Rough Set Approximations Research Articles

Near open generalizations of rough sets and their applications

BackgroundThe concept of near open sets is a potent tool that empowers researchers to achieve a more encompassing approximation of rough sets, thereby enhancing the accuracy of measurements. The evolution of rough set theory into various generalized forms, based on topological structures, has emerged as a significant approach in the realm of knowledge discovery within databases.ResultsThis paper’s primary contribution lies in the introduction of a novel category of generalized near open sets, termed “inverse simply open sets,” within the context of the j\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{j}$$\\end{document}-neighborhood space. The paper proposes diverse methods for extending the Pawlak’s rough approximations leading to the definition of new approximations in the j\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{j}$$\\end{document}-neighborhood space. By employing these newly introduced generalizations, we establish fresh connections between two pivotal theories, namely “general topology and rough set theory”. Through a comprehensive investigation, we conduct multiple comparisons between our methodology and classical approaches. Furthermore, we showcase practical applications of these techniques within real-life scenarios, demonstrating their utility in decision-making processes.ConclusionsWe reduced the data’s ambiguity while increasing its accuracy measure. As a result, we may conclude that the proposed approximations were more precise than earlier techniques and contributed to the elimination of data ambiguity in real-world scenarios requiring accurate decisions.

On Proximity Spaces Constructed on Rough Sets

Based on equivalence relation R on X, equivalence class [x] of a point and equivalence class [A] of a subset represent the neighborhoods of x and A, respectively. These neighborhoods play the main role in defining separation axioms, metric spaces, proximity relations and uniformity structures on an approximation space (X,R) depending on the lower approximation and the upper approximation of rough sets. The properties and the possible implications of these definitions are studied. The generated approximation topology τR on X is equivalent to the generated topologies associated with metric d, proximity δ and uniformity U on X. Separated metric spaces, separated proximity spaces and separated uniform spaces are defined and it is proven that both are associating exactly discrete topology τR on X.

Boundary-wise loss for medical image segmentation based on fuzzy rough sets

The loss function plays an important role in deep learning models as it determines the model convergence behavior and performance. In semantic segmentation, many methods utilize pixel-wise (e.g. cross-entropy) and region-wise (e.g. dice) losses while boundary-wise loss is underexplored. It is known that one of the key aims of semantic segmentation is to precisely delineate objects' boundaries. Hence, it is essential to design a loss function that measures the errors around objects' boundaries. Fuzzy rough sets are constituted by the fuzzy equivalence relation, which is commonly used to measure the difference between two sets. In this paper, the lower approximation of fuzzy rough sets is proposed to construct the boundary-wise loss in deep learning models for the first time. The experiments with various segmentation models and datasets have verified that the proposed fuzzy rough sets loss is superior to other boundary-wise losses in terms of segmentation accuracy and time complexity. Compared with the commonly used pixel-wise and region-wise losses, the proposed boundary-wise loss performs similarly in dice coefficient, pixel-wise accuracy, but has a better performance in Hausdorff distance and symmetric surface distance. It indicates that the proposed loss provides a better guidance for segmentation models in producing more accurate shapes of the target objects. Code is available online at Github: https://github.com/qiaolin1992/Boundary-Loss.

Increasing and Decreasing Soft Rough Set Approximations

Increasing and Decreasing Soft Rough Set Approximations

A Vague Rough Set Approach for Failure Mode and Effect Analysis in Medical Accident

Identifying the risk factor is a key to success for the Failure model. This paper presents a new FMEA (Failure Mode and Effect Analysis) in medical accident for the hospital management by using vague rough matrix information captured through vague rough set. Using β-product and β-complement product into fuzzy relation matrix and vague set to find the new lower and upper approximation of vague rough set. This method will provide an easy methodology to find the weight of risk factor. Using an example we validate our proposed approach.

