ABSTRACT

The common approach for regression analysis with compositional variables is to express compositions in log-ratio coordinates (coefficients) and then perform standard statistical processing in real space. Similar to working in real space, the problem is that the standard least squares regression fails when the number of parts of all compositional covariates is higher than the number of observations. The aim of this study is to analyze in detail the partial least squares (PLS) regression which can deal with this problem. In this paper, we focus on the PLS regression between more than one compositional response variable and more than one compositional covariate. First, we give the PLS regression model with log-ratio coordinates of compositional variables, then we express the PLS model directly in the simplex. We also prove that the PLS model is invariant under the change of coordinate system, such as the ilr coordinates with a different contrast matrix or the clr coefficients. Moreover, we give the estimation and inference for parameters in PLS model. Finally, the PLS model with clr coefficients is used to analyze the relationship between the chemical metabolites of Astragali Radix and the plasma metabolites of rat after giving Astragali Radix.

摘要

使用成分变量进行回归分析的常用方法是以对数比坐标（系数）表示成分，然后在实际空间中执行标准统计处理。与在真实空间中工作类似，问题在于，当所有组合协变量的部分数高于观察数时，标准最小二乘回归会失败。本研究的目的是详细分析可以解决此问题的偏最小二乘 (PLS) 回归。在本文中，我们关注多个组成响应变量和多个组成协变量之间的 PLS 回归。首先，我们给出组成变量的对数比坐标的PLS回归模型，然后我们直接用单纯形表示PLS模型。我们还证明了PLS模型在坐标系变化下是不变的，例如不同对比矩阵的ilr坐标或clr系数。此外，我们给出了PLS模型中参数的估计和推断。最后采用clr系数PLS模型分析黄芪化学代谢物与大鼠血浆代谢物给予黄芪后的关系。