协方差算子的基于迁移的泛函ANOVA和PCA
Transportation-Based Functional ANOVA and PCA for Covariance Operators
摘要：
我们考虑的问题比较几个样本的随机过程相对于他们的二阶结构，并描述在这个二阶结构的变化的主要模式，如果存在。这些任务可分别被视为协方差运算符的方差分析（ANOVA）和主成分分析（PCA）。它们在函数数据分析中自然出现，在函数数据分析中，几个总体相对于其平均值的分散性质进行对比，而不是相对于其平均值本身。我们贡献了一种新的方法的基础上最佳（多）运输，每个协方差可以确定一个中心的高斯过程相应的协方差。通过构建这些高斯过程的最佳同时耦合，我们对比（线性）映射，实现它的身份相对于一个范数诱导的距离。由此产生的检验统计量，校准排列，被认为是明显优于国家的最先进的，并提供相当大的权力，即使在当地的替代品。这种效应被认为是真正的功能性的，并且与无限维中完美辨别的潜力有关。在拒绝的原假设规定平等的情况下，几何解释的运输地图允许我们构建一个（切空间）PCA揭示的主要模式的变化。作为一个必要的步骤，发展我们的方法，我们证明结果的存在性和有界性的最佳multitransport映射。这些都是独立的利益在理论的运输高斯过程。运输方差分析和主成分分析说明了各种模拟和真实的例子。
英文摘要：
We consider the problem of comparing several samples of stochastic processes with respect to their second-order structure, and describing the main modes of variation in this second order structure, if present. These tasks can be seen as an Analysis of Variance (ANOVA) and a Principal Component Analysis (PCA) of covariance operators, respectively. They arise naturally in functional data analysis, where several populations are to be contrasted relative to the nature of their dispersion around their means, rather than relative to their means themselves. We contribute a novel approach based on optimal (multi)transport, where each covariance can be identified with a a centred Gaussian process of corresponding covariance. By means of constructing the optimal simultaneous coupling of these Gaussian processes, we contrast the (linear) maps that achieve it with the identity with respect to a norm-induced distance. The resulting test statistic, calibrated by permutation, is seen to distinctly outperform the state-of-the-art, and to furnish considerable power even under local alternatives. This effect is seen to be genuinely functional, and is related to the potential for perfect discrimination in infinite dimensions. In the event of a rejection of the null hypothesis stipulating equality, a geometric interpretation of the transport maps allows us to construct a (tangent space) PCA revealing the main modes of variation. As a necessary step to developing our methodology, we prove results on the existence and boundedness of optimal multitransport maps. These are of independent interest in the theory of transport of Gaussian processes. The transportation ANOVA and PCA are illustrated on a variety of simulated and real examples.
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