thomas.estpcf(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())optim
to control the optimisation algorithm. See Details.
pcf.ppp
to control the smoothing in the estimation of the
pair correlation function.
"minconfit". There are methods for printing
and plotting this object. It contains the following main components:pcf. The argument X can be either
The algorithm fits the Thomas point process to X,
by finding the parameters of the Thomas model
which give the closest match between the
theoretical pair correlation function of the Thomas process
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast.
The Thomas point process is described in
\Moller and Waagepetersen (2003, pp. 61--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 isotropically
Normally distributed around the parent point with standard deviation
$sigma$ which is equal to the parameter scale. The
named vector of stating values can use either sigma2
($sigma^2$) or scale as the name of the second
component, but the latter is recommended for consistency with other
cluster models.
The theoretical pair correlation function of the Thomas process is $$ g(r) = 1 + \frac 1 {4\pi \kappa \sigma^2} \exp(-\frac{r^2}{4\sigma^2})). $$ The theoretical intensity of the Thomas 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 $sigma^2$. 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 Thomas process can be simulated, using rThomas.
Homogeneous or inhomogeneous Thomas process 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.
\Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
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.
thomas.estK
mincontrast,
pcf,
rThomas to simulate the fitted model.
data(redwood)
u <- thomas.estpcf(redwood, c(kappa=10, scale=0.1))
u
plot(u, legendpos="topright")
u2 <- thomas.estpcf(redwood, c(kappa=10, scale=0.1),
pcfargs=list(stoyan=0.12))
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