NScluster (version 1.2.0)

PalmThomas: Non-Parametric and Parametric Estimate of the Palm Intensity of Thomas Model

Description

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

Usage

PalmThomas(xy.points, pars1 = NULL, pars2 = NULL, delta = 0.001, 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, sigma), where mu is an intensity of parents, nu is an expected number of descendants for each parent and sigma is a parameter of the dispersal kernel.

pars2

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

delta

a width for the non-parametric Palm intensity function.

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 'Details' in EstimateThomas.

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, sigma = 0.03)
z <- SimulateThomas(pars, seed = 117)

## estimation
## need long c.p.u time in the minimization procedure
# }
# NOT RUN {
init.pars <- c(mu = 40.0, nu = 40.0, sigma = 0.05)
z1 <- EstimateThomas(z$offspring$xy, init.pars)
# Parameter               mu            nu         sigma
# Initial value      40.0000       40.0000        0.0500
# MPLE               46.7657       27.2079        0.0279
# }
# NOT RUN {

## Palm intensity
par1 <- c(50.0, 30.0, 0.03)  # pars
par2 <- c(46.7657, 27.2079, 0.0279)  # z1$mple
PalmThomas(z$offspring$xy, par1, par2)
# }