Fit a homogeneous or inhomogeneous cluster process or Cox point process model to a point pattern.
kppm(X, …) # S3 method for formula
kppm(X,
clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"),
…,
data=NULL)
# S3 method for ppp
kppm(X,
trend = ~1,
clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"),
data = NULL,
...,
covariates=data,
subset,
method = c("mincon", "clik2", "palm"),
improve.type = c("none", "clik1", "wclik1", "quasi"),
improve.args = list(),
weightfun=NULL,
control=list(),
algorithm="Nelder-Mead",
statistic="K",
statargs=list(),
rmax = NULL,
covfunargs=NULL,
use.gam=FALSE,
nd=NULL, eps=NULL)
# S3 method for quad
kppm(X,
trend = ~1,
clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"),
data = NULL,
...,
covariates=data,
subset,
method = c("mincon", "clik2", "palm"),
improve.type = c("none", "clik1", "wclik1", "quasi"),
improve.args = list(),
weightfun=NULL,
control=list(),
algorithm="Nelder-Mead",
statistic="K",
statargs=list(),
rmax = NULL,
covfunargs=NULL,
use.gam=FALSE,
nd=NULL, eps=NULL)
A point pattern dataset (object of class "ppp" or
"quad") to which the model should be fitted, or a
formula in the R language defining the model. See Details.
An R formula, with no left hand side, specifying the form of the log intensity.
Character string determining the cluster model.
Partially matched.
Options are "Thomas", "MatClust",
"Cauchy", "VarGamma" and "LGCP".
The values of spatial covariates (other than the Cartesian coordinates) required by the model. A named list of pixel images, functions, windows, tessellations or numeric constants.
Additional arguments. See Details.
Optional.
A subset of the spatial domain,
to which the model-fitting should be restricted.
A window (object of class "owin")
or a logical-valued pixel image (object of class "im"),
or an expression (possibly involving the names of entries in data)
which can be evaluated to yield a window or pixel image.
The fitting method. Either
"mincon" for minimum contrast,
"clik2" for second order composite likelihood,
or "palm" for Palm likelihood.
Partially matched.
Method for updating the initial estimate of the trend.
Initially the trend is estimated as if the process
is an inhomogeneous Poisson process.
The default, improve.type = "none", is to use this initial estimate.
Otherwise, the trend estimate is
updated by improve.kppm, using information
about the pair correlation function.
Options are "clik1"
(first order composite likelihood, essentially equivalent to "none"),
"wclik1" (weighted first order composite likelihood) and
"quasi" (quasi likelihood).
Additional arguments passed to improve.kppm when
improve.type != "none". See Details.
Optional weighting function \(w\)
in the composite likelihood or Palm likelihood.
A function in the R language.
See Details.
List of control parameters passed to the optimization function
optim.
Name of the summary statistic to be used
for minimum contrast estimation: either "K" or "pcf".
Optional list of arguments to be used when calculating
the statistic. See Details.
Maximum value of interpoint distance to use in the composite likelihood.
Arguments passed to ppm when fitting the intensity.
An object of class "kppm" representing the fitted model.
There are methods for printing, plotting, predicting, simulating
and updating objects of this class.
To fit a log-Gaussian Cox model with non-exponential covariance,
specify clusters="LGCP" and use additional arguments
to specify the covariance structure. These additional arguments can
be given individually in the call to kppm, or they can be
collected together in a list called covmodel.
For example a Matern model with parameter \(\nu=0.5\) could be specified
either by kppm(X, clusters="LGCP", model="matern", nu=0.5) or by
kppm(X, clusters="LGCP", covmodel=list(model="matern", nu=0.5)).
The argument model specifies the type of covariance
model: the default is model="exp" for an exponential covariance.
Alternatives include "matern", "cauchy" and "spheric".
Model names correspond to functions beginning with RM in the
RandomFields package: for example model="matern"
corresponds to the function RMmatern in the
RandomFields package.
Additional arguments are passed to the
relevant function in the RandomFields package:
for example if model="matern" then the additional argument
nu is required, and is passed to the function
RMmatern in the RandomFields package.
Note that it is not possible to use anisotropic covariance models
because the kppm technique assumes the pair correlation function
is isotropic.
See ppm.ppp for a list of common error messages
and warnings originating from the first stage of model-fitting.
This function fits a clustered point process model to the
point pattern dataset X.
The model may be either a Neyman-Scott cluster process
or another Cox process.
The type of model is determined by the argument clusters.
Currently the options
are clusters="Thomas" for the Thomas process,
clusters="MatClust" for the Matern cluster process,
clusters="Cauchy" for the Neyman-Scott cluster process
with Cauchy kernel,
clusters="VarGamma" for the Neyman-Scott cluster process
with Variance Gamma kernel (requires an additional argument nu
to be passed through the dots; see rVarGamma for details),
and clusters="LGCP" for the log-Gaussian Cox process (may
require additional arguments passed through …; see
rLGCP for details on argument names).
The first four models are Neyman-Scott cluster processes.
The algorithm first estimates the intensity function
of the point process using ppm.
The argument X may be a point pattern
(object of class "ppp") or a quadrature scheme
(object of class "quad"). The intensity is specified by
the trend argument.
If the trend formula is ~1 (the default)
then the model is homogeneous. The algorithm begins by
estimating the intensity as the number of points divided by
the area of the window.
Otherwise, the model is inhomogeneous.
The algorithm begins by fitting a Poisson process with log intensity
of the form specified by the formula trend.
(See ppm for further explanation).
The argument X may also be a formula in the
R language. The right hand side of the formula gives the
trend as described above. The left hand side of the formula
gives the point pattern dataset to which the model should be fitted.
