标题： 基于数据的成分数据幂变换 显示英文标题

标题： A data-based power transformation for compositional data

摘要： 成分数据分析要么通过忽略成分约束并应用标准多元数据分析，要么通过对分量比的对数变换来转换数据。 在这项工作中，我们考察了一种更通用的变换，它包含了这两种方法作为特殊情况。 这是一种幂变换，并且涉及一个单一参数 {\alpha }。 这种变换有两个等价版本。 第一个是保持在单纯形中的版本，即 Aitchison 在 1986 年定义的幂变换。 第二个版本是这种幂变换的线性变换，是一种 Box-Cox 类型的变换。 我们讨论了一种估计 {\alpha } 值的参数化方法，即最大化其轮廓似然（假设变换后的数据服从多元正态分布），并展示了两种版本之间的等价性。 其他方法包括判别分析中的正确分类概率最大化以及线性回归中 Aitchison 在 1986 年定义的伪 \( R^2 \) 的最大化。 我们研究了 {\alpha }-变换、原始数据方法和等距对数比变换之间的关系。 此外，我们还定义了与 {\alpha }-变换族相对应的一组合适的度量，并考虑了相应的 Fréchet 均值族。 显示英文摘要

摘要： Compositional data analysis is carried out either by neglecting the compositional constraint and applying standard multivariate data analysis, or by transforming the data using the logs of the ratios of the components. In this work we examine a more general transformation which includes both approaches as special cases. It is a power transformation and involves a single parameter, {\alpha}. The transformation has two equivalent versions. The first is the stay-in-the-simplex version, which is the power transformation as defined by Aitchison in 1986. The second version, which is a linear transformation of the power transformation, is a Box-Cox type transformation. We discuss a parametric way of estimating the value of {\alpha}, which is maximization of its profile likelihood (assuming multivariate normality of the transformed data) and the equivalence between the two versions is exhibited. Other ways include maximization of the correct classification probability in discriminant analysis and maximization of the pseudo R-squared (as defined by Aitchison in 1986) in linear regression. We examine the relationship between the {\alpha}-transformation, the raw data approach and the isometric log-ratio transformation. Furthermore, we also define a suitable family of metrics corresponding to the family of {\alpha}-transformation and consider the corresponding family of Frechet means.