Rough Set Model Research Articles

Multi-granulation rough approximations under normal distribution

Multi-granulation decision-theoretic rough set effectively combines Bayesian decision approaches with multi-granulation rough set theory, and provides an important theoretical framework for studying rough set. In this paper, we explore several extensional models of multi-granulation decision-theoretic rough sets under the normal distribution of the decision loss function. Using the 3σ rule of normal distribution, we transform the decision loss of the multi-granulation decision-theoretic rough set into a set of interval values. We construct the upper and lower approximations from the optimistic, weakly optimistic, pessimistic, weakly pessimistic, optimistic-pessimistic, weakly optimistic-pessimistic, pessimistic-optimistic, and weakly pessimistic-optimistic viewpoints, and provide the decision rules of the proposed rough set models. The work in this paper brings the decision behavior based on a multi-granulation decision-theoretic rough set closer to the actual situation.

Hemimetric-based λ-valued fuzzy rough sets

A λ-subset, or a [0,λ]-valued fuzzy subset, is a mapping from a nonempty set to the interval [0,λ]. In this paper, we use the notion of hemimetrics, a kind of distance functions, as the basic structure to define and study fuzzy rough set model of λ-subsets by using the usual addition and subtraction of real numbers. We define a pair of fuzzy upper/lower approximation operators and investigate their properties and interrelations. These two operators have nice logical descriptions by using the related Lukasiewicz logical systems. We show that upper definable sets, lower definable sets and definable sets are equivalent, and they form an Alexandrov fuzzy topology. A processing of a λ-subset via fuzzy upper/lower approximation operators can actually considered as a processing of the related image, and thus has potential applications in image processing.

A multigranulation rough set model based on variable precision neighborhood and its applications

A multigranulation rough set model based on variable precision neighborhood and its applications

An interval rough number variable precision rough sets model and its attribute reduction

The interval rough number rough sets model is the generalization of the classical rough sets. Since the lower approximation condition of interval rough number rough sets model is a full inclusion relation which is too strict to tolerate noisy data, strict conditions increase the possibility of a sample classified into a wrong class. To overcome the above shortcomings, an interval rough number variable precision rough sets model is proposed in this paper, which is combined with interval rough number similarity and the concept of variable precision rough sets. The model introduces the error parameter and can improve the tolerance of noise data. Then the related properties of the model are also proved. Moreover, we construct a maximal positive domain attribute reduction method based on the proposed model, which can process the data type of interval rough number without discretization. Finally, numerical examples are given to verify the rationality of the model.

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.

A bi-variable precision rough set model and its application to attribute reduction

Significant indicators for measuring algorithms for attribute reduction include their fault tolerance. The majority of existing algorithms rely on the incorporation of fault tolerance at the upper and lower approximation operators. However, information granules partitioned by these algorithms are not inherently fault-tolerant. To enhance the fault tolerance of attribute reduction algorithms, this study introduces a new model based on variable precision rough sets, comprising of three parameters: the inclusion of fault tolerance in the comparison of attribute values between two objects, the partition of information granules, and the calculation of upper and lower approximation operators. As a result, the model tolerates errors, enhances the fault-tolerance ability of information granules, and improves its inclusivity. Using this new model, we design a novel algorithm that deletes both redundant and noise attributes to improve the classification accuracy. Through analysis of classification accuracy and robustness, we confirm that the novel algorithm's fault tolerance and robustness are maintained even after adding random Gaussian noise. The parameters included as part of the model are also found to be mostly unchanged after the addition of random Gaussian noise.

A new multi-objective decision-making method with diversified weights and Pythagorean fuzzy rough sets

In reality, there are many multi-objective decision-making problems with fuzziness and uncertainty, such as the choice of clinical treatment schemes. To effectively cope with such problems, it is urgent to scientifically handle fuzzy and uncertain information. Given the fact that the Pythagorean fuzzy set is a useful tool to tackle fuzzy information, and the rough set theory is superior in dealing with uncertain data, this paper aims to establish a novel decision-making method for scheme selection from the perspective of Pythagorean fuzzy rough sets. Pythagorean fuzzy numbers (PFNs) are first employed to express fuzzy preference information, and then multi-objective Pythagorean fuzzy information systems are introduced. Subsequently, we develop a new fuzzy rough set using an improved correlation coefficient method to deal with the uncertainty of PFNs. Furthermore, considering the diversified weights, we present a weight calculation method based on approximate accuracies. Meanwhile, a weighted fuzzy rough set model is constructed to achieve multi-objective information fusion. Accordingly, a multi-objective decision-making method is put forward with the help of overlap functions. Finally, we apply the built method to three cases from different decision scenarios to illustrate the applicability of the method. By some comparative analyses, the effectiveness and superiority of the method are also exhibited. Moreover, we discuss the establishment of decision support systems regarding scheme selection. On the whole, the work of this paper is helpful to support the solution of complex multi-objective decision-making problems.

