A fuzzy similarity-based rough set approach for attribute selection in set-valued information systems
一种基于模糊相似性的粗糙集方法，用于集合值信息系统中的属性选择

Published online: 23 July 2019
网络发布日期：2019 年 7 月 23 日

Springer-Verlag GmbH Germany, part of Springer Nature 2019
德国 Springer-VerlagGmbH，Springer Nature2019 的一部分

Abstract
抽象

Databases obtained from different search engines, market data, patients’ symptoms and behaviours, etc., are some common examples of set-valued data, in which a set of values are correlated with a single entity. In real-world data deluge, various irrelevant attributes lower the ability of experts both in speed and in predictive accuracy due to high dimension and insignificant information, respectively. Attribute selection is the concept of selecting those attributes that ideally are necessary as well as sufficient to better describe the target knowledge. Rough set-based approaches can handle uncertainty available in the real-valued information systems after the discretization process. In this paper, we introduce a novel approach for attribute selection in set-valued information system based on tolerance rough set theory. The fuzzy tolerance relation between two objects using a similarity threshold is defined. We find reducts based on the degree of dependency method for selecting best subsets of attributes in order to obtain higher knowledge from the information system. Analogous results of rough set theory are established in case of the proposed method for validation. Moreover, we present a greedy algorithm along with some illustrative examples to clearly demonstrate our approach without checking for each pair of attributes in set-valued decision systems. Examples for calculating reduct of an incomplete information system are also given by using the proposed approach. Comparisons are performed between the proposed approach and fuzzy rough- assisted attribute selection on a real benchmark dataset as well as with three existing approaches for attribute selection on six real benchmark datasets to show the supremacy of proposed work.
从不同搜索引擎获得的数据库、市场数据、患者的症状和行为等，是集值数据的一些常见例子，其中一组值与单个实体相关联。在现实世界的数据洪流中，由于高维度和不重要的信息，各种不相关的属性分别降低了专家在速度和预测准确性方面的能力。属性选择是选择那些理想情况下是必要且足以更好地描述目标知识的属性的概念。基于粗糙集合的方法可以处理离散化过程后实值信息系统中可用的不确定性。在本文中，我们介绍了一种基于公差粗糙集理论的集合值信息系统中的属性选择新方法。使用相似性阈值定义两个对象之间的模糊容差关系。我们根据依赖程度方法找到归约，用于选择最佳属性子集，以便从信息系统获得更高的知识。在所提出的验证方法的情况下，建立了粗糙集理论的类似结果。此外，我们提出了一个贪婪算法以及一些说明性示例，以清楚地演示我们的方法，而无需检查设定值决策系统中的每一对属性。使用所提出的方法还给出了计算不完全信息系统还原的示例。 在真实基准数据集上对所提出的方法和模糊粗略辅助属性选择进行了比较，并在六个真实基准数据集上与三种现有的属性选择方法进行了比较，以表明所提出工作的至高无上。

Keywords Set-valued data Rough set Fuzzy tolerance relation Degree of dependency Attribute selection
关键字设置值数据粗糙集模糊容忍关系依赖度属性选择

Introduction
介绍

Shivani Singh shivanithakur030@gmail.com
希瓦尼·辛格 shivanithakur030@gmail.com

Tanmoy Som tsom.apm@itbhu.ac.in

Gaurav Somani gaurav.somani.mat14@itbhu.ac.in

¹ DST-Centre for Interdisciplinary Mathematical Sciences, Institute of Science, BHU, Varanasi 221005, India
¹ 印度瓦拉纳西 221005 BHU 科学研究所 DST-跨学科数学科学中心

² Department of Mathematical Sciences, IIT (BHU), Varanasi 221005, India
² 印度理工学院 （BHU） 数学科学系，印度瓦拉纳西221005

Many real applications in the area of machine learning and data mining consist of set-valued data, i.e. the data where the attribute value of an object is not unique but a set of values, for example, in a venture investment company, the set of evaluation results given by expert (Qian et al. 2010b), the set of languages for each person from the foreign language ability test (Qian et al. 2009), and in a medical database, the set of patients symptoms and activ- ities (He and Naughton 2009), etc. These kinds of infor- mation systems are called set-valued information systems, which are another important type of data tables and are generalized models of single-valued information systems. An incomplete information system (Kryszkiewicz 1998, 1999; Dai 2013; Dai and Xu 2012; Dai et al. 2013; Leung and Li 2003; Yang et al. 2011, 2012) can be con- verted into a set-valued information system by replacing all missing values with the set of all possible values of each
机器学习和数据挖掘领域的许多实际应用都是由定值数据组成的，即对象的属性值不唯一而是一组值的数据，例如，在一家风险投资公司中，专家给出的一组评估结果（Qian et al.2010b），外语能力测试中每个人的语言集（Qian et al. 2009 年），以及在医学数据库中，患者的症状和活动集（He 和 Naughton 2009 年）等。这类信息系统被称为 集值信息系统，它是另一种重要的数据表类型，是单值信息系统的泛化模型。不完整的信息系统（Kryszkiewicz 1998， 1999;Dai 2013;Dai 和 Xu 2012;Dai 等人。 2013 年;Leung 和 Li 2003;Yang 等人。 2011， 2012） 可以通过将所有缺失值替换为每个值的所有可能值的集合来转换为一个集值信息系统

