Random sets and large deviations principle as a foundation for possibility measures

Using the concept of selectors of random sets, we provide an interpretation for numerical degrees of possibility. The axioms (and hence the calculus of possibilities) of possibility measures are justified, in the context of random sets, on the basis that possibility distributions, as covering functions, lead to maxitive capacity functionals of random closed sets. Also, possibility measures appear as limits of probability measures in the study of large deviations principle, and as such, the idempotent operator max is justified. The problem of admissibility of possibility measures is also discussed.This work was partially carried out while H. T. Nguyen was at the University of Paris VI, October 2001, as an Invited Professor, and on the Visiting Distinguished Lukacs Professorship at Bowling Green State University, Bowling Green, Ohio, Spring 2002.     

A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread

One of the most important aspects of the (statistical) analysis of imprecise data is the usage of a suitable distance on the family of all compact, convex fuzzy sets, which is not too hard to calculate and which reflects the intuitive meaning of fuzzy sets. On the basis of expressing the metric of Bertoluzza et al. [C. Bertoluzza, N. Corral, A. Salas, On a new class of distances between fuzzy numbers, Mathware Soft Comput. 2 (1995) 71-84] in terms of the mid points and spreads of the corresponding intervals we construct new families of metrics on the family of all d-dimensional compact convex sets as well as on the family of all d-dimensional compact convex fuzzy sets. It is shown that these metrics not only fulfill many good properties, but also that they are easy to calculate and easy to manage for statistical purposes, and therefore, useful from the practical point of view.     

Strong convergence for weighted sums of fuzzy random sets

In this paper, we establish some results on strong convergence for sums and weighted sums of uniformly integrable fuzzy random sets taking values in the space of upper-semicontinuous fuzzy sets in R^p.     

犹豫模糊软集

犹豫模糊集是对模糊集的一种推广,它是一类关于域中每个元素所含隶属度的集合,常应用于群决策中,但由于其本身在参数工具上的缺乏使得难于处理不确定数据。为了提高决策的精确性,将软集与犹豫模糊集结合起来,提出犹豫模糊软集的概念,并给出犹豫模糊软集的基本运算法则和性质。     

贝叶斯数据融合中“模糊”数据随机集模型的探讨

基于随机集理论运动物体轨迹的检测、提取、识别和跟踪是比较前沿的研究课题之一;有限集统计理论的核心是多源、多目标微积分计算。随着GPS、GIS的快速发展,为GPS、GIS提出了更高的要求;怎样处理具有多源、多目标传感器＂模糊＂数据是当前工程技术人员急需解决的问题。将贝叶斯方法论进行推广,利用有限集统计理论的专家系统方法（如模糊理论、D-S证据理论）,可以使得不完整特性的传感器数据得到处理。     

Random set framework for multiple instance learning

Multiple instance learning (MIL) is a technique used for learning a target concept in the presence of noise or in a condition of uncertainty. While standard learning techniques present the learner with individual samples, MIL alternatively presents the learner with sets of samples. Although sets are the primary elements used for analysis in MIL, research in this area has focused on using standard analysis techniques. In the following, a random set framework for multiple instance learning (RSF-MIL) is proposed that can directly perform analysis on sets. The proposed method uses random sets and fuzzy measures to model the MIL problem, thus providing a more natural mathematical framework, a more general MIL solution, and a more versatile learning tool. Comparative experimental results using RSF-MIL are presented for benchmark data sets. RSF-MIL is further compared to the state-of-the-art in landmine detection using ground penetrating radar data.     

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。     

On the influence of using fuzzy extensions in linguistic fuzzy rule-based regression systems

Fuzzy Rule-Based Systems, FRBSs, are powerful tools to address regression problems. They can model the relationship between inputs and outputs by linguistic concepts. However, those FRBSs which are based on the conventional Type-1 fuzzy sets may not be able to handle some difficulties of real-world applications. In such situations, using novel representations of fuzzy sets seems like a good idea. Different extensions of fuzzy sets usually help to provide more precise models in the real-world problems. In this study, the influence of using fuzzy extensions in improving the efficiency of linguistic fuzzy rule-based regression models is investigated. For this purpose, a conventional Type-1 Mamdani FRBS is adapted to the three extensions of fuzzy sets, namely Interval Type-2, Intuitionistic, and Interval Type-2 Intuitionistic fuzzy sets. A two-pass method is proposed to define membership (non-membership) functions of these fuzzy sets; this method is based on the 3-tuples representation of the standard Type-1 membership functions. Wang and Mendel’s rule learning method is adapted to extract fuzzy rules from regression data. In order to tune the membership functions up to different extents, three evolutionary extensions are also presented for each type of the proposed FRBSs. Individual, internal, and external comparisons of the proposed FRBSs were done using 22 real-world regression datasets and statistical tests. Experimental results confirm that all the three proposed FRBSs outperform the classical Type-1 framework; furthermore, the Interval Type-2 Intuitionistic FRBS is the superior system so that an appropriate tuning of its parameters makes it the most accurate model.     

