stpp (version 2.0-8)

stdcpp: Generate double-cluster point pattern

Description

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

Usage

stdcpp(lambp, a, b, c, 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 parent process. Can be either a numeric value, a function, or a 3d-array (see rpp).

a

Length of the semi-axes \(x\) of ellipsoid.

b

Length of the semi-axes \(y\) of ellipsoid.

c

Length of the semi-axes \(y\) of ellipsoid.

mu

Average number of daughter per parent. (a single positive number).

Author

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

Details

We consider the straightforward extension of the classical Matern cluster process on the \(R^3\) case (with ellipsoid or balls) by considering the \(z\)-coordiantes as times.

Consider a Poisson point process in the plane with intensity \(\lambda_p\) as cluster centres for all times 'parent', as well as a ellipsoid (or ball) where the semi-axes are of lengths \(a\), \(b\) and \(c\), around of each Poisson point under a random general rotation. The scatter uniformly in all ellipsoid (or ball) 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

# Ellipsoid
Xe <- stdcpp(lambp=20,a=0.5,b=0.09,c=0.07,mu=100)
plot(Xe$xyt)

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

# Balls
Xb <- stdcpp(lambp=20,a=0.07,b=0.07,c=0.07,mu=100)
plot(Xb$xyt)

# \donttest{
# Spatio-temporal 3D scatter plot
par(mfrow=c(1,1))
plot(Xb$xyt,type="mark",style="elegant")

# Northcumbria
data(northcumbria)
Northcumbria <- northcumbria/1000
X <- stdcpp(lambp=0.00004,a=10,b=10,c=10,mu=120,
s.region=Northcumbria,t.region=c(0,200))
plot(X$xyt,s.region=Northcumbria, cex=0.5)

# Spatio-temporal 3D scatter plot
par(mfrow=c(1,1))
plot(X$xyt,type="scatter",theta=45,phi=30,cex=0.1,
ticktype="detailed",col="black",style="elegant")
# }