate=expression(rMaternI(100,0.02)))) ## End(Not run) # Envelope under random toroidal shifts data(amacrine) ## Not run: plot(envelope(amacrine, Kcross, i="on", j="off", simulate=expression(rshift(amacrine, radius=0.25)))) ## End(Not run) # Envelope under random shifts with erosion ## Not run: plot(envelope(amacrine, Kcross, i="on", j="off", simulate=expression(rshift(amacrine, radius=0.1, edge="erode")))) ## End(Not run) # Envelope of INHOMOGENEOUS K-function with fitted trend ## Not run: trend <- density.ppp(X, 1.5) plot(envelope(X, Kinhom, lambda=trend, simulate=expression(rpoispp(trend)))) ## End(Not run) # Precomputed list of point patterns X <- rpoispp(30) PatList <- list() for(i in 1:19) PatList[[i]] <- runifpoint(npoints(X)) E <- envelope(X, Kest, nsim=19, simulate=PatList) if(interactive()) plot(E) # re-using the same point patterns EK <- envelope(X, Kest, nsim=10, savepatterns=TRUE) EG <- envelope(X, Kest, nsim=10, simulate=EK)