LagNet: Lag-Structured Operators for Long-Term Time Series Forecasting

Zhenan Yu; Guangxin Jiang; Jiuyun Jiang; Jin Yang

cs.LGOct 16, 2026

LagNet: Lag-Structured Operators for Long-Term Time Series Forecasting

Authors: Zhenan Yu, Guangxin Jiang, Jiuyun Jiang, Jin Yang

Organizations: Harbin Institute of Technology

Abstract

Long-term time series forecasting maps a finite history window to an extended future horizon. Many existing models focus primarily on the design of input representations and model structures, while paying less attention to the structural form of the history-to-future forecasting operator itself. We examine this operator directly and find that, across model families, sensitivity matrices exhibit recurring diagonal patterns whose diagonals correspond to relative temporal lags. Motivated by this observation, we develop a framework for direct multi-step forecasting based on lag-structured temporal operators. Its basic building block, LagLayer, constructs a Toeplitz matrix by tying history-to-future connections with the same relative lag and incorporates horizon-wise and history-wise scaling coefficients to relax exact sharing. This design enables LagLayer to serve as an effective replacement for temporal linear layers in existing forecasting models. Stacking LagLayer modules with activations and multiple feature maps yields LagNet. Experiments on long-term time series forecasting benchmarks show that our method is competitive with strong baselines, while sensitivity matrix diagnostics and limited-data experiments demonstrate that LagLayer provides an effective and sample-efficient alternative to a linear layer.

Figures & tables

Figure 1: History-to-future sensitivity matrices for six forecasting settings. Each panel averages input–output Jacobians over test windows and channels.

Figure 2: Linear, lag-structured, and adaptively scaled forecasting matrices.

Figure 3: Two-layer LagNet architecture. Deeper models repeat a LagLayer followed by a pointwise activation, and hidden feature maps allow the network to represent multiple lag-response patterns.

Operator	Element form	Parameters
Linear	$w_{h,\ell}$	$HL$
Lag-structured matrix	$k_{L+h-\ell}$	$L+H-1$
Horizon scale only	$\alpha_{h}k_{L+h-\ell}$	$L+2H-1$
History scale only	$\beta_{\ell}k_{L+h-\ell}$	$2L+H-1$
LagLayer	$\alpha_{h}\beta_{\ell}k_{L+h-\ell}$	$2L+2H-1$

Table 1: Parameter complexity of forecasting operators, with bias terms omitted for clarity.

	Baselines										Proposed
Dataset	DLinear (2023)		PatchTST (2023)		iTransformer (2024)		TimeMixer (2024)		TimeKAN (2025)		LagLayer (ours)		LagNet (ours)		Lag-Time Mixer (ours)		Lag-Time KAN (ours)
	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	0.463	0.458	0.516	0.484	0.454	0.448	0.464	0.448	0.422	0.429	0.449	0.435	0.461	0.443	0.459	0.444	0.421	0.428
ETTh2	0.512	0.491	0.391	0.412	0.383	0.407	0.376	0.404	0.376	0.401	0.378	0.399	0.384	0.408	0.373	0.401	0.386	0.409
ETTm1	0.405	0.409	0.406	0.408	0.407	0.410	0.384	0.398	0.381	0.399	0.410	0.404	0.388	0.402	0.383	0.397	0.387	0.402
ETTm2	0.321	0.371	0.290	0.334	0.288	0.332	0.279	0.325	0.284	0.329	0.285	0.326	0.281	0.325	0.277	0.324	0.292	0.339

Table 2: Four-horizon average forecasting results for

L=96

. All results are averaged over five random seeds. The best and second-best values are shown in bold and underlined, respectively; complete horizon-wise results and seed statistics are provided in the supplementary material.

