Fuzzy Approximation Space Research Articles

Multigranulation variable-scale fuzzy neighborhood measures and corresponding Choquet-like integrals for feature selection

Choquet-like integrals, a nonlinear fuzzy aggregation function, are diffusely applied in several real problems. However, the corresponding fuzzy measures in them are provided by human intervention and not by data-driven methods. As an effective knowledge representation tool, rough set models in different approximation spaces are established for data processing. Particularly, the multigranulation fuzzy β-covering approximation space has been concerned with only a single scale β∈(0,1]. However, different granulation structures should have different scales. Therefore, establishing a multigranulation approximation space with a variable scale by inducing fuzzy measures through data-driven methods and using corresponding Choquet-like integrals in practical problems is a significant challenge. To address this challenge, the notion of multigranulation variable-scale fuzzy Θ-covering group approximation space is presented here, as well as multigranulation variable-scale fuzzy neighborhood measures in it. Furthermore, Choquet-like integrals with the presented measures are constructed to solve the issue of feature selection (i.e., attribute reduction) under the new approximation spaces. Firstly, the concept of multigranulation variable-scale fuzzy Θ-covering group approximation space is presented, where different fuzzy β-coverings have different scales β∈Θ. Moreover, multigranulation variable-scale fuzzy neighborhood measures in it, as fuzzy measures, are presented. Subsequently, Choquet-like integrals with multigranulation variable-scale fuzzy neighborhood measures are constructed. A novel attribute reduction method under Choquet-like integrals with multigranulation variable-scale fuzzy neighborhood measures is proposed for any decision information table. Finally, the presented method is selected for solving problems of classification and diagnosis of rolling bearing faults. Several public data sets are used to demonstrate the effectiveness and feasibility of the presented methods mentioned above.

Using the Degree r ∈ [0, 1] in Defining Rough Fuzzy Sets

This paper introduced a new definition of rough fuzzy sets based on a fuzzy ideal defined on a fuzzy approximation space (X, R). This definition considered the degree r ∈ [0, 1] in defining the after sets and the before sets of the arbitrary fuzzy relation R on the finite set of objects X. It is shown that this new type of roughness is a generalization to many of previous definitions in the fuzzy case and also in the ordinary case. As an application, it is given a rough fuzzy‐connected set notion depending on some degree r ∈ [0, 1].

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 neighborhood Choquet integrals with overlap functions and their applications in attribute reduction

In fuzzy rough sets, fuzzy neighborhood operators and fuzzy rough approximate operators are mainly used by traditional methods to deal with the problem of attribute reduction (i.e., feature selection). However, they are parametric linear, while the corresponding weights are additive measures. For complex data, parametric linear operators and additive measures do not reflect correlations among data (or attributes). Therefore, it is important to develop non-additive measures and nonlinear operators for attribute reduction, which can effectively find those attributes that are not needed to reach a decision from the viewpoint of correlations. Along these lines, in this work, OI-fuzzy β-neighborhood measures (i.e., non-additive measures) and generalized Choquet integrals (i.e., nonlinear integrals) are presented by using overlap functions to deal with the problem of attribute reduction under the fuzzy β-covering approximation space. First, four pairs of OI-fuzzy β-neighborhood operators are constructed based on the implementation of overlap functions and their residual implications in a fuzzy β-covering approximation space. Furthermore, four pairs of OI-fuzzy β-neighborhood measures, as non-additive measures, are introduced to instead of rough approximate operators in data mining. Then, a novel method with respect to OI-fuzzy β-neighborhood measures is presented to deal with the attribute reduction in a fuzzy β-covering information decision table. Third, four pairs of generalized Choquet integrals based on the employment of OI-fuzzy β-neighborhood measures are constructed. On top of that, another method with respect to these generalized Choquet integrals is proposed to deal with the attribute reduction in a fuzzy β-covering information decision table. Finally, the presented methods are used to deal with the problem of classification. Several public data sets are applied to illustrate the feasibility and effectiveness of the above-mentioned proposed methods.

Active Antinoise Fuzzy Dominance Rough Feature Selection Using Adaptive K-Nearest Neighbors

Feature selection methods with anti-noise performance are effective dimensionality reduction methods for classification tasks with noise. However, there are few studies on robust feature selection methods for monotonic classification tasks. The fuzzy dominance rough set (FDRS) model is a nontrivial knowledge acquisition tool, which is widely used in feature selection of monotonic classification tasks. Nonetheless, this model has been proved in practice to be generally poorly fault-tolerance, and only one noisy sample can cause huge interference in acquiring knowledge. In view of these two issues, this paper firstly designs an adaptive <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 neighbors strategy to calculate the density of samples. The noisy samples are identified according to their densities, and then an active anti-noise fuzzy dominance rough set model is proposed. Then, in the active anti-noise fuzzy dominance rough approximation space, the class-separability is evaluated by the approximation operators of the proposed model, and the feature-redundancy is evaluated by the fuzzy ranking conditional mutual information. On this basis, a feature evaluation index is designed comprehensively considering class-separability and feature-redundancy. Finally, a feature selection algorithm is designed to select the feature subset with the highest classification performance. The experimental results show that the proposed algorithm has better robustness and classification performance.

