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

LagNet: Lag-Structured Operators for Long-Term Time Series Forecasting Zhenan Yu1 , Guangxin Jiang1 , Jiuyun Jiang1 , Jin Yang1∗

Introduction Long-term time series forecasting (LTSF) supports decisionmaking in energy management, traffic planning, weather forecasting, finance, and industrial monitoring. Compared with short-horizon forecasting, LTSF maps a finite historical window to an extended forecast horizon, placing greater demands on temporal modeling, generalization, and computational efficiency. Recent methods largely differ in how they represent and mix historical observations. Transformer variants target long-sequence modeling (Zhou et al. 2021; Wu et al. 2021; Zhou et al. 2022), while PatchTST (Nie et al. 2023) and iTransformer (Liu et al. 2024) reorganize the input into temporal patches and variate tokens, respectively. Linear and MLP models provide simpler alternatives for temporal mixing (Zeng et al. 2023; Chen et al. 2023; Wang et al. 2024). Regardless of architecture, each forecaster induces a mapping from the historical window to the future horizon. A complementary question is whether the history-to-future forecasting operator contains reusable structure and, if so, how that structure should be parameterized. ∗

Corresponding author.

0

80 0

0

(c) Weather, H=336 (d) Electricity, H=336 2-layer MLP (d=256) 4-layer MLP (d=256)

-1 1

100 0

200 300 (e) Traffic, H=720 PatchTST

(f) Traffic, H=720 TimeMixer

-1 1

200 0

400 600 0

20 40 60 80 Historical position

0

20 40 60 80 Historical position

-1

Normalized J

40

1

Normalized J

(b) ETTm2, H=192 Linear

Normalized J

(a) ETTh1, H=96 Linear

0

Forecast horizon

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 sampleefficient alternative to a linear layer.

Forecast horizon

Abstract

Forecast horizon

arXiv:2610.23735v1 [cs.LG] 16 Oct 2026

1 Harbin Institute of Technology {25S010055@stu.hit.edu.cn, gxjiang@hit.edu.cn, jiuyunjiang@stu.hit.edu.cn, j.yang@hit.edu.cn}

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

To compare linear and nonlinear forecasters in a common form, we examine their input-to-output Jacobians, a standard gradient-based sensitivity measure (Simonyan, Vedaldi, and Zisserman 2014; Ismail et al. 2020). For a forecaster f , the Jacobian entry Jh,ℓ = ∂fh /∂xℓ measures the local influence of historical position ℓ on forecast step h. Averaging these sensitivities over test windows and channels produces an H × L matrix of influential history-to-future connections, where H and L denote the forecast horizon and history length, respectively. For a linear model, f (x) = W x + b and J = W , so this diagnostic recovers the learned forecasting matrix W ∈ RH×L , whose H × L coefficients are fitted independently. We refer to this one-layer model as Linear. A connection between historical position ℓ and forecast step h spans the relative lag d = L + h − ℓ. Connections with the same lag therefore lie on the same matrix diago-

nal. Figure 1(a) and (b) show pronounced diagonal bands in two Linear settings, with strong sensitivities recurring at approximately periodic lags. As shown in Figure 1 (c)–(f), similar lag-aligned patterns appear in the MLP, PatchTST, and TimeMixer settings, although their strength and axisaligned variation differ across panels. Their recurrence across architectures suggests that relative lag systematically influences which history-to-future connections receive high importance. This observation motivates our central hypothesis: connections with the same relative lag often carry similar predictive meaning, consistent with relative temporal invariance under valid time shifts. We encode this dominant organization in a lag-structured matrix by sharing one base coefficient among connections with the same relative lag. This yields a rectangular Toeplitz matrix (Gray 2006) with L + H − 1 lag weights instead of H × L independently learned coefficients. The empirical sensitivities, however, are not exactly Toeplitz: several panels also exhibit horizontal variation across forecast steps and vertical concentration at selected historical positions. These patterns suggest that relative lag provides the structural backbone, while forecast step and historical position further modulate connection strength. Accordingly, LagLayer modulates the lag-structured matrix with horizon-wise and history-wise scale coefficients. LagNet stacks these operators with pointwise nonlinearities and multiple hidden feature maps, adding capacity while retaining lag structure in every temporal mapping. Our contributions are summarized as follows: • We identify recurring lag-aligned structure in historyto-future sensitivity matrices across diverse forecasting architectures without explicit lag sharing. This evidence motivates a relative-lag sharing prior for direct multi-step forecasting. • We develop a model family around this prior. LagLayer combines a lag-structured matrix with horizon-wise and history-wise adaptation, while LagNet extends the operator through nonlinear depth and multiple hidden feature maps. • We demonstrate competitive performance against strong baselines on eight standard LTSF benchmarks. Ablations isolate the effects of adaptive scaling and nonlinear capacity, while matrix diagnostics and limited-data experiments provide complementary evidence for relative-lag sharing. Matched replacement experiments with existing architectures further examine operator transferability.

and variate tokens. TimesNet (Wu et al. 2023) converts discovered periods into two-dimensional representations, and convolutional models capture temporal patterns with hierarchical or large-receptive-field designs (Liu et al. 2022; Luo and Wang 2024). Direct forecasters provide a simpler alternative. N-BEATS and NHITS construct long-horizon predictions through deep basis expansion and hierarchical interpolation (Oreshkin et al. 2020; Challu et al. 2023), while DLinear (Zeng et al. 2023) demonstrates the competitiveness of a single linear temporal mapping. TiDE (Das et al. 2023), TSMixer (Chen et al. 2023; Ekambaram et al. 2023), and TimeMixer (Wang et al. 2024) develop MLP-based encoders or mixing architectures. Recent lightweight models further reduce cost through simplified mixing, sparse or frequency-domain parameterization, and factorized mappings (Chen et al. 2025; Lin et al. 2024; Xu, Zeng, and Xu 2024; Ma, Luo, and Sha 2026). These works mainly refine input representations, feature mixing, or compact parameterizations, while the structure of the direct history-to-future mapping remains less explored. We encode the recurring relative-lag structure observed in this operator’s sensitivity matrices through LagLayer and its nonlinear extension, LagNet, reducing the parameters required by each temporal mapping.

Structured Operators and Toeplitz Constraints Structured operators have long supported efficient sequence modeling. Temporal convolution reuses weights across relative local positions (Bai, Kolter, and Koltun 2018), while structured state-space models admit convolutional representations for long-range dependencies (Gu, Goel, and Ré 2022). Toeplitz matrices provide the corresponding matrix form for linear time-invariant filtering (Gray 2006). Within this broader family, our setting differs from local temporal convolution: rather than sliding a kernel within a sequence, our lag-structured matrix directly maps a history window to a distinct forecast horizon. Most closely related to our work, the Toeplitz Neural Network (TNN) employs relative-position-encoded square Toeplitz operators for within-sequence token mixing, focusing on computational efficiency and length extrapolation in general sequence modeling (Qin et al. 2023). We adopt a related sharing principle in a rectangular Toeplitz matrix, but derive it from lag-aligned patterns observed in longterm forecasters and use it to directly constrain the historyto-future mapping. TNN therefore studies positional mixing within a sequence, whereas we study the forecasting operator between history and future.

Related Work Representation and Architecture Design for Long-Term Forecasting Long-term forecasting models have primarily advanced through new representations and architectures. Transformerbased methods improve long-range modeling through sparse attention, decomposition, and frequency-domain processing (Zhou et al. 2021; Wu et al. 2021; Zhou et al. 2022), while PatchTST (Nie et al. 2023) and iTransformer (Liu et al. 2024) reorganize temporal and variate dimensions through patch

Method Consider a multivariate time series with C channels. For each channel c ∈ {1, . . . , C}, the historical sequence and future target are (c)

(c)

(c)

(c)

(c)

xt−L+1:t = [xt−L+1 , . . . , xt ]⊤ ∈ RL , (c)

xt+1:t+H = [xt+1 , . . . , xt+H ]⊤ ∈ RH , (c)

where xt ∈ R is the observation of channel c at time t, L is the history length, and H is the forecast horizon.

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

This parameterization uses L + H − 1 lag coefficients instead of the HL independently learned elements of W .

Lag-Structured Forecasting Operator

LagLayer with Adaptive Scaling

We derive the temporal operator for an arbitrary channel c and omit the superscript (c) for readability. Parameter sharing across channels is specified separately for each model configuration. A dense forecasting operator uses a matrix W ∈ RH×L , where wh,ℓ = [W ]h,ℓ is its (h, ℓ)-th element. Given the historical sequence, the dense prediction is

Exact lag sharing assigns one coefficient to all connections on the same diagonal. The sensitivity matrices in Figure 1 retain a diagonal organization but also vary across forecast steps and historical positions. Several panels also contain vertically concentrated regions of high sensitivity, including pronounced responses near the newest observations. This variation indicates that relative lag alone does not fully describe the forecasting matrix. To capture this variation without discarding the relative lag structure, LagLayer introduces a horizon-wise scale vector α ∈ RH and a history-wise scale vector β ∈ RL . For a connection from historical position ℓ to forecast step h, the scaled coefficient is

b t+1:t+H = W xt−L+1:t + b. x H

Here, b ∈ R , and all HL elements of W are learned independently. The element wh,ℓ connects the historical value xt−L+ℓ to the prediction x̂t+h , with relative lag (t + h) − (t − L + ℓ) = L + h − ℓ. A dense matrix learns these connections independently, even when they have the same relative lag. The recurring diagonal patterns in Figure 1 suggest that connections with the same lag can share a common base coefficient. We can therefore construct a lag-structured matrix. Let k = [k1 , . . . , kL+H−1 ]⊤ ∈ RL+H−1 contain one coefficient for each possible lag, where k1 corresponds to the shortest lag and kL+H−1 to the longest. We define wh,ℓ = kL+h−ℓ .

(1)

Let TH,L denote this construction. The resulting lagstructured matrix is W lag = TH,L (k)   kL kL−1 · · · k1 kL · · · k2   kL+1 (2) = . . . ..  ..   .. .. . . kL+H−1

kL+H−2

···

kH

Because its entries depend only on h − ℓ, W lag is a rectangular Toeplitz matrix. The corresponding unscaled operator applies this matrix to the historical sequence: b t+1:t+H = W lag xt−L+1:t + b. x

wh,ℓ = αh kL+h−ℓ wh,ℓ = βℓ kL+h−ℓ wh,ℓ = αh βℓ kL+h−ℓ

(horizon-wise), (history-wise), (both).