	DLinear $\rightarrow$ DLagLayer		TimeMixer $\rightarrow$ Lag-TimeMixer		TimeKAN $\rightarrow$ Lag-TimeKAN
Dataset	$\Delta$ MSE (%)	Params	$\Delta$ MSE (%)	Params	$\Delta$ MSE (%)	Params
ETT (Avg.)	$-1.3$	65.2K $\to$ 2.4K	$-0.6$	120.1K $\to$ 62.7K	$+1.7$	41.5K $\to$ 10.1K
Weather	$+2.2$	65.2K $\to$ 2.4K	$-0.1$	119.7K $\to$ 62.3K	$+1.8$	41.2K $\to$ 9.8K
Solar	$+1.5$	65.2K $\to$ 2.4K	$+1.0$	2.37M $\to$ 2.25M	$+10.4$	45.1K $\to$ 13.8K
Electricity	$+1.8$	65.2K $\to$ 2.4K	$+1.2$	122.1K $\to$ 64.7K	$+2.1$	46.6K $\to$ 15.2K
Traffic	$+0.9$	65.2K $\to$ 2.4K	$+1.8$	132.8K $\to$ 75.4K	$+2.7$	50.9K $\to$ 19.6K

Table 3: Paired operator replacement over four horizons. Negative values indicate reductions. Trainable parameter counts are reported before and after replacement. ETT (Avg.) averages the four ETT datasets.

Figure 4: Signed effective forecasting matrices for Linear and lag-structured variants. Heatmaps within each row share a symmetric color scale whose magnitude limit is the 99.5th percentile of absolute weight values.

Dataset	Model	1%	5%	20%	50%	100%	Avg.
ETT (Avg.)	Linear	0.432	0.399	0.385	0.383	0.381	0.396
	LagLayer	0.401	0.392	0.386	0.386	0.385	0.390
	LagNet	0.401	0.386	0.389	0.385	0.385	0.389
Weather	Linear	0.287	0.275	0.273	0.272	0.272	0.276
	LagLayer	0.285	0.277	0.274	0.273	0.272	0.276
	LagNet	0.281	0.276	0.270	0.265	0.265	0.271

Table 4: Four-horizon average MSE under reduced training data. ETT (Avg.) aggregates the four ETT datasets. The best result in each column is shown in bold.

Figure 5: Forecast-step MSE and learned adaptive scales on Traffic with

H=720

. Errors are averaged over test windows and channels.

Dataset	Unscaled	Horizon Only	History Only	LagLayer	LagNet
ETT (Avg.)	0.390	0.389	0.386	0.385	0.381
Weather	0.284	0.279	0.274	0.273	0.267
Solar	0.542	0.440	0.457	0.371	0.256
Electricity	0.229	0.223	0.224	0.219	0.191
Traffic	0.681	0.642	0.652	0.634	0.460

Table 5: Four-horizon average MSE for LagLayer variants and LagNet.

(a) Depth ( $M=8$ )
Dataset	$R=2$	$R=3$	$R=4$	$R=5$
Solar	0.298	0.266	0.256	0.260
Electricity	0.208	0.201	0.194	0.194
Traffic	0.523	0.481	0.470	0.464

Table 6: LagNet capacity analysis using four-horizon average MSE. Depth varies with fixed

M=8

, and width varies with fixed

R=3

. The minimum in each row is shown in bold.

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Variates	Frequency	Total points	Train	Validation	Test
ETTh1, ETTh2	7	1 hour	17,420	8,640	2,880	2,880
ETTm1, ETTm2	7	15 minutes	69,680	34,560	11,520	11,520
Weather	21	10 minutes	52,696	36,887	5,270	10,539
Solar	137	10 minutes	52,560	36,792	5,256	10,512
Electricity	321	1 hour	26,304	18,412	2,632	5,260
Traffic	862	1 hour	17,544	12,280	1,756	3,508

Appendix

Table 1: Dataset statistics and chronological core split sizes. The ETT files contain observations beyond the standard 12/4/4-month benchmark interval; these extra observations are not used.

