stpp (version 2.0-8)

sthpcpp: Spatio-temporal hot-spots cluster point process model

Description

Generate a realisation of the hot-spots cluster process in a region \(S\times T\).

Usage

sthpcpp(lambp, r, mu, s.region, t.region)

Value

The simulated spatio-temporal point pattern.

Arguments

s.region

Two-column matrix specifying polygonal region containing all data locations.If s.region is missing, the unit square is considered.

t.region

Vector containing the minimum and maximum values of the time interval. If t.region is missing, the interval \([0,1]\) is considered.

lambp

Intensity of the Poisson process of cluster centres. A single positive number, a function, or a pixel image.

r

Radius parameter of the clusters.

mu

Average number of daughter per parent (a single positive number) or reference intensity for the cluster points (a function or a pixel image).

Author

Francisco J. Rodriguez Cortes <[e-mail]>

Details

This function generates a realisation of spatio-temporal cluster process, which can be considered as generalisation of the classical Matern cluster process, inside the spatio-temporal window.

Consider a Poisson point process in the plane with intensity \(\lambda_{p}\) as cluster centres for all times 'parent', as well as a infinite cylinder of radius \(R\) around of each Poisson point, orthogonal to the plane. The scatter uniformly in all cylinders of all points which are of the form \((x,y,z)\), the number of points in each cluster being random with a Poisson (\(\mu\)) distribution. The resulting point pattern is a spatio-temporal cluster point process with \(t=z\). This point process has intensity \(\lambda_{p}\times\mu\).

References

Baddeley, A., Rubak, E., Turner, R. (2015). Spatial Point Patterns: Methodology and Applications with R. CRC Press, Boca Raton.

Chiu, S. N., Stoyan, D., Kendall, W. S., and Mecke, J. (2013). Stochastic Geometry and its Applications. John Wiley & Sons.

Gabriel, E., Rowlingson, B., Diggle P J. (2013) stpp: an R package for plotting, simulating and analyzing Spatio-Temporal Point Patterns. Journal of Statistical Software 53, 1-29.

Illian, J B., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley and Sons, London.

Stoyan, D., Rodriguez-Cortes, F. J., Mateu, J., and Gille, W. (2017). Mark variograms for spatio-temporal point processes. Spatial Statistics. 20, 125-147.

Examples

# First example
X <- sthpcpp(lambp=20, r=0.05, mu=100)
plot(X$xyt)

# Spatio-temporal 3D scatter plot
par(mfrow=c(1,1))
plot(X$xyt,type="scatter")

## Spatio-temporal hot-spots cluster point process model
data(northcumbria)
Northcumbria <- northcumbria/1000
Xo <- sthpcpp(lambp=0.0035, r=5, mu=200,
s.region=Northcumbria, t.region=c(28,198))
plot(Xo$xyt,s.region=Northcumbria)

# Spatio-temporal 3D scatter plot
par(mfrow=c(1,1))
plot(Xo$xyt,type="scatter",style="elegant",theta=45,phi=30,cex=0.05,
ticktype="detailed",col="black")