Here, αh modulates the corresponding forecast row, while βℓ modulates the corresponding historical column. The lag coefficient kL+h−ℓ retains the shared temporal backbone. Let D α = diag(α) ∈ RH×H and D β = diag(β) ∈ L×L R denote the corresponding diagonal matrices. When both scale vectors are enabled, the effective LagLayer matrix is W layer = D α TH,L (k)D β  β 1 kL  β1 kL+1 = Dα  ..  .

β2 kL−1 β2 kL .. .

··· ··· .. .

 βL k1 βL k2  . ..  . 

β1 kL+H−1

β2 kL+H−2

···

βL kH

(3) Unlike its Toeplitz backbone, W layer is generally not Toeplitz because entries on the same diagonal can differ through αh βℓ . The shared structure therefore applies to the

base lag coefficients, rather than requiring identical effective weights along each diagonal. The corresponding prediction is b t+1:t+H = W layer xt−L+1:t + b x = D α TH,L (k)D β xt−L+1:t + b.

(4)

The scale coefficients are initialized to one, so LagLayer initially recovers its unscaled lag-structured backbone. They subsequently provide controlled flexibility along the forecasthorizon and historical-position axes without removing that backbone. Throughout the main experiments, LagLayer denotes the complete single-layer operator with both scale vectors; configurations with either or both scales removed are used only for ablation. Figure 2 summarizes the difference between the dense, lag-structured, and adaptively scaled parameterizations.

Deep Lag-Structured Networks

1 ≤ r ≤ R, Mr = M, 1 ≤ r < R.

The input and output are treated as single feature maps. Each hidden layer contains M latent maps whose temporal axis has length H; these latent maps are distinct from the C observed channels. This choice aligns the hidden temporal positions with the forecast steps; the hidden maps are latent representations rather than individually supervised forecasts. The extension to a general hidden length H ′ is given in the supplementary material. For layer r, each input-output map pair has its own lag kernel and scale vectors:   (r) (r) (r) Tr ×Tr−1 W (r) . m,n = D α,m,n TTr ,Tr−1 km,n D β,m,n ∈ R (5) Here, m = 1, . . . , Mr and n = 1, . . . , Mr−1 . The lag kernel has length Tr + Tr−1 − 1. The corresponding rowand column-wise diagonal scale matrices satisfy D (r) α,m,n ∈ (r)

RTr ×Tr and D β,m,n ∈ RTr−1 ×Tr−1 , respectively. (r) Tr Let z m ∈ RTr denote map m at layer r, and let b(r) m ∈R

denote its bias. The network is computed as   Mr−1 X (r−1)  , z (r) W (r) + b(r) m = σr m,n z n m n=1

1 ≤ r ≤ R, (0)

(R)

Element form

Parameters

Linear wh,ℓ HL Lag-structured matrix kL+h−ℓ L+H −1 Horizon scale only αh kL+h−ℓ L + 2H − 1 History scale only βℓ kL+h−ℓ 2L + H − 1 LagLayer αh βℓ kL+h−ℓ 2L + 2H − 1 Table 1: Parameter complexity of forecasting operators, with bias terms omitted for clarity. M 2 pair-specific LagLayer operators, and the final layer returns one forecast. Depth composes structured transformations with nonlinearities, while the feature-map dimension allows the network to represent multiple lag-response patterns without changing the hidden temporal axis.

Parameter Complexity

A single LagLayer is linear. LagNet introduces nonlinear capacity by stacking R ≥ 2 LagLayer modules with pointwise activations, as shown in Figure 3. We use a unified notation for all layers. Let Tr and Mr denote the temporal length and number of feature maps at layer r, respectively: T0 = L, Tr = H, M0 = 1, MR = 1,

Operator

(6)

1 ≤ m ≤ Mr ,

b t+1:t+H . The hiddenwith z 1 = xt−L+1:t and z 1 = x layer functions σr , r < R, are pointwise activations, while σR is the identity map. The first layer maps the history to M future-aligned latent maps. Each intermediate layer mixes these maps through

Table 1 summarizes the parameter complexity of singlechannel forecasting operators, excluding biases. Compared with the HL independent parameters of a dense operator, LagLayer uses L+H −1 lag coefficients and H +L horizonand history-wise scales, totaling 2L + 2H − 1 parameters without recovering HL independent weights. The parameter reduction also applies to hidden-to-hidden transformations. A shape-matched dense MLP flattens an H ×M representation to width HM and requires O(M 2 H 2 ) parameters per hidden layer. The corresponding LagNet layer contains M 2 pair-specific LagLayer operators, each with O(H) kernel and scale parameters. Its total parameter count is therefore O(M 2 H). Lag structuring reduces the parameter dependence on the temporal length from quadratic to linear while retaining all-to-all mixing between feature maps.

Experiments Experimental Setup Datasets. We evaluate on eight standard LTSF benchmarks: ETTh1, ETTh2, ETTm1, ETTm2, Electricity, SolarEnergy, Traffic, and Weather. We follow their public chronological splits and established preprocessing (Zeng et al. 2023; Wu et al. 2023; Nie et al. 2023; Liu et al. 2024; Wang et al. 2024). Dataset statistics and exact split sizes are provided in the supplementary material. Baselines and Experimental Settings. We compare with DLinear (Zeng et al. 2023), PatchTST (Nie et al. 2023), iTransformer (Liu et al. 2024), TimeMixer (Wang et al. 2024), and TimeKAN (Huang et al. 2025). They cover linear, Transformer, multiscale MLP, and frequency-decomposition architectures. All experiments use L = 96 and H ∈ {96, 192, 336, 720} under a unified setting, with results averaged over five random seeds. We report MSE as the primary metric and MAE as a complementary metric; Params denotes the total number of trainable parameters. Proposed Models and Implementation. We evaluate LagLayer and LagNet, together with DLinear, TimeMixer, and TimeKAN transfer variants. Each variant replaces only

Baselines Dataset

DLinear (2023)

PatchTST (2023)

Proposed

iTransformer TimeMixer (2024) (2024)

TimeKAN (2025)

LagLayer (ours)

LagNet (ours)

Lag-Time Lag-Time Mixer (ours) 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 Weather 0.267 0.317 0.265 0.286 0.258 0.278 0.246 0.276 0.244 0.274 0.246 0.272 0.249 0.278 0.245 0.276 0.248 0.280 Solar 0.326 0.397 0.287 0.333 0.233 0.262 0.230 0.298 0.282 0.315 0.368 0.359 0.255 0.282 0.233 0.303 0.311 0.339 Electricity 0.210 0.296 0.216 0.318 0.178 0.270 0.183 0.273 0.199 0.289 0.219 0.295 0.191 0.278 0.186 0.278 0.203 0.292 Traffic 0.625 0.384 0.529 0.341 0.428 0.282 0.491 0.303 0.580 0.372 0.632 0.378 0.458 0.293 0.499 0.306 0.595 0.374

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. TimeMixer → Lag-TimeMixer

TimeKAN → Lag-TimeKAN

∆ MSE (%)

DLinear → DLagLayer Params

∆ MSE (%)

Params

∆ MSE (%)

Params

ETT (Avg.) Weather Solar Electricity Traffic

−1.3 +2.2 +1.5 +1.8 +0.9

65.2K → 2.4K 65.2K → 2.4K 65.2K → 2.4K 65.2K → 2.4K 65.2K → 2.4K

−0.6 −0.1 +1.0 +1.2 +1.8

120.1K → 62.7K 119.7K → 62.3K 2.37M → 2.25M 122.1K → 64.7K 132.8K → 75.4K

+1.7 +1.8 +10.4 +2.1 +2.7

41.5K → 10.1K 41.2K → 9.8K 45.1K → 13.8K 46.6K → 15.2K 50.9K → 19.6K

Average

+0.1

−96.3%

+0.2

−41.8%

+3.0

−72.8%

Dataset

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. the linear temporal forecasting mapping with LagLayer while retaining all other components and training settings, without specific tuning. We use PyTorch, Adam, and MSE loss; adaptive scales are initialized to one, and testing uses the checkpoint with the lowest validation MSE. The supplementary material provides the complete settings and results.

Main Results Table 2 compares the four proposed models with five baselines across eight datasets. At least one proposed model ranks first in MSE or MAE on five datasets and among the top two on seven. LagLayer achieves the lowest MAE on ETTh2 and Weather, while the transferred models take first place in at least one metric on every ETT dataset. Electricity is the only dataset on which no proposed model ranks among the top two. The largest gains from nonlinear composition occur on Solar, Electricity, and Traffic. Relative to LagLayer, LagNet reduces average MSE by 30.8%, 13.0%, and 27.6% on these datasets, respectively. It also ranks second in both MSE and MAE on Traffic. Thus, added depth can be particularly beneficial for high-dimensional multivariate datasets. Table 3 isolates the effect of replacing each backbone’s linear temporal forecasting mapping with LagLayer while leaving the other components unchanged. This replacement reduces parameter counts by 96.3% for DLinear, 41.8% for TimeMixer, and 72.8% for TimeKAN. The corresponding

increases in average MSE are 0.1%, 0.2%, and 3.0%. The largest accuracy cost occurs for TimeKAN on Solar, where MSE increases by 10.4%. The replacement therefore preserves average accuracy closely for DLinear and TimeMixer, but less consistently for TimeKAN. Results across these three models, which employ distinct architectures and mechanisms, further indicate that relative-lag sharing is a transferable operator prior rather than one tied to a particular forecasting backbone, substantially reducing parameter counts while retaining broadly comparable aggregate forecasting accuracy.

Sample Efficiency Analysis Relative-lag sharing provides an effective structural prior that reduces the number of forecasting coefficients that must be estimated independently and may therefore improve sample efficiency. We evaluate Linear, LagLayer, and LagNet using 1%, 5%, 20%, 50%, or 100% of the original training data, while keeping the validation and test sets and all other training settings unchanged. Table 4 reports the average MSE over the four prediction lengths. The main table aggregates the four ETT datasets, with complete results provided in the supplementary material. With 1% of the training data, LagNet obtains lower MSE than Linear on all eight datasets. Across all training fractions, it achieves or ties for the lowest MSE in 21 of the 25 settings. The LagLayer also improves or matches Linear on

(c) Traffic H=720

0.12 0.06 0.00 -0.06 -0.12 0.2 0.1 0.0 -0.1 -0.2 0.2 0.1 0.0 -0.1 -0.2

80 0

100 200 300 0 200 400 600

Signed W

(v) LagLayer

40 Forecast horizon

(b) Electricity H=336

(iii) Horizon Only (iv) History Only

Signed W

(a) ETTh1 H=96

(ii) Unscaled

Signed W

(i) Linear

0

0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 Historical position Historical position Historical position Historical position Historical position

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.

