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Representation and learning of concepts are critical problems in data science and cognitive science. However, the existing research about concept learning has one prevalent disadvantage: incomplete and complex cognitive. Meanwhile, as a practical mathematical tool for concept representation and concept learning, two-way learning (2WL) also has some issues leading to the stagnation of its related research: the concept can only learn from specific information granules and lacks a concept evolution mechanism. To overcome these challenges, we propose the two-way concept-cognitive learning (TCCL) method for enhancing the flexibility and evolution ability of 2WL for concept learning. We first analyze the fundamental relationship between two-way granule concepts in the cognitive system to build a novel cognitive mechanism. Furthermore, the movement three-way decision (M-3WD) method is introduced to 2WL to study the concept evolution mechanism via the concept movement viewpoint. Unlike the existing 2WL method, the primary consideration of TCCL is two-way concept evolution rather than information granules transformation. Finally, to interpret and help understand TCCL, an example analysis and some experiments on various datasets are carried out to demonstrate our method’s effectiveness. The results show that TCCL is more flexible and less time-consuming than 2WL, and meanwhile, TCCL can also learn the same concept as the latter method in concept learning. In addition, from the perspective of concept learning ability, TCCL is more generalization of concepts than the granule concept cognitive learning model (CCLM).

Formal concept analysis (FCA) extracts interpretable rules by using implication relationships between concepts, which is an effective method for acquiring knowledge. In this article, a fuzzy decision rule-based online classification algorithm called OFRCA is designed by combining online learning theory and FCA in a fuzzy formal decision context. First, the original weight vector of all attributes is obtained by fusing the original deterministic decision rules. Second, incremental fuzzy decision rules are obtained according to the added objects. Then, for the incremental fuzzy decision rules, OFRCA updates the weight vector based on minimizing the loss function. At the end of the process, all the rules are fused into an attribute-weighted classifier with the final weight vector. In this article, we employ the regret to obtain a learning guarantee for OFRCA. The growth rate of OFRCA's regret converges to 0 as the number of iterations increases in the ideal state, which provides an effective learning guarantee for OFRCA. To verify the effectiveness of the proposed algorithm, this article conducted numerical experiments and systematically analyzed the experimental results, which demonstrated that OFRCA achieved the highest advanced classification performance among the compared algorithms.

In multilabel learning, research on label correlation provides an effective solution to compress the hypothesis space of classifiers. However, this article focus on the label correlation adapted to overall data, while ignoring the locally targeted information presented by some instances. The lack exploration on the distribution of local label correlation in multilabel instance space undoubtedly limits the in-depth application of label correlation in multilabel learning. Based on the fuzzy rough set theory, the instance-level label correlation distribution is first proposed in this article and applied to design a novel multilabel learner. For each multilabel instance, the local importance of features to label is quantitatively analyzed, by considering the decisive influence of input information on decision making. According to coincidence degree between local feature weight distribution for different labels, the instance-level label correlation is constructed. In order to reflect the internal relationship between label variables objectively, the instance-level label correlation distribution is integrated into the empirical label relevance. On the basis, the label relevance matrix is used to define the constraints of the optimization function in a new form. The relative position of subseparating hyperplanes in input space is quantitatively characterized to reduce the complexity of the multilabel classifier and improve the learning performance. The experiment results on 18 multilabel datasets illustrate the effectiveness of our algorithm. The impact of core parameters on the performance is also dissected.

As an essential way of knowledge representation and knowledge acquisition in uncertainty theory, variable precision rough set (VPRS) has been widely considered by researchers. Firstly, the corresponding CVPRS model via boundary region is investigated based on an existing covering-based rough set model. Its algebraic structures and properties are systematically studied. This model mainly depicts a pair of boundary operators and a pair of approximation operators with a certain threshold from the perspective of single knowledge granularity. And it not only relaxes the complete inclusion relation but also retains the basic properties of the classical rough set. Secondly, an attribute reduction approach is given for a covering-based decision information systemusing the proposed CVPRS model, which keeps the lower or upper approximation unchanged. The different performances of boundary operators and their related three indices in these reduction methods are compared. The results show that boundary operators and distribution functions have outstanding advantages in these applications. Finally, an extraction method of necessity rules and possibility rules corresponding to decision classes are established. The validity and security of the above methods are verified theoretically. Numerical experiments further show that the proposed models and methods have a certain application value. [ABSTRACT FROM AUTHOR]

