Fuzzy Rough Approximation Research Articles

Generalized Fuzzy Rough Approximations on Hypergroups

In this paper, we define the fuzzy set-valued homomorphisms of the canonical hypergroups as a generalization of fuzzy congruences and investigate some of their features. This structure is an extension of the definition of set-valued homomorphism defined for groups to hypergroups. With this extension, it has become possible to study generalized fuzzy rough approximations in hyperalgebraic structures such as semihypergroups, polygroups, hyperrings, hypermodules, etc. This paper presents the generalized fuzzy rough approximations based on two-universe (I,T)-fuzzy model on canonical hypergroups.

Relational correspondences for L-fuzzy rough approximations defined on De Morgan Heyting algebras

We consider fuzzy rough sets defined on De Morgan Heyting algebras. We present a theorem that can be used to obtain several correspondence results between fuzzy rough sets and fuzzy relations defining them. We characterize fuzzy rough approximation operators corresponding to compositions of reflexive, transitive, mediate, Euclidean and adjoint fuzzy relations defined on De Morgan Heyting algebras using only a single axiom.

A Robust Multilabel Feature Selection Approach Based on Graph Structure Considering Fuzzy Dependency and Feature Interaction

The performance of multilabel learning depends heavily on the quality of the input features. A mass of irrelevant and redundant features may seriously affect the performance of multilabel learning, and feature selection is an effective technique to solve this problem. However, most multilabel feature selection methods mainly emphasize removing these useless features, and the exploration of feature interaction is ignored. Moreover, the widespread existence of real-world data with uncertainty, ambiguity, and noise limits the performance of feature selection. To this end, our work is dedicated to designing an efficient and robust multilabel feature selection scheme. First, the distribution character of multilabel data is analyzed to generate robust fuzzy multi-neighborhood granules. By exploring the classification information implied in the data under the granularity structure, a robust multilabel <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -nearest neighbor fuzzy rough set model is constructed, and the concept of fuzzy dependency is studied. Second, a series of fuzzy multi-neighborhood uncertainty measures in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -nearest neighbor fuzzy rough approximation spaces are studied to analyze the correlations of feature pairs, including interactivity. Third, by investigating the uncertainty measure between feature and label, between features, multilabel data is modeled as a complete weighted graph. Then, these vertices are assessed iteratively to guide the assignment of feature weights. Finally, a graph structure-based robust multilabel feature selection algorithm (GRMFS) is designed. The experiments are conducted on fifteen multilabel datasets. The results verify the superior performance of GRMFS as compared with nine representative feature selection methods.

Fuzzy rough sets based on fuzzy quantification

Classical (fuzzy) rough sets exhibit sensitivity to noise, which is particularly undesirable for machine learning applications. One approach to solve this issue is by making use of fuzzy quantifiers, as done by the vaguely quantified fuzzy rough set (VQFRS) model. While this idea is intuitive, the VQFRS model suffers from both theoretical flaws as well as from suboptimal performance in applications. In this paper, we improve on VQFRS by introducing fuzzy quantifier-based fuzzy rough sets (FQFRS), which proposes an intuitive fuzzy rough approximation operator that utilizes general unary and binary quantification models. We show how several existing models fit inside FQFRS, as well as how it inspires novel ones. Additionally, we propose several binary quantification models to be used with FQFRS. Furthermore, we conduct a theoretical study of their properties, and investigate their potential by applying them to classification problems. In particular, we highlight the effectiveness of Yager's Weighted Implication-based (YWI) binary quantification model, which induces a fuzzy rough set model that is both a significant improvement on VQFRS, as well as a worthy competitor to the popular ordered weighted averaging based fuzzy rough set (OWAFRS) model.

