lgcpPredictSpatialINLA(sd, ns, model.parameters=lgcppars(),
spatial.covmodel="exponential", covpars=c(), cellwidth, gridsize,
spatial.intensity, ext=2, optimverbose=FALSE, inlaverbose=TRUE,
generic0hyper=list(theta = list(initial = 0, fixed = TRUE)),
strategy="simplified.laplace", method="Nelder-Mead")lgcpPredictThe function lgcpPredictSpatialINLA performs spatial prediction for log-Gaussian Cox Processes using the integrated nested Laplace approximation.
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 $R(\cdot)$, $$X_{S} \sim \mbox{Poisson}\left{\int_S R(s)ds\right}$$ Following Brix and Diggle (2001) and Diggle et al (2005) (but ignoring temporal variation), the intensity is decomposed multiplicatively as $$R(s,t) = \lambda(s)\exp{\mathcal Y(s,t)}.$$ In the above, the fixed spatial component, $\lambda:R^2\mapsto R_{\geq 0}$, is a known function, proportional to the population at risk at each point in space and scaled so that $$\int_W\lambda(s)d s=1.$$
Before calling this function, the user must decide on the parameters, spatial covariance model, spatial discretisation, fixed spatial ($\lambda(s)$) component and whether or not any output is required. Note there is no autorotate option for this function.