In this study, the fusion of probabilistic hesitant fuzzy sets (PHFSs) and rough sets is explored and applied to uncertain multi-criteria decision-making (MCDM). First, to reduce the huge amount of computation of probabilistic hesitant fuzzy operations in the MCDM process, we introduce an advanced method to obtain normalized PHFSs (NPHFSs), in which each probabilistic hesitant fuzzy element (PHFE) has the same length and probability distribution. Second, based on NPHFSs, a novel probabilistic hesitant fuzzy rough set (PHFRS) model is established to solve the information aggregation problem in MCDM. Subsequently, we propose the concept of fuzziness of a PHFRS and develop a fuzziness-based objective weight determination method. Further, we construct an extended PROMETHEE II method based on a PHFRS for MCDM problems. Finally, we analyze the effectiveness of our proposed method through two MCDM problems along with experimental comparisons. The results illustrate that the proposed method is useful in numerous uncertain application fields. [ABSTRACT FROM AUTHOR]

Multi-label decision data serve as an important extension of decision data, where several labels associate with each object. In classical multi-granulation rough set models, granulation selection is always related to positive region. However, some positive region reducts are not appropriate for multi-label data due to the fact that the uncertainties implied by labels are not totally considered. To overcome this deficiency in this paper, we propose the optimal granulation selection in multi-granulation and multi-label decision table. First, two kinds of fine and coarse decision functions are introduced, which represent all certainly and all possibly associated labels for the objects in the same class from optimistic and pessimistic viewpoints, respectively. Some relevant propositions and theorems are also carefully examined. Based on granular significance, a novel approach of optimal granulation selection is discussed. Finally, a heuristic algorithm is designed and an illustrative example is given to show the effectiveness of this algorithm. The main contribution of this paper is to extend the attribute reduction of multi-label data based on rough set theory, and overcome the limitation of the positive region reducts, in which the uncertainties implied by labels are not totally considered.

Abstract Attribute reduction in formal decision contexts is an important issue in formal concept analysis, which can help us to discover the knowledge hidden in formal decision contexts. However, most reduction methods in formal decision contexts are very time-consuming due to two main problems. The first is that one needs to construct a discernibility matrix after generating all the formal concepts of formal decision contexts. This is not an easy task because it requires much more storage space and computation time. Another problem is that most reduction methods are based on the Boolean reasoning and the computational complexity of which is exponential in the worst case. To overcome these problems, we propose a new attribute reduction method for formal decision contexts in this paper. A more simplified discernibility matrix which does not need to generate all the formal concepts is first constructed. It shows that the storage space and computation time are far less than the original method. Furthermore, different from the Boolean reasoning method, an approximation algorithm for obtaining a minimum reduct in formal decision contexts based on graph theory is designed. Finally, experiments are carried out to verify the effectiveness of the proposed method. The results, on 22 large data sets, demonstrate that the proposed method not only produces smaller subset of attributes but also has better performance in both storage space and speed. [ABSTRACT FROM AUTHOR]

The construction of belief intervals is crucial for decision-making in multi-attribute group information integration. Based on multi-adjoint and evidence theory, an approach to multi-criteria group decision-making(MCGDM) in intuitionistic fuzzy information system is proposed. First, the upper and lower approximations of alternatives are calculated by multi-adjoint operators under the correlation matrices which were given by different experts. After that the belief and plausibility functions are gained by intuitionistic fuzzy probability formulas. Second, the belief intervals of alternatives are acquired by combining all experts’ evidence. Then the alternatives are ranked by comparing the belief intervals. Finally, the effectiveness of the method is verified by an application of business transaction. Compared with the existing model, the method introduced in this paper is more effective and accurate.

The approximation operators play an important role in rough set theory, which are mainly defined by means of neighborhood systems. In this paper, firstly, we try to propose a class of novel definitions of the approximation operators via a predefined boundary region based on a binary relation. Then we compare the proposed concepts with the originals, the necessary and sufficient conditions of their equipollence are investigated. Secondly, we give the definitions of boundary region based on a covering. By employing the boundary, a class of novel definitions of the approximation operators based on a covering are proposed. It is shown that the proposed operators are equivalent to a class of covering approximation operators introduced by Zakowski. Thirdly, the relationship between general binary relations and coverings based approximation operators is investigated with the aid of the novel boundary region. Finally, the more reasonable characterizations of the accuracy and the roughness are proposed by employing the boundary operators. Meanwhile, we study uncertainty measures of approximation spaces based on a partition and a covering. [ABSTRACT FROM AUTHOR]

One of the key issues in the theory of concept lattices is to extract the useful rules from the decision formal context. The maximal rules implicate the others, thus people are interested in them. This paper proposes two new kinds of attribute reduction in the decision formal context based on maximal rules. The reducts preserve all the condition extensions and the decision extensions related to the original maximal rules. The internal relationship between the original maximal rules and the maximal rules in the reduced decision formal context is derived. The reducts can make the maximal rules more concise and accurate. The mathematical property of the proposed attribute reduction is investigated and we construct the dis-cernibility matrix and function to compute all the reducts. Finally, all the attributes are classified into three types based on the maximal rules. The characteristics of these types of attributes are also analyzed.