On a Type of Generalized GS_β-Closed Set in Grill Topology

This work intends to propose the concept of generalized -closed set-in grill topological space. Further, some of the properties of such generalized -closed set are evaluated and provided with suitable illustrations. Many theorems based on generalized -closed set have also been proved in this work. Furthermore, the ideas of -regular space and -normal space are also studied. Also, the relationship between -regular and - normal spaces is discussed. The properties of -regular space and -normal space concepts have also been discussed.Introduction: Alo, R. A., and Shapiro, H. L defined the semi- normal spaces and considered some of their important inheritance. The authors further studied the compactness in grill topological spaces and studied some of their properties. Data granulation serves as a vital decision-making tool in numerous real-life applications. Mashhour, A. S and Abd, M. E introduced certain new topological tools employing rough set approximations for the purpose of data granulation. The introduction of -normality helps to separate closed subsets, continuous functions. It also extends the support to define the concepts of complete regularity and Tietze extension theorem. The -regular space studied in this article is regular, that is completely regular, which is a stronger condition.Objectives: I.Arockiarani and A. Karthika introduced a new concept such as generalized -closed set in grill topological space. The above paper motivates me to do the research in generalized -closed set. In this paper a new concept of “generalized -closed set in grill topological space” are introduced. In this article we are using stronger condition for proving some theorems.Methods: The present task ambitioned at formation of the extension of topological system applying the idea of grill. The importance of the theory of grill between the topological spaces have been elaborated. A set is countable collection based on elements, without integrity of form. At the same time some kinds of algebraic actions are applied on this set, indefinitely the elements of this set are correlated into a whole, so ultimately it grows into a space. So ultimately in the whole paper we are applying the concept of grill topological space.Results: In this article a new concept of generalized -closed set in a grill topological space are introduced. Using this definition we are proving many results based on operators on grill topological space. Furthermore, the implication such as “all closed set are generalized -closed sets” is proved and the converse may not be true are also proved with proper illustration. In addition the relationship between -normal space, -normal space, -regular space and -regular space are also studied. Further many equivalent conditions are also proved using -normal space and -regular space.Conclusions: The work proposes and investigates the concept of generalized -closed sets within the framework of grill topological spaces. It explores the properties of these sets, provides illustrative examples, and proves theorems related to their behavior. Additionally, the study delves into the concepts of -regular space and -normal space, discussing their properties and implications. The research contributes to the understanding of these concepts and their applications in the context of grill topological spaces.

GRRS: Accurate and Efficient Neighborhood Rough Set for Feature Selection

Feature selection is an important preprocessing step in data mining and pattern recognition. The neighborhood rough set (NRS) model is a widely-used rough set model for feature selection on continuous data. All currently known NRS models are defined on a distance metric — mostly the Euclidean distance metric — which invalidates the NRS models in scenarios wherein the Euclidean distance is ineffective, for example, while considering differing attribute weights. We first introduce the concept of space division of granular-rectangular, and then construct the neighborhood radius in our method by describing the relationship between child and parent spaces, which avoids the use of a distance metric and reduces the search space for the neighborhood radius. This greatly improves both the accuracy and efficiency of NRS. In addition, the upper and lower approximations of the granular-rectangular rough sets (GRRSs) comprise equivalence classes; this results in better performance of GRRS in knowledge representation compared with the traditional NRS. Experimental results on public benchmark datasets reveal that our method, GRRS, achieves higher accuracy than ten popular and state-of-the-art feature-selection methods, including two NRS algorithms. Moreover, GRRS outperforms the established NRS algorithms regarding efficiency, including the state-of-the-art NRS algorithm, GBNRS. All code has been released as an open libary called GRRS: https://github.com/syxiaa/GRRS.

A dynamic framework for updating approximations with increasing or decreasing objects in multi-granulation rough sets

The data we need to deal with are getting bigger and bigger in recent years, and the same happens to multi-granulation rough sets, so updated schemes have been proposed with the variation of attributes or attribute values in multi-granulation rough sets. This paper puts forward a dynamic mechanism to update the approximations in multi-granulation rough sets when increasing or decreasing objects. Firstly, the relationships between the original approximations and updated approximations are explored when adding or deleting objects and the dynamic processes of updating the lower and upper approximations in optimistic and pessimistic approximations are proposed. Secondly, two corresponding dynamic algorithms and their time complexity are given. Finally, the experimental evaluations show the effectiveness of the proposed dynamic updating algorithms compared with the static algorithm.

A rough-set and AI based approach for hierarchical cognitive processing of perceptions

Modelling the psychological and cognitive processes is an important issue in theoretical and clinical aspects. Most of the modellings concern much on qualitative and philosophical descriptions. This research resorts to quantitative modelling of human cognitive processing based on rough sets, AI, and mathematical approaches. We formalise psychological targets and properties by objects and features. A feature domain is associated with these objects and features. Based on the subjects’ perception of the values in the feature domain, the cognitive processing is proceeded via deterministic procedures and non-deterministic procedures. The non-deterministic procedures involve the setting of rough set approximations. To evaluate the parameters involved in the setting, we design several experiments and algorithm, in particular probabilistic structures and entropy, to evaluate the effects of these parameters. These provide a great degree of flexibility in practical applications. These approaches could be easily implemented by or combined with machine learning and artificial intelligence techniques. Our approach could be applied directly in clinical assessment and mental health probing.

