# envelope

0th

Percentile

##### Simulation envelopes of summary function

Computes simulation envelopes of a summary function.

Keywords
spatial
##### Usage
envelope(Y, fun=Kest, nsim=99, nrank=1, verbose=TRUE, ...)
##### Arguments
Y
Either a point pattern (object of class "ppp") or a fitted point process model (object of class "ppm").
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
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.
verbose
Logical flag indicating whether to print progress reports during the simulations.
...
Extra arguments passed to fun.
##### Details

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

If Y is a point pattern (an object of class "ppp") then this routine generates 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 fitted point process model (an object of class "ppm") then this routine generates nsim simulated realisations of that model. The summary statistic fun is applied to each of these simulated patterns. Typically fun is one of the functions Kest, Gest, Fest, Jest or pcf. It can also be a home-made function; it should return an object of class "fv".

Upper and lower pointwise envelopes are computed 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 significance level of the associated Monte Carlo test is alpha = 2 * nrank/(1 + nsim). The return value is an object of class "fv" containing the summary function for the data point pattern and the upper and lower simulation envelopes. It can be plotted using plot.fv.

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.

##### Value

• An object of class "fv", see fv.object, which can be plotted directly using plot.fv.

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

##### 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. 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.

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

• envelope
##### Examples
X <- rpoispp(42)

# Envelope of K function under CSR
plot(envelope(X))
<testonly>plot(envelope(X, nsim=5))</testonly>

# Translation edge correction (this is also FASTER):
plot(envelope(X, correction="translate"))
<testonly>plot(envelope(X, nsim=5, correction="translate"))</testonly>

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

# Envelope of G function under CSR
plot(envelope(X, Gest))
<testonly>plot(envelope(X, Gest, nsim=5))</testonly>
Documentation reproduced from package spatstat, version 1.6-5, License: GPL version 2 or newer

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