Reduction In Rough Set Theory Research Articles

Partially-defined equivalence relations: Relationship with orthopartitions and connection to rough sets

We introduce partially-defined equivalence relations as a type of equivalence relation that incorporates uncertainty. In these relations, certain pairs of objects are not definitively determined to be related or unrelated. The relationship with orthopartitions is put forward, providing the conditions under which an orthopartition can be transformed into an equivalent partially-defined equivalence relation and vice versa. Additionally, we explore their connection with reducts in rough set theory, offering insights into the characterization of similarity reducts in terms of orthopartitions.

Optimal granularity selection based on algorithm stability with application to attribute reduction in rough set theory

Optimal granularity selection is a key issue in rough set, by which decision rules can be generated to assign corresponding decision labels for new samples. The current evaluation criteria of optimal granularity mainly focus on the experimental effect of prediction algorithms, thus ignoring the theoretical relationship between granularity and the generalization ability of algorithms. In this paper, we study the issue overlooked above and propose a novel framework for optimal granularity selection with theoretical guarantee. In our framework, a rough set-based prediction algorithm that incorporates a global confidence as scoring function is introduced, which is characterized by the granularity-based loss function. On the bias, the relationship between granularity and the generalization ability of algorithm is studied by introducing the stability theory in machine learning. The generalization error bound of algorithm is derived as a theoretical guarantee for optimal granularity selection. Finally, a novel optimal granularity selection strategy with theoretical guarantee is proposed, which is combined with existing attribute reduction methods to design the attribute re-reduction algorithms for generate optimal granularity by reducing the relatively unimportant attributes. Numerical experiments verify the rationality of optimal granularity selection framework and prove the effectiveness of the optimal granularity generated by the attribute re-reduction algorithms.

A random approximate reduct-based ensemble learning approach and its application in software defect prediction

Software defect prediction (SDP) is an important research topic in software engineering. It can optimize the allocation of testing resources by indicating the defect-prone software modules. In recent years, ensemble learning has been frequently used in SDP. Ensemble learning can effectively improve the generalization ability of individual learners, but how to increase the diversity of individual learners is still an open issue. In this paper, we propose a random approximate reduct-based ensemble algorithm called ERAR. To increase the diversity of learners generated by ERAR, a novel technique, called random approximate reduct (RAR), is proposed to perturb the attribute space, and the resampling technique is used to perturb the instance space. RAR is derived from the random subspace method and the granular decision entropy-based reduct in rough set theory. We then investigate the application of ERAR in SDP. In order to alleviate the class imbalance issue in SDP, we further design a hybrid mechanism by combining ERAR with the Synthetic Minority Over-sampling Technique (SMOTE). Experimental results on 20 benchmark data sets show that the hybrid mechanism can provide competitive solutions for SDP.

Tri-level attribute reduction in rough set theory

Attribute reduction serves as a pivotal topic of rough set theory for data analysis. The ideas of tri-level thinking from three-way decision can shed new light on three-level attribute reduction. Existing classification-specific and class-specific attribute reducts consider only macro-top and meso-middle levels. This paper introduces a micro-bottom level of object-specific reducts. The existing two types of reducts apply to the global classification with all objects and a local class with partial objects, respectively. The new type applies to an individual object. These three types of reducts constitute tri-level attribute reducts. Their development and hierarchy are worthy of systematical explorations. Firstly, object-specific reducts are defined by object consistency from dependency, and they improve both classification-specific and class-specific reducts. Secondly, tri-level reducts are unified by tri-level consistency. Hierarchical relationships between object-specific reducts and class-specific, classification-specific reducts are analyzed, and relevant connections of three-way classifications of attributes are given. Finally, tri-level reducts are systematically analyzed, and two approaches, i.e., the direct calculation and hierarchical transition, are suggested for constructing a specific reduct. We build a framework of tri-level thinking and analysis of attribute reduction to enrich three-way granular computing. Tri-level reducts lead to the sequential development and hierarchical deepening of attribute reduction, and their results profit intelligence processing and system reasoning.