Rough set model based on axiomatic fuzzy set

Both fuzzy set and rough set are important mathematical tools to describe incomplete and uncertain information, and they are highly complementary to each other. What is more, most fuzzy rough sets are obtained by combining Zadeh fuzzy sets and Pawlak rough sets. There are few reports about the combination of axiomatic fuzzy sets and Pawlak rough sets. For this reason, we propose the axiomatic fuzzy rough sets (namely rough set model with respect to the axiomatic fuzzy set) establishing on fuzzy membership space. In this paper, we first present a similarity description method based on vague partitions. Then the concept of similarity operator is proposed to describe uncertainty in the fuzzy approximation space. Finally, some characterizations concerning upper and lower approximation operators are shown, including basic properties. Furthermore, we give a algorithm to verify the effectiveness and efficiency of the model.

TI-fuzzy neighborhood measures and generalized Choquet integrals for granular structure reduction and decision making

Granular structures are mathematical representations of knowledge used in granular computing. Any fuzzy information table can be seen as a fuzzy granular structure, which is called a fuzzy β-covering approximation space. Hence, corresponding granular computing methods are widely used in data analysis, granular reduction and decision making. Currently, these methods mainly use rough approximate operators to process data where attributes are independent of each other in rough approximate operators. However, attributes are often associated with each other in real-life problems. As a parametric nonlinear aggregation function, the Choquet integral (CI) with respect to a fuzzy measure (FM) solves the problem of attribute association well. In this paper, we present TI-fuzzy β-neighborhood measures, which are FMs and generalized CIs, to deal with granularity reduction and decision making in a fuzzy β-covering approximation space. Firstly, four pairs of TI-fuzzy β-neighborhood measures under the t-norm “T” and its residual implication “IT”, as FMs, are presented for use in granular computing instead of rough approximation operators. Then, a novel method with TI-fuzzy β-neighborhood measures is presented to deal with granularity reduction in the fuzzy β-covering approximation space. Thirdly, four pairs of generalized CIs based on the TI-fuzzy β-neighborhood measures are constructed. By combining the presented CIs with the fuzzy PROMETHEE method, we propose a new method to solve the problem of decision making. Finally, several numerical examples and UCI data sets are used to illustrate the feasibility and effectiveness of our proposed methods.

Precipitation prediction by integrating Rough Set on Fuzzy Approximation Space with Deep Learning techniques

Artificial intelligence has evolved into a mechanism that can predict environmental changes and investigate climatic conditions. In recent days, extreme precipitation and flooding have resulted in huge damage to the society of human beings and farmers. In real-life, the climatic conditions were predicted using huge collected data. However, the collected data are not utilized properly due to the presence of uncertainty. Most of the research was carried out in time series data for classification and prediction. However uncertain data leads to erroneous predictions. Thus, an integrated model is required to overcome the uncertainty and to improve the prediction accuracy. The paper aims in integrating the Rough Set on Fuzzy Approximation Space (RSFAS) with a Deep Learning (DL) techniques to predict the precipitation level in the southern coastal areas of India in a seasonal way. The proposed model can handle uncertainty using an RSFAS whereas the DL technique is utilized for classification and prediction. The developed model is estimated based on various evaluation metrics and optimizers, benchmarked with existing Deep Learning and Machine Learning (ML) techniques to demonstrate the prediction accuracies obtained with minimized error rate. Consequently, the proposed model assists in understanding the climatic conditions in extreme events to provide a prior warning about heavy rainfall to the people who live near the coastal areas.

On L-Fuzzy Topological Spaces Determined by Implicators

This paper is towards the study of [Formula: see text]-fuzzy topological spaces, [Formula: see text]-fuzzy pretopological spaces and their corresponding operators determined by general implicators. We show that the results regarding one-to-one correspondence between [Formula: see text]-fuzzy reflexive approximation spaces and [Formula: see text]-fuzzy pretopological spaces as well as [Formula: see text]-fuzzy preorder approximation spaces and [Formula: see text]-fuzzy topological spaces heavily depend on implicators. Finally, we establish a relationship between category of [Formula: see text]-fuzzy pretopological spaces and category of [Formula: see text]-fuzzy topological spaces by using category of [Formula: see text]-fuzzy approximation spaces.

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.

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.

