The reduction and fusion of fuzzy covering systems based on the evidence theory

This paper studies reduction of a fuzzy covering and fusion of multi-fuzzy covering systems based on the evidence theory and rough set theory. A novel pair of belief and plausibility functions is defined by employing a method of non-classical probability model and the approximation operators of a fuzzy covering. Then we study the reduction of a fuzzy covering based on the functions we presented. In the case of multiple information sources, we present a method of information fusion for multi-fuzzy covering systems, by which objects can be well classified in a fuzzy covering decision system. Finally, by using the method of maximum flow, we discuss under what conditions, fuzzy covering approximation operators can be induced by a fuzzy belief structure.     

Evidence-theory-based numerical algorithms of attribute reduction with neighborhood-covering rough sets

Covering rough sets generalize traditional rough sets by considering coverings of the universe instead of partitions, and neighborhood-covering rough sets have been demonstrated to be a reasonable selection for attribute reduction with covering rough sets. In this paper, numerical algorithms of attribute reduction with neighborhood-covering rough sets are developed by using evidence theory. We firstly employ belief and plausibility functions to measure lower and upper approximations in neighborhood-covering rough sets, and then, the attribute reductions of covering information systems and decision systems are characterized by these respective functions. The concepts of the significance and the relative significance of coverings are also developed to design algorithms for finding reducts. Based on these discussions, connections between neighborhood-covering rough sets and evidence theory are set up to establish a basic framework of numerical characterizations of attribute reduction with these sets.     

A concept of fuzzy transition probabilities

Starting from fuzzy real numbers with an arbitrary lattice of belief and following the extension principle, we develop concepts of fuzzy probabilities, transition probabilities and random variables and of their combinations, and show that these concepts are consistent. We derive some results on fuzzy real numbers, on the expectation of fuzzy random variables and on fuzzy stochastic processes. To sketch the range of applications of fuzzy stochastics, we give two examples that show how real-world problems may be modeled by means of fuzzy probabilities and that give small numerical examples. Moreover, we give a brief outlook for a possible expansion of our theory to fuzzy Markovian decision processes by means of a partial order on the set of all fuzzy real numbers.     

Reduction about approximation spaces of covering generalized rough sets

The introduction of covering generalized rough sets has made a substantial contribution to the traditional theory of rough sets. The notion of attribute reduction can be regarded as one of the strongest and most significant results in rough sets. However, the efforts made on attribute reduction of covering generalized rough sets are far from sufficient. In this work, covering reduction is examined and discussed. We initially construct a new reduction theory by redefining the approximation spaces and the reducts of covering generalized rough sets. This theory is applicable to all types of covering generalized rough sets, and generalizes some existing reduction theories. Moreover, the currently insufficient reducts of covering generalized rough sets are improved by the new reduction. We then investigate in detail the procedures to get reducts of a covering. The reduction of a covering also provides a technique for data reduction in data mining.     

An incremental attribute reduction approach based on knowledge granularity under the attribute generalization

Attribute reduction is a key step to discover interesting patterns in the decision system with numbers of attributes available. In recent years, with the fast development of data processing tools, the information system may increase quickly in attributes over time. How to update attribute reducts efficiently under the attribute generalization becomes an important task in knowledge discovery related tasks since the result of attribute reduction may alter with the increase of attributes. This paper aims for investigation of incremental attribute reduction algorithm based on knowledge granularity in the decision system under the variation of attributes. Incremental mechanisms to calculate the new knowledge granularity are first introduced. Then, the corresponding incremental algorithms are presented for attribute reduction based on the calculated knowledge granularity when multiple attributes are added to the decision system. Finally, experiments performed on UCI data sets and the complexity analysis show that the proposed incremental methods are effective and efficient to update attribute reducts with the increase of attributes.     

