The log-ratio approach to compositional data (CoDa) analysis has now entered a mature phase. The principles and statistical tools introduced by J. Aitchison in the eighties have proven successful in solving a number of applied problems. The algebraic–geometric structure of the sample space, tailored to those principles, was developed at the beginning of the millennium. Two main ideas completed the J. Aitchison’s seminal work: the conception of compositions as equivalence classes of proportional vectors, and their representation in the simplex endowed with an interpretable Euclidean structure. These achievements allowed the representation of compositions in meaningful coordinates (preferably Cartesian), as well as orthogonal projections compatible with the Aitchison distance introduced two decades before. These ideas and concepts are reviewed up to the normal distribution on the simplex and the associated central limit theorem. Exploratory tools, specifically designed for CoDa, are also reviewed. To illustrate the adequacy and interpretability of the sample space structure, a new inequality index, based on the Aitchison norm, is proposed. Most concepts are illustrated with an example of mean household gross income per capita in Spain.

中文翻译：

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成分数据：样本空间及其结构

对数比值方法（CoDa）分析现已进入成熟阶段。事实证明，J。Aitchison在八十年代引入的原理和统计工具已成功解决了许多应用问题。根据这些原理量身定制的示例空间的代数-几何结构是在千禧年初开发的。J. Aitchison的开创性工作有两个主要思想：组成作为比例矢量的等价类的概念，以及它们在具有可解释欧氏结构的单纯形中的表示。这些成就使作品能够以有意义的坐标（最好是笛卡尔坐标）表示，并且与二十年前引入的Aitchison距离兼容的正交投影成为可能。对这些思想和概念进行了复习，直到单纯形上的正态分布和相关的中心极限定理。还审查了专门为CoDa设计的探索性工具。为了说明样本空间结构的充分性和可解释性，提出了一种基于Aitchison规范的新的不平等指数。以西班牙的平均家庭人均总收入为例，说明了大多数概念。