# Generation of points associated (in a Type I fashion) with a given set of points An object of class `"Patterns"`. Generates `n` points uniformly in the support for Type I association in the convex hull of set of points, `Yp`. `delta` is the parameter of association (that is, only $\delta 100$ % area around each vertex in each Delaunay triangle is allowed for point generation). `delta` corresponds to `eps` in the standard equilateral triangle $T_e$ as $delta=4eps^2/3$ (see `rseg.std.tri` function). If `Yp` consists only of 3 points, then the function behaves like the function `rassoc.tri`. `DTmesh` must be the Delaunay triangulation of `Yp` and `DTr` must be the corresponding Delaunay triangles (both `DTmesh` and `DTr` are `NULL` by default). If `NULL`, `DTmesh` is computed via `tri.mesh` and `DTr` is computed via `triangles` function in `interp` package. `tri.mesh` function yields the triangulation nodes with their neighbours, and creates a triangulation object, and `triangles` function yields a triangulation data structure from the triangulation object created by `tri.mesh` (the first three columns are the vertex indices of the Delaunay triangles). See (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:arc-density-PE,ceyhan:arc-density-CS,ceyhan:dom-num-NPE-Spat2011;textual",package="pcds",cached_env=.Rdpack.currefs) ) for more on the association pattern. Also, see (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="okabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textual",package="pcds",cached_env=.Rdpack.currefs) ) for more on Delaunay triangulation and the corresponding algorithm. ```r rassoc.multi.tri(n, Yp, delta, DTmesh = NULL, DTr = NULL) ``` ## Arguments - `n`: A positive integer representing the number of points to be generated. - `Yp`: A set of 2D points from which Delaunay triangulation is constructed. - `delta`: A positive real number in $(0,1)$. `delta` is the parameter of association (that is, only $\delta 100$ % area around vertices of each Delaunay triangle is allowed for point generation). - `DTmesh`: Delaunay triangulation of `Yp`, default is `NULL`, which is computed via `tri.mesh` function in `interp` package. `tri.mesh` function yields the triangulation nodes with their neighbours, and creates a triangulation object. - `DTr`: Delaunay triangles based on `Yp`, default is `NULL`, which is computed via `tri.mesh` function in `interp` package. `triangles` function yields a triangulation data structure from the triangulation object created by `tri.mesh`. ## Returns A `list` with the elements - **type**: The type of the pattern from which points are to be generated - **mtitle**: The `"main"` title for the plot of the point pattern - **parameters**: Attraction parameter, `delta`, of the Type I association pattern. `delta` is in $(0,1)$ and only $\delta 100$ % of the area around vertices of each Delaunay triangle is allowed for point generation. - **ref.points**: The input set of points `Yp`; reference points, i.e., points with which generated points are associated. - **gen.points**: The output set of generated points associated with `Yp` points. - **tri.Y**: Logical output, `TRUE` if triangulation based on `Yp` points should be implemented. - **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 ranges of the $x$- and $y$-coordinates of the reference points, which are the `Yp` points ## Examples ```r #nx is number of X points (target) and ny is number of Y points (nontarget) nx<-100; ny<-4; #try also nx<-40; ny<-10 or nx<-1000; ny<-10; set.seed(1) Yp<-cbind(runif(ny),runif(ny)) del<-.4 Xdt<-rassoc.multi.tri(nx,Yp,del) Xdt summary(Xdt) plot(Xdt) #or use DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove") #Delaunay triangulation based on Y points TRY<-interp::triangles(DTY)[,1:3]; Xp<-rassoc.multi.tri(nx,Yp,del,DTY,TRY)$g #data under CSR in the convex hull of Ypoints Xlim<-range(Yp[,1]) Ylim<-range(Yp[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] #plot of the data in the convex hull of Y points together with the Delaunay triangulation DTY<-interp::tri.mesh(Yp[,1],Yp[,2],duplicate="remove") #Delaunay triangulation based on Y points plot(Xp,main="Points from Type I Association \n in Multipe Triangles", xlab=" ", ylab=" ",xlim=Xlim+xd*c(-.05,.05), ylim=Ylim+yd*c(-.05,.05),type="n") interp::plot.triSht(DTY, add=TRUE, do.points=TRUE,col="blue") points(Xp,pch=".",cex=3) ``` ## 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`, and `rseg.multi.tri` ## Author(s) Elvan Ceyhan

rassoc.multi.tri function