Representations of L-fuzzy rough approximation operators

In recent years, there has been tremendous interest in developing lattice-valued rough set theory. In this framework, we primarily take into account the following two problems. Firstly, we include the well-known intuitionistic, interval-valued, neutrosophic and Pythagorean fuzzy rough sets into the framework of lattice-valued rough sets. Specifically, the four kinds of rough sets can be considered as special lattice-valued rough sets. Secondly, based on a completely distributive lattice L, we provide representations of the upper and lower L-fuzzy rough approximation operators by using four kinds of cut sets of an L-fuzzy set and an L-fuzzy relation, which generalizes the existing results in the case that L=[0,1]. In particular, we show that representations of intuitionistic and interval-valued fuzzy rough approximation operators provided by Zhou and Sun are special examples of our proposed representations.

Novel results on G -lower L -fuzzy rough approximation operator in ( O , G ) -fuzzy rough set model1

Lately, Jiang and Hu (H.B. Jiang, B.Q. Hu, On ( O , G ) -fuzzy rough sets based on overlap and grouping functions over complete lattices, Int. J. Approx. Reason. 144 (2022) 18-50.) put forward ( O , G ) -fuzzy rough sets via overlap and grouping functions over complete lattices. Meanwhile, they showed the characterizations of O -upper and G -lower L -fuzzy rough approximation operators in ( O , G ) -fuzzy rough set model based on some of specific L -fuzzy relations and studied the topological properties of the proposed model. Nevertheless, we discover that the partial results given by Jiang and Hu could be further optimized. So, as a replenish of the above article, in this paper, based on G -lower L -fuzzy rough approximation operator in ( O , G ) -fuzzy rough set model, we further explore several new conclusions on the relationship between G -lower L -fuzzy rough approximation operator and different L -fuzzy relations. In particular, the equivalent descriptions of relationship between G -lower L -fuzzy rough approximation operator and O -transitive ( O -Euclidean) L -fuzzy relations are investigated, which are not involved in above literature and can make the theoretical results of this newly fuzzy rough set model more perfect.

Axiomatic characterizations of (𝔾, 𝕆)-fuzzy rough approximation operators via overlap and grouping functions on a complete lattice

Recently, Jiang, H. B., and B. Q. Hu. [2022. “On (O,G)-Fuzzy Rough Sets Based on Overlap and Grouping Functions Over Complete Lattices.” International Journal of Approximate Reasoning 144: 18–50. doi:10.1016/j.ijar.2022.01.012] constructed a -fuzzy rough set model with the logical connectives–a grouping function and an overlap function on a complete lattice, which provided a new constructive approach to fuzzy rough set theory. The axiomatic approach is as important as the constructive approach in rough set theory. In this paper, we continue to study axiomatic characterizations of -fuzzy rough set. Traditionally, the associativity of the logical connectives plays a vital role in the axiomatic research of existing fuzzy rough set models. However, a grouping function and an overlap function lack the associativity. So we explore a novel axiomatic approach to -upper and -lower fuzzy rough approximation operators without associativity. Further, we provide single axioms to characterize -upper and -lower fuzzy rough approximation operators instead of sets of axioms. Finally, we use single axioms to characterize fuzzy rough approximation operators generated by various kinds of fuzzy relations including serial, reflexive, symmetric, -transitive, -transitive fuzzy relations as well as their compositions.

Relative Fuzzy Rough Approximations for Feature Selection and Classification.

Fuzzy rough set (FRS) theory is generally used to measure the uncertainty of data. However, this theory cannot work well when the class density of a data distribution differs greatly. In this work, a relative distance measure is first proposed to fit the mentioned data distribution. Based on the measure, a relative FRS model is introduced to remedy the mentioned imperfection of classical FRSs. Then, the positive region, negative region, and boundary region are defined to measure the uncertainty of data with the relative FRSs. Besides, a relative fuzzy dependency is defined to evaluate the importance of features to decision. With the proposed feature evaluation, we propose a feature selection algorithm and design a classifier based on the maximal positive region. The classification principle is that an unlabeled sample will be classified into the class corresponding to the maximal degree of the positive region. Experimental results show the relative fuzzy dependency is an effective and efficient measure for evaluating features, and the proposed feature selection algorithm presents better performance than some classical algorithms. Besides, it also shows the proposed classifier can achieve slightly better performance than the KNN classifier, which demonstrates that the maximal positive region-based classifier is effective and feasible.

