SimulateIP: Simulation of the Inverse-Power Type Model

Description Usage Arguments Details Value References Examples

View source: R/NScluster.R

Description

Simulation of the Inverse-power type model.

Usage

1
  SimulateIP(pars, seed = NULL, plot = TRUE)

Arguments

pars

a named vector of containing the values of the model 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.

seed

a positive integer, which is the seed for a sequence of uniform random numbers. The default seed is based on the current time.

plot

logical. If TRUE (default), simulated parent points and offspring points are plotted.

Details

Let random variable U be independently and uniformly distributed in [0,1].

For all r>=0,

Q_{p,c}(r) := integral_0^r q_{p,c}(t)dt

= c^{p-1} (p-1) {(r+c)^{1-p} - c^{1-p}}/(1-p)

= 1 - c^{p-1} (r+c)^{1-p}.

Here, we put Q_{p,c}(r) = U. From this, we have

r = c {(1-U)^{1/(1-p)} - 1}.

Let (x_i^p, y_i^p), i=1,2,…, I be a coordinate of each parent point where the integer I is generated from the Poisson random variable Poisson(μ) with mean μ from now on. Then, for each i, the number of offspring J_i is generated by the random variable Poisson(ν) with mean ν. Then, using series of different uniform random numbers {U} for different i and j, each of the offspring coordinates (x_j^i, y_j^i), j=1,2,…,J_i is given by

x_j^i = x_i^p + r cos(2πU),

y_j^i = y_i^p + r sin(2πU),

owing to the isotropy condition of the distribution.

Given a positive number ν and let a sequence of a random variable {Uk} be independently and uniformly distributed in [0,1], the Poisson random number M is the smallest integer such that

∑_{k=1}^{M+1} - log Uk > ν,

where log represents natural logarithm.

Value

parents

a list containing two components named "n" and "xy" giving the number and the matrix of (x,y) coordinates of simulated parents points respectively.

offspring

a list containing two components named "n" and "xy" giving the number and the matrix of (x,y) coordinates of simulated offspring points respectively.

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

1
2
pars <- c(mu = 50.0, nu = 30.0, p = 1.5, c = 0.005)
SimulateIP(pars, seed = 353)

NScluster documentation built on March 19, 2018, 9:03 a.m.