Dataset	Temporal operator	Batch size	LagLayer $(R,M)$	LagNet $(R,M)$	Adaptive scales
ETTh1	Shared	256	$(1,\text{--})$	$(2,4)$	Both
ETTh2	Shared	64	$(1,\text{--})$	$(2,4)$	Both
ETTm1	Per-variable	256	$(1,\text{--})$	$(2,4)$	Both
ETTm2	Per-variable	64	$(1,\text{--})$	$(2,4)$	Both
Weather	Per-variable	16	$(1,\text{--})$	$(2,4)$	Both
Solar	Shared	64	$(1,\text{--})$	$(4,12)$	Both

Appendix

Table 2: Standalone lag-structured configurations used in the main forecasting table.

R

is the number of lag-structured layers and

M

the hidden feature-map width. Both denotes horizon-wise and history-wise scaling; LagNet applies the corresponding row-wise and column-wise mechanisms in every layer.

Dataset	Batch size	$\eta_{96}$	$\eta_{192}$	$\eta_{336}$	$\eta_{720}$	Epochs	Patience	Individual
ETTh1	32	$5{\times}10^{-3}$	$5{\times}10^{-3}$	$5{\times}10^{-3}$	$5{\times}10^{-3}$	10	3	No
ETTh2	32	$5{\times}10^{-2}$	$5{\times}10^{-2}$	$5{\times}10^{-2}$	$5{\times}10^{-2}$	10	3	No
ETTm1	8	$1{\times}10^{-4}$	$1{\times}10^{-4}$	$1{\times}10^{-4}$	$1{\times}10^{-4}$	10	3	No
ETTm2	32	$1{\times}10^{-3}$	$1{\times}10^{-3}$	$1{\times}10^{-2}$	$1{\times}10^{-2}$	10	3	No
Weather	16	$1{\times}10^{-4}$	$1{\times}10^{-4}$	$1{\times}10^{-4}$	$1{\times}10^{-4}$	10	3	No
Solar	16	$1{\times}10^{-3}$	$1{\times}10^{-3}$	$1{\times}10^{-3}$	$1{\times}10^{-3}$	10	3	No

Appendix

Table 3: DLinear and DLagLayer configurations. Here

\eta_{H}

denotes the learning rate at forecast horizon

H

. The shared-variable DLinear architecture, moving-average kernel 25, MSE loss, Adam optimizer, official type-1 learning-rate schedule, and dataset-specific settings follow the official paper and repository. All runs use

L=96

. Solar is absent from the official DLinear benchmark scripts and therefore uses the complete official Electricity training setting.

Dataset	$M$	PDM blocks	$d_{\mathrm{model}}$	$d_{\mathrm{ff}}$	Batch size	Epochs	Patience	Channel independent	Normalize
ETTh1	3	2	16	32	128	10	10	Yes	Yes
ETTh2	3	2	16	32	128	10	10	Yes	Yes
ETTm1	3	2	16	32	128	10	10	Yes	Yes
ETTm2	3	2	32	32	128	10	10	Yes	Yes
Weather	3	2	16	32	128	20	10	Yes	Yes
Solar	3	2	128	2048	32	20	3	No	No

Appendix

Table 4: TimeMixer and Lag-TimeMixer configurations used in the main forecasting table. The core settings

M

, number of PDM blocks,

d_{\mathrm{model}}

, learning rate, loss, batch size, and epochs follow Table 7 of the official TimeMixer paper; unreported model and optimization options are taken from the released implementation. All configurations use

L=96

, four prediction lengths, MSE loss, Adam with learning rate

10^{-2}

, OneCycleLR with pct_start =0.2, average-pooling downsampling with window 2, moving-average decomposition with kernel 25, and dropout 0.1. Here

M=3

denotes three downsampling stages and hence four temporal scales.

