Theoretical Properties of Closed Frequent Itemsets in Frequent Pattern Mining

Closed frequent itemsets (CFIs) play a crucial role in frequent pattern mining by providing a compact and complete representation of all frequent itemsets (FIs). This study systematically explores the theoretical properties of CFIs by revisiting closure operators and their fundamental definitions. A series of formal properties and rigorous proofs are presented to improve the theoretical understanding of CFIs. Furthermore, we propose confidence interval-based closed frequent itemsets (CICFIs) by integrating frequent pattern mining with probability theory. To evaluate the stability, three classical confidence interval (CI) estimation methods of relative support (rsup) based on the Wald CI, the Wilson CI, and the Clopper–Pearson CI are introduced. Extensive experiments on both an illustrative example and two real datasets are conducted to validate the theoretical properties. The results demonstrate that CICFIs effectively enhance the robustness and interpretability of frequent pattern mining under uncertainty. These contributions not only reinforce the solid theoretical foundation of CFIs but also provide practical insights for the development of more efficient algorithms in frequent pattern mining.