NScluster (version 1.1.0)

SimulateIP: Simulation of the Inverse-Power Type Model

Description

Simulation of the Inverse-Power Type Model.

Usage

SimulateIP(seeds, pa, ty=1, pmax=100, omax=3000, plot=TRUE)

Arguments

seeds

the three positive integer variables, which are initial seeds for a sequence of uniform random numbers.

pa

the parameter values $(mu, nu, p, c)$.

ty

the variable $Ty$ for the standardized coordinates of points in the rectangular region $[0,1]*[0,Ty]$ without loss of generality except for the scaling.

pmax

maximum number of parent points.

omax

maximum number of offspring points.

plot

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

Value

Details

Let random variable $U$ be independently and uniformly distributed in [0,1]. For any $r>=0$,

$$Q_{p,c}(r) := \int_0^r q_{p,c}(t)dt$$ $$= c^{p-1}(p-1) \frac{(r+c)^{1-p} - c^{1-p}}{1-p}$$ $$= 1 - c^{p-1} (r+c)^{1-p}.$$

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

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

Similarly, coordinate of the offspring points $(x_j^i, y_j^i), j=1,2,...,Poisson(nu)$ with the inverse-power type is given for each $i=1,2,...,Poisson(mu)$,

$$x_j^i = x_i^p + r \cos(2 \pi U),$$ $$y_j^i = y_i^p + r \sin(2 \pi U),$$

using series of different uniform random number ${U}$ for different $i$ and $j$.

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

  seeds <- c(822, 913, 905)
  mu <- 50;  nu <- 30;  p <- 1.5;  c <- 0.005
  SimulateIP(seeds, c(mu,nu,p,c))