Generalization of Canonical Correlation Analysis from Multivariate to Functional Cases and its related problems

In multivariate cases, the aim of canonical correlation analysis (CCA) for two sets of variables x and y is to obtain linear combinations of them so that they have the largest possible correlation. However, when x and y are continouse functions of another variable (generally time) in nature, these two functions belong to function spaces which are of infinite dimension, and CCA for them should be carried out by using some tools provided by functional data analysis. In this paper we first review definitions and concepts of CCA for multivariate data, and then express those of CCA for functional data with considering the problems occur when generalizing the concepts from multivariate to functional cases. We have also treated a real functional data set and interpreted the obtained results.