IndTestPP (version 3.0)

DepNHPPMarked: Generating dependent point processes from a marked Poison Process

Description

This function generates \(d\) dependent (homogeneous or nonhomogeneous) point processes using a marked Poisson process, where the marks are generated by a Markov chain process defined by a transition matrix.

Usage

DepNHPPMarked(lambdaTot, MarkovM, inival = 1, dplot=TRUE, fixed.seed=NULL,...)

Arguments

lambdaTot

Numeric vector. Intensity values of the Poisson process used to generate the dependent processes.

MarkovM

Matrix. Trasition probabilities of the d-state Markov chain used to generate the marks of the process.

inival

Optional. Initial mark value used to generate the series of marks.

dplot

Optional. A logical flag. If it is TRUE, the marginal processes are plotted.

fixed.seed

Optional. An integer or NULL. 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

posNH

A list of \(d\) vectors, containing the occurrence points in each marginal point process. The name of the elements of the list are N1, N2,..., Nd.

posNHG

Numeric vector of the occurrences times of the generated Poisson process.

mark

Vector of the generated marks.

lambdaTot

Input argument.

MarkovM

Input argument.

Details

Points of the marked Poisson process are generated in continuous time, using the following procedure: First, a trajectory of the underlying Poisson process is generated. Then, the mark series is generated using a d-state Markov chain. The mark series takes values in 1,2,...,d and determines in which of the d processes the points occur.

The marginal processes defined by the marks are not Poisson unless the generated marks are independent observations, see Isham (1980).

A transition matrix \(P = (p_{ij})\) with equal rows leads to \(d\) independent point processes, and the more similar the rows of P, the less dependent the resulting processes. The spectral gap (SpecGap) measures the dependence between the generated processes, see Abaurrea et al. (2014).

Tha marginal processes of the marked process can be optionally plotted using dplot=TRUE.

References

Abaurrea, J. Asin, J. and Cebrian, A.C. (2015). A Bootstrap Test of Independence Between Three Temporal Nonhomogeneous Poisson Processes and its Application to Heat Wave Modeling. Environmental and Ecological Statistics, 22(1), 127-144.

Isham, V. (1980). Dependent thinning of point processes. J. Appl. Probab., 17(4), 987-95.

See Also

DepNHPPqueue, DepNHNeyScot, DepNHCPSP, IndNHPP, SpecGap

Examples

# NOT RUN {
# Generation of three dependent  point processes using a marked PP
set.seed(123)
lambdaTot<-runif(1000)/10
aux<-DepNHPPMarked(lambdaTot=lambdaTot,
	MarkovM=cbind(c(0.3,0.1,0.6), c(0.1, 0.6, 0.3), c(0.6, 0.3,0.1)),fixed.seed=123)
print(cbind(aux$posNH, aux$mark))
print(aux$posNHs)
# }