pc.estK | 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, inspired by some code of Philip M. Dixon
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.
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")
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