# kppm

##### Fit Cluster or Cox Point Process Model

Fit a homogeneous or inhomogeneous cluster process or Cox point process model to a point pattern.

##### Usage

`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)

##### Arguments

- 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
Character string determining the mathematical optimisation algorithm to be used by

`optim`

. This argument is passed to`optim`

as the argument`method`

.- 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.

##### Details

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.

- 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`clusterfit`

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
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.

##### Value

An object of class `"kppm"`

representing the fitted model.
There are methods for printing, plotting, predicting, simulating
and updating objects of this class.

##### Log-Gaussian Cox Models

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.

##### Error and warning messages

See `ppm.ppp`

for a list of common error messages
and warnings originating from the first stage of model-fitting.

##### References

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.

##### See Also

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`

##### Examples

```
# 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")
# }
```

*Documentation reproduced from package spatstat, version 1.62-2, License: GPL (>= 2)*