Dataset	$H=96$	$H=192$	$H=336$	$H=720$	Batch size	Epochs	Patience
ETTh1	$(16,2,2,0)$	$(16,2,1,0)$	$(16,2,2,0)$	$(16,2,3,1)$	128	10	10
ETTh2	$(16,2,3,0)$	$(16,1,1,0)$	$(32,1,3,0)$	$(16,2,3,1)$	128	10	10
ETTm1	$(16,2,2,1)$	$(16,2,2,1)$	$(16,3,2,1)$	$(16,2,3,1)$	128	10	10
ETTm2	$(32,3,2,1)$	$(32,3,2,1)$	$(32,1,1,1)$	$(16,3,3,1)$	128	10	10
Weather	$(16,3,3,1)$	$(16,3,2,1)$	$(16,3,2,1)$	$(16,3,2,1)$	128	20	10
Solar	$(16,3,3,1)$	$(16,3,3,1)$	$(16,3,3,1)$	$(16,3,3,1)$	32	20	3

Appendix

Table 5: TimeKAN and Lag-TimeKAN configurations. Each horizon-specific tuple is

(d_{\mathrm{model}},E,D,O)

, where

E

is the number of encoder layers,

D

the number of downsampling layers, and

O

the beginning decomposition order. The model configurations follow the official TimeKAN paper and released main-experiment scripts; Solar and Traffic use the released Electricity configuration as a fixed extension. All runs use

L=96

d_{\mathrm{ff}}=32

, downsampling window 2, moving-average kernel 25, dropout 0.1, normalization, MSE loss, Adam with learning rate

10^{-2}

, and OneCycleLR with pct_start =0.2. The dataset-specific batch sizes, epoch limits, and patience values form the unified training protocol shared by TimeKAN and Lag-TimeKAN.

Dataset	$H$	iTransformer		DLinear		PatchTST		TimeMixer		TimeKAN
		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.386	0.405	0.390	0.404	0.460	0.447	0.380	0.401	0.370	0.397
	192	0.441	0.436	0.448	0.443	0.512	0.477	0.446	0.434	0.416	0.421
	336	0.487	0.458	0.499	0.475	0.546	0.496	0.490	0.455	0.446	0.437
	720	0.503	0.491	0.515	0.509	0.544	0.517	0.538	0.500	0.456	0.460
ETTh2	96	0.297	0.349	0.374	0.410	0.308	0.355	0.297	0.348	0.296	0.345

Appendix

Table 6: Horizon-wise baseline results for

L=96

. The PatchTST values are copied from the unified TimeMixer evaluation, and iTransformer values from its official full forecasting table. DLinear, TimeMixer, and TimeKAN entries are five-seed means from our reproductions under the corresponding protocols. Each entry reports MSE/MAE.

Dataset	$H$	iTransformer	DLinear	PatchTST	TimeMixer	TimeKAN
ETTh1	96	841.6K	18.6K	35.2K	75.4K	12.8K
	192	866.2K	37.2K	53.7K	93.1K	20.5K
	336	3.38M	65.2K	81.5K	119.6K	36.1K
	720	3.58M	139.7K	155.6K	190.2K	77.6K
ETTh2	96	224.2K	18.6K	35.2K	75.4K	15.0K
	192	236.6K	37.2K	53.7K	93.1K	19.6K

Appendix

Table 7: Audited horizon-wise trainable parameter counts for the public baselines. Counts use the model definitions and configurations underlying the reported

L=96

results; DLinear, TimeMixer, and TimeKAN counts come from our reproduced implementations.

Dataset	$H$	DLinear			DLagLayer
		Params	MSE	MAE	Params	MSE	MAE
ETTh1	96	18.6K	0.382	0.393	958	0.392	0.407
	192	37.2K	0.438	0.430	1.5K	0.434	0.428
	336	65.2K	0.513	0.488	2.4K	0.482	0.457
	720	139.7K	0.507	0.502	4.7K	0.505	0.501
ETTh2	96	18.6K	0.470	0.474	958	0.317	0.375

Appendix

Table 8: Paired DLinear and DLagLayer results. Both models use the same official shared-variable DLinear protocol. DLagLayer preserves the moving-average decomposition and replaces only the independent seasonal and trend temporal projections with dimension-matched LagLayer operators. Solar uses the complete official Electricity training setting because it is absent from the official DLinear benchmark scripts. Params denotes total trainable parameters. Results are reported from one run with random seed 2027.

