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• The existing methods may lead to a contradictory in the transitivity. • A novel neighborhood geometric similarity was firstly defined. • Four new kinds of neighborhood similarities were proposed. • A new VPNRS model is constructed, and a novel decision tree algorithm is proposed. The decision tree algorithm has been widely used in data mining and machine learning due to its high accuracy, low computational cost and high interpretability. However, when dealing with the continuous data, the classical decision tree algorithm needs to replace continuous attributes with discretized attributes by the strategy of discretization. Discretization may cause a loss of information structure, which will affect the performance of classification. To tackle this problem, many researchers have proposed different decision tree methods based on variable precision neighborhood rough sets. However, these methods do not consider the geometric structure of neighborhood systems, which may lead to a contradiction in the transitivity of the equivalence relation. In this paper, we first define a novel neighborhood geometric similarity in a neighborhood system from the perspective of geometry. Second, by combining the neighborhood geometric similarity and the neighborhood algebraic similarity, we propose four new kinds of neighborhood similarities, which can solve the contradictory transitivity of the equivalence relation. Third, a variable precision neighborhood rough set model is constructed using the new similarities, and a novel decision tree algorithm is proposed based on this model, where the degree of attribute dependence is used as the partition measure. Experimental results on 14 selected datasets from the UCI Machine Learning Repository show that our algorithm is effective. The average accuracy of our algorithm is over 90%, which is 10% higher than the classical decision tree algorithms, and the number of leaf nodes increases slightly. [ABSTRACT FROM AUTHOR]

Clustering, as an essential technique in unsupervised learning, plays a pivotal role in the fields of data mining and machine learning. However, the classic K -means clustering algorithm has intrinsic drawbacks such as sensitivity to initial cluster centers, susceptibility to a local optimal solution, and challenges in handling data uncertainty. To address these problems, this paper proposes an artificial hummingbird algorithm (AHA)-based three-way K -means clustering algorithm, called AHA-3WKM. First, AHA is introduced to address the problems of sensitivity to initial cluster centers and local optima. Second, a fitness function of AHA is specifically constructed to find the best initial clustering centers so that the hummingbirds can search for high-quality food sources, i.e., the global optimum cluster centers. Third, a three-way clustering approach is utilized to capture information about data uncertainty. In this way, the results of clustering are divided into three distinct regions based on the relationship between objects and clusters. The experimental results demonstrate that AHA-3WKM has good performance, and enhances the stability and the accuracy of clustering results. • AHA is introduced to address the problems of the sensitivity to initial cluster centers and the proneness to local optima. Hummingbirds are treated as data points, which dynamically update their strategies and effectively find the optimal cluster centers during multiple iterations. • A fitness function is designed based on the clustering principle of "birds of a feather flock together", with the aim of simplifying calculations, which enhances the specificity and practicality of K-means algorithm. • An AHA-based three-way K-means clustering algorithm (i.e., AHA-3WKM) is proposed. The clustering process is initialized with cluster centers optimized by AHA, and the results are represented in three regions, which can capture the uncertainty within the datasets. [ABSTRACT FROM AUTHOR]

Minimal and maximal descriptions of concepts are two important notions in covering-based rough sets. Many issues in covering-based rough sets (e.g., reducts, approximations, etc.) are related to them. It is well known that, it is time-consuming and error-prone when set representations are used to compute minimal and maximal descriptions in a large scale covering approximation space. To address this problem, matrix-based methods have been proposed in which calculations can be conveniently implemented by computers. In this paper, motivated by the need for knowledge discovery from large scale covering information systems and inspired by the previous research work, we present two novel matrix-based approaches to compute minimal and maximal descriptions in covering-based rough sets, which can reduce the computational complexity of traditional methods. First, by introducing the operation "sum" into the calculation of matrix instead of the operation " ⊕ ", we propose a new matrix-based approach, called approach-1, to compute minimal and maximal descriptions, which does not need to compare the elements in two matrices. Second, by using the binary relation of inclusion between elements in a covering, we propose another approach to compute minimal and maximal descriptions. Finally, we present experimental comparisons showing the computational efficiency of the proposed approaches on six UCI datasets. Experimental results show that the proposed approaches are promising and comparable with other tested methods. [ABSTRACT FROM AUTHOR]

Three-way decisions provide a trisecting-and-acting framework for complex problem solving. For a cost-sensitive decision-making problem under multiple levels of granularity, sequential three-way decisions have come into being. Within this framework, how to act upon the three pair-wise disjoint regions is the most important issue. To this end, we propose a generalized model of sequential three-way decisions via multi-granularity in this paper. Subsequently, we adopt the typical aggregation strategies to implement the following five kinds of multigranulation sequential three-way decisions—the weighted arithmetic mean multigranulation sequential three-way decisions, the optimistic multigranulation sequential three-way decisions, the pessimistic multigranulation sequential three-way decisions, the pessimistic-optimistic multigranulation sequential three-way decisions and the optimistic-pessimistic multigranulation sequential three-way decisions. Furthermore, we discuss the rightness and rationality of the five kinds of multigranulation sequential three-way decisions and also analyze the relationships and differences between them. Finally, the experimental results demonstrate that the first four different multigranulation sequential three-way decisions are effective. These models will accelerate and enrich the development of multigranulation three-way decisions. [ABSTRACT FROM AUTHOR]