attribute. Set-valued information systems are always used to portray the inexact and lost information in a given dataset, in which the attribute set may vary with time as new information is added.
属性。集合值信息系统始终用于描绘给定数据集中的不精确和丢失信息，其中属性集可能会随着新信息的添加而随时间而变化。

Dataset dimensionality is the main hurdle for the com- putational application in pattern recognition and other machine learning tasks. In many real-world applications, the generation and expansion of data occur continuously and thousands of attributes are stored in databases. Gathering useful information and mining-required knowl- edge from an information system is the most difficult task in the area of knowledge-based system. Not all attributes are relevant to the learning tasks as they reduce the real performance of proposed algorithms and increase the training and testing times. In order to enhance the classi- fication accuracy and knowledge prediction, attribute sub- set selection (Hu et al. 2008; Yang and Li 2010; Qian et al. 2010a; Qian et al. 2011; Shi et al. 2011; Jensen and Shen 2009; Jensen et al. 2009) plays a key role via eliminating redundant and inconsistent attributes. Attribute selection is the process of selecting the most informative attributes of a given information system to reduce the classification time, complexity and overfitting.
数据集维数是模式识别和其他机器学习任务中计算应用的主要障碍。在许多实际应用程序中，数据的生成和扩展是连续发生的，并且数据库中存储了数千个属性。从信息系统中收集有用的信息和挖掘所需的知识是知识系统领域中最困难的任务。并非所有属性都与学习任务相关，因为它们会降低所建议算法的实际性能并增加训练和测试时间。为了提高分类准确性和知识预测，属性子集选择（胡等人。 2008Yang 和Li 2010Qian et al. 2010 年a;Qian 等人。 2011 年;Shi et al. 2011 年;Jensen 和 Shen 2009;Jensen 等人。 2009 年）通过消除冗余和不一致的属性发挥了关键作用。属性选择是选择给定信息系统信息量最大的属性以减少分类时间、复杂性和过度拟合的过程。

Rough set approximations (proposed by Pawlak 1991 and Pawlak and Skowron 2007a, b, c) are the central point of approaches to knowledge discovery. Rough set theory (RST) uses only internal information and does not depend on prior model conventions, which can be used to extract and signify the hidden knowledge available in the infor- mation systems. It has many applications in the fields of decision support, document analysis, data mining, pattern recognition, knowledge discovery and so on. In rough sets, several discrete partitions are needed in order to tackle real-valued attributes and then dependency of decision attribute over conditional attributes is calculated. The intrinsic error due to this discretization process is the main issue while computing the degree of dependency of real- valued attributes.
粗略集近似值（由 Pawlak 1991和 Pawlak 和 Skowron 2007a、b、c 提出）是知识发现方法的中心点。粗糙集论 （RST） 仅使用内部信息，不依赖于先前的模型约定，这些信息可用于提取和表示信息系统中可用的隐藏知识。它在决策支持、文档分析、数据挖掘、模式识别、知识发现等领域有许多应用。在粗略集中，需要几个离散分区来处理实值属性，然后计算决策属性对条件属性的依赖性。在计算实值属性的依赖程度时，由于这种离散化过程导致的内在误差是主要问题。

Dubois and Prade (1992) combines a fuzzy set (Zadeh 1996) with rough set and proposed a fuzzy rough set to provide an important tool in reasoning with uncertainty for real-valued datasets. Fuzzy rough sets combine distinct concepts of indiscernibility (for rough sets) and vagueness (for fuzzy sets) available in the datasets and successfully applied to many fields. However, very few researchers are working in the area of set-valued information systems under the framework of rough set model in fuzzy environment.
Dubois 和 Prade （1992） 将模糊集 （Zadeh 1996） 与粗糙集相结合，并提出了模糊粗糙集，为实值数据集的不确定性推理提供了重要工具。模糊粗糙集结合了数据集中可用的不可辨别性（对于粗糙集）和模糊性（对于模糊集）的不同概念，并成功应用于许多领域。然而，很少有研究人员在模糊环境下的粗糙集模型框架下从事集值信息系统领域的工作。