Class-specific information measures and attribute reducts for hierarchy and systematicness

Attribute reduction of rough set theory underlies knowledge acquisition and has two hierarchical types (classification-based and class-specific attribute reducts) and two perspectives from algebra and information theory; thus, there are four combined modes in total. Informational class-specific reducts are fundamental but lacking and are thus investigated by correspondingly constructing class-specific information measures. First, three types of information measures (i.e., information entropy, conditional entropy, and mutual information) are novelly established at the class level by hierarchical decomposition to acquire their hierarchical connection, systematical relationship, uncertainty semantics, and granulation monotonicity. Second, three types of informational class-specific reducts are correspondingly proposed to acquire their internal relationship, basic properties, and heuristic algorithm. Third, the informational class-specific reducts achieve their transverse connections, including the strength feature and consistency degeneration, with the algebraic class-specific reducts and their hierarchical connections, including the hierarchical strength and balance, with the informational classification-based reducts. Finally, relevant information measures and attribute reducts are effectively verified by decision tables and data experiments. Class-specific information measures deepen existing classification-based information measures by a hierarchical isomorphism, while the informational class-specific reducts systematically perfect attribute reduction by level and viewpoint isomorphisms; these results facilitate uncertainty measurement and information processing, especially at the class level.

Quantitative three-way class-specific attribute reducts based on region preservations

Attribute reduction of rough set theory is effective for intelligent information processing, and class-specific attribute reducts are beneficial for pattern recognition and rule reasoning. According to three-way decisions, class-specific reducts already have three-way types (namely, positive, negative, and positive-negative) of qualitative optimization, which adhere to classical rough sets. In terms of region preservations, there are no corresponding three-way types of quantitative optimization that match probabilistic rough sets. Thus, this paper constructs and investigates quantitative three-way class-specific attribute reducts based on region preservations. First, the uncertainty/nonmonotonicity of quantitative region change is revealed, and it naturally induces the reduction criteria of quantitative region preservations. Then, quantitative three-way class-specific reducts are constructed, and their basic properties regarding their necessary conditions, attribute cores, and reduct algorithms are achieved. Furthermore, their interrelation regarding equivalence and strengthening/balance are obtained for consistent and inconsistent decision classes, respectively, and their expansions for qualitative three-way class-specific reducts are proved. Finally, relevant concepts and obtained results are effectively verified by decision tables and data experiments. By virtue of sure region preservations, quantitative three-way class-specific attribute reducts robustly extend the existing qualitative three-way class-specific reducts and facilitate the optimal identification and quantitative reasoning of class-specific patterns.

Min-max attribute-object bireducts: On unifying models of reducts in rough set theory

A decision table describes a finite set of objects OB by using a finite set of condition attributes C and a finite set of decision attributes D. Pawlak defines attribute reducts by considering the entire decision table. As a generalization, we introduce the notion of min-max attribute-object bireducts of a sub-table restricted by a pair (B, X) of a subset of condition attributes B and a subset of objects X. A pair (R, Z) is a min-max attribute-object bireduct of (B, X) if and only if R is a minimal subset of B such that R and B make the same correct decisions for objects in X and Z is a maximal subset of X for which B can make the correct decisions. We propose the notion of the decidability of objects and introduce the decidable region of a set of objects as a generalization of the positive region of the set. We define and interpret a min-max attribute-object bireduct based on the decidable region. Min-max attribute-object bireducts offer a general model and existing models of attribute reducts are special cases. The results lead to a unified framework for studying four types of attribute reducts.

