Class Cover Catch Digraphs Research Articles

Law of large numbers for a two-dimensional class cover problem

We prove a Law of Large Numbers (LLN) for the domination number of class cover catch digraphs (CCCD) generated by random points in two (or higher) dimensions. DeVinney and Wierman (2002) proved the Strong Law of Large Numbers (SLLN) for the uniform distribution in one dimension, and Wierman and Xiang (2008) extended the SLLN to the case of general distributions in one dimension. In this article, using subadditive processes, we prove a SLLN result for the domination number generated by Poisson points in ℝ2. From this we obtain a Weak Law of Large Numbers (WLLN) for the domination number generated by random points in [0, 1]2 from uniform distribution first, and then extend these result to the case of bounded continuous distributions. We also extend the results to higher dimensions. The domination number of CCCDs and related digraphs have applications in statistical pattern classification and spatial data analysis.

Classification using proximity catch digraphs

We employ random geometric digraphs to construct semi-parametric classifiers. These data-random digraphs belong to parameterized random digraph families called proximity catch digraphs (PCDs). A related geometric digraph family, class cover catch digraph (CCCD), has been used to solve the class cover problem by using its approximate minimum dominating set and showed relatively good performance in the classification of imbalanced data sets. Although CCCDs have a convenient construction in $${\mathbb {R}}^d$$, finding their minimum dominating sets is NP-hard and their probabilistic behaviour is not mathematically tractable except for $$d=1$$. On the other hand, a particular family of PCDs, called proportional-edge PCDs (PE-PCDs), has mathematically tractable minimum dominating sets in $${\mathbb {R}}^d$$; however their construction in higher dimensions may be computationally demanding. More specifically, we show that the classifiers based on PE-PCDs are prototype-based classifiers such that the exact minimum number of prototypes (equivalent to minimum dominating sets) is found in polynomial time on the number of observations. We construct two types of classifiers based on PE-PCDs. One is a family of hybrid classifiers that depends on the location of the points of the training data set, and another type is a family of classifiers solely based on class covers. We assess the classification performance of our PE-PCD based classifiers by extensive Monte Carlo simulations, and compare them with that of other commonly used classifiers. We also show that, similar to CCCD classifiers, our classifiers tend to be robust to the class imbalance in classification as well.

Classification of imbalanced data with a geometric digraph family

We use a geometric digraph family called class cover catch digraphs (CCCDs) to tackle the class imbalance problem in statistical classification. CCCDs provide graph theoretic solutions to the class...

Extension of one-dimensional proximity regions to higher dimensions

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.

Class cover catch digraphs

AbstractThe 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. Copyright © 2010 John Wiley & Sons, Inc.This article is categorized under: Data: Types and Structure > Graph and Network Data

A CLT for a one-dimensional class cover problem

The central limit theorem is proved for the domination number of random class cover catch digraphs generated by uniform data in one dimension. The class cover problem is motivated by applications in statistical pattern classification.

The distribution of the domination number of class cover catch digraphs for non-uniform one-dimensional data

For two or more classes of points in R d with d ≥ 1 , the class cover catch digraphs (CCCDs) can be constructed using the relative positions of the points from one class with respect to the points from one or all of the other classes. The CCCDs were introduced by Priebe et al. [C.E. Priebe, J.G. DeVinney, D.J. Marchette, On the distribution of the domination number of random class catch cover digraphs. Statistics and Probability Letters 55 (2001) 239–246] who investigated the case of two classes, X and Y . They calculated the exact (i.e., finite sample) distribution of the domination number of the CCCDs based on X points relative to Y points both of which were uniformly distributed on a bounded interval. We investigate the distribution of the domination number of the CCCDs based on data from non-uniform X points on an interval with end points from Y . Then we extend these calculations for multiple Y points on bounded intervals.