Linear 0.432 0.399 0.385 0.383 0.381 0.396 ETT (Avg.) 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

Solar

Linear 0.431 0.382 0.363 0.358 0.356 0.378 LagLayer 0.540 0.444 0.389 0.373 0.368 0.423 LagNet 0.309 0.289 0.269 0.261 0.246 0.275

Electricity

Linear 0.309 0.238 0.218 0.215 0.215 0.239 LagLayer 0.268 0.230 0.221 0.219 0.219 0.232 LagNet 0.235 0.209 0.199 0.193 0.191 0.206

Traffic

Linear 0.949 0.701 0.632 0.625 0.624 0.706 LagLayer 0.790 0.680 0.647 0.635 0.632 0.677 LagNet 0.606 0.505 0.477 0.465 0.457 0.502

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. four of the five groups, supporting the effectiveness of its relative-lag structural constraint. It is less effective on Solar, where the additional nonlinear capacity of LagNet produces a clear improvement. Overall, the proposed models retain their effectiveness as the amount of training data decreases, with LagNet showing the strongest sample efficiency on the larger multivariate datasets.

Forecasting Operator Analysis Forecasting matrix visualization. Figure 4 compares the signed effective matrices learned by Linear and four lagstructured variants. The Linear matrix on ETTh1 contains prominent repeated diagonal ridges, while Electricity and Traffic additionally exhibit variation across historical posi-

tions and forecast steps, including vertically concentrated responses at selected Electricity history positions and horizontal bands in the longer-horizon cases. These patterns suggest that relative lag provides a common organization, but exact sharing along each diagonal is too restrictive. The remaining columns show how the proposed parameterization represents this structure. The unscaled variant enforces a shared coefficient along each diagonal. The horizon-only and history-only variants then introduce variation across matrix rows and columns, respectively. The complete LagLayer combines both forms of adaptation while retaining the diagonal lag structure. Rather than reproducing every Linear coefficient independently, it represents the observed matrices through a shared lag backbone and two lower-dimensional scale vectors. Complete comparisons are provided in the supplementary material. Forecast-step errors and adaptive scaling. Standard LTSF evaluation typically reports MSE and MAE aggregated over all forecast steps. Figure 5(a) reports the test MSE at each of the 720 forecast steps. All three models exhibit recurring error patterns at approximately 168-step intervals, corresponding to one week in the hourly Traffic data. The periodic residuals of PatchTST and TimeMixer suggest that these temporal regularities are not fully captured. LagNet obtains the lowest MSE at almost all steps, consistent with its relative-lag parameterization better representing recurring dependencies. The horizon-wise profiles in panel (b) show a similar piecewise-periodic pattern. Their main transitions align with shifts between intervals of higher and lower forecast error, while larger relative scales generally occur in regions where the models consistently produce higher MSE. This correspondence suggests that horizon-wise scaling adapts the operator rows to forecast-step difficulty. The history-wise profiles in panel (c) cannot be compared pointwise with forecast-step MSE because they operate

(a) Horizon-wise Test MSE Dataset

MSE

0.60 0.50 0.40 0.30 0

Scale

2.00

PatchTST TimeMixer LagNet 100 200 300 400 500 600 700 (b) Horizon-wise Scale Coefficients Horizon Only Both

ETT (Avg.) Weather Solar Electricity Traffic

Horizon History Only Only LagLayer LagNet

0.390 0.284 0.542 0.229 0.681

0.389 0.279 0.440 0.223 0.642

0.386 0.274 0.457 0.224 0.652

0.385 0.273 0.371 0.219 0.634

0.381 0.267 0.256 0.191 0.460

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

1.50 1.00

(a) Depth (M = 8)

0.50

Dataset

0 2.00 Scale

Unscaled

100

200

300 400 500 600 Forecast horizon (c) History-wise Scale Coefficients History Only Both

700

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 (b) Width (R = 3)

1.50 Dataset

1.00 0.50 0

20

40 60 Historical position

80

100

M = 1 M = 2 M = 4 M = 8 M = 16 M = 32

Solar 0.318 Electricity 0.221 Traffic 0.576

0.287 0.211 0.522

0.279 0.204 0.494

0.266 0.201 0.481

0.260 0.193 0.470

0.267 0.192 0.466

Figure 5: Forecast-step MSE and learned adaptive scales on Traffic with H = 720. Errors are averaged over test windows and channels.

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.

on historical positions. Both variants nevertheless emphasize the earliest and most recent parts of the input window while assigning smaller scales to many intermediate positions. Thus, α adapts the operator across recurring forecast regimes, whereas β captures nonuniform historical-position importance. Similar periodic error variation and structured scale responses recur across multiple datasets. Together, they add two-axis adaptation while retaining the relative-lag backbone.

improvements reflect the contribution of additional hidden feature maps rather than greater depth. The gains diminish at larger widths, with Solar slightly degrading from M = 16 to M = 32, suggesting that the benefit of additional capacity eventually saturates. Overall, depth enables repeated nonlinear composition, while width allows the model to represent multiple lag response patterns in parallel.

Ablation and Capacity Analysis

This paper studies the structure of history-to-future forecasting operators. LagLayer builds on a lag-structured matrix that shares parameters by relative lag and adapts along the forecast and history axes. LagNet extends this operator through nonlinear depth and multiple hidden feature maps. Matched operator-replacement studies on three established forecasters further demonstrate that relative-lag sharing transfers across decomposition-based linear, multiscale mixing, and frequency-decomposition architectures. Across standard benchmarks, the proposed models achieve competitive accuracy; ablations, matrix diagnostics, and limited-data experiments further support the value of relative-lag sharing. These results establish forecasting-operator structure as a useful design axis for long-term time series forecasting.

Component ablation. Table 5 isolates lag sharing, adaptive scaling, and nonlinear composition. The complete LagLayer matches or improves its unscaled counterpart on all eight datasets and improves it on seven; the one-sided variants further show that both horizon-wise and historywise adaptation are useful. Adaptive scaling therefore relaxes exact sharing while preserving the structured backbone. The additional gains from LagNet, especially on the large datasets, support nonlinear composition and multiple hidden lag-response maps. Depth and width. Table 6 examines the effects of LagNet depth and width on Solar, Electricity, and Traffic. The depth sweep fixes M = 8, while the width sweep fixes R = 3. Increasing depth generally lowers MSE, although the gains level off after four layers on Solar and Electricity. At fixed depth, increasing the width from M = 1 to its best setting reduces MSE by 13.0% to 19.1% across the three datasets. Because M = 1 already retains the nonlinear stack, these

Conclusion

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Supplementary Material for LagNet: Lag-Structured Operators for Long-Term Time Series Forecasting Zhenan Yu1 , Guangxin Jiang1 , Jiuyun Jiang1 , Jin Yang1∗ 1 Harbin Institute of Technology {25S010055@stu.hit.edu.cn, gxjiang@hit.edu.cn, jiuyunjiang@stu.hit.edu.cn, j.yang@hit.edu.cn}

A

Additional Method Details

This document provides the element-level derivation of the proposed operators, complete reproducibility details, horizon-wise forecasting results, full ablations, and additional structural diagnostics. Section A expands the singlelayer operator, Section B details its deep extension, and Section C specifies the experimental protocol used by the remaining sections.

A.1

Indexing Conventions

The main paper uses one chronological indexing convention throughout. The historical input is indexed by ℓ = 1, . . . , L, where ℓ = 1 denotes the oldest observation and ℓ = L denotes the most recent observation. The forecast steps are indexed by h = 1, . . . , H. The relative temporal lag between these positions is therefore L + h − ℓ. Accordingly, k1 represents the nearest history-to-future relation, while kL+H−1 represents the farthest relation covered by the two windows.

A.2

Full Matrix Form

For history length L and forecast horizon H, the dense forecasting matrix is   w1,1 w1,2 · · · w1,L  w2,1 w2,2 · · · w2,L  W dense =  , (1) .. ..  ..  ... . . .  wH,1

wH,2

···

wH,L

where all entries are independently learned. The lagstructured matrix replaces these H × L independently parameterized entries with a lag kernel:   kL kL−1 · · · k1 kL · · · k2   kL+1 W lag =  . (2) .. .. ..  ..  . . . .  kL+H−1

kL+H−2

···

kH

Each row and each column of W lag is a contiguous segment of the same lag kernel, with rows traversing the segment in reverse order. Elements on the same top-left-to-bottom-right diagonal share one parameter. The equality pattern is the key structural constraint. It should not be interpreted as a rank ∗

Corresponding author.

constraint. Rectangular Toeplitz matrices can be full rank, while low-rank factorization imposes a different assumption by forcing the matrix to pass through a low-dimensional latent factorization.

A.3

Adaptive Scaling

Horizon-wise and history-wise scaling can be expressed compactly as W α = D α W lag , (3) W β = W lag D β , where D α = diag(α) ∈ RH×H and D β = diag(β) ∈ RL×L are diagonal matrices generated from the horizonwise and history-wise scale vectors. When both are enabled, W layer = D α W lag D β .

(4)

In all cases, the lag kernel remains the backbone of the operator. The scale vectors only modulate row-wise or columnwise magnitude. The complete single-layer operator with both scale vectors is denoted LagLayer; versions with either or both vectors removed are used only as ablations.

B

Deep LagNet Construction

A one-layer LagLayer is a structured linear model. When the number of layers is one, no hidden feature maps or interlayer activations are introduced; the model degenerates to a lag-structured linear forecaster. The nonlinear LagNet stacks R ≥ 2 LagLayer modules. The main paper presents the inputto-hidden, hidden-map mixing, and hidden-to-output matrices together with a compact layer-wise recursion, with the reported hidden temporal length fixed at H. Here we introduce layer-wise dimensions only to give a unified element-level expansion and cover the implementation’s optional hidden temporal length H ′ . Let Tr be the output temporal length of layer r and Mr its number of output feature maps, with T0 = L, TR = H, M0 = 1, MR = 1,

Tr = H ′ , Mr = M,

1 ≤ r < R, 1 ≤ r < R.