Similarity-based attribute reduction in rough set theory: a clustering perspective

Attribute reduction is one of the most important research issues in the rough set theory. The purpose of attribute reduction is to find a minimal attribute subset that satisfies some specific criteria, while the minimal attribute subset is called attribute reduct. In this paper, we define a similarity-based attribute reduct based on a clustering perspective. Each decision class is treated as a cluster, and the defined similarity-based attribute reduct can maintain or increase the discriminating ability of different clusters in the case of removing redundant attributes. In view of this, firstly, we define the intra-class similarity for objects in the same decision class and the inter-class similarity for objects between different decision classes. Secondly, we define a similarity-based attribute reduct by maximizing intra-class similarity and minimizing inter-class similarity in the rough set model. Thirdly, by considering the heuristic search strategy, we also design a corresponding reduction method for the proposed attribute reduct. The experimental results indicate that compared with other representative attribute reducts, our proposed attribute reduct can significantly improve the classification performance.

Ensemble selector for attribute reduction

Through abstracting commonness from the existing heuristic algorithms, control strategies bring us higher level understandings of building reducts in rough set theory. To further improve the performances and strengthen the applicabilities of the addition control strategy, an ensemble selector is introduced into such framework. This ensemble selector is realized through using a set of the fitness functions which may be constructed by homogenous or heterogeneous evaluations of the candidate attributes. Based on the neighborhood rough set model, the experimental results tell us that by comparing the traditional addition control strategy, ensemble selector is effective in improving the stabilities of the reducts, the stabilities of the classification results and the AUC values from the viewpoints of KNN and SVM classifiers. This study suggests new trends for considering attribute reduction problems and provides guidelines for designing new algorithms in rough set theory.

Class-specific attribute reducts in rough set theory

The concept of attribute reducts plays a fundamental role in rough set analysis. There are at least two possibilities to define an attribute reduct. A classification-based or global attribute reduct is a minimal subset of condition attributes that preserves the positive region of the decision classification, namely, the positive regions of all decision classes, in a decision table. A class-specific, class-dependent, or local attribute reduct is a minimal subset of condition attributes that preserves the positive region of a particular decision class. While a classification-based reduct may not work equally well for every decision class, a class-specific attribute reduct is optimally tailored to a particular decision class. However, studies in rough set theory are dominated by classification-based reducts; there is very limited investigation on class-specific reducts. An objective of this paper is to draw attention to class-specific reducts. We systematically compare the two types of reducts and investigate their relationships with respect to both individual reducts and families of all reducts. It is possible to derive a class-specific reduct from a classification-based reduct and to derive a classification-based reduct from a family of class-specific reducts. The families of all class-specific reducts provide a pair of lower and upper bounds of the family of all classification-based reducts. Based on a three-way classification of attributes into the pair-wise disjoint sets of core, marginal, and nonuseful attributes, we examine relationships between the corresponding classes of classification-based and class-specific attributes. The union of the sets of class-specific core attributes is the set of classification-based core attributes. It is only possible to obtain an upper bound for the set of classification-based marginal attributes and a lower bound for the set of classification-based nonuseful attributes from the family of the class-specific corresponding sets of attributes.

Three-layer granular structures and three-way informational measures of a decision table

Attribute reduction in rough set theory serves as a fundamental topic for information processing, and its basis is usually a decision table (D-Table). D-Table attribute reduction concerns three hierarchical types, and only classification-based reduction is related to information-theoretic representation. Aiming at inducing comprehensive D-Table attribute reduction with hierarchies and information, this paper concretely constructs a D-Table’s three-layer granular structures and three-way informational measures via granular computing and Bayes’ theorem. With regard to the D-Table, the micro-bottom, meso-middle, and macro-top are hierarchically organized according to the formal structure and systematic granularity. Then, different layers produce different three-way informational measures by developing Bayes’ theorem. Thus, three-way weighted entropies originate from three-way probabilities at the micro-bottom and further evolve from the meso-middle to the macro-top, and their granulation monotonicity and evolution systematicness are acquired. Furthermore, three-way informational measures are analyzed by three-layer granular structures to achieve their hierarchical evolution, superiority, and algorithms. Finally, structural and informational results are effectively illustrated by a D-Table example. This study establishes D-Table’s hierarchical structures to reveal constructional mechanisms and systematic relationships of informational measures. The obtained results underlie the D-Table’s hierarchical, systematic, and informational attribute reduction, and they also enrich the three-way decisions theory.

