ecespa (version 1.1-17)

sim.poissonc: Simulate Poisson Cluster Process

Description

Generate a random point pattern, a simulated realisation of the Poisson Cluster Process

Usage

sim.poissonc(x.ppp, rho, sigma)

Value

The simulated point pattern (an object of class "ppp").

Arguments

x.ppp

Point pattern whose window and intensity will be simulated. An object with the ppp format of spatstat.

rho

Parameter \(rho\) of the Poisson Cluster process.

sigma

Parameter \(sigma\) of the Poisson Cluster process.

Author

Marcelino de la Cruz Rot

Warning

This implementation simulates only point patterns within rectangular windows. Use ipc.estK to fit and rIPCP (or the spatstat functions) to simulate point patterns within irregular windows.

Details

The Poisson cluster processes are defined by the following postulates (Diggle 2003):

PCP1	Parent events form a Poisson process with intensity \(rho\).
PCP2	Each parent produces a random number of offspring, according to a probability distribution
	\(p[s]: s = 0, 1, 2, ...\)
PCP3	The positions of the offspring relative to their parents are distributed according to a bivariate pdf \(h\).

This implementation asumes that the probability distribution \(p[s]\) of offspring per parent is a Poisson distribution and that the position of each offspring relative to its parent follows a radially symetric Gaussian distribution with pdf

$$h(x, y) = [1/(2*pi*sigma^2)]* exp[-(x^2+y^2)/(2*sigma^2)]$$

References

Diggle, P.J. 2003. Statistical analysis of spatial point patterns. Arnold, London.

See Also

rIPCP to simulate inhomogeneous PCP; rNeymanScott and rThomas in spatstat

Examples

data(gypsophylous)

# set the number of simulations (nsim=199 or larger for real analyses)
nsim<- 39

## Estimate K function ("Kobs").
gyps.env <- envelope(gypsophylous, Kest, correction="iso", nsim=nsim)

plot(gyps.env, sqrt(./pi)-r~r, legend=FALSE)

## Fit Poisson Cluster Process. The limits of integration 
## rmin and rmax are setup to 0 and 60, respectively. 
cosa.pc <- pc.estK(Kobs = gyps.env$obs[gyps.env$r<=60],
		           r = gyps.env$r[gyps.env$r<=60])

## Add fitted Kclust function to the plot.
lines(gyps.env$r,sqrt(Kclust(gyps.env$r, cosa.pc$sigma2,cosa.pc$rho)/pi)-gyps.env$r,
       lty=2, lwd=3, col="purple")

## A kind of pointwise test of the pattern gypsophilous been a realisation
## of the fitted model, simulating with sim.poissonc and using function J (Jest).

gyps.env.sim <- envelope(gypsophylous, Jest,  nsim=nsim,
                    simulate=expression(sim.poissonc(gypsophylous,
		    sigma=sqrt(cosa.pc$sigma2), rho=cosa.pc$rho)))

plot(gyps.env.sim,  main="")