Utilizing Genetic Algorithms to Optimize Membership Functions for Fuzzy Weighted Association Rules Mining

It is not an easy task to know a priori the most appropriate fuzzy sets that cover the domains of quantitative attributes for fuzzy association rules mining. In general, it is unrealistic that experts can always provide such sets. And finding the most appropriate fuzzy sets becomes a more complex problem when items are not considered to have equal importance and the support and confidence parameters required for the association rules mining process are specified as linguistic terms. Existing clustering based automated methods are not satisfactory because they do not consider the optimization of the discovered membership functions. In order to tackle this problem, we propose Genetic Algorithms (GAs) based clustering method, which dynamically adjusts the fuzzy sets to provide maximum profit based on user specified linguistic minimum support and confidence terms. This is achieved by tuning the base values of the membership functions for each quantitative attribute with respect to two different evaluation functions maximizing the number of large itemsets and the average of the confidence intervals of the generated rules. To the best of our knowledge, this is the first effort in this direction. Experiments conducted on 100 K transactions from the adult database of United States census in year 2000 demonstrate that the proposed clustering method exhibits good performance in terms of the number of produced large itemsets and interesting association rules.     

Importance–Performance Analysis by Fuzzy C-Means Algorithm

Traditional Importance–Performance Analysis assumes the distribution of a given set of attributes in four sets, “Keep up the good work”, “Concentrate here”, “Low priority” and “Possible overkill”, corresponding to the four possibilities, high–high, low–high, low–low and high–low, of the pair performance–importance. This can lead to ambiguities, contradictions or non-intuitive results, especially because the most real-world classes are fuzzy rather than crisp. The fuzzy clustering is an important tool to identify the structure in data, therefore we apply the Fuzzy C-Means Algorithm to obtain a fuzzy partition of a set of attributes. A membership degree of every attribute to each of the sets mentioned above is determined, against to the forcing categorization in traditional Importance–Performance Analysis. The main benefit is related with the deriving of the managerial decisions which become more refined due to the fuzzy approach. In addition, the development priorities and the directions in which the effort of an economic or non-economic entity would be useless or even dangerous are identified on a rigorous basis and taking into account only the internal structure of the input data.     

基于RPCL的模糊关联规则挖掘

在模糊关联规则的挖掘过程中,很难预先知道每个属性合适的模糊集。针对该问题,提出基于次胜者受罚竞争学习的模糊关联规则挖掘算法,无需先验知识,即可根据每个属性的性质找出对应的模糊集,并确定模糊集的数目。实验结果表明,与同类算法相比,该算法可以挖掘出更多有趣的关联规则。     

Handling linguistic web information based on a multi‐agent system

Much information over the Internet is expressed by natural languages. The management of linguistic information involves an operation of comparison and aggregation. Based on the Ordered Weighted Averaging (OWA) operator and modifying indexes of linguistic terms (their indexes are fuzzy numbers on [0,T] ? R⁺), new linguistic aggregating methods are presented and their properties are discussed. Also, based on a multi‐agent system and new linguistic aggregating methods, gathering linguistic information over the Internet is discussed. Moreover, by fixing the threshold α, “soft filtering information” is proposed and better Web pages (or documents) that the user needs are obtained. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 435–453, 2007.     

Fuzzy multi‐attribute cost–benefit analysis of e‐services

E‐service evaluation is a complex problem in which many qualitative attributes must be considered. These kinds of attributes make the evaluation process hard and vague. Cost–benefit analyses applied to various areas are usually based on the data under certainty or risk. In case of uncertain, vague, and/or linguistic data, the fuzzy set theory can be used to handle the analysis. In this article, after the evaluation attributes of e‐services and the fuzzy multi‐attribute decision‐making methods are introduced, a fuzzy hierarchical TOPSIS model is developed and applied to an e‐service provider selection problem with some sensitivity analyses. The developed model is a useful tool for the companies that prefer outsourcing for e‐activities. It is shown that service systems can be effectively evaluated by the proposed method. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 547–565, 2007.     

