A Wavelet-Based Independence Test for Functional Data With an Application to MEG Functional Connectivity
A Wavelet-Based Independence Test for Functional Data With an Application to MEG Functional Connectivity
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DOI：
10.1080/01621459.2021.2020126
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发表时间：
2020-09
影响因子：
3.7
通讯作者：
Rui Miao;Xiaoke Zhang;Raymond K. W. Wong
中科院分区：
文献类型：
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作者：
Rui Miao;Xiaoke Zhang;Raymond K. W. Wong
测量和测试多个随机函数之间的相关性是函数数据分析中的一项重要任务。在文献中，基于模型的方法依赖于模型，存在模型错误规范的风险，而无模型的方法仅提供相关性度量，不足以检验独立性。在本文中，我们采用Hilbert-Schmidt独立准则（HSIC）来度量两个随机函数之间的相关性。我们开发了一个两步程序，首先根据其离散和噪声测量对每个函数进行预平滑，然后将HSIC应用于恢复函数。为了确保这两个步骤之间的兼容性，使得当数据密集测量时，预平滑误差对后续HSIC的影响逐渐可以忽略不计，我们提出了一种新的小波阈值法进行预平滑，并使用besov -范数诱导核进行HSIC。并给出了相应的渐近分析。仿真结果表明，该方法具有较好的数值性能。此外，在脑磁图（MEG）数据应用中，所提出的方法识别的功能连接模式比现有方法更具解剖学意义。
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.