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Abstract:
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Thanks to technological advances, functional data with high dimensionality emerge increasingly from various fields such as neuroimaging analysis. Functional principal component analysis (FPCA) has become an important tool for dimension reduction, which however is scarcely researched for high-dimensional functional data where each variable features an infinite-dimensional process. Our goal is to perform multivariate FPCA simultaneously for $p$ random processes where $p$ is comparable to, or ever much larger than the sample size. While sparsity assumptions are necessary to deal with high-dimensional problems, there is no existing notion for high-dimensional functional data. In this work, we formulate two sensible sparsity regimes, namely the $l_0$ and weak $l_q$ sparsity. Accordingly we propose sparse FPCA by a thresholding rule that is easy to implement/compute without performing smoothing. The theoretical properties are investigated under two sparsity regimes for the resulting estimators. Simulated and real data examples are provided to offer empirical support for the proposed method which also performs well in subsequent analysis such as classification.
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