Attribute reduction for hybrid data based on fuzzy rough iterative computation model

A hybrid information system (HIS) means an information system (IS) with categorical attributes, real-valued attributes and missing values. It is more difficult to deal with an HIS than an IS with a type of attributes. This paper studies attribute reduction in an HIS by means of fuzzy rough iterative computation model (FRIC-model). To more accurately express the difference between objects in an HIS, a novel distance function on the object set is first constructed. Then, fuzzy symmetric relations on the object set based on this function are established and fuzzy rough approximations are defined. Next, some evaluation functions that represent the classification ability of attribute subsets, such as fuzzy positive regions, dependency functions and attribute importance functions are introduced. After then, FRIC-model is given and an attribute reduction algorithm for hybrid data based on this model is designed. At last, numerical experiments are carried out to estimate the performance of the designed algorithm. The experimental results show that the designed algorithm is more effective than some existing algorithms.

Robust fuzzy rough approximations with kNN granules for semi-supervised feature selection

Fuzzy rough set theory has attracted much attention because of its successful application in uncertainty measurement. To improve the efficiency and robustness of uncertainty measure using the theory, robust fuzzy rough approximations have become popular research. In this work, it first designs a novel uncertainty measure based on kNN granules. By combining the relative distance, the method is effectively applicable to the datasets with large density variation. Using the measure a robust FRS model is studied and simply denoted KNN-FRS model. The new model takes a compromise between the maximum lower approximation degree and the minimum classification loss in computing fuzzy rough approximations, which reduces the sensitivity of the classical FRS model to noise. Besides, some properties of KNN-FRS model are summarized. Next, KNN-FRS model is used to design a feature selection algorithm which is tested to validate the feasibility of the KNN-FRS model. And then the KNN-FRS model is applied to design a semi-supervised feature selection algorithm which selects features while labeling samples and the labeling method is designed based on the proposed uncertainty measure. Some experiments are conducted to validate the KNN-FRS model and the designed algorithms. The results illustrate the proposed KNN-FRS model is feasible, effective and robust. And the designed semi-supervised feature selection algorithm performs excellent performance.

Fuzzy approximation based on $ \tau- \mathfrak{K} $ fuzzy open (closed) sets

Rough set theory can be generalized by induced topology through equivalence relations. Motivated by the work of generalization of the rough set via topology, the concept and properties of $ \tau-\mathfrak{K} $-fuzzy open (closed) sets are proposed. Considering the $ \tau-\mathfrak{K} $-fuzzy open (closed) sets, we have obtained the $ \tau-\mathfrak{K} $-fuzzy lower and upper approximations and also proved their properties. $ \tau-\mathfrak{K} $-fuzzy open sets can be represented as $ \tau-\mathfrak{K} $-open sets by $ \alpha $ level sets. The properties of $ \tau-\mathfrak{K} $-fuzzy approximations and fuzzy rough approximations on the basis of binary fuzzy relation are compared. Finally, an example and the decision method's algorithm to illustrate the $ \tau-\mathfrak{K} $-fuzzy approximation-based approach to decision making are presented.

A Further Study on the Single Axiom Characterization of (∇,▲)-Fuzzy Rough Approximation Operators

A Further Study on the Single Axiom Characterization of (∇,▲)-Fuzzy Rough Approximation Operators

Feature Grouping and Selection With Graph Theory in Robust Fuzzy Rough Approximation Space

Most extant feature selection works neglect interactive features in the form of groups, leading to the omission of some important discriminative information. Moreover, the prevalence of data with uncertainty, fuzziness, and noise poses a certain obstacle to feature selection. Driven by these two issues, a Feature Grouping and Selection approach in Robust Fuzzy Rough Approximation Space using graph theory (FGS-RFRAS) is proposed in this study. First, a robust fuzzy rough approximation space is constructed by a neighborhood adaptive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> -precision fuzzy rough set model to enhance the robustness and antinoise ability of the fuzzy rough set model. Second, uncertainty measures in robust fuzzy rough approximation space are defined to analyze the interactivity and redundancy of pairwise features on graph structure. Then, a strategy of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Interactive Retainment, Weakly Correlated Removal, and Max-Dependent Selection</i> is devised to guide feature grouping and selection. Experiments are performed on 21 datasets to evaluate the performance of FGS-RFRAS and demonstrate its significance. The robustness test indicates that it is antinoise for mislabeling.

