# DistShift: Generates by translation a vector of independent processes,... In IndTestPP: Tests of Independence and Analysis of Dependence Between Point Processes in Time

## Description

This function generates a vector of two (or three) independent processes, conditionally on the first one, by shifting the second (and the third) process.

It also calculates the set of close points and the mean distance in the generated vector, for each point t_{x_i} in the first process.

## Usage

 `1` ```DistShift(posx,posy,posz=NULL, T, shii1, shii2=NULL, PA = FALSE, info=FALSE,...) ```

## Arguments

 `posx` Numeric vector. Position of the occurrence points in the first process. `posy` Numeric vector. Position of the occurrence points in the second process. `posz` Optional. Numeric vector. Position of the occurrence points in the third process. `T` Numeric value. Length of the observed period of the processes. `shii1` Numeric value. Distance used to shift the points in the second process. It must be a positive value lower than T. `shii2` Optional. Numeric value. Distance used to shift the points in the third process. It must be NULL, if there are two processes, and a positive value lower than T, if there are three. `PA` Optional. Logical flag. If it is TRUE, the close point relation is broadened by including the previous and the following points to the overlapping intervals. `info` Optional. Logical flag. If it is TRUE, information about the generated points is shown on the screen and dotcharts and bivariate charts of the occurrence points of the three processes are displayed. `...` Further arguments to be passed to the functions `plot` and `dotchart` if the argument `info=T`

## Details

This function is mainly used in the application of the Lotwicck- Silverman approach, see Lotwick and Silverman (1982), to generate a pair of independent processes with the same marginal distributions than the observed ones. These processes are used for example to build a test to assess independence between two or three processes, see `TestIndLS`.

The key idea is to wrap the processes onto a circumference by identifying the opposite sides of the time interval where they are observed. The first process is fixed, while the others are shifted over the circumference a given amount. The idea of this translation is to keep the marginal distribution of the processes but to break any dependence between them, without the need of parametric models to describe the marginal patterns.

The function also calculates the set of close points and the mean distance for each point t_{x_i} in the first process, in the new shifted vector of processes.

## Value

 `DistTri` The vector of the mean distances of points t_{x_i} in the shifted processes.

## References

Lotwick, H.W. and Silverman, B.W. (1982). Methods for analysing Spatial processes of several types of points. J.R. Statist. Soc. B, 44(3), pp. 406-13

`TestIndLS`, `DistSim`
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```set.seed(123) lambdax<-runif(200, 0.01,0.17) set.seed(124) lambday<-runif(200, 0.015,0.15) set.seed(125) lambdaz<-runif(200, 0.005,0.1) posx<-simNHPc(lambda=lambdax, fixed.seed=123)\$posNH posy<-simNHPc(lambda=lambday, fixed.seed=123)\$posNH posz<-simNHPc(lambda=lambdaz, fixed.seed=123)\$posNH aux<-DistShift(posx=posx, posy=posy, posz=posz, T=200, shii1=59, shii2=125 ) ```