In this paper, we introduce a novel approach for attribute selection in set-valued information system based on toler- ance rough set theory. We define a fuzzy relation between two objects of a set-valued information system. A fuzzy tolerance relation is introduced by using a similarity
在本文中，我们介绍了一种基于容差粗糙集理论的集合值信息系统中的属性选择新方法。我们定义了一个集合值信息系统的两个对象之间的模糊关系。通过使用相似度引入模糊容差关系

threshold to avoid misclassification and perturbation in order to tackle uncertainty in a much better way. Based on this relation, we calculate tolerance classes of each object to determine lower and upper approximations of any subset of the universe of discourse. Positive region of decision attribute over a subset of conditional attributes can be calculated using lower approximations. Degree of depen- dency of decision attribute over a subset of conditional attributes is the ratio of cardinality of positive region and cardinality of the universe of discourse. Analogous results of rough set theory are established in case of our proposed method for validation. Moreover, we present a greedy algorithm to clearly demonstrate our approach without calculating degree of dependencies for each pair of attri- butes. Illustrative example datasets are given for better understanding of our proposed approach. We compare the proposed approach with other existing approaches on real datasets and test the statistical significance of the obtained results.
threshold 来避免错误分类和扰动，以便以更好的方式解决不确定性。基于这种关系，我们计算每个对象的容忍等级，以确定话语宇宙的任何子集的下限和上限近似值。可以使用较低的近似值来计算条件属性子集上的决策属性的正区域。决策属性对条件属性子集的依赖程度是正区域基数与话语宇宙基数的比率。在我们提出的验证方法的情况下，建立了粗糙集理论的类似结果。此外，我们提出了一种贪婪算法来清楚地演示我们的方法，而无需计算每对属性的依赖程度。给出了说明性示例数据集，以便更好地理解我们提出的方法。我们在真实数据集上将所提出的方法与其他现有方法进行比较，并测试所获结果的统计意义。

The rest of the paper is structured as follows. Related works are given in Sect. 2. In Sect. 3, basic definitions related to incomplete and set-valued information systems are given. The proposed concept for set-valued datasets is presented and thoroughly studied in Sect. 4. In Sect. 5, analogous results of rough set theory are verified for the new proposed approach. An algorithm for attribute selec- tion of set-valued information system is presented in Sect. 6. Illustrative examples with comparative analysis are given to demonstrate proposed model in Sect. 7. In Sect. 8, experimental analysis is performed on six real benchmark datasets. Section 9 concludes our work.
本文的其余部分结构如下。相关著作在 Sect.2.在第3、给出了与不完备和集值信息系统相关的基本定义。在 Sect 中提出并深入研究了集值数据集的拟议概念。4.在第5，为新提出的方法验证了粗糙集理论的类似结果。Sect. 中介绍了一种用于设置值信息系统属性选择的算法。 6给出了带有比较分析的说明性示例，以演示Sect. 7在第 7 节中。 8、对 6 个 Real Benchmark 数据集进行实验分析。第 9 节结束了我们的工作。

Related works
相关作品

Nowadays, set-valued datasets are generated through many sources. Dimensionality reduction is a key issue for such type of datasets in order to reduce complexity, time and cost. Different criteria have been proposed by a few researchers to deal with set-valued datasets and to evaluate the best suitable attributes in the process of attribute selection. Lipski (1979, 1981) gave the idea of representing an incomplete information system as a set-valued infor- mation system and studied their basic properties. He also investigated the semantic and logical problems often occur in an incomplete information system. Concepts of internal and external interpretations are introduced in the paper. Internal interpretation is shown to lead towards the notion of topological Boolean algebra and a modal logic, whereas external interpretation is related to referring queries directly to reality leads towards Boolean algebra and classical logic. Orlowska and Pawlak (1984) established a method to deal with non-deterministic information system
如今，集值数据集是通过许多来源生成的。降维是此类数据集的一个关键问题，以降低复杂性、时间和成本。一些研究人员提出了不同的标准来处理设定值数据集，并在属性选择过程中评估最合适的属性。Lipski（1979， 1981） 提出了将不完备信息系统表示为固定值信息系统的想法，并研究了它们的基本性质。他还研究了不完整信息系统中经常出现的语义和逻辑问题。本文介绍了内部和外部解释的概念。内部解释被证明会导致拓扑布尔代数和模态逻辑的概念，而外部解释与将查询直接引用到现实有关，导致布尔代数和经典逻辑。Orlowska 和 Pawlak （1984） 建立了一种处理非确定性信息系统的方法

which is considered as set-valued data. They defined a language in order to define non-deterministic information and introduced the concept of knowledge representation system.
被视为集值数据。他们定义了一种语言来定义非确定性信息，并引入了知识表示系统的概念。

