NScluster (version 1.2.0)

PalmTypeB: Non-Parametric and Parametric Estimate of the Palm Intensity of Type B Model

Description

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

Usage

PalmTypeB(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 (mu1, mu2, nu, sigma1, sigma2), where (mu\(i\), nu, sigma\(i\)) is an intensity of parents, an expected number of descendants, a parameter of the dispersal kernel for superposed component \(i\) (\(i = 1,2\)), respectively.

pars2

a named vector of MPLEs (the maximum Palm likelihood estimates) (mu1, mu2, nu, sigma1, sigma2).

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 EstimateTypeB.

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(mu1 = 10.0, mu2 = 40.0, nu = 30.0, sigma1 = 0.01, sigma2 = 0.03)
z <- SimulateTypeB(pars, seed = 257)

## estimation
## need very long c.p.u time in the minimization procedure
# }
# NOT RUN {
init.pars <- c(mu1 = 20.0, mu2 = 30.0, nu = 30.0, sigma1 = 0.02, sigma2 = 0.02)
z1 <- EstimateTypeB(z$offspring$xy, init.pars)
# Parameter             mu1          mu2          nu        sigma1       sigma2
# Initial value     20.0000      30.0000      30.0000       0.0200       0.0200
# MPLE              16.1778      44.3974      28.3942       0.0101       0.0312
# }
# NOT RUN {
## Palm intensity
par1 <- c(10.0, 40.0, 30.0, 0.01, 0.03) # pars
par2 <- c(16.1778, 44.3974, 28.3942, 0.0101, 0.0312) # z1$mple
PalmTypeB(z$offspring$xy, par1, par2)
# }