Variable Precision Fuzzy Rough Set Research Articles

Covering variable precision fuzzy rough sets based on overlap functions and the application to multi-label classification

Fuzzy rough sets (FRSs) play a significant role in the field of data analysis, and one of the common methods for constructing FRSs is the use of the fuzzy logic operators. To further extend FRSs theory to more diverse information backgrounds, this article proposes a covering variable precision fuzzy rough set model based on overlap functions and fuzzy β-neighbourhood operators (OCVPFRS). Some necessary properties of OCVPFRS have also been studied in this work. Furthermore, multi-label classification is a prevalent task in the realm of machine learning. Each object (sample or instance) in multi-label data is associated with various labels (classes), and there are numerous features or attributes that need to be taken into account within the attribute space. To enhance various performance metrics in the multi-label classification task, attribute reduction is an essential pre-processing step. Therefore, according to overlap functions and fuzzy rough sets’ excellent work on applications: such as image processing and multi-criteria decision-making, we establish an attribute reduction method suitable for multi-label data based on OCVPFRS. Through a series of experiments and comparative analysis with existing multi-label attribute reduction methods, the effectiveness and superiority of the proposed method have been verified.

Variable precision fuzzy rough sets based on overlap functions with application to tumor classification

Overlap functions, which can be characterized as a type of non-associative binary aggregation operators, have emerged as one of the most extensively utilized aggregation operators in numerous applications, including image processing, information fusion, and classification problems. At the same time, fuzzy rough sets have also been widely used in these fields due to their excellent ability to handle continuous and uncertain information. However, the variable precision fuzzy rough set model based on overlap functions and its applications have not been fully studied. For instance, some basic properties are invalid and there is a lack of practical applications. In this paper, both overlap functions and precision parameters are introduced into the fuzzy rough sets, namely the overlap function-based variable precision fuzzy rough set (OVPFRS), which is then used in the practical problem of tumor classification. First, considering the existing overlap function-based rough set models, the OVPFRS model is established, and some underlying properties of this model are explored. Second, on the basis of the proposed model, a method for attribute reduction is developed. Finally, the new method is applied to the classification of tumor data from the real world. Through experimentation and comparison with other attribute reduction methods, it has been demonstrated that our model is flexible and the algorithm is viable and effective.

Generalized fuzzy neighborhood system-based multigranulation variable precision fuzzy rough sets with double TOPSIS method to MADM

Multigranulation fuzzy rough sets (MFRSs), variable precision fuzzy rough sets (VPFRSs) and their combinations with traditional decision-making methods are fundamental in the research of multi-attribute decision-making (MADM) problems. However, the existing theories and methods have two deficiencies: (1) many existing MFRSs are based on fuzzy relations or fuzzy coverings, which may not effectively solve some MADM problems; (2) VPFRSs usually do not meet the inclusion property, which hinders their further development in theory and application. Noted that both fuzzy relations and fuzzy coverings can generate fuzzy neighborhood systems in a natural way, so the fuzzy neighborhood system provides a broader multigranulation framework. Inspired by this, by using the generalized fuzzy neighborhood system, we propose three multigranulation variable precision fuzzy rough set models, namely optimistic, pessimistic and compromise, respectively. As expected, these models contain many existing models as special cases, thus broadening the framework of rough set. Additionally, we verify that these models fulfill many basic algebra properties, particularly the generalized inclusion properties, which overcome the limitation of VPFRSs. Finally, by combining the proposed fuzzy rough set models with the TOPSIS method, we develop a new method to MADM. A series of experiments show that our method has some obvious advantages.

The axiomatic characterization on fuzzy variable precision rough sets based on residuated lattice

Axiomatization is a lively research direction in fuzzy rough set theory. Fuzzy variable precision rough set (FVPRS) incorporates fault-tolerant factors to fuzzy rough set, so its axiomatic description becomes more complicated and difficult to realize. In this paper, we present an axiomatic approach to FVPRSs based on residuated lattice (L-fuzzy variable precision rough set (LFVPRS)). First, a pair of mappings with three axioms is utilized to characterize the upper (resp., lower) approximation operator of LFVPRS. This is distinct from the characterization on upper (resp., lower) approximation operator of fuzzy rough set, which consists of one mapping with two axioms. Second, utilizing the notion of correlation degree (resp., subset degree) of fuzzy sets, three characteristic axioms are grouped into a single axiom. At last, various special LFVPRS generated by reflexive, symmetric and transitive L-fuzzy relation and their composition are also characterized by axiomatic set and single axiom, respectively.