This paper discusses the reductions of a fuzzy covering decision system. Firstly, we define a new induced fuzzy covering by a given fuzzy covering and give the algorithm to compute a reduction of the induced fuzzy covering. Then, we discuss the most complete fuzzy description of every element with respect to a fuzzy covering system and the upper and lower approximation defined by it. And then, we introduce information entropy and conditional entropy of the fuzzy covering decision system and define the reduction of fuzzy covering system by means of conditional entropy (special conditional entropy) in consistent (β-consistent and inconsistent) fuzzy covering decision systems. Algorithms are designed to compute the approximate reduction of consistent, β-consistent and inconsistent fuzzy covering decision systems, respectively.

As a complicated cognitive process, multi-attribute decision-making usually focuses on the decision-making issue of seeking the optimal alternative or ranking alternatives under the framework of multiple attributes. The three-way decision approach with the delayed decision can more effectively reduce decision risks than traditional two-way counterparts for multi-attribute decision-making. In this article, we aim to put forward a novel three-way multi-attribute decision-making model in light of a probabilistic dominance relation with intuitionistic fuzzy sets. First, we investigate the three-way multi-attribute decision-making in light of a probabilistic dominance relation in an intuitionistic fuzzy information system. Second, we derive the conditional probability of the intuitionistic fuzzy set. Third, we evaluate the part supplier selection via the constructed model. At last, for the sake of showing the validity and applicability of the constructed three-way intuitionistic fuzzy multi-attribute decision-making model, we further perform extensive comparative analysis along with experimental analysis from diverse perspectives. [ABSTRACT FROM AUTHOR]

Generally, every label has its own indispensable features, and it is reasonable to assume higher repetitiveness of indispensable features represents higher correlations among labels. Inspired by discernibility matrix and granular computing, this paper draws conclusions as follows: • Essential elements for any label are defined and characterized to reflect the internalconnection between features and label. • Label correlation and relevance judgement matrix for label set are described considering overlap of essential element sets. • Labels with strong correlation are assigned to a relevant label group. • The compact subset of indispensable features for any relevant label group is effectively captured from original feature set. • A novel multi-label learning method, called CLSF-MK, is proposed to increase effectiveness and practicability of prediction. Each example of multi-label data is represented in an object with its feature vector (i.e. an instance) while being related to multiple labels. Learning label correlation can effectively reduce the labels needed to be predicted and optimize the classification performance. For this reason, label correlation plays a crucial role in multi-label learning and has been explored by many existing algorithms. Generally, every label has its own indispensable features, and it is reasonable to assume that a higher repetitiveness of indispensable features represents higher correlations among labels. Inspired by this fact, the essential elements for each label, which are composed of indispensable features, are constructed in this paper. A method is proposed for learning label correlation and applying it to feature selection of multi-label data based on the overlap of different families of essential elements related to label. Firstly, the essential elements of each label are defined and characterized to reflect the internal connection between features and label. In addition, a process for calculating the essential elements of a single label is provided. Secondly, by considering the overlap of the essential element collections that are determined by the different labels, relevancy of label and corresponding relevance judgement matrix with the label set are described. Therefore, several labels with strong relationships are assigned to a relevant group of labels. Meanwhile, local and global label correlations can be computed. Thus a novel multi-label learning algorithm, called CLSF, is presented to select a compact subset of indispensable features for each relevant group of labels by applying local label correlation to feature selection of multi-label data. Finally, comprehensive experiments on eleven benchmark data sets clearly illustrate the effectiveness and efficiency of CLSF against five other multi-label learning algorithms. [ABSTRACT FROM AUTHOR]

The axiomatic approach is an important one to the development of rough set theory. This paper proposes some independent and minimal axiom sets for characterizing the classical rough set. First, we investigate the correlations among some arbitrary binary relations in rough sets. Then, we provide several forms of equivalence relations which are different from the original ones. Finally, to well characterize the classical rough set, we propose fifteen new minimal axiom sets of rough sets based on the new equivalence relations. By employing relative tables, the independence of axiom sets for characterizing the approximation operators is also examined. [ABSTRACT FROM AUTHOR]