gte {stpp}

Temporal mark variogram function

Description

Computes an estimator of the temporal mark variogram function.

Usage

gte(xyt,t.region,t.lambda,dt,kt="epanech",ht,correction="none",approach="simplified")

Arguments

xyt

Spatial coordinates and times (x,y,t) of the point pattern.

t.region

Vector containing the minimum and maximum values of the time interval.

t.lambda

Vector of values of the temporal intensity function evaluated at the points t in T. If t.lambda is missing, the estimate of the temporal mark correlation function is computed as for the homogeneous case, i.e. considering n/|T| as an estimate of the temporal intensity under the parameter approach="standardised".

dt

A vector of times v at which gte(v) is computed.

kt

A kernel function for the temporal distances. The default is the "epanech" kernel. It can also be "box" for the uniform kernel, or "biweight".

ht

A bandwidth of the kernel function kt.

correction

A character vector specifying the edge-correction(s) to be applied among "isotropic", "border", "modified.border", "translate", "setcovf" and "none". The default is "none".

approach

A character vector specifying the approach to use for the estimation to be applied among "simplified" or "standardised". If approach is missing, "simplified" is considered by default.

Details

By default, this command calculates an estimate of the temporal mark variogram function \gamma_[te](t) for a spatio-temporal point pattern.

Value

egte

A vector containing the values of \gamma_[te](v) estimated.

dt

Parameter passed in argument. If dt is missing, a vector of temporal distances v at which gte(v) is computed from 0 to until quarter of the maximum distance between the times in the temporal pattern.

kernel

Parameters passed in argument. A vector of names and bandwidth of the spatial kernel.

gtetheo

Value under the Poisson case is calculated considering the side lengths of the bounding box of xyt[,1:2].

Author(s)

Francisco J. Rodriguez Cortes <[e-mail]> https://fjrodriguezcortes.wordpress.com

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

## Not run:
#################

# A realisation of spatio-temporal homogeneous Poisson point processes
hpp <- rpp(lambda = 100, replace = FALSE)$xyt

# R plot
plot(hpp)

# This function provides an kernel estimator of the temporal mark variograma function
out <- gte(hpp)

# R plot - Temporal mark variogram function
par(mfrow=c(1,1))
xl <- c(0,0.25)
yl <- c(0,max(out$egte,out$gtetheo))
plot(out$dt,out$egte,type="l",xlab="t = time",ylab=expression(gamma[te](t)),
                 xlim=xl,ylim=yl,col=1,cex.lab=1.5,cex.axis=1.5)
lines(out$dt,rep(out$gtetheo,length(out$dt)),col=11)

## End(Not run)