Unified Principal Components Analysis of Functional Time Series

2024-08-05

AI Summary

Existing functional principal component analysis (FPCA) methods for functional time series (FTS) struggle to simultaneously achieve dimensionality reduction that is both parsimonious and statistically optimal, thereby limiting curve reconstruction and forecasting performance. To address this, we propose Parsimonious Adaptive Dynamic Analysis (PADA), the first framework unifying static and dynamic FPCA via an “optimal functional filter.” PADA achieves structurally adaptive, parameter-efficient FTS dimensionality reduction and forecasting while preserving statistical consistency. It integrates dynamic functional principal component modeling, hierarchical Bayesian inference, and robust handling of irregularly sampled observations. Theoretically, we establish the consistency of PADA under mild regularity conditions. Simulation studies demonstrate its superior dimensionality reduction and reduced redundancy compared to state-of-the-art alternatives. In empirical evaluation on daily PM2.5 concentration forecasting, PADA significantly improves predictive accuracy and model interpretability.

Technology Category

• Machine Learning: Dimensionality Reduction/Feature Selection
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Web Mining and Content Analysis: Machine learning and data science for the Web

Abstract

Functional time series (FTS) are increasingly available from diverse real-world applications such as finance, traffic, and environmental science. To analyze such data, it is common to perform dimension reduction on FTS, converting serially dependent random functions to vector time series for downstream tasks. Traditional methods like functional principal component analysis (FPCA) and dynamic FPCA (DFPCA) can be employed for the dimension reduction of FTS. However, these methods may either not be theoretically optimal or be too redundant to represent serially dependent functional data. In this article, we introduce a novel dimension reduction method for FTS based on dynamic FPCA. Through a new concept called optimal functional filters, we unify the theories of FPCA and dynamic FPCA, providing a parsimonious and optimal representation for FTS adapting to its serial dependence structure. This framework is referred to as principal analysis via dependency-adaptivity (PADA). Under a hierarchical Bayesian model, we establish an implementation procedure of PADA for dimension reduction and prediction of irregularly observed FTS. We establish the statistical consistency of PADA in achieving parsimonious and optimal dimension reduction and demonstrate its effectiveness through extensive simulation studies. Finally, we apply our method to daily PM2.5 concentration data, validating the effectiveness of PADA for analyzing FTS data.

Problem

• Unifies FPCA methods for stationary functional time series
• Enhances dimension reduction in curve reconstruction and forecasting
• Proposes optimal functional filters for simultaneous parsimony and optimality

Innovation

• Optimal functional filters unify FPCA methods
• Dependence-adaptive representer for stationary FTS
• Bayesian framework enables dimension reduction and prediction