A new approach for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers

In this paper, we present a new method for fuzzy risk analysis based on similarity measures between generalized fuzzy numbers. First, we present a new similarity measure between generalized fuzzy numbers. It combines the concepts of geometric distance, the perimeter and the height of generalized fuzzy numbers for calculating the degree of similarity between generalized fuzzy numbers. We also prove some properties of the proposed similarity measure. We make an experiment to use 15 sets of generalized fuzzy numbers to compare the experimental results of the proposed method with the existing similarity measures. The proposed method can overcome the drawbacks of the existing similarity measures. Based on the proposed similarity measure between generalized fuzzy numbers, we present a new fuzzy risk analysis algorithm for dealing with fuzzy risk analysis problems, where the values of the evaluating items are represented by generalized fuzzy numbers. The proposed method provides a useful way to deal with fuzzy risk analysis problems.     

Fuzzy risk analysis based on similarity measures of generalized fuzzy numbers

In this paper, we present a new method for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers. Firstly, we present a method called the simple center of gravity method (SCGM) to calculate the center-of-gravity (COG) points of generalized fuzzy numbers. Then, we use the SCGM to propose a new method to measure the degree of similarity between generalized fuzzy numbers. The proposed similarity measure uses the SCGM to calculate the COG points of trapezoidal or triangular generalized fuzzy numbers and then to calculate the degree of similarity between generalized fuzzy numbers. We also prove some properties of the proposed similarity measure and use an example to compare the proposed method with the existing similarity measures. The proposed similarity measure can overcome the drawbacks of the existing methods. We also apply the proposed similarity measure to develop a new method to deal with fuzzy risk analysis problems. The proposed fuzzy risk analysis method is more flexible and more intelligent than the existing methods due to the fact that it considers the degrees of confidence of decisionmakers' opinions.     

A new approach to fuzzy distance measure and similarity measure between two generalized fuzzy numbers

In this paper a fuzzy distance measure between two generalized fuzzy numbers is developed. The metric properties of this distance measure are also studied. The new distance measure is compared with the other fuzzy distance measures proposed by Voxman [W. Voxman, Some remarks on distances between fuzzy numbers, Fuzzy Sets and Systems 100 (1998) 353–365] and Chakraborty and Chakraborty [C. Chakraborty, D. Chakraborty, A theoretical development on fuzzy distance measure for fuzzy numbers, Mathematical and Computer Modelling 43 (2006) 254–261] and turned out to be more reasonable. A new similarity measure is also developed with the help of the fuzzy distance measure. Examples are given to compare this similarity measure with the other similarity measure previously proposed. A decision making scheme is proposed using this similarity measure and this scheme is found to be more acceptable than the existing methods due to the fact that it considers the degrees of confidence of the experts’ opinion.     

Answers to queries concerning uncertain and imprecise information in fuzzy relational databases

A model of an extended fuzzy relational database was proposed to accommodate uncertain and imprecise information. We use two supplementary measurements, satisfactory degree and extra degree, for determining the quality of answers to Select‐Project‐Join (SPJ) queries. The method of measurement determines how much satisfactory information is provided and how much truth information is required for a query. The answers to the query thus contain sure answers and maybe answers. The core of this study is the detailed discussion on the quality of answers in an extended fuzzy relation to query processing. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 647–668, 2005.     

Analyzing fuzzy risk based on similarity measures between interval-valued fuzzy numbers

In this paper, we present a new method for handling fuzzy risk analysis problems based on the proposed new similarity measure between interval-valued fuzzy numbers. First, we present a new similarity measure between interval-valued fuzzy numbers. It considers the degrees of closeness between interval-valued fuzzy numbers on the X-axis and the degrees of differences between the shapes of the interval-valued fuzzy numbers on the X-axis and the Y-axis, respectively. We also prove three properties of the proposed similarity measure. Then, we make an experiment to compare the experimental results of the proposed method with the existing similarity measures between interval-valued fuzzy numbers. The proposed method can overcome the drawbacks of the existing methods. Finally, based on the proposed similarity measure between interval-valued fuzzy numbers, we present a new fuzzy risk analysis algorithm for dealing with fuzzy risk analysis problems. Because the proposed method allows the evaluating values of sub-components to be represented by interval-valued fuzzy numbers, it is more flexible than Chen and Chen’s method (2003).     

