A Frequency-Domain Approach for Integrating Multiple Functional Time Series

2026-03-23

AI Summary

This work addresses the challenge of modeling multivariate functional time series, which exhibit complex multidirectional dependencies—including within-curve structures, cross-curve temporal dynamics, and inter-individual interactions. The paper proposes the first unified frequency-domain framework tailored to such data. Building upon marginal dynamic Karhunen–Loève expansions, the approach constructs marginal spectral operators by integrating the spectral densities of individual series and then derives optimal functional filters from their eigenfunctions. These filters transform the original functional observations into structured multivariate time series, enabling joint modeling and efficient estimation. Theoretical analysis and simulations demonstrate superior performance of the proposed method, and its practical efficacy is validated through applications to imputation and forecasting of air pollutant concentration trajectories in China.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Integrative analysis of multivariate functional time series (MFTS) is both critical and challenging across many scientific domains. Such data often exhibit complex multi-way dependencies arising from within-curve structures, temporal correlations across curves, and cross-subject interactions, underscoring the need for efficient methods that can jointly capture these dependencies and support accurate downstream analyses. In this work, we propose a novel frequency-domain framework based on a marginal dynamic Karhunen--Loève expansion. The key idea is to integrate individual spectral densities of the MFTS to construct a marginal spectral operator, whose eigenfunctions yield optimal functional filters. These filters transform complex functional observations into a structured multivariate time series representation, providing a powerful foundation for joint modeling and estimation. Through extensive simulation studies, we demonstrate the superior performance of the proposed approach. We further validate its practical utility through an application to the imputation and forecasting of air pollutant concentration trajectories in China.

Problem

• multivariate functional time series
• integrative analysis
• multi-way dependencies
• spectral density
• functional data

Innovation

• frequency-domain
• multivariate functional time series
• spectral operator
• functional filtering
• Karhunen–Loève expansion