提出一种兼具理论严谨与计算高效的新信息度量，用于决策树构建。
A Hybrid Tsallis-Polarization Impurity Measure for Decision Trees: Theoretical Foundations and Empirical Evaluation
- 融合Tsallis熵与极化项，实现信息论基础与分布不对称敏感性的互补
- 参数α=0.5时平均准确率达91.17%，IQC变体达88.38%-89.16%
- 理论可证明凹性，适合对数学严谨性要求高的场景
我们引入集成Tsallis组合（ITC），一种用于决策树学习的混合不纯度度量，结合归一化Tsallis熵与指数极化成分。现有度量常在理论严谨性与计算效率间取舍，而ITC提供兼顾两者的数学严谨框架。核心创新在于Tsallis熵的信息论基础与极化成分对分布不对称性的敏感性之间的互补性。我们建立了关键理论性质——在特定参数条件下保持凹性、满足合理边界条件，并与经典度量建立联系，为混合策略提供严格论证。在七个基准数据集上进行五折重复的23种不纯度度量对比评估显示，简单参数设置（Tsallis α=0.5）取得最高平均准确率（91.17%），而ITC变体表现优异（88.38%-89.16%），且统计检验（Friedman检验：χ²=3.89，p=0.692）表明顶级方法间无显著差异，说明多数应用中可视为等效。ITC的价值在于其坚实的理论基础——在适当条件下可证明凹性、灵活参数化（α, β, γ）及O(K)计算复杂度，使其成为理论保障优先场景下的可靠通用替代方案。我们提供基于应用优先级的度量选择指南，并开源实现以促进可复现性与后续研究。
原文摘要 · Abstract (English)
We introduce the Integrated Tsallis Combination (ITC), a hybrid impurity measure for decision tree learning that combines normalized Tsallis entropy with an exponential polarization component. While many existing measures sacrifice theoretical soundness for computational efficiency or vice versa, ITC provides a mathematically principled framework that balances both aspects. The core innovation lies in the complementarity between Tsallis entropy's information-theoretic foundations and the polarization component's sensitivity to distributional asymmetry. We establish key theoretical properties-concavity under explicit parameter conditions, proper boundary conditions, and connections to classical measures-and provide a rigorous justification for the hybridization strategy. Through an extensive comparative evaluation on seven benchmark datasets comparing 23 impurity measures with five-fold repetition, we show that simple parametric measures (Tsallis $α=0.5$) achieve the highest average accuracy ($91.17\%$), while ITC variants yield competitive results ($88.38-89.16\%$) with strong theoretical guarantees. Statistical analysis (Friedman test: $χ^2=3.89$, $p=0.692$) reveals no significant global differences among top performers, indicating practical equivalence for many applications. ITC's value resides in its solid theoretical grounding-proven concavity under suitable conditions, flexible parameterization ($α$, $β$, $γ$), and computational efficiency $O(K)$-making it a rigorous, generalizable alternative when theoretical guarantees are paramount. We provide guidelines for measure selection based on application priorities and release an open-source implementation to foster reproducibility and further research.
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