(5)

The reported model sets H ′ = H. Thus, layer r maps Mr−1 input feature maps of temporal length Tr−1 to Mr output feature maps of temporal length Tr . For the reported setting H ′ = H, this factorization aligns the ordered hidden temporal axis with the H forecast steps while using the separate

feature-map axis to represent multiple learned lag-response patterns. For output feature-map index m, input feature-map index n, output temporal position q, and input temporal position p, the corresponding scalar element is (r)

(r) (r) (r) wm,n,q,p = αm,n,q βm,n,p km,n,Tr−1 +q−p .

(6)

Here m = 1, . . . , Mr , n = 1, . . . , Mr−1 , q = 1, . . . , Tr , and p = 1, . . . , Tr−1 . Accordingly, the pair-specific lag kernel has length Tr−1 + Tr − 1, while the row-wise and columnwise scale vectors have lengths Tr and Tr−1 , respectively. The full element-level recurrence is   Mr−1 Tr−1 X X (r) (r) (r−1) . zm,q = σr  wm,n,q,p zn,p + b(r) (7) m,q n=1 p=1 (0)

The boundary conditions are z 1 = xt−L+1:t and (R) b t+1:t+H = z 1 . The hidden-layer functions σr are pointx wise activations, while σR is the identity map because no activation follows the final layer. Equations (6) and (7) are the element-level expansion of the compact matrix form in the main paper and match the pair-specific kernel, rowwise scale, column-wise scale, and bias parameters used in the implementation. For the reported setting H ′ = H, the first-layer row-wise and column-wise scales are precisely the horizon-wise and history-wise scale coefficients defined for LagLayer. For subsequent layers, both vectors act on the rows and columns of the future-aligned hidden maps, respectively. Choosing H ′ ̸= H changes only the temporal dimensions of the intermediate maps; the kernel construction and layer-wise recurrence remain unchanged. The organization of feature maps is analogous to a multi-channel convolutional layer because every output map aggregates a separate kernel response from each input map. Repeated composition of linear maps and activations gives the network its MLP depth. We avoid adding additional branches, gates, attention modules, or decomposition blocks so that the model remains focused on the lag-structured operator.

C C.1

Complete Experimental Settings

Datasets and Chronological Splits

We use eight public long-term forecasting benchmarks and their established chronological protocols (Wu et al. 2023; Liu et al. 2024). Throughout the supplementary material, Solar denotes the Solar-Energy dataset named in the main paper. The ETT datasets use the standard 12/4/4-month train/validation/test partition. The remaining datasets use chronological 70%/10%/20% partitions. Table 1 reports the exact core split sizes used by the data loader. Validation and test arrays prepend the final L observations of the preceding split as input context; targets remain entirely inside their respective chronological partition. All datasets use look-back length L = 96 and prediction lengths H ∈ {96, 192, 336, 720}.

C.2

Preprocessing, Evaluation, and Reporting

Feature-wise standardization statistics are fitted only on the training partition and then applied to validation and test data.

The standalone Linear, LagLayer, and LagNet models additionally use RevIN without affine parameters. Training windows are shuffled, whereas validation and test windows retain chronological order. For a test set containing Ntest windows and C variables, let ŷn,c,h and yn,c,h denote the standardized prediction and target, respectively, for window n, variable c, and forecast step h. The reported metrics in standardized space are N

MSE =

C

H

test X X X 1 (ŷn,c,h − yn,c,h )2 , Ntest CH n=1 c=1

h=1

N C X H test X X

1 MAE = Ntest CH n=1 c=1

(8)

|ŷn,c,h − yn,c,h |.

h=1

Following standard LTSF evaluation protocols, we report both metrics because MSE penalizes large deviations more strongly, whereas MAE provides a complementary measure of absolute error. The checkpoint with the lowest validation MSE is evaluated once on the test partition. Unless a diagnostic experiment explicitly states otherwise, reported reproduced and proposed results use seeds {2027, 2026, 2025, 2024, 2023}. Tables report the arithmetic mean and sample standard deviation over P the five runs. Trainable parameter counts are computed as θ: requires_grad |θ| for the instantiated model, including normalization and bias parameters.

C.3

Computing Environment

Independent runs were distributed across two Windows workstations for parallel execution. Workstation A ran Windows 10 Enterprise LTSC (build 19044) with two Intel Xeon Gold 6430 CPUs, 255.4 GB RAM, and four NVIDIA GeForce RTX 4090 GPUs (24 GB each; driver 576.02). Its software environment used Python 3.11.5, PyTorch 2.2.0 with CUDA 12.1 and cuDNN 8.8.1, NumPy 1.26.4, pandas 2.2.2, and Matplotlib 3.8.4. Workstation B ran Windows 10 (build 19045) with an Intel Xeon Gold 6248R CPU, 255.6 GB RAM, and one NVIDIA RTX A6000 GPU (48 GB; driver 595.95). It used Python 3.11.11, PyTorch 2.1.0 with CUDA 12.1 and cuDNN 8.8.1, NumPy 1.26.4, pandas 2.2.3, and Matplotlib 3.10.9. Environment manifests for both workstations are included in the code supplement.

C.4

Standalone Lag-Structured Models

Table 2 gives the exact architecture, parameter sharing mode, and batch size used for the standalone LagLayer and LagNet columns. A shared operator applies one channel-independent temporal model to every variable; a per-variable setting learns a separate temporal model for each variable. Both proposed models use Adam with learning rate 10−3 , at most 20 epochs, early-stopping patience 5, gradient clipping at 1.0, and automatic mixed precision. The global and lag-kernel weight decays are 10−5 ; scale coefficients and biases are not decayed. Adaptive scales are initialized to one, biases are enabled, and hidden temporal length equals H. During development of the standalone lag-structured models, batch size was the only tuned optimization hyperparameter. We tried

Dataset ETTh1, ETTh2 ETTm1, ETTm2 Weather Solar Electricity Traffic

Variates Frequency Total points 7 7 21 137 321 862

1 hour 15 minutes 10 minutes 10 minutes 1 hour 1 hour

17,420 69,680 52,696 52,560 26,304 17,544

Train

Validation

Test

8,640 34,560 36,887 36,792 18,412 12,280

2,880 11,520 5,270 5,256 2,632 1,756

2,880 11,520 10,539 10,512 5,260 3,508

Table 1: Dataset statistics and chronological core split sizes. The ETT files contain observations beyond the standard 12/4/4month benchmark interval; these extra observations are not used. batch sizes {16, 64, 256} and selected the dataset-specific value by validation MSE; all other settings were fixed as reported above.

C.5

Operator-Transfer Configurations

DLinear, TimeMixer, and TimeKAN, together with their lagstructured variants, follow the model architectures and implementation details in their official papers and released repositories. Their configurations are summarized in Tables 3, 4, and 5. For a controlled comparison, each reproduced model and its lag-structured variant use identical datasetspecific training settings. Within each pair, the data pipeline, optimization settings, random seed, and all components outside the replaced temporal mapping remain identical. DLagLayer preserves the moving-average decomposition and branch summation, replacing only the independent seasonal and trend Linear(L, H) projections with dimensionmatched LagLayer operators. Lag-TimeMixer replaces each FMM temporal predictor Linear(Ti , H) with a dimensionmatched LagLayer. Lag-TimeKAN preserves TimeKAN’s frequency decomposition, M-KAN blocks, depthwise convolutions, normalization, and output projection, replacing only the final Linear(L, H) temporal predictor with a dimensionmatched LagLayer using both adaptive scales.

D

Complete Forecasting Results

The main paper reports four-horizon average MSE and MAE together with a compact paired operator-replacement comparison. The tables below provide the corresponding horizon-wise metrics, seed statistics, and parameter audits. All reproduced and proposed results are averaged over five random seeds.

D.1

Public Baselines and Parameter Audit

Table 6 gives the full public baseline results. The PatchTST column is copied from the unified L = 96 appendix of TimeMixer (Wang et al. 2024), while the iTransformer column is copied from the official L = 96 full forecasting table of iTransformer (Liu et al. 2024). For TimeMixer, we reproduce all eight datasets under the main-experiment configurations reported in Table 7 of the official paper, using the released implementation to determine model and optimization options not specified by that table, and report the mean over the same five seeds as our methods. This additional reproduction is used because the published results do not provide statistics under the same protocol with five

random seeds. For TimeKAN, we preserve the official model architecture and horizon-specific configurations while applying the unified dataset-specific training protocol in Table 5 to the same five random seeds. Table 7 reports the audited public-baseline parameter counts used in the main table.

D.2

Operator-Transfer Results

Table 8 reports the paired DLinear and DLagLayer results. Both variants follow the settings in Table 3; DLagLayer changes only the seasonal and trend temporal projections. Table 9 reports the paired TimeMixer and Lag-TimeMixer seed statistics. Lag-TimeMixer preserves the complete TimeMixer PDM feature extractor and multiscale aggregation and follows the same paired training protocol, replacing only each FMM temporal predictor Linear(Ti , H) with a LagLayer operator of the same input and output lengths. Table 10 reports the corresponding TimeKAN and Lag-TimeKAN results and run counts. Lag-TimeKAN preserves TimeKAN’s complete frequency-decomposition feature extractor and replaces only its final linear temporal predictor with the complete LagLayer operator. For the TimeMixer and TimeKAN paired studies, all 32 dataset–horizon settings contain complete records for the same five random seeds in each model. The paired operator-replacement table in the main paper uses fourhorizon averages. The DLinear–DLagLayer, TimeMixer– Lag-TimeMixer, and TimeKAN–Lag-TimeKAN blocks use the corresponding paired results reported below. MSE change is 100(MSElag −MSEbase )/MSEbase . The body of that table reports the four-horizon average trainable parameter counts before and after replacement, while its final row reports the mean relative changes in MSE and parameter count across the underlying datasets. The final average treats each underlying dataset equally; in particular, the four ETT datasets contribute four separate terms rather than a single aggregated term. These tables are separated for readability and to make the source of each number explicit.

D.3

Standalone Lag-Structured Results

Table 11 gives the complete LagLayer and LagNet batch results. Both columns use the final selected lag-structured runs with grouped lag regularization; their dataset-specific batch sizes are listed in Table 2.