A generalized decision logic, which is an extension of decision logic studied by Pawlak, in interval set-valued information system is presented by Yao and Liu (1999). They introduced two types of satisfiabilities of a formula, namely interval degree truth and interval-level truth. They also proposed generalized decision logic DGL and inter- preted this concept based on two types of satisfiabilities. A detailed discussion on inference rules is also presented. Yao (2001) presented a concept of granulation for a uni- verse of discourse in set-valued information systems and reviewed the corresponding approximation structure. The concept of ordered granulation and approximation struc- tures are used in defining stratified rough set approxima- tions. He first defined a nested sequence of granulations and then corresponding nested sequence of rough set approximations, which leads to a more general approxi- mation structure.
Yao 和 Liu （1999） 提出了一种广义决策逻辑，它是 Pawlak 研究的决策逻辑的扩展，在区间集值信息系统中。他们引入了公式的两种满足性，即区间度真值和区间级真值。他们还提出了广义决策逻辑 DGL，并根据两种类型的满足性对这一概念进行了解释。此外，还详细讨论了推理规则。Yao （2001） 提出了一个用于集合值信息系统中单宇宙话语的颗粒化概念，并回顾了相应的近似结构。有序粒化和近似结构的概念用于定义分层粗集近似值。他首先定义了一个嵌套的颗粒序列，然后定义了相应的粗略集近似的嵌套序列，这导致了更通用的近似结构。

Shoemaker and Ruiz (2003) introduced an extension of apriori algorithm that is able to mine association rules from a set-valued data. They introduced two different algorithms for mining association rules from set-valued data and compared their outcomes. They established a system based on one of these algorithms and applied it on some bio- logical datasets for justification. Set-valued information systems were presented by Guan and Wang (2006). To classify the universe of discourse, they proposed a toler- ance relation and used maximal tolerance classes. They introduced the concept of relative reduct of maximum tolerance classes and used Boolean reasoning technique for calculating relative reduct by defining a discernibility function. The concepts of E-lower, A-upper and A-lower relative reducts for set-valued decision systems are also discussed in details.
Shoemaker 和 Ruiz （2003） 引入了 apriori 算法的扩展，该算法能够从集合值数据中挖掘关联规则。他们引入了两种不同的算法，用于从设定值数据中挖掘关联规则，并比较了它们的结果。他们基于其中一种算法建立了一个系统，并将其应用于一些生物数据集以进行合理性验证。集合值信息系统由 Guan 和 Wang （2006） 提出。为了对话语世界进行分类，他们提出了一种容忍关系并使用了最大容忍类别。他们引入了最大公差等级的相对还原的概念，并使用布尔推理技术通过定义可辨别函数来计算相对还原。还详细讨论了设定值决策系统的 E-lower、A-upper 和 A-lower 相对还原的概念。

For the conjunctive/disjunctive type of set-valued ordered information systems, a dominance-based rough set approach was introduced by Qian et al. (2010b). This model is based on substitution of indiscernibility relation by a dominance relation. They also developed a new approach to sorting for objects in disjunctive set-valued ordered information systems. This approach is useful in simplifying a disjunctive set-valued ordered information system. Criterion reduction for a set-valued ordered information system is also discussed. Based on variable precision relation, Yang et al. (2010) generalized the notion of Qian et al. by defining an extended rough set model and propounded variable precision dominance relation for set- valued ordered information systems. They presented an attribute reduction method based on the discenibility matrix approach by using their proposed relation. 

Zhang et al. (2012) proposed matrix approaches based on rough set theory with dynamic variation of attributes in set-valued information systems. In this paper, they defined the lower and upper approximations directly by using basic vector generated by the relation matrix in the set-valued information system. The concept of updation of the lower and upper approximations is also introduced by use of the variation of the relation matrix. Luo et al. (2013) investi- gated the updating mechanisms for computing lower and upper approximations with the variation of the object set. Authors proposed two incremental algorithms for the updation of the defined approximations in disjunc- tive/conjunctive set-valued information systems. After experiments on several datasets for checking the perfor- mance of the proposed algorithms, they showed that the incremental approaches are way better than the non-in- cremental approaches. 

