# Number of arcs of Central Similarity Proximity Catch Digraphs (CS-PCDs) and related quantities of the induced subdigraphs for points in the partition intervals - multiple interval case An object of class `"NumArcs"`. Returns the number of arcs and various other quantities related to the partition intervals for Central Similarity Proximity Catch Digraph (CS-PCD) whose vertices are the data points in `Xp` in the multiple interval case. For this function, CS proximity regions are constructed data points inside or outside the intervals based on `Yp` points with expansion parameter $t \ge 0$ and centrality parameter $c \in (0,1)$. That is, for this function, arcs may exist for points in the middle or end-intervals. Range (or convex hull) of `Yp` (i.e., the interval $(\min(Yp),\max(Yp))$) is partitioned by the spacings based on `Yp` points (i.e., multiple intervals are these partition intervals based on the order statistics of `Yp` points whose union constitutes the range of `Yp` points). If there are duplicates of `Yp` points, only one point is retained for each duplicate value, and a warning message is printed. For the number of arcs, loops are not counted. ```r num.arcsCS1D(Xp, Yp, t, c = 0.5) ``` ## Arguments - `Xp`: A set or `vector` of 1D points which constitute the vertices of the CS-PCD. - `Yp`: A set or `vector` of 1D points which constitute the end points of the partition intervals. - `t`: A positive real number which serves as the expansion parameter in CS proximity region; must be $> 0$. - `c`: A positive real number in $(0,1)$ parameterizing the center inside the middle (partition) intervals with the default `c=.5`. For an interval, `int`$=(a,b)$, the parameterized center is $M_c=a+c(b-a)$. ## Returns A `list` with the elements - **desc**: A short description of the output: number of arcs and related quantities for the induced subdigraphs in the partition intervals - **num.arcs**: Total number of arcs in all intervals (including the end-intervals), i.e., the number of arcs for the entire CS-PCD - **num.in.range**: Number of `Xp` points in the range or convex hull of `Yp` points - **num.in.ints**: The vector of number of `Xp` points in the partition intervals (including the end-intervals) based on `Yp` points - **weight.vec**: The `vector` of the lengths of the middle partition intervals (i.e., end-intervals excluded) based on `Yp` points - **int.num.arcs**: The `vector` of the number of arcs of the components of the CS-PCD in the partition intervals (including the end-intervals) based on `Yp` points - **part.int**: A list of partition intervals based on `Yp` points - **data.int.ind**: A `vector` of indices of partition intervals in which data points reside, i.e., column number of `part.int` is provided for each `Xp` point. Partition intervals are numbered from left to right with 1 being the left end-interval. - **tess.points**: Tessellation points, i.e., points on which the tessellation of the study region is performed, here, tessellation is the partition intervals based on `Yp` points. - **vertices**: Vertices of the digraph, `Xp`. ## Examples ```r tau<-1.5 c<-.4 a<-0; b<-10; int<-c(a,b); #nx is number of X points (target) and ny is number of Y points (nontarget) nx<-20; ny<-4; #try also nx<-40; ny<-10 or nx<-1000; ny<-10; set.seed(1) xf<-(int[2]-int[1])*.1 Xp<-runif(nx,a-xf,b+xf) Yp<-runif(ny,a,b) Narcs = num.arcsCS1D(Xp,Yp,tau,c) Narcs summary(Narcs) plot(Narcs) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `num.arcsCSint`, `num.arcsCSmid.int`, `num.arcsCSend.int`, and `num.arcsPE1D` ## Author(s) Elvan Ceyhan

num.arcsCS1D function