Rudiments of rough sets

doi:10.1016/j.ins.2006.06.003

Information Sciences

Volume 177, Issue 1, 1 January 2007, Pages 3-27

https://doi.org/10.1016/j.ins.2006.06.003 Get rights and content

Abstract

Worldwide, there has been a rapid growth in interest in rough set theory and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on rough sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. In addition, many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs. Rough set theory has led to many interesting applications and extensions. It seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences, especially in research areas such as machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, knowledge discovery, decision analysis, and expert systems. In the article, we present the basic concepts of rough set theory and point out some rough set-based research directions and applications.

Introduction

The basic ideas of rough set theory and its extensions as well as many interesting applications can be found in a number of books (see, e.g., [28], [33], [44], [48], [74], [97], [112], [113], [131], [182], [194], [195], [206], [237], [241], [244], [245], [272], [305], [374]), issues of the Transactions on Rough Sets [225], [226], [227], [228], special issues of other journals (see, e.g., [25], [129], [218], [193], [281], [306], [377], [378]), proceedings of international conferences (see, e.g., [1], [88], [130], [243], [280], [293], [300], [301], [331], [336], [337], [350], [376], [380]), tutorials (see, e.g., [110]). For more information one can also visit web pages www.roughsets.org and logic.mimuw.edu.pl.

The basic notions of rough sets and approximation spaces were introduced during the early 1980s (see, e.g., [199], [201], [202]). In this paper, the basic concepts of rough set theory are presented. We also point out some research directions and applications based on rough sets. In articles [210], [214], we discuss in more detail two selected topics, namely, extensions of the rough set approach and the combination of rough sets and Boolean reasoning with applications in pattern recognition, machine learning, data mining and conflict analysis.

Section snippets

Sets and vague concepts

In this section, we give some general remarks on the concept of a set and the place of rough sets in set theory.

The concept of a set is fundamental for the whole of mathematics. Modern set theory was formulated by Cantor [19]. Bertrand Russell discovered that the intuitive notion of a set proposed by Cantor leads to antinomies [266]. Two kinds of remedy for this problem have been proposed: axiomatization of Cantorian set theory and alternative set theories.

Another issue discussed in connection

Rough sets

This section briefly delineates basic concepts in rough set theory.

Rough sets and logic

The father of contemporary logic is a German mathematician Gottlob Frege (1848–1925). He thought that mathematics should not be based on the notion of set but on the notions of logic. He created the first axiomatized logical system but it was not understood by the logicians of those days.

During the first three decades of the 20th century, there was a rapid development in logic bolstered to a great extent by Polish logicians, especially Alfred Tarski (1901–1983) (see,e.g., [330]).

Development of

Exemplary research directions and applications

In the article we have discussed some basic issues and methods related to rough sets. For more detail the reader is referred to the literature cited at the beginning of this article (see also rsds.wsiz.rzeszow.pl).

We are now observing a growing research interest in the foundations of rough sets (see, e.g., [13], [17], [62], [86], [93], [94], [104], [116], [117], [118], [119], [143], [153], [176], [171], [197], [203], [209], [211], [212], [213], [235], [240], [249], [273], [277], [278], [283],

Conclusions

In this article we have presented basic concepts of rough set theory. We have also listed some research directions and exemplary applications based on the rough set approach.

A variety of methods for decision rules generation, reducts computation and continuous variable discretization are very important issues not discussed here. We have only mentioned the methodology based on discernibility and Boolean reasoning for efficient computation of different entities including reducts and decision

Acknowledgements

The research of Andrzej Skowron has been supported by the grant 3 T11C 002 26 from Ministry of Scientific Research and Information Technology of the Republic of Poland.

Many thanks to Professors James Peters and Dominik Śle¸zak for their incisive comments and for suggesting many helpful ways to improve this article.