(O, G)-granular variable precision fuzzy rough sets based on overlap and grouping functions

Since Bustince et al. introduced the concepts of overlap and grouping functions, these two types of aggregation functions have attracted a lot of interest in both theory and applications. In this paper, the depiction of (O, G)-granular variable precision fuzzy rough sets ((O, G)-GVPFRSs for short) is first given based on overlap and grouping functions. Meanwhile, to work out the approximation operators efficiently, we give another expression of upper and lower approximation operators by means of fuzzy implications and co-implications. Furthermore, starting from the perspective of construction methods, (O, G)-GVPFRSs are represented under diverse fuzzy relations. Finally, some conclusions on the granular variable precision fuzzy rough sets (GVPFRSs for short) are extended to (O, G)-GVPFRSs under some additional conditions.

Novel variable precision fuzzy rough sets and three-way decision model with three strategies

Variable precision (fuzzy) rough sets are interesting generalizations of Pawlak rough sets and can handle uncertain and imprecise information well due to their error tolerance capability. The comparable property (CP for short), i.e., the lower approximation is included in the upper approximation, is fundamental in Pawlak rough sets since many theories and applications rely on this property. However, the CP is lost in many existing variable precision (fuzzy) rough sets. Therefore, a novel variable precision fuzzy rough set model with CP is proposed and an associated three-way decision model is developed. Firstly, a pair of variable precision fuzzy upper and lower approximation operators are defined and studied through fuzzy implication and co-implication operators. We show that this pair of approximation operators satisfy various properties (including CP) that the existing variable precision fuzzy approximation operators do not possess. Secondly, new methods for constructing loss functions and conditional probabilities are given by using CP, and then a new three-way decision model with three strategies (optimistic, pessimistic, and compromise strategies) is established. Finally, an explanatory example is provided to examine validity, stability and sensitivity of the proposed three-way decision model.

A novel granular variable precision fuzzy rough set model and its application in fuzzy decision system

A novel granular variable precision fuzzy rough set model and its application in fuzzy decision system

Generalized Fuzzy Neighborhood System-Based Multigranulation Variable Precision Fuzzy Rough Sets with Double Topsis Method to Madm

Generalized Fuzzy Neighborhood System-Based Multigranulation Variable Precision Fuzzy Rough Sets with Double Topsis Method to Madm

Covering-based generalized variable precision fuzzy rough set

Zhan and Jiang defined covering-based compact and loose variable precision fuzzy rough set models. Soon after, they proposed a reflexive fuzzy β-neighborhood operator and defined a covering-based generalized variable precision fuzzy rough set. Based on them, in this paper, we use the reflexive fuzzy β-neighborhood operator to establish two covering-based generalized variable precision fuzzy rough set models, which are called covering-based generalized compact and loose variable precision fuzzy rough set models, respectively. Then, we investigate the important properties of the two rough set models and their relationship to the original models. Finally, we apply the covering-based generalized compact variable precision fuzzy rough set model to decision-making problems. A simple example is given to verify the validity of the model and compare the results with other models.

Generalized fuzzy variable precision rough sets based on bisimulations and the corresponding decision-making.

Recently, the classical rough set has been extended in many ways. However, some of them are based on binary relations which only excavate “one step” information to distinguish objects. The “one step” in the binary relation means that the ordered pair of the starting and end points of the step belongs to the relation. Faced with some complex data sets, the “one step” information may be not feasible. Motivated by the notion of bisimulation in computer science, three types of bisimulation-based generalized fuzzy variable precision rough set (BGFVPRS) models are constructed. Different from many existed rough set models which are based on binary relations, the BGFVPRS models can distinguish objects by excavating the “multi-step” information of underlying relations. The related properties and relationships of BGFVPRS models are investigated. The uncertainty measure of BGFVPRS models and the reduction of fuzzy bisimulations are also discussed. Furthermore, learning from the PROMETHEE II method and combining it with our presented BGFVPRS models, a novel multiple-attribute decision-making method is provided. This method can effectively deal with complex problems including attribute data and relational data. The flexibility and effectiveness of our decision-making method are illustrated by comparative analysis and sensitivity analysis in the Zachary karate club network.

