# A test of segregation/association based on domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) for 2D data - Normal Approximation 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 i.e., cluster around the centers of the Delaunay triangles) and association (where `Xp` points cluster around `Yp` points) based on the normal approximation to the binomial distribution of the domination number of PE-PCD for uniform 2D data in the convex hull of `Yp` points 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 $Pr($domination number$\le 2)$), 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, probability of success (i.e., $Pr($domination number$\le 2)$) equals to its expected value under the uniform distribution) and `alternative` could be two-sided, or right-sided (i.e., data is accumulated around the `Yp` points, or association) or left-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 $M$-vertex regions where M is a center that yields non-degenerate asymptotic distribution of the domination number. The test statistic is based on the normal approximation to the binomial distribution, when success is defined as domination number being less than or equal to 2 in the one triangle case (i.e., number of failures is equal to number of times restricted domination number = 3 in the triangles). That is, the test statistic is based on the domination number for `Xp` points inside convex hull of `Yp` points for the PE-PCD and default convex hull correction, `ch.cor`, is `FALSE` where `M` is the center that yields nondegenerate asymptotic distribution for the domination number. For this approximation to work, number of `Yp` points must be at least 5 (i.e., about 7 or more Delaunay triangles) and number of `Xp` points must be at least 7 times more than the number of `Yp` points. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:dom-num-NPE-Spat2011;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r PEdom.num.norm.test( Xp, Yp, r, ch.cor = FALSE, ndt = NULL, 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 in $(1,1.5]$. - `ch.cor`: A logical argument for convex hull correction, default `ch.cor=FALSE`, recommended when both `Xp` and `Yp` have the same rectangular support. - `ndt`: Number of Delaunay triangles based on `Yp` points, default is `NULL`. - `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 domination number of PE-PCD whose vertices are 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 domination number at the given level `conf.level` and depends on the type of `alternative`. - **estimate**: A `vector` with two entries: first is the domination number, and second is the estimate of the parameter, i.e., $Pr($Domination Number$=3)$ - **null.value**: Hypothesized value for the parameter, i.e., the null value for expected domination number - **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<-100; ny<-5 #try also nx<-1000; ny<-10 r<-1.5 #try also r<-2 or r<-1.25 set.seed(1) Xp<-cbind(runif(nx,0,1),runif(nx,0,1)) 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="") PEdom.num.norm.test(Xp,Yp,r) #try also PEdom.num.norm.test(Xp,Yp,r, alt="l") PEdom.num.norm.test(Xp,Yp,1.25,ch=TRUE) #or try ndt<-num.delaunay.tri(Yp) PEdom.num.norm.test(Xp,Yp,r,ndt=ndt) #values might differ due to the random of choice of the three centers M1,M2,M3 #for the non-degenerate asymptotic distribution of the domination number ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `PEdom.num.binom.test` ## Author(s) Elvan Ceyhan

PEdom.num.norm.test function