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.
Highlights
In recent years, with the rapid popularization of intelligent terminals and sensors, massive data have been rapidly accumulated, and the processing technology of intelligent big data has attracted more and more attention
Because the L2-norm enlarges the influence of outliers, the traditional functional principal components analysis method is sensitive to outliers
In order to compare the robustness to outliers of L1-norm functional principal components (L1-FPCs) that are proposed in this paper and the classical L2-norm functional principal components (L2-FPCs), we performed this simulation
Summary
With the rapid popularization of intelligent terminals and sensors, massive data have been rapidly accumulated, and the processing technology of intelligent big data has attracted more and more attention. In contrast to the conventional L2-PCA, the solutions of the minimization of the L1-norm reconstruction error might not be same as the solutions of the maximization of the L1-norm deviation of projected data Inspired by these pieces of research on L1-PCA for multivariable data, in this paper, we try to construct a robust L1-norm principal component analysis method for functional data We build a functional data L1-norm maximized principal component optimization model, and a corresponding algorithm for solving the L1-norm maximized optimization model is extended to functional data based on the idea of a multivariate data L1-norm principal component analysis method [30]. Numerical experiments show that the L1-norm functional principal component analysis method provides a more robust estimation of principal components than the traditional. By comparing the reconstruction errors of the L1-norm FPCA and L2-norm FPCA, it is found that the reconstruction ability of the L1-norm principal components to the original uncontaminated functional data is as good as that of the L2-norm functional principal components
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