Mathematical models of uncertainty with a regard to membrane systems

A brief review is presented of the known mathematical models of uncertainty taking into account its grounds such as randomness, indiscernibility, andvagueness. Then, one discusses the models of the uncertainty caused byindiscernibility and random indiscernibility with a regard to membrane systems.The discussed models include rough sets, probabilistic rough sets, andprobabilistic fuzzy sets. An algebraic characterization of P systems ispresented, which makes possible to ``transfer' the methods of Petri net theoryto P system theory including the approach of the first theory to models ofuncertainty.     

粗糙support-intuitionistic模糊集及其聚类分析应用

粗糙集和模糊集理论已经被用于各种类型的不确定性建模中。Dubois和Prade研究了将模糊集和粗糙集结合的问题。提出了粗糙support-intuitionistic模糊集。介绍了粗糙集、粗糙直觉模糊集和support-intuitionistic模糊集等的概念;定义了在Pawlak近似空间中的support-intuitionistic模糊集的上下近似,讨论了一些粗糙support-intuitionistic模糊集近似算子的性质,给出了其相似度表达式;将其应用到聚类分析问题中,并通过一个实例验证其合理性。     

直觉模糊集的熵及其一般形式

在综合分析目前直觉模糊集熵的各种公理化定义的基础上,认为直觉模糊集合熵应该是直觉模糊集合的模糊性与不确定性的综合性度量,并且直觉模糊集合退化为模糊集合时,其熵应该与模糊集合熵是一致的。根据这种思想,给出了新的直觉模糊集合熵的公理化定义,并给出了几个直觉模糊集合熵的构造形式。     

Parameterized attribute reduction with Gaussian kernel based fuzzy rough sets

Fuzzy rough sets are considered as an effective tool to deal with uncertainty in data analysis, and fuzzy similarity relations are used in fuzzy rough sets to calculate similarity between objects. On the other hand in kernel tricks, a kernel maps data into a higher dimensional feature space where the resulting structure of the learning task is linearly separable, while the kernel is the inner product of this feature space and can also be viewed as a similarity function. It has been reported there is an overlap between family of kernels and collection of fuzzy similarity relations. This fact motivates the idea in this paper to use some kernels as fuzzy similarity relations and develop kernel based fuzzy rough sets. First, we consider Gaussian kernel and propose Gaussian kernel based fuzzy rough sets. Second we introduce parameterized attribute reduction with the derived model of fuzzy rough sets. Structures of attribute reduction are investigated and an algorithm with discernibility matrix to find all reducts is developed. Finally, a heuristic algorithm is designed to compute reducts with Gaussian kernel fuzzy rough sets. Several experiments are provided to demonstrate the effectiveness of the idea.     

Knowledge acquisition in incomplete fuzzy information systems via the rough set approach

Abstract: Machine learning can extract desired knowledge from training examples and ease the development bottleneck in building expert systems. Most learning approaches derive rules from complete and incomplete data sets. If attribute values are known as possibility distributions on the domain of the attributes, the system is called an incomplete fuzzy information system. Learning from incomplete fuzzy data sets is usually more difficult than learning from complete data sets and incomplete data sets. In this paper, we deal with the problem of producing a set of certain and possible rules from incomplete fuzzy data sets based on rough sets. The notions of lower and upper generalized fuzzy rough approximations are introduced. By using the fuzzy rough upper approximation operator, we transform each fuzzy subset of the domain of every attribute in an incomplete fuzzy information system into a fuzzy subset of the universe, from which fuzzy similarity neighbourhoods of objects in the system are derived. The fuzzy lower and upper approximations for any subset of the universe are then calculated and the knowledge hidden in the information system is unravelled and expressed in the form of decision rules.     

