Independence test and canonical correlation analysis based on the alignment between kernel matrices for multivariate functional data

In the case of vector data, Gretton et al. (Algorithmic learning theory. Springer, Berlin, pp 63-77, 2005) defined Hilbert-Schmidt independence criterion, and next Cortes et al. (J Mach Learn Res 13:795-828, 2012) introduced concept of the centered kernel target alignment (KTA). In this paper we generalize these measures of dependence to the case of multivariate functional data. In addition, based on these measures between two kernel matrices (we use the Gaussian kernel), we constructed independence test and nonlinear canonical variables for multivariate functional data. We show that it is enough to work only on the coefficients of a series expansion of the underlying processes. In order to provide a comprehensive comparison, we conducted a set of experiments, testing effectiveness on two real examples and artificial data. Our experiments show that using functional variants of the proposed measures, we obtain much better results in recognizing nonlinear dependence.