Enriched Fuzzy  -Algebra and Fuzzy RoughApproximation Operators

In this paper, we introduce the concept of enriched fuzzy -algebra and establish its relation to fuzzy rough approximation operators. Firstly, the concept of enriched fuzzy  -algebra is proposed as an extension of existing fuzzy -algebras, and several concrete examples are provided. Particularly, it has been proven that each - algebra can naturally generate an enriched fuzzy  -algebra. Secondly, the methods for constructing enriched fuzzy  -algebra using reflexive fuzzy rough approximation operators and serial rough fuzzy approximation operators are given, respectively. Finally, it is shown that enriched fuzzy  -algebras generated by -algebras must be induced by some rough fuzzy approximation operator.

On two new types of fuzzy rough sets via overlap functions and corresponding applications to three-way approximations

Fuzzy rough sets (FRSs) can be served as a useful technique to cope with inconsistent data by providing a pair of lower and upper fuzzy rough approximations of a fuzzy concept. Diversified FRS models have been proposed over the years. Meanwhile, t-norms and t-conorms, as two binary aggregation operators, play a critical part in the construction of fuzzy rough approximation operators. Nevertheless, one of common properties of t-norms and t-conorms is that they all fulfill associativity. Additionally, the associativity plays an important role in discussing properties of FRSs. Therefore, if the binary aggregation function does not satisfy associativity, how to define FRS models and discuss the relevant properties is a topic worthy of further investigation. Overlap functions, as a new kind of not necessary associative binary aggregation operator, have been continuously investigated by many scholars in theory and practical applications. Therefore, based on the advantages of overlap functions, and as the extension models for FRSs, we propose two new types of FRSs via overlap functions and their residual implications (or called IO-implications) in this paper. More specifically, we first give the definition of the 1st type of FRSs and study some essential properties. Subsequently, the definition of the 2nd type of FRSs is presented and the relevant properties are also investigated. Furthermore, the relationship between the 1st type of FRSs and the 2nd type of FRSs is described. That is, in a fuzzy approximation space, the 2nd type of lower approximation is contained in the 1st type of lower approximation, while the 1st type of upper approximation is included in the 2nd type of upper approximation when the overlap function has 1 as the identity element. At the end, a novel three-way decision (3WD) method is presented on an information table by using the new type of FRSs, and its practicality and superiority are verified via a numerical example and a series of experimental analyses.

Hierarchical Fuzzy Rough Approximations With Three-Way Multigranularity Learning

The approximation learning of fuzzy concepts associated with fuzzy rough sets and three-way decisions is a useful technology for the representation, learning, and transformation of uncertain knowledge. However, how to integrate the advantages of fuzzy rough approximations and three-way approximations has rarely been carefully investigated so far. In this article, we focus on exploring the connection and interplay of these two methods, and further propose the hierarchical fuzzy rough approximations in the dynamic fuzzy open-world environment. We utilize the temporal-spatial perspectives of three-way decisions to construct multigranularity structures and implement multigranularity learning. The temporality of data and the spatiality of model parameters are both considered in such frameworks. Subsequently, we discuss the interpretation and representation of fuzzy three-way regions in fuzzy rough sets with some definitions and properties. The max, double, and aggregated evaluation-based models of three-way approximations are proposed to gradually transform a fuzzy concept to a crisp concept by the time-evolving attributes. Finally, the comparative experimental results between static and dynamic approaches demonstrate the effectiveness of our proposed hierarchical approximation learning models.

