# lgcp.estK

##### 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.

##### Usage

```
lgcp.estK(X, startpar=c(sigma2=1,alpha=1), 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.
- 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 model to a point pattern dataset by the Method of Minimum Contrast, using the K 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 $K$ 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 with exponential covariance (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 $$c(r) = \sigma^2 e^{-r/\alpha}$$ where $\sigma^2$ and $\alpha$ are parameters. 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 $K$-function of the LGCP is $$K(r) = \int_0^r 2\pi s \exp(\sigma^2 \exp(-s/\alpha)) \, {\rm d}s.$$ The theoretical intensity of the LGCP is $$\lambda = \exp(\mu + \frac{\sigma^2}{2}).$$ 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 $\lambda$.

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: par Vector of fitted parameter values. fit 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.

##### References

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

##### Examples

```
data(redwood)
u <- lgcp.estK(redwood, c(sigma2=0.1, alpha=1))
u
plot(u)
```

*Documentation reproduced from package spatstat, version 1.20-2, License: GPL (>= 2)*