pc.estK {ecespa} | R Documentation |
Fits the Poisson Cluster point process to a point pattern dataset by the Method of Minimum Contrast.
pc.estK(Kobs, r, sigma2 = NULL, rho = NULL) Kclust(r, sigma2, rho)
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. |
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
.
sigma2 |
Parameter sigma^2. |
rho |
Parameter rho. |
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
.
Marcelino de la Cruz Rot marcelino.delacruz@upm.es, inspired by some code of Philip M. Dixon http://www.public.iastate.edu/~pdixon/
Diggle, P. J. 2003. Statistical analysis of spatial point patterns. Arnold, London.
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.
## Not run: data(gypsophylous) ## Estimate K function ("Kobs"). gyps.env <- envelope(gypsophylous, Kest, correction="iso", nsim=99) plot(gyps.env, sqrt(./pi)-r~r) ## 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$r, lty=2, lwd=3, col="purple") ## A kind of pointwise test of the gypsophylous pattern been a realisation ## of the fitted model, simulating with sim.poissonc and using function J (Jest). gyps.env.sim <- envelope(gypsophylous, Jest, nsim=99, simulate=expression(sim.poissonc(gypsophylous, sigma=sqrt(cosa.pc$sigma2), rho=cosa.pc$rho))) plot(gyps.env.sim, main="") ## End(Not run)