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The theories of rough sets (RSs) and soft sets (SSs) are practical mathematical techniques to accommodate data uncertainty. On the other hand, fuzzy bipolar soft sets (FBSSs) can address uncertainty and bipolarity in various situations. The key objective of this study is to establish the notions of rough fuzzy bipolar soft ideals in semigroup (SG), which is an extension of the idea of rough fuzzy bipolar soft sets in a SG. Also, we have analyzed the roughness in the bipolar fuzzy subsemigroup (BF-SSG) by employing a congruence relation (CR) defined on the SG and investigating several related characteristics. Further, the idea is expanded to the rough fuzzy bipolar soft ideal, rough fuzzy bipolar soft interior ideal, and rough fuzzy bipolar soft bi-ideal in SGs. Moreover, it is observed that CR and complete CR (CCR) are critical in developing rough approximations of fuzzy bipolar soft ideals. Therefore, their related characteristics are studied via CRs and CCRs.

The well-known computing models, classical/fuzzy/rough automata, in spite of their multifarious applications, fail to model those complex real-world systems that contain both vagueness and incomplete information in datasets of real-world complex systems. The computing model, $L$-fuzzy rough automata, incorporates both vagueness and incomplete information in datasets of such systems but fails to incorporate the weights of fuzzy attributes. To overcome this issue, we first introduce the concept weighted hesitant fuzzy finite automaton (WHFFA), as a generalized notion of the hesitant $L$-fuzzy automaton (HLFA), where weights show that the decision maker has distinct confidence in providing the possible valuation of the membership degree. We used the concept of weighted hesitant fuzzy rough set (WHFRS) as a hybrid concept of hesitant fuzzy rough set (HFRS) and weighted hesitant fuzzy set (WHFS) to introduce a novel computing model weighted hesitant fuzzy finite rough automaton (WHFFRA). The introduced WHFFRA is efficient for dealing with vagueness and incomplete information inherent in our natural languages and in datasets of the real-world complex systems. Finally, we discuss determinization of WHFFRA and demonstrated the application of introduced WHFFRA in decision-making scenarios of medical diagnosis problems.

In this paper, a fast pre-classified-based super-resolution model has been proposed to overcome the problems of degraded imaging in weak-target real-time detection system, specialized to copper defect detection. To accurately characterize the defected image, textural features based on the statistical function of gray-gradient are presented. Furthermore, to improve the effectiveness and practicality of the online detection, a concept of pre-classified learning is introduced and an edge smoothness rule is designed. Some experiments are carried out on defect images in different environments and the experimental results show the efficiency and effectiveness of the algorithm.

Due to non-stationary nature of Indian summer monsoon rainfall (ISMR), analysis of its patterns and behaviors is a very tedious task. Advance prediction and behaviors play a significant role in various domains. Literature review reveals that researchers’ works are limited to design predictive models but not on inherited patterns and behaviors for the ISMR. In this study, a novel method based on the hybridization of two computational techniques, viz., fuzzy and rough sets is proposed for patterns and behaviors. The proposed method initially classifies the information into the four distinct regions, as fuzzy positive region, fuzzy negative region, completely fuzzy region, and gray fuzzy region. Based on four regions, four different patterns of decision rules are explored. Further, a method is discussed to represent such decision rules in terms of graph, which helps to analyze the patterns of ISMR by discovering new knowledge. The proposed method is validated by performing various statistical analyses.

The purpose of this paper is to start a conceptual investigation of approximation rule based on VPRS as a result of the certainty degree of rules in complete information system that cannot exactly express the uncertainty of those in incomplete information system, and then an efficient approximation rule induction algorithm under the rough set framework is presented. Instead of focusing on the minimal rule set, this algorithm hierarchically extracts rules in multistages from data sets to suit changing environments in learning and classification. In addition, a heuristic strategy is employed in the algorithm to improve its performance and reduce the time consumed in inducing. Experiments are carried out, and the results show that the proposed algorithm is effective in inducing rules which can enhance their adaptive capacities.

The increasing diverse demand for image feature recognition and complicated relationships among image pixels cannot be fully and effectively handled by traditional single image recognition methods. In order to effectively improve classification accuracy in image processing, a deep belief network (DBN) classification model based on probability measure rough set theory is proposed in our research.

First, the incomplete and inaccurate fuzzy information in the original image is preprocessed by the rough set method based on probability measure. Second, the attribute features of the image information are extracted, the attribute feature set is reduced to generate the classification rules, and key components are extracted as the input of the DBN. Third, the network structure of the DBN is determined by the extracted classification rules, and the importance of the rough set attributes is integrated and the weights of the neuronal nodes are corrected by the backpropagation (BP) algorithm. Last, the DBN is trained to classify images. The experimental analysis of the proposed method for medical imagery shows that it is more effective than current single rough set approach or the taxonomy of deep learning.

