# lgcpPredictSpatial: lgcpPredictSpatial function In lgcp: Log-Gaussian Cox Process

## Description

The function lgcpPredictSpatial performs spatial prediction for log-Gaussian Cox Processes

## Usage

 1 2 3 4 5 lgcpPredictSpatial(sd, model.parameters = lgcppars(), spatial.covmodel = "exponential", covpars = c(), cellwidth = NULL, gridsize = NULL, spatial.intensity, spatial.offset = NULL, mcmc.control, output.control = setoutput(), gradtrunc = Inf, ext = 2, inclusion = "touching") 

## Arguments

 sd a spatial point pattern object, see ?ppp model.parameters values for parameters, see ?lgcppars spatial.covmodel correlation type see ?CovarianceFct covpars vector of additional parameters for certain classes of covariance function (eg Matern), these must be supplied in the order given in ?CovarianceFct cellwidth width of grid cells on which to do MALA (grid cells are square) in same units as observation window. Note EITHER gridsize OR cellwidthe must be specified. gridsize size of output grid required. Note EITHER gridsize OR cellwidthe must be specified. spatial.intensity the fixed spatial component: an object of that can be coerced to one of class spatialAtRisk spatial.offset Numeric of length 1. Optional offset parameter, corresponding to the expected number of cases. NULL by default, in which case, this is estimateed from teh data. mcmc.control MCMC paramters, see ?mcmcpars output.control output choice, see ?setoutput gradtrunc truncation for gradient vector equal to H parameter Moller et al 1998 pp 473. Default is Inf, which means no gradient truncation. Set to NULL to estimate this automatically (though note that this may not necessarily be a good choice). The default seems to work in most settings. ext integer multiple by which grid should be extended, default is 2. Generally this will not need to be altered, but if the spatial correlation decays slowly, increasing 'ext' may be necessary. inclusion criterion for cells being included into observation window. Either 'touching' or 'centroid'. The former, the default, includes all cells that touch the observation window, the latter includes all cells whose centroids are inside the observation window.

## Details

The following is a mathematical description of a log-Gaussian Cox Process, it is best viewed in the pdf version of the manual.

Let \mathcal Y(s) be a spatial Gaussian process and W\subset R^2 be an observation window in space. Cases occur at spatial positions x \in W according to an inhomogeneous spatial Cox process, i.e. a Poisson process with a stochastic intensity R(x), The number of cases, X_{S}, arising in any S \subseteq W is then Poisson distributed conditional on R(\cdot),

X_{S} \sim \mbox{Poisson}≤ft{\int_S R(s)ds\right}.

Following Brix and Diggle (2001) and Diggle et al (2005) (but ignoring temporal variation), the intensity is decomposed multiplicatively as

R(s,t) = λ(s)Exp{\mathcal Y(s,t)}.

In the above, the fixed spatial component, λ:R^2\mapsto R_{≥q 0}, is a known function, proportional to the population at risk at each point in space and scaled so that

\int_Wλ(s)d s=1.

Before calling this function, the user must decide on the parameters, spatial covariance model, spatial discretisation, fixed spatial (λ(s)) component, mcmc parameters, and whether or not any output is required. Note there is no autorotate option for this function.

## Value

the results of fitting the model in an object of class lgcpPredict

## References

1. Benjamin M. Taylor, Tilman M. Davies, Barry S. Rowlingson, Peter J. Diggle (2013). Journal of Statistical Software, 52(4), 1-40. URL http://www.jstatsoft.org/v52/i04/

2. Brix A, Diggle PJ (2001). Spatiotemporal Prediction for log-Gaussian Cox processes. Journal of the Royal Statistical Society, Series B, 63(4), 823-841.

3. Diggle P, Rowlingson B, Su T (2005). Point Process Methodology for On-line Spatio-temporal Disease Surveillance. Environmetrics, 16(5), 423-434.

4. Wood ATA, Chan G (1994). Simulation of Stationary Gaussian Processes in [0,1]d. Journal of Computational and Graphical Statistics, 3(4), 409-432.

5. Moller J, Syversveen AR, Waagepetersen RP (1998). Log Gaussian Cox Processes. Scandinavian Journal of Statistics, 25(3), 451-482.