A Rough Based Hybrid Binary PSO Algorithm for Flat Feature Selection and Classification in Gene Expression Data

Feature selection in high dimensional data, particularly, in gene expression data, is one of the challenging task in bioinformatics due to the curse of dimensionality, data redundancy and noise values. In gene expression data, insignificant features causes poor classification, hence feature selection reduces feature subset, improving classification accuracy. Feature selection algorithms in gene expression data(such as filter based, wrapper based and hybrid methods) performing poor accuracy, where as few methods takes too much time to converge for an acceptable results. For example, in NSGA-II, over 10,000 generations, on an average, to converge in the search space. where it incurs increased computational time. Proposed rough based hybrid binary PSO algorithm, which uses a heuristic based fast processing strategy to reduce crude domain features by statistical elimination of redundant features and then discretized subsequently into a binary table, known as distinction table, in rough set theory. This distinction table is later used as input to evaluate and optimize the objectives functions i.e., to generate reduct in rough set theory. The proposed hybrid binary PSO is then used to tune the objective functions, to choose the most important features (i:e:reduct). The fitness function is used in such a way that it can reduce the cardinality of the features and at the same time, improve the classification performance as well. Results have been demonstrated to show the effectiveness of the proposed method, on existing three benchmark datasets (i.e. colon cancer, lymphoma and leukemia data), from literature.

Fuzzy Population-Based Meta-Heuristic Approaches for Attribute Reduction in Rough Set Theory

Fuzzy Population-Based Meta-Heuristic Approaches for Attribute Reduction in Rough Set Theory

ラフ集合におけるGeneralized Dynamic Reductの抽出手法の改良

ラフ集合におけるGeneralized Dynamic Reductの抽出手法の改良

A new method for constructing granular neural networks based on rule extraction and extreme learning machine

This paper introduces a framework of granular neural networks named rough rule granular extreme learning machine (RRGELM), and develops its comprehensive design process. The proposed granular neural networks are formed on the basis of rough decision rules extracted from training samples through rough set theory. Firstly, Sample data are reduced by the algorithms of attributes reduction and attributes values reduction in rough set theory, and then they are compressed to an irredundant data set. In this data set, each sample can represent a rough rule, and is expressed as an If-Then rule which indicates the relationship between the input and output pattern. Moreover, the confidence level and the coverage level of each rule are calculated. Secondly, granular-neurons can be constructed through the If-Then rules, and all the granular-neurons constitute rule matching layer which is regarded as the hidden layer of the RRGELM. The linked weights between the input neurons and granular-neurons can be determined by the confidences of rough decision rules, while the linked weights between the output neurons and granular-neurons can be initialized as the contributions of the rough rules to the classification. Finally, the extreme learning machine (ELM) algorithm is introduced to improve the learning speed of the RRGELM, rather than the BP algorithm used by other traditional GNN models. Good performance of the proposed RRGELM is demonstrated on several well-known benchmark data sets.

Generalized attribute reduct in rough set theory

Attribute reduction plays an important role in the areas of rough sets and granular computing. Many kinds of attribute reducts have been defined in previous studies. However, most of them concentrate on data only, which result in the difficulties of choosing appropriate attribute reducts for specific applications. It would be ideal if we could combine properties of data and user preference in the definition of attribute reduct. In this paper, based on reviewing existing definitions of attribute reducts, we propose a generalized attribute reduct which not only considers the data but also user preference. The generalized attribute reduct is the minimal subset which satisfies a specific condition defined by users. The condition is represented by a group of measures and a group of thresholds, which are relevant to user requirements or real applications. For the same data, different users can define different reducts and obtain their interested results according to their applications. Most current attribute reducts can be derived from the generalized reduct. Several reduction approaches are also summarized to help users to design their appropriate reducts.

