从H到H+L-1参数：一种用于长期时间序列预测的Hankel-Toeplitz预测器
From HL to H+L-1 Parameters: A Hankel-Toeplitz Forecaster for Long-Term Time Series Forecasting
- Indiana University Bloomington（印第安纳大学伯明顿分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出Hankel-Toeplitz预测器（HTF），利用经典平稳预测理论实现参数共享，仅用H+L-1个参数在七个基准上达到与Dense Linear相当精度，参数减少75-229倍。
AI中文摘要：
线性预测器在长期时间序列预测中已展现出与基于Transformer的模型相当的竞争性精度。我们研究经典平稳预测理论如何指导参数共享，以构建更紧凑的线性预测器。对于具有非奇异历史协方差的中心化二阶平稳过程，最小均方误差的有限窗口线性预测器可分解为一个Hankel互协方差矩阵和一个逆Toeplitz协方差矩阵。共享滞后与尺度消除使得该预测器在回看长度L和预测视界H下仅需H+L-1个自相关系数。基于创新表示，我们的Hankel-Toeplitz预测器（HTF）学习一个脉冲响应，该响应同时定义逆滤波器和预测映射。我们刻画了有限历史修正，并在可和性假设下，界定了截断真实滤波器的超额风险。HTF使用H+L-1个可训练系数，同时允许满秩的预测矩阵。在L=336的七个基准数据集上，其视界平均MSE在每个数据集上均与Dense Linear相差在1.2%以内，而可训练参数减少了75至229倍。
英文摘要：
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.
