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=list(sigma2=1,alpha=1), lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL)

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.

References

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

Waagepetersen, R. (2006). An estimation function approach to inference for inhomogeneous Neyman-Scott processes. Submitted.

See Also

lgcp.estK, matclust.estK, mincontrast, Kest

Aliases

• lgcp.estK

Examples

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