An efficient rough feature selection algorithm with a multi-granulation view

Feature selection is a challenging problem in many areas such as pattern recognition, machine learning and data mining. Rough set theory, as a valid soft computing tool to analyze various types of data, has been widely applied to select helpful features (also called attribute reduction). In rough set theory, many feature selection algorithms have been developed in the literatures, however, they are very time-consuming when data sets are in a large scale. To overcome this limitation, we propose in this paper an efficient rough feature selection algorithm for large-scale data sets, which is stimulated from multi-granulation. A sub-table of a data set can be considered as a small granularity. Given a large-scale data set, the algorithm first selects different small granularities and then estimate on each small granularity the reduct of the original data set. Fusing all of the estimates on small granularities together, the algorithm can get an approximate reduct. Because of that the total time spent on computing reducts for sub-tables is much less than that for the original large-scale one, the algorithm yields in a much less amount of time a feature subset (the approximate reduct). According to several decision performance measures, experimental results show that the proposed algorithm is feasible and efficient for large-scale data sets.     

多粒度模糊粗糙集研究

本文研究了模糊粗糙集中属性约简问题.利用模糊粗糙集和多粒度粗糙集各自优点的结合,提出了两类多粒度模糊粗糙集模型,使得两类粗糙集中的上下近似算子关于负算子对偶.同时研究了多粒度模糊粗糙集的性质及与单粒度模糊粗糙集的关系.并通过构造区分函数的方法提出了一类多粒度模糊粗糙集模型的近似约简方法.最后用一个实例核对了该类多粒度模糊粗糙决策系统近似约简方法的有效性.     

Dependence-space-based attribute reductions in inconsistent decision information systems

In rough set theory, attribute reduction is an important mechanism for knowledge discovery. This paper mainly deals with attribute reductions of an inconsistent decision information system based on a dependence space. Through the concept of inclusion degree, a generalized decision distribution function is first constructed. A decision distribution relation is then defined. On the basis of this decision distribution relation, a dependence space is proposed, and an equivalence congruence based on the indiscernibility attribute sets is also obtained. Applying the congruences on a dependence space, new approaches to find a distribution consistent set are formulated. The judgement theorems for judging distribution consistent sets are also established by using these congruences and the decision distribution relation.     

A relational perspective of attribute reduction in rough set-based data analysis

Attribute reduction is very important in rough set-based data analysis (RSDA) because it can be used to simplify the induced decision rules without reducing the classification accuracy. The notion of reduct plays a key role in rough set-based attribute reduction. In rough set theory, a reduct is generally defined as a minimal subset of attributes that can classify the same domain of objects as unambiguously as the original set of attributes. Nevertheless, from a relational perspective, RSDA relies on a kind of dependency principle. That is, the relationship between the class labels of a pair of objects depends on component-wise comparison of their condition attributes. The larger the number of condition attributes compared, the greater the probability that the dependency will hold. Thus, elimination of condition attributes may cause more object pairs to violate the dependency principle. Based on this observation, a reduct can be defined alternatively as a minimal subset of attributes that does not increase the number of objects violating the dependency principle. While the alternative definition coincides with the original one in ordinary RSDA, it is more easily generalized to cases of fuzzy RSDA and relational data analysis.     

Updating attribute reduction in incomplete decision systems with the variation of attribute set

In rough set theory, attribute reduction is a challenging problem in the applications in which data with numbers of attributes available. Moreover, due to dynamic characteristics of data collection in decision systems, attribute reduction will change dynamically as attribute set in decision systems varies over time. How to carry out updating attribute reduction by utilizing previous information is an important task that can help to improve the efficiency of knowledge discovery. In view of that attribute reduction algorithms in incomplete decision systems with the variation of attribute set have not yet been discussed so far. This paper focuses on positive region-based attribute reduction algorithm to solve the attribute reduction problem efficiently in the incomplete decision systems with dynamically varying attribute set. We first introduce an incremental manner to calculate the new positive region and tolerance classes. Consequently, based on the calculated positive region and tolerance classes, the corresponding attribute reduction algorithms on how to compute new attribute reduct are put forward respectively when an attribute set is added into and deleted from the incomplete decision systems. Finally, numerical experiments conducted on different data sets from UCI validate the effectiveness and efficiency of the proposed algorithms in incomplete decision systems with the variation of attribute set.     