E

Complete Ablation Results

This section provides the complete ablation results corresponding to the compact ablation table in the main paper.

Dataset ETTh1 ETTh2 ETTm1 ETTm2 Weather Solar Electricity Traffic

Temporal operator Batch size LagLayer (R, M ) LagNet (R, M ) Adaptive scales Shared Shared Per-variable Per-variable Per-variable Shared Shared Shared

256 64 256 64 16 64 64 64

(1, –) (1, –) (1, –) (1, –) (1, –) (1, –) (1, –) (1, –)

(2, 4) (2, 4) (2, 4) (2, 4) (2, 4) (4, 12) (4, 12) (4, 12)

Both Both Both Both Both Both Both Both

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

η96

η192

η336

η720

ETTh1 ETTh2 ETTm1 ETTm2 Weather Solar Electricity Traffic

32 32 8 32 16 16 16 16

5×10−3 5×10−2 1×10−4 1×10−3 1×10−4 1×10−3 1×10−3 5×10−2

5×10−3 5×10−2 1×10−4 1×10−3 1×10−4 1×10−3 1×10−3 5×10−2

5×10−3 5×10−2 1×10−4 1×10−2 1×10−4 1×10−3 1×10−3 5×10−2

5×10−3 5×10−2 1×10−4 1×10−2 1×10−4 1×10−3 1×10−3 5×10−2

Epochs Patience Individual 10 10 10 10 10 10 10 10

3 3 3 3 3 3 3 3

No No No No No No No No

Table 3: DLinear and DLagLayer configurations. Here ηH 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 datasetspecific 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. The single-layer variants first isolate relative-lag sharing and each scale mechanism. The deep entry is a separate capacity diagnostic in which both scale vectors are disabled, so that its difference from the unscaled single-layer backbone isolates nonlinear composition and multiple feature maps. This diagnostic setting should be distinguished from the complete LagNet used in the main forecasting results, where every layer includes adaptive scaling. All variants in this diagnostic ablation use the fixed diagnostic seed 2027. All entries use a shared channel-independent operator, batch size 64, Adam with learning rate 10−3 , at most 20 epochs, early-stopping patience 5, and the preprocessing and evaluation protocol in Section C. The main paper reports dataset-wise averages for compactness, while Table 13 reports the full horizonwise MSE and MAE. The purpose of this ablation is not to tune a different model for each dataset, but to show how each structural component changes the behavior of the same lag-structured forecasting family.

F

Depth and Width Analysis

This section reports the full depth/width sweep for the deep LagNet architecture on Solar, Electricity, and Traffic. The sweep and its associated qualitative forecasting case use the fixed diagnostic seed 2027. The depth sweep fixes the hidden width to M = 8 and varies the number of lagstructured layers. The width sweep fixes the depth to R = 3 and varies the hidden width, where M denotes the number of hidden feature maps and R denotes the number of

lag-structured layers. Both scale mechanisms are disabled throughout this sweep, while batch size 64 and the remaining standalone optimization settings are fixed. This protocol isolates whether the lag-structured-layer–activation–lagstructured-layer construction provides useful nonlinear capacity rather than merely increasing parameters. Table 14 reports the horizon-wise MSE and MAE. Figure 1 provides a qualitative complement to the capacity sweeps using a fixed Electricity test window with H = 336. In this case, the unscaled deep configuration follows changes in level and amplitude more closely than Linear and the single-layer lag-structured variants, illustrating the additional fitting capacity provided by depth and hidden feature maps.

G

Sample-Efficiency Protocol and Results

The sample-efficiency experiment trains representative models using fractions of the original training windows. Validation and test splits remain unchanged, and only the training windows are subsampled. We sample without replacement using a deterministic permutation for each fraction and fixed seed 2027; no minimum subset size is imposed. The training fractions are 1%, 5%, 20%, 50%, and 100%. All models use look-back length L = 96, prediction lengths H ∈ {96, 192, 336, 720}, RevIN normalization, scaled metric space, batch size 64, Adam optimization with learning rate 10−3 , at most 20 epochs, early-stopping patience 5, gradient clipping at 1.0, and automatic mixed precision. The compared models are Linear, LagLayer, and LagNet. Lin-

Dataset

M PDM blocks dmodel

dff

ETTh1 ETTh2 ETTm1 ETTm2 Weather Solar Electricity Traffic

3 3 3 3 3 3 3 3

32 32 32 32 32 2048 32 64

2 2 2 2 2 2 2 2

16 16 16 32 16 128 16 32

Batch size Epochs Patience Channel independent Normalize 128 128 128 128 128 32 32 8

10 10 10 10 20 20 20 20

10 10 10 10 10 3 10 15

Yes Yes Yes Yes Yes No Yes Yes

Yes Yes Yes Yes Yes No Yes Yes

Table 4: TimeMixer and Lag-TimeMixer configurations used in the main forecasting table. The core settings M , number of PDM blocks, dmodel , 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

ETTh1 ETTh2 ETTm1 ETTm2 Weather Solar Electricity Traffic

(16, 2, 2, 0) (16, 2, 3, 0) (16, 2, 2, 1) (32, 3, 2, 1) (16, 3, 3, 1) (16, 3, 3, 1) (16, 3, 3, 1) (16, 3, 3, 1)

(16, 2, 1, 0) (16, 1, 1, 0) (16, 2, 2, 1) (32, 3, 2, 1) (16, 3, 2, 1) (16, 3, 3, 1) (16, 3, 3, 1) (16, 3, 3, 1)

(16, 2, 2, 0) (32, 1, 3, 0) (16, 3, 2, 1) (32, 1, 1, 1) (16, 3, 2, 1) (16, 3, 3, 1) (16, 3, 3, 1) (16, 3, 3, 1)

(16, 2, 3, 1) (16, 2, 3, 1) (16, 2, 3, 1) (16, 3, 3, 1) (16, 3, 2, 1) (16, 3, 3, 1) (16, 3, 3, 1) (16, 3, 3, 1)

Batch size Epochs Patience 128 128 128 128 128 32 32 8

10 10 10 10 20 20 20 20

10 10 10 10 10 3 10 15

Table 5: TimeKAN and Lag-TimeKAN configurations. Each horizon-specific tuple is (dmodel , 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, dff = 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. ear is included as an unconstrained free-matrix reference; LagLayer tests the complete single-layer lag-structured operator with horizon-wise and history-wise scaling; LagNet tests its nonlinear deep extension. For LagNet, the five small datasets use two layers and four hidden feature maps, while Solar, Electricity, and Traffic use four layers and 12 hidden feature maps, matching the corresponding main-experiment architectures. The global and lag-kernel weight decays are 10−5 , while scale coefficients and biases are not decayed. Table 15 reports the complete horizon-wise MSE results for all eight datasets.

H H.1

Extended Structural Diagnostics

Metrics and Visualization Protocol

This paper uses diagnostics to connect forecasting accuracy with learned operator structure. Unless otherwise stated, the structural visualizations and adaptive-scaling diagnostics use the fixed diagnostic seed 2027. For the cross-architecture visualization in the main paper, we define the mean history-

to-future Jacobian J̄ ∈ RH×L as j̄h,ℓ =

1

NJ X C X ∂ ŷn,c,h

NJ C n=1 c=1 ∂xn,c,ℓ

,

(9)

where NJ sampled test windows and C channels are averaged. For nonlinear models, this input–output Jacobian captures local, input-dependent sensitivity rather than a fixed global forecasting matrix. The sampled windows are evenly spaced over the test set, with NJ = min(64, ⌈512/C⌉), targeting approximately 512 channel traces while capping the number of sampled windows at 64. Each mean Jacobian is divided by the 99.5th percentile of its absolute entries, clipped to [−1, 1], and displayed with a symmetric logarithmic color normalization. This panel-wise normalization is used only to expose spatial structure; values are not compared across panels. For an induced forecasting matrix W ∈ RH×L with scalar elements wh,ℓ , history-position importance is computed as H

I(ℓ) =

1 X |wh,ℓ |. H h=1

(10)

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 0.345 192 0.380 0.400 0.468 0.465 0.393 0.405 0.380 0.400 0.380 0.396 336 0.428 0.432 0.510 0.495 0.427 0.436 0.402 0.424 0.396 0.419 720 0.427 0.445 0.693 0.596 0.436 0.450 0.423 0.443 0.431 0.444

ETTm1

96 0.334 0.368 0.345 0.372 0.352 0.374 0.321 0.359 0.330 0.367 192 0.377 0.391 0.383 0.392 0.390 0.393 0.363 0.383 0.359 0.384 336 0.426 0.420 0.415 0.416 0.421 0.414 0.395 0.407 0.388 0.405 720 0.491 0.459 0.476 0.455 0.462 0.449 0.456 0.444 0.448 0.439

ETTm2

96 0.180 0.264 0.188 0.284 0.183 0.270 0.177 0.259 0.175 0.257 192 0.250 0.309 0.270 0.342 0.255 0.314 0.242 0.302 0.243 0.305 336 0.311 0.348 0.346 0.393 0.309 0.347 0.298 0.339 0.305 0.344 720 0.412 0.407 0.480 0.467 0.412 0.404 0.398 0.399 0.413 0.411

Weather

96 0.174 0.214 0.197 0.257 0.186 0.227 0.164 0.211 0.164 0.211 192 0.221 0.254 0.238 0.295 0.234 0.265 0.211 0.253 0.208 0.251 336 0.278 0.296 0.284 0.333 0.284 0.301 0.265 0.292 0.263 0.291 720 0.358 0.347 0.348 0.384 0.356 0.349 0.343 0.346 0.339 0.343

Solar

96 0.203 0.237 0.285 0.372 0.265 0.323 0.210 0.284 0.239 0.287 192 0.233 0.261 0.316 0.394 0.288 0.332 0.225 0.300 0.279 0.316 336 0.248 0.273 0.349 0.412 0.301 0.339 0.243 0.303 0.310 0.334 720 0.249 0.275 0.354 0.410 0.295 0.336 0.244 0.304 0.300 0.322

Electricity 96 0.148 0.240 0.195 0.278 0.190 0.296 0.155 0.246 0.176 0.269 192 0.162 0.253 0.194 0.280 0.199 0.304 0.167 0.258 0.182 0.273 336 0.178 0.269 0.207 0.296 0.217 0.319 0.184 0.275 0.198 0.289 720 0.225 0.317 0.242 0.329 0.258 0.352 0.228 0.314 0.241 0.324 Traffic

96 0.395 0.268 0.649 0.397 0.526 0.347 0.463 0.289 0.585 0.380 192 0.417 0.276 0.599 0.371 0.522 0.332 0.482 0.299 0.559 0.363 336 0.433 0.283 0.606 0.374 0.517 0.334 0.493 0.304 0.571 0.364 720 0.467 0.302 0.646 0.396 0.552 0.352 0.525 0.321 0.603 0.381

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. Forecast-step error is computed over all Ntest test windows as MSE(h) =

1

N C test X X

Ntest C n=1 c=1

2

(ŷn,c,h − yn,c,h ) .

