2.7. Functional Canonical Correlation Analysis

Canonical correlation analysis (CCA) [12,28] quantifies the associations between two sets of variables by transforming both of them into a common lower dimensional space with maximum correlations. Given two random vectors X and Y, the CCA finds two weight vectors u and v such that the two linear transformations uTX and vTY, also called the canonical variables, are maximally correlated. In 1993, Leurgans et al. [36] adapted the CCA for functional data. Given two random functions, X(t) and Y(t), the functional correlation analysis (FCCA) seeks two weigh functions u(t) and v(t) such that ∫0TuX dt and ∫0TvY dt, called functional correlation variables, are maximally correlated. In our analysis, we used the cca.fd function from the fda R package [37] to conduct FCCA with roughness penalties on the second derivative of each curve.