Inclusion measures,similarity measures,and the fuzziness of fuzzy sets and their relations

The inclusion measure, the similarity measure, and the fuzziness of fuzzy sets are three important measures in fuzzy set theory. In this article, we investigate the relations among inclusion measures, similarity measures, and the fuzziness of fuzzy sets, prove eight theorems that inclusion measures, similarity measures, and the fuzziness of fuzzy sets can be transformed by each other based on their axiomatic definitions, and propose some new formulas to calculate inclusion measures, similarity measures, and the fuzziness of fuzzy sets. These results can be applied in many fields, such as pattern recognition, image processing, fuzzy neural networks, fuzzy reasoning, and fuzzy control. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 639–653, 2006.     

基于指数型模糊数的多属性决策模型及其应用

针对模糊多属性决策问题,给出一种基于指数型模糊数的多属性决策模型。一方面,通过定义指数型模糊数的期望,以实现属性权重向量的解模糊化处理;另一方面,根据三元区间数理论和指数型模糊数的截集信息,定义指数型模糊数上一种新的距离度量,以计算各备选方案与正、负理想方案之间的距离。根据模糊理想点思想,基于指数型模糊数的期望和距离的定义,给出一种指数型模糊数上的Topsis多属性决策方法。将该模型应用于一个具体实例,其结果证实了该方法的有效性。     

Abductive case‐based reasoning

This article introduces abductive case‐based reasoning (CBR) and attempts to show that abductive CBR and deductive CBR can be integrated in clinical process and problem solving. Then it provides a unified formalization for integration of abduction, abductive CBR, deduction, and deductive CBR. This article also investigates abductive case retrieval and deductive case retrieval using similarity relations, fuzzy similarity relations, and similarity metrics. The proposed approach demonstrates that the integration of deductive CBR and abductive CBR is of practical significance in problem solving such as system diagnosis and analysis, and will facilitate research of abductive CBR and deductive CBR. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 957–983, 2005.     

Fuzzy entropy on intuitionistic fuzzy sets

In this article we exploit the concept of probability for defining the fuzzy entropy of intuitionistic fuzzy sets (IFSs). We then propose two families of entropy measures for IFSs and also construct the axiom definition and properties. Two definitions of entropy for IFSs proposed by Burillo and Bustince in 1996 and Szmidt and Kacprzyk in 2001 are used. The first one allows us to measure the degree of intuitionism of an IFS, whereas the second one is a nonprobabilistic‐type entropy measure with a geometric interpretation of IFSs used in comparison with our proposed entropy of IFSs in the numerical comparisons. The results show that the proposed entropy measures seem to be more reliable for presenting the degree of fuzziness of an IFS. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 443–451, 2006.     

基于包含度的直觉模糊相似度量方法

针对现有文献中涉及到直觉模糊相似度量的计算公式大多是基于直觉模糊距离测度的现状,提出一种基于包含度的直觉模糊相似度量方法。借助模糊蕴涵算子和集合基数,建立了IFS包含度的一系列具体公式,揭示了IFS包含度与相似度的关系,提出了一种基于包含度的IFS相似度量方法,并以具体算例表明该方法不仅可以解决中部分存在的问题,而且为研究IFS相似度量理论开辟一个新的途径。     

