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

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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.