IndTestPP (version 3.0)

NHD: Estimating the D-function

Description

This function estimates the cross nearest neighbour distance distribution function, D, between two sets, \(C\) and \(D\), of (homogenous or nonhomogeneous) point processes. The D-function is evaluated in a grid of values \(r\), and it can be optionally plotted.

It calls the auxiliary functions NHDaux and other functions, not intended for users.

Usage

NHD(lambdaC, lambdaD, T=NULL,Ptype='inhom', posC, typeC=1, posD, typeD=1, 
	r = NULL, dplot = TRUE, tit = "D(r)",...)

Arguments

lambdaC

A matrix of positive values. Each column is the intensity vector of one of the point processes in \(C\). If there is only one process in \(C\), it can be a vector or even a numeric value if the process is homogeneous.

lambdaD

A matrix of positive values. Each column is the intensity vector of one of the point process in \(D\). If there is only one process in \(D\), it can be a vector or even a numeric value if the process is homogeneous.

T

Numeric value. Length of the observed period. It only must be specified if the number of rows in lambdaC and lambdaD is 1.

Ptype

Optional. Label: "hom" or "inhom". The first one indicates that all the point processes in sets \(C\) and \(D\) are homogeneous.

posC

Numeric vector. Occurrence times of the points in all the point processes in \(C\).

typeC

Numeric vector with the same length as posC. Code of the point process in \(C\) where points in posC have occurred. See Details.

posD

Numeric vector. Occurrence times of the points in all the point processes in \(D\).

typeD

Numeric vector with the same length as posD. Code of the point process in \(D\) where points in posD have occurred.

r

Numeric vector. Values where the D-function must be evaluated. If it is NULL, a default vector is used, see Details.

dplot

Optional. A logical flag. If it is TRUE, the D-function is plotted.

tit

Optional. The title to be used in the plot of the D-function.

...

Further arguments to be passed to the function plot.

Value

A list with elements:

r

Vector of values \(r\) where the D-function is estimated.

NHDr

Estimated values of \(D_{CD}(r)\).

T

Length of the observed period.

Details

The information about the processes is provided by arguments posC, the vector of all the occurrence times in the processes in \(C\), and typeC, the vector of the code of the point process in set \(C\) where each point in posC has occurred; the second set D is characterized analogously by typeD and posD.

This function estimates the D-function between two sets, \(C\) and \(D\), of (homogenous or nonhomogeneous) point processes, see Cebrian et al (2020) for details of the estimation. The D-function is the distribution function of the distances from a point in a process in \(C\) to the nearest point in a process \(D\). In homogeneous proceesses, it estimates the probability that at least one point in a process in set \(D\) occurs at a distance lower than \(r\) of a given point in a process in set \(C\). If the processes are nonhomogenous, the inhomogenous version of the function, adjusted for time varying intensities, is used. It is calculated using the Hanisch estimator, see Van Lieshout (2006) Small values of the D-function suggest few points in processes in \(D\) in the r-neighbourhood of points of processes in \(C\). Large values indicate that points in processes in \(D\) are attracted by those of processes in \(C\).

For inference about independence of the processes, K and J-functions should be used.

If argument r is NULL, the following grid is used to evaluate the function

r1<-max(20, floor(T/20))

r<-seq(1,r1,by=2)

if (length(r)>200) r<-seq(1,r1,length.out=200)

References

Cebrian, A.C., Abaurrea, J. and Asin, J. (2020). Testing independence between two point processes in time. Journal of Simulation and Computational Statistics.

Van Lieshout, M.N.M. (2006) A J-function for marked point patterns. AISM, 58, 235-259. DOI 10.1007/s10463-005-0015-7

See Also

NHK, NHJ, NHF

Examples

# NOT RUN {
#Sets C and D with  independent NHPPs
set.seed(123)
lambda1<-runif(500, 0.05, 0.1)
set.seed(124)
lambda2<-runif(500, 0.01, 0.2)
pos1<-simNHPc(lambda=lambda1, fixed.seed=123)$posNH
pos2<-simNHPc(lambda=lambda2, fixed.seed=123)$posNH
aux<-NHD(lambdaC=lambda1, lambdaD=lambda2, posC=pos1, typeC=1, posD=pos2, typeD=1)
aux$NHDr

#Example with  independent NHPPs
#pos3<-simNHPc(lambda=lambda1, fixed.seed=321)$posNH
#pos4<-simNHPc(lambda=lambda2, fixed.seed=321)$posNH
#aux<-NHD(lambdaC=cbind(lambda1,lambda2),lambdaD=cbind(lambda1,lambda2),posC=c(pos1,pos2),
# typeC=c(rep(1, length(pos1)), rep(2, length(pos2))), posD=c(pos3, pos4),
# typeD=c(rep(1, length(pos3)), rep(2, length(pos4))))
#aux$NHDr

# }