Entropies and Co-Entropies of Coverings with Application to Incomplete Information Systems

  • Authors:

  • Daniela Bianucci;Gianpiero Cattaneo;Davide Ciucci

  • Affiliations:

  • Dip. di Informatica Sistemistica e Comunicazione, Università di Milano - Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italia. E-mail: {bianucci,cattang,ciucci}@disco.unimib.it;Dip. di Informatica Sistemistica e Comunicazione, Università di Milano - Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italia. E-mail: {bianucci,cattang,ciucci}@disco.unimib.it;Dip. di Informatica Sistemistica e Comunicazione, Università di Milano - Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italia. E-mail: {bianucci,cattang,ciucci}@disco.unimib.it

  • Venue:
  • Fundamenta Informaticae - New Frontiers in Scientific Discovery - Commemorating the Life and Work of Zdzislaw Pawlak
  • Year:
  • 2007

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Abstract

Different generalizations to the case of coverings of the standard approach to entropy applied to partitions of a finite universe X are explored. In the first approach any covering is represented by an identity resolution of fuzzy sets on X and a corresponding probability distribution with associated entropy is defined. A second approach is based on a probability distribution generated by the covering normalizing the standard counting measure. Finally, the extension to a generic covering of the Liang-Xu approach to entropy is investigated, both from the "global" and the "local" point of view. For each of these three possible entropies the complementary entropy (or co-entropy) is defined showing in particular that the Liang-Xu entropy is a co-entropy.