(11)

The main-paper adaptive-scaling diagnostic compares this forecast-step-resolved error with the learned horizon-wise coefficient αh and separately examines the history-wise profile {βℓ }L ℓ=1 . Because h indexes future steps whereas ℓ indexes historical positions, β is not interpreted through pointwise alignment with forecast-step MSE. Instead, we examine whether its local response shape recurs qualitatively across repeated segments of the error curve, revealing nonuniform relative historical-position importance. Related periodic error variation and structured scale responses were observed in several dataset–horizon settings; Traffic with H = 720 is shown in the main paper as a representative case be-

cause these patterns are especially visible. Parameter-matrix heatmaps are generated from learned parameters: dense operators are visualized directly, while lag-structured operators are reconstructed from lag kernels and optional scale coefficients. Within each parameter-matrix comparison group, heatmaps use a shared color scale; if clipping is used for readability, the clipping percentile is reported in the figure caption.

H.2

Complementary Magnitude-Only Visualizations

For completeness, Figures 2 and 3 provide complementary magnitude-only visualizations for the same settings as the two signed visualizations in the main paper. These views discard coefficient signs and emphasize the strength of the corresponding history-to-future connections.

Dataset

H

ETTh1

96 192 336 720

841.6K 866.2K 3.38M 3.58M

18.6K 37.2K 65.2K 139.7K

35.2K 53.7K 81.5K 155.6K

75.4K 93.1K 119.6K 190.2K

12.8K 20.5K 36.1K 77.6K

ETTh2

96 192 336 720

224.2K 236.6K 255.2K 304.7K

18.6K 37.2K 65.2K 139.7K

35.2K 53.7K 81.5K 155.6K

75.4K 93.1K 119.6K 190.2K

15.0K 19.6K 43.5K 77.6K

ETTm1

96 192 336 720

224.2K 236.6K 255.2K 304.7K

18.6K 37.2K 65.2K 139.7K

548.7K 696.3K 917.6K 1.51M

75.4K 93.1K 119.6K 190.2K

14.4K 23.7K 40.1K 77.6K

ETTm2

96 192 336 720

224.2K 236.6K 255.2K 304.7K

18.6K 37.2K 65.2K 139.7K

548.7K 696.3K 917.6K 1.51M

77.6K 95.3K 121.8K 192.5K

38.1K 47.4K 38.2K 81.4K

Weather

96 192 336 720

4.83M 4.88M 4.96M 5.15M

18.6K 37.2K 65.2K 139.7K

548.7K 696.3K 917.6K 1.51M

75.5K 93.2K 119.7K 190.4K

20.9K 26.2K 40.2K 77.4K

Solar

96 192 336 720

3.26M 3.31M 3.38M 3.58M

18.6K 37.2K 65.2K 139.7K

548.7K 696.3K 917.6K 1.51M

2.28M 2.32M 2.37M 2.51M

21.9K 31.2K 45.1K 82.4K

Electricity 96 192 336 720

4.83M 4.88M 4.96M 5.15M

18.6K 37.2K 65.2K 139.7K

548.7K 696.3K 917.6K 1.51M

77.9K 95.6K 122.1K 192.8K

23.3K 32.6K 46.6K 83.9K

Traffic

6.41M 6.46M 6.53M 6.73M

18.6K 37.2K 65.2K 139.7K

548.7K 696.3K 917.6K 1.51M

88.6K 106.3K 132.8K 203.5K

27.7K 37.0K 50.9K 88.2K

96 192 336 720

iTransformer DLinear PatchTST TimeMixer TimeKAN

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.

H.3

Additional Dense-Operator Motivation Visualizations

The main paper uses representative signed history-to-future sensitivity matrices across diverse datasets and forecasting architectures to motivate the lag-structured forecasting operator. To provide broader evidence at the parameter level, the Linear columns of Figures 5–8 show the complete absolute dense forecasting matrices on all eight benchmark datasets and all four prediction lengths {96, 192, 336, 720}. All models are trained with RevIN normalization, matching the model configuration used for the main-paper visualization. Figure 4 summarizes the corresponding history-position importance curves, computed as H

1 X |wh,ℓ |, I(ℓ) = H h=1

without normalization, so the vertical axis reflects the actual average weight magnitude within each dataset.

Across datasets, these visualizations reveal diverse yet repeatedly structured and nonuniform allocations across the history-to-future matrix. Together, they support using relative lag as an explicit structural prior, while adaptive scaling accommodates dataset- and position-dependent variation in connection strength.

H.4

Additional Forecasting Matrix Comparisons

The main paper visualizes representative signed forecasting matrices for three dataset–horizon pairs. Figures 5–8 provide the complete set of matrix comparisons for all eight benchmark datasets and all four prediction lengths. Each figure fixes one prediction length and compares five operator variants: Linear, LagLayer without scaling, the two one-sided scale ablations, and the complete LagLayer with both scales. Within each row, the five heatmaps use the same color scale, whose upper limit is the 99.5th percentile of the absolute weights pooled across the five variants. Weight magnitudes are therefore comparable within the same dataset and

Dataset H

DLinear

DLagLayer

Dataset

H

DLinear

Params MSE MAE Params MSE MAE

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

Weather

96 18.6K 0.197 0.256 958 0.204 0.267 192 37.2K 0.238 0.294 1.5K 0.244 0.303 336 65.2K 0.283 0.332 2.4K 0.291 0.344 720 139.7K 0.348 0.385 4.7K 0.350 0.387

ETTh2 96 18.6K 0.470 0.474 958 0.317 0.375 192 37.2K 0.449 0.451 1.5K 0.434 0.446 336 65.2K 0.532 0.505 2.4K 0.575 0.532 720 139.7K 0.651 0.580 4.7K 0.754 0.620

Solar

96 18.6K 0.284 0.372 958 0.286 0.373 192 37.2K 0.316 0.394 1.5K 0.320 0.387 336 65.2K 0.349 0.411 2.4K 0.355 0.404 720 139.7K 0.354 0.411 4.7K 0.362 0.405

ETTm1 96 18.6K 0.345 0.373 958 0.350 0.375 192 37.2K 0.385 0.395 1.5K 0.386 0.394 336 65.2K 0.414 0.413 2.4K 0.417 0.417 720 139.7K 0.474 0.453 4.7K 0.476 0.454

Electricity 96 18.6K 0.195 0.278 958 0.199 0.282 192 37.2K 0.194 0.280 1.5K 0.198 0.284 336 65.2K 0.207 0.296 2.4K 0.210 0.299 720 139.7K 0.243 0.329 4.7K 0.246 0.332

ETTm2 96 18.6K 0.186 0.280 958 0.185 0.274 192 37.2K 0.288 0.360 1.5K 0.275 0.352 336 65.2K 0.356 0.405 2.4K 0.337 0.386 720 139.7K 0.476 0.457 4.7K 0.459 0.462

Traffic

96 18.6K 0.650 0.398 958 0.653 0.398 192 37.2K 0.599 0.371 1.5K 0.605 0.374 336 65.2K 0.605 0.374 2.4K 0.613 0.377 720 139.7K 0.646 0.395 4.7K 0.651 0.398

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. (b) ETTm2, H=192 Linear

40

0.5

80 (d) Electricity, H=336 4-layer MLP (d=256)

0

0.0 1.0

100 0.5

200 300

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

Forecast horizon

(e) Traffic, H=720 PatchTST

0

1.0

0.5

400 600

0.0

20

40

60

80

Historical position

I

Limitations and Broader Impact

Limitations. LagNet is based on a relative-lag sharing prior. This prior is natural for many sliding-window forecasting settings, but it may be less suitable when forecasting rela-

0.0

200

0

horizon. These complete supplementary panels intentionally show absolute effective forecasting weights, complementing the representative signed comparison in the main paper. Across dataset–horizon settings, the scaled operators repeatedly exhibit horizon-wise bands and nonuniform historywise modulation, providing broader matrix-level context for the representative coefficient profiles analyzed in the main paper.

(f) Traffic, H=720 TimeMixer

Normalized |J|

Forecast horizon

(c) Weather, H=336 2-layer MLP (d=256)

Normalized |J|

1.0

Normalized |J|

Forecast horizon

(a) ETTh1, H=96 Linear 0

0

20

40

60

80

Historical position

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.

tions depend strongly on absolute calendar positions, regime changes, or non-repeating events. Adaptive scaling alleviates part of this rigidity through row-wise and column-wise scale modulation, but it does not fully replace explicit calendar modeling or regime-aware modeling. The present study focuses on temporal operator structure; integrating LagLayer with dedicated cross-variate interaction modules remains a

Dataset

H

TimeMixer Params Runs

MSE

Lag-TimeMixer MAE

Params Runs

MSE

MAE

ETTh1

96 75.4K 192 93.1K 336 119.6K 720 190.2K

5/5 5/5 5/5 5/5

0.380±0.008 0.401±0.002 0.446±0.006 0.434±0.003 0.490±0.010 0.455±0.009 0.538±0.026 0.500±0.015

59.3K 60.4K 62.2K 66.8K

5/5 5/5 5/5 5/5

0.378±0.007 0.397±0.004 0.435±0.006 0.428±0.002 0.484±0.013 0.448±0.007 0.539±0.046 0.501±0.023

ETTh2

96 75.4K 192 93.1K 336 119.6K 720 190.2K

5/5 5/5 5/5 5/5

0.297±0.002 0.348±0.002 0.380±0.003 0.400±0.003 0.402±0.011 0.424±0.007 0.423±0.006 0.443±0.004

