| In this dissertation, we introduce proximity catch digraphs (PCDs) based on two sets of points, Xn and Ym from classes X and Y , respectively, in Rd with d > 1, investigate their properties, and present examples in R2 for illustrative purposes. PCDs are a generalization of the class cover catch digraphs (CCCDs) introduced by Priebe et al. in [35]; a slight modification of the CCCD yields a special case of PCDs. The PCDs are constructed based on the relative positions of members of, say, class X points, with respect to the Delaunay tessellation of class Y points. Our main motivation for introducing PCDs is that a direct extension of the CCCD to multidimensional data lacks mathematical tractability of the distribution of the domination number, moments of relative density, and geometry invariance for uniform Xn in the convex hull of Ym . We investigate two major aspects of the PCDs, namely, the distribution of the domination number and the relative density. In R2 , we analyze these concepts for Xn in one triangle (formed by Y3 ) and then generalize the analyses for Xn in multiple triangles from the Delaunay triangulation (assumed to exist) of Ym with m > 3. Our PCDs make tractable the mathematics in multiple dimensions, thereby enhancing the applicability of the methodology to statistical hypothesis testing and pattern classification. We compute the asymptotic distribution of the domination number of one of the PCDs. Furthermore, the relative density of the PCDs is shown to be a U-statistic which avails the asymptotic normality of the relative density. The domination number and relative density are both used to test a type of complete spatial randomness against spatial point patterns of segregation and association. The power of the tests is investigated by using asymptotic efficacy methods such as Pitman asymptotic efficacy, Hodges-Lehman asymptotic efficacy, asymptotic power function analysis. Finite sample power is analyzed by Monte Carlo simulations. The methods are illustrated in the two dimensional case, but are applicable to higher dimensions as well as to other types of proximity maps. |