A dominance intuitionistic fuzzy-rough set approach and its applications

Although the rough set and intuitionistic fuzzy set both capture the same notion, imprecision, studies on the combination of these two theories are rare. Rule extraction is an important task in a type of decision systems where condition attributes are taken as intuitionistic fuzzy values and those of decision attribute are crisp ones. To address this issue, this paper makes a contribution of the following aspects. First, a ranking method is introduced to construct the neighborhood of every object that is determined by intuitionistic fuzzy values of condition attributes. Moreover, an original notion, dominance intuitionistic fuzzy decision tables (DIFDT), is proposed in this paper. Second, a lower/upper approximation set of an object and crisp classes that are confirmed by decision attributes is ascertained by comparing the relation between them. Third, making use of the discernibility matrix and discernibility function, a lower and upper approximation reduction and rule extraction algorithm is devised to acquire knowledge from existing dominance intuitionistic fuzzy decision tables. Finally, the presented model and algorithms are applied to audit risk judgment on information system security auditing risk judgement for CISA, candidate global supplier selection in a manufacturing company, and cars classification.     

Homomorphisms between fuzzy information systems

The information system is one of the most important mathematical models in the field of artificial intelligence, and the concept of mapping is a useful tool for studying the communication between two information systems. In this work, the concepts of fuzzy relation mapping and inverse fuzzy relation mapping are first introduced and their properties are studied. Then, the notions of homomorphisms of information systems based on fuzzy relations are proposed, and it is proved that attribute reductions in the original system and image system are equivalent to each other under the condition of homomorphism.     

A multi-attribute group decision-making method based on weighted geometric aggregation operators of interval-valued trapezoidal fuzzy numbers

With respect to the multiple attribute group decision making problems in which the attribute values take the form of generalized interval-valued trapezoidal fuzzy numbers (GITFN), this paper proposed a decision making method based on weighted geometric aggregation operators. First, some operational rules, the distance and comparison between two GITFNs are introduced. Second, the generalized interval-valued trapezoidal fuzzy numbers weighted geometric aggregation (GITFNWGA) operator, the generalized interval-valued trapezoidal fuzzy numbers ordered weighted geometric aggregation (GITFNOWGA) operator, and the generalized interval-valued trapezoidal fuzzy numbers hybrid geometric aggregation (GITFNHGA) operator are proposed, and their various properties are investigated. At the same time, the group decision methods based on these operators are also presented. Finally, an illustrate example is given to show the decision-making steps and the effectiveness of this method.     

Neighborhood based decision-theoretic rough set models

As an extension of Pawlak rough set model, decision-theoretic rough set model (DTRS) adopts the Bayesian decision theory to compute the required thresholds in probabilistic rough set models. It gives a new semantic interpretation of the positive, boundary and negative regions by using three-way decisions. DTRS has been widely discussed and applied in data mining and decision making. However, one limitation of DTRS is its lack of ability to deal with numerical data directly. In order to overcome this disadvantage and extend the theory of DTRS, this paper proposes a neighborhood based decision-theoretic rough set model (NDTRS) under the framework of DTRS. Basic concepts of NDTRS are introduced. A positive region related attribute reduct and a minimum cost attribute reduct in the proposed model are defined and analyzed. Experimental results show that our methods can get a short reduct. Furthermore, a new neighborhood classifier based on three-way decisions is constructed and compared with other classifiers. Comparison experiments show that the proposed classifier can get a high accuracy and a low misclassification cost.     

A group decision making approach based on aggregating interval data into interval-valued intuitionistic fuzzy information

Group decision making is one of the most important problems in decision making sciences. The aim of this article is to aggregate the interval data into the interval-valued intuitionistic fuzzy information for multiple attribute group decision making. In this model, the decision information is provided by decision maker, which is characterized by interval data. Based on the idea of mean and variance in statistics, we first define the concepts of satisfactory and dissatisfactory intervals of attribute vector against each alternative. Using these concepts, we develop an approach to aggregate the attribute vector into interval-valued intuitionistic fuzzy number under group decision making environment. A practical example is provided to illustrate the proposed method. To show the validity of the reported method, comparisons with other methods are also made.     