Local double quantitative fuzzy rough sets over two universes

As an important expanded quantification fuzzy rough set model, the local fuzzy rough set model is used to measure relative quantitative information between the fuzzy similarity classes and the basic concept. Although many studies have taken on this topic, the existing local fuzzy rough sets ignore the absolute quantitative information in the fuzzy information system over two universes. Therefore, based on this observation, we propose a local double quantitative fuzzy rough set model over two universes, which considers the problems from the double quantization form of the local fuzzy rough sets over two universes. Furthermore, the corresponding properties and decision rules of the local double quantitative “logical and” fuzzy rough set model over two universes are studied. Then, in order to improve the applicability of the model, an effective reduction method is introduced. Finally, we conduct experimental comparisons showing the computational efficiency and approximate accuracy of the model in concept approximation and reduction.

Bi-directional adaptive neighborhood rough sets based attribute subset selection

In fields such as pattern recognition and computational intelligence, attribute subset selection has gradually attracted the attention of researchers as a challenging issue, particularly in big data. In rough sets community, many of the selection methods until now have been proposed based on one-directional and non-adaptive rough approximation sets, which increase the uncertainty of granular computing. Based on a series of bi-directional adaptive neighborhood rough sets models, a novel attribute subset selection method is proposed in this paper. In contrast to classical one-directional k-nearest neighborhood relations, the proposed model not only allocates different numbers of neighbors to each sample, but also takes the support information of the neighbors into account in the construction of information granularity. On the basis of set operators, two specific bi-directional neighborhood relations as well as optimistic and pessimistic rough set models are discussed in detail. Several propositions and uncertainty measures not only show the relationship between the traditional and the proposed models, but also present the intrinsic connection of three discussed models. In addition, we propose an optimal neighbor selection scheme, in which justifiable granularity is fused with evidence theory to obtain an optimal k value for each sample. Finally, in the attribute selection process, we prove non-monotonic property of the neighborhood entropy and describe an upgraded forward greedy algorithm. The proposed algorithm is compared with existing algorithms using a ten-fold cross-validation method on several UCI datasets. The experimental results show the advantages of our models in terms of computational performance and classification accuracy, respectively.

Optimizing group-oriented multiple attribute decision-making method based on dominance-based rough set model with advantaged (disadvantaged) neighborhoods

With the prominence of multi-attribute decision-making methods in the problem of intra-group interaction and inconsistency of ranking results in the problem of program group evaluation and ranking, group-based evaluation and ranking are receiving active attention. Although the group-oriented multi-attribute decision-making based on mixed advantage–disadvantage degree (GOMADMMADD) method innovatively proposes the idea of evaluating and ranking the group as a whole, there is an exponential explosion of the number of groups to be evaluated. Aiming at the above problems, this paper improves and optimizes the GOMADMMADD method. Firstly, this paper designs the advantaged neighborhood operator and disadvantaged neighborhood operator of the group from the perspective of the advantage relationship and disadvantage relationship between the data objects, designs the advantage–disadvantage neighborhood degree (ADND) according to the neighborhood operator, and studies the related properties. Secondly, based on ADND, a group-oriented multi-attribute decision-making method based on dominance-based rough set (GMADMDRS) is designed and optimized by using the properties of ADND. Thirdly, in the experiments of numerical examples, the experimental results of multiple traditional MADM methods show that GMADMDRS is reasonable. By comparing the experimental results of GMADMDRS before and after optimization on numerical examples, it is shown that the optimized GMADMDRS is consistent in the experimental results, and the time performance is greatly improved. Finally, the effectiveness of the optimized GMADMDRS method is verified on the UCI real dataset. The experimental results show that the GMADMDRS proposed in this paper can effectively deal with the problem of intra-group interaction and inconsistent ranking results in group-oriented evaluation and ranking, which has theoretical significance; it has practical significance for solving the exponential explosion of the number of groups to be evaluated. GMADMDRS provides a novel perspective and efficient method for multi-attribute decision-making problems.

Asymmetric models of intuitionistic fuzzy rough sets and their applications in decision-making

Asymmetric models of intuitionistic fuzzy rough sets and their applications in decision-making

Hesitant fuzzy β-covering ( T , I ) rough set models: An application to multi-attribute decision-making

Covering rough sets have been successfully applied to decision analysis because of the strong representing capability for uncertain information. As a research hotspot in decision analysis, hesitant fuzzy multi-attribute decision-making (HFMADM) has received increasing attention. However, the existing covering rough sets cannot handle hesitant fuzzy information, which limits its application. To tackle this problem, we set forth hesitant fuzzy β-covering rough set models and discuss their application to HFMADM. Specifically, we first construct four types of hesitant fuzzy β-covering ( T , I ) rough set models via hesitant fuzzy logic operators and hesitant fuzzy β-neighborhoods, which can handle hesitant fuzzy information without requiring any prior knowledge other than the data sets. Then, some intriguing properties of these models and their relationships are also discussed. In addition, we design a new method to deal with HFMADM problems by combining the merits of the proposed models and the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method. In this method, we not only consider the risk preferences of decision-makers, but also present a new hesitant fuzzy similarity measure expressed by hesitant fuzzy elements to measure the degree of closeness between two alternatives. Finally, an enterprise project investment problem is applied to illustrate the feasibility of our proposed method. Meanwhile, the stability and effectiveness of our proposed method are also verified by sensitivity and comparative analyses.

