单纯形上的对数比传播：成分数据细胞污染的理论
Log-Ratio Propagation on the Simplex: A Theory of Cellwise Contamination for Compositional Data
AI总结：
本文提出单纯形上细胞污染的理论，通过乘法扰动和传播定理证明单个成分污染导致对数比向量秩一偏移，并揭示欧几里得细胞方法在单纯形上的失效与降维现象。
摘要：
成分数据必须通过对数比进行分析：尺度不变性，该领域的定义公理，别无选择。中心化对数比除以每个部分的几何平均值，因此单个受污染成分会同时移动所有中心化对数比坐标，将对数比向量位移一个固定量，任何坐标选择都无法减少。我们围绕这一观察发展了单纯形上细胞污染的理论。基于乘法扰动的尺度不变污染模型与传播定理相结合，表明单个原始部分的腐败会导致对数比向量的秩一偏移，方向由对比矩阵决定。由此产生的扰动模式不等同于对数比坐标中的任何独立细胞污染模型——因此，应用于对数比的标准欧几里得细胞方法在单纯形污染机制下是不适定的。对于其欧几里得细胞崩溃由列集中配置见证的估计量——包括MCD、$S$-、$τ$-和坐标$M$-估计量的位置和散度——单纯形上的细胞崩溃值相对于其欧几里得对应值减少了因子$(D-1)/D$，这种减少是紧的，并且纯粹源于$nD$个原始细胞与$n(D-1)$个ilr细胞之间的归一化不匹配。变异矩阵的细胞影响函数携带诊断指纹：单个部分的污染恰好膨胀一行和一列，从而识别出责任成分。这些结果为单纯形上的细胞鲁棒方法奠定了理论基础；一篇配套论文开发了一种利用传播几何的细胞鲁棒PCA估计器，并在模拟和地球化学数据上进行了演示。
英文摘要：
Compositional data must be analysed through log-ratios: scale invariance, the defining axiom of the field, leaves no alternative. The centred log-ratio divides by the geometric mean of every part, so a single contaminated component shifts every centred-log-ratio coordinate at once, displacing the log-ratio vector by a fixed amount that no choice of coordinates can reduce. We develop a theory of cellwise contamination on the simplex around this observation. A scale-invariant contamination model built from multiplicative perturbation combines with a propagation theorem showing that corruption of a single raw part induces a rank-one shift of the log-ratio vector, with direction determined by the contrast matrix. The resulting perturbation pattern is not equivalent to any independent cellwise contamination model in log-ratio coordinates -- so standard Euclidean cellwise methods applied to log-ratios are ill-posed under the simplex contamination mechanism. For estimators whose Euclidean cellwise breakdown is witnessed by a column-concentrated configuration -- a class including MCD, $S$-, $τ$-, and coordinate-wise $M$-estimators of location and scatter -- the cellwise breakdown value on the simplex is reduced by the factor $(D-1)/D$ relative to its Euclidean counterpart, a reduction that is tight and arises purely from the normalisation mismatch between $nD$ raw cells and $n(D-1)$ ilr cells. The cellwise influence function for the variation matrix carries a diagnostic fingerprint: contamination of a single part inflates exactly one row and column, identifying the responsible component. These results form the theoretical foundation for cellwise-robust methods on the simplex; a companion paper develops a cellwise-robust PCA estimator that exploits the propagation geometry and demonstrates it on simulated and geochemical data.
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