TimeES: Probabilistic and Deterministic Time Series Forecasting via Evolutionary Spectra

2026-09-26

AI Summary

This study addresses the limitation of existing methods that fail to explicitly model the Fourier spectral evolution and stochasticity of non-stationary time series, resulting in inaccurate prediction trajectories and uncertainty estimates. To overcome this, we propose a general framework grounded in evolutionary spectral theory, which reformulates non-stationary stochastic processes as evolutionary representation learning modulated by random variables. Furthermore, we introduce a parameterizable evolutionary spectrum formulation that exploits Hermitian symmetry and energy sparsity to reduce computational complexity from O(NM) to O(NK). By integrating linear backbone networks with a frequency selection algorithm, the proposed approach achieves state-of-the-art performance on both deterministic and probabilistic forecasting tasks, offering high computational efficiency alongside strong physical interpretability.

Abstract

Real-world time series are inherently non-stationary, with trends, periodic patterns, and uncertainty evolving over time. While the Fourier domain offers a natural lens to model time series, current deep learning approaches do not explicitly model evolution and randomness in the Fourier spectra, which limits their ability to accurately predict both the expected trajectory and its uncertainty in non-stationary time series. Motivated by Evolutionary Spectra (ES) theory, we propose TimeES, a general framework that enables probabilistic and deterministic forecasting via the evolutionary spectra theory. Specifically, we derive a parameterizable evolutionary spectra formulation, recasting non-stationary random process modeling as learning an evolving representation modulated by random variables. Furthermore, we reduce the complexity of the estimated spectra from O(NM) to O(NK), where K<<M/2, by exploiting Hermitian symmetry and spectral energy sparsity for frequency selection. Based on a simple linear backbone, our proposed TimeES achieves consistent state-of-the-art performance across both deterministic and probabilistic forecasting tasks, with high efficiency and interpretability. Code is available at: https://github.com/wwy155/TimeES.

Problem

• Time Series Forecasting
• Non-stationary Time Series
• Probabilistic Forecasting
• Deterministic Forecasting
• Evolutionary Spectra

Innovation

• Evolutionary Spectra
• Probabilistic Forecasting
• Non-stationary Time Series
• Spectral Sparsity
• Linear Backbone