The function lgcpPredict performs spatiotemporal prediction for log-Gaussian Cox Processes
lgcpPredict(
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"
)a spatio-temporal point pattern object, see ?stppp
time point of interest
specifies lag window, so that data from and including time (T-laglength) to time T is used in the MALA algorithm
values for parameters, see ?lgcppars
correlation type see ?CovarianceFct
vector of additional parameters for certain classes of covariance function (eg Matern), these must be supplied in the order given in ?CovarianceFct
width of grid cells on which to do MALA (grid cells are square) in same units as observation window. Note EITHER gridsize OR cellwidth must be specified.
size of output grid required. Note EITHER gridsize OR cellwidthe must be specified.
the fixed spatial component: an object of that can be coerced to one of class spatialAtRisk
the fixed temporal component: either a numeric vector, or a function that can be coerced into an object of class temporalAtRisk
MCMC paramters, see ?mcmcpars
output choice, see ?setoutput
a list of owin objects (of length laglength+1) which has xyt$window as a base window, but with polygonal holes specifying spatial areas where there is missing data.
logical: whether or not to automatically do MCMC on optimised, rotated grid.
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.
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 very slowly (compared withe the size of hte observation window), increasing 'ext' may be necessary.
criterion for cells being included into observation window. Either 'touching' or 'centroid'. The former includes all cells that touch the observation window, the latter includes all cells whose centroids are inside the observation window. further notes on autorotate argument: If set to TRUE, and the argument spatial is not NULL, then the argument spatial must be computed in the original frame of reference (ie NOT in the rotated frame). Autorotate performs bilinear interpolation (via interp.im) on an inverse transformed grid; if there is no computational advantage in doing this, a warning message will be issued. Note that best accuracy is achieved by manually rotating xyt and then computing spatial on the transformed xyt and finally feeding these in as arguments to the function lgcpPredict. By default autorotate is set to FALSE.
the results of fitting the model in an object of class lgcpPredict
The following is a mathematical description of a log-Gaussian Cox Process, it is best viewed in the pdf version of the manual.
Let \(\mathcal Y(s,t)\) 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.
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/
Brix A, Diggle PJ (2001). Spatiotemporal Prediction for log-Gaussian Cox processes. Journal of the Royal Statistical Society, Series B, 63(4), 823-841.
Diggle P, Rowlingson B, Su T (2005). Point Process Methodology for On-line Spatio-temporal Disease Surveillance. Environmetrics, 16(5), 423-434.
Wood ATA, Chan G (1994). Simulation of Stationary Gaussian Processes in [0,1]d. Journal of Computational and Graphical Statistics, 3(4), 409-432.
Moller J, Syversveen AR, Waagepetersen RP (1998). Log Gaussian Cox Processes. Scandinavian Journal of Statistics, 25(3), 451-482.
KinhomAverage, ginhomAverage, lambdaEst, muEst, spatialparsEst, thetaEst, spatialAtRisk, temporalAtRisk, lgcppars, CovarianceFct, mcmcpars, setoutput print.lgcpPredict, xvals.lgcpPredict, yvals.lgcpPredict, plot.lgcpPredict, meanfield.lgcpPredict, rr.lgcpPredict, serr.lgcpPredict, intens.lgcpPredict, varfield.lgcpPredict, gridfun.lgcpPredict, gridav.lgcpPredict, hvals.lgcpPredict, window.lgcpPredict, mcmctrace.lgcpPredict, plotExceed.lgcpPredict, quantile.lgcpPredict, identify.lgcpPredict, expectation.lgcpPredict, extract.lgcpPredict, showGrid.lgcpPredict