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2022 Functional principal subspace sampling for large scale functional data analysis
Shiyuan He, Xiaomeng Yan
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Electron. J. Statist. 16(1): 2621-2682 (2022). DOI: 10.1214/22-EJS2010

Abstract

Functional data analysis (FDA) methods have computational and theoretical appeals for some high dimensional data, but lack the scalability to modern large sample datasets. To tackle the challenge, we develop randomized algorithms for two important FDA methods: functional principal component analysis (FPCA) and functional linear regression (FLR) with scalar response. The two methods are connected as they both rely on the accurate estimation of functional principal subspace. The proposed algorithms draw subsamples from the large dataset at hand and apply FPCA or FLR over the subsamples to reduce the computational cost. To effectively preserve subspace information in the subsamples, we propose a functional principal subspace sampling probability, which removes the eigenvalue scale effect inside the functional principal subspace and properly weights the residual. Based on the operator perturbation analysis, we show the proposed probability has precise control over the first order error of the subspace projection operator and can be interpreted as an importance sampling for functional subspace estimation. Moreover, concentration bounds for the proposed algorithms are established to reflect the low intrinsic dimension nature of functional data in an infinite dimensional space. The effectiveness of the proposed algorithms is demonstrated upon synthetic and real datasets.

Funding Statement

Shiyuan He’s research was supported by National Natural Science Foundation of China (No.11801561). This research was supported by Public Computing Cloud, Renmin University of China.

Acknowledgments

The authors thank the editor, the associate editor, and an anonymous reviewer for constructive comments that helped significantly improve this work. The authors also thank Professor Jianhua Z. Huang, Professor Xiangyu Chang and Professor Yixuan Qiu for comments on the initial draft of this work.

Citation

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Shiyuan He. Xiaomeng Yan. "Functional principal subspace sampling for large scale functional data analysis." Electron. J. Statist. 16 (1) 2621 - 2682, 2022. https://doi.org/10.1214/22-EJS2010

Information

Received: 1 April 2021; Published: 2022
First available in Project Euclid: 12 April 2022

arXiv: 2109.03397
Digital Object Identifier: 10.1214/22-EJS2010

Subjects:
Primary: 62R10
Secondary: 62D99 , 62H25

Keywords: Functional linear regression , functional principal component analysis , operator perturbation theory , randomized algorithm

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