Many supervised machine learning algorithms require a discrete feature space. In this paper, we review previous work on continuous feature discretization, identify defining characteristics of the methods, and conduct an empirical evaluation of several methods. We compare binning, an unsupervised discretization method, to entropy-based and purity-based methods, which are supervised algorithms. We found that the performance of the Naive-Bayes algorithm significantly improved when features were discretized using an entropy-based method. In fact, over the 16 tested datasets, the discretized version of Naive-Bayes slightly outperformed C4.5 on average. We also show that in some cases, the performance of the C4.5 induction algorithm significantly improved if features were discretized in advance; in our experiments, the performance never significantly degraded, an interesting phenomenon considering the fact that C4.5 is capable of locally discretizing features.

Abstract. Real-life data usually are presented in databases by real numbers. On the other hand, most inductive learning methods require small number of attribute values. Thus it is necessary to convert input data sets with continuous attributes into input data sets with discrete attributes. Methods of discretization restricted to single continuous attributes will be called local, while methods that simultaneously convert all continuous attributes will be called global. In this paper, a method of transforming any local discretization method into a global one is presented. A global discretization method, based on cluster analysis, is presented and compared experimentally with three known local methods, transformed into global. Experiments include ten-fold cross validation and leaving-one-out methods for ten real-life data sets.

s. The quantization of real value attributes is one of the main problem to be solved in synthesis of decision rules from data tables with real value attributes. We present an approach to this problem based on rough set methods and Boolean reasoning. The main result states that the problem of optimal quantization of real value attributes is polynomially reducible to the problem of minimal reduct finding, so it is NP-hard. We construct efficient heuristics for finding suboptimal quantization of real value attributes. 1 INTRODUCTION A great effort has been made (see e.g. [5], [7], [9], [17], [18]) to find effective methods for real value attributes quantization (discretization). Our approach is based on the rough set methods and Boolean reasoning. We discuss the computational complexity of the quantization problems and we show that they can be solved by Boolean reasoning [1]. We prove that the main quantization problems are either NP -complete or NP - hard. We show that the problem of o...

The task of extracting knowledge from databases is quite often performed by machine leaming algorithms. The majority of these algorithms can be applied only to data described by discrete numerical or nominal attributes (features). In the case of continuous attributes, there is a need for a discretization algorithm that transforms continuous attributes into discrete ones. This paper describes such an algorithm, called CAIM (class-attribute interdependence maximization), which is designed to work with supervised data. The goal of the CAlM algorithm is to maximize the class-attribute interdependence and to generate a (possibly) minimal number of discrete intervals. The algorithm does not require the user to predefine the number of intervals, as opposed to some other discretization algorithms. The tests performed using CALM, and six other state-of-the-art discretization algorithms, show that discrete attributes generated by the CAlM algorithm almost always have the lowest number of intervals and the highest class-attribute interdependency. Two machine learning algorithms, the CLIP4 rule algorithm and the decision tree algorithm, are used to generate classification rules from data discretized by CALM. For both the CLIP4 and decision tree algorithms, the accuracy of the generated rules is higher and the number of the rules is lower for data discretized using the CAlM algorithm when compared to data discretized using six other discretization algorithms. The highest classification accuracy was always achieved for datasets discretized using the CAlM algorithm, as compared with the other six algorithms. Of four supervised algorithms used for comparison, the CAlM algorithm is comparable in speed to the two fastest.

Discretization is indispensable in preprocessing of data analysis. Any discretization process is defined by a set of cuts [8, 16, 3, 1, 6, 4, 18, 11] over domains of attributes. Almost all existing methods do not discern between equivalent cuts i.e. cuts discerning the same pairs of objects with different decisions. Usually an "intermediate" cut is chosen as a representive of the whole family of cuts. However it is not true that in the family of indiscernible cuts the "median" cuts are the best ones. This fact can be justified using probability theory. The main goal of this paper is to propose some methods for reconstructing the obtained cut set after completing the discretization process. 1 INTRODUCTION The decision tables with real value attributes are usually discretized in the preprocesing step before some synthesis strategies for decision rules are initiated. We list some well known groups of discretization methods. Among them are: equal width and equal frequency intervals, one r...

: In previous papers we have presented several methods of searching for semi-optimal real value attributes discretization. The problem of searching for optimal discretization of the given decision table is NP \Gamma hard [15]. We have proposed also some efficient heuristic searching for the semi-optimal set of cuts in two cases, namely cuts defined by hyperplanes parallel to axes [10] and cuts defined by hyperplanes not necessarily parallel to axes [9]. In the paper we present some properties of the set of cuts determined by our methods and an approach to a semi-optimal decision tree construction. Keywords: rough set theory, boolean reasoning, discretization, oblique hyperplanes, decision tree. 1 Introduction A decision tree is optimal if it satisfies the following conditions: 1. The number of its leaves is small (this implies that the numbers of objects supporting decision rules defined by that tree should be relatively large and in the consequence these rules are strong). 2. The lo...