If improve.type="none" this is the final estimate of the
intensity. Otherwise, the intensity estimate is updated, as explained in
improve.kppm. Additional arguments to
improve.kppm are passed as a named list in
improve.args.
The clustering parameters of the model are then fitted either by minimum contrast estimation, or by maximising a composite likelihood.
If method = "mincon" (the default) clustering parameters of
the model will be fitted
by minimum contrast estimation, that is, by matching the theoretical
\(K\)-function of the model to the empirical \(K\)-function
of the data, as explained in mincontrast.
For a homogeneous model ( trend = ~1 )
the empirical \(K\)-function of the data is computed
using Kest,
and the parameters of the cluster model are estimated by
the method of minimum contrast.
For an inhomogeneous model,
the inhomogeneous \(K\) function is estimated
by Kinhom using the fitted intensity.
Then the parameters of the cluster model
are estimated by the method of minimum contrast using the
inhomogeneous \(K\) function. This two-step estimation
procedure is due to Waagepetersen (2007).
If statistic="pcf" then instead of using the
\(K\)-function, the algorithm will use
the pair correlation function pcf for homogeneous
models and the inhomogeneous pair correlation function
pcfinhom for inhomogeneous models.
In this case, the smoothing parameters of the pair correlation
can be controlled using the argument statargs,
as shown in the Examples.
Additional arguments … will be passed to
clusterfit to control the minimum contrast fitting
algorithm.
If method = "clik2" the clustering parameters of the
model will be fitted by maximising the second-order composite likelihood
(Guan, 2006). The log composite likelihood is
$$
\sum_{i,j} w(d_{ij}) \log\rho(d_{ij}; \theta)
- \left( \sum_{i,j} w(d_{ij}) \right)
\log \int_D \int_D w(\|u-v\|) \rho(\|u-v\|; \theta)\, du\, dv
$$
where the sums are taken over all pairs of data points
\(x_i, x_j\) separated by a distance
\(d_{ij} = \| x_i - x_j\|\)
less than rmax,
and the double integral is taken over all pairs of locations
\(u,v\) in the spatial window of the data.
Here \(\rho(d;\theta)\) is the
pair correlation function of the model with
cluster parameters \(\theta\).
The function \(w\) in the composite likelihood
is a weighting function and may be chosen arbitrarily.
It is specified by the argument weightfun.
If this is missing or NULL then the default is
a threshold weight function,
\(w(d) = 1(d \le R)\), where \(R\) is rmax/2.
If method = "palm" the clustering parameters of the
model will be fitted by maximising the Palm loglikelihood
(Tanaka et al, 2008)
$$
\sum_{i,j} w(x_i, x_j) \log \lambda_P(x_j \mid x_i; \theta)
- \int_D w(x_i, u) \lambda_P(u \mid x_i; \theta) {\rm d} u
$$
with the same notation as above. Here
\(\lambda_P(u|v;\theta\) is the Palm intensity of
the model at location \(u\) given there is a point at \(v\).
In all three methods, the optimisation is performed by the generic
optimisation algorithm optim.
The behaviour of this algorithm can be modified using the
arguments control and algorithm.
Useful control arguments include
trace, maxit and abstol
(documented in the help for optim).
Fitting the LGCP model requires the RandomFields package, except in the default case where the exponential covariance is assumed.
Guan, Y. (2006) A composite likelihood approach in fitting spatial point process models. Journal of the American Statistical Association 101, 1502--1512.
Jalilian, A., Guan, Y. and Waagepetersen, R. (2012) Decomposition of variance for spatial Cox processes. Scandinavian Journal of Statistics 40, 119--137.
Tanaka, U. and Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for Neyman-Scott point processes. Biometrical Journal 50, 43--57.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252--258.
Methods for kppm objects:
plot.kppm,
fitted.kppm,
predict.kppm,
simulate.kppm,
update.kppm,
vcov.kppm,
methods.kppm,
as.ppm.kppm,
as.fv.kppm,
Kmodel.kppm,
pcfmodel.kppm.
Minimum contrast fitting algorithm:
higher level interface clusterfit;
low-level algorithm mincontrast.
Alternative fitting algorithms:
thomas.estK,
matclust.estK,
lgcp.estK,
cauchy.estK,
vargamma.estK,
thomas.estpcf,
matclust.estpcf,
lgcp.estpcf,
cauchy.estpcf,
vargamma.estpcf,
Summary statistics:
Kest,
Kinhom,
pcf,
pcfinhom.
See also ppm
# NOT RUN {
# method for point patterns
kppm(redwood, ~1, "Thomas")
# method for formulas
kppm(redwood ~ 1, "Thomas")
# different models for clustering
kppm(redwood ~ x, "MatClust")
kppm(redwood ~ x, "MatClust", statistic="pcf", statargs=list(stoyan=0.2))
kppm(redwood ~ x, cluster="Cauchy", statistic="K")
kppm(redwood, cluster="VarGamma", nu = 0.5, statistic="pcf")
# log-Gaussian Cox process (LGCP) models
kppm(redwood ~ 1, "LGCP", statistic="pcf")
if(require("RandomFields")) {
# Random Fields package is needed for non-default choice of covariance model
kppm(redwood ~ x, "LGCP", statistic="pcf",
model="matern", nu=0.3,
control=list(maxit=10))
}
# Different fitting techniques
kppm(redwood ~ 1, "Thomas", method="c")
kppm(redwood ~ 1, "Thomas", method="p")
# composite likelihood method
kppm(redwood ~ x, "VarGamma", method="clik2", nu.ker=-3/8)
# quasi-likelihood method
kppm(redwood ~ x, "Thomas", improve.type = "quasi")
# }
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