Fuzzy rough dimensionality reduction: A feature set partition-based approach

Dimensionality reduction is considered in many learning methods using discriminative features to obtain optimal performance. In general, feature extraction and feature selection are two independent methods that cherry-pick the informative features in the mapping data space or original feature space. Fuzzy rough set theory is widely used in feature selection, which well keeps the interpretability of features. How to comprehensively consider the intrinsic information contained in features and combine the feature extraction with feature selection is a challenging and meaningful task. Most existing fuzzy rough-based feature selection methods select features from the whole feature set. However, some nonsignificant features may lead to a decline in useful information. Meanwhile, these existing methods directly remove redundant and irrelevant features. This study scrutinizes features to distinguish the significant degree of features and deal with them in different ways. The feature selection and feature extraction methods are combined to obtain compound features for the learning method. A feature set partition-based approach to fuzzy rough dimensionality reduction (FSPFRdr) by simultaneously considering feature selection and fuzzy similarity relation-based supervised locally linear embedding (FRSLLE) is proposed. First, the ϑ-fuzzy rough set model is defined to avoid the affection of noise, and approximation space is constructed based on fuzzy approximations and granular structure. The normalized independent classification information (NICI) is defined in the approximation space. According to NICI, the original feature set is divided into a nonsignificant feature set, a weak significant feature set, and a significant feature set. Then, the nonsignificant feature set is deleted before dimensionality reduction. Next, the proposed FRSLLE is applied to a weak significant feature set, and the embedded feature set is obtained. Furthermore, a feature evaluation function is proposed to select features from the significant and embedded feature sets. Finally, the performance of FSPFRdr is compared with thirteen dimensionality reduction algorithms on twenty-three public datasets. Experimental results on four different classifiers show that FSPFRdr has higher classification performance.

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.

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.

Probabilistic Hesitant Fuzzy Decision-Theoretic Rough Set Model and Its Application in Supervision of Shared Parking

A sophisticated three-way decision model utilizing a multi-granularity probabilistic hesitant fuzzy rough set is introduced to mitigate the issue of information loss arising from hesitant fuzzy sets when characterizing decision information. Initially, the properties of probabilistic hesitant fuzzy sets are examined, subsequently proposing a distance measure and loss function tailored to these sets. Following this, a multi-attribute group decision-making model incorporating probabilistic hesitant fuzzy information is established, and pertinent decision rules that satisfy minimal risk criteria are presented. Building on this foundation, a series of critical steps for resolving a category of multi-attribute group decision-making problems that involve probabilistic hesitant fuzzy information is proposed. Lastly, the multi-attribute group decision-making model with probabilistic hesitant fuzzy information is applied to the supervision of urban shared parking platforms. The results indicate that the decision-making process based on probabilistic hesitant fuzzy sets is more reliable, and the decision-making outcome aligns with the actual situation, thus providing valuable decision-making references for managers.

A Collaborative Multi-Granularity Architecture for Multi-Source IoT Sensor Data in Air Quality Evaluations

Air pollution (AP) is a significant environmental issue that poses a potential threat to human health. Its adverse effects on human health are diverse, ranging from sensory discomfort to acute physiological reactions. As such, air quality evaluation (AQE) serves as a crucial process that involves the collection of samples from the environment and their analysis to measure AP levels. With the proliferation of Internet of Things (IoT) devices and sensors, real-time and continuous measurement of air pollutants in urban environments has become possible. However, the data obtained from multiple sources of IoT sensors can be uncertain and inaccurate, posing challenges in effectively utilizing and fusing this data. Meanwhile, differences in opinions among decision-makers regarding AQE can affect the outcome of the final decision. To tackle these challenges, this paper systematically investigates a novel multi-attribute group decision-making (MAGDM) approach based on hesitant trapezoidal fuzzy (HTrF) information and discusses its application to AQE. First, by combining HTrF sets (HTrFSs) with multi-granulation rough sets (MGRSs), a new rough set model, named HTrF MGRSs, on a two-universe model is proposed. Second, the definition and property of the presented model are studied. Third, a decision-making approach based on the background of AQE is constructed via utilizing decision-making index sets (DMISs). Lastly, the validity and feasibility of the constructed approach are demonstrated via a case study conducted in the AQE setting using experimental and comparative analyses. The outcomes of the experiment demonstrate that the presented architecture owns the ability to handle multi-source IoT sensor data (MSIoTSD), providing a sensible conclusion for AQE. In summary, the MAGDM method presented in this article is a promising scheme for solving decision-making problems, where HTrFSs possess excellent information description capabilities and can adequately describe indecision and uncertainty information. Meanwhile, MGRSs serve as an outstanding information fusion tool that can improve the quality and level of decision-making. DMISs are better able to analyze and evaluate information and reduce the impact of disagreement on decision outcomes. The proposed architecture, therefore, provides a viable solution for MSIoTSD facing uncertainty or hesitancy in the AQE environment.