A method of relational fuzzy clustering based on producing feature vectors using FastMap

The first stage of organizing objects is to partition them into groups or clusters. The clustering is generally done on individual object data representing the entities such as feature vectors or on object relational data incorporated in a proximity matrix.This paper describes another method for finding a fuzzy membership matrix that provides cluster membership values for all the objects based strictly on the proximity matrix. This is generally referred to as relational data clustering. The fuzzy membership matrix is found by first finding a set of vectors that approximately have the same inter-vector Euclidian distances as the proximities that are provided. These vectors can be of very low dimension such as 5 or less. Fuzzy c-means (FCM) is then applied to these vectors to obtain a fuzzy membership matrix. In addition two-dimensional vectors are also created to provide a visual representation of the proximity matrix. This allows comparison of the result of automatic clustering to visual clustering. The method proposed here is compared to other relational clustering methods including NERFCM, Rouben’s method and Windhams A-P method. Various clustering quality indices are also calculated for doing the comparison using various proximity matrices as input. Simulations show the method to be very effective and no more computationally expensive than other relational data clustering methods. The membership matrices that are produced by the proposed method are less crisp than those produced by NERFCM and more representative of the proximity matrix that is used as input to the clustering process.     

Generalized weighted conditional fuzzy clustering

Fuzzy clustering helps to find natural vague boundaries in data. The fuzzy c-means method is one of the most popular clustering methods based on minimization of a criterion function. Among many existing modifications of this method, conditional or context-dependent c-means is the most interesting one. In this method, data vectors are clustered under conditions based on linguistic terms represented by fuzzy sets. This paper introduces a family of generalized weighted conditional fuzzy c-means clustering algorithms. This family include both the well-known fuzzy c-means method and the conditional fuzzy c-means method. Performance of the new clustering algorithm is experimentally compared with fuzzy c-means using synthetic data with outliers and the Box-Jenkins database.     

Multi-objective genetic algorithms based automated clustering for fuzzy association rules mining

Researchers realized the importance of integrating fuzziness into association rules mining in databases with binary and quantitative attributes. However, most of the earlier algorithms proposed for fuzzy association rules mining either assume that fuzzy sets are given or employ a clustering algorithm, like CURE, to decide on fuzzy sets; for both cases the number of fuzzy sets is pre-specified. In this paper, we propose an automated method to decide on the number of fuzzy sets and for the autonomous mining of both fuzzy sets and fuzzy association rules. We achieve this by developing an automated clustering method based on multi-objective Genetic Algorithms (GA); the aim of the proposed approach is to automatically cluster values of a quantitative attribute in order to obtain large number of large itemsets in less time. We compare the proposed multi-objective GA based approach with two other approaches, namely: 1) CURE-based approach, which is known as one of the most efficient clustering algorithms; 2) Chien et al. clustering approach, which is an automatic interval partition method based on variation of density. Experimental results on 100 K transactions extracted from the adult data of USA census in year 2000 showed that the proposed automated clustering method exhibits good performance over both CURE-based approach and Chien et al.’s work in terms of runtime, number of large itemsets and number of association rules.     

Boosting fuzzy rules in classification problems under single‐winner inference

In previous studies, we have shown that an Adaboost‐based fitness can be successfully combined with a Genetic Algorithm to iteratively learn fuzzy rules from examples in classification problems. Unfortunately, some restrictive constraints in the implementation of the logical connectives and the inference method were assumed. Alas, the knowledge bases Adaboost produces are only compatible with an inference based on the maximum sum of votes scheme, and they can only use the t‐norm product to model the “and” operator. This design is not optimal in terms of linguistic interpretability. Using the sum to aggregate votes allows many rules to be combined, when the class of an example is being decided. Because it can be difficult to isolate the contribution of individual rules to the knowledge base, fuzzy rules produced by Adaboost may be difficult to understand linguistically. In this point of view, single‐winner inference would be a better choice, but it implies dropping some nontrivial hypotheses. In this work we introduce our first results in the search for a boosting‐based genetic method able to learn weighted fuzzy rules that are compatible with this last inference method. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 1021–1034, 2007.     

