# Arc density of Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case Returns the arc density of PE-PCD whose vertex set is the given 2D numerical data set, `Xp`, (some of its members are) in the triangle `tri`. PE proximity regions is defined with respect to `tri` with expansion parameter $r \ge 1$ and vertex regions are based on center $M=(m_1,m_2)$ in Cartesian coordinates or $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of the triangle `tri` or based on circumcenter of `tri`; default is $M=(1,1,1)$, i.e., the center of mass of `tri`. The function also provides arc density standardized by the mean and asymptotic variance of the arc density of PE-PCD for uniform data in the triangle `tri` only when `M` is the center of mass. For the number of arcs, loops are not allowed. `in.tri.only` is a logical argument (default is `FALSE`) for considering only the points inside the triangle or all the points as the vertices of the digraph. if `in.tri.only=TRUE`, arc density is computed only for the points inside the triangle (i.e., arc density of the subdigraph induced by the vertices in the triangle is computed), otherwise arc density of the entire digraph (i.e., digraph with all the vertices) is computed. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:arc-density-PE;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r PEarc.dens.tri(Xp, tri, r, M = c(1, 1, 1), in.tri.only = FALSE) ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of the PE-PCD. - `tri`: A $3 \times 2$ matrix with each row representing a vertex of the triangle. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$. - `M`: A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle `tri` or the circumcenter of `tri` which may be entered as "CC" as well; default is $M=(1,1,1)$, i.e., the center of mass of `tri`. - `in.tri.only`: A logical argument (default is `in.tri.only=FALSE`) for computing the arc density for only the points inside the triangle, `tri`. That is, if `in.tri.only=TRUE` arc density of the induced subdigraph with the vertices inside `tri` is computed, otherwise otherwise arc density of the entire digraph (i.e., digraph with all the vertices) is computed. ## Returns A `list` with the elements - **arc.dens**: Arc density of PE-PCD whose vertices are the 2D numerical data set, `Xp`; PE proximity regions are defined with respect to the triangle `tri` and `M`-vertex regions - **std.arc.dens**: Arc density standardized by the mean and asymptotic variance of the arc density of PE-PCD for uniform data in the triangle `tri`. This will only be returned, if `M` is the center of mass. ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); n<-10 #try also n<-20 set.seed(1) Xp<-runif.tri(n,Tr)$g M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.0) num.arcsPEtri(Xp,Tr,r=1.5,M) PEarc.dens.tri(Xp,Tr,r=1.5,M) PEarc.dens.tri(Xp,Tr,r=1.5,M,in.tri.only = TRUE) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `ASarc.dens.tri`, `CSarc.dens.tri`, and `num.arcsPEtri` ## Author(s) Elvan Ceyhan

PEarc.dens.tri function