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TStruct: Learning Shared Temporal Structures for Long-Term Time-Series Forecasting

Yangze Li, Zhengyang Zhou, Qihe Huang, Pengkun Wang, Binwu Wang, Yang Wang

arXiv:2610.17864v1cs.LGcs.AI

TL;DR

Noisy, non-stationary time series make it difficult to learn temporal dependencies that remain stable under distribution shifts. TStruct combines shared dependency structures with sample- and variable-specific weights, and the paper reports consistent gains over strong baselines alongside generalization, robustness, and efficiency. The approach is tied to a particular patch resolution, and sharing structures across resolutions remains non-trivial.

  • Problem

    The central question is how to identify and model temporal dependencies that remain stable across shifting distributions despite noisy, non-stationary series and incidental training interactions.

    Source: §1 · p. 1

  • Method

    TStruct learns shared temporal dependency matrices and combines them using normalized weights generated from dynamic sample information and static variable priors.

    Source: §1 · pp. 2–3

  • Results

    TStruct consistently outperforms strong baselines on standard long-term forecasting benchmarks and remains effective under diverse generalization and robustness evaluations, with high computational efficiency.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The findings support treating temporal dependencies as reusable structures across samples and variables rather than as fully input-specific interactions.

    Source: §1 · p. 3, §5 · p. 9

    Learned temporal structures are tied to a specific patch resolution, and directly sharing them across patch resolutions remains non-trivial.

    Source: F Limitations and Future Directions · p. 30

Abstract

from arXiv · show

Long-term time series forecasting requires temporal dependency modeling that remains stable and generalizable under temporal distribution shifts. However, real-world time series are often noisy and non-stationary, making deep learning models prone to fitting incidental temporal interactions in the training data. A key question is whether there exist temporal relationship structures that recur across different temporal positions and variables. We analyze temporal relationship matrices on commonly used time series datasets and find that samples from different temporal positions and different variables exhibit shared dominant temporal relationship structures. This indicates that temporal relationship structures contain patterns that recur stably across samples and variables. To learn such shared structures, we propose \textbf{T}emporal \textbf{Struct}ure Network (\textbf{TStruct}), a simple yet effective network for directly learning shared and generalizable temporal dependencies. Specifically, TStruct directly parameterizes a set of shared temporal dependency matrices and generates normalized combination weights using dynamic sample information and static variable priors. These weights represent a soft allocation of each sample-variable pair over shared temporal dependency structures, enabling the model to generate adaptive temporal dependency matrices under shared structural constraints, rather than fitting incidental interactions in a fully unconstrained manner. Notably, this simple structure-aware design consistently outperforms several strong baselines across all datasets, achieving superior accuracy while maintaining high computational efficiency.

1 Introduction

Long-term forecasting must model dependencies that remain effective across shifting distributions, despite noisy, non-stationary data and potentially incidental training relationships. The paper identifies recurring temporal structures and proposes TStruct to combine shared structures with sample- and variable-specific adaptation; it reports strong benchmark, generalization, robustness, and efficiency results.

1 Introduction

  • The central challenge is learning dependencies that generalize across future temporal distributions when noisy, non-stationary data can make incidental training interactions unreliable.Flexible models may fit incidental relationships, while fixed transformations may miss heterogeneous dependencies.Source: §1 · pp. 1–2
  • Real-data patch-level relationship matrices show shared dominant patterns across samples and variables, with higher spectral concentration than patch-shuffled and IID Gaussian controls.The shuffled control disrupts alignment of relative patch positions; its reduced concentration supports recurrence at corresponding positions.Source: §1 · p. 2, Figure 1 · p. 2
  • TStruct learns shared temporal dependency matrices and uses dynamic sample information and static variable priors to form sample- and variable-specific dependencies.Normalized combination weights softly allocate each sample-variable pair across the shared structures.Source: §1 · pp. 2–3
  • TStruct consistently outperforms strong baselines on standard long-term forecasting benchmarks and remains effective across generalization and robustness evaluations, with high computational efficiency.The paper characterizes the design as simple.Source: §1 · p. 3

2 Related Work

Prior forecasting methods trade off adaptive temporal interactions against fixed, efficient mixing, while explicit temporal-structure methods have largely focused on periodic patterns. TStruct instead targets shared temporal dependency structures.

2.1 Temporal Dependency Modeling for Time Series Forecasting

  • TStruct uses shared temporal dependency structures with sample-variable-conditioned weights, between attention’s flexible input-dependent interactions and fixed shared mixing.Attention can capture incidental relations in low-SNR series, while fixed mixing has limited adaptability to heterogeneous dependencies.Source: §2.1 · p. 3

2.2 Explicit Temporal Structure Parameterization

  • Earlier explicit parameterizations mainly represent periodic patterns in inputs or temporal representations; TStruct parameterizes dependency structures, which can capture broader patch-level relationships.Temporal dependency structures may contain periodicity but are not equivalent to periodicity itself.Source: §2.2 · p. 3

3 Methodology

TStruct maps historical observations to future values while focusing on temporal dependencies that remain stable across samples and variables. Its pipeline embeds patches, learns shared yet adaptive dependency patterns, and uses a parameter-efficient variable-aware prediction head.

