Fits the Thomas point process to a point pattern dataset by the Method of Minimum Contrast using the K function.
thomas.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)Data to which the Thomas model will be fitted. Either a point pattern or a summary statistic. See Details.
Vector of starting values for the parameters of the Thomas process.
Optional. An estimate of the intensity of the point process.
Optional. Exponents for the contrast criterion.
Optional. The interval of \(r\) values for the contrast criterion.
Optional arguments passed to optim
to control the optimisation algorithm. See Details.
An object of class "minconfit". There are methods for printing
and plotting this object. It contains the following main components:
Vector of fitted parameter values.
Function value table (object of class "fv")
containing the observed values of the summary statistic
(observed) and the theoretical values of the summary
statistic computed from the fitted model parameters.
This algorithm fits the Thomas 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 Thomas point process to X,
by finding the parameters of the Thomas model
which give the closest match between the
theoretical \(K\) function of the Thomas process
and the observed \(K\) 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 \(K\)-function of the Thomas process is $$ K(r) = \pi r^2 + \frac 1 \kappa (1 - \exp(-\frac{r^2}{4\sigma^2})). $$ 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.
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.
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.
kppm,
lgcp.estK,
matclust.estK,
mincontrast,
Kest,
rThomas to simulate the fitted model.
# NOT RUN {
data(redwood)
u <- thomas.estK(redwood, c(kappa=10, scale=0.1))
u
plot(u)
# }
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