Feature Selection With Fuzzy-Rough Minimum Classification Error Criterion

Classical fuzzy rough set often uses fuzzy rough dependency as an evaluation function of feature selection. However, this function only retains the maximum membership degree of a sample to one decision class, it cannot describe the classification error. Therefore, in this article, a novel criterion function for feature selection is proposed to overcome this weakness. To characterize the classification error rate, we first introduce a class of irreflexive and symmetric fuzzy binary relations to redefine the concepts of fuzzy rough approximations. Then, we propose a novel concept of dependency: inner product dependency to describe the classification error and construct a criterion function to evaluate the importance of candidate features. The proposed criterion function not only can maintain a maximum dependency function, but also guarantees the minimum classification error. The experimental analysis shows that the proposed criterion function is effective for datasets with a large overlap between different categories.

On (GO, O)-fuzzy rough sets derived from overlap and grouping functions

Rough sets, as a powerful tool to deal with uncertainties and inaccuracies in data analysis, have been continuously concerned and studied by many scholars since it was put forward, especially the research on various rough set models. On the other hand, overlap and grouping functions, as two newly aggregation operators and mathematical model to handle the problems involving in information fusion, have been successfully applied in many real-life problems. In this paper, based on overlap and grouping functions, we propose a new fuzzy rough set model named (GO, O)-fuzzy rough sets and consider its characterizations along with topological properties. Properly speaking, firstly, we utilize QL-operators (and also QL-implications) constructed from overlap and grouping functions and fuzzy negations to define the lower approximation operator in (GO, O)-fuzzy rough set model named GO-lower fuzzy rough approximation operator and the upper approximation operator in (GO, O)-fuzzy rough set model is considered as the O-upper fuzzy rough approximation operator in (IO, O)-fuzzy rough set model proposed by Qiao recently. Secondly, we discuss lots of basic properties of (GO, O)-fuzzy rough sets, especially for the properties of GO-lower fuzzy rough approximation operator. Thirdly, we focus on the relationship between (GO, O)-fuzzy rough sets and concrete fuzzy relations. Finally, we give the topological properties of the upper and lower approximation operators in (GO, O)-fuzzy rough set model.

On (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices

In rough set theory, upper and lower approximation operators are two crucial concepts. The study of these two approximation operators in the framework of lattice theory is an important generalization from the mathematical point of view. On the other hand, overlap and grouping functions, as two types of not necessarily associative binary aggregation functions different from the common binary aggregation functions triangular norms and triangular conorms, have not only rich theoretical results but also a wide range of practical applications. Therefore, based on overlap and grouping functions over complete lattices, this paper is devoted to proposing (O,G)-fuzzy rough sets as a further generalization of the notion of rough sets. Firstly, we define a pair of O-upper and G-lower L-fuzzy rough approximation operators and investigate basic properties of them. Then, the characterizations of (O,G)-fuzzy rough approximation operators are discussed by using different kinds of L-fuzzy relations. Meanwhile, we investigate the topological properties of (O,G)-fuzzy rough sets. Furthermore, we show a brief comparison of the (O,G)-fuzzy rough sets with other common rough set models. At the end of this paper, we further propose multigranulation (O,G)-fuzzy rough sets over complete lattices from the viewpoint of multigranulation structure.

Feature selection for a set-valued decision information system based on fuzzy rough iterative computation model

To deal with set-valued data, one usually exploits the information value similarity, which is fed back to the object set. Alternatively, it is feasible but scarcely considered to handle set-valued data from the perspective of the information-value similarity which is fed back to the feature set. Based on this perspective, this paper studies feature selection for a set-valued decision information system (SVDIS) by means of fuzzy rough iterative computation model. In order to describe the similarity between objects, the fuzzy symmetric relations in an SVDIS are first defined, and a variable parameter to control the similarity is introduced. Then, new fuzzy rough set model for set valued data is proposed, and this model employs the iterative computation strategy to define fuzzy rough approximations and dependency functions. Next, fuzzy rough iterative computation model is established by using the iteration of fuzzy positive region and fuzzy dependency. Moreover, fuzzy rough iterative computation model is applied to the feature selection for set-valued data, and the corresponding algorithm is given. Finally, experiments are carried out to evaluate the performance of the given algorithm. The experimental results indicate that the given algorithm is more effective than some existing algorithms.