A Method of L1-Norm Principal Component Analysis for Functional Data

Abstract

Recently, with the popularization of intelligent terminals, research on intelligent big data has been paid more attention. Among these data, a kind of intelligent big data with functional characteristics, which is called functional data, has attracted attention. Functional data principal component analysis (FPCA), as an unsupervised machine learning method, plays a vital role in the analysis of functional data. FPCA is the primary step for functional data exploration, and the reliability of FPCA plays an important role in subsequent analysis. However, classical L2-norm functional data principal component analysis (L2-norm FPCA) is sensitive to outliers. Inspired by the multivariate data L1-norm principal component analysis methods, we propose an L1-norm functional data principal component analysis method (L1-norm FPCA). Because the proposed method utilizes L1-norm, the L1-norm FPCs are less sensitive to the outliers than L2-norm FPCs which are the characteristic functions of symmetric covariance operator. A corresponding algorithm for solving the L1-norm maximized optimization model is extended to functional data based on the idea of the multivariate data L1-norm principal component analysis method. Numerical experiments show that L1-norm FPCA proposed in this paper has a better robustness than L2-norm FPCA, and the reconstruction ability of the L1-norm principal component analysis to the original uncontaminated functional data is as good as that of the L2-norm principal component analysis.

Lagged covariance and cross-covariance operators of processes in Cartesian products of abstract Hilbert spaces

“…FPCA in L 2 [0, 1] is also extensively discussed in the existing literature. In [7,28,31,35,37] one finds asymptotic upper bounds of the estimation errors for FPCs, estimated seperately and uniformly, in second mean and almost surely (a.s.), and [61] introduced L 1 -norm FPCA.…”

Lagged Covariance and Cross-Covariance Operators of Processes in Cartesian Products of Abstract Hilbert Spaces

“…In [5], [19], [22], [26], [28] one finds asymptotic upper bounds for the principle components, both estimated seperately and uniformly, in sense of convergence in the second mean as well as almost surely. Moreover, [49] introduced L 1 -norm FPCA.…”

Lagged Covariance and Cross-Covariance Operators of Processes in Cartesian Products of Abstract Hilbert Spaces

“…FPCA in L 2 [0, 1] is extensively discussed in the existing literature, both from a probabilistic and statistical point of view. In [4; 17; 20; 24; 26] one finds asymptotic upper bounds for the principal components, estimated both seperately and uniformly in sense of convergence in second mean and almost surely (a.s.), and [45] introduced L 1 -norm FPCA. A comprehensive study of lag-h-cross-covariance operators C X,Y;h of L 2 [0, 1]valued processes X = (X k ) k∈Z , Y = (Y k ) k∈Z can be found in Rice & Shum [33].…”