Pinele Julianna, Strapasson João E, Costa Sueli I R
Center of Exact and Technological Sciences, University of Reconcavo of Bahia, Cruz das Almas 44380-000, Brazil.
School of Applied Sciences, University of Campinas, Limeira 13484-350, Brazil.
Entropy (Basel). 2020 Apr 1;22(4):404. doi: 10.3390/e22040404.
The Fisher-Rao distance is a measure of dissimilarity between probability distributions, which, under certain regularity conditions of the statistical model, is up to a scaling factor the unique Riemannian metric invariant under Markov morphisms. It is related to the Shannon entropy and has been used to enlarge the perspective of analysis in a wide variety of domains such as image processing, radar systems, and morphological classification. Here, we approach this metric considered in the statistical model of normal multivariate probability distributions, for which there is not an explicit expression in general, by gathering known results (closed forms for submanifolds and bounds) and derive expressions for the distance between distributions with the same covariance matrix and between distributions with mirrored covariance matrices. An application of the Fisher-Rao distance to the simplification of Gaussian mixtures using the hierarchical clustering algorithm is also presented.
费希尔 - 拉奥距离是一种衡量概率分布之间差异的度量，在统计模型的某些正则性条件下，它在乘以一个比例因子后是马尔可夫态射下唯一的黎曼度量不变量。它与香农熵相关，并已被用于拓展图像处理、雷达系统和形态分类等广泛领域的分析视角。在此，我们针对多元正态概率分布统计模型中所考虑的这种度量进行研究，由于一般情况下不存在显式表达式，我们通过收集已知结果（子流形的封闭形式和边界），推导出具有相同协方差矩阵的分布之间以及具有镜像协方差矩阵的分布之间的距离表达式。还给出了费希尔 - 拉奥距离在使用层次聚类算法简化高斯混合模型方面的一个应用。