基于信息测度的勾股模糊决策模型及其应用

相对于直觉模糊集，勾股模糊集能够更为全面和有效地表达描述复杂问题中的不确定和非一致信息，使其受到了广泛研究。对于属性评价值为勾股模糊数并且属性指标权重信息数据完全未知的多属性决策问题，以提出的勾股模糊信息测度为基础，设计了新的多属性决策模型。该模型运用对数函数设计了一种新的勾股模糊数信息熵计算方法；引入了勾股模糊相似度概念，并结合对数行数提出勾股模糊数相似度的衡量方法，随后挖掘出勾股模糊数的信息熵和相似度之间的内在联系；运用提出的勾股模糊熵和相似度计算方法，构建新的多属性决策模型，并进行应用研究。实验结果表明，提出的模型合理有效，同时拓展了模型的使用范围。     

Fuzzy coherence measures

Coherence measures are a tool to compare those fuzzy sets that are sensitive to their own similarity as well as to their fuzzy nature. Within this article we can find three generalizations made about the definition of coherence measures: a first one for any fuzzy set, a second one for any definition about strong negation, and a final one for an extension in those coherence measures that, as a result, do not cause a value in the unit interval, but a fuzzy set in that interval. Tools and properties are offered to create coherence measures. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1–11, 2005.     

An improved similarity measure for generalized fuzzy numbers and its application to fuzzy risk analysis

This paper presents an improved method to compute the degree of similarity between generalized trapezoidal fuzzy numbers. The proposed similarity measure contains many features of fuzzy numbers such as geometric distance, center of gravity (COG), area, perimeter, and height. The previous methods are criticized via presenting some examples. In addition, the performance of the proposed methods is compared by the existing similarity measures using twenty different sets of generalized trapezoidal fuzzy numbers. Furthermore, the proposed method is used for fuzzy risk analysis based on similarity measures. Finally, an example is introduced to illustrate the fuzzy risk analysis.     

Fuzzy risk analysis based on similarity measures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operators

In this paper, we present a new method for fuzzy risk analysis based on a new similarity measure between interval-valued fuzzy numbers and new interval-valued fuzzy number arithmetic operators. First, we present a new similarity measure between interval-valued fuzzy numbers. The proposed similarity measure considers the similarity of the gravities on the X-axis between upper fuzzy numbers, the difference of the spreads between upper fuzzy numbers, the heights of the upper fuzzy numbers, the degree of similarity on the X-axis between interval-valued fuzzy numbers, and the gravities on the Y-axis between interval-valued fuzzy numbers. We also present three properties of the proposed similarity measure between interval-valued fuzzy numbers. Then, we present new interval-valued fuzzy number arithmetic operators. Finally, we apply the proposed similarity measure between interval-valued fuzzy numbers and the proposed interval-valued fuzzy number arithmetic operators to propose a fuzzy risk analysis algorithm to deal with fuzzy risk analysis problems. The proposed method provides a useful way for handling fuzzy risk analysis problems based on interval-valued fuzzy numbers.     

A new approach to mining fuzzy databases using nearest neighbor classification by exploiting attribute hierarchies

Data classification is a well‐organized operation in the field of data mining. This article presents an application of the k‐nearest neighbor classification technique for mining a fuzzy database. We consider a data set in which attribute values have certain similarities in nature and analyze the observations for the domain of each attribute, on the basis of fuzzy similarity relations. The proposed technique is general and the presented case study demonstrates the suitability of using this fuzzy approach for mining fuzzy databases, especially when the database contains various levels of abstraction. © 2004 Wiley Periodicals, Inc. Int J Int Syst 19: 1277–1290, 2004.     

Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications

In this paper we propose an entropy measure for interval-valued intuitionistic fuzzy sets, which generalizes three entropy measures defined independently by Szmidt, Wang and Huang, for intuitionistic fuzzy sets. We also give an approach to construct similarity measures using entropy measures for interval-valued intuitionistic fuzzy sets. In particular, the proposed entropy measure for interval-valued intuitionistic fuzzy sets can yield a similarity measure. Several illustrative examples are given to demonstrate the practicality and effectiveness of the proposed formulas. We apply the similarity measure to solve problems on pattern recognitions, multi-criteria fuzzy decision making and medical diagnosis.     

