正态模糊集合——Fuzzy集理论的新拓展

直觉模糊集（intuitionistic fuzzy sets）、区间值模糊集（interval-valued fuzzy sets）以及Vague集对普通fuzzy集的扩展是给出了隶属度的上下限,把隶属度从[0,1]区间中的一个单值推广到了[0,1]的子区间。但是该子区间犹如一个黑洞,隶属度在其内部的分布情况我们无从知晓,即这个子区间中的每一个值是等可能地作为元素的隶属度还是区间中的某些值较另外的值有更大的可能性呢？为了清晰的刻画出元素的隶属度在[0,1]区间中的分布情况,本文通过对投票模型的分析及正态分布理论,提出了一种新的模糊集合——正态模糊集合,同时对正态模糊集合的交、并、补等基本运算性质进行了讨论,文章最后对正态模糊集与fuzzy集、直觉模糊集的相互关系也作出了详细阐述。正态模糊集合是模糊集合理论的进一步推广,为我们处理模糊信息提供了一种全新的思想方法。     

Nonstationary Fuzzy Sets

In this paper, the notion termed a “nonstationary fuzzy set” is introduced, and the concept of a perturbation function that is used for generating nonstationary fuzzy sets is presented. Definitions of the basic set operators (the union, the intersection, and the complement) for nonstationary fuzzy sets are given, together with proofs of selected properties of these operators. Two case studies were carried out in order to illustrate the use of nonstationary fuzzy sets in a nonstationary fuzzy inference, and to provide an initial insight into the relationships between nonstationary and interval type-2 fuzzy sets.      

粗糙集与直觉模糊特殊集

粗集理论和直觉模糊特殊集理论都是近年来发展起来的一种有效的信息处理理论,尤其在不确定信息处理中各有优势。本文首先讨论了直觉模糊特殊集及其算子的一些性质,并给出了其海明距离的定义和计算。同时研究了粗糙近似空间上一类粗代数结构与直觉模糊特殊集的同态关系。     

广义粗糙模糊集

粗糙集理论的主要思想是不精确的概念(被近似集)如何用可利用的知识库中的已知知识(近似空间中的可定义集全体)来近似描述[2].粗糙集理论与应用的核心基础是从近似空间导出的一对非数值型算子--上近似算子和下近似算子,并且这对算子与证据理论中的一对数值型算子--似然测度与信任测度--的关系非常密切,集合的上、下近似被看成是对该集合近似好坏的定性描述,而集合的似然测度与信任测度又可以看成是对该集合近似好坏的定量描述,甚至从一定程度上可以将粗糙集理论看成是证据理论的基础[4].     

Counting with Fuzzy Sets

The notion of fuzzy set cardinality is examined. Zadeh's suggested measure of fuzzy cardinality, the sigma-count, is adopted and shown to generalize classical counting measure. This allows many combinatorial structures and counting techniques to be fuzzified, and hence used in knowledge representation and pattern recognition models. A fuzzy set review is found in the Appendix.     

Rough Sets Model Based on Random Fuzzy Sets and Fuzzy Logic Operators

As knowledge block in knowledge base is fuzzy and obtained randomly, we propose a random fuzzy rough set model based on random fuzzy sets and fuzzy logic operators. We give some properties of the random fuzzy rough set. We investigate the relationship between fuzzy measures defined by lower approximation and upper approximation of fuzzy set and fuzzy probability measures.     

多值直觉模糊集定义

基于直觉模糊集的基本概念,考虑其隶属度与非隶属度两个因素的影响,定义了一种多值直觉模糊集,并给出了五种多值直觉模糊集的隶属度与非隶属度的综合评判准则,即算术平均法、几何平均法、去掉最大最小值算术平均法、隶属度中值法、非隶属度中值法,从而使直觉模糊得到了拓广和应用.     

