Functional partial canonical correlation

TL;DR

This paper rigorously derives canonical and partial canonical correlations for Hilbert space-indexed stochastic processes using a novel congruence mapping and orthogonal sum methodology.

Contribution

It introduces a new theoretical framework for deriving canonical correlations in Hilbert space processes with a novel congruence mapping approach.

Findings

Provides a rigorous derivation of canonical correlations for Hilbert space processes.

Introduces a congruence mapping between process space and a Hilbert function space.

Utilizes orthogonal direct sum construction for theoretical development.

Abstract

A rigorous derivation is provided for canonical correlations and partial canonical correlations for certain Hilbert space indexed stochastic processes. The formulation relies on a key congruence mapping between the space spanned by a second order, $\mathcal{H}$-valued, process and a particular Hilbert function space deriving from the process' covariance operator. The main results are obtained via an application of methodology for constructing orthogonal direct sums from algebraic direct sums of closed subspaces.