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Cited by (1973)

Shortest-length and coarsest-granularity constructs vs. reducts: An experimental evaluation
2024, International Journal of Approximate Reasoning
In the domain of rough set theory, super-reducts represent subsets of attributes possessing the same discriminative power as the complete set of attributes when it comes to distinguishing objects across distinct classes in supervised classification problems. Within the realm of super-reducts, the concept of reducts holds significance, denoting subsets that are irreducible.
Contrastingly, constructs, while serving the purpose of distinguishing objects across different classes, also exhibit the capability to preserve certain shared characteristics among objects within the same class. In essence, constructs represent a subtype of super-reducts that integrates information both inter-classes and intra-classes. Despite their potential, constructs have garnered comparatively less attention than reducts.
Both reducts and constructs find application in the reduction of data dimensionality. This paper exposes key concepts related to constructs and reducts, providing insights into their roles. Additionally, it conducts an experimental comparative study between optimal reducts and constructs, considering specific criteria such as shortest length and coarsest granularity, and evaluates their performance using classical classifiers.
The outcomes derived from employing seven classifiers on sixteen datasets lead us to propose that both coarsest granularity reducts and constructs prove to be effective choices for dimensionality reduction in supervised classification problems. Notably, when considering the optimality criterion of the shortest length, constructs exhibit clear superiority over reducts, which are found to be less favorable.
Moreover, a comparative analysis was conducted between the results obtained using the coarsest granularity constructs and a technique from outside of rough set theory, specifically correlation-based feature selection. The former demonstrated statistically superior performance, providing further evidence of its efficacy in comparison.
Prediction of liquefaction of gravelly soils based on a cost-sensitive Bayesian network combined with rough set weighting
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Existing liquefaction prediction approaches for gravelly soils rely on in situ test data, which neglects the in-suit test cost sensitivity. Furthermore, these approaches do not consider the effect of the weights of the factors on the model performance. Therefore, in this paper, a cost-sensitive Bayesian network with rough set theory (CSW-BN) and model hierarchies and factor weighting are combined to address the two aforementioned challenges in research on gravelly soil liquefaction prediction. A two-layer model is constructed using the proposed CSW-BN approach. The first layer of the model reduces the data costs by utilizing conventional soil parameters, without necessitating specific in situ tests such as shear wave velocity tests. In the second layer of the model, variable weights calculated using rough set theory are incorporated into the parameter learning to enhance the generalization of the model. The results show that the learning and prediction accuracies of the new approach are 0.968 and 0.95, respectively. Compared with existing models, the proposed method not only reduces the number of required in situ tests but also improves the prediction accuracy and interpretability. Furthermore, the effectiveness of the CSW-BN model is evaluated using new cases for the 2008 Wenchuan earthquake, with a prediction accuracy of 0.89. Finally, a software tool is developed using Visual Basic for Applications programming to facilitate its application in engineering practice.
Intensions and extensions of granules: A two-component treatment
2024, International Journal of Approximate Reasoning
The concept of a granule (of knowledge) originated from Zadeh, where granules appeared as references to words (phrases) of a natural (or an artificial) language. According to Zadeh's program, “a granule is a collection of objects drawn together by similarity or functionality and considered therefore as a whole”. Pawlak's original theory of rough sets and its different generalizations have a common property: all systems rely on a given background knowledge represented by the system of base sets. Since the members of a base set have to be treated similarly, base sets can be considered as granules. The background knowledge has a conceptual structure, and it contains information that does not appear on the level of base granules, so such information cannot be taken into consideration in approximations. A new problem arises: is there any possibility of constructing a system modeling the background knowledge better? A two-component treatment can be a solution to this problem. After giving the formal language of granules involving the tools for approximations, a logical calculus containing approximation operators is introduced. Then, a two-component semantics (treating intensions and extensions of granule expressions) is defined. The authors show the connection between the logical calculus and the two-component semantics.
Fermatean fuzzy similarity measures based on Tanimoto and Sørensen coefficients with applications to pattern classification, medical diagnosis and clustering analysis
2024, Engineering Applications of Artificial Intelligence
Fermatean fuzzy sets (FFSs) have emerged as a powerful tool for handling uncertain information and have been successfully applied in various domains. However, the existing similarity measures for FFSs often face challenges in accurately capturing the similarity or difference between FFSs, leading to counterintuitive results. To address this issue, we propose sixteen new similarity measures for FFSs inspired by the Tanimoto and Sørensen coefficients. We analyze the properties of these measures and demonstrate their effectiveness through comparative examples with existing measures for FFSs, showing that our proposed measures outperform them in processing fuzzy information derived from FFSs. Additionally, we apply the proposed similarity measures to pattern recognition, medical diagnosis and clustering analysis to demonstrate their practical applicability. The results indicate that the proposed similarity measures can yield more meaningful and effective outcomes.
Incremental information fusion in the presence of object variations for incomplete interval-valued data based on information entropy
2024, Information Sciences
Information fusion technology plays a crucial role in integrating data from multiple sources or sensors to generate comprehensive representation, which can eliminate uncertainty in multi-source information systems (Ms-IS). Incomplete interval-valued data, a generalized form of single-valued data, is commonly encountered in real-world scenarios and effectively represents uncertain information. This paper introduces a novel information entropy specifically designed to quantify the uncertainty in incomplete interval-valued data. Based on the proposed entropy, a new unsupervised fusion approach is developed. Additionally, two dynamic update mechanisms are established to obtain fusion results efficiently when collecting new objects and removing obsolete ones. The relevant static and dynamic fusion algorithms are provided, and a detailed analysis and comparison of their time complexities are conducted. Finally, the effectiveness analysis reveals that the proposed method achieves higher average classification accuracy (5% to 8.7% improvement) compared to three common fusion methods (MAXF, MEANF, and MINF), as well as the state-of-the-art entropy-based supervised fusion method (ESF). The efficiency analysis demonstrates that the average running time of dynamic fusion algorithms is significantly lower (66.9% to 85.6% reduction) compared to the static fusion algorithm, and this difference is statistically significant.
A four-stage branch local search algorithm for minimal test cost attribute reduction based on the set covering[Formula presented]
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Attribute reduction is a fundamental problem in rough set theory and serves as an effective data reduction technique. The minimal test cost attribute reduction problem poses a more challenging task built upon the classical reduction problem and finds widespread applications in the real world. The development of efficient heuristic algorithms is crucial for effectively addressing this problem. However, research efforts utilizing local search algorithms for solving the minimal test cost attribute reduction problem remain limited. This paper aims to combine the graph-based rough set attribute reduction method with the effectiveness of local search algorithms known for solving the set covering problem. It proposes a method to transform the minimal test cost attribute reduction problem into a set covering problem and a four-stage branch local search algorithm based on the set covering is designed to solve the minimal test cost attribute reduction problem, which includes three core algorithmic techniques to further improve the practical performance of the algorithm. Extensive experiments are conducted to evaluate the effectiveness of the proposed algorithms, comparing them against various existing algorithms. The experimental results demonstrate that the proposed algorithms achieve fewer attributes and lower test costs within an acceptable running time, while maintaining competitive classification accuracy.

