# DepNHNeyScot: Generating a multivariate Neyman-Scott cluster process In IndTestPP: Tests of Independence and Analysis of Dependence Between Point Processes in Time

## Description

This function generates a multivariate Neyman-Scott cluster process, that is a vector of d dependent (homogeneous or nonhomogeneous) point processes which are Neyman-Scott processes with the same trajectory of cluster centers.

It calls the auxiliary function `GenSons` (not intended for the users), see Details.

## Usage

 ```1 2``` ```DepNHNeyScot(lambdaParent, d, lambdaNumP = 1, dist = "normal", sigmaC = 1, minC = -1, maxC = 1, dplot=TRUE, fixed.seed=NULL,...) ```

## Arguments

 `lambdaParent` Numeric vector. Intensity values of the Poisson process used to generate the centers of the clusters of the Neyman-Scott process. `d` Integer. Number of dependent processes to be generated. `lambdaNumP` Optional. Numeric vector. Mean values of the number of sons of each dependent process. If its length is equal to 1, the same value is used to generate all the dependent processes. `dist` Optional. Label "normal" or "uniform". Distribution used to generate the points of each cluster. `sigmaC` Optional. Numeric vector. Only used if dist="normal". Standard deviation of the normal distribution. If its length is equal to 1, the same value is used in the d processes. `minC` Optional. Numeric vector. Only used if dist="uniform". Lower bounds of the Uniform distribution. If its length is equal to 1, the same value is used in the d processes. `maxC` Optional. Numeric vector. Only used if dist="uniform". Upper bounds of the Uniform distribution. If its length is equal to 1, the same value is used in the d processes. `dplot` Optional. A logical flag. If it is TRUE, the generated 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

A Neyman-Scott process is a Poisson cluster process where the points in each cluster are randomly distributed around the cluster center, see Neyman and Scott (1958) and Entekhabi et al. (1989).

Homogeneous or NH Neyman-Scott processes in continuous time and with the same trajectory of cluster centers are generated, so that d dependent processe are obtained. First, the Poisson process of the cluster centers is generated. Then, the number of points in each cluster is generated using a Poisson distribution with means which can be different in each process. The distances of each point in the cluster to its centre can be generated using two distributions a N(0, sigmaC) or a Uniform(minC, maxC).

It is noteworthy that high values of `sigmaC` or the range `maxC-minC` lead to a high variability around the centre and to a low dependence between the processes.

The marginal processes of the generated vector can be optionally plotted.

## Value

A list with elements:

 `posNH` A list of d vectors, containing the occurrence points of the d dependent processes. The name of the elements of the list are N1, N2,..., Nd. `sizeCL` A list of d vectors. Each vector contains the size (number of points) of each cluster in a processes. The name of the elements of the list are size1, size2,..., sized.

## References

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

Neyman, J., & Scott, E. L. (1958). Statistical approach to problems of cosmology. Journal of the Royal Statistical Society. Series B (Methodological), 1-43.

Entekhabi, D., Rodriguez-Iturbe, I., & Eagleson, P. S. (1989). Probabilistic representation of the temporal rainfall process by a modified Neyman-Scott Rectangular Pulses Model: Parameter estimation and validation. Water Resources Research, 25(2), 295-302.

`IndNHNeyScot`, `DepNHPPqueue`, `DepNHPPMarked`, `DepNHCPSP`
 ```1 2 3 4 5 6``` ```# Generation of three dependent Neyman-Scott processes with normal distances set.seed(123) lambdaParent<-runif(100,0,0.1) DepNHNeyScot(lambdaParent=lambdaParent, d=3, lambdaNumP = c(2,3,2), dist = "normal", sigmaC = c(3,2,2),fixed.seed=123) ```