R: Simulation Envelopes of Summary Function

envelope {spatstat.core} | 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, ...,
      funargs=list(), funYargs=funargs,
      simulate=NULL,  fix.n=FALSE, fix.marks=FALSE,
      verbose=TRUE, clipdata=TRUE,
      transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL,
      alternative=c("two.sided", "less", "greater"),
      scale=NULL, clamp=FALSE,
      savefuns=FALSE, savepatterns=FALSE,
      nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL,
      maxnerr=nsim, rejectNA=FALSE, silent=FALSE,
      do.pwrong=FALSE, envir.simul=NULL)

      ## S3 method for class 'ppm'
    envelope(Y, fun=Kest, nsim=99, nrank=1, ...,
      funargs=list(), funYargs=funargs,
      simulate=NULL, fix.n=FALSE, fix.marks=FALSE,
      verbose=TRUE, clipdata=TRUE,
      start=NULL, control=update(default.rmhcontrol(Y), nrep=nrep), nrep=1e5,
      transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL,
      alternative=c("two.sided", "less", "greater"),
      scale=NULL, clamp=FALSE,
      savefuns=FALSE, savepatterns=FALSE,
      nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL,
      maxnerr=nsim, rejectNA=FALSE, silent=FALSE,
      do.pwrong=FALSE, envir.simul=NULL)

      ## S3 method for class 'kppm'
    envelope(Y, fun=Kest, nsim=99, nrank=1, ...,
      funargs=list(), funYargs=funargs,
      simulate=NULL,
      verbose=TRUE, clipdata=TRUE,
      transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL,
      alternative=c("two.sided", "less", "greater"),
      scale=NULL, clamp=FALSE,
      savefuns=FALSE, savepatterns=FALSE,
      nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL,
      maxnerr=nsim, rejectNA=FALSE, silent=FALSE,
      do.pwrong=FALSE, envir.simul=NULL)

      ## S3 method for class 'slrm'
    envelope(Y, fun=Kest, nsim=99, nrank=1, ...,
      funargs=list(), funYargs=funargs,
      simulate=NULL,
      verbose=TRUE, clipdata=TRUE,
      transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL,
      alternative=c("two.sided", "less", "greater"),
      scale=NULL, clamp=FALSE,
      savefuns=FALSE, savepatterns=FALSE,
      nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL,
      maxnerr=nsim, rejectNA=FALSE, silent=FALSE,
      do.pwrong=FALSE, envir.simul=NULL)

### Arguments

`Y` | Object containing point pattern data. A point pattern (object of class `"ppp"`) or a fitted point process model (object of class `"ppm"`, `"kppm"` or `"slrm"`).
`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`.
`funargs` | A list, containing extra arguments to be passed to `fun`.
`funYargs` | Optional. A list, containing extra arguments to be passed to `fun` when applied to the original data `Y` only.
`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 function, then this function will be repeatedly applied to the data pattern `Y` to obtain `nsim` simulated patterns. 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.
`fix.n` | Logical. If `TRUE`, simulated patterns will have the same number of points as the original data pattern. This option is currently not available for `envelope.kppm`.
`fix.marks` | Logical. If `TRUE`, simulated patterns will have the same number of points and the same marks as the original data pattern. In a multitype point pattern this means that the simulated patterns will have the same number of points of each type as the original data. This option is currently not available for `envelope.kppm`.
`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`.
`use.theory` | Logical value indicating whether to use the theoretical value, computed by `fun`, as the reference value for simultaneous envelopes. Applicable only when `global=TRUE`. Default is `use.theory=TRUE` if `Y` is a point pattern, or a point process model equivalent to Complete Spatial Randomness, and `use.theory=FALSE` otherwise.
`alternative` | Character string determining whether the envelope corresponds to a two-sided test (`side="two.sided"`, the default) or a one-sided test with a lower critical boundary (`side="less"`) or a one-sided test with an upper critical boundary (`side="greater"`).
`scale` | Optional. Scaling function for global envelopes. A function in the R language which determines the relative scale of deviations, as a function of distance r, when computing the global envelopes. Applicable only when `global=TRUE`. Summary function values for distance `r` will be divided by `scale(r)` before the maximum deviation is computed. The resulting global envelopes will have width proportional to `scale(r)`.
`clamp` | Logical value indicating how to compute envelopes when `alternative="less"` or `alternative="greater"`. Deviations of the observed summary function from the theoretical summary function are initially evaluated as signed real numbers, with large positive values indicating consistency with the alternative hypothesis. If `clamp=FALSE` (the default), these values are not changed. If `clamp=TRUE`, any negative values are replaced by zero.
`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 a fatal 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.
`rejectNA` | Logical value specifying whether to reject a simulated pattern if the resulting values of `fun` are all equal to `NA`, `NaN` or infinite. If `FALSE` (the default), then simulated patterns are only rejected when `fun` gives a fatal error.
`silent` | Logical value specifying whether to print a report each time a simulated pattern is rejected.
`do.pwrong` | Logical. If `TRUE`, the algorithm will also estimate the true significance level of the “wrong” test (the test that declares the summary function for the data to be significant if it lies outside the pointwise critical boundary at any point). This estimate is printed when the result is printed.
`envir.simul` | Environment in which to evaluate the expression `simulate`, if not the current environment.