一类基于指数型模糊数的模糊多属性topsis决策

针对指数型模糊数上的模糊多属性决策问题,根据模糊理想点法的思想,给出两种多属性topsis决策方法。通过定义指数型模糊数的期望值,实现属性权重向量的解模糊化处理;定义指数型模糊数之间的距离测度,以计算各方案与理想方案之间的距离。基于期望值和距离测度的定义,从两种不同的角度出发,给出了两种模糊多属性topsis决策方法。实例验证两种方法的可行性和有效性,并对这两种方法进行比较和分析。     

Fast indexing and visualization of metric data sets usingslim-trees

Many recent database applications need to deal with similarity queries. For such applications, it is important to measure the similarity between two objects using the distance between them. Focusing on this problem, this paper proposes the slim-tree, a new dynamic tree for organizing metric data sets in pages of fixed size. The slim-tree uses the triangle inequality to prune the distance calculations that are needed to answer similarity queries over objects in metric spaces. The proposed insertion algorithm uses new policies to select the nodes where incoming objects are stored. When a node overflows, the slim-tree uses a minimal spanning tree to help with the splitting. The new insertion algorithm leads to a tree with high storage utilization and improved query performance. The slim-tree is a metric access method that tackles the problem of overlaps between nodes in metric spaces and that allows one to minimize the overlap. The proposed "fat-factor" is a way to quantify whether a given tree can be improved and also to compare two trees. We show how to use the fat-factor to achieve accurate estimates of the search performance and also how to improve the performance of a metric tree through the proposed "slim-down" algorithm. This paper also presents a new tool in the slim-tree's arsenal of resources, aimed at visualizing it. Visualization is a powerful tool for interactive data mining and for the visual tracking of the behavior of a tree under updates. Finally, we present a formula to estimate the number of disk accesses in range queries. Results from experiments with real and synthetic data sets show that the new slim-tree algorithms lead to performance improvements. These results show that the slim-tree outperforms the M-tree by up to 200% for range queries. For insertion and splitting, the minimal-spanning-tree-based algorithm achieves up to 40 times faster insertions. We observed improvements of up to 40% in range queries after applying the slim-down algorithm     

Tuning the matching function for a threshold weighting semantics in a linguistic information retrieval system

Information retrieval is an activity that attempts to produce documents that better fulfill user information needs. To achieve this activity an information retrieval system uses matching functions that specify the degree of relevance of a document with respect to a user query. Assuming linguistic‐weighted queries we present a new linguistic matching function for a threshold weighting semantics that is defined using a 2‐tuple fuzzy linguistic approach (Herrera F, Martínez L. IEEE Trans Fuzzy Syst 2000;8:746–752). This new 2‐tuple linguistic matching function can be interpreted as a tuning of that defined in “Modelling the Retrieval Process for an Information Retrieval System Using an Ordinal Fuzzy Linguistic Approach” (Herrera‐Viedma E. J Am Soc Inform Sci Technol 2001;52:460–475). We show that it simplifies the processes of computing in the retrieval activity, avoids the loss of precision in final results, and, consequently, can help to improve the users' satisfaction. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 921–937, 2005.     

Multiple similarity queries: a basic DBMS operation for mining inmetric databases

Metric databases are databases where a metric distance function is defined for pairs of database objects. In such databases, similarity queries in the form of range queries or k-nearest-neighbor queries are the most important query types. In traditional query processing, single queries are issued independently by different users. In many data mining applications, however, the database is typically explored by iteratively asking similarity queries for answers of previous similarity queries. We introduce a generic scheme for such data mining algorithms and we investigate two orthogonal approaches, reducing I/O cost as well as CPU cost, to speed-up the processing of multiple similarity queries. The proposed techniques apply to any type of similarity query and to an implementation based on an index or using a sequential scan. Parallelization yields an additional impressive speed-up. An extensive performance evaluation confirms the efficiency of our approach