# lgcp.estK

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

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

##### Usage

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

##### 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.

##### Details

This algorithm fits a log-Gaussian Cox point process (LGCP) model to a point pattern dataset by the Method of Minimum Contrast, using the K 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 \(K\) function of the point pattern will be computed using`Kest`

, 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 \(K\) function, and this object should have been obtained by a call to`Kest`

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 \(K\) function of the LGCP model
and the observed \(K\) 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 \(K\)-function of the LGCP is $$ K(r) = \int_0^r 2\pi s \exp(C(s)) \, {\rm d}s. $$ The intensity of the LGCP is $$ \lambda = \exp(\mu + \frac{C(0)}{2}). $$

The covariance function \(C(r)\) is parametrised in 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 as the values
`var`

and `scale`

respectively.
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.

##### Value

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.

##### Note

This function is considerably slower than `lgcp.estpcf`

because of the computation time required for the integral
in the \(K\)-function.

Computation can be accelerated, at the cost of less accurate results,
by setting `spatstat.options(fastK.lgcp=TRUE)`

.

##### References

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.

##### See Also

`lgcp.estpcf`

for alternative method of fitting LGCP.

`matclust.estK`

,
`thomas.estK`

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.

`Kest`

for the \(K\) function.

##### Examples

```
# NOT RUN {
if(interactive()) {
u <- lgcp.estK(redwood)
} else {
# slightly faster - better starting point
u <- lgcp.estK(redwood, c(var=1, scale=0.1))
}
u
plot(u)
# }
# NOT RUN {
if(FALSE) {
## takes several minutes!
lgcp.estK(redwood, covmodel=list(model="matern", nu=0.3))
}
# }
```

*Documentation reproduced from package spatstat, version 1.56-1, License: GPL (>= 2)*