3 Methodology

  • The forecasting task maps L historical observations across C variables to the next H values, with a focus on dependencies stable across samples and variables.Source: §3 · p. 4

3.1 Overview

  • TStruct normalizes each instance, embeds patches, encodes shared-adaptive dependencies, and uses PCAHead to generate variable-aware forecasting projections.The architecture includes instance-wise normalization before encoding and restores the original scale after prediction.Source: §3.1 · p. 4, Figure 2 · p. 4

3.2 Patch Embedding

  • The model segments each variable’s input into N non-overlapping patches of length P and projects them into D-dimensional representations for temporal modeling.The projection is shared across variables; N = L/P when L is divisible by P.Source: §3.2 · pp. 4–5

3.3 Temporal Structure Encoder

  • TSEncoder learns K shared patch-position dependency patterns and combines them using static variable priors and sample-derived queries into normalized, pair-specific aggregation weights.Each shared structure is a learnable patch-level dependency score pattern.Source: §3.3 · p. 5

3.4 Parameter-efficient Channel Adaptation Head

  • PCAHead balances variable-specific forecasting patterns with parameter efficiency using a shared base projection and variable-specific low-rank residual adaptation.This reduces head complexity from O(CDinH) to O(DinH + Kr(Din + H) + CK).Source: §3.4 · p. 6

4 Experiment

Across seven forecasting benchmarks, TStruct leads on nearly all dataset-metric comparisons. Ablation, generalization, look-back, and interpretability analyses further examine its components and learned structures.

4.1 Experiment Setup

  • Experiments use seven benchmarks, MSE and MAE, prediction horizons of 96, 192, 336, and 720, and eight forecasting baselines.The datasets span different domains, variable scales, and temporal patterns.Source: §4.1 · p. 6

4.2 Main results

  • On ETT and Weather, TStruct’s gains are associated with stable structures plus adaptability; on Electricity and Traffic, it has the best average MSE on both.It also has the best average MAE on Electricity and the second-best on Traffic.Source: §4.2 · pp. 6–7
  • TStruct ranks first in 13 of 14 dataset-metric comparisons, with the best average MSE on all seven benchmarks and best average MAE on six.Table 1 averages results over all prediction horizons.Source: §4.2 · p. 6, Table 1 · p. 7

4.3 Ablation Study

  • On ETT and Weather, TStruct outperforms both attention and shared-linear mixers; on Electricity and Traffic, its gap over the linear mixer narrows, but it still beats attention.Stronger spectral concentration on Electricity and Traffic is associated with more pronounced shared temporal dependencies, making simple shared mixing competitive.Source: §4.3 · pp. 7–8

4.4 Generalization Analysis

  • With timestamp-ordered training, TSM beats Linear from 40% of the earliest samples onward; with random training samples, it performs best at 20% and converges around 40%.Figure 3(a) evaluates temporal distribution shift, while Figure 3(b) evaluates sample efficiency; MHA performs worse than Linear and TSM in the shift setting.Source: Figure 3 · p. 8, §4.4 · p. 8

4.3 Ablation Study

  • Replacing PCAHead with a shared linear head causes moderate, consistent degradation, supporting variable-aware prediction under variable heterogeneity.The ablation compares the full model with component-replacement variants under the same objective and training protocol.Source: §4.3 · pp. 7–8, Table 2 · p. 7

4.5 Effect of Look-back Length

  • As look-back length increases from 48 to 720, TStruct benefits consistently, unlike global-periodic baselines whose gains can flatten or degrade.The authors attribute this pattern to TStruct’s patch-level dependency modeling, which can use richer historical context.Source: Figure 4 · p. 8, §4.5 · pp. 8–9

4.6 Interpretability of Learned Temporal Structures

  • On Weather, Var 16’s periodic autocorrelation aligns with higher weight on recurrent Structure #1, while Var 14’s rapidly decaying autocorrelation aligns with diagonal-dominant Structure #2.The learned static prior assigns these variables different preferences over shared structures.Source: Figure 5 · p. 9, §4.6 · p. 9

5 Conclusion

The conclusion identifies shared dominant temporal structures across samples and variables and presents TStruct as a way to model adaptive dependencies within that shared space.