### 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"`, `"kppm"` and `"slrm"` described here. There are also methods for the classes `"pp3"`, `"lpp"` and `"lppm"` which are described separately under `envelope.pp3` and `envelope.lpp`. Envelopes can also be computed from other envelopes, using `envelope.envelope`.

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"` or `"slrm"`) 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 function in the R language, typically containing a call to a random generator. This function will be applied repeatedly to the original data pattern `Y` to yield `nsim` point patterns. For example if `simulate=rlabel` then each simulated pattern was generated by evaluating `rlabel(Y)` and consists of a randomly-relabelled version of `Y`.

  - 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:

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)`.

  This test can also be performed using `mad.test`.

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 means that you simply specify these arguments in the call to `envelope`, and they will be passed to `fun`. 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.

Arguments can also be passed to the function `fun` through the list `funargs`. This mechanism is typically used if an argument of `fun` has the same name as an argument of `envelope`. The list `funargs` should contain entries of the form `name=value`, where each `name` is the name of an argument of `fun`.

There is also an option, rarely used, in which different function arguments are used when computing the summary function for the data `Y` and for the simulated patterns. If `funYargs` is given, it will be used when the summary function for the data `Y` is computed, while `funargs` will be used when computing the summary function for the simulated patterns. This option is only needed in rare cases: usually the basic principle requires that the data and simulated patterns must be treated equally, so that `funargs` and `funYargs` should be identical.

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. Such transforms are useful if `global=TRUE` or `VARIANCE=TRUE`. The `transform` must be an expression object using the symbol `.` to represent the function value (and possibly other symbols recognised by `with.fv`). 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)/π) and this is implemented by setting `transform=expression(sqrt(./pi))`.

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 `"envelope"` and `"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 summary function of the point process, use `lohboot`. See also `varblock`.

### Edge corrections

It is common to apply a correction for edge effects when calculating a summary function such as the K function. Typically the user has a choice between several possible edge corrections. In a call to `envelope`, the user can specify the edge correction to be applied in `fun`, using the argument `correction`. See the Examples below.

Summary functions in spatstat

  Summary functions that are available in spatstat, such as `Kest`, `Gest` and `pcf`, have a standard argument called `correction` which specifies the name of one or more edge corrections.

  The list of available edge corrections is different for each summary function, and may also depend on the kind of window in which the point pattern is recorded. In the case of `Kest` (the default and most frequently used value of `fun`) the best edge correction is Ripley's isotropic correction if the window is rectangular or polygonal, and the translation correction if the window is a binary mask. See the help files for the individual functions for more information.

  All the summary functions in spatstat recognise the option `correction="best"` which gives the “best” (most accurate) available edge correction for that function.

  In a call to `envelope`, if `fun` is one of the summary functions provided in spatstat, then the default is `correction="best"`. This means that by default, the envelope will be computed using the “best” available edge correction.

  The user can override this default by specifying the argument `correction`. For example the computation can be accelerated by choosing another edge correction which is less accurate than the “best” one, but faster to compute.

User-written summary functions

  If `fun` is a function written by the user, then `envelope` has to guess what to do.

  If `fun` has an argument called `correction`, or has `...` arguments, then `envelope` assumes that the function can handle a correction argument. To compute the envelope, `fun` will be called with a `correction` argument. The default is `correction="best"`, unless overridden in the call to `envelope`.

  Otherwise, if `fun` does not have an argument called `correction` and does not have `...` arguments, then `envelope` assumes that the function cannot handle a correction argument. To compute the envelope, `fun` is called without a correction argument.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### 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 84 (3) 477–489.

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. There are also methods for `print` and `summary`.