59.3K 60.4K 62.2K 66.8K

5/5 5/5 5/5 5/5

0.295±0.004 0.346±0.004 0.374±0.006 0.395±0.003 0.388±0.010 0.413±0.006 0.434±0.010 0.448±0.008

ETTm1

96 75.4K 192 93.1K 336 119.6K 720 190.2K

5/5 5/5 5/5 5/5

0.321±0.007 0.359±0.005 0.363±0.003 0.383±0.002 0.395±0.003 0.407±0.004 0.456±0.004 0.444±0.004

59.3K 60.4K 62.2K 66.8K

5/5 5/5 5/5 5/5

0.321±0.003 0.359±0.002 0.363±0.003 0.384±0.003 0.393±0.003 0.406±0.002 0.454±0.003 0.442±0.002

ETTm2

96 77.6K 192 95.3K 336 121.8K 720 192.5K

5/5 5/5 5/5 5/5

0.177±0.001 0.259±0.001 0.242±0.002 0.302±0.002 0.298±0.003 0.339±0.002 0.398±0.005 0.399±0.005

61.5K 62.6K 64.4K 69.0K

5/5 5/5 5/5 5/5

0.175±0.000 0.258±0.001 0.240±0.002 0.301±0.003 0.300±0.001 0.340±0.002 0.395±0.004 0.396±0.002

Weather

96 75.5K 192 93.2K 336 119.7K 720 190.4K

5/5 5/5 5/5 5/5

0.164±0.001 0.211±0.001 0.211±0.001 0.253±0.002 0.265±0.002 0.292±0.001 0.343±0.004 0.346±0.003

59.4K 60.5K 62.3K 66.9K

5/5 5/5 5/5 5/5

0.163±0.001 0.210±0.001 0.211±0.001 0.254±0.002 0.265±0.001 0.293±0.000 0.342±0.002 0.346±0.001

Solar

96 192 336 720

2.28M 2.32M 2.37M 2.51M

5/5 5/5 5/5 5/5

0.210±0.011 0.284±0.004 0.225±0.013 0.300±0.014 0.243±0.012 0.303±0.007 0.244±0.009 0.304±0.007

2.25M 2.25M 2.25M 2.26M

5/5 5/5 5/5 5/5

0.202±0.010 0.289±0.009 0.235±0.014 0.306±0.012 0.247±0.010 0.310±0.004 0.247±0.010 0.308±0.008

96 77.9K 192 95.6K Electricity 336 122.1K 720 192.8K

5/5 5/5 5/5 5/5

0.155±0.000 0.246±0.000 0.167±0.001 0.258±0.001 0.184±0.000 0.275±0.000 0.228±0.003 0.314±0.003

61.8K 62.9K 64.7K 69.3K

5/5 5/5 5/5 5/5

0.157±0.000 0.251±0.000 0.170±0.001 0.264±0.001 0.188±0.001 0.281±0.002 0.228±0.002 0.314±0.003

96 88.6K 192 106.3K 336 132.8K 720 203.5K

5/5 5/5 5/5 5/5

0.463±0.006 0.289±0.002 0.482±0.011 0.299±0.011 0.493±0.011 0.304±0.011 0.525±0.010 0.321±0.011

72.5K 73.6K 75.4K 80.0K

5/5 5/5 5/5 5/5

0.470±0.006 0.290±0.003 0.487±0.014 0.305±0.015 0.505±0.006 0.311±0.014 0.534±0.009 0.319±0.006

Traffic

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. direction for future work. Broader impact. This work studies a general forecasting architecture and does not target a specific high-risk application. Potential benefits include more efficient and interpretable forecasting models, which may help deployment in resourceconstrained environments. As with other forecasting systems, inappropriate use can lead to poor decisions when the data distribution shifts, when uncertainty is ignored, or when forecasts are treated as deterministic guarantees. Applications in safety-critical domains should therefore include uncertainty estimation, domain validation, and human oversight.

Dataset

H

TimeKAN Params Runs

MSE

Lag-TimeKAN MAE

Params Runs

MSE

MAE

ETTh1

96 192 336 720

12.8K 20.5K 36.1K 77.6K

5/5 5/5 5/5 5/5

0.370±0.001 0.397±0.001 4.0K 0.416±0.002 0.421±0.001 2.7K 0.446±0.003 0.437±0.001 4.7K 0.456±0.006 0.460±0.007 10.1K

5/5 5/5 5/5 5/5

0.367±0.003 0.395±0.002 0.420±0.006 0.422±0.002 0.443±0.004 0.438±0.002 0.456±0.014 0.455±0.007

ETTh2

96 192 336 720

15.0K 19.6K 43.5K 77.6K

5/5 5/5 5/5 5/5

0.296±0.005 0.345±0.004 6.2K 0.380±0.005 0.396±0.004 1.8K 0.396±0.006 0.419±0.004 12.1K 0.431±0.008 0.444±0.005 10.1K

5/5 5/5 5/5 5/5

0.306±0.005 0.356±0.004 0.383±0.005 0.397±0.004 0.410±0.005 0.430±0.003 0.444±0.012 0.454±0.009

ETTm1

96 192 336 720

14.4K 23.7K 40.1K 77.6K

5/5 5/5 5/5 5/5

0.330±0.003 0.367±0.002 5.5K 0.359±0.003 0.384±0.002 5.8K 0.388±0.003 0.405±0.003 8.7K 0.448±0.003 0.439±0.002 10.1K

5/5 5/5 5/5 5/5

0.333±0.008 0.370±0.005 0.370±0.008 0.390±0.004 0.390±0.004 0.405±0.002 0.455±0.006 0.441±0.004

ETTm2

96 192 336 720

38.1K 47.4K 38.2K 81.4K

5/5 5/5 5/5 5/5

0.175±0.001 0.257±0.001 29.3K 0.243±0.003 0.305±0.002 29.6K 0.305±0.004 0.344±0.002 6.8K 0.413±0.010 0.411±0.009 13.9K

5/5 5/5 5/5 5/5

0.182±0.003 0.268±0.004 0.255±0.004 0.317±0.003 0.316±0.007 0.355±0.006 0.415±0.010 0.415±0.007

Weather

96 192 336 720

20.9K 26.2K 40.2K 77.4K

5/5 5/5 5/5 5/5

0.164±0.001 0.211±0.001 12.1K 0.208±0.001 0.251±0.000 8.4K 0.263±0.001 0.291±0.001 8.8K 0.339±0.001 0.343±0.002 10.0K

5/5 5/5 5/5 5/5

0.167±0.002 0.216±0.002 0.214±0.001 0.260±0.003 0.268±0.001 0.298±0.003 0.342±0.002 0.347±0.002

Solar

96 192 336 720

21.9K 31.2K 45.1K 82.4K

5/5 5/5 5/5 5/5

0.239±0.008 0.287±0.003 0.279±0.009 0.316±0.004 0.310±0.019 0.334±0.012 0.300±0.002 0.322±0.003

13.0K 13.3K 13.8K 14.9K

5/5 5/5 5/5 5/5

0.252±0.017 0.305±0.015 0.305±0.019 0.334±0.010 0.342±0.016 0.355±0.006 0.346±0.024 0.361±0.007

Electricity 96 192 336 720

23.3K 32.6K 46.6K 83.9K

5/5 5/5 5/5 5/5

0.176±0.001 0.269±0.002 0.182±0.001 0.273±0.001 0.198±0.001 0.289±0.001 0.241±0.002 0.324±0.002

14.5K 14.8K 15.2K 16.4K

5/5 5/5 5/5 5/5

0.182±0.002 0.274±0.002 0.187±0.001 0.277±0.001 0.201±0.001 0.291±0.001 0.244±0.002 0.325±0.002

Traffic

27.7K 37.0K 50.9K 88.2K

5/5 5/5 5/5 5/5

0.585±0.012 0.380±0.003 0.559±0.006 0.363±0.001 0.571±0.005 0.364±0.002 0.603±0.007 0.381±0.002

18.8K 19.1K 19.6K 20.7K

5/5 5/5 5/5 5/5

0.591±0.006 0.378±0.008 0.579±0.006 0.364±0.007 0.589±0.006 0.366±0.004 0.622±0.018 0.390±0.004

96 192 336 720

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

LagNet

MSE

MAE

Params Runs

MSE

MAE

ETTh1

96 192 336 720

479 767 1.2K 2.4K

5/5 5/5 5/5 5/5

0.390±0.001 0.396±0.001 3.5K 0.440±0.001 0.425±0.000 6.3K 0.483±0.001 0.447±0.000 10.5K 0.482±0.000 0.470±0.000 21.6K

5/5 5/5 5/5 5/5

0.389±0.002 0.402±0.002 0.438±0.004 0.430±0.002 0.487±0.005 0.448±0.002 0.531±0.020 0.493±0.009

ETTh2

96 192 336 720

479 767 1.2K 2.4K

5/5 5/5 5/5 5/5

0.295±0.002 0.341±0.001 3.5K 0.378±0.001 0.391±0.001 6.3K 0.417±0.001 0.426±0.000 10.5K 0.421±0.000 0.439±0.000 21.6K

5/5 5/5 5/5 5/5

0.301±0.009 0.348±0.006 0.369±0.004 0.391±0.004 0.419±0.004 0.432±0.002 0.447±0.008 0.460±0.005

ETTm1

96 3.4K 192 5.4K 336 8.4K 720 16.5K

5/5 5/5 5/5 5/5

0.352±0.001 0.374±0.001 24.8K 0.389±0.001 0.390±0.000 44.3K 0.418±0.000 0.409±0.000 73.5K 0.481±0.001 0.442±0.001 151.5K

5/5 5/5 5/5 5/5

0.324±0.002 0.363±0.001 0.369±0.004 0.389±0.003 0.397±0.005 0.409±0.004 0.461±0.005 0.446±0.005

ETTm2

96 3.4K 192 5.4K 336 8.4K 720 16.5K

5/5 5/5 5/5 5/5

0.179±0.001 0.261±0.001 24.8K 0.245±0.001 0.302±0.001 44.3K 0.307±0.001 0.342±0.000 73.5K 0.409±0.001 0.398±0.000 151.5K

5/5 5/5 5/5 5/5

0.178±0.001 0.259±0.001 0.242±0.001 0.301±0.001 0.300±0.001 0.340±0.002 0.403±0.003 0.400±0.002

