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

这项研究一方面涉及函数数据分析中的一个统计问题，另一方面涉及一个来自微扰理论的数学工具，该工具可能也是独立感兴趣的。更具体地说，统计问题涉及适当定义的泛函正则关联及其对应的正则变量的极限分布。函数正则关联本身是有意义的，但也可能有助于组织高维数据集。所要推导的数学工具是紧致、非负的厄米特算子(在泛函演算意义下)的解析函数的Frechet导数，该导数与所有紧埃尔米特算子的空间相切。这种Frechet导数允许人们通过应用随后的增量方法来获得例如协方差算子的函数的渐近分布。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例子是步态示例，其中信号的一部分表示膝角周期中的可变性，而第二部分表示髋角周期中的可变性。正则关联及其对应的正则变量可能捕捉到两个周期之间关系的重要部分。当应用于这个例子时，建议的研究应该得到这些对象的渐近分布，特别是正则关联的置信度区间。

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.