A general SLLN for the one-dimensional class cover problem

The class cover catch digraph (CCCD) is motivated by applications in statistical pattern classification. For the special case of uniformly distributed data in one dimension, Priebe et al. [Priebe, C.E., DeVinney, J.G., Marchette, D.J., 2001. On the distribution of the domination number for random class cover catch digraphs. Statist. Probab. Lett. 55, 239–246] found the exact distribution of the domination number of the random data-induced CCCD, and DeVinney and Wierman [DeVinney, J.G., Wierman, J.C., 2002. A SLLN for a one-dimensional class cover problem. Statist. Probab. Lett. 59, 425–435] proved the Strong Law of Large Numbers (SLLN). This paper proves the generalized SLLN for the domination number of CCCDs for non-uniform data in one dimension.

A new family of proximity graphs: Class cover catch digraphs

Motivated by issues in machine learning and statistical pattern classification, we investigate a class cover problem (CCP) with an associated family of directed graphs—class cover catch digraphs (CCCDs). CCCDs are a special case of catch digraphs. Solving the underlying CCP is equivalent to finding a smallest cardinality dominating set for the associated CCCD, which in turn provides regularization for statistical pattern classification. Some relevant properties of CCCDs are studied and a characterization of a family of CCCDs is given.

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

Priebe et al. (2001. Statist. Prob. Lett. 55, 239–246) introduced the class cover catch digraphs and computed the distribution of the domination number of such digraphs for one-dimensional data. In higher dimensions these calculations are extremely difficult due to the geometry of the proximity regions; and only upper-bounds are available. In this article, we introduce a new type of data-random proximity map and the associated (di)graph in R d . We find the asymptotic distribution of the domination number and use it for testing spatial point patterns of segregation and association.

Classification Using Class Cover Catch Digraphs

based on proximity between training observations. Performance comparisons are presented on synthetic and real examples versus k-nearest neighbors, Fisher's linear discriminant and support vector machines. We demonstrate that the proposed semiparametric classifier has performance approaching that of the optimal parametric classifier in cases for which the optimal is available for comparison.

Class cover catch digraphs for latent class discovery in gene expression monitoring by DNA microarrays

The purpose of this article is to introduce a data visualization technique for class cover catch digraphs which allows for the discovery of latent subclasses. We illustrate the technique via a pedagogical example and an application to data sets from artificial nose chemical sensing and gene expression monitoring by DNA microarrays. Of particular interest is the discovery of latent subclasses representing chemical concentration in the artificial nose data and two subtypes of acute lymphoblastic leukemia in the gene expression data and the associated conjectures pertaining to the geometry of these subclasses in their respective high-dimensional observation spaces.

A SLLN for a one-dimensional class cover problem

Class cover catch digraphs arise in classification problems in statistical pattern recognition. We prove a strong law of large numbers for the domination number in a random one-dimensional model of class cover catch digraphs. The proof avoids complicated computations due to the dependence of random variables by considering a related Poisson process problem where we may apply classical strong law results and Chernoff exponential probability bounds. Complete convergence in the Poisson representation establishes the desired result for the original problem.

A VISUALIZATION FRAMEWORK FOR THE ANALYSIS OF HYPERDIMENSIONAL DATA

The purpose of this article is to describe a new visualization framework for the analysis of hyperdimensional data. This framework was developed in order to facilitate the study of a new class of classifiers designated class cover catch digraphs. The class cover catch digraph is an original random graph technique for the construction of classifiers on high dimensional data. This framework allows the user to study the geometric structure of hyperdimensional data sets via the reduction of the original hyperdimensional space to a cover with a small number of balls. The framework allows for the elicitation of geometric and other structures through the visualization of the relationships between the balls and each other and the observations they cover.

On the distribution of the domination number for random class cover catch digraphs

In this article we initiate the study of class cover catch digraphs, a special case of intersection digraphs motivated by applications in machine learning and statistical pattern recognition. Our main result is the exact distribution of the domination number for a data-driven model of random interval catch digraphs.