Frechet differentiation of functions of operators with application in functional data analysis

算子函数的 Frechet 微分及其在函数数据分析中的应用

• 批准号: 0605167
• 负责人: Frits Ruymgaart
• 金额: $ 3.75万
• 依托单位: Texas Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0605167&HistoricalAwards=false
• 关键词: Frechet; differentiation; functions; operators; application

This research deals on the one hand with a statistical problem in functional data analysis and on the otherwith a mathematical tool from perturbation theory that might also be of independent interest. More specifically the statistical problem concerns the limiting distribution of properly defined functional canonical correlations and their corresponding canonical variates. Functional canonical correlations are of interest in their own right, but may also help organize high-dimensional data sets. The mathematical tool to be derived is the Frechet derivative of an analytic function of a compact, nonnegative, Hermitian operator (in the sense of functional calculus), tangentially to the space of all compact Hermitian operators. This Frechet derivative allows one to obtain, for instance, the asymptotic distribution of a function of the covariance operator by applying the ensuing delta-method. This type of result provides the major ingredient for deriving the aforementioned asymptotic distributions. In practice the theory applies to data that are independent copies of a certain infinite dimensional signal. A well-known instance of such a signal in the literature is the gait example where one part of the signal represents the variability in the knee angle cycle and the second part that in the hip angle cycle. Canoncial correlations and their corresponding canonical variates may capture a significant part of the relation between the two cycles. When applied to this example the proposed research should lead to the asymptotic distributions of these objects and in particular to confidence intervals for the canonical correlations.

本研究一方面涉及函数数据分析中的统计问题，另一方面涉及可能也有独立兴趣的微扰理论的数学工具。 更具体地说，统计问题涉及适当定义的函数典型相关及其相应的典型变量的极限分布。 函数典型相关性本身就很有趣，但也可能有助于组织高维数据集。 数学工具是导出一个紧凑的，非负的，埃尔米特算子（在功能微积分的意义上），切向所有紧凑埃尔米特算子空间的解析函数的弗雷歇导数。 这Frechet导数允许一个获得，例如，渐近分布的函数的协方差算子通过应用随后的δ-方法。 这种类型的结果提供了推导上述渐近分布的主要成分。在实践中，该理论适用于作为某个无限维信号的独立副本的数据。 文献中这种信号的一个众所周知的实例是步态示例，其中信号的一部分表示膝关节角度周期中的可变性，第二部分表示髋关节角度周期中的可变性。 典型相关性及其相应的典型变量可以捕捉两个周期之间的关系的重要部分。 当应用到这个例子中，拟议的研究应导致这些对象的渐近分布，特别是典型相关的置信区间。