# Generation of points associated (in a radial or circular fashion) with a given set of points An object of class `"Patterns"`. Generates `n` 2D points uniformly in $(a_1-e,a_1+e) \times (a_1-e,a_1+e) \cap U_i B(y_i,e)$ ($a_1$ and $b1$ are denoted as `a1` and `b1` as arguments) 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`. `e` must be positive and very large values of `e` provide patterns close to CSR. `a1` is defaulted to the minimum of the $x$-coordinates of the `Yp` points, `a2` is defaulted to the maximum of the $x$-coordinates of the `Yp` points, `b1` is defaulted to the minimum of the $y$-coordinates of the `Yp` points, `b2` is defaulted to the maximum of the $y$-coordinates of the `Yp` points. This function is also very similar to `rassoc.matern`, where `rassoc.circular` needs the study window to be specified, while `rassoc.matern` does not. ```r rassoc.circular( n, Yp, e, a1 = min(Yp[, 1]), a2 = max(Yp[, 1]), b1 = min(Yp[, 2]), b2 = max(Yp[, 2]) ) ``` ## 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 circular or radial 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). - `a1, a2`: Real numbers representing the range of $x$-coordinates in the region (default is the range of $x$-coordinates of the `Yp` points). - `b1, b2`: Real numbers representing the range of $y$-coordinates in the region (default is the range of $y$-coordinates of the `Yp` points). ## 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 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 range of the $x$- and $y$-coordinates of the generated points. ## Examples ```r nx<-100; ny<-4; #try also nx<-1000; ny<-10; e<-.15; #with default bounding box (i.e., unit square) Y<-cbind(runif(ny),runif(ny)) Xdt<-rassoc.circular(nx,Y,e) Xdt summary(Xdt) plot(Xdt,asp=1) Xdt<-Xdt$gen.points Xlim<-range(Xdt[,1],Y[,1]); Ylim<-range(Xdt[,2],Y[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Y,asp=1,xlab="x",ylab="y", main="Circular Association of X points with Y Points", xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01), pch=16,col=2,lwd=2) points(Xdt) #with default bounding box (i.e., unit square) Xlim<-range(Xdt[,1],Y[,1]); Ylim<-range(Xdt[,2],Y[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Y,asp=1,xlab="x",ylab="y", main="Circular Association of X points with Y Points", xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01),pch=16, col=2,lwd=2) points(Xdt) #with a rectangular bounding box a1<-0; a2<-10; b1<-0; b2<-5; e<-1.1; #try also e<-5; #pattern very close to CSR! Y<-cbind(runif(ny,a1,a2),runif(ny,b1,b2)) #try also Y<-cbind(runif(ny,a1,a2/2),runif(ny,b1,b2/2)) Xdt<-rassoc.circular(nx,Y,e,a1,a2,b1,b2)$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="Circular 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) ``` ## See Also `rseg.circular`, `rassoc.std.tri`, `rassocII.std.tri`, `rassoc.matern`, and `rassoc.multi.tri` ## Author(s) Elvan Ceyhan

rassoc.circular function