Wang et al. (2013) defined a new fuzzy preference relation and fuzzy rough set technique for disjunctive-type interval and set-valued information systems. They dis- cussed the concept of relative significance measure of conditional attributes in interval and set-valued decision systems by using the degree of dependency approach. In this paper, authors mainly focused on semantic interpre- tation of disjunctive type only. They also presented an algorithm for calculating fuzzy positive region in interval and set-valued decision systems. 

An incremental algorithm was designed to reduce the size of dynamic set-valued information systems by Lang et al. (2014). They presented three different relations and investigated their basic properties. Two types of discerni- bility matrices based on these relations for set-valued decision systems are also introduced. Furthermore, using the proposed relations and information system homomor- phisms, a large-scale set-valued information system is compressed into a smaller information system. They addressed the compression updating via variations of fea- ture set, immigration and emigration of objects and alter- ations of attribute values. 

In set-valued ordered decision systems, Luo et al. (2014) worked on maintaining approximations dynamically and studied the approximations of decision classes by defining the dominant and dominated matrices via a dominance relation. The updating properties for dynamic maintenance of approximations were also introduced, when the evolu- tion of the criteria values with time occurs in the set-valued decision system. Firstly, they constructed a matrix-based approach for computing lower and upper approximations of upward and downward unions of decision classes. Fur- thermore, incremental approaches for updating approxi- mations are presented by modifying relevant matrices without retraining from the start on all accumulated train- ing data. 

Shu and Qian (2014) presented an attribute selection method for set-valued data based on mutual information of the unmarked objects. Mutual information-based feature selection methods use the concept of dependency among features. Unlike the traditional approaches, here the mutual information is calculated on the unmarked objects in the set-valued data. Furthermore, mutual information-based feature selection algorithm is developed and implemented on an universe of discourse to fasten the feature selection process. Due to the dynamic variation of criteria values in the set-valued information systems, Luo et al. (2015) pre- sented the properties for dynamic maintenance of approx- imations. Two incremental algorithms for modernizing the approximations in disjunctive/conjunctive set-valued information system are presented corresponding to the addition and removal of criteria values, respectively. 

Most of the above approaches are based on classical and rough set techniques, which have their own limitations of discretization, which leads to information loss. Rough set in fuzzy environment-based methods deal with uncertainty as well as noise available in information system in a much better way as compared to classical and rough set-based approaches without requirement of any discretization pro- cess. Dai et al. (2013) defined a fuzzy relation between two objects and constructed a fuzzy rough set model for attri- bute reduction in set-valued information systems based on discernibility matrices. In this paper, the similarity of two objects in set-valued information system is taken up to a threshold value in order to avoid misclassification and perturbation and a tolerance rough set-based attribute selection is presented by using degree of dependency approach. 

Preliminaries 

In this section, we describe some basic concepts, symbol- ization and examples of set-valued information system. 

AT and h : U AT ! V is an information function that assigns particular values to the objects against attribute set such that 8a 2 AT,8u 2 U; hðu; aÞ 2 V_a
AT和h ： UAT ！V是一个信息函数，它根据属性集为对象分配特定值，以便 8a 2 AT，8 u 2 U;hðu;aþ 2 伏_a.

Definition 3.2 (Huang 1992): In an information system, if each attribute has a single entity as attribute value, then it is called single-valued information system, otherwise it is known as set-valued information system. Set-valued
定义3.2（Huang1992）：在一个信息系统中，如果每个属性都有一个实体作为属性值，那么它被称为单值信息系统，否则称为集合值信息系统。Set-value （设置值）

information system is a generalization of the single-valued information system, in which an object can have more than one attribute values. Table 1 illustrates a set-valued infor- mation system.
信息系统是单值信息系统的泛化，其中对象可以具有多个属性值。表 1说明了一个固定值信息系统。

Definition 3.3 (Guan and Wang 2006): An information system IS ¼ ðU; AT; V ; hÞ is said to be a set-valued deci- sion system if AT ¼ C [ D where C is a non-empty finite set of conditional attributes and D is a non-empty collec- tion of decision attributes with C \ D ¼ ;. Here V ¼ V_C [ V_D with V_C and V_D as the set of conditional attribute values and decision attribute values, respectively. h be a mapping from U C [ Dto V such that h : U C ! 2^VC is a set-valued mapping and h : U C ! V_D is a single- valued mapping. Table 2 exemplifies a set-valued decision system.
定义 3.3（Guan 和 Wang 2006）：信息系统IS1/4ðU;AT;五;如果 AT 1/4C[D，则称 hÞ为集值决定系统，其中 C 是非空的有限条件属性集，D 是 C\D 为 1/4 的非空决策属性集合;这里V1/4 V_C[V_D，其中 V_C和 V_D分别作为条件属性值和决策属性值的集合。 h 是从UC [D到V 的映射，使得h：UC！2^VC是固定值映射，h ： UC ！V_D是单值映射。表 2举例说明了一个设定值决策系统。

