John W. T. Lee

On the generalization of fuzzy rough sets

Rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty, it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets. The existing generalizations of fuzzy rough sets are all based on special fuzzy relations (fuzzy similarity relations, T-similarity relations), it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches. In this paper, from the viewpoint of constructive approach, we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them, the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined. In axiomatic approach, we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms. The lattice and topological structures of fuzzy rough sets are also proposed. In order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets, a special lower approximation operator is applied to a fuzzy reasoning system, which coincides with the Mamdani algorithm.

Attributes Reduction Using Fuzzy Rough Sets

Fuzzy rough sets are the generalization of traditional rough sets to deal with both fuzziness and vagueness in data. The existing researches on fuzzy rough sets are mainly concentrated on the construction of approximation operators. Less effort has been put on the attributes reduction of databases with fuzzy rough sets. This paper mainly focuses on the attributes reduction with fuzzy rough sets. After analyzing the previous works on attributes reduction with fuzzy rough sets, we introduce formal concepts of attributes reduction with fuzzy rough sets and completely study the structure of attributes reduction. An algorithm using discernibility matrix to compute all the attributes reductions is developed. Based on these lines of thought, we set up a solid mathematical foundation for attributes reduction with fuzzy rough sets. The experimental results show that the idea in this paper is feasible and valid.

Refinement of generated fuzzy production rules by using a fuzzy neural network

Fuzzy production rules (FPRs) have been used for years to capture and represent fuzzy, vague, imprecise and uncertain domain knowledge in many fuzzy systems. There have been a lot of researches on how to generate or obtain FPRs. There exist two methods to obtain FPRs. One is by painstakingly, repeatedly and time-consuming interviewing domain experts to extract the domain knowledge. The other is by using some machine learning techniques to generate and extract FPRs from some training samples. These extracted rules, however, are found to be nonoptimal and sometimes redundant. Furthermore, these generated rules suffer from the problem of low accuracy of classifying or recognizing unseen examples. The reasons for having these problems are: 1) the FPRs generated are not powerful enough to represent the domain knowledge, 2) the techniques used to generate FPRs are pre-matured, ad-hoc or may not be suitable for the problem, and 3) further refinement of the extracted rules has not been done. In this paper we look into the solutions of the above problems by 1) enhancing the representation power of FPRs by including local and global weights, 2) developing a fuzzy neural network (FNN) with enhanced learning algorithm, and 3) using this FNN to refine the local and global weights of FPRs. By experimenting our method with some existing benchmark examples, the proposed method is found to have high accuracy in classifying unseen samples without increasing the number of the FPRs extracted and the time required to consult with domain experts is greatly reduced.

Rule Induction Based on Fuzzy Rough Sets

In this paper, we propose one method of rule induction based on fuzzy rough set. First, the consistence degree is proposed as the basic concept to induce rules based on fuzzy rough sets. The concepts of rule induction, such as value reduct, reduct rule and so on, are then proposed based on the definition of consistence degree. Second, a discernibility array is constructed, and then an algorithm to find the reduct rule using the discernibility array is designed. Finally, the numerical experimental results demonstrate that the method of rule induction proposed in this paper is feasible. The key idea of this paper is that the value reduct (i.e. reduct rule) keeps the consistence degree invariant. The main contribution of this paper is introduction of rule induction based on fuzzy rough sets using the concept of fuzzy lower and upper approximation.

Information retrieval based on semantic query on RDF annotated resources

The development of the semantic Web, where Web resources would be semantically annotated based on RDF and shared ontology, provides an opportunity for more effective meaning-based search and retrieval of information available through the Web. In such environment, a free text query can be posted and resolved into its semantics, which direct the information retrieval. One important factor in the performance of such semantic query is finding the right interpretation for the query words (i.e. word sense disambiguation). In this paper, we examine the performance of a new approach to such semantic query in the context of semantic Web.

