Simulate Thomas Process
Generate a random point pattern, a realisation of the Thomas cluster process.
rThomas(kappa, scale, mu, win = owin(c(0,1),c(0,1)),
nsim=1, drop=TRUE,
saveLambda=FALSE, expand = 4*scale, ...,
poisthresh=1e-6, saveparents=TRUE)kappa |
Intensity of the Poisson process of cluster centres. A single positive number, a function, or a pixel image. |
scale |
Standard deviation of random displacement (along each coordinate axis) of a point from its cluster centre. |
mu |
Mean number of points per cluster (a single positive number) or reference intensity for the cluster points (a function or a pixel image). |
win |
Window in which to simulate the pattern.
An object of class |
nsim |
Number of simulated realisations to be generated. |
drop |
Logical. If |
saveLambda |
Logical. If |
expand |
Numeric. Size of window expansion for generation of parent points. Has a sensible default. |
... |
Passed to |
poisthresh |
Numerical threshold below which the model will be treated as a Poisson process. See Details. |
saveparents |
Logical value indicating whether to save the locations of the parent points as an attribute. |
This algorithm generates a realisation of the (‘modified’)
Thomas process, a special case of the Neyman-Scott process,
inside the window win.
In the simplest case, where kappa and mu
are single numbers, the algorithm
generates a uniform Poisson point process of “parent” points
with intensity kappa. Then each parent point is
replaced by a random cluster of “offspring” points,
the number of points per cluster being Poisson (mu)
distributed, and their
positions being isotropic Gaussian displacements from the
cluster parent location. The resulting point pattern
is a realisation of the classical
“stationary Thomas process” generated inside the window win.
This point process has intensity kappa * mu.
The algorithm can also generate spatially inhomogeneous versions of the Thomas process:
The parent points can be spatially inhomogeneous.
If the argument kappa is a function(x,y)
or a pixel image (object of class "im"), then it is taken
as specifying the intensity function of an inhomogeneous Poisson
process that generates the parent points.
The offspring points can be inhomogeneous. If the
argument mu is a function(x,y)
or a pixel image (object of class "im"), then it is
interpreted as the reference density for offspring points,
in the sense of Waagepetersen (2007).
For a given parent point, the offspring constitute a Poisson process
with intensity function equal to mu * f,
where f is the Gaussian probability density
centred at the parent point. Equivalently we first generate,
for each parent point, a Poisson (mumax) random number of
offspring (where M is the maximum value of mu)
with independent Gaussian displacements from the parent
location, and then randomly thin the offspring points, with
retention probability mu/M.
Both the parent points and the offspring points can be spatially inhomogeneous, as described above.
Note that if kappa is a pixel image, its domain must be larger
than the window win. This is because an offspring point inside
win could have its parent point lying outside win.
In order to allow this, the simulation algorithm
first expands the original window win
by a distance expand and generates the Poisson process of
parent points on this larger window. If kappa is a pixel image,
its domain must contain this larger window.
The intensity of the Thomas process is kappa * mu
if either kappa or mu is a single number. In the general
case the intensity is an integral involving kappa, mu
and f.
The Thomas process with homogeneous parents
(i.e. where kappa is a single number)
can be fitted to data using kppm.
Currently it is not possible to fit the Thomas model
with inhomogeneous parents.
If the pair correlation function of the model is very close
to that of a Poisson process, deviating by less than
poisthresh, then the model is approximately a Poisson process,
and will be simulated as a Poisson process with intensity
kappa * mu, using rpoispp.
This avoids computations that would otherwise require huge amounts
of memory.
A point pattern (an object of class "ppp") if nsim=1,
or a list of point patterns if nsim > 1.
Additionally, some intermediate results of the simulation are returned
as attributes of this point pattern (see
rNeymanScott). Furthermore, the simulated intensity
function is returned as an attribute "Lambda", if
saveLambda=TRUE.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
and Rolf Turner r.turner@auckland.ac.nz
Diggle, P. J., Besag, J. and Gleaves, J. T. (1976) Statistical analysis of spatial point patterns by means of distance methods. Biometrics 32 659–667.
Thomas, M. (1949) A generalisation of Poisson's binomial limit for use in ecology. Biometrika 36, 18–25.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
#homogeneous
X <- rThomas(10, 0.2, 5)
#inhomogeneous
Z <- as.im(function(x,y){ 5 * exp(2 * x - 1) }, owin())
Y <- rThomas(10, 0.2, Z)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.