The generalized dependency degree between attributes

Abstract

Inspired by the dependency degree ␥, a traditional measure in Rough Set Theory, we propose a generalized dependency degree, ⌫, between two given sets of attributes, which counts both deterministic and indeterministic rules while ␥ counts only deterministic rules. We first give its definition in terms of equivalence relations and then interpret it in terms of minimal rules, and further describe the algorithm for its computation. To understand ⌫ better, we investigate its various properties. We further extend ⌫ to incomplete information systems. To show its advantage, we make a comparative study with the conditional entropy and ␥ in a number of experiments. Experimental results show that the speed of the new C4.5 using ⌫ is greatly improved when compared with the original C4.5R8 using conditional entropy, while the prediction accuracy and tree size of the new C4.5 are comparable with the original one. Moreover, ⌫ achieves better results on attribute selection than ␥. The study shows that the generalized dependency degree is an informative measure in decision trees and in attribute selection. IntroductionAs one of several models that are used to extract previously unknown and potentially useful information from the databases, Rough Set Theory provides an effective tool for mining deterministic rules from a database. In recent years, it has received considerable attention. For example, Yao, Li, Lin, and Liu (1994) makes it possible to obtain the upper and lower bounds by eliminating transitivity, reflexivity, and symmetry axioms. In other works (Kryszkiewicz, 1998(Kryszkiewicz, , 1999Leung & Li, 2003;Lingras & Yao, 1998), various extensions of the Rough Set Theory to incomplete information systems are considered.The objective of this article is to generalize the dependency degree g that is widely used in the rough set theory. Besides, we aim to develop a deeper understanding of the generalized dependency degree and to justify it both theoretically and empirically. On the theoretical side, we give its various forms and describe its properties; on the empirical side, we show its effectiveness in decision trees and in attribute selection. For this sake, we first need to introduce the following basic concepts concerning the dependency degree.An information system is represented by an attributevalue table in which rows are labeled by objects of the universe and columns by their attributes. Denote the universe of objects by U, the set of attributes or features by A, and the set of all possible values of attribute a by V a . Let P be a subset of A, that is, P is a subset of attributes. The P-indiscernibility relation, denoted by IND(P), is defined as is an equivalence relation. The set of equivalence classes is denoted by or by and the equivalence class in is called the P-class. For , let P(x) denote the P-class containing x.Let C and D be two subsets of A. The dependency degree g(C, D) is defined (Pawlak, 1999) as where, C(X) ϭ and denote the cardinality of the set U and the cardinality of the set C(X), respectively. denot...