rLGCP {spatstat.random}

Simulate Log-Gaussian Cox Process

Description

Generate a random point pattern, a realisation of the log-Gaussian Cox process.

Usage

 rLGCP(model=c("exponential", "gauss", "stable", "gencauchy", "matern"),
       mu = 0, param = NULL,
       ...,
       win=NULL, saveLambda=TRUE, nsim=1, drop=TRUE)

Arguments

model

character string (partially matched) giving the name of a covariance model for the Gaussian random field.

mu

mean function of the Gaussian random field. Either a single number, a function(x,y, ...) or a pixel image (object of class "im").

param

List of parameters for the covariance. Standard arguments are var and scale.

...

Additional parameters for the covariance, or arguments passed to as.mask to determine the pixel resolution.

win

Window in which to simulate the pattern. An object of class "owin".

saveLambda

Logical. If TRUE (the default) then the simulated random intensity will also be saved, and returns as an attribute of the point pattern.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details

This function generates a realisation of a log-Gaussian Cox process (LGCP). This is a Cox point process in which the logarithm of the random intensity is a Gaussian random field with mean function \mu and covariance function c(r). Conditional on the random intensity, the point process is a Poisson process with this intensity.

The string model specifies the covariance function of the Gaussian random field, and the parameters of the covariance are determined by param and ....

All models recognise the parameters var for the variance at distance zero, and scale for the scale parameter. Some models require additional parameters which are listed below.

The available models are as follows:

model="exponential":

the exponential covariance function

C(r) = \sigma^2 \exp(-r/h)

where \sigma^2 is the variance parameter var, and h is the scale parameter scale.

model="gauss":

the Gaussian covariance function

C(r) = \sigma^2 \exp(-(r/h)^2)

where \sigma^2 is the variance parameter var, and h is the scale parameter scale.

model="stable":

the stable covariance function

C(r) = \sigma^2 \exp(-(r/h)^\alpha)

where \sigma^2 is the variance parameter var, h is the scale parameter scale, and \alpha is the shape parameter alpha. The parameter alpha must be given, either as a stand-alone argument, or as an entry in the list param.

model="gencauchy":

the generalised Cauchy covariance function

C(r) = \sigma^2 (1 + (x/h)^\alpha)^{-\beta/\alpha}

where \sigma^2 is the variance parameter var, h is the scale parameter scale, and \alpha and \beta are the shape parameters alpha and beta. The parameters alpha and beta must be given, either as stand-alone arguments, or as entries in the list param.

model="matern":

the Whittle-Matern covariance function

C(r) = \sigma^2 \frac{1}{2^{\nu-1} \Gamma(\nu)} (\sqrt{2 \nu} \, r/h)^\nu K_\nu(\sqrt{2\nu}\, r/h)

where \sigma^2 is the variance parameter var, h is the scale parameter scale, and \nu is the shape parameter nu. The parameter nu must be given, either as a stand-alone argument, or as an entry in the list param.

The algorithm uses the circulant embedding technique to generate values of a Gaussian random field, with the specified mean function mu and the covariance specified by the arguments model and param, on the points of a regular grid. The exponential of this random field is taken as the intensity of a Poisson point process, and a realisation of the Poisson process is then generated by the function rpoispp in the spatstat.random package.

If the simulation window win is missing or NULL, then it defaults to Window(mu) if mu is a pixel image, and it defaults to the unit square otherwise.

The LGCP model can be fitted to data using kppm.

Value

A point pattern (object of class "ppp") or a list of point patterns.

Additionally, the simulated intensity function for each point pattern is returned as an attribute "Lambda" of the point pattern, if saveLambda=TRUE.

Warning: new implementation

The simulation algorithm for rLGCP has been completely re-written in spatstat.random version 3.2-0 to avoid depending on the package RandomFields which is now defunct (and is sadly missed).

It is no longer possible to replicate results that were obtained using rLGCP in previous versions of spatstat.random.

The current code is a new implementation and should be considered vulnerable to new bugs.

Author(s)

Abdollah Jalilian and Rasmus Waagepetersen. Modified by Adrian Baddeley [e-mail], Rolf Turner [e-mail] and Ege Rubak [e-mail].

References

Moller, J., Syversveen, A. and Waagepetersen, R. (1998) Log Gaussian Cox Processes. Scandinavian Journal of Statistics 25, 451–482.

See Also

rpoispp, rMatClust, rGaussPoisson, rNeymanScott.

For fitting the model, see kppm, lgcp.estK.

Examples

    online <- interactive()

    # homogeneous LGCP with exponential covariance function
    X <- rLGCP("exp", 3, var=0.2, scale=.1)

    # inhomogeneous LGCP with Gaussian covariance function
    m <- as.im(function(x, y){5 - 1.5 * (x - 0.5)^2 + 2 * (y - 0.5)^2}, W=owin())
    X <- rLGCP("gauss", m, var=0.15, scale =0.1)

    if(online) {
     plot(attr(X, "Lambda"))
     points(X)
    }

    # inhomogeneous LGCP with Matern covariance function
    X <- rLGCP("matern", function(x, y){ 1 - 0.4 * x},
               var=2, scale=0.7, nu=0.5,
               win = owin(c(0, 10), c(0, 10)))
    if(online) plot(X)