# SimulateIP: Simulation of the Inverse-Power Type Model In NScluster: Simulation and Estimation of the Neyman-Scott Type Spatial Cluster Models

## 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.