Fundamenta Informaticae - Volume 142, issue 1-4

Article Type: Other

DOI: 10.3233/FI-2015-1281

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. v-viii, 2015

Authors: Ciucci, Davide | Dubois, Didier | Prade, Henri

Article Type: Research Article

Abstract: The square of opposition is as old as logic. There has been a recent renewal of interest on this topic, due to the emergence of new structures (hexagonal and cubic) extending the square. They apply to a large variety of representation frameworks, all based on the notions of sets and relations. After a reminder about the structures of opposition, and an introduction to their gradual extensions (exemplified on fuzzy sets), the paper more particularly studies fuzzy rough sets and rough fuzzy sets in the setting of gradual structures of opposition.

Keywords: square of opposition, fuzzy set, fuzzy relation, rough set

DOI: 10.3233/FI-2015-1282

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 1-19, 2015

Authors: Inuiguchi, Masahiro

Article Type: Research Article

Abstract: In this paper we focus on generalizations of the classical rough set approach to fuzzy environments. There are two aspects of rough set approaches: classification and approximation. In the classification aspect, by rough set approaches we can classify objects into positive and negative examples of a class. On the other hand, in the approximation aspect, by rough set approaches we obtain the lower and upper approximations of a class. The former model works better in the attribute reduction while the latter model works better in the rule induction. In the setting of the classical rough set approach, the lower approximation …is nothing but the set of positive examples and the upper approximation is the complementary set of negative examples. However, these equalities do not always hold in the generalized settings. Most of fuzzy rough set models proposed earlier are defined in the classification aspect. The approaches based on those models do not always work well in approximating fuzzy subsets. In this paper we define the fuzzy rough set models in the approximation aspect. We investigate their fundamental properties and demonstrate the advantages of fuzzy set approximation. Finally we consider attribute reduction based on the proposed fuzzy rough set models. Show more

Keywords: rough sets, fuzzy sets, inclusion degree, certainty qualification, implication function

DOI: 10.3233/FI-2015-1283

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 21-51, 2015

Authors: Vluymans, Sarah | D’eer, Lynn | Saeys, Yvan | Cornelis, Chris

Article Type: Research Article

Abstract: Data used in machine learning applications is prone to contain both vague and incomplete information. Many authors have proposed to use fuzzy rough set theory in the development of new techniques tackling these characteristics. Fuzzy sets deal with vague data, while rough sets allow to model incomplete information. As such, the hybrid setting of the two paradigms is an ideal candidate tool to confront the separate challenges. In this paper, we present a thorough review on the use of fuzzy rough sets in machine learning applications. We recall their integration in preprocessing methods and consider learning algorithms in the supervised, …unsupervised and semi-supervised domains and outline future challenges. Throughout the paper, we highlight the interaction between theoretical advances on fuzzy rough sets and practical machine learning tools that take advantage of them. Show more

Keywords: fuzzy sets, rough sets, fuzzy rough sets, machine learning

DOI: 10.3233/FI-2015-1284

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 53-86, 2015

Authors: Wu, Wei-Zhi | Li, Tong-Jun | Gu, Shen-Ming

Article Type: Research Article

Abstract: Axiomatic characterizations of approximation operators are important in the study of rough set theory. In this paper, axiomatic characterizations of relation-based fuzzy rough approximation operators determined by a fuzzy implication operator ℐ are investigated. We first review the constructive definitions and properties of lower and upper ℐ-fuzzy rough approximation operators. We then propose an operator-oriented characterization of ℐ-fuzzy rough sets. We show that the lower and upper ℐ-fuzzy rough approximation operators generated by an arbitrary fuzzy relation can be described by single axioms. We further examine that ℐ-fuzzy rough approximation operators corresponding to some special types of fuzzy relations, such …as serial, reflexive, and 𝒯-transitive ones, can also be characterized by single axioms. Show more

Keywords: Approximation operators, fuzzy implication operators, fuzzy rough sets, rough sets, triangular norms

DOI: 10.3233/FI-2015-1285

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 87-104, 2015

Authors: Liu, Guilong

Article Type: Research Article

Abstract: This paper studies quasi-discrete closure spaces and fuzzy closure spaces. We show that any topological closure cT induced by a closure c is the smallest extension from a closure space to a topological closure space in both crisp and fuzzy environment, in addition, a characterization of the continuous mappings in quasi-discrete closure spaces is obtained. We propose the concept of quasi-discrete fuzzy closure spaces in the context of fuzzy sets and establish a one to one correspondence between quasi-discrete fuzzy closure spaces and reflexive fuzzy relations. We also discuss the relationship between topological closure cT …and closure c in quasi-discrete fuzzy closure spaces and show that the process from closure c to topological closure cT can be realized via the process from a reflexive fuzzy relation to its transitive closure. Show more