Learning capability in fuzzy Petri nets

Petri nets (PNs) have been widely used in modeling and analyzing many real applications such as computers, automatic control and management information systems, etc. The power of PNs comes from their ability to model and analyze the behaviors and states of systems (events) concurrently. Neural networks (NNs), on the other hand, were developed to handle and solve many linear and nonlinear complex problems by forming an association (relationship) between its input and output training patterns. It will be advantageous if a learning capability is incorporated into a fuzzy Petri net (FPN) which has the capability of both systems. In this paper, a FPN model which has learning capability is proposed. The purpose of including a learning facility in FPNs is that many parameters of a fuzzy expert system, included in fuzzy production rules (FPRs), once when it has been modeled by a FPN could be tuned. These parameters, including membership values, weights (local and global) and certainty factors etc., play important roles in capturing and representing complex domain expert knowledge. By comparing the artificial neural networks (ANN) with FPNs having learning capability, we have advantages such as: a) FPNs provide a transparent modeling and analyzing capability whereas ANN provides a black-box learning and no-analysis capability; b) FPNs representing a fuzzy expert system could be used to analyze the different inference states step-by-step; c) FPNs could tune parameters in a fuzzy expert system so that the overall system performance is improved.

Induction of ordinal decision trees

In many disciplines, such as social and behavioral sciences, we often have to do ordinal classification by assigning objects to ordinal classes. The fundamental objective of ordinal classification is to create an ordering in the universe of discourse. As such, a decision tree for ordinal classification should aim at producing an ordering which is most consistent with the implicit ordering in the input data. Ordinal classification problems are often dealt with by treating ordinal classes as nominal classes, or by representing the classes as values on a quantitative scale. Such approaches may not lead to the most desirable results since the methods do not fit the type of data, viz. ordinal data, concerned. In this paper, we propose a new measure for assessing the quality of output from an ordinal classification approach. We also propose an induction method to generate an ordinal decision tree for ordinal classification based on this quality perspective. We demonstrate the advantage of our method using results from a set of experiments.

Hierarchical clustering based on ordinal consistency

Hierarchical clustering is the grouping of objects of interest according to their similarity into a hierarchy, with different levels reflecting the degree of inter-object resemblance. It is an important area in data analysis and pattern recognition. In this paper, we propose a new approach for robust hierarchical clustering based on possibly incomplete and noisy similarity data. Our approach uses a novel perspective in finding the object hierarchy by trying to optimize ordinal consistency between the available similarity data and the hierarchical structure. Using experiments we show that our approach is able to perform more effectively than similar algorithms when there are substantial noises in the data. Furthermore, when similarity-ordering information is only available in the form of incomplete pairwise similarity comparisons, our approach can still be applied directly. We illustrate this by applying our approach to randomly generated hierarchies and phylogenetic tree construction from quartets, an important area in computational biology.

Tuning certainty factor and local weight of fuzzy production rules by using fuzzy neural network

Approximate reasoning in a fuzzy system is concerned with inferring an approximate conclusion from fuzzy and vague inputs. There are many ways in which different forms of conclusions can be drawn. Fuzzy sets are usually represented by fuzzy membership functions. These membership functions are assumed to have a clearly defined base. For other fuzzy sets such as intelligent, smart, or beautiful, etc., it would be difficult to define clearly its base because its base may consist of several other fuzzy sets or unclear nonfuzzy bases. A method to handle this kind of fuzzy set is proposed. A fuzzy neural network (FNN) is also proposed to tune knowledge representation parameters (KRPs). The contributions are that we are able to handle a broader range of fuzzy sets and build more powerful fuzzy systems so that the conclusions drawn are more meaningful, reliable, and accurate. An experiment is presented to demonstrate how our method works.

Ordinal decomposability and fuzzy connectives

The benefit of computing with linguistic terms is now generally accepted. Fuzzy set theory provides us the conceptual tool for the interpretation and evaluation of linguistic concepts and expressions. It constitutes a quantification of the compatibility degree of objects with the associated linguistic concept through a membership function. When we make computation using fuzzy membership values such as in the evaluation of fuzzy rules confidence, the implicit assumptions are that the membership values have quantitative semantics (the extensive scale assumption) and that the numeric values are commensurate among the different fuzzy sets generated by the different concepts involved (the common scale assumption). In most situations these assumptions are difficult to justify and may lead to various anomalies. The membership values are more suitably interpreted only as ordinal scales where the numeric representations reflect compatibility orderings. In this paper, we examine the concept of fuzzy intersection and union from the perspective of decomposability and ordinal conjoint structure in measurement theory. We determine conditions under which a weak order, induced by a fuzzy set or otherwise, can be decomposed into other weak orders. We show particular cases of ordinal decomposability which correspond naturally to our concept of fuzzy intersection and union. This perspective of fuzzy connectives help us resolve some of the difficulties related to the above assumptions.