envelope {spatstat}R Documentation

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, ..., 
  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)

Arguments

Y

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 taken as the simulated patterns from which the envelopes are computed. If 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 simulate may be an object produced by the envelope command: see Details.

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 and simulations are properly comparable.

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".

nrep

Number of iterations in the Metropolis-Hastings 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 and a new random point pattern will be generated. If this happens more than maxnerr times, the algorithm will give up.

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.

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:

pointwise:

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).

simultaneous:

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).

based on sample moments:

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.

Value

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 fun has been estimated

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.

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 mean of the data pattern, use varblock.

Author(s)

Adrian Baddeley Adrian.Baddeley@csiro.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner r.turner@auckland.ac.nz

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.envelope, plot.fv, envelope.envelope, pool.envelope, Kest, Gest, Fest, Jest, pcf, ppp, ppm, default.expand

Examples

 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)

[Package spatstat version 1.25-3 Index]