`Kest`, `Gest`, `Fest`, `Jest`, `pcf`, `ppp`, `ppm`, `default.expand`

### Examples

     X <- simdat
     online <- interactive()
     Nsim <- if(online) 19 else 3

     # Envelope of K function under CSR
     plot(envelope(X, nsim=Nsim))

     # Translation edge correction (this is also FASTER):
     if(online) {
       plot(envelope(X, correction="translate"))
     } else {
      E <- envelope(X, nsim=Nsim, correction="translate")
     }

    # Global envelopes
     if(online) {
       plot(envelope(X, Lest, global=TRUE))
       plot(envelope(X, Kest,  global=TRUE, scale=function(r) { r }))
     } else {
      E <- envelope(X, Lest, nsim=Nsim, global=TRUE)
      E <- envelope(X, Kest, nsim=Nsim, global=TRUE, scale=function(r) { r })
      E
      summary(E)
     }

     # Envelope of K function for simulations from Gibbs model
     if(online) {
       fit <- ppm(cells ~1, Strauss(0.05))
       plot(envelope(fit))
       plot(envelope(fit, global=TRUE))
     } else {
      fit <- ppm(cells ~1, Strauss(0.05), nd=20)
      E <- envelope(fit, nsim=Nsim, correction="border", nrep=100)
      E <- envelope(fit, nsim=Nsim, correction="border", global=TRUE, nrep=100)
     }

     # Envelope of K function for simulations from cluster model
     fit <- kppm(redwood ~1, "Thomas")
     if(online) {
       plot(envelope(fit, Gest))
       plot(envelope(fit, Gest, global=TRUE))
     } else {
      E <- envelope(fit, Gest, correction="rs", nsim=Nsim, global=TRUE, nrep=100)
     }

     # Envelope of G function under CSR
     if(online) {
       plot(envelope(X, Gest))
     } else {
      E <- envelope(X, Gest, correction="rs", nsim=Nsim)
     }

     # Envelope of L function under CSR
     #  L(r) = sqrt(K(r)/pi)
     if(online) {
      E <- envelope(X, Kest)
     } else {
      E <- envelope(X, Kest, correction="border", nsim=Nsim)
     }
     plot(E, sqrt(./pi) ~ r)

     # Simultaneous critical envelope for L function
     # (alternatively, use Lest)
     if(online) {
      plot(envelope(X, Kest, transform=expression(sqrt(./pi)), global=TRUE))
     } else {
      E <- envelope(X, Kest, nsim=Nsim, correction="border",
                   transform=expression(sqrt(./pi)), global=TRUE)
     }

     ## One-sided envelope
     if(online) {
      plot(envelope(X, Lest, alternative="less"))
     } else {
      E <- envelope(X, Lest, nsim=Nsim, alternative="less")
     }

     # 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'

     if(online) {
     plot(envelope(demopat, Jcross, i="A", j="B"))
     } else {
     plot(envelope(demopat, Jcross, correction="rs", i="A", j="B", nsim=Nsim))
     }

     # Use of `simulate' expression
     if(online) {
     plot(envelope(cells, Gest, simulate=expression(runifpoint(42))))
     plot(envelope(cells, Gest, simulate=expression(rMaternI(100,0.02))))
     } else {
      plot(envelope(cells, Gest, correction="rs", simulate=expression(runifpoint(42)), nsim=Nsim))
        plot(envelope(cells, Gest, correction="rs", simulate=expression(rMaternI(100, 0.02)),
    nsim=Nsim, global=TRUE))
     }

     # Use of `simulate' function
     if(online) {
       plot(envelope(amacrine, Kcross, simulate=rlabel))
     } else {
       plot(envelope(amacrine, Kcross, simulate=rlabel, nsim=Nsim))
     }

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

     # Envelope under random shifts with erosion
     if(online) {
     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")
     if(online) {
        plot(envelope(fit, Kinhom, lambda=fit, nsim=19))
     } else {
        envelope(fit, Kinhom, lambda=fit, nsim=Nsim)
     }

     # 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.

     if(online) {
     red.dens <- density(redwood, sigma=bw.diggle, positive=TRUE)
     plot(envelope(redwood, Kinhom, sigma=bw.diggle,
             simulate=expression(rpoispp(red.dens))))
     }

     # Precomputed list of point patterns
     if(online) {
      nX <- npoints(X)
      PatList <- list()
      for(i in 1:Nsim) PatList[[i]] <- runifpoint(nX)
      E <- envelope(X, Kest, nsim=19, simulate=PatList)
     } else {
      PatList <- list()
      for(i in 1:Nsim) PatList[[i]] <- runifpoint(10)
     }
     E <- envelope(X, Kest, nsim=Nsim, simulate=PatList)

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

     if(online) {
       EK <- envelope(X, Kest, nsim=Nsim, savepatterns=TRUE)
       EG <- envelope(X, Gest, nsim=Nsim, simulate=EK)
     }

[Package spatstat.core version 2.3-0 Index]