5 Conclusion

  • TStruct learns shared temporal dependency structures and converts sample-variable-specific scores into normalized soft-allocation weights for adaptive dependency modeling.The conclusion frames this as adaptive modeling under shared structural constraints.Source: §5 · p. 9
  • Experiments show that TStruct consistently improves forecasting accuracy over strong baselines.Ablation and generalization analyses further support shared-adaptive modeling; look-back and interpretability analyses indicate use of longer contexts and meaningful learned structures.Source: §5 · pp. 9–10

A.2 Temporal Structure Analysis

The analysis tests whether temporal relationship patterns recur across windows and variables, then connects that evidence to shared temporal dependency parameterization for forecasting.

A.2 Temporal Structure Analysis

  • Patch-level relationship matrices represent relationships among relative temporal positions in normalized, non-overlapping patches from each sample-variable sequence.The analysis stacks vectorized matrices across samples or variables and normalizes rows before SVD.Source: §A.2 · p. 15
  • Spectral concentration measures how many singular components explain 95% of spectral energy; the rank compression factor contextualizes that concentration against the reference dimension.The analysis compares real data with patch-shuffle and IID Gaussian controls because Gram construction can itself induce concentration.Source: §A.2 · p. 16
  • Across all seven datasets, real data has substantially greater spectral concentration than shuffled data on both sample and variable sides, supporting recurrence at aligned relative patch positions.Shuffling disrupts alignment of relative patch positions across matrices.Source: §A.2 · pp. 16–17, Table 4 · p. 17
  • Real data also exceeds the IID Gaussian reference in spectral concentration across all datasets on both sides, supporting compact representation through shared patterns.The IID control replaces observed windows with independent Gaussian noise.Source: §A.2 · pp. 16–17
  • Electricity and Traffic show larger gains over both controls than ETT and Weather; their shuffled data also remains more concentrated than the IID reference.The larger Real/Shuffle gains indicate greater dependence on original patch-position alignment.Source: §A.2 · p. 17
  • Under an idealized shared temporal evolution, historical relationship directions map to shared directions in the history–future relation family, with the same coefficients when residuals are small.This provides a conditional link between reusable historical relationships and a compact description of forecasting-relevant relationships.Source: §A.2 · pp. 17–18
  • TStruct parameterizes temporal aggregation with shared structure matrices learned under forecasting supervision, forming operators that aggregate projected historical features.The connection motivates shared parameterization without requiring equality with the analyzed historical or history–future relation matrices.Source: §A.2 · p. 18

B Implementation Details

The implementation fixes the main look-back length at 720 and uses a hybrid Huber-plus-FFT training objective; reported analyses cover seed stability, initialization, and objective choice.

B Implementation Details

  • Main experiments use a fixed look-back length of 720, Adam for at most 25 epochs, and early stopping with patience 10.The best checkpoint is selected by validation loss; configurations for baselines retain their native objectives and allow look-back selection where specified.Source: B Implementation Details · p. 18
  • TStruct is trained with a hybrid objective combining Huber loss and FFT loss; MSE and MAE are the evaluation metrics.The FFT loss measures prediction discrepancy in the frequency domain.Source: B Implementation Details · pp. 18–19

C.1 Random Seed Stability

  • Across five random seeds, TStruct’s standard deviations are mostly 0.000–0.002 for both MSE and MAE on ETTh1 and Weather.The authors report that performance is not sensitive to random initialization or stochastic optimization.Source: §C.1 · p. 19, Table 5 · p. 19

C.2 Initialization Study

  • Random initialization yields noticeably higher errors than Orthogonal, Xavier, and Kaiming, which achieve consistently strong and comparable performance on ETTh1 and Weather.The authors use Orthogonal initialization by default, reporting diverse initial directions at a common scale.Source: §C.2 · p. 20, Table 6 · p. 20

C.3 Training Objective Comparison

  • Under both Huber+FFT and plain MSE, TStruct ranks first in MSE on all seven datasets and in MAE on six, while PhaseFormer leads MAE on Traffic.Huber+FFT improves TStruct’s MSE and MAE on all seven datasets compared with plain MSE.Source: §C.3 · p. 20

C.4 Noise Injection Robustness

On ETTh1, TSM has lower errors than MHA under tested local-noise conditions; sensitivity analyses also identify useful ranges for shared-structure count and patch resolution.

