Fit the Matern Cluster Point Process by Minimum Contrast
Fits the Matern Cluster point process to a point pattern dataset by the Method of Minimum Contrast.
matclust.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)X |
Data to which the Matern Cluster model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the Matern Cluster process. |
lambda |
Optional. An estimate of the intensity of the point process. |
q,p |
Optional. Exponents for the contrast criterion. |
rmin, rmax |
Optional. The interval of r values for the contrast criterion. |
... |
Optional arguments passed to |
This algorithm fits the Matern Cluster point process model to a point pattern dataset by the Method of Minimum Contrast, using the K function.
The argument X can be either
An object of class "ppp"
representing a point pattern dataset.
The K function of the point pattern will be computed
using Kest, and the method of minimum contrast
will be applied to this.
An object of class "fv" containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the K function,
and this object should have been obtained by a call to
Kest or one of its relatives.
The algorithm fits the Matern Cluster point process to X,
by finding the parameters of the Matern Cluster model
which give the closest match between the
theoretical K function of the Matern Cluster process
and the observed K function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast.
The Matern Cluster point process is described in Moller and Waagepetersen
(2003, p. 62). It is a cluster process formed by taking a
pattern of parent points, generated according to a Poisson process
with intensity kappa, and around each parent point,
generating a random number of offspring points, such that the
number of offspring of each parent is a Poisson random variable with mean
mu, and the locations of the offspring points of one parent
are independent and uniformly distributed inside a circle of radius
R centred on the parent point, where R is equal to
the parameter scale. The named vector of stating values can use
either R or scale as the name of the second component,
but the latter is recommended for consistency with other cluster models.
The theoretical K-function of the Matern Cluster process is
K(r) = pi r^2 + h(r/(2*R))/kappa
where the radius R is the parameter scale and
h(z) = 2 + (1/pi) * ((8 * z^2 - 4) * arccos(z) - 2 * arcsin(z) + 4 * z * sqrt((1 - z^2)^3) - 6 * z * sqrt(1 - z^2))
for z <= 1, and h(z) = 1 for z > 1. The theoretical intensity of the Matern Cluster process is lambda=kappa* mu.
In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters kappa and R. Then the remaining parameter mu is inferred from the estimated intensity lambda.
If the argument lambda is provided, then this is used
as the value of lambda. Otherwise, if X is a
point pattern, then lambda
will be estimated from X.
If X is a summary statistic and lambda is missing,
then the intensity lambda cannot be estimated, and
the parameter mu will be returned as NA.
The remaining arguments rmin,rmax,q,p control the
method of minimum contrast; see mincontrast.
The Matern Cluster process can be simulated, using
rMatClust.
Homogeneous or inhomogeneous Matern Cluster models can also be
fitted using the function kppm.
The optimisation algorithm can be controlled through the
additional arguments "..." which are passed to the
optimisation function optim. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B" to select an optimisation
algorithm that respects box constraints, and use the arguments
lower and upper to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit". There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
Rasmus Waagepetersen rw@math.auc.dk Adapted for spatstat by Adrian Baddeley Adrian.Baddeley@curtin.edu.au
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
kppm,
lgcp.estK,
thomas.estK,
mincontrast,
Kest,
rMatClust to simulate the fitted model.
data(redwood)
u <- matclust.estK(redwood, c(kappa=10, scale=0.1))
u
plot(u)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.