symmetry studies and decompositions of entropy对称熵的研究和分解.pdfVIP

Entropy 2006, 8[2], 88-109 Entropy ISSN 1099-4300 /entropy/ Full paper Symmetry Studies and Decompositions of Entropy Marlos Viana The University of Illinois at Chicago Eye Center, 1855 W. Taylor St., Chicago, IL 60612, USA E-mail: viana@ Received: 13 January 2006 / Accepted: 24 April 2006 / Published: 14 June 2006 Abstract: This paper describes a group-theoretic method for decomposing the entropy of a finite ensemble when symmetry considerations are of interest. The cases in which the elements in the ensemble are indexed by {1, 2, . . . , n} and by the permutations of a finite set are considered in detail and interpreted as particular cases of ensembles with elements indexed by a set subject to the actions of a finite group. Decompositions for the entropy in binary ensembles and in ensembles indexed by short DNA words are discussed. Graphical descriptions of the decompositions of the entropy in geological samples are illustrated. The decompositions derived in the present cases follow from a systematic data analytic tool to study entropy data in the presence of symmetry considerations. Keywords: Group representation; characters; canonical decomposition; invariance; Kullback’s divergence; visual acuity, Sloan fonts. MSC 2000 codes: 94A17, 54C70, 37A35 Entropy 2006, 8[2], 88-109 89 1 Introduction The entropy H = − p log p of a finite set of n mutually exclusive events with corresponding j j j probabilities p1 , . . . , pn measures the amount of uncertainty associated with