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Graphical Principal Component Analysis of Multivariate Functional Time Series

Jianbin Tan, Decai Liang, Yongtao Guan, Hui Huang

TL;DR

This work addresses two-way dependencies in multivariate functional time series by introducing dynamic weak separability on the spectral density kernel $f(t,s|\theta)$ and linking it to a partial correlation graph among infinite-dimensional curves. It then develops Graphical Dynamic Functional Principal Component Analysis (GDFPCA) that uses a common set of functional filters across series, with scores that retain the graph structure, and a two-step reconstruction procedure for contaminated data employing a joint graphical Lasso and Whittle likelihood. Theoretical results establish the spectral and score representations, optimality of the truncated dynamic KL expansion, and consistency of estimators, while simulations show that GDFPCA outperforms non-graphical and static methods, especially when graph information is informative. A PM2.5 data example demonstrates practical gains in both reconstruction quality and interpretable connectivity patterns. Overall, the framework unifies graphical models with dynamic FPCA for high-dimensional, temporally dependent functional data, enabling more efficient dimension reduction and robust signal reconstruction.

Abstract

In this paper, we consider multivariate functional time series with a two-way dependence structure: a serial dependence across time points and a graphical interaction among the multiple functions within each time point. We develop the notion of dynamic weak separability, a more general condition than those assumed in literature, and use it to characterize the two-way structure in multivariate functional time series. Based on the proposed weak separability, we develop a unified framework for functional graphical models and dynamic principal component analysis, and further extend it to optimally reconstruct signals from contaminated functional data using graphical-level information. We investigate asymptotic properties of the resulting estimators and illustrate the effectiveness of our proposed approach through extensive simulations. We apply our method to hourly air pollution data that were collected from a monitoring network in China.

Graphical Principal Component Analysis of Multivariate Functional Time Series

TL;DR

This work addresses two-way dependencies in multivariate functional time series by introducing dynamic weak separability on the spectral density kernel and linking it to a partial correlation graph among infinite-dimensional curves. It then develops Graphical Dynamic Functional Principal Component Analysis (GDFPCA) that uses a common set of functional filters across series, with scores that retain the graph structure, and a two-step reconstruction procedure for contaminated data employing a joint graphical Lasso and Whittle likelihood. Theoretical results establish the spectral and score representations, optimality of the truncated dynamic KL expansion, and consistency of estimators, while simulations show that GDFPCA outperforms non-graphical and static methods, especially when graph information is informative. A PM2.5 data example demonstrates practical gains in both reconstruction quality and interpretable connectivity patterns. Overall, the framework unifies graphical models with dynamic FPCA for high-dimensional, temporally dependent functional data, enabling more efficient dimension reduction and robust signal reconstruction.

Abstract

In this paper, we consider multivariate functional time series with a two-way dependence structure: a serial dependence across time points and a graphical interaction among the multiple functions within each time point. We develop the notion of dynamic weak separability, a more general condition than those assumed in literature, and use it to characterize the two-way structure in multivariate functional time series. Based on the proposed weak separability, we develop a unified framework for functional graphical models and dynamic principal component analysis, and further extend it to optimally reconstruct signals from contaminated functional data using graphical-level information. We investigate asymptotic properties of the resulting estimators and illustrate the effectiveness of our proposed approach through extensive simulations. We apply our method to hourly air pollution data that were collected from a monitoring network in China.
Paper Structure (15 sections, 6 theorems, 36 equations, 6 figures, 3 tables)

This paper contains 15 sections, 6 theorems, 36 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A Unified Framework for Graphical Models and FPCA
  3. Dynamic Weak Separability
  4. Graphical Model for MFTS
  5. Dynamic Weakly-Separable KL Expansion
  6. Graphical Functional Principal Component Analysis
  7. Graphical DFPCA for Contaminated Data
  8. First-Step Estimation
  9. Conditional Expectation for Score Extraction
  10. Statistical Properties
  11. Simulation Study
  12. Simulation Setup and Data Generation
  13. Results
  14. Data Illustration
  15. Discussion

Key Result

Theorem 1

Under the dynamic weak separability (ws), we assume that for all $\theta\in[-\pi,\pi]$ and $k\geq 1$, $\bm{\eta}_k(\theta)$ is a nonsingular matrix. Then for $i_1\neq i_2$, with $\sigma_{i_1,i_2,k}(\theta)=-\frac{[\bm{\Phi}_k(\theta)]_{i_1,i_2}}{[\bm{\Phi}_k(\theta)]_{i_1,i_1}[\bm{\Phi}_k(\theta)]_{i_2,i_2}-[\bm{\Phi}_k(\theta)]_{i_1,i_2}[\bm{\Phi}_k(\theta)]_{i_2,i_1}}$, where $\bm{\Phi}_k(\thet

Figures (6)

  • Figure 1: A. Hourly readings of PM2.5 concentrations for seven days from three selected monitoring stations in Beijing, Tianjin, and Langfang, respectively. Each row is a functional time series for a specific station. B. Locations of 24 monitoring stations in Beijing, Tianjin, and Langfang.
  • Figure 2: One-day lag (hourly) autocovariance of the data from two selected stations: Changping and Aotizhongxin.
  • Figure 3: Examples of generated graphs with $p=30$ (first row) and $p=60$ (second row).
  • Figure 4: Reconstructed curves on the last seven days for three stations in Beijing, Tianjin, and Langfang, respectively. The black dots denote the observed data, whereas the solid and dashed lines denote reconstructed curves using static and dynamic FPCAs and a spatiotemporal model, respectively.
  • Figure 5: A. Scatter plot of partial mutual information versus geographical distances between monitoring stations. The blue line is a local polynomial fit for data points. B. Partial correlation graph for the stations in three cities, where the blue edge between two stations is connected if their partial mutual information is larger than 0.05.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6