C.4 Noise Injection Robustness

  • The ETTh1 perturbation test injects Gaussian noise into a randomly positioned continuous segment, varying the segment-length ratio and noise strength.TSM has lower MSE and MAE than MHA throughout both tested noise variations.Source: §C.4 · p. 20
  • At noise strength 8, TSM obtains MSE 0.404 versus 0.479 for MHA, a gap of 0.075.As the noise ratio increases from 0.05 to 0.30, TSM’s MSE ranges from 0.392 to 0.400, remaining 0.042–0.047 below MHA.Source: §C.4 · pp. 20–21, Table 8 · p. 21

C.5 Hyperparameter Sensitivity

  • TStruct remains stable across a wide range of shared-structure counts; very small counts may limit diversity, while overly large counts may add redundancy and optimization difficulty.The authors use K = 4 as the default in main experiments.Source: §C.5 · p. 21
  • Performance is robust across a broad middle range of patch counts, especially N ∈[10, 30]; too few patches may underfit detail, while too many may heighten noise sensitivity and overfitting risk.Patch count controls the resolution of the patch-level temporal dependency matrix.Source: §C.5 · pp. 21–22

C.6 Efficiency and Complexity Analysis

TStruct reduces temporal-score construction cost by composing shared structures instead of estimating fully input-dependent attention scores. On Electricity, it improves MSE over patch-based Transformer baselines while using fewer parameters and MACs; the appendix also examines efficiency conditions and hyperparameter sensitivity.

Empirical efficiency

  • On Electricity, models were compared with input length L = 720, patch size and stride P = S = 48, and hidden dimension D = 512.The reported comparison includes accuracy, parameters, MACs, GPU memory, training time per step, and inference time per batch.Source: §C.6 · p. 22

Complexity analysis

  • TSM constructs temporal dependency scores at O(N^2K), compared with O(N^2D) for attention-based patch-level mixing, with K ≪ D.Patch-level feature aggregation remains quadratic in N; the efficiency difference is in score estimation.Source: §C.6 · p. 22

Empirical efficiency

  • 0.151 MSE versus PatchTST’s 0.171, with 3.324M versus 5.496M parameters and 11.779G versus 23.134G MACs, for TStruct on Electricity.The comparison uses the same evaluation setting for all models.Source: Table 9 · p. 22, §C.6 · p. 22

Hyperparameter sensitivity

  • Figure 6 plots MSE as bars and MAE as lines across prediction horizons to examine sensitivity to the number of shared structures K.The analysis uses the ETT benchmarks.Source: Figure 6 · p. 23
  • TStruct remains stable across a broad middle range of patch counts N, while very small or large N may cause underfitting or overfitting.Figure 7 presents the sensitivity analysis on the ETT benchmarks.Source: Figure 7 · p. 23

D Additional Interpretability Visualizations

The learned temporal structures show reusable patterns across benchmark datasets, including local continuity, recurring interactions, and broader transitions. The appendix also points to detailed forecasting results across prediction lengths.

D Additional Interpretability Visualizations

  • Diagonal and near-diagonal motifs indicate local continuity, while stripe-like or checkerboard patterns suggest recurrent interactions between relative patch positions.Smooth transitions across rows or columns also show broader dependencies; the structures are not limited to a predefined periodicity prior.Source: Figure 8 · p. 24, D Additional Interpretability… · p. 24
  • ETT structures mix local continuity, alternating patterns, and smooth transitions, while Weather, Electricity, and Traffic show more regular banded or repeated patterns.The visualizations are presented as qualitative evidence that learned structures are interpretable and adapt to dataset-specific regularities.Source: D Additional Interpretability… · p. 24, Figure 13 · p. 26

E.1 Detailed Forecasting Results

The appendix provides horizon-specific forecasting results across seven benchmarks and reports that the benefits of TSM and PCAHead remain consistent across prediction horizons.

E.1 Detailed Forecasting Results

  • Table 10 reports full forecasting results for TStruct and all baseline methods across seven benchmarks, including MSE and MAE under each prediction horizon, complementing the averaged results in the main paper.Source: §E.1 · p. 28

E.2 Detailed Ablation Results

  • The benefits of TSM and PCAHead remain consistent across forecasting horizons in the detailed ablation results.Table 11 compares TSM→MHA, TSM→Linear, and PCAHead→Linear variants.Source: §E.2 · p. 28, Table 11 · p. 29

F Limitations and Future Directions

The limitations identify patch-resolution dependence and leave transfer across datasets and domains at scale untested. Resolution-adaptive structures and structure-based foundation models are proposed as directions for further work.

Patch resolution dependency

  • Resolution-adaptive structures that transfer, interpolate, or compose across patch scales are proposed for future exploration.TStruct currently uses a compact set of shared matrices with controlled initialization and softmax-normalized composition and aggregation.Source: F Limitations and Future Directions · p. 30
  • Temporal structures are parameterized over N patch positions, so changing patch size or look-back length can alter matrix dimensions or coordinate meaning.Although TStruct remains effective under different look-back lengths, sharing structures directly across patch resolutions remains non-trivial.Source: F Limitations and Future Directions · p. 30

Temporal structure foundation models

  • Transfer and generalization across datasets and domains at scale remain untested, despite evidence of reusable relationships across temporal positions and variables.A proposed direction is a foundation model built around a pretrained dictionary of shared dependency structures.Source: F Limitations and Future Directions · p. 30

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