基于空间等价性的成分数据变换方法比较研究

A Comparative Study of Compositional Data Transformation Methods Based on Spatial Equivalence

摘要: 单形空间的定和约束使得传统统计分析方法对成分数据失效，通常需要采用适当的变换方法将成分数据转化到欧氏空间后再进行统计分析。本文以非对称对数比变换、中心化对数比变化、等距对数比变换等三种常用的变换方法为研究对象，基于成分数据代数体系，从能否实现单形空间到欧氏空间等价转换的角度，比较研究了三种变换方法的合理性，为成分数据变换技术的选择提供理论依据。并选取岩石判别分类问题，分别采用以上方法对原始成分数据进行变换后建立判别模型，比较判别结果的可靠性。实证结果表明，等距对数比变换既克服了非对称对数比变换改变内积及距离等几何概念的缺陷，又避免了中心化对数比变换导致的多重共线性给多元分析方法带来的影响，在保持样本空间形态不发生变化的前提下解除了定和约束，是一种合理的变换方法。

Abstract: Traditional statistical analysis method in Euclidean space is not suitable for compositional data, due to its unit-sum constraint in Simplex space. A common solution is to firstly transform compositional data in Simplex space into data in Euclidean space and then perform statistical analysis on the transformed data. This paper proposes to compare three commonly used method, i.e., additive logratiotransformation (alr), centered logratio transformation (clr), and isometric logratio transformation (ilr). Based on Aitchison’s algebra, the comparison is carried out to examine whether a transformation method satisfies the properties of linearity and orthogonality. A real dataset, namely the rock data, is used to verify the comparison results. Three transformation methods are used to relax the unit-sum constraint of the rock data, respectively, and a discriminant model is then established on the transformed data. Comparison results from both theory and real-data studies indicate that isometric logratio transformation is superior to the other two transformation methods in two points. First, isometric logratio transformation does not change the geometry concepts, i.e., inner product and distance, which is inevitably caused by additive logratio transformation. Second, isometric logratio transformation successfully relaxes the unit-sum constraint and avoids multicolinearity, which cannot be solved by centered logratio transformation.