The use of domination number of a random proximity catch digraph for testing spatial patterns of segregation and association

Ceyhan, Elvan; Priebe, Carey E.

doi:10.1016/j.spl.2005.02.012

Cited by 20 publications

(56 citation statements)

References 9 publications

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“…Furthermore, this article forms the foundation of the extensions of the methodology to higher dimensions. The domination number has other applications, e.g., in testing spatial point patterns (see, e.g., Ceyhan and Priebe (2005)) and our results can help make the power comparisons possible for a large family of alternative patterns in such a setting. Some trivial proofs regarding PICDs are omitted, while others are mostly deferred to the Supplementary Materials Section.…”

Section: Introductionmentioning

confidence: 83%

“…The domination number approach is easily adaptable to testing nonuniform distributions as well (see Remark S6.4 for more detail). PICDs have other applications, e.g., as in Ceyhan and Priebe (2005), we can use the domination number in testing one-dimensional spatial point patterns and our results can help make the power comparisons possible for a large family of distributions (see, e.g., Section 6.2 for a brief treatment of this issue). PICDs can also be employed in pattern classification as well (see, e.g., Priebe et al (2003) and Manukyan and Ceyhan (2016)).…”

Section: Discussionmentioning

confidence: 99%

“…Extension of this approach to higher dimensions is a challenging problem, since there is no such ordering for point in R d with d > 1. However, we can use the Delaunay tessellation based on Y m to partition the space as in Ceyhan and Priebe (2005). Furthermore, for most of the article and for all non-trivial results (i.e., for the exact and asymptotic distributions of the domination number), we assumed Y m is given; removing the conditioning on Y m is a topic of ongoing research along various directions, namely: (i) X and Y both have uniform distribution, (ii) X and Y both have the same (absolutely) continuous distribution, and (iii) X is distributed as F X and Y is distributed as F Y (where F X = F Y and both F X and F Y are absolutely continuous).…”

Section: Discussionmentioning

confidence: 99%

“…The PICDs are closely related to the proximity graphs of Jaromczyk and Toussaint (1992) and might be considered as one-dimensional versions of proportional-edge proximity catch digraphs of Ceyhan and Priebe (2005). Furthermore, when r = 2 and c = 1/2 (i.e., M c,i = Y (i) + Y (i+1) /2) we have N (x, r, c) = B(x, r(x)) where B(x, r(x)) is the ball centered at x with radius r(x) = d(x, Y m ) = min y∈Ym d(x, y).…”

Section: Relation Of Picds With Other Graph Familiesmentioning

confidence: 99%

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Domination number of an interval catch digraph family and its use for testing uniformity

Ceyhan

2020

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We consider a special type of interval catch digraph (ICD) family for one-dimensional data in a randomized setting and propose its use for testing uniformity. These ICDs are defined with an expansion and a centrality parameter, hence we will refer to this ICD as parameterized ICD (PICD). We derive the exact (and asymptotic) distribution of the domination number of this PICD family when its vertices are from a uniform (and non-uniform) distribution in one dimension for the entire range of the parameters; thereby determine the parameters for which the asymptotic distribution is non-degenerate. We observe jumps (from degeneracy to non-degeneracy or from a non-degenerate distribution to another) in the asymptotic distribution of the domination number at certain parameter combinations. We use the domination number for testing uniformity of data in real line, prove its consistency against certain alternatives, and compare it with two commonly used tests and three recently proposed tests in literature and also arc density of this ICD and of another ICD family in terms of size and power. Based on our extensive Monte Carlo simulations, we demonstrate that domination number of our PICD has higher power for certain types of deviations from uniformity compared to other tests.

