pc.estK {ecespa}

Fit the Poisson Cluster Point Process by Minimum Contrast

Description

Fits the Poisson Cluster point process to a point pattern dataset by the Method of Minimum Contrast.

Usage

pc.estK(Kobs, r, sigma2 = NULL, rho = NULL)
Kclust(r, sigma2, rho)

Arguments

Kobs

Empirical K-function.

r

Sequence of distances at which function K has been estimated.

sigma2

Optional. Starting value for the parameter sigma2 of the Poisson Cluster process.

rho

Optional. Starting value for the parameter rho of the Poisson Cluster process.

Details

The algorithm fits the Poisson cluster point process to a point pattern, by finding the parameters of the Poisson cluster model which give the closest match between the theoretical K function of the Poisson cluster process and the observed K function. For a more detailed explanation of the Method of Minimum Contrast, see mincontrast in spatstat or Diggle (2003: 86).

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)]

The theoretical K-function of this Poisson cluster process is (Diggle, 2003):

pi*r^2 + [1- exp(-r^2/4*sigma^2)]/rho

The command Kclust computes the theoretical K-function of this Poisson cluster process and can be used to find some initial estimates of rho and sigma^2. In any case, the optimization usually finds the correct parameters even without starting values for these parameters.

This Poisson cluster process can be simulated with sim.poissonc.

Value

sigma2

Parameter sigma^2.

rho

Parameter rho.

Note

The exponents p and q of the contrast criterion (see mincontrast) are fixed respectively to p = 2 and q = 1/4. The rmin and rmax limits of integration of the contrast criterion are set up by the sequence of values of r and Kobs passed to pc.estK.

Author(s)

Marcelino de la Cruz Rot, inspired by some code of Philip M. Dixon

References

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

See Also

ipc.estK for fitting the inhomogeneous Poisson cluster process; some functions in spatstat ( matclust.estK and lgcp.estK) fit other appropriate processes for clustered patterns; mincontrast performs a more general implementation of the method of mimimum contrast.

Examples

data(gypsophylous)


# set the number of simulations (nsim=199 or larger for real analyses)
nsim<- 19

## 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 gypsophylous pattern 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="",legendpos="bottomleft")