Symmetry Studies and Decompositions of Entropy

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.

[1]  M. L. Eaton.  Group invariance applications in statistics , 1989 .

[2]  Jean-Pierre Serre,et al.  Linear representations of finite groups , 1977, Graduate texts in mathematics.

[3]  John Aitchison,et al.  The Statistical Analysis of Compositional Data , 1986 .

[4]  R. Wijsman.  Invariant measures on groups and their use in statistics , 1990 .

[5]  E. Hannan.  Group representations and applied probability , 1965, Journal of Applied Probability.

[6]  F L Ferris,et al.  Relative letter and position difficulty on visual acuity charts from the Early Treatment Diabetic Retinopathy Study. , 1993, American journal of ophthalmology.

[7]  M. Viana,et al.  Dihedral representations and statistical geometric optics II: elementary optical instruments , 2007 .

[8]  Invariance Conditions for Random Curvature Models , 2003 .

[9]  M. L. Eaton.  Multivariate statistics : a vector space approach , 1985 .

[10]  L. Nachbin,et al.  The Haar integral , 1965 .

[11]  V. Lakshminarayanan,et al.  Dihedral representations and statistical geometric optics. I. Spherocylindrical lenses. , 2005, Journal of the Optical Society of America. A, Optics, image science, and vision.

[12]  Michael D. Perlman,et al.  [A Review of Multivariate Analysis]: Comment: Group Symmetry Covariance Models , 1987 .

[13]  R. A. Bailey,et al.  Strata for Randomized Experiments , 1991 .

[14]  A. Dawid,et al.  Symmetry models and hypotheses for structured data layouts , 1988 .

[15]  Robert B. Ash,et al.  Information Theory , 2020, The SAGE International Encyclopedia of Mass Media and Society.

[16]  U. Grenander.  Probabilities on Algebraic Structures , 1964 .

[17]  Solomon Kullback,et al.  Information Theory and Statistics , 1970, The Mathematical Gazette.

[18]  L. Goddard.  Information Theory , 1962, Nature.

[19]  Steen A. Andersson,et al.  Invariant Normal Models , 1975 .

[20]  A. James,et al.  THE RELATIONSHIP ALGEBRA OF AN EXPERIMENTAL DESIGN , 1957 .

[21]  H. Doi.  Importance of purine and pyrimidine content of local nucleotide sequences (six bases long) for evolution of the human immunodeficiency virus type 1. , 1991, Proceedings of the National Academy of Sciences of the United States of America.

[22]  Stephen E. Fienberg,et al.  Discrete Multivariate Analysis: Theory and Practice , 1976 .

[23]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.