Hankel-Toeplitz method drastically reduces parameters in long-term forecasts
From HL to H+L-1 Parameters: A Hankel-Toeplitz Forecaster for Long-Term Time Series Forecasting
Machine Learning
Summary
Predicting future values of time-based data is important but can require very complex models. The researchers found a clever mathematical way to significantly reduce the number of parameters needed in linear forecasting models without losing accuracy. They used classical stationary theory to share parameters compactly, resulting in a model called the Hankel-Toeplitz Forecaster. This model performs nearly as well as much larger models on multiple datasets but with far fewer parameters to train.
What this means in practice
- •For data scientists: Create compact long-term forecasting models for large time series datasets with drastically fewer parameters while maintaining similar accuracy.
- •For financial analysts: Build efficient models to forecast economic indicators or asset prices over extended horizons using fewer computation resources.
Authors
Chaoqi Zhang, Yu Wang, Haixu Tang
Abstract
Linear forecasters have shown competitive accuracy against Transformer-based models in long-term time series forecasting. We study how classical stationary prediction theory can guide parameter sharing for more compact linear forecasters. For centered second-order stationary processes with nonsingular history covariance, the minimum-MSE finite-window linear predictor factors into a Hankel cross-covariance matrix and an inverse Toeplitz covariance matrix. Shared lags and scale cancellation specify this predictor using $H+L-1$ autocorrelations for lookback $L$ and horizon $H$. Building on the innovations representation, our Hankel-Toeplitz Forecaster (HTF) learns one impulse response that defines both an inverse filter and a forecast map. We characterize the finite-history correction and, under summability assumptions, bound the excess risk of truncating the true filters. HTF uses $H+L-1$ trainable coefficients while allowing a full-rank forecasting matrix. Across seven benchmarks at $L=336$, its horizon-averaged MSE is within 1.2% of Dense Linear on each dataset with 75-229 times fewer trainable parameters.