Shannon, Rényi, Tsallis, and Beyond: A Unified Sharma–Mittal Framework for Parameter-Integrated Feature Selection
Entropy, cilt.28, sa.11, ss.1-21, 2026 (Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 28 Sayı: 11
- Basım Tarihi: 2026
- Doi Numarası: 10.3390/e28111214
- Dergi Adı: Entropy
- Derginin Tarandığı İndeksler: Scopus
- Sayfa Sayıları: ss.1-21
- Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
- Yıldız Teknik Üniversitesi Adresli: Evet
Özet
Entropy originated in thermodynamics and has since become a central tool for quantifying uncertainty in probabilistic and information-theoretic systems. This study provides a unified mathematical synthesis of the Shannon, Rényi, Tsallis, and Sharma–Mittal entropies, focusing on their axiomatic foundations, functional structures, and limiting relationships. We show that Rényi entropy reduces to Shannon entropy as α → 1, Tsallis entropy reduces to Shannon entropy as q → 1, and the two-parameter Sharma–Mittal entropy encompasses both Rényi (as β → 1) and Tsallis (when β = α), converging to Shannon entropy in the joint limit (α, β) → (1, 1). Additivity, pseudo-additivity, concavity, and stability are compared across the four families, consolidating the unifying role of Sharma–Mittal entropy noted in earlier work rather than treating it merely as a more flexible formulation. Building on this synthesis, we connect it to a previously developed parameter-integrated framework that addresses the practical difficulty introduced by the Sharma–Mittal entropy’s two-parameter structure: fixing (α, β) at a single, arbitrary pair makes any derived quantity—and, in feature selection, the resulting variable ranking—dependent on that choice. The framework integrates the conditional Sharma–Mittal entropy and information-gain surfaces volumetrically over a bounded region of the α-β plane, yielding three parameter-integrated indices—PICSME, PIGSME, and NIGSME—that are exact monotone transformations of one another and therefore induce an identical variable ranking for a fixed dataset and integration region. This illustrates how the unifying position of Sharma–Mittal entropy can be operationalized into a practical feature-ranking tool that does not require the selection of a single predefined (α, β) pair.