thomas.estpcf(X, startpar=c(kappa=1,sigma2=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:"fv")
containing the observed values of the summary statistic
(observed) and the theoretical values of the summary
statistic computed from the fitted model parameters.pcf. The argument X can be either
[object Object],[object Object]
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 $\sigma$.
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.
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, sigma2=0.1))
u
plot(u, legendpos="topright")
u2 <- thomas.estpcf(redwood, c(kappa=10, sigma2=0.1),
pcfargs=list(stoyan=0.12))Run the code above in your browser using DataLab