Fuzzy classifier system architectures for mobile robotics: An experimental comparison

We present an experimental comparison between two approaches to optimization of the rules for a fuzzy controller. More specifically, the problem is autonomous acquisition of an “investigative” obstacle avoidance competency for a mobile robot. We report on results from investigating two alternative approaches to the use of a Learning Classifier System (LCS) to optimize the fuzzy rule base. One approach operates at the level of whole rule bases, the “Pittsburgh” LCS. The other approach operates at the level of individual rules, the “Michigan” LCS. In this work, both of these Fuzzy Classifier Systems were designed to operate only on the rules of fuzzy controllers, with predefined fuzzy membership functions. There are two main results from this work. First, both approaches were capable of producing fuzzy controllers with subtle interactions between rules leading to competencies exceeding that of the hand‐coded fuzzy controller presented in this article. Second, the Michigan approach suffered more seriously than the Pittsburgh approach from the well‐known LCS “cooperation/competition” problem, which is accentuated here by the structural combination of Evolutionary Computation and a fuzzy system. This problem was alleviated a little by the combination of a clustered subpopulation niche system and a fitness‐sharing scheme applied to the Michigan approach, but still remains. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 993–1019, 2007.     

Clustering categorical data sets using tabu search techniques

Clustering methods partition a set of objects into clusters such that objects in the same cluster are more similar to each other than objects in different clusters according to some defined criteria. The fuzzy k-means-type algorithm is best suited for implementing this clustering operation because of its effectiveness in clustering data sets. However, working only on numeric values limits its use because data sets often contain categorical values. In this paper, we present a tabu search based clustering algorithm, to extend the k-means paradigm to categorical domains, and domains with both numeric and categorical values. Using tabu search based techniques, our algorithm can explore the solution space beyond local optimality in order to aim at finding a global solution of the fuzzy clustering problem. It is found that the clustering results produced by the proposed algorithm are very high in accuracy.     

A model of an information retrieval system with unbalanced fuzzy linguistic information

Most information retrieval systems based on linguistic approaches use symmetrically and uniformly distributed linguistic term sets to express the weights of queries and the relevance degrees of documents. However, to improve the system–user interaction, it seems more adequate to express these linguistic weights and degrees by means of unbalanced linguistic scales, that is, linguistic term sets with different discrimination levels on both sides of the middle linguistic term. In this contribution we present an information retrieval system that accepts weighted queries whose weights are expressed using unbalanced linguistic term sets. Then, the system provides the retrieved documents classified in linguistic relevance classes assessed on unbalanced linguistic term sets. To do so, we propose a methodology to manage unbalanced linguistic information and we use the linguistic 2‐tuple model as the representation base of the unbalanced linguistic information. Additionally, the linguistic 2‐tuple model allows us to increase the number of relevance classes in the output and also to improve the performance of the information retrieval system. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 1197–1214, 2007.     

GROUP DECISION-MAKING WITH A FUZZY MAJORITY VIA LINGUISTIC QUANTIFIERS. PART II: A COMPETITIVE-LIKE POOLING

Abstract

Starting from individual fuzzy preference relations, some (sets of) socially best acceptable options are determined, directly or via a social fuzzy preference relation. An assumed fuzzy majority rule is given by a fuzzy linguistic quantifier, e.g., “most.” Here, as opposed to Part I, where we used a consensory-like pooling of individual opinions, we use an approach to linguistic quantifiers that leads to a competitive-like pooling. Some solution concepts are considered: cores, minimax (opposition) sets, consensus winners, and so forth,     

Performance evaluation of fuzzy classifier systems formultidimensional pattern classification problems

We examine the performance of a fuzzy genetics-based machine learning method for multidimensional pattern classification problems with continuous attributes. In our method, each fuzzy if-then rule is handled as an individual, and a fitness value is assigned to each rule. Thus, our method can be viewed as a classifier system. In this paper, we first describe fuzzy if-then rules and fuzzy reasoning for pattern classification problems. Then we explain a genetics-based machine learning method that automatically generates fuzzy if-then rules for pattern classification problems from numerical data. Because our method uses linguistic values with fixed membership functions as antecedent fuzzy sets, a linguistic interpretation of each fuzzy if-then rule is easily obtained. The fixed membership functions also lead to a simple implementation of our method as a computer program. The simplicity of implementation and the linguistic interpretation of the generated fuzzy if-then rules are the main characteristic features of our method. The performance of our method is evaluated by computer simulations on some well-known test problems. While our method involves no tuning mechanism of membership functions, it works very well in comparison with other classification methods such as nonfuzzy machine learning techniques and neural networks.     