模糊数决策粗糙集

考虑到实际决策问题中损失函数的不确定性特征,从贝叶斯理论出发,将模糊数损失函数引入决策粗糙集,提出模糊数决策粗糙集模型。首先,讨论在贝叶斯期望风险最小决策的语义下模糊数决策粗集理论基本模型的构建过程。其次,分析模糊数决策粗集理论的相关数学性质和准则。最后,通过一个企业信用评佑问题来阐明模糊数决策粗糙集模型的应用过程。     

模糊信息处理中Vague集向模糊集转化的一种方法

在模糊信息处理中,针对模糊信息未知度的度量,结合Fuzzy集和Vague集的特点,以投票模型为例,对Vague集向Fuzzy集转化的方法进行分析,说明均值法与比例法存在的问题,提出一种新的有效转化方法,并通过实例说明其有效性。     

Equivalent Structures of Interval Sets and Fuzzy Interval Sets

Ideas of interval sets come from the lower and upper approximations of rough sets to study a unified structure of rough sets and their generalizations. Starting from the interval sets and their operations, this paper summarizes and analyzes other sets that have similarities with the interval sets or fuzzy interval sets. Our conclusions are that interval sets are mathematically equivalent to shadowed sets and flou sets, respectively, and fuzzy interval sets are mathematically equivalent to interval‐valued fuzzy sets and intuitionistic fuzzy sets, respectively.     

Higher Order Vagueness in Geographical Information: Empirical Geographical Population of Type n Fuzzy Sets

Fuzzy set theory has been suggested as a means for representing vague spatial phenomena, and is widely known for directly addressing some of the issues of vagueness such as the sorites paradox. Higher order vagueness is widely considered a necessary component of any theory of vagueness, but it is not so well known that it too is competently modelled by Type n Fuzzy sets. In this paper we explore the fuzzy representation of higher order vagueness with respect to spatial phenomena. Initially we relate the arguments on philosophical vagueness to Type n Fuzzy sets. As an example, we move on to an empirical generation of spatial Type 2 Fuzzy sets examining the spatial extent of mountain peaks in Scotland. We show that the Type 2 Fuzzy sets can be populated by using alternative parameterisations of a peak detection algorithm. Further ambiguities could also be explored using other parameters of this and other algorithms. We show some novel answers to interrogations of the mountain peaks of Scotland. The conclusion of this work is that higher order vagueness can be populated for Type 2 and higher fuzzy sets. It does not follow that it is always necessary to examine these higher order uncertainties, but a possible advantage in terms of the results of spatial inquiry is demonstrated.

基于公理化模糊集合的模糊推理方法

以公理化模糊集合理论作为基础,把模糊推理看成两个模糊隶属空间之间的映射,利用输入模糊集合在模糊隶属空间中的构成方式,给出了模糊推理输出结果的3种基本形式.对于强否定算子、t-模算子、t-余模算子,利用Minkowski积分形式的距离讨论了这些算子在模糊隶属空间中的扰动性,并在此基础之上分析所提模糊推理方法的连续性.     

Fuzzy Sets and Neural Networks

Abstract

It is possible that a better model for the behavior of a nerve cell may be provided by what might be called a fuzzy neuron, which is a generalization of the McCulloch-Pitts model. The concept of a fuzzy neuron employs some of the concepts and techniques of the theory of fuzzy sets which was introduced by Zadeh [2, 3] and applied to the theory of automaton by Wee and Fu [6], Tanaka et al. [7], Santo [8] and others. In effect, the introduction of fuzziness into the model of a neuron makes it better adapted to the study of the behavior of systems which are imprecisely defined by virtue of their high degree of complexity. Many of the biological systems, economic systems, urban systems and more generally, large-scale systems fall into this category.

In the nearly three decades since its publication, the pioneering work of McCulloch and Pitts [1], has had a profound influence on the development of the theory of neural nets, in addition to stimulating much of the early work in automata theory and regular events.

Although the McCulloch-Pitts model of a neuron has contributed a great deal to the understanding of the behavior of neural-like systems, it fails to reflect the fact that the behavior of even the simplest type of nerve cell exhibits not only randomness but, more importantly, a type of imprecision which is associated with the lack of sharp transition from the occurrence of an event to its non-occurrence.