View all citing articles on Scopus

^✠: Professor Zdzisław Pawlak passed away on 7 April 2006.

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Rudiments of rough sets

Abstract

Introduction

Section snippets

Sets and vague concepts

Rough sets

Rough sets and logic

Exemplary research directions and applications

Conclusions

Acknowledgements

Information Sciences

International Journal of Approximate Reasoning

Pattern Recognition Letters

Neurocomputing

Fuzzy Sets and Systems

Theoretical Computer Science

Artificial Intelligence

International Journal of Approximate Reasoning

European Journal of Operational Research

Engineering Applications of Artificial Intelligence

International Journal of Approximate Reasoning

A modal logic for indiscernibility and complementarity in information systems

Fundamenta Informaticae

Logic for rough truth

Fundamenta Informaticae

Vagueness: an exercise in logical analysis

Philosophy of Science

Boolean Reasoning

Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen

Crelle’s Journal für Mathematik

Grundlagen einer allgemeinen Mannigfaltigkeitslehre

Algebraic structures related to many valued logical systems. Part I: Heyting–Wajsberg algebras

Fundamenta Informaticae

Algebraic structures related to many valued logical systems. Part II: Equivalence among some widespread structures

Fundamenta Informaticae

Computational Intelligence: An International Journal

Data Mining Methods for Knowledge Discovery

An algebraic approach to the approximation of information

Fundamenta Informaticae

Automata-theoretic decision procedures for information logics

Fundamenta Informaticae

Computational complexity of multimodal logics based on rough sets

Fundamenta Informaticae

Knowledge Engineering: A Rough Set Approach

Rough fuzzy sets and fuzzy rough sets

International Journal of General Systems

Pattern Classification