To give a semantic interpretation of the set-valued data, many ways are given (Guan and Wang 2006), here we encapsulate them as two types. In Type1, hðu; aÞ is inter- preted conjunctively, and in Type2, hðu; aÞ is interpreted disjunctively. For example, if a is an attribute, ‘‘speaking a language’’, then hðu; aÞ = {Chinese, Spanish, English} can be inferred as: u speaks Chinese, Spanish and English in case of Type1 and u speaks Chinese or Spanish or English,
为了给出集合值数据的语义解释，给出了许多方法（Guan 和 Wang 2006），这里我们将它们封装为两种类型。在 Type1 中，hðu;aÞ 是连词，在 Type2 中，hðu;aÞ 是析取解释的。例如，如果a是一个属性，''说一种语言''，那么hðu;aÞ={中文，西班牙语，英文}可以推断为：在 Type1 的情况下，u 会说中文、西班牙语和英语，而u会说中文、西班牙语或英语，

i.e. u can speak only one of them in case of Type2. Incomplete information systems with some unknown attribute values or partially known attribute values are of Type2 set-valued information system.
即， 在 Type2 的情况下，您只能说其中一种。具有一些未知属性值或部分已知属性值的不完整信息系统属于 Type2 集值信息系统。

In many real-world application problems, lots of missing data existed in the information system due to ambiguity and incompleteness. All missing values presented in
在许多实际应用问题中，由于模糊性和不完整性，信息系统中存在大量缺失数据。中显示的所有缺失值

Table 2 Set-valued decision system
表2设置值决策系统

For B AT, a tolerance relation is defined as
对于BAT，公差关系定义为

2 _ \ _

T_B ¼ u_i; u_j jbðu_iÞ \ b u_j 6¼ /; 8b 2 B ¼ T_b ð2Þ
T_B1/4uubðuÞbu61/4/8b2B1/4T_bð2Þ

4

5

information system can be characterized by the set of all possible values of each attribute. This type of information systems can also be considered as a special case of set-
信息系统可以通过每个属性的所有可能值的集合来表征。这种类型的信息系统也可以被视为set-

where u_i; u_j 2 T_B implies that u_i and u_j are indiscernible (tolerant) with respect to a set of attributes B:
其中u;u2 T_B意味着 u和 u对于一组属性 B 是不可辨别的（宽容的）：

Example 4.1 Let ðU; AT; V; hÞ be a set-valued information system with b 2 AT and u₁; u₂; u₃ 2 U such that bðu₁Þ ¼ fv₁; v₂; v₃; v₂g; bðu₂Þ ¼ fv₄; v₅; v₆; v₇g. and bðu₃Þ ¼ fv₁; v₂; v₃g: Then, by Definition 4.1, we say that both ðu₁; u₂Þ and ðu₁; u₃Þ belong to T_b, that is, u₁; u₂ are indiscernible with respect to attribute b and u₁; u₃ are indiscernible with respect to attribute b simultaneously.
例4.1设ðu;AT;五;hÞ是一个集合值信息系统，b 2 AT和u₁;u₂;u₃2 U使得 bðu₁Þ1/4fv₁; 第₂ 节; 第₃ 节; v₂克; bðu₂Þ1/4fv₄; 第_{5 节}; 第₆ 节; v₇克。 和 bðu₃Þ1/4fv₁;第₂ 节;v₃g：那么，根据定义 4。1，我们说两者都是 ðu₁;u₂Þ 和 ðu₁;u₃Þ 属于 T_b，即 u₁;u₂ 对于属性 b 和 u₁ 是无法区分的;u₃ 同时对于属性 B 是无法辨别的。

It is obvious from above example that discernibility of u₁ and u₃ is more difficult than discernibility of u₁ and u₂, but Definition 4.1 is not able to describe the extent to which two objects are related. To overcome this issue, we define a fuzzy relation for a set-valued dataset.
从上面的例子中可以明显看出，u₁ 和 u₃ 的可辨别性比 u₁ 和 u₂ 的可辨别性更困难，但定义4.1无法描述其程度哪个两个对象相关。为了克服这个问题，我们为一个集值数据集定义了一个模糊关系。

Definition 4.2 Let SVIS ¼ ðU; AT; V; hÞ; 8b 2 AT be a set-valued information system, then we define a fuzzy
定义4.2设 SVIS1/4ðU;AT;五;hÞ; 8b2AT 为一个集合值信息系统，则定义一个模糊