Dataset	$H$	TimeMixer				Lag-TimeMixer
		Params	Runs	MSE	MAE	Params	Runs	MSE	MAE
ETTh1	96	75.4K	5/5	0.380 $\pm$ 0.008	0.401 $\pm$ 0.002	59.3K	5/5	0.378 $\pm$ 0.007	0.397 $\pm$ 0.004
	192	93.1K	5/5	0.446 $\pm$ 0.006	0.434 $\pm$ 0.003	60.4K	5/5	0.435 $\pm$ 0.006	0.428 $\pm$ 0.002
	336	119.6K	5/5	0.490 $\pm$ 0.010	0.455 $\pm$ 0.009	62.2K	5/5	0.484 $\pm$ 0.013	0.448 $\pm$ 0.007
	720	190.2K	5/5	0.538 $\pm$ 0.026	0.500 $\pm$ 0.015	66.8K	5/5	0.539 $\pm$ 0.046	0.501 $\pm$ 0.023
ETTh2	96	75.4K	5/5	0.297 $\pm$ 0.002	0.348 $\pm$ 0.002	59.3K	5/5	0.295 $\pm$ 0.004	0.346 $\pm$ 0.004

Appendix

Table 9: Paired TimeMixer and Lag-TimeMixer results over five seeds. TimeMixer follows the main-experiment configurations in its official paper; options not specified there follow the released implementation. Lag-TimeMixer uses the identical protocol without a separate search, preserves PDM and all components outside FMM, and replaces only the FMM temporal predictors with LagLayer operators. Params denotes total trainable parameters. Runs reports the number of completed seeds out of five.

Dataset	$H$	TimeKAN				Lag-TimeKAN
		Params	Runs	MSE	MAE	Params	Runs	MSE	MAE
ETTh1	96	12.8K	5/5	0.370 $\pm$ 0.001	0.397 $\pm$ 0.001	4.0K	5/5	0.367 $\pm$ 0.003	0.395 $\pm$ 0.002
	192	20.5K	5/5	0.416 $\pm$ 0.002	0.421 $\pm$ 0.001	2.7K	5/5	0.420 $\pm$ 0.006	0.422 $\pm$ 0.002
	336	36.1K	5/5	0.446 $\pm$ 0.003	0.437 $\pm$ 0.001	4.7K	5/5	0.443 $\pm$ 0.004	0.438 $\pm$ 0.002
	720	77.6K	5/5	0.456 $\pm$ 0.006	0.460 $\pm$ 0.007	10.1K	5/5	0.456 $\pm$ 0.014	0.455 $\pm$ 0.007
ETTh2	96	15.0K	5/5	0.296 $\pm$ 0.005	0.345 $\pm$ 0.004	6.2K	5/5	0.306 $\pm$ 0.005	0.356 $\pm$ 0.004

Appendix

Table 10: Paired TimeKAN and Lag-TimeKAN results under the unified training protocol. Each entry reports mean

\pm

standard deviation and total trainable parameters. Lag-TimeKAN uses the identical data, optimization, and random-seed setting as TimeKAN and replaces only the final linear temporal predictor with the complete LagLayer operator. Runs reports the number of completed seeds out of five.

Dataset	$H$	LagLayer				LagNet
		Params	Runs	MSE	MAE	Params	Runs	MSE	MAE
ETTh1	96	479	5/5	0.390 $\pm$ 0.001	0.396 $\pm$ 0.001	3.5K	5/5	0.389 $\pm$ 0.002	0.402 $\pm$ 0.002
	192	767	5/5	0.440 $\pm$ 0.001	0.425 $\pm$ 0.000	6.3K	5/5	0.438 $\pm$ 0.004	0.430 $\pm$ 0.002
	336	1.2K	5/5	0.483 $\pm$ 0.001	0.447 $\pm$ 0.000	10.5K	5/5	0.487 $\pm$ 0.005	0.448 $\pm$ 0.002
	720	2.4K	5/5	0.482 $\pm$ 0.000	0.470 $\pm$ 0.000	21.6K	5/5	0.531 $\pm$ 0.020	0.493 $\pm$ 0.009
ETTh2	96	479	5/5	0.295 $\pm$ 0.002	0.341 $\pm$ 0.001	3.5K	5/5	0.301 $\pm$ 0.009	0.348 $\pm$ 0.006

