Research Project: YENİ RASSAL YAKINLIK ÇİZGELERİ VE DESEN ANALİZİNDE UYGULAMALARI
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Contributors
Funders
ID
TB.00005
Authors
Ceyhan, Elvan
Faculty Member
Publications
Spatial clustering tests based on the domination number of a new random digraph family
(Taylor and Francis , 2011) Ceyhan, Elvan; Department of Mathematics; Yes; College of Sciences
We use the domination number of a parametrized random digraph family called proportional-edge proximity catch digraphs (PCDs) for testing multivariate spatial point patterns. This digraph family is based on relative positions of data points from various classes. We extend the results on the distribution of the domination number of proportional-edge PCDs, and use the domination number as a statistic for testing segregation and association against complete spatial randomness. We demonstrate that the domination number of the PCD has binomial distribution when size of one class is fixed while the size of the other (whose points constitute the vertices of the digraph) tends to infinity and has asymptotic normality when sizes of both classes tend to infinity. We evaluate the finite sample performance of the test by Monte Carlo simulations and prove the consistency of the test under the alternatives. We find the optimal parameters for testing each of the segregation and association alternatives. Furthermore, the methodology discussed in this article is valid for data in higher dimensions also.
An investigation of new graph invariants related to the domination number of random proximity catch digraphs
(Springer, 2012) Ceyhan, Elvan; Department of Mathematics; Yes; College of Sciences
Proximity catch digraphs (PCDs) are a special type of proximity graphs based on proximity maps which yield proximity regions. PCDs are defined using the relative allocation of points from two or more classes in a region of interest and have applications in various fields. We introduce some auxiliary tools for PCDs and graph invariants related to the domination number of the PCDs and investigate their probabilistic properties. We consider the cases in which the vertices of the PCDs come from uniform and non-uniform distributions in the region of interest. We also provide some of the newly defined proximity maps as illustrative examples.
Extension of one-dimensional proximity regions to higher dimensions
(Elsevier, 2010) Ceyhan, Elvan; Department of Mathematics; Yes; College of Sciences
Proximity regions (and maps) are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which gave rise to class cover catch digraph (CCCD) and was applied to pattern classification. In this article, we note some appealing properties of the spherical proximity map in compact intervals on the real line, thereby introduce the mechanism and guidelines for defining new proximity maps in higher dimensions. For non-spherical PCDs, Delaunay tessellation (triangulation in the real plane) is used to partition the region of interest in higher dimensions. We also introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of these maps and the resulting PCDs. We provide the distribution of graph invariants, namely, domination number and relative density, of the PCDs and characterize the geometry invariance of the distribution of these graph invariants for uniform data and provide some newly defined proximity maps in higher dimensions as illustrative examples. (C) 2010 Elsevier B.V. All rights reserved.
The distribution of the relative arc density of a family of interval catch digraph based on uniform data
(Springer, 2012) Ceyhan, Elvan; Department of Mathematics; Yes; College of Sciences
We study a family of interval catch digraph called proportional-edge proximity catch digraph (PCD) which is also a special type of intersection digraphs parameterized with an expansion and a centrality parameter. PCDs are random catch digraphs that have been developed recently and have applications in classification and spatial pattern analysis. We investigate a graph invariant of the PCDs called relative arc density. We demonstrate that relative arc density of PCDs is a U-statistic and using the central limit theory of U-statistics, we derive the (asymptotic) distribution of the relative arc density of proportional-edge PCD for uniform data in one dimension. We also determine the parameters for which the rate of convergence to asymptotic normality is fastest.
