DepNHPPMarked: Generating dependent point processes from a marked Poison... In IndTestPP: Tests of Independence and Analysis of Dependence Between Point Processes in Time

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

 `1` ```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`.

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`.

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.

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.

`DepNHPPqueue`, `DepNHNeyScot`, `DepNHCPSP`, `IndNHPP`, `SpecGap`
 ```1 2 3 4 5 6 7``` ```# 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) ```