sim.cppm  R Documentation 
Simulation for the Thomas and Inversepower type models, and the extended Thomas models of type A, B, and C.
sim.cppm(model = "Thomas", pars, seed = NULL)
## S3 method for class 'sim.cpp'
print(x, ...)
## S3 method for class 'sim.cpp'
plot(x, parents.distinct = FALSE, ...)
model 
a character string indicating each cluster point process model:

pars 
a named vector giving the values of each parameter. See 'Details'. 
seed 
arbitrary positive integer to generate a sequence of uniform random numbers. The default seed is based on the current time. 
x 
an object of class 
parents.distinct 
logical. If 
... 
further graphical parameters from 
We consider the five cluster point process models: the Thomas and Inversepower type models, and the extended Thomas models of type A, B, and C.
"Thomas"
(Thomas model)
The parameters of the model are as follows:
mu
: the intensity of parent points.
nu
: the expectation of a random number of descendant points
of each parent point.
sigma
: the parameter set of the dispersal kernel.
Let a random variable U
be independently and uniformly distributed
in [0,1].
Consider
U = \int_0^r q_\sigma(t)dt =
1  \exp \left( \frac{r^2}{2\sigma^2} \right),
where r
is the random variable of the distance between each parent
point and the descendant points associated with the given parent. The
distance is distributed independently and identically according to the
dispersal kernel.
We have
r = \sigma \sqrt{2 \log(1U)}.
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.
"IP"
(Inversepower type model)
The parameters of the model are as follows:
mu
: the intensity of parent points.
nu
: the expectation of a random number of descendant points
of each parent point.
p
, c
: the set of parameters of the dispersal kernel,
where p
> 1 and c
> 0.
Let U
be as above.
For all r \ge 0
,
Q_{p,c}(r) := \int_0^r q_{p,c}(t)dt
= c^{p1}(p1) \frac{(r+c)^{1p}  c^{1p}}{1p}
= 1  c^{p1} (r+c)^{1p}.
Here, we put Q_{p,c}(r) = U
. From this, we have
r = c\{(1U)^{1/(1p)}  1\}.
The parent points and their descendant points are generated the same as the Thomas model.
"TypeA"
(Type A model)
The parameters of the model are as follows:
mu
: the intensity of parent points.
nu
: the expectation of a random number of descendant points
of each parent point.
a
, sigma1
, sigma2
: the set of parameters of
the dispersal kernel, where where a
is a mixture ratio
parameter with 0 < a
< 1.
Let each random variable U_k, k=1,2
, be independently and
uniformly distributed in [0,1].
Then r
satisfies as follows:
r = \sigma_1 \sqrt{2 \log(1U_1)}, \quad U_2 \le a ,
r = \sigma_2 \sqrt{2 \log(1U_1)}, \quad \mathrm{otherwise.}
The parent points and their descendant points are generated the same as the Thomas model.
"TypeB"
(Type B model)
The TypeB is a superposed Thomas model. The parameters of the model are as follows:
mu1
, mu2
: the corresponding intensity of parent
points of each Thomas model.
nu
: the expectation of a random number of descendant points
of each parent point.
sigma1
, sigma2
: the corresponding set of parameters
of the dispersal kernel of each Thomas model.
Consider the two types of the Thomas model with parameters
(\mu_1, \nu, \sigma_1)
and
(\mu_2, \nu, \sigma_2)
.
Parents' configuration and numbers of the descendant 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 descendant coordinates
(x_j^{k,i}, y_j^{k,i})
of the parents (x_i^k, y_i^k)
,
k=1,2
, j=1,2,\dots,Poisson(\nu)
, 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 (1U_k)}, \quad k = 1, 2,
with different random numbers \{U_k, U\}
for different
k, i
, and j
.
"TypeC"
(Type C model)
The TypeC is a superposed Thomas model. The parameters of the model are as follows:
mu1
, mu2
: the corresponding intensity of parent
points of each Thomas model.
nu1
, nu2
: the corresponding expectation of a random
number of descendant points of each Thomas model.
sigma1
, sigma2
: the corresponding set of parameters
of the dispersal kernel of each Thomas model.
The parent points and their descendant points are generated the same as the Type B model.
sim.cppm
returns an object of class "sim.cpp"
containing the
following components which has print
and plot
methods.
parents 
a list containing two components named " 
offspring 
a list containing two components named " 
Tanaka, U., Ogata, Y. and Katsura, K. (2008) Simulation and estimation of the NeymanScott type spatial cluster models. Computer Science Monographs 34, 144. The Institute of Statistical Mathematics, Tokyo. https://www.ism.ac.jp/editsec/csm/.
## Thomas Model
pars < c(mu = 50.0, nu = 30.0, sigma = 0.03)
t.sim < sim.cppm("Thomas", pars, seed = 117)
t.sim
plot(t.sim)
## InversePower Type Model
pars < c(mu = 50.0, nu = 30.0, p = 1.5, c = 0.005)
ip.sim < sim.cppm("IP", pars, seed = 353)
ip.sim
plot(ip.sim)
## Type A Model
pars < c(mu = 50.0, nu = 30.0, a = 0.3, sigma1 = 0.005, sigma2 = 0.1)
a.sim < sim.cppm("TypeA", pars, seed = 575)
a.sim
plot(a.sim)
## Type B Model
pars < c(mu1 = 10.0, mu2 = 40.0, nu = 30.0, sigma1 = 0.01, sigma2 = 0.03)
b.sim < sim.cppm("TypeB", pars, seed = 257)
b.sim
plot(b.sim, parents.distinct = TRUE)
## Type C Model
pars < c(mu1 = 5.0, mu2 = 9.0, nu1 = 30.0, nu2 = 150.0,
sigma1 = 0.01, sigma2 = 0.05)
c.sim < sim.cppm("TypeC", pars, seed = 555)
c.sim
plot(c.sim, parents.distinct = FALSE)
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