Optimal scale selection based on three-way decisions with decision-theoretic rough sets in multi-scale set-valued decision tables

Optimal scale selection (OSS) is a fundamental topic in the studies of multi-scale decision tables (MSDTs). Multi-scale set-valued decision tables (MSSVDTs) widely exist in practical applications, and the attribute value is a linguistic set-value. Existing studies of OSS in the MSDT with cost-sensitive learning have constructed total cost mainly from two aspects: test cost and delay cost. Moreover, they are given subjectively, resulting in a lack of objectivity in the construction of total cost. Therefore, constructing a relatively objective and comprehensive total cost for OSS based on cost-sensitive learning is worthwhile in MSSVDTs. In this paper, we firstly propose a quantization method to reasonably transform the linguistic set-value into a numerical value according to the granular structures. Then, based on three-way decisions with decision-theoretic rough sets, loss functions of every object on different scales are constructed, and uncertainty is quantified. Afterwards, loss functions are introduced into the construction of total cost with regard to OSS. This helps us obtain relatively objective total cost, including test cost, delay cost, and misclassification cost. Furthermore, in light of the idea of Technique for Order Preferences by Similarity to an Ideal Solution, we design an OSS algorithm to select the optimal scale according to the ordered change of uncertainty and total cost. Finally, the feasibility and effectiveness of the proposed algorithm are verified through experiments on UCI data sets.