Fits a log-Gaussian Cox point process model to a point pattern dataset by the Method of Minimum Contrast using the pair correlation function.

```
lgcp.estpcf(X,
startpar=c(var=1,scale=1),
covmodel=list(model="exponential"),
lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
```

X

Data to which the model will be fitted. Either a point pattern or a summary statistic. See Details.

startpar

Vector of starting values for the parameters of the log-Gaussian Cox process model.

covmodel

Specification of the covariance model for the log-Gaussian field. See Details.

lambda

Optional. An estimate of the intensity of the point process.

q,p

Optional. Exponents for the contrast criterion.

rmin, rmax

Optional. The interval of \(r\) values for the contrast criterion.

…

Optional arguments passed to `optim`

to control the optimisation algorithm. See Details.

pcfargs

Optional list containing arguments passed to `pcf.ppp`

to control the smoothing in the estimation of the
pair correlation function.

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 a log-Gaussian Cox point process (LGCP) model to a point pattern dataset by the Method of Minimum Contrast, using the estimated pair correlation function of the point pattern.

The shape of the covariance of the LGCP must be specified: the default is the exponential covariance function, but other covariance models can be selected.

The argument `X`

can be either

- a point pattern:
An object of class

`"ppp"`

representing a point pattern dataset. The pair correlation function of the point pattern will be computed using`pcf`

, and the method of minimum contrast will be applied to this.- a summary statistic:
An object of class

`"fv"`

containing the values of a summary statistic, computed for a point pattern dataset. The summary statistic should be the pair correlation function, and this object should have been obtained by a call to`pcf`

or one of its relatives.

The algorithm fits a log-Gaussian Cox point process (LGCP)
model to `X`

, by finding the parameters of the LGCP model
which give the closest match between the
theoretical pair correlation function of the LGCP model
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see `mincontrast`

.

The model fitted is a stationary, isotropic log-Gaussian Cox process (Moller and Waagepetersen, 2003, pp. 72-76). To define this process we start with a stationary Gaussian random field \(Z\) in the two-dimensional plane, with constant mean \(\mu\) and covariance function \(C(r)\). Given \(Z\), we generate a Poisson point process \(Y\) with intensity function \(\lambda(u) = \exp(Z(u))\) at location \(u\). Then \(Y\) is a log-Gaussian Cox process.

The theoretical pair correlation function of the LGCP is $$ g(r) = \exp(C(s)) $$ The intensity of the LGCP is $$ \lambda = \exp(\mu + \frac{C(0)}{2}). $$

The covariance function \(C(r)\) takes the form $$ C(r) = \sigma^2 c(r/\alpha) $$ where \(\sigma^2\) and \(\alpha\) are parameters controlling the strength and the scale of autocorrelation, respectively, and \(c(r)\) is a known covariance function determining the shape of the covariance. The strength and scale parameters \(\sigma^2\) and \(\alpha\) will be estimated by the algorithm. The template covariance function \(c(r)\) must be specified as explained below.

In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters \(\sigma^2\) and \(\alpha\). Then the remaining parameter \(\mu\) is inferred from the estimated intensity \(\lambda\).

The template covariance function \(c(r)\) is specified
using the argument `covmodel`

. This should be of the form
`list(model="modelname", …)`

where
`modelname`

is a string identifying the template model
as explained below, and `…`

are optional arguments of the
form `tag=value`

giving the values of parameters controlling the
*shape* of the template model.
The default is the exponential covariance
\(c(r) = e^{-r}\)
so that the scaled covariance is
$$
C(r) = \sigma^2 e^{-r/\alpha}.
$$
To determine the template model, the string `"modelname"`

will be
prefixed by `"RM"`

and the code will search for
a function of this name in the RandomFields package.
For a list of available models see
`RMmodel`

in the
RandomFields package. For example the
Matern covariance with exponent \(\nu=0.3\) is specified
by `covmodel=list(model="matern", nu=0.3)`

corresponding
to the function `RMmatern`

in the RandomFields package.

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 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.

Moller, J.,
Syversveen, A. and Waagepetersen, R. (1998)
Log Gaussian Cox Processes.
*Scandinavian Journal of Statistics* **25**, 451--482.

Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.

Waagepetersen, R. (2007)
An estimating function approach to inference for
inhomogeneous Neyman-Scott processes.
*Biometrics* **63**, 252--258.

`lgcp.estK`

for alternative method of fitting LGCP.

`matclust.estpcf`

,
`thomas.estpcf`

for other models.

`mincontrast`

for the generic minimum contrast
fitting algorithm, including important parameters that affect
the accuracy of the fit.

`RMmodel`

in the
RandomFields package, for covariance function models.

`pcf`

for the pair correlation function.

# NOT RUN { data(redwood) u <- lgcp.estpcf(redwood, c(var=1, scale=0.1)) u plot(u) if(require(RandomFields)) { lgcp.estpcf(redwood, covmodel=list(model="matern", nu=0.3)) } # }