lgcp.estpcf

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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.

Keywords
models, spatial
Usage
lgcp.estpcf(X, startpar=c(sigma2=1,alpha=1), 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.
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 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 pair correlation function of the LGCP is $$g(r) = \exp(\sigma^2 \exp(-r/\alpha))$$ 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:
• parVector of fitted parameter values.
• fitFunction 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.

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.

lgcp.estK, thomas.estpcf, matclust.estpcf, mincontrast, pcf

• lgcp.estpcf
Examples
data(redwood)
u <- lgcp.estpcf(redwood, c(sigma2=0.1, alpha=1))
u
plot(u)
Documentation reproduced from package spatstat, version 1.21-0, License: GPL (>= 2)

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