成分数据三种对数比变换的研究

A Study on Three Kinds of Log-Ratio Transformations for Compositional Data

摘要: 成分数因受“闭合性”约束天然存在于$D-1$ 维单纯形空间，难以直接应用传统多维统计分析方法。非对称对数比变换(ALR)、中心化对数比变换(CLR)与等距对数比变换(ILR)通过对数比映射将成分数据转化至欧氏空间，是成分数据统计分析的核心工具。本文从变换基、保距性、保角性、矩阵表示、几何意义及应用场景六个维度，系统对比三种变换的差异，明确了各变换的适用边界与优势，为地质学、生态学、医学等多领域的成分数据分析实践提供了理论依据。

Abstract: Compositional data are naturally constrained by “closure” and exist in a $D-1$ dimensional simplex space, making it difficult to directly apply traditional multivariate statistical methods. Additive log-ratio transformation (ALR), centered log-ratio transformation (CLR), and isometric log-ratio transformation (ILR) convert compositional data to Euclidean space through log-ratio mapping, and they are the core tools for compositional data analysis. This paper systematically compares the differences among the three transformations from six dimensions: transformation basis, distance preservation, angle preservation, matrix representation, geometric meaning, and application scenarios. The research clarifies the applicable boundaries and advantages of each transformation, providing a theoretical basis for the practice of compositional data analysis in multiple fields such as geology, ecology, and medicine.