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

2025/07/10 by Liu, Haojie, Lin, Zihan
#Econometrics (econ.EM) #FOS: Computer and information sciences #FOS: Economics and business #Machine Learning (cs.LG) #Machine Learning (stat.ML)

paper · doi:10.48550/arxiv.2507.07469

Abstract

We introduce Galerkin-ARIMA, a novel time-series forecasting framework that integrates Galerkin projection techniques with the classical ARIMA model to capture potentially nonlinear dependencies in lagged observations. By replacing the fixed linear autoregressive component with a spline-based basis expansion, Galerkin-ARIMA flexibly approximates the underlying relationship among past values via ordinary least squares, while retaining the moving-average structure and Gaussian innovation assumptions of ARIMA. We derive closed-form solutions for both the AR and MA components using two-stage Galerkin projections, establish conditions for asymptotic unbiasedness and consistency, and analyze the bias-variance trade-off under basis-size growth. Complexity analysis reveals that, for moderate basis dimensions, our approach can substantially reduce computational cost compared to maximum-likelihood ARIMA estimation. Through extensive simulations on four synthetic processes-including noisy ARMA, seasonal, trend-AR, and nonlinear recursion series-we demonstrate that Galerkin-ARIMA matches or closely approximates ARIMA's forecasting accuracy while achieving orders-of-magnitude speedups in rolling forecasting tasks. These results suggest that Galerkin-ARIMA offers a powerful, efficient alternative for modeling complex time series dynamics in high-volume or real-time applications.

Citations

• A hybrid method of exponential smoothing and recurrent neural networks for time series forecasting
• Manifold Markov chain Monte Carlo methods for Bayesian inference in diffusion models
• N-BEATS: Neural basis expansion analysis for interpretable time series forecasting
• Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks

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