Galerkin-ARIMA: A Two-Stage Polynomial Regression Framework for Fast Rolling One-Step-Ahead Forecasting

2025-07-10

AI Summary

Traditional time-series models such as ARIMA suffer from restrictive linearity assumptions, low computational efficiency, and inability to capture nonlinear dependencies among lagged variables. To address these limitations, this paper proposes the Galerkin-ARIMA framework: retaining the moving-average (MA) structure and Gaussian noise assumption, it introduces a nonparametric autoregressive component constructed via spline basis functions and—novelly in time-series modeling—incorporates Galerkin projection to efficiently approximate nonlinear dynamics. Orthogonal projection yields a closed-form ordinary least squares (OLS) estimator, ensuring asymptotic unbiasedness and consistency. This two-stage polynomial regression approach balances interpretability with modeling flexibility. Empirically, it achieves significant accuracy gains in one-step-ahead rolling forecasts while substantially reducing computational overhead on large-scale datasets.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Time-series models like ARIMA remain widely used for forecasting but limited to linear assumptions and high computational cost in large and complex datasets. We propose Galerkin-ARIMA that generalizes the AR component of ARIMA and replace it with a flexible spline-based function estimated by Galerkin projection. This enables the model to capture nonlinear dependencies in lagged values and retain the MA component and Gaussian noise assumption. We derive a closed-form OLS estimator for the Galerkin coefficients and show the model is asymptotically unbiased and consistent under standard conditions. Our method bridges classical time-series modeling and nonparametric regression, which offering improved forecasting performance and computational efficiency.

Problem

• Extends ARIMA to capture nonlinear dependencies in time-series data
• Reduces computational cost for large and complex datasets
• Combines classical time-series modeling with nonparametric regression

Innovation

• Spline-based function via Galerkin projection
• Closed-form OLS estimator for coefficients
• Combines ARIMA with nonparametric regression