# A graphical test to assess independence between two point processes This function applies the Diggle's randomization testing procedure extended by Dutilleul(2011), and performs a plot which graphically assesses the independence between two point proceses. It is implemented for homogenous and non homogenous Poisson processes. ```r DutilleulPlot(posx, posy, lambday, nsim = 1000, lenve = c(0.025, 0.975), ...) ``` ## Arguments - `posx`: Numeric vector. Occurrence times of the points in the first point process. - `posy`: Numeric vector. Occurrence times of the points in the second point process. - `lambday`: Numeric vector. Intensity vector of the second point process. If the process is homogeneous, a vector of length $T$, with equal values must be provided; see Details. - `nsim`: Optional. Positive integer. Number of simulations to calculate the confidence band. - `lenve`: Optional. Numeric vector. The order of the lower and the upper percentiles to build the confidence band. - `...`: Further arguments to be passed to the function `plot`. ## Details This graphical approach is based on the comparison of the cumulative relative frequency of the nearest neighbour distances between the points in the two observed processes, with their counterpart in two independent processes with the same marginal distributions, which are obtained by simulation. The function plots the cumulative relative frequency of the observed processes and a confidence band calculated from **nsim** simulated independent processes. The length of the observed period $T$ is determined by the length of the argument `lambday`. ## Returns A list with the elements: - **quantobs**: Vector of the observed percentiles of the nearest neighbour distances. - **enve1**: Vector of the lower bounds of the confidence band. - **enve2**: Vector of the upper bounds of the confidence band. ## References Dutilleul, P. (2011), **Spatio-temporal heterogeneity: Concepts and analyses**, Cambridge University Press. ## See Also `TestIndNH`, `CondTest`,`nearestdist` ## Examples ```r #Two independent NHPPs set.seed(123) lambdax<-runif(200, 0.01,0.1) set.seed(124) lambday<-runif(200, 0.015,0.15) posx<-simNHPc(lambdax,fixed.seed=123)$posNH posy<-simNHPc(lambday, fixed.seed=123)$posNH aux<-DutilleulPlot(posx, posy, lambday, nsim = 100) #Two dependent Neyman Scott processes #set.seed(123) #lambdaParent<-runif(200)/10 #DepPro<-DepNHNeyScot(lambdaParent=lambdaParent, d=2, lambdaNumP = 3, # dist = "normal", sigmaC = 3,fixed.seed=123) #posx<-DepPro$PP1 #posy<-DepPro$PP2 #aux<-DutilleulPlot(posx, posy, lambday, nsim = 100) ```

DutilleulPlot function