On Covering-Based Rough Intuitionistic Fuzzy Sets

Intuitionistic Fuzzy Sets (IFSs) and rough sets depending on covering are important theories for dealing with uncertainty and inexact problems. We think the neighborhood of an element is more realistic than any cluster in the processes of classification and approximation. So, we introduce intuitionistic fuzzy sets on the space of rough sets based on covering by using the concept of the neighborhood. Three models of intuitionistic fuzzy set approximation space based on covering are defined by using the concept of neighborhood. In the first and second model, we approximate IFS by rough set based on one covering (C) by defining membership and non-membership degree depending on the neighborhood. In the third mode, we approximate IFS by rough set based on family of covering (Ci) by defining membership and non-membership degree depending on the neighborhood. We employ the notion of the neighborhood to prove the definitions and the features of these models. Finlay, we give an illustrative example for the new covering rough IF approximation structure.

Three-way multi-criteria group decision-making method in a fuzzy [formula omitted]-covering group approximation space

At present, some researchers have studied the decision-making methods in a fuzzy β-covering approximation space, which not only can play the advantages of rough set theory in dealing with inaccurate data, but also inherit the ranking function of traditional decision-making methods. However, these methods merely consider the ranking problem in a single decision-maker environment and most of the decision-making problems in reality are group decision-making problems that need to consider multiple opinions. In light of this, in this paper, we propose the concept of fuzzy β-covering group approximation spaces and establish a three-way multi-criteria group decision-making method, which can solve some ranking and classification problems of objects under a group decision-making environment. Based on a fuzzy β-covering group approximation space, we firstly propose two fuzzy β-fitting neighborhoods with pessimistic and optimistic attitudes to construct a fuzzy binary relation between any two objects. Secondly, we introduce an overall loss function to estimate the risk loss of all objects when they choose different behaviors in different states under a group decision-making environment. Subsequently, based on the conditional probability estimation formula and the overall loss function, we propose a three-way group decision-making idea in a fuzzy β-covering group approximation space, which contains eight different decision-making attitudes to meet the preferences of decision-makers. Furthermore, for the ranking and classification performance of our method, we use numerical analysis, comparative analysis and Spearman analysis to illustrate the feasibility and superiority of the method, and take experimental analysis to test the stability of our method.

Measures of uncertainty for a fuzzy probabilistic approximation space

An approximation space (A-space) is the base of rough set theory and a fuzzy approximation space (FA-space) can be seen as an A-space under the fuzzy environment. A fuzzy probability approximation space (FPA-space) is obtained by putting probability distribution into an FA-space. In this way, it combines three types of uncertainty (i.e., fuzziness, probability and roughness). This article is devoted to measuring the uncertainty for an FPA-space. A fuzzy relation matrix is first proposed by introducing the probability into a given fuzzy relation matrix, and on this basis, it is expanded to an FA-space. Then, granularity measurement for an FPA-space is investigated. Next, information entropy measurement and rough entropy measurement for an FPA-space are proposed. Moreover, information amount in an FPA-space is considered. Finally, a numerical example is given to verify the feasibility of the proposed measures, and the effectiveness analysis is carried out from the point of view of statistics. Since three types of important theories (i.e., fuzzy set theory, probability theory and rough set theory) are clustered in an FPA-space, the obtained results may be useful for dealing with practice problems with a sort of uncertainty.

Linear Diophantine Fuzzy Rough Sets: A New Rough Set Approach with Decision Making

In this article, a new hybrid model named linear Diophantine fuzzy rough set (LDFRS) is proposed to magnify the notion of rough set (RS) and linear Diophantine fuzzy set (LDFS). Concerning the proposed model of LDFRS, it is more efficient to discuss the fuzziness and roughness in terms of linear Diophantine fuzzy approximation spaces (LDFA spaces); it plays a vital role in information analysis, data analysis, and computational intelligence. The concept of (&lt;p,p′&gt;,&lt;q,q′&gt;)-indiscernibility of a linear Diophantine fuzzy relation (LDF relation) is used for the construction of an LDFRS. Certain properties of LDFA spaces are explored and related results are developed. Moreover, a decision-making technique is developed for modeling uncertainties in decision-making (DM) problems and a practical application of fuzziness and roughness of the proposed model is established for medical diagnosis.

