组合数据分析的信息几何视角
The Information-Geometric Perspective of Compositional Data Analysis
摘要：
信息几何使用微分几何的形式工具来描述空间的概率分布作为一个黎曼流形与额外的对偶结构。成分数据与离散概率分布的形式等价性使得将相同的描述应用于成分数据分析（CoDA）的样本空间成为可能。后者已被正式描述为一个欧几里得空间的规范正交基的特点是组件的原始部分的适当组合。相对于欧几里德度量，信息几何描述挑选出费舍尔信息度量作为唯一一个保持流形的几何结构不变的等价表示下的基础随机变量。在欧几里德坐标中有效的众所周知的概念，例如，勾股定理，由信息几何推广到适用于更一般坐标相应概念。在简要回顾欧几里德CoDA，更详细地说，信息几何的方法，我们展示了后者如何证明使用距离的措施和分歧，到目前为止，很少受到关注的CoDA，因为他们不适合欧几里德几何青睐当前的思维。我们还展示了熵和相对熵如何以简单的方式描述合并，而艾奇森距离需要使用几何平均来获得更简洁的关系。我们继续证明Aitchison距离的信息单调性。最后，我们对CoDA的新方向进行了一些思考，其中可以利用信息几何提供的丰富结构。
英文摘要：
Information geometry uses the formal tools of differential geometry to describe the space of probability distributions as a Riemannian manifold with an additional dual structure. The formal equivalence of compositional data with discrete probability distributions makes it possible to apply the same description to the sample space of Compositional Data Analysis (CoDA). The latter has been formally described as a Euclidean space with an orthonormal basis featuring components that are suitable combinations of the original parts. In contrast to the Euclidean metric, the information-geometric description singles out the Fisher information metric as the only one keeping the manifold's geometric structure invariant under equivalent representations of the underlying random variables. Well-known concepts that are valid in Euclidean coordinates, e.g., the Pythogorean theorem, are generalized by information geometry to corresponding notions that hold for more general coordinates. In briefly reviewing Euclidean CoDA and, in more detail, the information-geometric approach, we show how the latter justifies the use of distance measures and divergences that so far have received little attention in CoDA as they do not fit the Euclidean geometry favored by current thinking. We also show how entropy and relative entropy can describe amalgamations in a simple way, while Aitchison distance requires the use of geometric means to obtain more succinct relationships. We proceed to prove the information monotonicity property for Aitchison distance. We close with some thoughts about new directions in CoDA where the rich structure that is provided by information geometry could be exploited.
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