Keywords: Fuzzy relation, Topology, Fuzzy topological closure, Quasi-discreteness, Fuzzy set

DOI: 10.3233/FI-2015-1286

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 105-115, 2015

Authors: Lang, Guangming | Yang, Tian

Article Type: Research Article

Abstract: In practical situations, interval-valued fuzzy sets are of interest because fuzzy sets of this kind are frequently encountered. In this paper, motivated by the needs for solving imprecise problems, we generalize the concept of shadowed sets for understanding interval-valued fuzzy sets and provide a solution to compute a pair of thresholds by searching for a balance of uncertainty. Then we present three-way approximations of interval-valued fuzzy sets and a formulation for calculating the pair of thresholds using single-valued loss functions. We also compute three-way approximations of interval-valued fuzzy sets using interval-valued loss functions. Afterwards, we employ several examples to illustrate …that how to take an action for an object with an interval-valued membership grade using an interval-valued loss function. Show more

Keywords: Decision-theoretic rough sets, Interval-valued fuzzy sets, Interval-valued loss function, Shadowed sets

DOI: 10.3233/FI-2015-1287

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 117-143, 2015

Authors: Ma, Jian-Min | Yao, Yiyu

Article Type: Research Article

Abstract: Pawlak’s rough set model considers the rough approximations based on an equivalence relation. Multi-granulation rough set models concern rough approximations based on multiple equivalence relations. In this paper, we examine six types of rough set approximations in multi-granulation fuzzy approximation spaces (MGFASs). We construct a partition of the given universe based on a fuzzy binary relation in a fuzzy approximation space. Based on the partition, we introduce a pair of rough set approximations. In a multi-granulation fuzzy approximation space, by a family of fuzzy binary relations, we introduce two kinds of rough set approximations in terms of the union and …intersection of fuzzy relations, respectively. A pair of rough set approximations based on the family of fuzzy binary relations is also discussed. Furthermore, the optimistic and pessimistic multi-granulation rough set approximations are investigated due to the fuzzy binary relations in aMGFAS. Properties of these rough set approximations are demonstrated. Finally, we examine relationships of them. It is proved that the lower and upper approximations generated by a family of fuzzy binary relations are the pair nearest to the undefinable set, and the pessimistic multi-granulation lower and upper approximations are the pair farthest to the undefinable set. Show more

Keywords: multi-granulation, multi-granulation fuzzy approximation space, optimisticmulti-granulation approximation, pessimistic multi-granulation approximation

DOI: 10.3233/FI-2015-1288

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 145-160, 2015

Authors: Ma, Zhou-Ming | Mi, Ju-Sheng

Article Type: Research Article

Abstract: Multi-granulation rough set(MGRS), as a kind of fusion mechanism of different information or data, is an useful development of Pawlak rough set theory. Firstly, this paper gives an introduction for various types of MGRS, their properties and axiomatization characterizations are studied. We show that, except for the optimistic one, each of the existing MGRS means a single granulation rough set. Then, we made a comparative analysis on the different uncertainty measures among the various multi-granulation approximation spaces. At the basis of investigating for the existing uncertainty measures, we discuss their limitations via some examples, and propose a total ordered relation …among approximation spaces, even in the more general covering ones. It will be better than the original partial relation in revealing uncertainty, which conceal in the approximation space or covering one. Finally, based on the total ordered relation, we present improved information entropy, rough entropy, knowledge granulation and axiomatic definition of the knowledge granulation measures. It is proved that they are more reasonable than the original ones. Then, some novel uncertainty measures and improved fusion uncertainty measures about various granulations are also proposed. By employing these measures, granulation measures of various MGRSs are defined and studied. Show more

Keywords: Rough set, Multi-granulation, Total ordered relation, Uncertainty measure

DOI: 10.3233/FI-2015-1289

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 161-181, 2015

Authors: Qin, Keyun | Luo, Junfang | Pei, Zheng

Article Type: Research Article

Abstract: Rough set approach for knowledge discovery in incomplete information systems has been extensively studied. This paper conduct a further study of valued tolerance relation based rough approximations. We make an analysis of the existing rough approximabilities and propose a new approach for lower (upper) approximability, which is a generalization of Pawlak approximation operators for complete information system. The approach has also been generalized to fuzzy cases. Some basic properties of the approximation operators are examined.

