Mining fuzzy β-certain and β-possible rules from quantitative data based on the variable precision rough-set model

Abstract

The rough-set theory proposed by Pawlak, has been widely used in dealing with data classification problems. The original rough-set model is, however, quite sensitive to noisy data. Ziarko thus proposed the variable precision rough-set model to deal with noisy data and uncertain information. This model allowed for some degree of uncertainty and misclassification in the mining process. Conventionally, the mining algorithms based on the rough-set theory identify the relationships among data using crisp attribute values; however, data with quantitative values are commonly seen in real-world applications. This paper thus deals with the problem of producing a set of fuzzy certain and fuzzy possible rules from quantitative data with a predefined tolerance degree of uncertainty and misclassification. A new method, which combines the variable precision rough-set model and the fuzzy set theory, is thus proposed to solve this problem. It first transforms each quantitative value into a fuzzy set of linguistic terms using membership functions and then calculates the fuzzy β-lower and the fuzzy β-upper approximations. The certain and possible rules are then generated based on these fuzzy approximations. These rules can then be used to classify unknown objects. The paper thus extends the existing rough-set mining approaches to process quantitative data with tolerance of noise and uncertainty.

Introduction

Machine learning and data mining techniques have recently been developed to find implicitly meaningful patterns and ease the knowledge-acquisition bottleneck. Among these approaches, deriving inference or association rules from training examples is the most common (Kodratoff and Michalski, 1983, Michalski et al., 1983, Michalski et al., 1983). Given a set of examples and counterexamples of a concept, the learning program tries to induce general rules that describe all or most of the positive training instances and none or few of the counterexamples (Hong & Tseng, 1997). If the training instances belong to more than two classes, the learning program tries to induce general rules that describe each class.

Recently, the rough-set theory has been used in reasoning and knowledge acquisition for expert systems (Grzymala-Busse, 1988, Orlowska, 1993). It was proposed by Pawlak in 1982, with the concept of equivalence classes as its basic principle. Several applications and extensions of the rough-set theory have also been proposed. Examples are Orlowska’s reasoning with incomplete information, Germano and Alexandre’s (1996) knowledge-base reduction, Lingras and Yao’s (1998) data mining, Zhong, Dong, Ohsuga, and Lin’s (1998) rule discovery. Due to the success of the rough-set theory to knowledge acquisition, many researchers in database and machine learning fields are interested in this new research topic because it offers opportunities to discover useful information in training examples.

Ziarko (1993) mentioned that the main issue in the rough-set approach was the formation of good rules. He compared the rough-set approach with some other classification approaches. The main characteristic of the rough-set approach lies in that it can use the notion of inadequacy of available information to perform classification of objects (Ziarko, 1993). It can also form an approximation space for analysis of information systems. Partial classification may be formed from the given objects. Ziarko also mentioned the limitations of the rough-set model. For example, the classification with a controlled degree of uncertainty or misclassification error is outside the realm of the approach. Overgeneralization is another limitation to the rough-set approach. Ziarko thus proposed the variable precision rough-set model to solve the above problems.

The variable precision rough-set model has however only shown how binary or crisp valued training data may be handled. Training data in real-world applications usually consist of quantitative values. Although the variable precision rough-set model can also manage the quantitative values by taking each quantitative value as an attribute value, the rules formed in this way may be too specific. It may also cause humans hard to interpret them. Extending the variable precision rough-set model to effectively dealing with quantitative values is thus important to real applications of the model.

Since the fuzzy set concepts are often used to represent quantitative data by linguistic terms and membership functions because of their simplicity and similarity to human reasoning (Graham & Jones, 1988), we thus attempt to combine the variable precision rough-set model and the fuzzy set theory to solve the above problems. The rules mined are expressed in linguistic terms, which are more natural and understandable for human beings. Since the number of linguistic terms is much less than that of possible quantitative values, the over-specialization problem can be avoided.

In Hong, Wang, and Wang (2000), we have successfully proposed a mining algorithm to find fuzzy rules based on the rough-set model. The variable precision rough-set model can be thought of as a generalization of the rough-set model. It allows for some degree of uncertainty and misclassification in the mining process. In this paper, we thus further uses the fuzzy concepts in the variable precision rough-set model to manage quantitative data. The problem to be solved in this paper is stated as follows. Given a quantitative data set with n objects, each with m attribute values, and a predefined tolerance degree β of noise and misclassification, a set of maximally general fuzzy β-certain and fuzzy β-possible rules is to be found.

A new rough-set-based fuzzy mining algorithm is then proposed to solve the above problem. Quantitative data are first transferred into fuzzy values. The proposed fuzzy mining algorithm then deals with these fuzzy values and induces certain and possible rules respectively. These rules can then be used to classify unknown objects. The applications of the variable precision rough-set model can thus be broader in this way. The paper thus extends the existing rough-set mining approaches to process quantitative data with tolerance of noise and uncertainty.

The remaining parts of this paper are organized as follows. In Section 2, the variable precision rough-set model is reviewed. In Section 3, the notation used in this paper is described. In Section 4, a fuzzy mining algorithm based on the variable precision rough-set model is proposed to induce certain and possible rules from quantitative values. In Section 5, an example is given to illustrate the proposed algorithm. Experimental results are shown in Section 6. Finally, discussion and conclusion are given in Section 7.