In this chapter, we examine the concept of canonical correlation. The idea was introduced in Chapter 1 as a problem of finding maximally correlated linear combinations of two random vectors. Our goal here is to extend this notion to a sufficiently general setting where it becomes applicable to fda and other related abstract data analysis problems.

From our work in Chapter 7, we know that functional data can be viewed from two perspectives: namely, as realizations of Hilbert space valued random elements or of second-order, continuous time, stochastic processes. These two views overlap; but, they are not, in general, equivalent. In particular, we saw how, depending on which perspective one employs, slightly different definitions are obtained for the covariance operator (Sections 7.2 and 7.3) and linear span (Section 7.6) while different considerations also arise for estimating the mean and covariance functions (Chapter 8). In this chapter, we, for the most part, adhere to the random element viewpoint; but, we also mention the key differences that result when data is collected from a second-order process. Thus, unless otherwise stated, we consider random elements of some separable Hilbert space defined on a common probability space . Both ...

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