# Generation of points segregated (in a radial or circular fashion) from a given set of points An object of class `"Patterns"`. Generates `n` 2D points uniformly in $(a_1-e,a_1+e) \times (a_1-e,a_1+e) \setminus B(y_i,e)$ ($a_1$ 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 segregation pattern and $B(y_i,e)$ is the ball centered at $y_i$ with radius `e`. Positive values of `e` yield realizations from the segregation pattern and nonpositive values of `e` provide a type of complete spatial randomness (CSR), `e` should not be too large to make the support of generated points empty, `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. ```r rseg.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 segregated (in a circular or radial fashion) from these points. - `e`: A positive real number representing the radius of the balls centered at `Yp` points. These balls are forbidden for the generated points (i.e., generated points would be in the complement of 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 (i.e., circular) exclusion parameter of the segregation pattern - **ref.points**: The input set of reference points `Yp`, i.e., points from which generated points are segregated. - **gen.points**: The output set of generated points segregated from `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 reference (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; #a negative e provides a CSR realization #with default bounding box (i.e., unit square) Y<-cbind(runif(ny),runif(ny)) Xdt<-rseg.circular(nx,Y,e) Xdt summary(Xdt) plot(Xdt,asp=1) #with default bounding box (i.e., unit square) Y<-cbind(runif(ny),runif(ny)) Xdt<-Xdt$gen.points Xlim<-range(Xdt[,1],Y[,1]); Ylim<-range(Xdt[,2],Y[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Y,asp=1,pch=16,col=2,lwd=2, xlab="x",ylab="y", main="Circular Segregation of X points from Y Points", xlim=Xlim+xd*c(-.01,.01),ylim=Ylim+yd*c(-.01,.01)) points(Xdt) #with a rectangular bounding box a1<-0; a2<-10; b1<-0; b2<-5; e<-1.5; Y<-cbind(runif(ny,a1,a2),runif(ny,b1,b2)) #try also Y<-cbind(runif(ny,a1,a2/2),runif(ny,b1,b2/2)) Xdt<-rseg.circular(nx,Y,e,a1,a2,b1,b2)$gen.points Xlim<-range(Xdt[,1],Y[,1]); Ylim<-range(Xdt[,2],Y[,2]) plot(Y,pch=16,asp=1,col=2,lwd=2, xlab="x",ylab="y", main="Circular Segregation of X points from Y Points", xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) points(Xdt) ``` ## See Also `rassoc.circular`, `rseg.std.tri`, `rsegII.std.tri`, and `rseg.multi.tri` ## Author(s) Elvan Ceyhan

rseg.circular function