Theoretical Considerations for Multivariate Functional Data Analysis
Yoshiharu Sato · 2013
Multivariate functional data is defined as an element of a direct sum of Hilbert spaces, H(p) = H1⊕H2⊕ · · ·⊕Hp, where each Hk (k = 1; 2; : : : ; p) is a real separable Hilbert space. In this paper, we consider a Gaussian measures on H(p) as its probability structure. That is, H(p)-valued Gaussian random variable is defined for a measureable space. Under the joint Gaussian probability measure, we will discuss the multivariate analysis just like the classical multivariate analysis. For this purpose, we shall extend the theory of finite dimensional multivariate normal distribution to H(p)-valued random variables. Using the properties of H(p)-valued Gaussian measure, it is shown that a concept of regression can be given by the conditional expectation and the concept of principal components is given by the use of eigen structure of covariance operator.