In this paper, some basic properties of fuzzy neural networks as well as their applications to the synthesis of fuzzy automata are investigated. It is shown that any n-state minimal fuzzy automaton can be realized by a network of m fuzzy neurons, where ┌log₂ n┐ ? m ? 2n. Examples are given to illustrate the procedure. As an example of application, a realization of λ-fuzzy language recognizer using a fuzzy neural network is presented. The techniques described in this paper may be of use in the study of neural networks as well as in formal languages, pattern recognition, and learning.     

直觉模糊集的模糊蕴含式运算方法

与经典模糊集相比,直觉模糊集具有更强的表达能力和灵活性.针对直觉模糊集的模糊推理,将经典的模糊集的模糊蕴含式拓展到直觉模糊集中,提出基于扩展二值逻辑的直觉模糊集下各种模糊蕴含式运算方法,通过实例验证直觉模糊集模糊蕴含式运算方法的有效性和正确性.     

A New Fuzzy Connectivity Measure for Fuzzy Sets

Fuzzy set theory constitutes a powerful representation framework that can lead to more robustness in problems such as image segmentation and recognition. This robustness results to some extent from the partial recovery of the continuity that is lost during digitization. In this paper we deal with connectivity measures on fuzzy sets. We show that usual fuzzy connectivity definitions have some drawbacks, and we propose a new definition that exhibits better properties, in particular in terms of continuity. This definition leads to a nested family of hyperconnections associated with a tolerance parameter. We show that corresponding connected components can be efficiently extracted using simple operations on a max-tree representation. Then we define attribute openings based on crisp or fuzzy criteria. We illustrate a potential use of these filters in a brain segmentation and recognition process.

关于模糊集的粗糙度

模糊集粗糙度的研究对于粗糙集和模糊集的理论和应用都有着重要作用。最近，Huynh等改进了由Baner—jee等给出的模糊集的粗糙度，提出了一种新的关于模糊集粗糙度的度量方法。本文对Huynh和Banerjee提出的粗糙度量进行了分析，指出了其中的不足，提出了一种新的改进方法。对Huynh度量方法存在缺陷的分析以及对新提出的改进方法性质的研究证明了新改进方法的合理性。     

CNS-HLS Mapping Using Fuzzy Sets

Existing specification systems, such as the hue/ lighness/saturation system, use triples of numbers to spec1fy color, which can be tedious to work with. The Color Naming system allows users to specify color in natural lanuage phrases or words and is a siperior way for humans to select colors. the problems addressed in this article is how to map the linguistic values of the CNS to the numerical values of the HLS system so that the CNS can be part of an integrated graphics system. The algorithm used is based on fuzzy set theory and is implemented for the Tektronix 4027 color display terminal.     

Fuzzy Sets and Social Choice

Choosing between alternative courses of action presupposes preference. Individual preference appears as a primitive notion with little need for elaboration. Mathematically, it is simply expressed by a complete (or, sometimes, only partial) reflexive antisymmetric and transitive ordering over a set (A) of ‘objects’ which are available for choice. If we wish, we can even consider the derived linear ordering over the quotient set formed by partitioning (A) into a set of ‘indifference classes.’ Even though individuals do seem to contradict some of their own previous choices at times, we do not wish to examine such refinements here. In a static context, it is not an oversimplification to say that individual choices are of the {Yes.No} (0,1) type. Can we, safely, assume that group choices are also that clear-cut? Clearly not. A cursory examination of the history of collective decisions should suffice to convince us of the fuzziness of group preferences.     

Cluster Validity with Fuzzy Sets

Abstract

Given a finite, unlabelled set of real vectors X, one often presumes the existence of (c) subsets (clusters) in X, the members of which somehow bear more similarity to each other than to members of adjoining clusters. In this paper, we use membership function matrices associated with fuzzy c-partitions of X, together with their values in the Euclidean (matrix) norm, to formulate an a posteriori method for evaluating algorithmically suggested clusterings of X. Several numerical examples are offered in support of the proposed technique.