统计粗糙集

现有的模糊粗糙集方法,由于其基础理论复杂度的桎梏,无法应用到大规模数据集上.考虑到随机抽样是一种可以极大地减少运算量的统计学方法,本文将随机抽样引入到经典的模糊粗糙集理论中,建立了一种统计粗糙集模型.首先,我们提出了统计上、下近似的概念,它相比经典模糊粗糙集模型的优势在于, 以随机抽样得到的小容量样本代替大规模全集,从而显著降低了计算量.而且,随着全集数量增大,抽样样本数量并不会显著增大.这是本文的主要贡献.此外,我们还讨论了统计上下近似的性质,揭示统计上下近似和经典上下近似之间的关系.并且,我们提出了一个定理,该定理保证了统计下近似与经典下近似的取值统计误差在允许的范围内.最后,通过数值实验验证了统计下近似在计算时间上的显著优势.     

基于一种新的核函数的模糊粗糙集

模糊粗糙集作为模糊集与粗糙集的结合体,能够有效处理数据的复杂性和不确定性。由模糊相似关系产生的模糊粒结构可以对模糊粗糙集中不确定性的概念进行近似。核函数和模糊相似关系分别是机器学习和模糊粗糙集的核心因素,因此借助模糊相似关系和核函数之间的关系,构造了一种新的核函数,并定义了相应的核模糊粗糙集。最后通过实例说明新构造的核函数具有一定的推广性。     

基于阴影集的粗糙模糊可能性C均值聚类算法

相对于硬聚类算法,软聚类算法可以更好地表示具有不精确边界的类簇。粗糙集和模糊集均是用于描述不确定数据的有效的数学工具,二者互为补充。研究人员已经将粗糙集和模糊集的概念相结合,并应用到聚类算法中,提出了粗糙模糊可能性C均值聚类算法。而文中通过引入阴影集,有效地解决了粗糙模糊可能性C均值聚类算法中的阈值选择问题。     

Industrial applications of type-2 fuzzy sets and systems: A concise review

Data, as being the vital input of system modelling, contain dissimilar level of imprecision that necessitates different modelling approaches for proper analysis of the systems. Numbers, words and perceptions are the forms of data that has varying levels of imprecision. Existing approaches in the literature indicate that, computation of different data forms are closely linked with the level of imprecision, which the data already have. Traditional mathematical modelling techniques have been used to compute the numbers that have the least imprecision. Type-1 fuzzy sets have been used for words and type-2 fuzzy sets have been employed for perceptions where the level of imprecision is relatively high. However, in many cases it has not been easy to decide whether a solution requires a traditional approach, i.e., type-1 fuzzy approach or type-2 fuzzy approach. It has been a difficult matter to decide what types of problems really require modelling and solution either with type-1 or type-2 fuzzy approach. It is certain that, without properly distinguishing differences between the two approaches, application of type-1 and type-2 fuzzy sets and systems would probably fail to develop robust and reliable solutions for the problems of industry. In this respect, a review of the industrial applications of type-2 fuzzy sets, which are relatively novel to model imprecision has been considered in this work. The fundamental focus of the work has been based on the basic reasons of the need for type-2 fuzzy sets for the existing studies. With this purpose in mind, type-2 fuzzy sets articles have been selected from the literature using the online databases of ISI-Web of Science, ScienceDirect, SpringerLink, Informaworld, Engineering Village, Emerald and IEEE Xplore. Both the terms “type-2 fuzzy” and “application” have been searched as the main keywords in the topics of the studies to retrieve the relevant works. The analysis on the industrial applications of type-2 fuzzy sets/systems (FSs) in different topics allowed us to summarize the existing research areas and therefore it is expected be useful to prioritize future research topics. This review shows that there are still many opportunities for application of type-2 FSs for several different problem domains. Shortcomings of type-1 FSs can also be considered as an opportunity for the application of type-2 FSs in order to provide a better solution approach for industrial problems.     

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.     

A new extension of fuzzy sets using rough sets: R-fuzzy sets

This paper presents a new extension of fuzzy sets: R-fuzzy sets. The membership of an element of a R-fuzzy set is represented as a rough set. This new extension facilitates the representation of an uncertain fuzzy membership with a rough approximation. Based on our definition of R-fuzzy sets and their operations, the relationships between R-fuzzy sets and other fuzzy sets are discussed and some examples are provided.