envelope: Simulation Envelopes of Summary Function

Description

Computes simulation envelopes of a summary function.

Usage

envelope(Y, fun, ...)
  ## S3 method for class 'ppp':
envelope(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)
  ## S3 method for class 'ppm':
envelope(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)
  ## S3 method for class 'kppm':
envelope(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

Object containing point pattern data. A point pattern (object of class "ppp") or a fitted point process model (object of class "ppm" or "kppm").

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 nsim simulated values. A rank of 1 means that the minimum and maximum simulated values will be used.

...

Extra arguments passed to fun.

simulate

Optional. Specifies how to generate the simulated point patterns. If simulate is an expression in the R language, then this expression will be evaluated nsim times, to obtain nsim point patterns which are

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 TRUE to ensure that the data a

start,control

Optional. These specify the arguments start and control of rmh, giving complete control over the simulation algorithm. Applicable only when Y is a fitted model of class "ppm".

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 (global=FALSE) or simultaneous (global=TRUE).

ginterval

Optional. A vector of length 2 specifying the interval of $r$ values for the simultaneous critical envelopes. Only relevant if global=TRUE.

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 global=TRUE and the simulations are not based on CSR.

VARIANCE

Logical. If TRUE, critical envelopes will be calculated as sample mean plus or minus nSD times sample standard deviation.

nSD

Number of estimated standard deviations used to determine the critical envelopes, if VARIANCE=TRUE.

Yname

Character string that should be used as the name of the data point pattern Y when printing or plotting the results.

maxnerr

Maximum number of rejected patterns. If fun yields an error when applied to a simulated point pattern (for example, because the pattern is empty and fun requires at least one point), the pattern will be rejected a

Value

An object of class "fv", see fv.object, which can be printed and plotted directly.
Essentially a data frame containing columns
rthe vector of values of the argument $r$ at which the summary function fun has been estimated
obsvalues of the summary function for the data point pattern
lolower envelope of simulations
hiupper envelope of simulations
and either
theotheoretical value of the summary function under CSR (Complete Spatial Randomness, a uniform Poisson point process) if the simulations were generated according to CSR
mmeanestimated 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.

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 fun.

Details

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.

IfYis a point pattern (an object of class"ppp") andsimulate=NULL, then we generatensimsimulations of Complete Spatial Randomness (i.e.nsimsimulated point patterns each being a realisation of the uniform Poisson point process) with the same intensity as the patternY. (IfYis 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 inY.)
IfYis a fitted point process model (an object of class"ppm"or"kppm") andsimulate=NULL, then this routine generatesnsimsimulated realisations of that model.
Ifsimulateis 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 evaluatednsimtimes to yieldnsimpoint patterns. For example ifsimulate=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 callingenvelopewith the argumentsavepatterns=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: [object Object],[object Object],[object Object] 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: [object Object],[object Object]

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 link{plot.fv} for information about how to plot the envelopes.

References

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.

Examples

Run this code

data(simdat)
 X <- simdat

 # Envelope of K function under CSR
 plot(envelope(X))
 <testonly>plot(envelope(X, nsim=4))</testonly>

 # Translation edge correction (this is also FASTER):
 plot(envelope(X, correction="translate"))
 <testonly>plot(envelope(X, nsim=4, correction="translate"))</testonly>

 # Envelope of K function for simulations from Gibbs model 
 data(cells)
 fit <- ppm(cells, ~1, Strauss(0.05))
 plot(envelope(fit))
 plot(envelope(fit), global=TRUE)
 <testonly>plot(envelope(fit, nsim=4))
  plot(envelope(fit, nsim=4, global=TRUE))</testonly>

 # Envelope of K function for simulations from cluster model 
 data(redwood)
 fit <- kppm(redwood, ~1, "Thomas")
 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))</testonly>

 # Envelope of G function under CSR
 plot(envelope(X, Gest))
 <testonly>plot(envelope(X, Gest, nsim=4))</testonly>

 # Envelope of L function under CSR
 #  L(r) = sqrt(K(r)/pi)
 E <- envelope(X, Kest)
  plot(E, sqrt(./pi) ~ r)
 <testonly>E <- envelope(X, Kest, nsim=4)
  plot(E, sqrt(./pi) ~ r)</testonly>

 # Simultaneous critical envelope for L function
 # (alternatively, use Lest)
 plot(envelope(X, Kest, transform=expression(sqrt(./pi)), global=TRUE))
 <testonly>plot(envelope(X, Kest, nsim=4,transform=expression(sqrt(./pi)), global=TRUE))</testonly>

 # 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)
 plot(envelope(demopat, Jcross, i="A", j="B"))
 <testonly>plot(envelope(demopat, Jcross, i="A", j="B", nsim=4))</testonly>
 
 # Use of `simulate'
 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))</testonly>

 # Envelope under random toroidal shifts
 data(amacrine)
 plot(envelope(amacrine, Kcross, i="on", j="off",
               simulate=expression(rshift(amacrine, radius=0.25))))

 # Envelope under random shifts with erosion
 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
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)

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