palm.cppm: Non-parametric and Parametric Estimation for Palm Intensity

Description

Compute the non-parametric and the parametric Palm intensity function of the Neyman-Scott cluster point process models and their extensions.

Usage

palm.cppm(mple, pars = NULL, delta = 0.001, uplimit = 0.3)
# S3 method for Palm
print(x, ...)

Value

An object of class "Palm" containing the following components:

r: the distance \(r=j\Delta\), where \(j=1,2,\dots,[R/\Delta]\), \([\cdot]\) is the Gauss' symbol and \(R=1/2\) in the program.
np.palm: the corresponding values of the non-parametric Palm intensity function, which is normalized by the total intensity estimate (the mean number of points in \(W\)) of a given point pattern data.
norm.palm: the corresponding values of the normalized Palm intensity function, i.e., \(\lambda_{\bm{o}}(r)/\hat{\lambda}\), where \(\lambda_{\bm{o}}(r)\) is the Palm intensity and \(\lambda\) is an intensity of a cluster point process model. See 'Details' in mple.cppm.

There is another method plot.Palm for this class.

Arguments

mple: an object of class "mple".
pars: a named vector of the true parameters, if any.
delta: a width for the non-parametric Palm intensity function.
uplimit: upper limit in place of \(\infty\) of the integral in the probability distribution function relative to the random distance between two descendant points within the same cluster. The uplimit is valid for "IP" and "TypeA".
x: an object of class "Palm".
...: ignored.

References

Tanaka, U., Ogata, Y. and Katsura, K. (2008) Simulation and estimation of the Neyman-Scott type spatial cluster models. Computer Science Monographs 34, 1-44. The Institute of Statistical Mathematics, Tokyo. https://www.ism.ac.jp/editsec/csm/.

Examples

Run this code

if (FALSE) {
# The computation of MPLEs takes a long CPU time in the minimization procedure,
# especially for the Inverse-power type and the Type A models.

### Thomas Model
 #simulation
 pars <- c(mu = 50.0, nu = 30.0, sigma = 0.03)
 t.sim <- sim.cppm("Thomas", pars, seed = 117)
 ## estimation => Palm intensity
 init.pars <- c(mu = 40.0, nu = 40.0, sigma = 0.05)
 t.mple <- mple.cppm("Thomas", t.sim$offspring$xy, init.pars)
 t.palm <- palm.cppm(t.mple, pars)
 plot(t.palm)

### Inverse-Power Type Model
 # simulation
 pars <- c(mu = 50.0, nu = 30.0, p = 1.5, c = 0.005)
 ip.sim <- sim.cppm("IP", pars, seed = 353)
 ## estimation => Palm intensity
 init.pars <- c(mu = 55.0, nu = 35.0, p = 1.0, c = 0.01)
 ip.mple <- mple.cppm("IP", ip.sim$offspring$xy, init.pars, skip = 100)
 ip.palm <- palm.cppm(ip.mple, pars)
 plot(ip.palm)

### Type A Model
# simulation
 pars <- c(mu = 50.0, nu = 30.0, a = 0.3, sigma1 = 0.005, sigma2 = 0.1)
 a.sim <- sim.cppm("TypeA", pars, seed=575)
 ## estimation => Palm intensity
 init.pars <- c(mu=60.0, nu=40.0, a=0.5, sigma1=0.01, sigma2=0.1)
 a.mple <- mple.cppm("TypeA", a.sim$offspring$xy, init.pars, skip=100)
 a.palm <- palm.cppm(a.mple, pars)
 plot(a.palm)

### Type B Model
 # simulation
 pars <- c(mu1 = 10.0, mu2 = 40.0, nu = 30.0, sigma1 = 0.01, sigma2 = 0.03)
 b.sim <- sim.cppm("TypeB", pars, seed = 257)
 ## estimation => Palm intensity
 init.pars <- c(mu1 = 20.0, mu2 = 30.0, nu = 30.0, sigma1 = 0.02, sigma2 = 0.02)
 b.mple <- mple.cppm("TypeB", b.sim$offspring$xy, init.pars)
 b.palm <- palm.cppm(b.mple, pars)
 plot(b.palm)


### Type C Model
 # simulation
 pars <- c(mu1 = 5.0, mu2 = 9.0, nu1 = 30.0, nu2 = 150.0,
           sigma1 = 0.01, sigma2 = 0.05)
 c.sim <- sim.cppm("TypeC", pars, seed = 555)
 ## estimation => Palm intensity
 init.pars <- c(mu1 = 10.0, mu2 = 10.0, nu1 = 30.0, nu2 = 120.0,
                sigma1 = 0.03, sigma2 = 0.03)
 c.mple <- mple.cppm("TypeC", c.sim$offspring$xy, init.pars)
 c.palm <- palm.cppm(c.mple, pars)
 plot(c.palm)
}