lgcpPredict performs spatiotemporal
prediction for log-Gaussian Cox ProcesseslgcpPredict(xyt, T, laglength, model.parameters = lgcppars(),
spatial.covmodel = "exponential", covpars = c(), cellwidth = NULL,
gridsize = NULL, spatial.intensity, temporal.intensity, mcmc.control,
output.control = setoutput(), missing.data.areas = NULL,
autorotate = FALSE, gradtrunc = Inf, ext = 2, inclusion = "touching")lgcpPredictLet $\mathcal Y(s,t)$ be a spatiotemporal Gaussian process, $W\subset R^2$ be an observation window in space and $T\subset R_{\geq 0}$ be an interval of time of interest. Cases occur at spatio-temporal positions $(x,t) \in W \times T$ according to an inhomogeneous spatio-temporal Cox process, i.e. a Poisson process with a stochastic intensity $R(x,t)$, The number of cases, $X_{S,[t_1,t_2]}$, arising in any $S \subseteq W$ during the interval $[t_1,t_2]\subseteq T$ is then Poisson distributed conditional on $R(\cdot)$, $$X_{S,[t_1,t_2]} \sim \mbox{Poisson}\left{\int_S\int_{t_1}^{t_2} R(s,t)d sd t\right}$$ Following Brix and Diggle (2001) and Diggle et al (2005), the intensity is decomposed multiplicatively as $$R(s,t) = \lambda(s)\mu(t)\exp{\mathcal Y(s,t)}.$$ 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 $$\int_W\lambda(s)d s=1,$$ whilst the fixed temporal component, $\mu:R_{\geq 0}\mapsto R_{\geq 0}$, is also a known function with $$\mu(t) \delta t = E[X_{W,\delta t}],$$ for $t$ in a small interval of time, $\delta t$, over which the rate of the process over $W$ can be considered constant.
NOTE: the xyt stppp object can be recorded in
continuous time, but for the purposes of prediciton,
discretisation must take place. For the time dimension,
this is achieved invisibly by as.integer(xyt$t) and
as.integer(xyt$tlim). Therefore, before running an
analysis please make sure that this is commensurate with
the physical inerpretation and requirements of your output.
The spatial discretisation is chosen with the argument
cellwidth (or gridsize). If the chosen discretisation in
time and space is too coarse for a given set of parameters
(sigma, phi and theta) then the proper correlation
structures implied by the model will not be captured in the
output.
Before calling this function, the user must decide on the time point of interest, the number of intervals of data to use, the parameters, spatial covariance model, spatial discretisation, fixed spatial ($\lambda(s)$) and temporal ($\mu(t)$) components, mcmc parameters, and whether or not any output is required.