无限维空间中的数据深度
On data depth in infinite dimensional spaces
摘要：
数据深度的概念导致了多元数据的中心向外排序，它已被有效地用于开发各种数据分析工具。虽然不同的深度概念最初是针对有限维数据开发的，但最近已经有一些尝试针对无限维空间中的数据开发深度函数。本文考虑了无限维空间中深度的一些概念，并在各种随机模型下研究了它们的性质。我们的分析表明，在文献中的一些深度函数退化的行为，一些常用的概率分布在无穷维空间的序列和功能。因此，它们对于分析满足这种无限维概率模型的数据不是很有用。然而，这些深度函数的一些修改版本以及空间深度的无限维扩展不会遭受这种退化，并且可以方便地用于分析无限维数据。
英文摘要：
The concept of data depth leads to a center-outward ordering of multivariate data, and it has been effectively used for developing various data analytic tools. While different notions of depth were originally developed for finite dimensional data, there have been some recent attempts to develop depth functions for data in infinite dimensional spaces. In this paper, we consider some notions of depth in infinite dimensional spaces and study their properties under various stochastic models. Our analysis shows that some of the depth functions available in the literature have degenerate behaviour for some commonly used probability distributions in infinite dimensional spaces of sequences and functions. As a consequence, they are not very useful for the analysis of data satisfying such infinite dimensional probability models. However, some modified versions of those depth functions as well as an infinite dimensional extension of the spatial depth do not suffer from such degeneracy, and can be conveniently used for analyzing infinite dimensional data.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
