论文标题
从多元设置到功能案例的主要点和椭圆分布
Principal points and elliptical distributions from the multivariate setting to the functional case
论文作者
论文摘要
随机向量$ \ mathbf {x} $的$ k $主点定义为一组点，可最大程度地减少$ \ mathbf {x} $和集合中最近点之间的预期平方距离。他们在Flury（1990，1993），Tarpey（1995）和Tarpey，Li and Flury（1995）中进行了详尽的研究。对于他们的治疗，检查通常仅限于椭圆形分布的家族。在本文中，我们向功能椭圆分布案例介绍了先前结果的扩展，即在可分离的希尔伯特空间上处理随机元素$ {\ cal H} $时。 Tarpey和Kineder（2003）定义了高斯流程的主要点。在本文中，我们概括了主要点，自洽点和椭圆分布的概念，以便将它们适合在此功能框架中。在此新设置中重新考虑了链接自隔离和协方差操作员的特征向量的结果，以及$ k = 2 $ case的显式公式，以便在$ {\ cal H} $中包括椭圆形的随机元素。
The $k$ principal points of a random vector $\mathbf{X}$ are defined as a set of points which minimize the expected squared distance between $\mathbf{X}$ and the nearest point in the set. They are thoroughly studied in Flury (1990, 1993), Tarpey (1995) and Tarpey, Li and Flury (1995). For their treatment, the examination is usually restricted to the family of elliptical distributions. In this paper, we present an extension of the previous results to the functional elliptical distribution case, i.e., when dealing with random elements over a separable Hilbert space ${\cal H}$. Principal points for gaussian processes were defined in Tarpey and Kinateder (2003). In this paper, we generalize the concepts of principal points, self-consistent points and elliptical distributions so as to fit them in this functional framework. Results linking self-consistency and the eigenvectors of the covariance operator are re-obtained in this new setting as well as an explicit formula for the $k=2$ case so as to include elliptically distributed random elements in ${\cal H}$.
