  Measuring and testing the dependency between multiple random functions is
often an important task in functional data analysis. In the literature, a
model-based method relies on a model which is subject to the risk of model
misspecification, while a model-free method only provides a correlation measure
which is inadequate to test independence. In this paper, we adopt the
Hilbert-Schmidt Independence Criterion (HSIC) to measure the dependency between
two random functions. We develop a two-step procedure by first pre-smoothing
each function based on its discrete and noisy measurements and then applying
the HSIC to recovered functions. To ensure the compatibility between the two
steps such that the effect of the pre-smoothing error on the subsequent HSIC is
asymptotically negligible, we propose to use wavelet soft-thresholding for
pre-smoothing and Besov-norm-induced kernels for HSIC. We also provide the
corresponding asymptotic analysis. The superior numerical performance of the
proposed method over existing ones is demonstrated in a simulation study.
Moreover, in an magnetoencephalography (MEG) data application, the functional
connectivity patterns identified by the proposed method are more anatomically
interpretable than those by existing methods.
