# A test of segregation/association based on arc density of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data An object of class `"htest"` (i.e., hypothesis test) function which performs a hypothesis test of complete spatial randomness (CSR) or uniformity of `Xp` points in the convex hull of `Yp` points against the alternatives of segregation (where `Xp` points cluster away from `Yp` points) and association (where `Xp` points cluster around `Yp` points) based on the normal approximation of the arc density of the PE-PCD for uniform 2D data. The function yields the test statistic, $p$-value for the corresponding `alternative`, the confidence interval, estimate and null value for the parameter of interest (which is the arc density), and method and name of the data set used. Under the null hypothesis of uniformity of `Xp` points in the convex hull of `Yp` points, arc density of PE-PCD whose vertices are `Xp` points equals to its expected value under the uniform distribution and `alternative` could be two-sided, or left-sided (i.e., data is accumulated around the `Yp` points, or association) or right-sided (i.e., data is accumulated around the centers of the triangles, or segregation). PE proximity region is constructed with the expansion parameter $r \ge 1$ and $CM$-vertex regions (i.e., the test is not available for a general center $M$ at this version of the function). **Caveat:** This test is currently a conditional test, where `Xp` points are assumed to be random, while `Yp` points are assumed to be fixed (i.e., the test is conditional on `Yp` points). Furthermore, the test is a large sample test when `Xp` points are substantially larger than `Yp` points, say at least 5 times more. This test is more appropriate when supports of `Xp` and `Yp` have a substantial overlap. Currently, the `Xp` points outside the convex hull of `Yp` points are handled with a convex hull correction factor, `ch.cor`, which is derived under the assumption of uniformity of `Xp` and `Yp` points in the study window, (see the description below and the function code.) However, in the special case of no `Xp` points in the convex hull of `Yp` points, arc density is taken to be 1, as this is clearly a case of segregation. Removing the conditioning and extending it to the case of non-concurring supports is an ongoing topic of research of the author of the package. `ch.cor` is for convex hull correction (default is `"no convex hull correction"`, i.e., `ch.cor=FALSE`) which is recommended when both `Xp` and `Yp` have the same rectangular support. 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) ) for more on the test based on the arc density of PE-PCDs. ```r PEarc.dens.test( Xp, Yp, r, ch.cor = FALSE, alternative = c("two.sided", "less", "greater"), conf.level = 0.95 ) ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of the PE-PCD. - `Yp`: A set of 2D points which constitute the vertices of the Delaunay triangles. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$. - `ch.cor`: A logical argument for convex hull correction, default `ch.cor=FALSE`, recommended when both `Xp` and `Yp` have the same rectangular support. - `alternative`: Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"`. - `conf.level`: Level of the confidence interval, default is `0.95`, for the arc density of PE-PCD based on the 2D data set `Xp`. ## Returns A `list` with the elements - **statistic**: Test statistic - **p.value**: The $p$-value for the hypothesis test for the corresponding `alternative` - **conf.int**: Confidence interval for the arc density at the given confidence level `conf.level` and depends on the type of `alternative`. - **estimate**: Estimate of the parameter, i.e., arc density - **null.value**: Hypothesized value for the parameter, i.e., the null arc density, which is usually the mean arc density under uniform distribution. - **alternative**: Type of the alternative hypothesis in the test, one of `"two.sided"`, `"less"`, `"greater"` - **method**: Description of the hypothesis test - **data.name**: Name of the data set ## Examples ```r #nx is number of X points (target) and ny is number of Y points (nontarget) nx<-100; ny<-5; #try also nx<-40; ny<-10 or nx<-1000; ny<-10; set.seed(1) Xp<-cbind(runif(nx),runif(nx)) Yp<-cbind(runif(ny,0,.25), runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1)) #try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1)) plotDelaunay.tri(Xp,Yp,xlab="",ylab="") PEarc.dens.test(Xp,Yp,r=1.25) PEarc.dens.test(Xp,Yp,r=1.25,ch=TRUE) #since Y points are not uniform, convex hull correction is invalid here ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `CSarc.dens.test` and `PEarc.dens.test1D` ## Author(s) Elvan Ceyhan

PEarc.dens.test function