A data-adaptive dimension reduction for functional data via penalized low-rank approximation

• Published in: Statistics and computing Vol. 34; no. 1
• Main Authors: Park, Yeonjoo, Oh, Hee-Seok, Lim, Yaeji
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2024; Springer Nature B.V
• Subjects: Advanced metering infrastructure; Approximation; Artificial Intelligence; Basis functions; Computer Science; Mathematical analysis; Measurement; Operators (mathematics); Original Paper; Probability and Statistics in Computer Science; Reduction; Skewed distributions; Smoothness; Spatial data; Statistical Theory and Methods; Statistics and Computing/Statistics Programs; Outliers; Asymmetric errors; Dimension reduction; Functional data; Sparsity; Submatrix completion; FPCA; Robustness; Missing segments
• ISSN: 0960-3174; 1573-1375
• DOI: 10.1007/s11222-023-10348-7

We introduce a data-adaptive nonparametric dimension reduction tool to obtain a low-dimensional approximation of functional data contaminated by erratic measurement errors following symmetric or asymmetric distributions. We propose to apply robust submatrix completion techniques to matrices consisting of coefficients of basis functions calculated by projecting the observed trajectories onto a given orthogonal basis set. In this process, we use a composite asymmetric Huber loss function to accommodate domain-specific erratic behaviors in a data-adaptive manner. We further incorporate the L 1 penalty to regularize the smoothness of latent factor curves. The proposed method can also be applied to partially observed functional data, where each trajectory contains individual-specific missing segments. Moreover, since our method does not require estimating the covariance operator, the extension to any dimensional functional data observed over a continuum is straightforward. We demonstrate the empirical performance in estimating lower-dimensional space and reconstruction of trajectories of the proposed method through simulation studies. We then apply the proposed method to two real datasets, one-dimensional Advanced Metering Infrastructure (AMI) data in South Korea and two-dimensional max precipitation spatial data collected in North America and South America.