Redefined fuzzy rough set models in fuzzy β-covering group approximation spaces

The research purpose of this paper is to explore some new fuzzy rough set models in the fuzzy β-covering group approximation space (FβCGAS) and study the related properties, thereby provide new thinking directions for the theoretical development and application expansion of fuzzy rough set models. Firstly, in the fuzzy β-covering approximation space (FβCAS), in light of some existing fuzzy covering rough set (FCRS) models which do not satisfy the inclusion relations between the fuzzy upper and lower approximations, we define the concepts of parameterized fuzzy β-neighborhoods and parameterized fuzzy complementary β-neighborhoods to eliminate part of the noise data and establish four types of parameterized FCRS models that satisfy the inclusion relationship between the fuzzy lower and upper approximations. Meanwhile, the properties of the models are discussed. Further, in the FβCGAS, we define the concepts of the parameterized pessimistic (optimistic) fuzzy β-fitting neighborhoods and the parameterized pessimistic (optimistic) fuzzy complementary β-fitting neighborhoods and construct the parameterized pessimistic and optimistic FCRS models that satisfy the inclusion relationship between the fuzzy lower and upper approximations. At the same time, we likewise explore the properties of the models. Besides, we further use the newly proposed parameterized fuzzy neighborhoods to study the variable precision multi-granulation fuzzy decision-theoretic rough set models in the FβCGAS. Finally, for all the models mentioned above, we recommend them for solving the criteria weight determination problem, ranking problem, optimal selection problem and qualitative classification problem in multi-criteria decision-making. Meanwhile, we complete the relationships of all the new models mentioned in this paper and discuss them.

Spherical Linear Diophantine Fuzzy Soft Rough Sets with Multi-Criteria Decision Making

Modeling uncertainties with spherical linear Diophantine fuzzy sets (SLDFSs) is a robust approach towards engineering, information management, medicine, multi-criteria decision-making (MCDM) applications. The existing concepts of neutrosophic sets (NSs), picture fuzzy sets (PFSs), and spherical fuzzy sets (SFSs) are strong models for MCDM. Nevertheless, these models have certain limitations for three indexes, satisfaction (membership), dissatisfaction (non-membership), refusal/abstain (indeterminacy) grades. A SLDFS with the use of reference parameters becomes an advanced approach to deal with uncertainties in MCDM and to remove strict limitations of above grades. In this approach the decision makers (DMs) have the freedom for the selection of above three indexes in [0,1]. The addition of reference parameters with three index/grades is a more effective approach to analyze DMs opinion. We discuss the concept of spherical linear Diophantine fuzzy numbers (SLDFNs) and certain properties of SLDFSs and SLDFNs. These concepts are illustrated by examples and graphical representation. Some score functions for comparison of LDFNs are developed. We introduce the novel concepts of spherical linear Diophantine fuzzy soft rough set (SLDFSRS) and spherical linear Diophantine fuzzy soft approximation space. The proposed model of SLDFSRS is a robust hybrid model of SLDFS, soft set, and rough set. We develop new algorithms for MCDM of suitable clean energy technology. We use the concepts of score functions, reduct, and core for the optimal decision. A brief comparative analysis of the proposed approach with some existing techniques is established to indicate the validity, flexibility, and superiority of the suggested MCDM approach.

Relationships between fuzzy probabilistic approximation spaces and their entropy measurement

A fuzzy probability approximation space (FPA-space) is a approximation space (A-space) where three types of uncertainty (probability, fuzziness and roughness) are combined, which is obtained by putting probability distribution into a fuzzy approximation space (FA-space). This paper studies relationships between FPA-spaces and and their entropy measurement. Two types of fuzzy relation matrices are first defined by introducing the probability into a given fuzzy relation matrix in two ways, and on this basis, they are extended to two FA-spaces. Then, equality, dependence and independence between FPA-spaces are studied. Next, the distance between FPA-spaces is discussed. Moreover, the uncertainty for an FPA-space is measured by means of information entropy. Finally, the proposed information entropy is applied in the selection of classifier systems. Since fuzzy set theory, probability theory and rough set theory are aggregated together in an FPA-space, the obtained results of this paper may be helpful for dealing with practice problems with a sort of uncertainty.

An approach to the study of fuzzy rough prime ideal extensions of semigroups

Fuzzy rough set is a pair of lower and upper approximations of fuzzy set in a fuzzy approximation space. This paper studies the extension of fuzzy rough prime and semi-prime ideals in semigroups. Then we introduce the composition theory on fuzzy rough set. Finally some new characterization of semigroups by the properties of composition on fuzzy rough ideal extension are given.

Topological properties of multigranular rough sets on intuitionistic fuzzy approximation spaces

Two of the extensions of the basic rough set model introduced by Pawlak in 1982 is the rough sets on fuzzy approximation spaces introduced by De et al. in 2003 and its extension rough sets on intuitionistic fuzzy approximation spaces in 2006 by Tripathy. The model of Tripathy is dependent upon an intuitionistic fuzzy proximity relation. In an attempt to handle multiple granulations simultaneously two types of multigranulations called the optimistic and pessimistic multigranulation were introduced by Qian et al. in 2006 and 2010 respectively. The notion of multigranular rough sets over intuitionistic fuzzy approximation spaces were introduced by Tripathy et al. in 2018. Topological properties of rough sets, introduced by Pawlak have been studied extensively for different types of rough sets by Tripathy et al. Continuing this, here we introduce and study the concept of topological properties of multigranular rough sets on intuitionistic fuzzy approximation spaces.