Cancer prediction from gene expression data is a very challenging area of research in the field of computational biology and bioinformatics. Conventional classifiers are often unable to achieve desired accuracy due to the lack of ‘sufficient’ training patterns in terms of clinically labeled samples. Active learning technique, in this respect, can be useful as it automatically finds only few most informative (or confusing) samples to get their class labels from the experts and those are added to the training set, which can improve the accuracy of the prediction consequently. A novel active learning technique using fuzzy-rough nearest neighbor classifier (ALFRNN) is proposed in this paper for cancer classification from microarray gene expression data. The proposed ALFRNN method is capable of dealing with the uncertainty, overlapping and indiscernibility often present in cancer subtypes (classes) of the gene expression data. The performance of the proposed method is tested using different real-life microarray gene expression cancer datasets and its performance is compared with five other state-of-the-art techniques (out of which three are active learning-based and two are traditional classification methods) in terms of percentage accuracy, precision, recall, $F_1$-measures and kappa. Superiority of the proposed method over the other counterpart algorithms is established from experimental results for cancer prediction and results of the paired $t$-test confirm statistical significance of the results in favor of the proposed method for almost all the datasets.

Rough set based methods have been applied successfully in many real world applications such as data mining, knowledge discovery, machine learning, and control. The rough set theory is used to deal with imperfect data and to eliminate dispensable, superfluous and redundant information as to obtain a simplified set of decision rules. Thus, several approaches and methods have been proposed to find minimal coverings, from which the decision rules can be induced. In many of these approaches, an improvement in the utilization of computational resources is encouraged.

In this paper, a binary encoding for attribute sets and a discernibility matrix is proposed. Such a binary representation of sets and sets operations in the implementation of algorithms provides a machine-oriented approach to the utilization of computational memory and allow parallel processing among groups of attributes. The discernibility matrix is reduced to its minimal size through the identification of main patterns in order to eliminate redundancies. Bit-wise operations replace sets operations, thus the search for minimal coverings is performed in an efficient way. Resulting improvement is shown in the analysis of medium-sized data sets using two generic methods to obtain minimal coverings.

In this paper, associated with dominance relation, lattice theory and intuitionistic fuzzy sets theory, the lattice-valued information systems with interval-valued intuitionistic fuzzy decision are proposed and some of its properties are investigated carefully. And, an approach to knowledge reduction based on discernibility matrix in consistent lattice-valued information systems with interval-valued intuitionistic fuzzy decision is constructed and an illustrative example is applied to show its validity. Moreover, extended from the idea of knowledge reduction in consistent information systems, four types of reductions and approaches to obtaining the knowledge reductions of the inconsistent lattice-valued information systems with interval-valued intuitionistic fuzzy decision are formulated via the use of discernibility matrix. Furthermore, examples are considered to show that the approaches are useful and effective. One can obtain that the research is meaningful both in theory and in application for the issue of knowledge reduction in complex information systems.

In this paper, we would like to present some logics with semantics based on rough set theory and related notions. These logics are mainly divided into two classes. One is the class of modal logics and the other is that of quantifier logics. For the former, the approximation space is based on a set of possible worlds, whereas in the latter, we consider the set of variable assignments as the universe of approximation. In addition to surveying some well-known results about the links between logics and rough set notions, we also develop some new applied logics inspired by rough set theory.

Theories of fuzzy sets and rough sets have emerged as two major mathematical approaches for managing uncertainty that arises from inexact, noisy, or incomplete information. They are generalizations of classical set theory for modelling vagueness and uncertainty. Some integrations of them are expected to develop a model of uncertainty stronger than either. The present work may be considered as an attempt in this line, where we would like to study fuzziness in probabilistic rough set model, to portray probabilistic rough sets by fuzzy sets. First, we show how the concept of variable precision lower and upper approximation of a probabilistic rough set can be generalized from the vantage point of the cuts and strong cuts of a fuzzy set which is determined by the rough membership function. As a result, the characters of the (strong) cut of fuzzy set can be used conveniently to describe the feature of variable precision rough set. Moreover we give a measure of fuzziness, fuzzy entropy, induced by roughness in a probabilistic rough set and make some characterizations of this measure. For three well-known entropy functions, including the Shannon function, we show that the finer the information granulation is, the less the fuzziness (fuzzy entropy) in a rough set is. The superiority of fuzzy entropy to Pawlak's accuracy measure is illustrated with examples. Finally, the fuzzy entropy of a rough classification is defined by the fuzzy entropy of corresponding rough sets. and it is shown that one possible application of it is lies in measuring the inconsistency in a decision table.

In some probabilistic problems, complete information about the probability model may not exist. In this article, we obtain a lower and upper probability for an arbitrary event by using rough set theory and then a measurement for inclusiveness of events is introduced.

