NScluster (version 1.2.0)

PalmIP: Non-Parametric and Parametric Estimate of the Palm Intensity of Inverse-Power Type Model

Description

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

Usage

PalmIP(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 true parameters (mu, nu, p, c), where mu is an intensity of parents, nu is an expected number of descendants for each parent, p is the decay order and c is the scaling parameter.

pars2

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

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, p = 1.5, c = 0.005)
z <- SimulateIP(pars, seed = 353)

## estimation
## need very long c.p.u time in the minimization procedure
# }
# NOT RUN {
init.pars <- c(mu = 55.0, nu = 35.0, p = 1.0, c = 0.01)
z1 <- EstimateIP(z$offspring$xy, init.pars, skip = 100)
# Parameter               mu            nu             p             c
# Initial value      55.0000       35.0000        1.0000        0.0100
# MPLE               48.3143       30.5047        1.6467        0.0079
# }
# NOT RUN {
## Palm intensity
par1 <- c(50.0, 30.0, 1.5, 0.005)  # pars
par2 <- c(48.3143, 30.5047, 1.6467, 0.0079) # z1$mple
PalmIP(z$offspring$xy, par1, par2)
# }