NScluster (version 1.2.0)

PalmTypeA: Non-Parametric and Parametric Estimate of the Palm Intensity of Type A Model

Description

Calculate the non-parametric and parametric Palm intensity function of Type A model estimated directly from a set of point pattern data.

Usage

PalmTypeA(xy.points, pars1 = NULL, pars2 = NULL, delta = 0.001, uplimit = 0.3,
            plot = TRUE)

Arguments

xy.points

a matrix containing the coordinates (x,y) of points in a unit square: \(W=[0,1]\times[0,1]\).

pars1

a named vector of the true parameters (mu, nu, a, sigma1, sigma2), where mu is an intensity of parents, nu is an expected number of descendants for each parent, a is a mixture parameter, sigma1 and sigma2 are parameters of the dispersal kernel for each component.

pars2

a named vector of MPLEs (the maximum Palm likelihood estimates) (mu, nu, a, sigma1, sigma2).

delta

a width for the non-parametric Palm intensity function.

uplimit

upper limit value in place of \(\infty\).

plot

logical. If TRUE (default), the non-parametric estimate and the curves of true parameters and MPLEs are shown.

Value

r

the distance \(r=j\Delta\), where \(j=1,2,\dots,[R/\Delta]\), where [ ] is the Gauss' symbol and \(R=1/2\) is given in the program for the normalized rectangular region for the point pattern.

np.palm

the corresponding values of the non-parametric Palm intensity function of \(r\), which is normalized by the total intensity estimate of the point pattern data.

palm.normal

the normalized Palm intensity functions \(\lambda_{\bm{o}}(r)/\hat{\lambda}\) calculated from the given sets of parameter values. See ../doc/NScluster-guide.pdf.

References

U. Tanaka, Y. Ogata and K. Katsura, Simulation and estimation of the Neyman-Scott type spatial cluster models, Computer Science Monographs No.34, 2008, 1-44. The Institute of Statistical Mathematics.

Examples

# NOT RUN {
## simulation
pars <- c(mu = 50.0, nu = 30.0, a = 0.3, sigma1 = 0.005, sigma2 = 0.1)
z <- SimulateTypeA(pars, seed=575)

## estimation
## need very long c.p.u time in the minimization procedure
# }
# NOT RUN {
init.pars <- c(mu=60.0, nu=40.0, a=0.5, sigma1=0.01, sigma2=0.1)
z1 <- EstimateTypeA(z$offspring$xy, init.pars, skip=100)
# Parameter              mu           nu            a       sigma1       sigma2
# Initial value     60.0000      40.0000       0.5000       0.0100       0.1000
# MPLE              51.2441      25.1439       0.3431       0.0054       0.0824
# }
# NOT RUN {
## Palm intensity
par1 <- c(50.0, 30.0, 0.3, 0.005, 0.1)  # pars
par2 <- c(51.2441, 25.1439, 0.3431, 0.0054, 0.0824)  # z1$mple
PalmTypeA(z$offspring$xy, par1, par2)
# }