Weather

96 192 336 720

10.1K 16.1K 25.2K 49.4K

5/5 5/5 5/5 5/5

0.164±0.001 0.208±0.001 74.4K 0.210±0.001 0.249±0.000 132.9K 0.265±0.001 0.290±0.001 220.6K 0.344±0.001 0.340±0.001 454.4K

5/5 5/5 5/5 5/5

0.165±0.001 0.213±0.001 0.212±0.002 0.256±0.002 0.269±0.002 0.296±0.002 0.350±0.004 0.348±0.003

Solar

96 192 336 720

479 767 1.2K 2.4K

5/5 5/5 5/5 5/5

0.321±0.002 0.341±0.002 123.0K 0.358±0.002 0.360±0.002 244.1K 0.397±0.001 0.371±0.001 425.7K 0.397±0.001 0.362±0.001 909.9K

5/5 5/5 5/5 5/5

0.223±0.005 0.263±0.004 0.253±0.017 0.279±0.010 0.273±0.004 0.294±0.005 0.270±0.004 0.291±0.002

Electricity 96 192 336 720

479 767 1.2K 2.4K

5/5 5/5 5/5 5/5

0.203±0.000 0.279±0.001 123.0K 0.201±0.000 0.280±0.001 244.1K 0.215±0.000 0.294±0.000 425.7K 0.257±0.000 0.327±0.001 909.9K

5/5 5/5 5/5 5/5

0.167±0.001 0.255±0.001 0.174±0.001 0.262±0.001 0.189±0.000 0.278±0.001 0.232±0.002 0.316±0.001

Traffic

479 767 1.2K 2.4K

5/5 5/5 5/5 5/5

0.653±0.000 0.386±0.001 123.0K 0.607±0.001 0.366±0.001 244.1K 0.616±0.000 0.369±0.001 425.7K 0.653±0.000 0.389±0.001 909.9K

5/5 5/5 5/5 5/5

0.439±0.003 0.287±0.003 0.444±0.002 0.284±0.002 0.456±0.001 0.292±0.002 0.491±0.004 0.309±0.002

96 192 336 720

Table 11: Complete LagLayer and LagNet main results. Metrics are reported as mean±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 Linear LagLayer w/o Scaling LagLayer w/o History Scale LagLayer w/o Horizon Scale LagLayer LagNet capacity diagnostic

Relative-lag Row Column Nonlinear Extra sharing scale scale depth modules No Yes Yes Yes Yes Yes

No No Yes No Yes No

No No No Yes Yes No

No No No No No Yes

No No No No No No

Diagnostic free matrix lag sharing row adaptation column adaptation row/column adaptation nonlinear responses

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.

w/o Scaling

w/o History w/o Horizon Scale Scale LagLayer

Dataset

H

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 0.369/0.390 336 0.418/0.427 0.425/0.430 0.425/0.429 0.416/0.426 0.416/0.426 0.409/0.427 720 0.422/0.440 0.427/0.441 0.427/0.441 0.421/0.440 0.421/0.439 0.446/0.455

ETTm1

96 0.352/0.372 0.377/0.390 0.375/0.388 0.376/0.389 0.371/0.383 0.341/0.373 192 0.391/0.391 0.407/0.400 0.404/0.398 0.405/0.401 0.402/0.397 0.381/0.392 336 0.423/0.413 0.436/0.420 0.435/0.419 0.436/0.420 0.434/0.418 0.411/0.415 720 0.487/0.448 0.496/0.452 0.495/0.453 0.496/0.453 0.495/0.453 0.476/0.447

ETTm2

96 0.181/0.263 0.193/0.273 0.190/0.269 0.183/0.267 0.187/0.267 0.182/0.265 192 0.246/0.304 0.256/0.311 0.253/0.308 0.249/0.307 0.248/0.306 0.246/0.307 336 0.307/0.342 0.316/0.347 0.314/0.346 0.311/0.347 0.311/0.346 0.304/0.344 720 0.407/0.398 0.414/0.401 0.413/0.400 0.412/0.401 0.411/0.401 0.417/0.410

Weather

96 0.193/0.233 0.208/0.251 0.202/0.241 0.193/0.237 0.195/0.235 0.190/0.230 192 0.238/0.269 0.252/0.284 0.247/0.275 0.244/0.278 0.242/0.272 0.236/0.270 336 0.291/0.307 0.302/0.317 0.298/0.311 0.294/0.312 0.291/0.307 0.285/0.302 720 0.365/0.354 0.372/0.361 0.369/0.356 0.366/0.357 0.364/0.353 0.357/0.350

Solar

96 0.307/0.330 0.522/0.454 0.408/0.369 0.378/0.425 0.325/0.348 0.218/0.263 192 0.347/0.351 0.549/0.467 0.436/0.378 0.468/0.477 0.360/0.363 0.255/0.282 336 0.387/0.365 0.561/0.471 0.463/0.390 0.497/0.482 0.399/0.376 0.287/0.300 720 0.389/0.359 0.534/0.454 0.452/0.381 0.488/0.467 0.400/0.368 0.262/0.285

Electricity

96 0.198/0.275 0.215/0.290 0.209/0.280 0.208/0.286 0.203/0.279 0.168/0.255 192 0.198/0.278 0.211/0.290 0.205/0.282 0.207/0.287 0.201/0.280 0.173/0.261 336 0.212/0.292 0.224/0.304 0.219/0.296 0.221/0.302 0.215/0.295 0.190/0.278 720 0.254/0.325 0.267/0.336 0.261/0.329 0.262/0.333 0.257/0.328 0.234/0.315

Traffic

96 0.646/0.385 0.705/0.425 0.668/0.395 0.653/0.387 0.653/0.388 0.439/0.288 192 0.599/0.362 0.654/0.405 0.614/0.369 0.627/0.384 0.608/0.367 0.444/0.284 336 0.606/0.364 0.661/0.405 0.623/0.370 0.643/0.396 0.620/0.369 0.458/0.291 720 0.644/0.384 0.702/0.424 0.660/0.390 0.684/0.417 0.654/0.389 0.498/0.313

Linear

LagNet

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.

(a) MSE Dataset

H

R2-M8 R3-M8 R4-M8 R5-M8 R3-M1 R3-M2 R3-M4 R3-M16 R3-M32

Solar

96 192 336 720

0.256 0.289 0.331 0.317

0.230 0.268 0.280 0.285

0.224 0.263 0.273 0.265

0.226 0.261 0.284 0.269

0.287 0.318 0.345 0.322

0.245 0.295 0.302 0.306

0.240 0.278 0.306 0.293

0.227 0.260 0.284 0.268

0.213 0.257 0.289 0.310

96 192 Electricity 336 720

0.188 0.192 0.205 0.246

0.180 0.186 0.197 0.240

0.176 0.178 0.193 0.231

0.174 0.179 0.191 0.234

0.206 0.206 0.218 0.254

0.196 0.195 0.207 0.248

0.187 0.188 0.202 0.241

0.172 0.176 0.193 0.233

0.169 0.173 0.194 0.233

96 192 336 720

0.527 0.505 0.514 0.544

0.475 0.470 0.476 0.504

0.460 0.458 0.469 0.493

0.459 0.452 0.459 0.487

0.633 0.577 0.546 0.549

0.543 0.517 0.505 0.522

0.492 0.486 0.488 0.507

0.456 0.455 0.468 0.500

0.440 0.448 0.469 0.509

Traffic

(b) MAE Dataset

H

R2-M8 R3-M8 R4-M8 R5-M8 R3-M1 R3-M2 R3-M4 R3-M16 R3-M32

Solar

96 192 336 720

0.296 0.315 0.338 0.328

0.275 0.294 0.299 0.304

0.265 0.293 0.296 0.288

0.267 0.280 0.299 0.284

0.327 0.342 0.360 0.343

0.287 0.325 0.317 0.318

0.280 0.307 0.319 0.312

0.275 0.291 0.308 0.298

0.257 0.285 0.310 0.322

Electricity

96 192 336 720

0.271 0.275 0.289 0.322

0.267 0.272 0.286 0.318

0.262 0.265 0.281 0.317

0.260 0.266 0.279 0.316

0.285 0.288 0.300 0.328

0.280 0.278 0.291 0.324

0.270 0.272 0.288 0.319

0.259 0.264 0.279 0.314

0.255 0.260 0.281 0.313

Traffic

96 192 336 720

0.344 0.328 0.331 0.347

0.313 0.301 0.299 0.322

0.300 0.294 0.299 0.311

0.303 0.289 0.294 0.310

0.426 0.388 0.363 0.353

0.365 0.339 0.326 0.335

0.325 0.318 0.312 0.325

0.301 0.293 0.298 0.317

0.286 0.284 0.298 0.320

Table 14: Complete horizon-wise depth and width sweeps for the unscaled capacity diagnostic. Rr-Mm denotes r lag-structured layers and m hidden feature maps.

(c) Traffic H=720

(iii) w/o History Scale

(iv) w/o Horizon Scale

(v) LagLayer 0.08

40

0.04

80

|Weight|

0.12

0.00 0.2

0 100

0.1

200 300 0

|Weight|

(b) Electricity H=336

(ii) w/o Scaling

0.0 0.2

200 0.1

400

|Weight|

(a) ETTh1 H=96

Forecast horizon Forecast horizon Forecast horizon

(i) Linear 0

600 0.0

0

20 40 60 80 Historical position

0

20 40 60 80 Historical position

0

20 40 60 80 Historical position

0

20 40 60 80 Historical position

0

20 40 60 80 Historical position

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.

Dataset H

50% 100%

Dataset

H

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

ETTm1

96

ETTh2

96

ETTm2

96

Model

1%

5%

20%

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

1%

5%

20%

50% 100%

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 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

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

Electricity 96

Solar

96

Traffic

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

Model

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

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

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 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 PH 1 I(ℓ) = H h=1 |wh,ℓ |, 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.

References Liu, Y.; Hu, T.; Zhang, H.; Wu, H.; Wang, S.; Ma, L.; and Long, M. 2024. iTransformer: Inverted Transformers Are Effective for Time Series Forecasting. In International Conference on Learning Representations. Wang, S.; Wu, H.; Shi, X.; Hu, T.; Luo, H.; Ma, L.; Zhang, J. Y.; and Zhou, J. 2024. TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting. In International Conference on Learning Representations. Wu, H.; Hu, T.; Liu, Y.; Zhou, H.; Wang, J.; and Long, M. 2023. TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis. In International Conference on Learning Representations.

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