Method and Application of Air Traffic Risk Quantitative Identification Based on Rough Sets

The qualitative method, which can be divided into three types: the technocrat type, the chart type and the system type, is mainly adopted to identify the risk of ATC operation units but has the disadvantages in low work efficiency, poor visibility and easily influenced by subjective factors. In order to overcome its drawbacks, the quantitative risk identification model, which based on the idea of attribute reduction in rough set theory, is established to eliminate redundant risk indexes and improve the accuracy of risk identification on the condition of without losing key risk indexes. This mode will be applied to identify risk consisting in the control mode changing of ATC operation unit. The result show that 4 key risk indexes are extracted from 24 basic risk indexes which is premised on the consistency evaluation of expert evaluation results used the Kendall harmonious coefficient, and then provide the basis for the control decision.

Forward tentative selection with backward propagation of selection decision algorithm for attribute reduction in rough set theory

In rough set theory, attribute reduction is an important application. Many approaches to attribute reduction were developed using rough set theory. These approaches used various greedy heuristics to find a reduct approximation. In this paper, we propose forward tentative selection with backward propagation of selection decision (FTSBPSD) algorithm to find a reduct. The proposed algorithm is based on the principle of indiscernibility of rough set theory. It finds one of the prime implicants of the discernibility function as the reduct. The proposed algorithm works for various types of reducts, defined in the rough set theory. In this work, we have analysed the performance of distribution reduct, maximum distribution reduct, positive region reduct and possible reduct. The proposed algorithm was tested on various datasets found in University of California, machine learning repository. It has given good results for classification accuracy during tests performed on the datasets. Experimental results obtained by FTSBPSD algorithm have been found to give better classification accuracy when tested using C4.5 classifier in comparison to the results obtained by the Q-MDRA algorithm described in the literature.

A Rough Set-Based Method for Updating Decision Rules on Attribute Values’ Coarsening and Refining

Rule induction method based on rough set theory (RST) has received much attention recently since it may generate a minimal set of rules from the decision system for real-life applications by using of attribute reduction and approximations. The decision system may vary with time, e.g., the variation of objects, attributes and attribute values. The reduction and approximations of the decision system may alter on Attribute Values' Coarsening and Refining (AVCR), a kind of variation of attribute values, which results in the alteration of decision rules simultaneously. This paper aims for dynamic maintenance of decision rules w.r.t. AVCR. The definition of minimal discernibility attribute set is proposed firstly, which aims to improve the efficiency of attribute reduction in RST. Then, principles of updating decision rules in case of AVCR are discussed. Furthermore, the rough set-based methods for updating decision rules in the inconsistent decision system are proposed. The complexity analysis and extensive experiments on UCI data sets have verified the effectiveness and efficiency of the proposed methods.

An Innovative Approach for Attribute Reduction in Rough Set Theory

The Rough Sets Theory is used in data mining with emphasis on the treatment of uncertain or vague information. In the case of classification, this theory implicitly calculates reducts of the full set of attributes, eliminating those that are redundant or meaningless. Such reducts may even serve as input to other classifiers other than Rough Sets. The typical high dimensionality of current databases precludes the use of greedy methods to find optimal or suboptimal reducts in the search space and requires the use of stochastic methods. In this context, the calculation of reducts is typically performed by a genetic algorithm, but other metaheuristics have been proposed with better performance. This work proposes the innovative use of two known metaheuristics for this calculation, the Variable Neighborhood Search, the Variable Neighborhood Descent, besides a third heuristic called Decrescent Cardinality Search. The last one is a new heuristic specifically proposed for reduct calculation. Considering some databases commonly found in the literature of the area, the reducts that have been obtained present lower cardinality, i.e., a lower number of attributes.