Fuzzy rough feature selection using a robust non-linear vague quantifier for ordinal classification

Ordinal classification is a common classification problem, which widely exists in multi-attribute decision making problems. The dominance-based rough set approach (DRSA) is a knowledge acquisition tool for ordinal classification tasks. Nevertheless, the existence of noise information in the collected data greatly reduces the accuracy of the DRSA and some extended model. An effective way to solve this issue is to improve the robustness of these models. In addition, most existing feature selection strategies based on dominance rough sets mainly focus on the monotonic classification consistency between features and decision, but ignoring the classification information provided by features combination. Motivated by these two issues, this paper proposes a robust fuzzy dominance rough set model to combat noise interference and develop a feature selection approach based on the proposed model for ordinal classification tasks. First, a non-linear vague quantifier is adopted to construct the robust model, and related dependency function is introduced. Second, the rank entropy based uncertainty measures are explored to characterize the contribution of features combination for ordinal classification. Based on the proposed uncertainty measures, a new feature evaluation index is presented. Meanwhile, the corresponding feature selection algorithm is designed. Finally, numerical comparative experiments are performed to indicate that the proposed model and feature selection algorithm have good capability.

Extended rough sets model based on fuzzy granular ball and its attribute reduction

Attribute reduction is one of the core steps of data analysis. The attribute reduction method based on neighborhood rough sets (NRS) is widely used. However, the time complexity of this method is excessively high because its radius selection relies on grid search. Granular ball neighborhood rough sets (GBNRS) can generate different neighborhoods adaptively and has more generality and flexibility than the NRS method. Nevertheless, due to the purity of granular ball is required to be 1, the GBNRS algorithm will generate many granular balls with the sample number of 1 at the boundaries of classes. These granular balls will be regarded as outliers and deleted, resulting in the loss of class boundary information. Moreover, due to the strict reduction conditions of GBNRS, the reduction effect cannot be achieved on some datasets. To solve the above problems, the fuzzy granular ball is defined by relaxing the generation conditions of the granular ball, and an extended rough sets model is proposed on the basis of the fuzzy granular ball (FGBERS). In this model, we retain as much class boundary information as possible by preserving the fuzzy granular ball of class boundary. A new forward heuristic attribute reduction algorithm is designed on the basis of this model. To demonstrate the performance of FGBERS, we conducted sufficient experiments on 18 real datasets from different fields. The experimental results show that FGBERS not only compensates for the shortcomings of GBNRS, but also demonstrates higher classification accuracy, especially on the KNN classifier, the classification accuracy of the FGBERS algorithm is 5% higher on average, and the highest improvement is 20% on high-dimensional dataset ALLAML. In addition, compared with the comparison algorithms, FGBERS has excellent reduction performance in both large-scale datasets and high-dimensional datasets.

Valued outranking relation-based heterogeneous multi-decision multigranulation probabilistic rough set and its use in medical decision-making

Regarding to group decision-making, decision makers may express different decision preferences for the same target. Since existing decision-making methods only consider a single decision preference at most, it is difficult to integrate the multi-decision preferences of multiple decision makers effectively. In this paper, we explore a class of heterogeneous multi-attribute group decision-making (MAGDM) problems with multiple decisions. Over the framework of granular computing, we present a new MAGDM method with the aid of the valued outranking relations in the ELECTRE III method. Considering the non-compensability among attributes, we first define a special valued outranking relation on a heterogeneous attribute multi-decision information system, and then present the notion of valued outranking classes. Subsequently, given the fact that the fault tolerance of probability approximations can reduce the vulnerability of decision-making methods, we construct multi-decision multigranulation probabilistic rough sets (MD-MGPRSs) form different perspectives, which extend existing rough set models to a more general situation. After that, we establish a novel MAGDM method, and apply the method to a real medical case to illustrate its applicability. Lastly, the experimental results on seven datasets demonstrate the effectiveness, robustness, and superiority of the method. Our study not only enriches the theoretical research of rough sets, but also extends the application scope of decision-making theory.