envelope

Simulation Envelopes of Summary Function

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,
  do.pwrong=FALSE, envir.simul=NULL)
  ## S3 method for class 'ppm':
envelope(Y, fun=Kest, nsim=99, nrank=1, \dots, 
  simulate=NULL, verbose=TRUE, clipdata=TRUE,
  start=NULL, control=update(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,
  do.pwrong=FALSE, envir.simul=NULL)
  ## 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,
  do.pwrong=FALSE, envir.simul=NULL)

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 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 .... In particular, the argument correction determines the edge correction to be used to calculate the summary statistic. See the section on Edge Corrections, and the Examples.

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 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.

Errors and warnings

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.

Confidence intervals

Simulation envelopes do not compute confidence intervals; they generate significance bands. If you really need a confidence interval for the true summary function of the point process, use lohboot. See also varblock.

References

Baddeley, A., Diggle, P.J., Hardegen, A., Lawrence, T., Milne, R.K. and Nair, G. (2014) On tests of spatial pattern based on simulation envelopes. Ecological Monographs, to appear. 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

dclf.test, mad.test for envelope-based tests. fv.object, plot.envelope, plot.fv, envelope.envelope, pool.envelope for handling envelopes. Kest, Gest, Fest, Jest, pcf, ppp, ppm, default.expand

Aliases

• envelope
• envelope.ppp
• envelope.ppm
• envelope.kppm

Examples

X <- simdat

 # Envelope of K function under CSR
 plot(envelope(X))
 <testonly>plot(envelope(X, nsim=3))</testonly>

 # Translation edge correction (this is also FASTER):
 plot(envelope(X, correction="translate"))
 <testonly>E <- envelope(X, nsim=3, correction="translate")</testonly>

 # Envelope of K function for simulations from Gibbs model 
 fit <- ppm(cells, ~1, Strauss(0.05))
 plot(envelope(fit))
 plot(envelope(fit), global=TRUE)
 <testonly>fit <- ppm(cells, ~1, Strauss(0.05), nd=20)
  E <- envelope(fit, nsim=3, correction="border", nrep=100)
  E <- envelope(fit, nsim=3, correction="border", global=TRUE, nrep=100)</testonly>

 # Envelope of K function for simulations from cluster model 
 fit <- kppm(redwood, ~1, "Thomas")
 plot(envelope(fit, Gest))
 plot(envelope(fit, Gest, global=TRUE))
 <testonly>E <- envelope(fit, Gest, correction="rs", nsim=3, global=TRUE, nrep=100)</testonly>

 # Envelope of G function under CSR
 plot(envelope(X, Gest))
 <testonly>E <- envelope(X, Gest, correction="rs", nsim=3)</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, correction="border", nsim=3)
  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>E <- envelope(X, Kest, nsim=3, correction="border",
               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, correction="rs", i="A", j="B", nsim=3))</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, correction="rs", simulate=expression(runifpoint(42)), nsim=3))
    plot(envelope(cells, Gest, correction="rs", simulate=expression(rMaternI(100, 0.02)),
nsim=3, 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

 # The following is valid.
 # Setting lambda=fit means that the fitted model is re-fitted to
 # each simulated pattern to obtain the intensity estimates for Kinhom.
 # (lambda=NULL would also be valid)

 fit <- kppm(redwood, ~1, clusters="MatClust")
 plot(envelope(fit, Kinhom, lambda=fit, nsim=19))
 <testonly>envelope(fit, Kinhom, lambda=fit, nsim=3)</testonly>

 # Note that the principle of symmetry, essential to the validity of
 # simulation envelopes, requires that both the observed and
 # simulated patterns be subjected to the same method of intensity
 # estimation. In the following example it would be incorrect to set the
 # argument 'lambda=red.dens' in the envelope command, because this
 # would mean that the inhomogeneous K functions of the simulated
 # patterns would be computed using the intensity function estimated
 # from the original redwood data, violating the symmetry.  There is
 # still a concern about the fact that the simulations are generated
 # from a model that was fitted to the data; this is only a problem in
 # small datasets.

red.dens <- density(redwood, sigma=bw.diggle)
 plot(envelope(redwood, Kinhom, sigma=bw.diggle,
         simulate=expression(rpoispp(red.dens))))

 # Precomputed list of point patterns
nX <- npoints(X)
 PatList <- list()
 for(i in 1:19) PatList[[i]] <- runifpoint(nX)
 E <- envelope(X, Kest, nsim=19, simulate=PatList)
<testonly>PatList <- list()
 for(i in 1:3) PatList[[i]] <- runifpoint(10)
 E <- envelope(X, Kest, nsim=3, simulate=PatList)</testonly>

# re-using the same point patterns
EK <- envelope(X, Kest, savepatterns=TRUE)
 EG <- envelope(X, Gest, simulate=EK)
<testonly>EK <- envelope(X, Kest, nsim=3, savepatterns=TRUE)
 EG <- envelope(X, Gest, nsim=3, simulate=EK)</testonly>