kmmt: Temporal mark correlation function

Description

Computes an estimator of the temporal mark correlation function.

Usage

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

Value

ekmmt: A vector containing the values of \(k_[mm](v)\) estimated.
dt: Parameter passed in argument. If dt is missing, a vector of temporal distances v at which kmmt(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.
kmmttheo: Value under the Poisson case is calculated considering the side lengths of the bounding box of xyt[,1:2].

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 kmmt(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.

Author

Francisco J. Rodriguez Cortes <frrodriguezc@unal.edu.co> https://fjrodriguezcortes.wordpress.com

Details

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

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 Wilfried, G. (2016). Mark variograms for spatio-temporal point processes, Submitted .

Examples

Run this code

    ## 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 correlation function
    out <- kmmt(hpp)
    
    # R plot - Temporal mark correlation function
    par(mfrow=c(1,1))
    xl <- c(0,0.25)
    plot(out$dt,out$ekmmt,type="l",xlab="t = time",ylab=expression(k[mm](t)),
         xlim=xl,col=1,cex.lab=1.5,cex.axis=1.5)
    lines(out$dt,rep(out$kmmttheo,length(out$dt)),col=11)
    
    ## End(Not run)

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