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)

X

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.

trend

An R formula, with no left hand side, specifying the form of the log intensity.

clusters

Character string determining the cluster model.
Partially matched.
Options are `"Thomas"`

, `"MatClust"`

,
`"Cauchy"`

, `"VarGamma"`

and `"LGCP"`

.

data,covariates

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.

subset

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.

method

The fitting method. Either
`"mincon"`

for minimum contrast,
`"clik2"`

for second order composite likelihood,
or `"palm"`

for Palm likelihood.
Partially matched.

improve.type

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).

improve.args

Additional arguments passed to `improve.kppm`

when
`improve.type != "none"`

. See Details.

weightfun

Optional weighting function \(w\)
in the composite likelihood or Palm likelihood.
A `function`

in the R language.
See Details.

control

List of control parameters passed to the optimization function
`optim`

.

algorithm

statistic

Name of the summary statistic to be used
for minimum contrast estimation: either `"K"`

or `"pcf"`

.

statargs

Optional list of arguments to be used when calculating
the `statistic`

. See Details.

rmax

Maximum value of interpoint distance to use in the composite likelihood.

covfunargs,use.gam,nd,eps

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 maximum composite likelihood.

- Minimum contrast:
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`mincontrast`

to control the minimum contrast fitting algorithm.- Composite likelihood:
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`

.- Palm likelihood:
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
argument `control`

.
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`

,
`Kmodel.kppm`

,
`pcfmodel.kppm`

.

Minimum contrast fitting 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")
kppm(redwood ~ 1, "Thomas", method="c")
kppm(redwood ~ 1, "Thomas", method="p")
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")
# LGCP models
kppm(redwood ~ 1, "LGCP", statistic="pcf")
if(require("RandomFields")) {
kppm(redwood ~ x, "LGCP", statistic="pcf",
model="matern", nu=0.3,
control=list(maxit=10))
}
# fit with composite likelihood method
kppm(redwood ~ x, "VarGamma", method="clik2", nu.ker=-3/8)
# fit intensity with quasi-likelihood method
kppm(redwood ~ x, "Thomas", improve.type = "quasi")
# }
```

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