Calculate simulation envelope for a Poisson Cluster Process

This function computes the envelope of Khat from simulations of a Poisson Cluster Process for a given polygon

Kenv.pcp(rho, m, s2, region.poly, larger.region=NULL, nsim, r, vectorise.loop=TRUE)

Arguments

rho

intensity of the parent process

m

average number of offsprings per parent

s2

variance of location of offsprings relative to their parent

region.poly

a polygon defining the region in which the process is to be generated

larger.region

a rectangle containing the region of interest given in the form (xl,xu,yl,yu), defaults to sbox() around region.poly

nsim

number of simulations required

r

vector of distances at which the K function has to be estimated

vectorise.loop

if TRUE, use new vectorised code, if FALSE, use loop as before

Value

ave

mean of simulations

upper

upper bound of envelope

lower

lower bound of envelope

References

Diggle, P. J. (1983) Statistical analysis of spatial point patterns, London: Academic Press, pp. 55-57 and 78-81; Bailey, T. C. and Gatrell, A. C. (1995) Interactive spatial data analysis, Harlow: Longman, pp. 106-109.

Author

Giovanni Petris <GPetris@uark.edu>, Roger.Bivand@nhh.no

See also

pcp, pcp.sim, khat

Examples

data(cardiff)
polymap(cardiff$poly)
pointmap(as.points(cardiff), add=TRUE)
title("Locations of homes of 168 juvenile offenders")

pcp.fit <- pcp(as.points(cardiff), cardiff$poly, h0=30, n.int=30)
pcp.fit
#> $par
#>         s2        rho 
#> 6.16109743 0.01136752 
#> 
#> $value
#> [1] 0.02734823
#> 
#> $counts
#> function gradient 
#>       77       NA 
#> 
#> $convergence
#> [1] 0
#> 
#> $message
#> NULL
#> 
m <- npts(as.points(cardiff))/(areapl(cardiff$poly)*pcp.fit$par[2])
r <- seq(2,30,by=2)
K.env <- Kenv.pcp(pcp.fit$par[2], m, pcp.fit$par[1], cardiff$poly,
           nsim=20, r=r)
L.env <- lapply(K.env, FUN=function(x) sqrt(x/pi)-r)
limits <- range(unlist(L.env))
plot(r, sqrt(khat(as.points(cardiff),cardiff$poly,r)/pi)-r, ylim=limits,
     main="L function with simulation envelopes and average", type="l",
     xlab="distance", ylab="")
lines(r, L.env$lower, lty=5)
lines(r, L.env$upper, lty=5)
lines(r, L.env$ave, lty=6)
abline(h=0)