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<p>Optimal scale selection represents a pivotal challenge in knowledge acquisition within multi-scale decision tables. In practical applications, most information tables rely on dominance relations with unknown or missing attribute values, rather than equivalence relations. This paper addresses optimal scale selection in incomplete multi-scale ordered data using the Dempster-Shafer theory of evidence (DSTE). We first formalize incomplete multi-scale ordered information tables (IMOITs) and incomplete multi-scale ordered decision tables (IMODTs). Subsequently, we construct dominance relations and their resultant dominance classes across scales in IMOITs. We further define upper and lower approximations of upward union of decision classes under decision dominance relations at different scales in IMODTs, and we also present fundamental relationships between these approximations. With reference to plausibility and belief functions in the DSTE, we then define the notions of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\ge \)</EquationSource> </InlineEquation>-plausibility/belief optimal scale in IMOITs, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\ge \)</EquationSource> </InlineEquation>-relative plausibility/belief optimal scale in IMODTs. We clarify relationships between these new concepts of optimal scales and existing ones. We demonstrate that, in an IMOIT, a scale is <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\ge \)</EquationSource> </InlineEquation>-belief optimal if and only if (iff) it is <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\ge \)</EquationSource> </InlineEquation>-optimal. In an IMODT, a scale is <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\textrm{Upp}^\ge \)</EquationSource> </InlineEquation> (respectively, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\textrm{Low}^\ge \)</EquationSource> </InlineEquation>) optimal scale iff it is <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\ge \)</EquationSource> </InlineEquation>-relative plausibility (respectively, belief) one. Furthermore, in a consistent IMODT, the notions of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\ge \)</EquationSource> </InlineEquation>-optimal and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\ge \)</EquationSource> </InlineEquation>-relative belief optimal are equivalent, while they are indeed different notions in an inconsistent IMODT. Finally, there is no fixed relationship between the concepts of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\ge \)</EquationSource> </InlineEquation>-belief optimal scale (respectively, <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\ge \)</EquationSource> </InlineEquation>-relative belief optimal scale) and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(\ge \)</EquationSource> </InlineEquation>-plausibility optimal scale (respectively, <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(\ge \)</EquationSource> </InlineEquation>-relative plausibility optimal scale) in IMOITs (respectively, IMODTs).</p>

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On evidence theory based optimal scale selection in incomplete multi-scale ordered decision tables

  • 期刊论文

  • 发表日期：

    2026年2月16日

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  • Han Bao,
  • Wei-Zhi Wu,
  • Tong-Jun Li,
  • An-Hui Tan

摘要

Optimal scale selection represents a pivotal challenge in knowledge acquisition within multi-scale decision tables. In practical applications, most information tables rely on dominance relations with unknown or missing attribute values, rather than equivalence relations. This paper addresses optimal scale selection in incomplete multi-scale ordered data using the Dempster-Shafer theory of evidence (DSTE). We first formalize incomplete multi-scale ordered information tables (IMOITs) and incomplete multi-scale ordered decision tables (IMODTs). Subsequently, we construct dominance relations and their resultant dominance classes across scales in IMOITs. We further define upper and lower approximations of upward union of decision classes under decision dominance relations at different scales in IMODTs, and we also present fundamental relationships between these approximations. With reference to plausibility and belief functions in the DSTE, we then define the notions of \(\ge \) -plausibility/belief optimal scale in IMOITs, and \(\ge \) -relative plausibility/belief optimal scale in IMODTs. We clarify relationships between these new concepts of optimal scales and existing ones. We demonstrate that, in an IMOIT, a scale is \(\ge \) -belief optimal if and only if (iff) it is \(\ge \) -optimal. In an IMODT, a scale is \(\textrm{Upp}^\ge \) (respectively, \(\textrm{Low}^\ge \) ) optimal scale iff it is \(\ge \) -relative plausibility (respectively, belief) one. Furthermore, in a consistent IMODT, the notions of \(\ge \) -optimal and \(\ge \) -relative belief optimal are equivalent, while they are indeed different notions in an inconsistent IMODT. Finally, there is no fixed relationship between the concepts of \(\ge \) -belief optimal scale (respectively, \(\ge \) -relative belief optimal scale) and \(\ge \) -plausibility optimal scale (respectively, \(\ge \) -relative plausibility optimal scale) in IMOITs (respectively, IMODTs).