Yule-Walker Estimation for Functional Time Series in Hilbert Space

2025-06-06

AI Summary

Estimating autoregressive operators in high-order functional time series modeling is ill-posed due to the compactness of covariance operators in infinite-dimensional Hilbert spaces, limiting existing approaches to low-order dependence assumptions or truncation-based approximations. Method: This paper introduces Tikhonov regularization into the functional Yule–Walker estimator for the first time, establishing a unified asymptotic theory. Contributions/Results: We identify a novel phenomenon—non-weak convergence of estimators under the operator norm—and propose a triple-decomposition framework for mean squared prediction error (MSPE) to disentangle prediction error sources. The resulting explicit estimator is computationally feasible; its regularized predictor is asymptotically normal. Both theoretical analysis and empirical studies demonstrate substantially improved predictive accuracy over conventional truncation methods, with validation on high-frequency wearable sensor data.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Intelligent Robots: State Estimation
• Search and Optimization: Non-convex Optimization

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Large pretrained models with web data
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

Recent advances in data collection technologies have led to the widespread availability of functional data observed over time, often exhibiting strong temporal dependence. However, existing methodologies typically assume independence across functions or impose restrictive low-order dependence structures, limiting their ability to capture the full dynamics of functional time series. To address this gap, we investigate higher-order functional autoregressive (FAR) models in Hilbert spaces, focusing on the statistical challenges introduced by infinite dimensionality. A fundamental challenge arises from the ill-posedness of estimating autoregressive operators, which stems from the compactness of the autocovariance operator and the consequent unboundedness of its inverse. We propose a regularized Yule-Walker-type estimation procedure, grounded in Tikhonov regularization, to stabilize the estimation. Specializing to $L^2$ spaces, we derive explicit and computationally feasible estimators that parallel classical finite-dimensional methods. Within a unified theoretical framework, we study the asymptotic properties of the proposed estimators and predictors. Notably, while the regularized predictors attain asymptotic normality, the corresponding estimators of the autoregressive operators fail to converge weakly in distribution under the operator norm topology, due to the compactness of the autocovariance operator. We further analyze the mean squared prediction error (MSPE), decomposing it into components attributable to regularization bias, truncation, and estimation variance. This decomposition reveals the advantages of our approach over traditional linear truncation schemes. Extensive simulations and an application to high-frequency wearable sensor data demonstrate the practical utility and robustness of the proposed methodology in capturing complex temporal structures in functional time series.

Problem

• Estimating higher-order functional autoregressive models in Hilbert spaces
• Addressing ill-posedness in autoregressive operator estimation
• Analyzing asymptotic properties and prediction error decomposition

Innovation

• Regularized Yule-Walker estimation for FAR models
• Tikhonov regularization to stabilize operator estimation
• Explicit estimators in L2 spaces for computational feasibility