Abstract Class imbalance is a substantial challenge in classifying many real-world cases. Synthetic over-sampling methods have been effective to improve the performance of classifiers for imbalance problems. However, most synthetic over-sampling methods generate synthetic instances within the convex hull formed by the existing minority instances as they only concentrate on the minority class and ignore the vast information provided by the majority class. They also often do not perform well for extremely imbalanced data, as fewer minority instances mean less information with which to generate synthetic instances. Moreover, existing methods that generate synthetic instances using the majority class distributional information cannot perform effectively when the majority class has a multi-modal distribution. We propose a new method to generate diverse and adaptable synthetic instances using Synthetic Over-sampling with Minority and Majority classes (SOMM). SOMM generates synthetic instances diversely within the minority data space. It updates the generated instances adaptively to the neighbourhood including both classes. Thus, SOMM performs well for imbalance problems. We examine the performance of SOMM for binary multiclass imbalance classification problems for different imbalance levels. The empirical results and nonparametric statistical testing show the superiority of SOMM compared to existing methods. We also discuss the strengths and limitations of SOMM through visualisations.

Bagging is a common approach in ensemble learning that generates a group of classifiers through bootstrapping for classification tasks. Despite its wide applications, generating redundant classifiers remains a central challenge in bagging. In recent years, many selective bagging models have been presented to deal with this challenge. These models mostly focused on the accuracy of classifiers and the diversity among them. Despite the importance of uncertainty in the performance of ensemble classifiers, this criterion has been neglected in selective bagging models. In this paper, we propose a two-stage selective bagging model. In the first stage, we formalize the selective bagging problem as a bi-objective optimization model considering both the uncertainty and accuracy of classifiers. We propose an adaptive evolutionary Two-Arch2 algorithm, named Diverse-Two-Arch2, to solve the bi-objective model. The output of this stage is a subset of classifiers that are diverse, certain about correct predictions, and uncertain about incorrect predictions. While most selective bagging models focus on the selection of a fixed subset of classifiers for all test samples (static approach), our proposed model has a dynamic approach to the selection process. So, in the second stage of the model, we select only certain classifiers to make an ensemble prediction for each test sample. Experimental results on twenty data sets and comparing with two ensemble models, and five state-of-the-art dynamic selective bagging models show the outperformance of the proposed model. We also compare the performance of the proposed Diverse-Two-Arch2 to alternative evolutionary computation methods.

This contribution reviews critically the existing entropy measures for probabilistic hesitant fuzzy sets (PHFSs), and demonstrates that these entropy measures fail to effectively distinguish a variety of different PHFSs in some cases. In the sequel, we develop a new axiomatic framework of entropy measures for probabilistic hesitant fuzzy elements (PHFEs) by considering two facets of uncertainty associated with PHFEs which are known as fuzziness and nonspecificity. Respect to each kind of uncertainty, a number of formulae are derived to permit flexible selection of PHFE entropy measures. Moreover, based on the proposed PHFE entropy measures, we introduce some entropy-based distance measures which are used in the portion of comparative analysis. Eventually, the proposed PHFE entropy measures and PHFE entropy-based distance measures are applied to decision making in the strategy initiatives where their reliability and effectiveness are verified.