Abstract Three-way decisions have become a representative of trisecting-and-acting model for uncertainty data analysis. With granular computing point of view, a more reasonable decision-making model should process data from multiview and multilevel. Classical sequential three-way decisions are based on a granular structure with a single threshold under multiple levels of granularity, which make the traditional models incapable of adapting to the case of multiview granular structures with multiple thresholds. To overcome this disadvantage, we propose a generalized multigranulation sequential three-way decision model based on multiple different thresholds. By controlling the number of multiview granular structures satisfying the corresponding two tolerance thresholds, we further adopt the optimistic, pessimistic, variable, weighted and weighted arithmetic mean aggregation strategies to construct five kinds of multigranulation sequential three-way decision models from the quantitative point of view. The corresponding relationships and the uncertainty measures of these multigranulation sequential three-way decisions are discussed. Finally, the experimental results demonstrate that different multigranulation sequential three-way decisions can solve the problem under multiple granular structures and have more flexible fault tolerance under different levels of granularity. This study will enrich and prompt the development of the multigranulation three-way decisions. [ABSTRACT FROM AUTHOR]

Abstract: Recently, much attention has been given to multi-granulation rough sets (MGRS) and different kinds of multi-granulation rough set models have been developed from various viewpoints. In this paper, we propose four types of multi-granulation covering rough set (MGCRS) models under covering approximation space, where a target concept is approximated by employing the maximal or minimal descriptors of objects in a given universe of discourse U. And then, we investigate a number of basic properties of the four types of MGCRS models, and discuss the relationships and differences among the classical MGRS model and our MGCRS models. Moreover, the conditions for two distinct MGCRS models which produce identical lower and upper approximations of a target concept in a covering approximation space are also studied. Finally, the relationships among the four types of MGCRS models are explored. We find that for any subset , the lower approximations of X and the upper approximations of X under the four types of MGCRS models can construct a lattice, if we consider the binary relation of inclusion. [Copyright &y& Elsevier]

As two important tools of information descriptions, minimal description (MinD) and maximal description (MaxD) are used to eliminate redundant information when describing an object in covering-based rough sets (CbRSs). Hence, there are many key issues (such as reductions, neighborhoods and approximations) have to do with them in CbRSs. However, it is complicated and time-consuming to use set approaches to compute them in a large cardinal covering approximation space. So the matrix approaches have been used to calculate them by which computers can implement the corresponding calculations easily. Based on the previous research work and inspired by the demand of knowledge discovery under large scale covering information systems, we propose two new matrix methods to calculate MinD and MaxD in CbRSs in this paper. Firstly, a note on the existing methods (matrix approach-1 and approach-2) is presented. Secondly, a new matrix, called grained matrix, is presented. After combining grained matrices and the mean of "sum", the new matrix approach-1 is presented to calculate MinD and MaxD. Thirdly, three new matrix operations and the notion of complementary matrix are given which can simplify the calculation of computers. Based on new operations, we present the new matrix approach-2 to compute MinD and MaxD. Finally, the computational efficiency of the presented approaches are shown and compared with other methods by experiments on several UCI data sets. [ABSTRACT FROM AUTHOR]

• We generalize the concepts of the conditional and decision attributes through concept hierarchy tree, and define a generalized hierarchical decision table. • We decompose and aggregate condition and decision attribute granules under different levels of granularity. • We propose the optimistic and pessimistic generalized multigranulation three-way decision models for hierarchical classification. • We compare the advantages under different sequential three-way decision models experimentally. Hierarchical classification is an important research hotspot in machine learning due to the widespread existence of data with hierarchical class structures. The existing sequential three-way decision models mainly constructed the hierarchical condition information granules via concept hierarchy tree to discuss the three probabilistic regions for flat classification. However, in real-world applications, one may face not only the tree-structured data with hierarchical condition attributes but also more often the multi-level data with hierarchical decision attribute (hierarchical class labels). How to obtain acceptable decisions under different levels of granularity is the most important issue within the multi-level and multi-view data. To this end, we construct a generalized hierarchical decision table and propose a generalized hierarchical multigranulation sequential three-way decision model by combining multi-granularity and sequential three-way decisions. Specifically, we first design a generalized hierarchical decision table using concept hierarchy trees of all conditional attributes and decision attribute, and explore some basic properties. Then we decompose and aggregate condition and decision granules under different levels of granularity, propose the optimistic and pessimistic generalized hierarchical multigranulation three-way decision models to update the three probabilistic regions for flat and hierarchical classification, and discuss the relationships between these two models. Finally, the experimental results demonstrate that the proposed models are more suitable for different applications. These models will provide a novel insight and enrich the development of multigranulation three-way decisions from the perspective of multi-level and multi-view. [ABSTRACT FROM AUTHOR]