-

relation R_b as:
关系R_b为：

¼ jbðu Þj þ b u ð3Þ
1/4bðuÞþbuð3Þ

of objects are missing. Table 4 illustrates the transforma-
的对象缺失。表4说明了transforma-

tion of incomplete information system into a set-valued information system.
将不完整的信息系统转化为固定值的信息系统。

For a set of attributes B
对于一组属性 B

defined as
定义为

-

A, a fuzzy relation R_B can be
A是模糊关系 _R，B可以是

Proposed methodology
建议的方法

In this section, we define a new kind of fuzzy relation
在本节中，我们定义了一种新的模糊关系

Example 4.2 (Continued from Example 4.1). After cal- culating degree of similarity by using fuzzy relation as defined in Eq. (3), we get
例 4.2（接例 4.1）。使用方程 （3） 中定义的模糊关系计算相似度后，我们得到

between two objects and supremacy over previously defined relation is shown using an example. Then, lower and upper approximations are defined using a threshold value on fuzzy similarity degree. Some basic properties of above-defined lower and upper approximations are also
Between two objects 和 prepremacy over previously defined relation 通过示例来显示。然后，使用模糊相似度的阈值定义下限和上限近似值。上面定义的lower和upper近似的一些基本属性也是

l_ ðu₁; u₂Þ ¼
lðu₁u₂Þ1/4

R_b

l_ ðu₁; u₃Þ ¼
lðu₁u₃Þ1/4

R_b

2jbðu₁Þ \ bðu₂Þj jbðu₁Þj þ jbðu₂Þj 2jbðu₁Þ \ bðu₃Þj jbðu₁Þj þ jbðu₃Þj
2bðu₁Þbðu₂Þj bðu₁Þj þ bðu₂Þj 2bðu₁Þbðu₃Þj jbðu₁Þjþbðu₃Þj

2

6

discussed.
讨论。

Fuzzy relation between objects
对象之间的模糊关系

In this subsection, first we present the definition of toler- ance relation available in the literature and then propose a new fuzzy relation between two objects. In continuation, we compared both the definition through an example.
在本节中，我们首先介绍了文献中可用的公差关系的定义，然后提出了两个对象之间的新模糊关系。接下来，我们通过一个例子比较了这两个定义。

Definition 4.1 (Orlowska (1985), Yao (2001)) For a set- valued information system IS ¼ ðU; AT; V ; hÞ; 8b 2 AT and u_i; u_j 2 U; tolerance relation is defined as
定义 4.1（Orlowska （1985）， Yao （2001）） 对于一个有值信息系统，IS1/4ðU;AT;五;hÞ;8b2AT 和 u;u2 u; tolerance 关系定义为

Now, we can easily calculate the degree to which two
现在，我们可以很容易地计算出两个

objects are discernible. h
物体是可辨别的。h

Fuzzy tolerance relation-assisted rough approximations
模糊容差关系辅助的粗略近似值

In this subsection, a fuzzy tolerance relation using a threshold value is defined and lower and upper approxi- mations of a set are presented
在本小节中，定义了使用阈值的模糊容差关系，并给出了一组的下限和上限近似值

If we ignore some misclassification and perturbation by using a threshold value on fuzzy relation between two objects as given in Eq. (3), then the involvement of fuzzy
如果我们使用方程（3）中给出的两个对象之间的模糊关系的阈值来忽略一些错误分类和扰动，那么模糊

sets in the computation of fuzzy lower approximation will increase and fuzzy positive region enlarges. Thus, the knowledge representation ability becomes much stronger with respect to misclassification.
模糊下限近似计算中的集合将增加，而模糊正区域将扩大。因此，知识表示能力在 错误分类方面变得更加强大。

So we define a new kind of binary relation using a
因此，我们使用

threshold value a as follows:
阈值A如下所示：

attribute b: So, our proposed definition gives more precise tolerance relation than previous ones.
属性 b：因此，我们提出的定义给出了比以前的定义更精确的公差关系。

a _{_}a
一个

½  ðu Þ ¼ u 2 Uj ð7Þ
1/2ðuÞ1/4u2Uj ð7Þ

Tb i

_j u_iTbu
乌特布_j

½_ a ðu Þ ¼ nu
1/2aðuÞ1/4nu

_-a

_j 2 Uju_iTbu
2UuTb u_j;