Appendix

Table 11: Complete LagLayer and LagNet main results. Metrics are reported as mean

\pm

standard deviation over five seeds. Params denotes the total number of trainable parameters in each instantiated model. LagLayer is the complete single-layer operator with both horizon-wise and history-wise scaling, while LagNet is its deep nonlinear extension.

Variant	Relative-lag sharing	Row scale	Column scale	Nonlinear depth	Extra modules	Diagnostic
Linear	No	No	No	No	No	free matrix
LagLayer w/o Scaling	Yes	No	No	No	No	lag sharing
LagLayer w/o History Scale	Yes	Yes	No	No	No	row adaptation
LagLayer w/o Horizon Scale	Yes	No	Yes	No	No	column adaptation
LagLayer	Yes	Yes	Yes	No	No	row/column adaptation
LagNet capacity diagnostic	Yes	No	No	Yes	No	nonlinear responses

Appendix

Table 12: Ablation variants and the component isolated by each diagnostic. The deep capacity diagnostic disables adaptive scaling; the complete LagNet used in the main forecasting table enables it in every layer. No extra decomposition, attention, gating, or residual branch is introduced.

Dataset	$H$	Linear	w/o Scaling	w/o History Scale	w/o Horizon Scale	LagLayer	LagNet
ETTh1	96	0.387/0.394	0.393/0.397	0.391/0.395	0.393/0.398	0.392/0.397	0.389/0.401
	192	0.438/0.424	0.439/0.424	0.438/0.423	0.440/0.425	0.440/0.424	0.441/0.434
	336	0.482/0.447	0.482/0.446	0.482/0.445	0.482/0.447	0.482/0.447	0.491/0.456
	720	0.482/0.471	0.480/0.469	0.480/0.469	0.481/0.471	0.482/0.470	0.498/0.477
ETTh2	96	0.290/0.338	0.304/0.345	0.303/0.344	0.294/0.341	0.294/0.340	0.301/0.345
	192	0.376/0.391	0.389/0.396	0.388/0.395	0.378/0.391	0.377/0.391	0.369/0.390

Appendix

Table 13: Complete horizon-wise MSE/MAE component diagnostics. Each entry reports MSE/MAE. Linear is the unconstrained free-matrix reference. For LagLayer, “w/o Scaling” removes both scale vectors, while each one-sided ablation removes the named scale. The LagNet capacity diagnostic adds nonlinear lag-structured composition to the unscaled backbone.

Figure 1: Forecasts on a fixed Electricity test case with $H=336$ , used as a qualitative complement to the unscaled depth/width diagnostic.