Covering Fuzzy Rough Sets via Variable Precision

Lately, covering fuzzy rough sets via variable precision according to a fuzzy γ -neighborhood were established by Zhan et al. model. Also, Ma et al. gave the definition of complementary fuzzy γ -neighborhood with reflexivity. In a related context, we used the concepts by Ma et al. to construct three new kinds of covering-based variable precision fuzzy rough sets. Furthermore, we establish the relevant characteristics. Also, we study the relationships between Zhan’s model and our three models. Finally, we introduce a MADM approach to make a decision on a real problem.

New results on granular variable precision fuzzy rough sets based on fuzzy (co)implications

Some fuzzy rough sets only consider relative error limit and others are still sensitive to mislabeled samples. Considering the absolute error limit, Fang and Hu [11] proposed granular variable precision fuzzy rough sets based on fuzzy (co)implications to remedy these defects, which are suitable to the databases with errors in conditional attribute(s) and decision making. However, there are some faults in the characterizations of granular variable precision fuzzy rough sets based on fuzzy (co)implications presented by Fang and Hu, such as false conclusions, insufficient and redundant conditions. In this paper, we further discuss the equivalent expressions of granular variable precision σ-lower and θ-upper approximation operators with the proper semicontinuity of fuzzy (co)implications. Based on those new equivalent expressions of granular variable precision σ-lower and θ-upper approximation operators, the composition of granular variable precision fuzzy rough sets based on fuzzy (co)implications is studied with respect to a general fuzzy relation. Moreover, we further study granular variable precision fuzzy rough sets based on fuzzy (co)implications to rectify those faults mentioned above.

PROMETHEE II method based on variable precision fuzzy rough sets with fuzzy neighborhoods

The model of covering-based fuzzy rough sets (CFRSs) can be regarded as a hybrid one by combining covering-based rough sets with fuzzy sets. In this paper, based on fuzzy neighborhoods, we propose two types of covering-based variable precision fuzzy rough sets (CVPFRSs) via fuzzy logical operators, i.e., type-I CVPFRSs and type-II CVPFRSs. Then, several basic properties of the two types of CVPFRSs are discussed. In addition, by virtue of the idea of PROMETHEE II methods, we construct a novel method to multi-attribute decision-making (MADM) in the context of medical diagnosis based on the proposed rough approximation operators. Finally, a test example for illustrating the proposed method is given. Meanwhile, a comparative analysis and an experimental evaluation are further discussed to interpret and evaluate the effectiveness and superiority of the proposed method. The proposed rough set model not only extends the theory of CFRSs, but also provides a new perspective for MADM with fuzzy evaluation information.

Covering-based variable precision fuzzy rough sets with PROMETHEE-EDAS methods

This paper proposes a reflexive fuzzy β-neighborhood operator by modifying Ma’s fuzzy β-neighborhood operators. Then, we use such an operator to build a covering-based variable precision fuzzy rough set (CVPFRS) model that can deal with the issue of misclassifications and perturbations in decision-making problems. By combining the CVPFRS model with two traditional decision-making methods (the PROMETHE method and the DEAS method), we introduce a novel method for addressing multi-attribute decision-making (MADM) problems. An illustrative example is utilized to demonstrate the practicality of the proposed method. The effectiveness of the proposed method is validated by comparing it with existing methods. By virtue of the cross-validation and hypothesis testing, we give an experimental analysis to interpret the validity and stability of the proposed method.

An MADM approach to covering-based variable precision fuzzy rough sets: an application to medical diagnosis

In medical diagnosis, how to select an optimal medicine from some medicines with similar efficacy values to treat diseases has become common problems between doctors and patients. To solve this problem, we describe it as a multi-attribute decision-making (MADM) in a finite fuzzy covering approximation space. This paper aims to propose two pairs of covering-based variable precision fuzzy rough sets. By combining the proposed rough set model with the VIKOR method, we construct a novel method to medicine selection MADM problems in the context of medical diagnosis. A real-life case study of selecting a proper medicine to treat Alzheimer’s disease is given to demonstrate the practicality of our proposed method. Through a comparative analysis and an experimental analysis, we further explore the effectiveness and stability of the established method.

