arXiv 10 Jul 2025 · Statistics — Machine Learning
arXiv:2507.07469 · PDF · DOI · OpenAlex · Extracted main text
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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Brockwell, Peter J. and Davis, Richard A (1991) Time Series: Theory and Methods | 0.737 | 3 | 2 | 100% |
| 2 | Box, George E. P. and Jenkins, Gwilym M (1970) Time Series Analysis: Forecasting and Control | 0.644 | 2 | 2 | 100% |
| 3 | Federal Reserve Bank of St. Louis (2026) FRED Economic Data | 0.585 | 3 | 1 | 100% |
| 4 | Zhang, G. Peter (2003) Time Series Forecasting Using a Hybrid ARIMA and Neural Network Model | 0.511 | 2 | 1 | 100% |
| 5 | Yahoo Finance (2026) Yahoo Finance: Historical Market Data | 0.511 | 2 | 1 | 100% |
| 6 | Chen, Rong and Tsay, Ruey S (1993) Functional-Coefficient Autoregressive Models | 0.405 | 1 | 1 | 100% |
| 7 | Dokuchaev, Mikhail and Zhou, Guanglu and Wang, Song (2022) A modification of Galerkin's method for option pricing | 0.405 | 1 | 1 | 100% |
| 8 | Doukhan, Paul (1994) Mixing: Properties and Examples | 0.405 | 1 | 1 | 100% |
| 9 | Franses, Philip Hans and van Dijk, Dick (2000) Non-Linear Time Series Models in Empirical Finance | 0.405 | 1 | 1 | 100% |
| 10 | Graham, Matthew M. and Thiery, Alexandre H. and Beskos, Alexandros (2023) Manifold Markov chain Monte Carlo methods for Bayesian inference in diffusion models | 0.405 | 1 | 1 | 100% |
Showing the top 10 of 22 scored citations.