NScluster (version 1.2.0)

PalmTypeC: Non-Parametric and Parametric Estimate of the Palm Intensity of Type C Model

Description

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

Usage

PalmTypeC(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, nu1, nu2, sigma1, sigma2), where (mu\(i\), nu\(i\), 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, nu1, nu2, 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 EstimateTypeC.

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 = 5.0, mu2 = 9.0, nu1 = 30.0, nu2 = 150.0,
          sigma1 = 0.01, sigma2 = 0.05)
z <- SimulateTypeC(pars, seed = 555)

## estimation
## need long c.p.u time in the minimization procedure
# }
# NOT RUN {
init.pars <- c(mu1 = 10.0, mu2 = 10.0, nu1 = 30.0, nu2 = 120.0,
               sigma1 = 0.03, sigma2 = 0.03)
z1 <- EstimateTypeC(z$offspring$xy, init.pars)
## Parameter          mu1        mu2        nu1        nu2     sigma1     sigma2
## Initial value  10.0000    10.0000    30.0000   120.0200     0.0300     0.0300
## MPLE            4.6481     5.1690    28.1250   296.7022     0.0088     0.0477
# }
# NOT RUN {
## Palm intensity
par1 <- c(5.0, 9.0, 30.0, 150.0, 0.01, 0.05)  # pars
par2 <- c(4.6481, 5.1690, 28.1250, 296.7022, 0.0088, 0.0477)  # z1$mple
PalmTypeC(z$offspring$xy, par1, par2)
# }