Geodesic PCA in the Wasserstein space by convex PCA
Geodesic PCA in the Wasserstein space by convex PCA
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DOI：
10.1214/15-aihp706
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发表时间：
2017-02-01
影响因子：
1.5
通讯作者：
Lopez, Alfredo
中科院分区：
文献类型：
--
作者：
Bigot, Jeremie;Gouet, Raul;Lopez, Alfredo
本文介绍了一种基于Wasserstein度量的有限秒矩线上概率测度空间的测地线主成分分析方法。我们讨论了这种方法的优点，与Hilbert空间中平方可积函数的概率密度的标准泛函PCA相比。我们通过证明经验GPCA收敛于其人口对应，当样本量趋于无穷大，建立了方法的一致性。研究GPCA的一个关键性质是Wasserstein空间与平方可积函数空间的闭凸子集在适当测度下的等距性。因此，我们在可分离Hilbert空间的闭凸子集中考虑PCA的一般问题，这是分析GPCA的基础，也是其本身的兴趣所在。我们提供了简单统计模型的说明性示例，以展示这种方法对数据分析的好处。该方法也适用于实际的人口金字塔数据集。
We introduce the method of Geodesic Principal Component Analysis (GPCA) on the space of probability measures on the line, with finite second moment, endowed with the Wasserstein metric. We discuss the advantages of this approach, over a standard functional PCA of probability densities in the Hilbert space of square-integrable functions. We establish the consistency of the method by showing that the empirical GPCA converges to its population counterpart, as the sample size tends to infinity. A key property in the study of GPCA is the isometry between the Wasserstein space and a closed convex subset of the space of square-integrable functions, with respect to an appropriate measure. Therefore, we consider the general problem of PCA in a closed convex subset of a separable Hilbert space, which serves as basis for the analysis of GPCA and also has interest in its own right. We provide illustrative examples on simple statistical models, to show the benefits of this approach for data analysis. The method is also applied to a real dataset of population pyramids.