# Generation of points associated (in a Matern-like fashion) to a given set of points An object of class `"Patterns"`. Generates `n` 2D points uniformly in $\cup B(y_i,e)$ where $Y_p=(y_1,y_2,\ldots,y_{n_y})$ with $n_y$ being number of `Yp` points for various values of `e` under the association pattern and $B(y_i,e)$ is the ball centered at $y_i$ with radius `e`. The pattern resembles the Matern cluster pattern (see `rMatClust` in the `spatstat.random` package for further information (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="baddeley:2005;textual",package="pcds",cached_env=.Rdpack.currefs) ). `rMatClust(kappa, scale, mu, win)` in the simplest case generates a uniform Poisson point process of "parent" points with intensity `kappa`. Then each parent point is replaced by a random cluster of "offspring" points, the number of points per cluster being Poisson(`mu`) distributed, and their positions being placed and uniformly inside a disc of radius scale centered on the parent point. The resulting point pattern is a realization of the classical "stationary Matern cluster process" generated inside the window `win`. The main difference of `rassoc.matern` and `rMatClust` is that the parent points are `Yp` points which are given beforehand and we do not discard them in the end in `rassoc.matern` and the offspring points are the points associated with the reference points, `Yp`; `e` must be positive and very large values of `e` provide patterns close to CSR. This function is also very similar to `rassoc.circular`, where `rassoc.circular` needs the study window to be specified, while `rassoc.matern` does not. ```r rassoc.matern(n, Yp, e) ``` ## Arguments - `n`: A positive integer representing the number of points to be generated. - `Yp`: A set of 2D points representing the reference points. The generated points are associated (in a Matern-cluster like fashion) with these points. - `e`: A positive real number representing the radius of the balls centered at `Yp` points. Only these balls are allowed for the generated points (i.e., generated points would be in the union of these balls). ## Returns A `list` with the elements - **type**: The type of the point pattern - **mtitle**: The `"main"` title for the plot of the point pattern - **parameters**: Radial (i.e., circular) attraction parameter of the association pattern. - **ref.points**: The input set of attraction points `Yp`, i.e., points with which generated points are associated. - **gen.points**: The output set of generated points associated with `Yp` points. - **tri.Yp**: Logical output for triangulation based on `Yp` points should be implemented or not. if `TRUE` triangulation based on `Yp` points is to be implemented (default is set to `FALSE`). - **desc.pat**: Description of the point pattern - **num.points**: The `vector` of two numbers, which are the number of generated points and the number of attraction (i.e., `Yp`) points. - **xlimit,ylimit**: The possible ranges of the $x$- and $y$-coordinates of the generated points. ## Examples ```r nx<-100; ny<-4; #try also nx<-1000; ny<-10; e<-.15; #try also e<-1.1; #closer to CSR than association, as e is large #Y points uniform in unit square Y<-cbind(runif(ny),runif(ny)) Xdt<-rassoc.matern(nx,Y,e) Xdt summary(Xdt) plot(Xdt,asp=1) Xdt<-Xdt$gen.points Xlim<-range(Xdt[,1],Y[,1]); Ylim<-range(Xdt[,2],Y[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Y,asp=1,xlab="x",ylab="y", main="Matern-like Association of X points with Y Points", xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01), pch=16,col=2,lwd=2) points(Xdt) a1<-0; a2<-10; b1<-0; b2<-5; e<-1.1; #Y points uniform in a rectangle Y<-cbind(runif(ny,a1,a2),runif(ny,b1,b2)) #try also Y<-cbind(runif(ny,a1,a2/2),runif(ny,b1,b2/2)) Xdt<-rassoc.matern(nx,Y,e)$gen.points Xlim<-range(Xdt[,1],Y[,1]); Ylim<-range(Xdt[,2],Y[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Y,asp=1,xlab="x",ylab="y", main="Matern-like Association of X points with Y Points", xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),pch=16,col=2,lwd=2) points(Xdt) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `rassoc.circular`, `rassoc.std.tri`, `rassocII.std.tri`, `rassoc.multi.tri`, `rseg.circular`, and `rMatClust` in the `spatstat.random` package ## Author(s) Elvan Ceyhan

rassoc.matern function