Theory of fuzzy set and theory of rough set are two useful means of describing and modeling of uncertainty in ill defined environment where precise mathematical analysis are not suitable. Classical rough set theory is based on equivalent relation. It has been indicated that it could be generated to case with a similarity relation. So far, there has been theoretical investigations on roughness measure of fuzzy set based on equivalent relation. The intention of this paper is to go further and propose a measure of roughness of a type-2 fuzzy set based on similarity relation and prove some properties of this novel measure.

Rough set theory is a relatively new mathematical tool for computer applications in circumstances characterized by vagueness and uncertainty. In this paper, we address uncertainty of rough sets for incomplete information systems. An axiom definition of knowledge granulation for incomplete information systems is obtained, under which a measure of uncertainty of a rough set is proposed. This measure has some nice properties such as equivalence, maximum and minimum. Furthermore, we prove that the uncertainty measure is effective and suitable for measuring roughness and accuracy of rough sets for incomplete information systems.

The purpose of this paper is to present a new rough set model for generating negative rules from the incomplete information system. A negative rule indicates that if an object does not satisfy the attribute-value pairs in the condition part, then we can exclude the decision part from such object. The proposed rough set model is constructed on the basis of a difference relation. Such difference relation is a binary relation without any constraints. Moreover, to simplify the negative rules generated from the difference relation-based rough approximations, the concepts of lower, upper approximate and rough reducts are also proposed. Some numerical examples are employed to substantiate the conceptual arguments.

The neighborhood system based rough set is a generalization of Pawlak's rough set model since the former uses the neighborhood system instead of the partition for constructing target approximation. In this paper, the neighborhood system based rough set approach is employed to deal with the incomplete information system. By the coverings induced by the maximal consistent blocks and the support sets of the descriptors, respectively, two neighborhood systems based rough sets are explored. By comparing with the original maximal consistent block and descriptor based rough sets, the neighborhood system based rough sets hold the same lower approximations and the smaller upper approximations. Furthermore, the concept of attribute reduction is introduced into the neighborhood systems and the corresponding rough sets. The judgement theorems and discernibility functions to compute reducts are also presented. Some numerical examples are employed to substantiate the conceptual arguments.

In this paper, we aim to study intuitionistic fuzzy ordered information systems. Firstly, the concept of intuitionistic fuzzy ordered information systems is proposed by introducing an intuitionistic fuzzy relation to ordered information systems. Meanwhile, two approximation operators are defined and a rough set approach is established in intuitionistic fuzzy ordered information systems. Secondly, a ranking approach for all objects is constructed in this system. Thirdly, approximation reduction is addressed in intuitionistic fuzzy ordered decision information system. These results will be helpful for decision-making analysis in intuitionistic fuzzy ordered information systems.

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a generalization of linear algebra and graph theory. Recently, a matroidal structure of rough sets is established and applied to the problem of attribute reduction which is an important application of rough set theory. In this paper, we propose a new matroidal structure of rough sets and call it a parametric matroid. On the one hand, for an equivalence relation on a universe, a parametric set family, with any subset of the universe as its parameter, is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore a matroid is generated, and we call it a parametric matroid of the rough set. Through the lower approximation operator, three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, partition-circuit matroids are well studied through the lower approximation number, and then we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.

In this paper, a new incremental knowledge acquisition method is proposed based on rough set theory, decision tree and granular computing. In order to effectively process dynamic data, describing the data by rough set theory, computing equivalence classes and calculating positive region with hash algorithm are analyzed respectively at first. Then, attribute reduction, value reduction and the extraction of rule set by hash algorithm are completed efficiently. Finally, for each new additional data, the incremental knowledge acquisition method is proposed and used to update the original rules. Both algorithm analysis and experiments show that for processing the dynamic information systems, compared with the traditional algorithms and the incremental knowledge acquisition algorithms based on granular computing, the time complexity of the proposed algorithm is lower due to the efficiency of hash algorithm and also this algorithm is more effective when it is used to deal with the huge data sets.

Microblogging platforms like Twitter, Tumblr and Plurk have radically changed our lives. The presence of millions of people has made these platforms a preferred channel for communication. A large amount of User Generated Content, on these platforms, has attracted researchers and practitioners to mine and extract information nuggets. For information extraction, clustering is an important and widely used mining operation. This paper addresses the issue of clustering of micro-messages and corresponding users based on the text content of micro-messages that reflect their primitive interest. In this paper, we performed modification of the Similarity Upper Approximation based clustering algorithm for clustering of micro-messages. We compared the performance of the modified Similarity Upper Approximation based clustering algorithm with state-of-the-art clustering algorithms such as Partition Around Medoids, Hierarchical Agglomerative Clustering, Affinity Propagation Clustering and DBSCAN. Experiments were performed on micro-messages collected from Twitter. Experimental results show the effectiveness of the proposed algorithm.