Simulation of the generalized Thomas model of type C.
SimulateTypeC(pars, seed = NULL, parents.distinct = FALSE, plot = TRUE)a named vector of containing the values of the model parameters
(mu1, mu2, nu1, nu2, sigma1,
sigma2), where (mu\(i\), nu\(i\),
sigma\(i\)) is an intensity of parents, an expected number of
descendants, a parameter of the dispersal kernel for superposed component
\(i\) (\(i = 1,2\)), respectively.
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, points are distinguished by two
groups specified by parameters (mu1, nu1, sigma1)
and (mu2, nu2, sigma2).
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. xy[1:n[1], 1:2] are
generated from parameters (mu1, nu1, sigma1) and the
remainder are generated from (mu2, nu2, sigma2).
a list containing two components named "n" and
"xy" giving the number and the matrix of (x,y) coordinates of
simulated offspring points respectively. xy[1:n[1], 1:2] are
generated from parameters (mu1, nu1, sigma1) and
the remainder are generated from (mu2, nu2, sigma2).
Consider the two types of the Thomas model with parameters \((\mu_1, \nu_1, \sigma_1)\) and \((\mu_2, \nu_2, \sigma_2)\). Parents' configuration and numbers of the offspring cluster sizes are generated by the two types of uniformly distributed parents \((x_i^k, y_i^k)\) with \(i=1,2,\dots,Poisson(\mu_k)\) for \(k=1,2\), respectively.
Then, using series of different uniform random numbers \(\{ U \}\) for different \(i\) and \(j\), each of the offspring coordinates \((x_j^{k,i}, y_j^{k,i})\), \(j=1,2,\dots,Poisson(\nu_k)\) of the parents \((k,i)\) with \(k=1,2\) is given by
$$x_j^{k,i} = x_i^k + r_k \cos(2 \pi U),$$ $$y_j^{k,i} = y_i^k + r_k \sin(2 \pi U),$$
where
$$r_k = \sigma_k \sqrt{-2 \log(1-U_k)}, \quad k = 1, 2,$$
with different random numbers \(\{ U_k, U \}\) for different \(k, i\) and \(j\).
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(mu1 = 5.0, mu2 = 9.0, nu1 = 30.0, nu2 = 150.0,
sigma1 = 0.01, sigma2 = 0.05)
SimulateTypeC(pars, seed = 555)
# }
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