Cost-Sensitive Multigranulation Approximation in Decision-Making Applications

A multigranulation rough set (MGRS) model is an expansion of the Pawlak rough set, in which the uncertain concept is characterized by optimistic and pessimistic upper/lower approximate boundaries, respectively. However, there is a lack of approximate descriptions of uncertain concepts by existing information granules in MGRS. The approximation sets of rough sets presented by Zhang provide a way to approximately describe knowledge by using existing information granules. Based on the approximation set theory, this paper proposes the cost-sensitive multigranulation approximation of rough sets, i.e., optimistic approximation and pessimistic approximation. Their related properties were further analyzed. Furthermore, a cost-sensitive selection algorithm to optimize the multigranulation approximation was performed. The experimental results show that when multigranulation approximation sets and upper/lower approximation sets are applied to decision-making environments, multigranulation approximation produces the least misclassification costs on each dataset. In particular, misclassification costs are reduced by more than 50% at each granularity on some datasets.

A three-way classification with fuzzy decision trees

This study is concerned with the design of a three-way classification mechanism realized through combing fuzzy decision trees and expressing uncertainty associated with classification results. The fuzzy decision tree used in this study is constructed through the generalization of the commonly used decision trees. The notion of three-way decision model first proposed for the interpretation of rules generated in rough set approximation has been widely used in many fields. This study proposes an efficient way to flag data with high level of uncertainty with the classification realized by fuzzy decision trees, which is the capability that commonly used fuzzy decision trees do not have. The data identified in this way are left to users’ judgments or more advanced classification techniques. The developed mechanism is formed as a two-stage construct where a fuzzy decision tree is built by generalizing the Boolean classification boundaries of a pre-constructed decision tree using fuzzy sets and then determining the level of uncertainty to identify instances to be rejected due to a lack of sufficient confidence in their belongingness. The rejected instances that are difficult to process are classified as non-commitment cases and left to some further analyses. The rejection quality of the developed three-way classifier is quantified in terms of the classification accuracy and rejection coefficient. We also elaborate on striking a sound tradeoff between these two performance indicators. Experimental studies demonstrate that the developed mechanism could effectively improve the classification accuracy at the cost of a small proportion of the rejected instances and achieve better performance in comparison with other three-way decision models when generating a three-way decision output.

Rough set approximations based on a matroidal structure over three sets.

Pawlak’s classical model of rough set approximations provides an efficient tool for extracting information exactly by employing available knowledge (i.e., known knowledge) in an information system, since many problems in rough set theory are NP-hard and their solution process is therefore greedy and approximate. Many extensions of Pawlak’s classical model have been proposed in recent years. Most of them are considered over one or two sets, that is, one- or two-dimensional space or one- or two-dimensional data. Aided by relation-based rough set models, a few of these extensions are considered over three sets. However, the real world is in three-dimensional space. Therefore, it is necessary to solve these problems with other models, such as covering rough set models. For this purpose, we propose the TP-matroid—a matroidal structure over three sets. Employing the family of feasible sets of a TP-matroid as the available knowledge, a pair of rough set approximations—lower and upper approximations—is provided. In addition, for an information system defined over three sets, assisted by formal concept analysis, we establish a pair of rough set approximations. Furthermore, two TP-matroids are established based on the above pair of rough set approximations. The integration between the two pairs of rough set approximations presented here is discussed. The results show that for an information system in three-dimensional space, the rough set approximations provided here can effectively explore unknown knowledge by using available knowledge based on the family of feasible sets of a TP-matroid.

Novel Bipolar Soft Rough-Set Approximations and Their Application in Solving Decision-Making Problems

Novel Bipolar Soft Rough-Set Approximations and Their Application in Solving Decision-Making Problems

Enhanced Feature Selection Based on Integration Containment Neighborhoods Rough Set Approximations and Binary Honey Badger Optimization.

This article appoints a novel model of rough set approximations (RSA), namely, rough set approximation models build on containment neighborhoods RSA (CRSA), that generalize the traditional notions of RSA and obtain valuable consequences by minifying the boundary areas. To justify this extension, it is integrated with the binary version of the honey badger optimization (HBO) algorithm as a feature selection (FS) approach. The main target of using this extension is to assess the quality of selected features. To evaluate the performance of BHBO based on CRSA, a set of ten datasets is used. In addition, the results of BHOB are compared with other well-known FS approaches. The results show the superiority of CRSA over the traditional RS approximations. In addition, they illustrate the high ability of BHBO to improve the classification accuracy overall the compared methods in terms of performance metrics.