基于特征属性分类改进的模糊聚类方法及其应用

模糊聚类分析是一种重要的分类方法。传统模糊聚类分析法着眼于全体属性,在对多属性数据集分类方面具有明显优势,对基于特定、重要属性的分类时显得不足。本文对传统方法进行改进,提出了一种基于特征属性分类的模糊聚类方法,利用特征属性进行分类,产生了较好的分类效果,展示了一个成用实例。改进的方法人人提高了特定分类问题的应用价值。     

The Representation of Fuzzy Knowledge

Abstract

A new AI programming language (called FUZZY) is introduced which provides a number of facilities for efficiently representing and manipulating fuzzy knowledge. A fuzzy associative net is maintained by the system, and procedures with associated “procedure demons” may be defined for the control of fuzzy processes. Such standard AI language features as a pattern-directed data access and procedure invocation mechanism and a backtrack control structure are also available.

This paper examines some general techniques for representing fuzzy knowledge in FUZZY, including the use of the associative net for the explicit representation of fuzzy sets and fuzzy relations, and the use of “deduce procedures” to implicitly define fuzzy sets, logical combinations of fuzzy sets, linguistic hedges, and fuzzy algorithms. The role of inference in a fuzzy environment is also discussed, and a technique for computing fuzzy inferences in FUZZY is examined.

The programming language FUZZY is implemented in LISP, and is currently running on a UNIVAC 1110 computer.     

针对混合型分类数据改进的[K]-modes算法距离公式

传统[K]-modes算法在分类属性聚类中有着广泛的应用,但是传统算法并不区分有序分类属性与无序分类属性。在区分这两种属性的基础上,提出了一种新的距离公式,并优化了算法流程。基于无序分类属性的距离数值,确定了有序分类属性相邻属性值之间距离数值的合理范围。借助有序分类属性蕴含的顺序关系,构建了有序分类属性的距离公式。计算样本点与质心距离之时,引入了簇内各属性值的比例作为总体距离公式的重要参数。综上,新的距离公式良好地刻画了有序分类属性的距离,并且平衡了两种不同分类属性距离公式之间的差异性。实验结果表明,提出的改进算法和距离公式在UCI真实数据集上比原始[K]-modes算法及其改进算法均有显著的效果。     

Fuzzy functions based ARX model and new fuzzy basis function models for nonlinear system identification

In this study, auto regressive with exogenous input (ARX) modeling is improved with fuzzy functions concept (FF-ARX). Fuzzy function with least squares estimation (FF-LSE) method has been recently developed and widely used with a small improvement with respect to least squares estimation method (LSE). FF-LSE is structured with only inputs and their membership values. This proposed model aims to increase the capability of the FF-LSE by widening the regression matrix with lagged input–output values. In addition, by using same idea, we proposed also two new fuzzy basis function models. In the first, basis of the fuzzy system and lagged input–output values are structured together in the regression matrix and named as “L-FBF”. Secondly, instead of using basis function, the membership values of the lagged input–output values are used in the regression matrix by using Gaussian membership functions, called “M-FBF”. Therefore, the power of the fuzzy basis function is also enhanced. For the corresponding models, antecedent part parameters for the input vectors are determined with fuzzy c-means (FCM) clustering algorithm. The consequent parameters of the all models are estimated with the LSE. The proposed models are utilized and compared for the identification of nonlinear benchmark problems.     

应用模糊集保护隐私信息

数据转换是保护数据隐私的一种有效方法。针对如何保持转换后数据的可用性问题,提出了一种基于模糊集的隐私保护方法。该方法把隐私属性值转换成模糊值,然后把转换后的数据及其模糊偏移度一起公开,既保护了数据隐私,也标示了数据的相对大小,很好地保持了数据的可用性。实验采用k-平均聚类方法对转换前后的数据进行聚类分析对比,结果表明,转换前后数据的聚类结果有很高的相似性,满足保护隐私和保持可用性的要求。     

European option pricing under fuzzy environments

The application of fuzzy sets theory to the Black–Scholes formula is proposed in this article. Owing to the vague fluctuation of financial markets from time to time, the risk‐free interest rate, volatility, and the price of underlying assets may occur imprecisely. In this case, it is natural to consider the fuzzy interest rate, fuzzy volatility, and fuzzy stock price. The form of “Resolution Identity” in fuzzy sets theory will be invoked to propose the fuzzy price of European options. Under these assumptions, the European option price at time t will turn into a fuzzy number. This will allow a financial analyst to choose the European price at his (her) acceptable degree of belief. To obtain the belief degree, the optimization problems have to be solved. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 89–102, 2005.