In this paper we build a set of parametric quotient Lie group structures on the probabilistic simplex that can be extended to real vector space structures. In particular, we rediscover the main mathematical objects generally used when treating compositional data as elements associated to the quotient Lie group with respect to the equivalence relation induced by the scale invariance principle. This perspective facilitates the adaptation of the statistical methods used for classical compositional data to data that follows a more general equivalence relation.

翻译：在本文中,我们根据可扩展到实际矢量空间结构的概率简单化简单法,建立了一套参数比值群结构,特别是,我们重新发现了在将组成数据作为与等级变差原则引起的等同关系相关要素处理时通常使用的主要数学对象。这一视角有助于将古典组成数据所用的统计方法与遵循更一般等同关系的数据相适应。