Keywords: Rough set, tolerance relation, valued tolerance relation, lower (upper) approximability

DOI: 10.3233/FI-2015-1290

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 183-194, 2015

Authors: Zhang, Xiaohong | Zhang, Yanning | Xue, Zhanao | Ma, Yingcang

Article Type: Research Article

Abstract: The relationships between T-rough sets and covering based rough sets are investigated, and two kinds of generated methods of rough approximation operators from existing rough sets are established. Moreover, applying the aforementioned generated methods of approximation operators, S-rough sets and some new covering-based rough sets are introduced and their basic properties are discussed.

Keywords: Rough approximation pair, Closeness, Covering, T-rough set, S-rough set

DOI: 10.3233/FI-2015-1291

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 195-212, 2015

Authors: Dutta, Soma | Skowron, Andrzej

Article Type: Research Article

Abstract: Looking back to Prof. Zadeh’s paradigm of Computing with Words (CWW) [28, 29, 30], one can notice that the initial attempt of such an endeavour was to set up a basic vocabulary of linguistic words, and fix their semantics based on fuzzy sets. Then a grammar was proposed to generate compound linguistic expressions based on the primitive ones, and simultaneously based on the semantic interpretations of those basic linguistic expressions a general scheme for the semantics of the rest of linguistic expressions were proposed. Sentences involving linguistic quantifiers and vague predicates constitute a fragment of natural language. In this paper, …we choose this fragment of the natural language, and explore the semantics from the perspective of rough sets [13, 14, 16, 17, 18, 21]. We fix a set of basic crisp quantifiers, mainly of proportional kind. A set of vague quantifiers are proposed to lie in a close vicinity of those crisp quantifiers in the sense that a particular vague quantifier can be visualized as a blurred, may be called rough, image of a set of crisp quantifiers. Semantics of the rest of the vague quantifiers can be obtained based on the subjective perception of the interrelations among the (vague) quantifiers. Show more

Keywords: Linguistic quantifiers, Rough set, Similarity relation, Rough membership function

DOI: 10.3233/FI-2015-1292

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 213-236, 2015

Authors: Maji, Pradipta | Roy, Shaswati

Article Type: Research Article

Abstract: One of the important problems in medical diagnosis is the segmentation and detection of brain tumor in MR images. The accurate estimation of brain tumor size is important for treatment planning and therapy evaluation. In this regard, this paper presents a new method, termed as SoBT-RFW, for segmentation of brain tumor from MR images. It integrates judiciously the merits of rough-fuzzy computing and multiresolution image analysis technique. The proposed method starts with a simple skull stripping algorithm to remove non-cerebral tissues such as skull, scalp, and dura from brain MR images. To extract the scale-space feature vector for each pixel …of brain region, the dyadic wavelet analysis is used, while an unsupervised feature selection method, based on maximum relevance-maximum significance criterion, is used to select relevant and significant textural features for brain tumor segmentation. To address the uncertainty problem of brain MR image segmentation, the proposed SoBT-RFW method uses the robust rough-fuzzy c -means algorithm. After the segmentation process, asymmetricity is analyzed by using the Zernike moments of each of the tissues segmented in the brain to identify the tumor. Finally, the location of the tumor is searched by a region growing algorithm based on the concept of rough sets. The performance of the proposed SoBT-RFW method, along with a comparison with related approaches, is demonstrated on a set of synthetic and real brain MR images using standard validity indices. Show more

Keywords: Brain tumor detection, segmentation, clustering, fuzzy set, rough sets, wavelets

DOI: 10.3233/FI-2015-1293

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 237-267, 2015

Authors: Polkowski, Lech | Semeniuk-Polkowska, Maria

Article Type: Research Article

Abstract: The title of this note announces an attempt at pointing to particular points where rough set and fuzzy set approaches to decision making are close to each other, the closeness meaning that formal approaches to behaviour of the two at those points can be given in an analogous form. This by no means implies that the two can be unified as theories dealing with uncertainty. As the notion of truth for rough set decision rules is well established, we propose a notion of truth for fuzzy decision rules and we seek an analogy between the two. In order to introduce …an analogous form of graded notion of truth for decision rules in both theories, we introduce a new context in which to set this notion. This context is based on our earlier results concerning rough mereological granular logics and their relevance for rough decision rules. To make our exposition satisfactorily complete, we recall our approach to granularity based on rough mereology. This note is partitioned into two parts. In Prologue, four sections present basic ideas on vagueness and ambiguity, rough sets, fuzzy sets and mereology along with rough mereology. Here also notions of truth for rough and fuzzy decision rules are presented. In Episode, main protagonists enter an analysis aimed at pointing to further close analogies between them, notably concerning notions of partial truth and dependency among attributes. To this end in four sections on similarity, granulation of knowledge, granular logics and dependencies, we give basic information on similarity, granulation and dependency and we point to analogies between the two theories with respect to those notions. We sum up the content of the note in Conclusions. Show more