Probabilistic variable precision fuzzy rough set technique for discovering optimal learning patterns in e-learning

Probabilistic variable precision fuzzy rough set technique for discovering optimal learning patterns in e-learning

A comparative study of variable precision fuzzy rough sets based on residuated lattices

Although Qiao and Hu intended to propose granular variable precision L-fuzzy rough sets based on residuated lattices, L actually denotes a doubly residuated lattice with an involutive negation. However, there exist doubly residuated lattices without involutive negations. This paper presents a comparative study of variable precision fuzzy rough sets based on residuated lattices, in which we compare variable precision L-fuzzy rough sets with the preexisting models. A variable precision upper L-fuzzy rough approximation operator is proposed with L-fuzzy granules based on L-fuzzy implication on a residuated lattice L instead of a doubly residuated lattice. The variable precision upper L-fuzzy rough approximation operator is investigated with respect to different L-fuzzy relations regardless of proposing an involutive negation on a residuated lattice.

Synergistic Effect Evaluation of Main and Auxiliary Industry of Power Grid Based on the Information Fusion Technology from the Perspective of Sustainable Development of Enterprises

With the development of the diversified economic union, the power grid enterprises’ auxiliary industries are continuing to grow and develop faster. However, in the process of development, power grid enterprises are still faced with the problem of low efficiency of the management of main and auxiliary and resource allocation and utilization. Thus, the synergistic development evaluation for power grid enterprises’ main and auxiliary industries has a strong practical significance. It can help the managers to find the short board of each industry and search for the exploration direction for improvement to promote the sustainable development of the main and auxiliary industries comprehensively. It can also provide the reference for improving the management level of power grid enterprises. Information fusion technology, as a process of information processing for decision making, can make use of multi-source information synergistically to get a more objective and more essential understanding of the same thing or the same goal. It has been applied to many fields in a mature way. Based on the characteristics of each industry, this paper constructs a synergistic effect evaluation index system of the main and auxiliary industries from the aspects of management foundation, resource integration, operational efficiency and effectiveness. The variable precision fuzzy rough set (VPFRS) is introduced to screen the index system, eliminate redundant indexes, retain key indexes and improve the efficiency and accuracy of evaluation effectively. Meanwhile, based on the characteristic of dealing with the imprecise problem of the vague set, this paper establishes the evaluation model based on information fusion technology of the variable precision fuzzy rough set and vague set (VPFRS-Vague). In order to verify the validity of the model, five typical companies belonging to power grid enterprises are selected as examples for analysis to prove the validity and applicability of the evaluation method.

Granular fuzzy rough sets based on fuzzy implicators and coimplicators

This paper introduces granular fuzzy rough sets in the view of fuzzy implicators and fuzzy coimplicators, and discusses the constructive and axiomatic approach to fuzzy granules based on fuzzy implicators and coimplicators. Moreover, we study the connection between fuzzy granules and fuzzy relations and discuss the relationship between existing granular fuzzy rough set models and that proposed in this paper. Considering the absolute error limit, we introduce the concept of the granular variable precision fuzzy rough sets based on fuzzy implicators and coimplicators. Then we present four propositions to ensure that the approximation operators can be efficiently calculated.

Probabilistic Variable Precision Fuzzy Rough Set Technique for Discovering Optimal Learning Patterns in E-learning

In e-learning environment, optimal learning patterns are discovered for realising and understanding the effective learning styles. The value of uncertain and imprecise knowledge collected has to be categorised into classes known as membership grades. Rough set theory is potential in categorising data into equivalent classes and fuzzy logic may be applied through soft thresholds for refining equivalence relation that quantifies correlation between each class of elucidated data. In this paper, probabilistic variable precision fuzzy rough set technique (PVPFRST) is proposed for deriving robust approximations and generalisations that handles the types of uncertainty namely stochastic, imprecision and noise in membership functions. The result infers that the degree of accuracy of PVPFRST is 21% superior to benchmark techniques. Result proves that PVPFRST improves effectiveness and efficiency in identifying e-learners styles and increases the performance by 27%, 22% and 25% in terms of discrimination rate, precision and recall value than the benchmark approaches.