A Wavelet-Based Independence Test for Functional Data With an Application to MEG Functional Connectivity

Abstract

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 when the data are densely measured, we propose a new wavelet thresholding method for pre-smoothing and to use 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 a magnetoencephalography (MEG) data application, the functional connectivity patterns identified by the proposed method are more anatomically interpretable than those by existing methods.

Keywords:

• Besov spaces
• Dense functional data
• Hilbert space
• Human connectome project
• Permutation test
• Reproducing kernel

Supplementary Material

The supplementary material includes background materials on distance-induced characteristic kernels and Besov spaces, technical proofs of Theorems 1–4 and additional simulations.

Acknowledgments

The authors thank the editor, an associate editor and two referees for their constructive comments and suggestions.

Notes

¹ The package is only for Windows platform. For the user-chosen parameters required by this package, we followed the recommendation in Section 4.1 of Patilea, Sánchez-Sellero, and Saumard (2016) and set the bandwidth $h=n^{-2/9}$, penalty coefficient α = 2, grid size n_q = 50 and number of FPCs which cumulatively account for 95% of the variation of the functional predictor.

Additional information

Funding

The research of Xiaoke Zhang is partially supported by National Science Foundation grant DMS-1832046. The research of Raymond K. W. Wong is partially supported by National Science Foundation grants DMS-1806063, DMS-1711952 and CCF-1934904.