Linear-combined rough vague sets and their three-way decision modeling and uncertainty measurement optimization

Rough sets (RSs) and vague sets (VSs) are fundamental uncertainty methodologies, and their integrated rough vague sets (RVSs) establish a robust platform for data analysis. For RVSs, double-approximate VSs exhibit the singleness and extreme, so improved models are worth constructing. Thus, linear-combined rough vague sets (LcRVSs) are proposed to perfect RVSs, and subsequent probabilistic-RSs (called LcRVSs-PRSs) are modeled by three-way decision (3WD); furthermore, relevant uncertainty measurement optimization is investigated. At first, LcRVSs are parametrically constructed by fusing and extending RVSs, and their integration algorithms and operation properties are discussed; the granulation-cognitive approximation from LcRVSs to VSs is optimally considered by three-way similarity measures, and thus the optimal parameter value is approximately solved by a discrete search algorithm. Furthermore, LcRVSs motivate the LcRVSs-PRSs model via 3WD, and related construction algorithms and operation properties are offered; for uncertainty measurement, the accuracy, roughness, and dependency are discussed, and their cognitive approximation and parametric optimization are revealed. Finally, the obtained results of models, properties, measures, and algorithms are thoroughly demonstrated by data examples and numerical experiments. By this study, LcRVSs systematically extend, balance, improve RVSs, and their 3WD model (i.e., LcRVSs-PRSs) facilitates roughness decision, so relevant uncertainty measurement and parameter optimization benefit cognitive learning.