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Section: Introductionmentioning

confidence: 83%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Relation Of Picds With Other Graph Familiesmentioning

confidence: 99%

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Domination number of an interval catch digraph family and its use for testing uniformity

Ceyhan

2020

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“…The Y points are at the vertices of the triangles and the X points are squares. Ceyhan & Priebe (2003) introduced an (unparameterized) version of the PCD we discuss in this article; Ceyhan & Priebe (2005) also introduced another parameterized family of PCDs and used the domination number (which is another statistic based on the number of arcs from the vertices) of this latter parameterized family for testing segregation and association. The domination number approach is appropriate when both classes are comparably large.…”

Section: Introductionmentioning

confidence: 99%

A new family of random graphs for testing spatial segregation

2007

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Abstract:The authors discuss a graph-based approach for testing spatial point patterns. This approach falls under the category of data-random graphs, which have been introduced and used for statistical pattern recognition in recent years. The authors address specifically the problem of testing complete spatial randomness against spatial patterns of segregation or association between two or more classes of points on the plane. To this end, they use a particular type of parameterized random digraph called a proximity catch digraph (PCD) which is based on relative positions of the data points from various classes. The statistic employed is the relative density of the PCD, which is a U -statistic when scaled properly. The authors derive the limiting distribution of the relative density, using the standard asymptotic theory of U -statistics. They evaluate the finite-sample performance of their test statistic by Monte Carlo simulations and assess its asymptotic performance via Pitman's asymptotic efficiency, thereby yielding the optimal parameters for testing. They further stress that their methodology remains valid for data in higher dimensions. Une nouvelle famille de graphes aléatoires utile pour tester la ségrégation spatialeRésumé : Les auteurs montrent comment on peut détecter des configurations de points dans l'espaceà l'aide de graphes. Leur approche s'appuie sur la notion de graphe aléatoire observé, récemment introduite et utilisée en statistique pour la reconnaissance de formes. Les auteurs cherchent plus précisémentà détecter la présence de ségrégation ou d'association entre deux ou plusieurs ensembles de points du plan en testant l'hypothèse d'absence complète de structure. Dans ce but, ils font appelà une classe paramétrique particulière de digraphes aléatoires appelés digraphes "à captation proximale" (DCP) qui tiennent compte de la disposition relative deséléments des diverses classes. Le test s'appuie sur la densité relative du DCP qui, une fois proprement normalisée, est une U -statistique. Les auteurs en déterminent la loi limite en invoquant la théorie asymptotique des U -statistiques. Ils enévaluent la performanceà taille finie au moyen de simulations de Monte-Carlo et enétudient aussi le comportement limite sous l'angle de l'efficacité asymptotique de Pitman, dont découlent des choix optimaux de paramètres aux fins de test. Ils soulignent de plus que leur méthodologie reste valide en dimensions supérieures.

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Class cover catch digraphs

Marchette

2010

WIREs Computational Stats

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The class cover problem is one of finding a small number of sets covering (containing) points from one class without covering any points from a second class. The class cover catch digraph provides a solution to the class cover problem, which can be extended to a nonparametric classifier, similar in flavor to a reduced nearest neighbor classifier. This article describes the class cover catch digraph and its application to classification.  R andom graphs occur in many pattern recognition problems. This article describes a particular type of random graph, the class cover catch digraph, which is a type of vertex random graph. Here, the graph is defined by a set of points and sets associated to the vertices. These points represent the training data in a classification problem, and the sets represent the 'coverage' or support of one of the classes.Using the sets, a simple classifier can be defined according to which of the sets contain a new observation. Instead of using all the training data, as in a standard nearest neighbor classifier, the graphs provide a simple method to reduce the complexity of the problem, resulting in a kind of reduced nearest neighbor classifier. We consider several variations of the resulting classifier, and also discuss some of the theoretical results that are known about the graphs.

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The use of domination number of a random proximity catch digraph for testing spatial patterns of segregation and association

Cited by 20 publications

References 9 publications

Domination number of an interval catch digraph family and its use for testing uniformity

Domination number of an interval catch digraph family and its use for testing uniformity

A new family of random graphs for testing spatial segregation

Class cover catch digraphs

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