AI 中文总结
研究极值理论与成分数据分析，通过斜投影等简单变换建立二者显式联系，可实现统计方法在两领域间转移，如引入基于极值模型的成分数据内在逻辑正态图形模型，探索成分数据分析降维方法用于多元极值分析。
AI 中文摘要
极值理论和成分数据分析都研究相对信息起核心作用的情况。在多元极值理论中，阈值超越极限满足同质性属性，将极端事件的径向大小与其相对轮廓分开。在成分数据分析中，正向量按乘法尺度分析，推断基于分量之间的比率或对数比率。两个领域都开发了几种潜在相对结构的协方差和依赖性表示。我们通过一小组简单变换建立了两个领域之间的显式联系，包括斜投影、许斯勒 - 赖斯逆变换和变差函数图。利用这种代数联系能在两个领域间转移统计方法，如为成分数据引入基于极值的许斯勒 - 赖斯图形模型的内在逻辑正态图形模型，反之探索成分数据分析的降维方法如何用于多元极值分析。
英文摘要
Extreme value theory and compositional data analysis both study settings where relative information plays a central role. In multivariate extreme value theory, threshold exceedance limits satisfy homogeneity properties that separate the radial size of an extreme event from its relative profile. In compositional data analysis, positive vectors are analysed up to multiplicative scale, and inference is based on ratios or log-ratios between components. Consequently, both fields have developed several covariance and dependence representations of the underlying relative structure. In the Hüsler-Reiss model for extremes, these include variogram, covariance, and precision parametrizations. In compositional data analysis, analogous representations arise from pairwise log-ratios, centred log-ratios, and additive log-ratios. We establish an explicit link between the two fields that relates these different representations by a small set of simple transformations, including oblique projections, Hüsler-Reiss inverses, and the variogram map. From a methodological perspective, leveraging this algebraic connection enables the transfer of statistical approaches from one field to the other. For instance, we introduce intrinsic logistic-normal graphical models for compositional data, which are based on Hüsler-Reiss graphical models for extremes. Conversely, we explore how dimensionality reduction methods from compositional data analysis can be applied to the analysis of multivariate extremes.
