envelope

Simulation envelopes of summary function

Computes simulation envelopes of a summary function.

Usage

envelope(Y, fun=Kest, nsim=99, nrank=1, ..., 
  simulate=NULL, verbose=TRUE, clipdata=TRUE,
  start=NULL,control=list(nrep=1e5,expand=1.5),
  transform=NULL,global=FALSE,ginterval=NULL,
  savefuns=FALSE, savepatterns=FALSE,
  nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, internal=NULL)

Details

Simulation envelopes can be used to assess the goodness-of-fit of a point process model to point pattern data. See the References.

This function first generates nsim random point patterns in one of the following ways.

• IfYis a point pattern (an object of class"ppp") andsimulate=NULL, then this routine generatesnsimsimulations 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 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.fv.

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.

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.

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.

See Also

fv.object, plot.fv, Kest, Gest, Fest, Jest, pcf, ppp, ppm

Aliases

• envelope

Examples

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)