From HL to H+L-1 Parameters: A Hankel-Toeplitz Forecaster for Long-Term Time Series Forecasting

Chaoqi Zhang; Yu Wang; Haixu Tang

cs.LGSep 27, 2026

From HL to H+L-1 Parameters: A Hankel-Toeplitz Forecaster for Long-Term Time Series Forecasting

Authors: Chaoqi Zhang, Yu Wang, Haixu Tang

Organizations: Indiana University Bloomington, IN, USA

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.

Figures & tables

Dataset	Dense Linear		DLinear		FITS		SparseTSF		DiPE-Linear		TimeBase		HTF (Ours)
	MSE	K	MSE	K	MSE	K	MSE	K	MSE	K	MSE	K	MSE	K
ETTh1	0.4088	112.90	0.4139	226.46	0.4112	40.35	0.4044	0.22	0.4053	1.87	0.4093	0.19	0.4075	0.67
ETTh2	0.3429	112.90	0.3436	226.46	0.3388	40.35	0.3475	0.22	0.3441	5.64	0.3535	0.19	0.3390	0.67
ETTm1	0.3614	112.90	0.3624	226.46	0.3606	16.99	0.3834	0.11	0.3635	1.87	0.3724	0.06	0.3595	0.67
ETTm2	0.2581	112.90	0.2581	226.46	0.2568	17.62	0.2842	0.11	0.2582	1.87	0.2940	0.06	0.2572	0.67
Weather	0.2481	112.90	0.2477	226.46	0.2483	7.80	0.2769	0.15	0.2249	7.58	0.2935	0.04	0.2491	0.67

Table 1: Test MSE and trainable parameters (K = thousands), averaged over four horizons at

L=336

. Bold marks the lowest value for each metric in each row.

All 28 settings, $H\in\{96,192,336,720\}$
Model	Coefficients	Difference
HTF (Ours)	$H+L-1$	$-0.12\%$
Ridge (zero bias)	$HL$	$-0.61\%$
Ridge-projected, rank 1	$H+L$	$+80.45\%$
Ridge-projected, rank 2	$2(H+L)$	$+39.19\%$
Ridge-projected, rank 8	$8(H+L)$	$+10.69\%$

Table 2: Zero-bias internal controls,

L=336

. Within each block, differences are mean relative MSE against Dense Linear (negative is better). Counts are fitted coefficients; COV-PACF denotes the covariance control parameterized by partial autocorrelations.

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How Good Can Linear Models Be for Time-Series Forecasting?

Lang Huang, Jinglue Xu, Luke Darlow

1Sakana AI, Tokyo, Japan · 2National Institute of Informatics, Japan

Time Series Forecasting Time Series Hyperparameter Horizon

Open paper AskPDF arXiv

Time-series forecasting research has been moving steadily toward larger architectures, from specialized transformers to general-purpose foundation models, on the assumption that capacity is what unlocks accuracy. We take the opposite position: most of the gap can be closed at far lower cost by tuning preprocessing rather than scaling models. We use Ridge regression as the testbed, since it has a closed-form solution and interpretable weights, which let the optimal hyperparameters be read off the search directly. We search over context length, local normalization, regularization, and augmentation on eight standard benchmarks and find three patterns. (1) Optimal lookback is strongly series-specific and often non-monotonic in forecast horizon, with fitted power-law exponents ranging from

+0.46

on ETTm2 to

-0.19

on Exchange and Traffic, challenging the convention that longer horizons need longer history. (2) Normalizing over a learned trailing fraction of the context, rather than its entirety, is almost universally preferred. (3) Series within the same dataset often disagree on hyperparameters; the optimal degree of cross-series sharing varies from fully shared to fully per-series. The resulting models beat prior linear forecasters on most dataset-horizon entries and exceed Transformer, MLP, and CNN baselines on six of eight benchmarks. The optimized hyperparameters also serve as a diagnostic on the data itself, revealing structures that larger models absorb silently into their learned parameters. We provide an accompanying interactive online demonstration and the code at https://sakanaai.github.io/SearchCast/.

Abstract

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