Dataset

H

Model 1% 5% 20% 50% 100% ETTh1 96 Linear 0.490 0.419 0.394 0.391 0.386 LagLayer 0.421 0.399 0.391 0.394 0.391 LagNet 0.429 0.404 0.391 0.385 0.388 192 Linear 0.546 0.468 0.446 0.440 0.441 LagLayer 0.474 0.447 0.441 0.441 0.441 LagNet 0.455 0.444 0.442 0.436 0.432 336 Linear 0.579 0.509 0.487 0.483 0.482 LagLayer 0.508 0.486 0.481 0.485 0.486 LagNet 0.516 0.481 0.502 0.489 0.498 720 Linear 0.571 0.513 0.486 0.486 0.482 LagLayer 0.489 0.479 0.479 0.485 0.482 LagNet 0.496 0.480 0.569 0.569 0.567 ETTh2 96 Linear 0.362 0.313 0.297 0.292 0.292 LagLayer 0.321 0.307 0.302 0.296 0.292 LagNet 0.350 0.299 0.295 0.296 0.314 192 Linear 0.439 0.401 0.387 0.378 0.375 LagLayer 0.403 0.391 0.384 0.378 0.377 LagNet 0.401 0.376 0.373 0.363 0.364 336 Linear 0.463 0.436 0.426 0.423 0.416 LagLayer 0.430 0.423 0.420 0.416 0.416 LagNet 0.427 0.419 0.415 0.424 0.424 720 Linear 0.471 0.458 0.428 0.427 0.423 LagLayer 0.428 0.432 0.420 0.423 0.421 LagNet 0.452 0.478 0.458 0.457 0.441 Weather 96 Linear 0.212 0.198 0.195 0.192 0.194 LagLayer 0.211 0.201 0.195 0.196 0.194 LagNet 0.205 0.199 0.192 0.184 0.184 192 Linear 0.254 0.243 0.239 0.241 0.239 LagLayer 0.253 0.247 0.243 0.240 0.240 LagNet 0.246 0.243 0.241 0.235 0.231 336 Linear 0.306 0.294 0.292 0.291 0.291 LagLayer 0.303 0.295 0.292 0.292 0.291 LagNet 0.300 0.295 0.288 0.285 0.286 720 Linear 0.377 0.367 0.364 0.365 0.365 LagLayer 0.373 0.366 0.364 0.363 0.364 LagNet 0.371 0.367 0.360 0.357 0.356 Solar 96 Linear 0.393 0.336 0.312 0.308 0.305 LagLayer 0.508 0.396 0.344 0.327 0.322 LagNet 0.273 0.254 0.241 0.225 0.214 192 Linear 0.426 0.373 0.352 0.345 0.343 LagLayer 0.546 0.435 0.382 0.365 0.357 LagNet 0.314 0.298 0.259 0.257 0.242 336 Linear 0.453 0.409 0.393 0.389 0.386 LagLayer 0.561 0.468 0.416 0.401 0.395 LagNet 0.324 0.294 0.286 0.295 0.260 720 Linear 0.452 0.408 0.394 0.391 0.388 LagLayer 0.545 0.476 0.413 0.401 0.399 LagNet 0.324 0.310 0.291 0.267 0.269

Dataset

H

Model 1% 5% 20% 50% 100% ETTm1 96 Linear 0.395 0.364 0.352 0.349 0.356 LagLayer 0.398 0.380 0.366 0.368 0.368 LagNet 0.376 0.345 0.344 0.335 0.342 192 Linear 0.421 0.396 0.392 0.393 0.391 LagLayer 0.419 0.411 0.405 0.402 0.404 LagNet 0.397 0.385 0.394 0.375 0.380 336 Linear 0.452 0.432 0.428 0.424 0.423 LagLayer 0.442 0.435 0.434 0.433 0.434 LagNet 0.436 0.414 0.416 0.408 0.409 720 Linear 0.512 0.495 0.488 0.487 0.487 LagLayer 0.498 0.496 0.495 0.494 0.496 LagNet 0.494 0.493 0.469 0.480 0.469 ETTm2 96 Linear 0.202 0.190 0.183 0.182 0.181 LagLayer 0.195 0.192 0.184 0.186 0.182 LagNet 0.199 0.192 0.184 0.184 0.181 192 Linear 0.265 0.253 0.248 0.246 0.247 LagLayer 0.259 0.254 0.250 0.248 0.248 LagNet 0.254 0.249 0.249 0.246 0.243 336 Linear 0.324 0.315 0.309 0.308 0.308 LagLayer 0.316 0.316 0.313 0.316 0.314 LagNet 0.314 0.314 0.308 0.309 0.302 720 Linear 0.422 0.416 0.410 0.410 0.408 LagLayer 0.417 0.413 0.412 0.413 0.411 LagNet 0.415 0.404 0.407 0.406 0.402 Electricity 96 Linear 0.293 0.220 0.200 0.198 0.198 LagLayer 0.248 0.213 0.204 0.203 0.203 LagNet 0.217 0.188 0.176 0.171 0.167 192 Linear 0.293 0.220 0.200 0.198 0.198 LagLayer 0.251 0.212 0.203 0.201 0.201 LagNet 0.209 0.191 0.181 0.177 0.174 336 Linear 0.306 0.235 0.215 0.212 0.212 LagLayer 0.266 0.226 0.217 0.215 0.215 LagNet 0.223 0.208 0.197 0.192 0.189 720 Linear 0.345 0.276 0.256 0.254 0.254 LagLayer 0.307 0.269 0.259 0.257 0.257 LagNet 0.293 0.250 0.244 0.234 0.233 Traffic 96 Linear 0.949 0.720 0.655 0.647 0.645 LagLayer 0.801 0.702 0.666 0.656 0.653 LagNet 0.633 0.502 0.461 0.451 0.442 192 Linear 0.940 0.675 0.607 0.600 0.599 LagLayer 0.776 0.650 0.619 0.608 0.607 LagNet 0.581 0.491 0.462 0.450 0.440 336 Linear 0.936 0.680 0.616 0.607 0.606 LagLayer 0.777 0.661 0.632 0.618 0.616 LagNet 0.573 0.498 0.478 0.459 0.457 720 Linear 0.968 0.728 0.653 0.645 0.644 LagLayer 0.805 0.708 0.671 0.656 0.654 LagNet 0.639 0.531 0.507 0.501 0.489