8b 2 Bo ð8Þ

classes.
类。

For a set of attributes B AT; we define binary relation
对于一组属性 BAT;我们定义二元关系

Then we propose, lower and upper approximations of any object set X U as:
然后我们提出，任何对象集 XU 的 lower和upper近似为：

as:
如：

a n h_ ai o
一个nhao

jl_

u_i; u
uu u_j

ao
一个O

ð6Þ

Tb # X ¼ u_i 2 Uj Tb
Tb#X1/4u2UTb

ðu_iÞ X

ð9Þ

B

where l
其中 l_

R_b

R_b

u_i; u
uu u_j

and l
和 l_

R_B

u_i; u
uu u_j

are defined by Eqs. (3)
由Eqs 定义。（3）

a " X ¼ nu
a“X1/4nu

_-a

Tb ðu_iÞ \ X 6¼ /
TbðuÞX61/4/

ð10Þ

and (4), respectively.
和 （4）。

Definition 4.3 A fuzzy binary relation R~ ðu ; u Þ between
定义4.3模糊二元关系R~ðu;uÞ之间

The tuple \Tb # X; Tb " X [ is called a tolerance rough set.
元组\Tb#X;Tb“X[称为公差粗略集。

b i j

_-a

Lemma 4.1 Tb is a tolerance relation.
引理4.1Tb是一个公差关系。

Proof
证明

Reflexive:
反身：

Properties of lower and upper approximations
下限和上限近似的属性

In this subsection, we will examine the results on lower and upper approximations equivalent to Dubois and Prade (1992) for our proposed approach
在本小节中，我们将检查我们提出的方法等效于 Dubois 和 Prade （1992） 的 lower 和 upper 近似的结果

Let ðU; C [ D; V ; hÞ be a set-valued decision system.
设ðU;C[D;五;hÞ是一个设定值决策系统。

Let B C and X U,a 2 ð0; 1Þ
设bc和xua2ð01þ

_-a _{_}a
一个

2 bðu_iÞ \ b u_j

Theorem 4.1 TB # X X TB " X
定理4.1TB#XX XTB“X

_a h _a
一个H_A i

h_ a
哈a i

jbðu_iÞj þ jbðu_iÞj jbðu_iÞj

_-a

Since x 2 TB ðxÞ ) x 2 X
自 x 2 TB ðxÞ ） x 2 x:

_-a

) ðu_i; u_iÞ 2 Tb
）ðuuÞ2Tb

Therefore, TB # X X
因此，TB # X X:

Now, let x 2 X, since x 2 h_ a iðxÞ ) h_ a iðxÞ\ X 6¼
现在，设x2Xsincex2haðxÞ）haðxÞX61/4

Symmetric:
对称：

Let u ; u 2 _ a
让你两个

_-a

u ) x 2 " X
u）x 2 英寸 X

TB TB

i j Tb

2jbðu_iÞ\bðu_jÞj a

TB

_-a

Now, l
现在，l_

u_j; u_i ¼
你你1/4

2jbðu_jÞ\bðu_iÞj

¼ l
1/4 升_

u_i; u_j a
你你

Hence, TB # X X TB " X h
因此，TB#XX TB“Xh

R_b

) u_j; u
你你_i

a

2 Tb
2吨b

jbðu_jÞjþjbðu_iÞj R_b
bðuÞþ bðuÞR_b

Theorem 4.2 Let B₁
定理4.2设B₁

B₂
乙₂

C, then
C，则

_-a _{_}a _{_}a
一个AA

Therefore, Tb is a fuzzy tolerance relation. h (i) TB1 # X TB2 # X
因此，Tb是一个模糊容差关系。h（i）TB# XTB# X

_-a _{_}a
一个

Example 4.3 If we take a ¼ 0:3 in Eq. (5) and apply it on
例4.3如果我们取方程（5）中的1/403 并将其应用于

Example 4.1, then we can see that only ðu₁; u₃Þ belongs to
例4.1那么我们可以看到只有ðu₁;u₃Þ属于

-

T_b, i.e. only u₁ and u₃ are indiscernible with respect to
T_b即只有u₁和u₃在

(ii) TB2 " X TB1 " X
（ii）TB“XTB”X

Proof
证明

_-a

ðiÞ Let x 2 T
ð Þ让x2T

_-a

8 *B₁

B₂
乙₂

B₁ B₁
乙₁乙₁

> h_ a i n o

l_ ðx; yÞ a; 8b 2 B₁

1 2 TB
1 2结核病2

h i
嗨I

TB
结核病1

h_ a i n o
哎呀

_-a

Thus; T
因此;吨

_-a

ðxÞ X ) x 2 T
ðxÞX）x2吨

# X:

TB
结核病2

ðxÞ ¼ y 2 Ujl_ ðx; yÞ a; 8b 2 B₂
ðxÞ1/4y2UlðxyÞa8b2B₂

R_b

B₂ B₂

>: h_ a i

h_ a
哈a i