The evidential reasoning approach for MADA under both probabilistic and fuzzy uncertainties

Many multiple attribute decision analysis (MADA) problems are characterised by both quantitative and qualitative attributes with various types of uncertainties. Incompleteness (or ignorance) and vagueness (or fuzziness) are among the most common uncertainties in decision analysis. The evidential reasoning (ER) approach has been developed in the 1990s and in the recent years to support the solution of MADA problems with ignorance, a kind of probabilistic uncertainty. In this paper, the ER approach is further developed to deal with MADA problems with both probabilistic and fuzzy uncertainties.In this newly developed ER approach, precise data, ignorance and fuzziness are all modelled under the unified framework of a distributed fuzzy belief structure, leading to a fuzzy belief decision matrix. A utility-based grade match method is proposed to transform both numerical data and qualitative (fuzzy) assessment information of various formats into the fuzzy belief structure. A new fuzzy ER algorithm is developed to aggregate multiple attributes using the information contained in the fuzzy belief matrix, resulting in an aggregated fuzzy distributed assessment for each alternative. Different from the existing ER algorithm that is of a recursive nature, the new fuzzy ER algorithm provides an analytical means for combining all attributes without iteration, thus providing scope and flexibility for sensitivity analysis and optimisation. A numerical example is provided to illustrate the detailed implementation process of the new ER approach and its validity and wide applicability.     

A theoretical investigation of regular equivalences for fuzzy graphs

The notion of regular equivalence or bisimulation arises in different applications, such as positional analysis of social networks and behavior analysis of state transition systems. The common characteristic of these applications is that the system under modeling can be represented as a graph. Thus, regular equivalence is a notion used to capture the similarity between nodes based on their linking patterns with other nodes. According to Borgatti and Everett, two nodes are regularly equivalent if they are equally related to equivalent others. In recent years, fuzzy graphs have also received considerable attention because they can represent both the qualitative relationships and the degrees of connection between nodes. In this paper, we generalize the notion of regular equivalence to fuzzy graphs based on two alternative definitions of regular equivalence. While the two definitions are equivalent for crisp graphs, they induce different generalizations for fuzzy graphs. The first generalization, called regular similarity, is based on the characterization of regular equivalence as an equivalence relation that commutes with the underlying graph edges. The regular similarity is then a fuzzy binary relation that specifies the degree of similarity between nodes in the graph. The second generalization, called generalized regular equivalence, is based on the definition of coloring. A coloring is a mapping from the set of nodes to a set of colors. A coloring is regular if nodes that are mapped to the same color, have the same colors in their neighborhoods. Hence, generalized regular equivalence is an equivalence relation that can determine a crisp partition of the nodes in a fuzzy graph.     

Extension of VIKOR method for decision making problem based on hesitant fuzzy set

The multiple criteria decision making (MCDM) methods VIKOR and TOPSIS are all based on an aggregating function representing “closeness to the ideal”, which originated in the compromise programming method. The VIKOR method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum of an “individual regret” for the “opponent”, which is an effective tool in multi-criteria decision making, particularly in a situation where the decision maker is not able, or does not know to express his/her preference at the beginning of system design. The TOPSIS method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative-ideal solution, but it does not consider the relative importance of these distances. And, the hesitant fuzzy set is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. In this paper, we develop the E-VIKOR method and TOPSIS method to solve the MCDM problems with hesitant fuzzy set information. Firstly, the hesitant fuzzy set information and corresponding concepts are described, and the basic essential of the VIKOR method is introduced. Then, the problem on multiple attribute decision marking is described, and the principles and steps of the proposed E-VIKOR method and TOPSIS method are presented. Finally, a numerical example illustrates an application of the E-VIKOR method, and the result by the TOPSIS method is compared.