DepCPSPNHK: Estimating cross K-function and envelopes for marginal processes of a CPSP

Description

This function estimates the cross K-function between two (homogenous or nonhomogeneous) point processes in time, \(N_{x}\) and \(N_{y}\). It is evaluated in a grid of distances \(r\) and plotted. An envelope built by simulation under the hypothesis that the processes are the marginal processes of a bivariate CPSP is also plotted.

It calls the auxiliary function DepCPSPKenv, not intended for users.

Usage

DepCPSPNHK(posx, posy, lambdaix, lambdaiy, lambdaixy, r=NULL, typeEst=1, 
                  nsim=1000, conf=0.95,tit=NULL, cores=1,fixed.seed=NULL,...)

Arguments

posx

Numeric vector. Occurrence times of the points in the first point process \(N_{x}\) .

posy

Numeric vector. Occurrence times of the points in the second point process \(N_{y}\) .

lambdaix

Numeric vector. Intensity values of \(N_{(x)}\).

lambdaiy

Numeric vector. Intensity values of \(N_{(y)}\).

lambdaixy

Numeric vector. Intensity values of \(N_{(xy)}\).

Optional. Numeric vector. Grid values where the K-function must be evaluated. If it is NULL, a default vector is used; see Details.

typeEst

Optional. Two possible values: 1 or 2. They determines which one of the two available estimators of the function \(K_{ij}\) has to be used; see Details.

nsim

Optional. Numeric value. Number of simulations to obtain the envelope.

conf

Optional. Numeric value in (0,1). Confidence level of the envelope for the K-function.

tit

Optional. Title to be used in the plot of the K-function.

cores

Optional. Number of cores of the computer to be used in the calculations.

fixed.seed

An integer or NULL. If it is an integer, that is the value used to set the seed in random generation processes. If it is NULL, a random seed is used.

...

Further arguments to be passed to the function plot.

Value

A list with elements:

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

NHKr

Estimated values of \(K_{ij}(r)\).

KenvL

Lower bounds of the envelope of \(K_{ij}(r)\).

KenvU

Upper bounds of the envelope of \(K_{ij}(r)\).

Length of the observed period of the processes.

Details

This function estimates the cross K function between two (homogenous or nonhomogeneous) point processes in time, \(N_{x}\) and \(N_{y}\). Two different estimators are available, see NHK for more details.

An envelope for the cross K function is built under the hypothesis that the processes are the marginal processes of a bivariate CPSP with intensities of the indicator processes lambdaxi, lambdayi and lambdaxyi. The envelope is based on the simulation of CPSPs, gnerated by function DepNHCPSP.

If argument r is NULL, the following r-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)

where T is the length of the observed period.

Examples

Run this code

# NOT RUN {

set.seed(123)
lambdai1<-runif(500,0.005,0.01)
set.seed(124)
lambdai2<-runif(500,0.005,0.01)
set.seed(125)
lambdai12<-runif(500,0.005,0.02)

#Observed process: independent Poisson processes
posx<-simNHPc(lambda = (lambdai1+lambdai12), fixed.seed = 13)$posNH
posy<-simNHPc(lambda = (lambdai2+lambdai12), fixed.seed = 14)$posNH
aux<-DepCPSPNHK(posx=posx, posy=posy, lambdaix=lambdai1, lambdaiy=lambdai2, 
	       lambdaixy=lambdai12, fixed.seed=123, r=c(1:10), nsim=500)

#Observed processes: dependent marginal processes of a CPSP
#pos<-DepNHCPSP(lambdaiM=cbind(lambdai1, lambdai2, lambdai12), d=2,
#	fixed.seed=123, dplot=F)$posNH
#aux<-DepCPSPNHK(posx=pos$N1, posy=pos$N2, lambdaix=lambdai1, 
#	lambdaiy=lambdai2, lambdaixy=lambdai12, fixed.seed=125, r=c(1:10))
# }