Dynamic Multivariate Functional Data Modeling via Sparse Subspace Learning

ABSTRACT

Multivariate functional data from a complex system are naturally high-dimensional and have a complex cross-correlation structure. The complexity of data structure can be observed as that (1) some functions are strongly correlated with similar features, while some others may have almost no cross-correlations with quite diverse features; and (2) the cross-correlation structure may also change over time due to the system evolution. With this regard, this article presents a dynamic subspace learning method for multivariate functional data modeling. In particular, we consider that different functions come from different subspaces, and only functions of the same subspace have cross-correlations with each other. The subspaces can be automatically formulated and learned by reformatting the problem as a sparse regression. By allowing but regularizing the regression change over time, we can describe the cross-correlation dynamics. The model can be efficiently estimated by the fast iterative shrinkage-thresholding algorithm, and the features of each subspace can be extracted using the smooth multi-channel functional principal component analysis. Some theoretical properties of the model are presented. Numerical studies, together with case studies, demonstrate the efficiency and applicability of the proposed methodology.

KEYWORDS:

• Dynamic correlation
• Functional data analysis
• Fused lasso
• Sparse subspace learning

Supplementary Materials

The supplementary files contain additional simulations results, figures, tables, algorithms, proofs of theorems, as well as the MATLAB code of DFSL.

Acknowledgments

The authors thank the editor, associate editor, and two anonymous reviewers for many constructive comments and suggestions that have improved the quality of this work significantly.

Additional information

Funding

This work was partially supported by NSFC 71901131, NSFC 71932006, NSF CCF 1740776, NSF DMS 1830363, NSF CMMI 1922739, and Tsinghua University Intelligent Logistics and Supply Chain Research Center grant THUCSL20182911756-001.