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.

翻译：成分数据必须通过对数比进行分析：尺度不变性——该领域的核心公理——没有其他选择。中心化对数比通过除以所有部分的几何平均值，因此单个受污染组分会同时移动所有中心化对数比坐标，使对数比向量产生一个固定的位移，且任何坐标选择都无法减小该位移。我们围绕这一观察发展了单纯形上单元污染的理论。一个基于乘法扰动的尺度不变污染模型与传播定理相结合，表明对单个原始部分的污染会导致对数比向量产生秩为一的位移，其方向由对比矩阵决定。由此产生的扰动模式与对数比坐标中的任何独立单元污染模型都不等价——因此，将对数比应用于标准欧几里得单元方法在单纯形污染机制下是不适定的。对于其欧几里得单元崩溃由列集中配置揭示的估计量——这一类别包括位置和散布的MCD、$S$、$τ$和坐标方向$M$估计量——其在单纯形上的单元崩溃值相对于欧几里得对应值减少了因子$(D-1)/D$，这一减少是紧的，且纯粹源于$nD$个原始单元与$n(D-1)$个ilr单元之间的归一化不匹配。变异矩阵的单元影响函数携带诊断标记：对单个部分的污染恰好使一行和一列膨胀，从而识别出导致污染的组分。这些结果为单纯形上的单元鲁棒方法奠定了理论基础；伴随论文开发了一个利用传播几何特性的单元鲁棒PCA估计量，并在模拟数据和地球化学数据上进行了验证。