Simulation of the Inverse-power type model.
SimulateIP(pars, seed = NULL, plot = TRUE)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.
a positive integer, which is the seed for a sequence of uniform random numbers. The default seed is based on the current time.
logical. If TRUE (default), simulated parent points and
offspring points are plotted.
a list containing two components named "n" and
"xy" giving the number and the matrix of (x,y) coordinates of
simulated parents points respectively.
a list containing two components named "n" and
"xy" giving the number and the matrix of (x,y) coordinates of
simulated offspring points respectively.
Let random variable \(U\) be independently and uniformly distributed in [0,1].
For all \(r \ge 0\),
$$Q_{p,c}(r) := \int_0^r q_{p,c}(t)dt$$ $$= c^{p-1}(p-1) \frac{(r+c)^{1-p} - c^{1-p}}{1-p}$$ $$= 1 - c^{p-1} (r+c)^{1-p}.$$
Here, we put \(Q_{p,c}(r) = U\). From this, we have
$$r = c\{(1-U)^{1/(1-p)} - 1\}.$$
Let \((x_i^p, y_i^p), i=1,2,\dots, I\) be a coordinate of each parent point where the integer \(I\) is generated from the Poisson random variable \(Poisson(\mu)\) with mean \(\mu\) from now on. Then, for each \(i\), the number of offspring \(J_i\) is generated by the random variable \(Poisson(\nu)\) with mean \(\nu\). Then, using series of different uniform random numbers \(\{U\}\) for different \(i\) and \(j\), each of the offspring coordinates \((x_j^i, y_j^i), j=1,2,\dots,J_i\) is given by
$$x_j^i = x_i^p + r \cos(2 \pi U),$$ $$y_j^i = y_i^p + r \sin(2 \pi U),$$
owing to the isotropy condition of the distribution.
Given a positive number \(\nu\) and let a sequence of a random variable \(\{U_k\}\) be independently and uniformly distributed in [0,1], the Poisson random number \(M\) is the smallest integer such that
$$\sum_{k=1}^{M+1} - \log U_k > \nu,$$
where \(\log\) represents natural logarithm.
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.
# NOT RUN {
pars <- c(mu = 50.0, nu = 30.0, p = 1.5, c = 0.005)
SimulateIP(pars, seed = 353)
# }
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