Problems in compositional data analysis and their solutions

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  • Ohta Tohru
    Department of Earth Sciences, School of Education, Waseda University
  • Arai Hiroyoshi
    Department of Earth Sciences, School of Education, Waseda University

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  • 組成データ解析の問題点とその解決方法
  • ソセイ データ カイセキ ノ モンダイテン ト ソノ カイケツ ホウホウ

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Abstract

Compositional data, represented as percent or parts per million, are subject to the constant-sum constraint that precludes compositional data from much of statistical analysis. Despite this constraint, a theory for statistically rigorous treatment of compositional data is currently under intense development. This paper reviews the utility of two main procedures for compositional data analysis, which will be termed "logratio analysis" and "simplicial analysis". Logratio analysis is a way to map compositional data from a simplex space to a Euclidean real space by transforming compositional data into logarithms of component ratios. This bijectional mapping allows the transformed data to be analyzed by many traditional statistical methods available in real space. On the other hand, simplicial analysis introduces proper classes of parametric distribution, translation operation, scalar multiplication operation, identity unit and metric function within the simplex space. These definitions permit the simplex space to be reviewed as a metric space and compositional data as an Abelian group. Moreover, simplicial analysis provides statistical methodologies for compositional data, which are analogous to those for data sets associated with real space. A brief overview of the constant-sum constraint is followed by mathematical descriptions of logratio and simplicial analyses. Practical analyses of real data sets based on logratio and simplicial analyses are provided to illustrate their potential and to encourage their use.<br>

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