The cost-sensitive approximation of neighborhood rough sets and granular layer selection

Neighborhood rough sets (NRS) are the extended model of the classical rough sets. The NRS describe the target concept by upper and lower neighborhood approximation boundaries. However, the method of approximately describing the uncertain target concept with existed neighborhood information granules is not given. To solve this problem, the cost-sensitive approximation model of the NRS is proposed in this paper, and its related properties are analyzed. To obtain the optimal approximation granular layer, the cost-sensitive progressive mechanism is proposed by considering user requirements. The case study shows that the reasonable granular layer and its approximation can be obtained under certain constraints, which is suitable for cost-sensitive application scenarios. The experimental results show that the advantage of the proposed approximation model, moreover, the decision cost of the NRS approximation model will monotonically decrease with granularity being finer.

Label-based Approach for Dynamic Updating Approximations in Incomplete Fuzzy Probabilistic Rough Sets over Two Universes

Label-based Approach for Dynamic Updating Approximations in Incomplete Fuzzy Probabilistic Rough Sets over Two Universes

Improved evolutionary-based feature selection technique using extension of knowledge based on the rough approximations

This paper establishes an innovative approach of rough set (RS) approximations, namely the extension of knowledge based on the rough approximation (EKRA), which generalizes the old concepts and gets preferable results by reducing the boundary regions. In contrast to the former RS methods that obtained upper and lower approximations by several methods for special cases of binary relations. In addition, to assess the applicability of this approach it is combined with LSHADE with semi-parameter adaptation combined with CMA-ES (LSHADE-SPACMA) as a feature selection method, where EKRA is used as an objective function. The developed FS approach, named, LSPEKRA, which depends on LSHADE-SPACMA and EKRA aims to find the relevant features. This leads to improving the classification of different datasets. The experimental results show the great performance of the presented method against other Evolutionary algorithms. In addition, the FS methods based on EKRA provide results better than traditional RS in terms of performance measures.

Fuzzy rough set models over two universes using textures

The purpose of this paper is twofold: firstly, to exhibit the existence of a reasonable connection between fuzzy relations and (textural) fuzzy direlations in terms of fuzzy logical connectives; and secondly, to give a new perspective with respect to approximation operators due to fuzzy rough set models over two universes using textures. To this end, we investigate the serialities of fuzzy relations in terms of fuzzy logic connectives using fuzzy direlations. For different fuzzy logic systems, we show that the textural serialities provide alternative descriptions of the serialities of a fuzzy relation in terms of parameters. Without considering the motivation of the semantic link with the classical Pawlak rough sets, we obtain and discuss the natural fuzzifications of rough set models over two universes. Further, we execute that a textural fuzzy rough set model is an efficient template for the basic properties of the various fuzzy rough set models over two universes. For instance, we prove that the revised fuzzy rough set approximations with two universes are the natural extensions of the loose and tight approximations of De Cock et al.

Two Different Views for Generalized Rough Sets with Applications

Rough set philosophy is a significant methodology in the knowledge discovery of databases. In the present paper, we suggest new sorts of rough set approximations using a multi-knowledge base; that is, a family of the finite number of general binary relations via different methods. The proposed methods depend basically on a new neighborhood (called basic-neighborhood). Generalized rough approximations (so-called, basic-approximations) represent a generalization to Pawlak’s rough sets and some of their extensions as confirming in the present paper. We prove that the accuracy of the suggested approximations is the best. Many comparisons between these approaches and the previous methods are introduced. The main goal of the suggested techniques was to study the multi-information systems in order to extend the application field of rough set models. Thus, two important real-life applications are discussed to illustrate the importance of these methods. We applied the introduced approximations in a set-valued ordered information system in order to be accurate tools for decision-making. To illustrate our methods, we applied them to find the key foods that are healthy in nutrition modeling, as well as in the medical field to make a good decision regarding the heart attacks problem.

A novel notion in rough set theory: Invariant subspace

All current work on calculating approximate sets of a rough set inevitably requires the participation of all elements in the universe. However, it is cumbersome and causes a huge waste of resources when the universe is quite big and the computed set is small enough. By introducing the notion of invariant subspace in rough set theory, we skillfully reduce the range of elements involved in the computations of approximations of a rough set from the whole universe U to a suitable invariant subspace V of U, and then give the modular method within the reduced range. Moreover, we present the modular Boolean matrix method, such that the calculation of upper and lower approximations of rough sets can be converted into operations on modular matrices. Finally, the results in covering approximation space are generalized to fuzzy β-covering approximation space to calculate the upper and lower approximations of the crisp sets. In particular, an algorithm for calculating the value of β is proposed to make the involved information has the highest distinction degree. This β is more reliable and meaningful than the empirical one, and an example about COVID-19 is put forward to simply illustrate its application.