Topological Properties of Incomplete Multigranulation Based on Rough Fuzzy Sets

Rough set theory for the interval-valued fuzzy information systems

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

This paper researched and constructed residuation implication operator of intuitionistic fuzzy T-norms.Based on this,the paper deducted the intuitionistic fuzzy rough sets model,and proved that the Pawlak rough sets,intuitionistic fuzzy sets,fuzzy rough sets,rough fuzzy sets and fuzzy T-norms rough sets were special cases of intuitionistic fuzzy rough sets.At last,some properties of intuitionistic fuzzy sets were presented and proved.

Fuzzy concepts are represented by fuzzy sets, so a fuzzy knowledge is defined to be a collection of fuzzy sets (concepts) which includes two constants, 1 (true) and 0 (false). Abstractly, a logical derivation is a finite series of logical operations acting on a given knowledge. In fuzzy world, logical operations are defined by mathematical operations. So fuzzy logical derivations are mathematical derivations, and vice versa In this paper, the closure of fuzzy reasoning is characterized by fuzzy rough sets, and by topologies (neighborhood systems). A new fuzzy concept F N is derivable from the old knowledge K iff the membership function of F N is continuous in K-topology, or equivalently, iff F N is a rough fuzzy set (concept) of I N D(K),or plainly, iff the membership function of F N is a “step” function in the sense that it takes a constant value in each equivalence class of I N D(K).A fuzzy set L partitions the universe into equivalence classes; each equivalence class consists of those elements which have the same degree of membership. The indiscernibility relation I N D(K)over K is the “intersection” of all the equivalence relations induced by the fuzzy sets in K. The membership function of a fuzzy set is a real valued function, so U can be given the minimal topology such that the membership function of each fuzzy set in K is a continuous function. Such topology is called K-topology.

Uncertainty about future developments constitutes the most important and most difficult challenge for mankind. Despite this fact, uncertainty is not a part of general science. General science assumes that the future development follows cause-effect relations which can be described by mathematical functions, where the argument represents the cause and the image represents the effect. Scientific theories have exactly this form and it is widely believed that these functions represent “truth”. Of course, this is nonsense as all the scientific theories are with certainty wrong and cannot describe the real evolution correctly. The inappropriate handling of uncertainty in science has produced a strange variety of “uncertainty theories” that causes confusion and helplessness. A friend of mine has expressed his confusion by the following words: ‘Crisp sets’, ‘fuzzy sets’, ‘rough sets’, ‘grey sets’, ‘fuzzy rough sets’, ‘rough fuzzy sets’, ‘fuzzy grey sets’, ‘grey fuzzy sets’, ‘rough grey sets’, ‘grey rough sets’, and now ‘affinity sets’. My goodness! Is there anybody around who can enlighten me, i.e., help me to see a clear pattern in this set of sets, allegedly providing powerful tools to model various kinds of uncertainty? This paper examines the role and the handling of uncertainty in quality control. How is uncertainty quantified in quality control for making decisions aiming at maintaining or improving quality of processes and products.KeywordsRandomnessIgnoranceKnowledgeFuzzynessQuantificationProbabilityCredibilityProbability spaceUncertainty spaceBernoulli space

In order to effectively handle the real-valued data sets in practice, it is valuable from theoretical and practical aspects to combine fuzzy rough set and variable precision rough set so that a powerful tool can be developed. That is, the model of fuzzy variable precision rough set, which not only can handle numerical data but also is less sensitive to misclassification and perturbation,In this paper, we propose a new variable precision rough fuzzy set by introducing the variable precision parameter to generalized rough fuzzy set, i.e., the variable precision rough fuzzy set based on general relation. We, respectively, define the variable precision rough lower and upper approximations of any fuzzy set and it level set with variable precision parameter by constructive approach. Also, we present the properties of the proposed model in detail. Meanwhile, we establish the relationship between the variable precision rough approximation of a fuzzy set and the rough approximation of the level set for a fuzzy set. Furthermore, we give a new approach to uncertainty measure for variable precision rough fuzzy set established in this paper in order to overcome the limitations of the traditional methods. Finally, some numerical example are used to illuminate the validity of the conclusions given in this paper.

Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes.

Based on the analysis of the rough set model on a tolerance relation and the fuzzy rough set, two types of fuzzy rough sets models on tolerance relations are constructed and researched. Then we propose the optimistic and pessimistic multi-granulation fuzzy rough sets models in a fuzzy tolerance approximation space with the point view of granular computing. In these models, the fuzzy lower and upper approximations of a fuzzy set are defined on multiple fuzzy tolerance relations. It follows the research on the properties of the fuzzy lower and upper approximations of the new models. The fuzzy rough set model, rough set model on a tolerance relation and multi-granulation rough sets models are special cases of the new ones from the perspective of the considered concepts, related relations and granular computing. The new models are meaningful generalizations of the classical rough sets.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft---rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.

On (IO,O)-fuzzy rough sets based on overlap functions

Rough sets and fuzzy sets are both theories for handling uncertain problems, and the viewpoints between them are different. They are complementary than competitive. Much seminal research has been carried out into combining them both; the results are rough fuzzy sets and fuzzy rough sets. In this paper, we introduce the idea of fuzzy rough sets in research of decision tables with quantitative data. Thus, we can handle the decision table directly without discretization. Combining the idea of variable precision rough sets, we make the form of the upper and lower approximation uniform.

On the granular representation of fuzzy quantifier-based fuzzy rough sets

Soft rough fuzzy sets and soft fuzzy rough sets

In this study, we establish a connection between rough soft set (Shabir et al., Knowl Base Syst 40:72–80, 2013) and fuzzy set. Based on the novel granulation structure called modified soft rough approximation space, fuzzy rough soft set is introduced. The important basic properties of fuzzy rough soft set are studied and supported by illustrative examples. Moreover lattice theory is studied on fuzzy rough soft set. The definitions and propositions presented in this paper enrich the soft set theory, rough set theory and fuzzy set theory, and also extend their application scopes. The paper ends with conclusions having future investigations of the study.

Fuzzy rough set theory can model uncertainty in data and has been applied to feature selection for machine learning tasks. The existence of noise in data is one of the reasons for data uncertainty. However, most classical fuzzy rough set models are often sensitive to the noise in data, which somewhat degrades their applicability to process uncertainty of data. Furthermore, a robust feature evaluation function is nontrivial in a fuzzy rough set model as a nonoptimal feature subsets may be selected due to the perturbations from redundant features. In this article, we delve into local density and indispensable features for fuzzy rough feature selection to address these challenges. We first propose a local density-based fuzzy rough set (LDFRS) model to tackle noisy data. Mutual information is then plugged into the proposed LDFRS model to evaluate uncertainty in data. A joint feature evaluation function on the indispensability and relevance of features is constructed to evaluate the significance of features. On this basis, a fuzzy rough feature selection algorithm is built upon the LDFRS model. Experimental results using four typical classifiers demonstrate the robustness and effectiveness of the proposed model including our feature selection algorithm and its superiority against baseline methods.