lgcp (version 1.8)

lgcpSim: lgcpSim function

Description

Approximate simulation from a spatiotemoporal log-Gaussian Cox Process. Returns an stppp object.

Usage

lgcpSim(
  owin = NULL,
  tlim = as.integer(c(0, 10)),
  spatial.intensity = NULL,
  temporal.intensity = NULL,
  cellwidth = 0.05,
  model.parameters = lgcppars(sigma = 2, phi = 0.2, theta = 1),
  spatial.covmodel = "exponential",
  covpars = c(),
  returnintensities = FALSE,
  progressbar = TRUE,
  ext = 2,
  plot = FALSE,
  ratepow = 0.25,
  sleeptime = 0,
  inclusion = "touching"
)

Arguments

owin

polygonal observation window

tlim

time interval on which to simulate data

spatial.intensity

object that can be coerced into a spatialAtRisk object. if NULL then uniform spatial is chosen

temporal.intensity

the fixed temporal component: either a numeric vector, or a function that can be coerced into an object of class temporalAtRisk

cellwidth

width of cells in same units as observation window

model.parameters

parameters of model, see ?lgcppars.

spatial.covmodel

spatial covariance function, default is exponential, see ?CovarianceFct

covpars

vector of additional parameters for spatial covariance function, in order they appear in chosen model in ?CovarianceFct

returnintensities

logigal, whether to return the spatial intensities and true field Y at each time. Default FALSE.

progressbar

logical, whether to print a progress bar. Default TRUE.

ext

how much to extend the parameter space by. Default is 2.

plot

logical, whether to plot intensities.

ratepow

power that intensity is raised to for plotting purposes (makes the plot more pleasign to the eye), defaul 0.25

sleeptime

time in seconds to sleep between plots

inclusion

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.

Value

an stppp object containing the data

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 $\mathcal{Y}(s,t)$ be a spatiotemporal Gaussian process, $W\subset R^2$ be an observation window in space and $T\subset R_{\ge 0}$ 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 $R(\cdot )$, $$X_{S,[t_1,t_2]}\sim \text{Poisson}\{{\int }_S{\int }_{t_1}^{t_2}R(s,t)dsdt\}$$ 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, $\lambda :R^2\mapsto R_{\ge 0}$, is a known function, proportional to the population at risk at each point in space and scaled so that $${\int }_W\lambda (s)ds=1,$$ whilst the fixed temporal component, $\mu :R_{\ge 0}\mapsto R_{\ge 0}$, 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.

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.

See Also

lgcpPredict, showGrid.stppp, stppp

Examples

# NOT RUN {
library(spatstat.core); library(spatstat.utils); xyt <- lgcpSim()
# }