# lgcp.estpcf

##### Fit a Log-Gaussian Cox Point Process by Minimum Contrast

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

##### Usage

```
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())
```

##### Arguments

- 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.

##### Details

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

The argument `X`

can be either
[object Object],[object Object]
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
(

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 `RMmodel`

in the
`covmodel=list(model="matern", nu=0.3)`

corresponding
to the function `RMmatern`

in the `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.
}
`"minconfit"`

. There are methods for printing
and plotting this object. It contains the following main components:
`"fv"`

)
containing the observed values of the summary statistic
(`observed`

) and the theoretical values of the summary
statistic computed from the fitted model parameters.
}*Scandinavian Journal of Statistics* **25**, 451--482.

Waagepetersen, R. (2007)
An estimating function approach to inference for
inhomogeneous Neyman-Scott processes.
*Biometrics* **63**, 252--258.
}
[object Object],[object Object],[object Object]
`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
`pcf`

for the pair correlation function.

*Documentation reproduced from package spatstat, version 1.41-1, License: GPL (>= 2)*