Three-way improved neighborhood entropies based on three-level granular structures

Neighborhood systems and their rough sets have robustness and adaptability, and relevant neighborhood information measures underlie uncertainty analysis and intelligent processing. The classical conditional neighborhood entropy becomes fundamental and representative for dependency measurement, but it has three limitations: interaction incompleteness, hierarchy lack, and inconclusive monotonicity/non-monotonicity. This paper aims to improve the conditional neighborhood entropy, and thus we establish three-way neighborhood entropies based on three-level granular structures. At first, three-level granular structures are proposed for neighborhood decision systems, and the conditional neighborhood entropy is improved to hierarchical conditional neighborhood entropies, mainly by information enrichment and hierarchical decomposition. According to simulation extension, three-way neighborhood entropies are then hierarchically constructed by logarithmic information function on three-way probabilities, and they acquire systematicness equations, monotonicity/non-monotonicity mechanisms, and integration algorithms. Finally, all concerned neighborhood information measures and their calculations, relationships, properties are effectively verified by both decision table examples and data set experiments. Three-way neighborhood entropies adhere to three levels and three modes to realize criss-cross informatization for neighborhood decision systems, and they achieve four improvement merits regarding accuracy, hierarchy, systematicness, and monotonicity. This study facilitates uncertainty measurement, information processing, and knowledge discovery.