DOI: 10.3233/FI-2015-1294

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 269-284, 2015

Authors: Cai, Mingjie | Li, Qingguo

Article Type: Research Article

Abstract: The notion of homomorphism, as an important tool for studying the relationship between information systems, has attracted a great deal of attention in recent years, and the authors tend to pay their attention to static information systems in the existing studies. In the present paper, we aim to study homomorphisms between fuzzy relation information systems (FRISs) in dynamic environments, where the terminology of dynamic refers to the fact that the involved information systems need to be updated with time due to the inflow of new information. To be more specific, we firstly examine properties of consistent functions with respect to …fuzzy relations and construct homomorphisms between FRISs. Then, we develop incremental mechanisms of computing homomorphisms between dynamic FRISs and illustrate how to construct relation reducts of dynamic FRISs using homomorphisms. Lastly, the experimental results are employed to demonstrate that compressing dynamic FRISs can be simplified significantly with the proposed algorithms. Show more

Keywords: Rough set, Consistent function, Fuzzy relation information system, Homomorphism

DOI: 10.3233/FI-2015-1295

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 285-306, 2015

Authors: Li, Mei-Zheng | Wang, Guo-Yin

Article Type: Research Article

Abstract: Knowledge reduction is a basic issue in knowledge representation and data mining. Although various methods have been developed to reduce the size of classical formal contexts, the reduction of formal fuzzy contexts based on fuzzy lattices remains a difficult problem owing to its complicated derivation operators. To address this problem, this paper proposes a method of knowledge reduction by reducing attributes in a formal fuzzy context based on the crisply generated fuzzy concept lattice. Employing the proposed approach, attributes which are non-essential to the structure of the crisply generated fuzzy concept lattice are removed. Discernibility matrix and Boolean function are …employed to compute the attribute reducts of the formal fuzzy contexts, by which all the attribute reducts of the formal fuzzy contexts are determined without changing the structure of the lattice. Further, all the attributes are classified into three types by their significance in constructing the crisply generated fuzzy concept lattice. The characteristics of these types of attributes are also analyzed. Finally, the proposed method is used to conduct knowledge reduction in the variable threshold concept lattices, which is a complement to the existing knowledge reduction methods. Show more

Keywords: Concept lattices, Discernibility matrix, Formal fuzzy contexts, Knowledge reduction, Attribute characteristics

DOI: 10.3233/FI-2015-1296

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 307-335, 2015

Authors: Wen, Liu-Ying | Min, Fan

Article Type: Research Article

Abstract: Symbolic value partitioning is a knowledge reduction technique in the field of data mining. In this paper, we propose a granular computing approach for the partitioning task that includes granule construction and granule selection algorithms. The granule construction algorithm takes advantage of local information associated with each attribute. A binary attribute value taxonomy tree is built to merge these attribute values in a bottom-up manner using information-loss heuristics. The use of a balancing technique enables us to control different nodes in the same level to have approximately the same size. The granule selection algorithm uses global information about all of …the attributes in the decision system. Hence, nodes across the taxonomy forest of all attributes are selected and expanded using information-gain heuristics. We present a series of experimental results that demonstrate the effectiveness of the proposed approach in terms of reducing the data size and improving the resulting classification accuracy. Show more

Keywords: Attribute taxonomy tree, granular computing, information entropy, symbolic value partition

DOI: 10.3233/FI-2015-1297

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 337-371, 2015

Authors: Yu, Jianhang | Xu, Weihua

Article Type: Research Article

Abstract: Rough set theory has been successfully used in formation system for classification analysis and knowledge discovery. The upper and lower approximations are fundamental concepts of this theory. The new information arrives continuously and redundant information may be produced with the time in real-world application. So, then incremental learning is an efficient technique for knowledge discovery in a dynamic database, which enables acquiring additional knowledge from new data without forgetting prior knowledge, which need to be updated incrementally while the object set get varies over time in the interval-valued ordered information system. In this paper, we analyzed the updating mechanisms for …computing approximations with the variation of the object set. Two incremental algorithms respectively for adding and deleting objects with updating the approximations are proposed in interval-valued ordered information system. Furthermore, extensive experiments are carried out on six UCI data sets to verify the performance of these proposed algorithms. And the experiments results indicate the incremental approaches significantly outperform non-incremental approaches with a dramatic reduction in the computational time. Show more

Keywords: Approximations, Dynamic database, Incremental learning, Interval-valued ordered information system

DOI: 10.3233/FI-2015-1298

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 373-397, 2015

Article Type: Other

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 399-400, 2015