Appendix

Table 15: Complete horizon-wise sample-efficiency MSE results. Training fractions are applied only to the training windows, while validation and test splits remain unchanged.

Figure 2: Complementary magnitude-only history-to-future sensitivity visualization for the main-paper settings. Each panel averages absolute input–output Jacobians over test windows and channels.

Figure 3: Complementary magnitude-only forecasting-matrix visualization for the main-paper settings. The panels show absolute effective forecasting weights, and heatmaps within each row share a color scale clipped at its 99.5th percentile.

Panel of Figure 4 — Figure 4: History-position importance curves for Linear across all eight datasets and four prediction lengths. The upper panel contains the ETT datasets, and the lower panel contains Weather, Solar, Electricity, and Traffic. Each curve is computed as $I(\ell)=\frac{1}{H}\sum_{h=1}^{H}|w_{h,\ell}|$ , without additional normalization. The nonuniform profiles show that historical positions can contribute with different overall strength.

Figure 5: Complete forecasting-matrix comparison for prediction length $H=96$ . Panels show absolute effective forecasting weights. Rows correspond to datasets and columns correspond to operator variants. The horizontal and vertical axes span historical positions 1–96 and forecast steps 1–96, respectively. Each row uses a shared color scale across the five variants.

Figure 6: Complete forecasting-matrix comparison for prediction length $H=192$ . Panels show absolute effective forecasting weights. Rows correspond to datasets and columns correspond to operator variants. The horizontal and vertical axes span historical positions 1–96 and forecast steps 1–192, respectively. Each row uses a shared color scale across the five variants.

Figure 7: Complete forecasting-matrix comparison for prediction length $H=336$ . Panels show absolute effective forecasting weights. Rows correspond to datasets and columns correspond to operator variants. The horizontal and vertical axes span historical positions 1–96 and forecast steps 1–336, respectively. Each row uses a shared color scale across the five variants.

Figure 8: Complete forecasting-matrix comparison for prediction length $H=720$ . Panels show absolute effective forecasting weights. Rows correspond to datasets and columns correspond to operator variants. The horizontal and vertical axes span historical positions 1–96 and forecast steps 1–720, respectively. Each row uses a shared color scale across the five variants.

Long-Term Time Series Forecasting

Long-term time series forecasting predicts distant future values from historical sequences in domains such as energy, weather, traffic, and health.

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

Chaoqi Zhang, Yu Wang, Haixu Tang

Indiana University Bloomington, IN, USA

Long-Term Time Series Forecasting Time Series Forecasting

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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.

Abstract

Long-Term Time Series Forecasting

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