Relative density of the random r-factor proximity catch digraph for testing spatial patterns of segregation and association

Ceyhan, Elvan; Priebe, Carey E.; Wierman, John C.

doi:10.1016/j.csda.2005.03.002

Cited by 18 publications

(46 citation statements)

References 18 publications

(29 reference statements)

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“…These statistics require keeping good track of the geometry of the relevant neighbourhoods, and the complicated computations of integrals are done in the symbolic computation package MAPLE. The methodology is similar to that given by Ceyhan, Priebe & Wierman (2006). However, the results are simplified by the deliberate choices we make.…”

Section: Discussionmentioning

confidence: 99%

“…By detailed geometric probability calculations provided in the Appendix and in Ceyhan, Priebe & Marchette (2004), the mean and the asymptotic variance of the relative density of our proximity catch digraph can be calculated explicitly. The central limit theorem for U -statistics then establishes the asymptotic normality under the null hypothesis.…”

Section: Asymptotic Normality Under the Null Hypothesismentioning

confidence: 99%

“…respectively, since µ ′ S (τ, ε = 0) = µ ′ A (τ, ε = 0) = 0. Equation (2) provides the denominator; the numerator requires µ S (τ, ε) and µ A (τ, ε) which are provided in Ceyhan, Priebe & Marchette (2004) where we only use the intervals of τ that do not vanish as ε → 0.…”

Section: Monte Carlo Power Analysismentioning

confidence: 99%

“…Thus asymptotic normality obtains provided that ν S (τ, ε) > 0 (ν A (τ, ε) > 0); otherwise ρ n (τ ) is degenerate. The explicit forms of µ S (τ, ε) and µ A (τ, ε) are given, defined piecewise, in Ceyhan, Priebe & Marchette (2004). Note that under H S ε ,…”

Section: Somentioning

confidence: 99%

“…The domination number approach is appropriate when both classes are comparably large. Ceyhan, Priebe & Wierman (2006) used the relative density of the same PCD for testing the spatial patterns. Our new parameterized family of PCDs has more geometric appeal, is simpler in distributional parameters in the asymptotics, and the range of the parameters is bounded.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 99%

Section: Asymptotic Normality Under the Null Hypothesismentioning

confidence: 99%

Section: Monte Carlo Power Analysismentioning

confidence: 99%

Section: Somentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A new family of random graphs for testing spatial segregation

2007

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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 distribution of the relative arc density of a family of interval catch digraph based on uniform data

Ceyhan

2011

Metrika

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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.

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Relative density of the random r-factor proximity catch digraph for testing spatial patterns of segregation and association

Cited by 18 publications

References 18 publications

A new family of random graphs for testing spatial segregation

A new family of random graphs for testing spatial segregation

Class cover catch digraphs

The distribution of the relative arc density of a family of interval catch digraph based on uniform data

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