Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators

Chapter 7Random elements in a Hilbert space

In this chapter, we give an overview of various foundational issues that arise in fda and related settings. There are two somewhat different perspectives of functional data. The first one is that functional data are realizations of random variables that take values in a Hilbert space; for convenience, we call this the random element perspective. The second view is that functional data are the sample paths of a (typically continuous time) stochastic process with smooth mean and covariance functions; we will refer to this as the stochastic process perspective. The differences between the two perspectives are subtle and worth exploring from a theoretical standpoint.

To develop the random element perspective, we need to lay a rigorous foundation for the study of Hilbert space valued “random variables” so that we can develop concepts for the mean and covariance in that abstract environment. The stochastic process perspective uses the covariance function of the stochastic process as the fundamental tool for assessing the variability. The classical theorem by Karhunen and Lòeve appears in this context.

For the purpose of developing the notions of prediction and canonical correlations, we define the concept of closed linear span corresponding to both the random element and stochastic process perspective and explore the properties of various associated congruence relations. This work will be useful in Chapter 10. Finally, we present some large ...