Routing Channel-Patch Dependencies in Time Series Forecasting with Graph Spectral Decomposition

Dongyuan Li; Shun Zheng; Chang Xu; Jiang Bian; Renhe Jiang

cs.LGMar 14, 2026

Routing Channel-Patch Dependencies in Time Series Forecasting with Graph Spectral Decomposition

Authors: Dongyuan Li, Shun Zheng, Chang Xu, Jiang Bian, Renhe Jiang

Organizations: The University of Tokyo · Microsoft Research Asia

Abstract

Time series forecasting has attracted significant attention in the field of AI. Previous works have revealed that the Channel-Independent (CI) strategy improves forecasting performance by modeling each channel individually, but it often suffers from poor generalization and overlooks meaningful inter-channel interactions. Conversely, Channel-Dependent (CD) strategies aggregate all channels, which may introduce irrelevant information and lead to oversmoothing. Despite recent progress, few existing methods offer the flexibility to adaptively balance CI and CD strategies in response to varying channel dependencies. To address this, we propose a generic plugin xCPD, that can adaptively model the channel-patch dependencies from the perspective of graph spectral decomposition. Specifically, xCPD first projects multivariate signals into the frequency domain using a shared graph Fourier basis, and groups patches into low-, mid-, and high-frequency bands based on their spectral energy responses. xCPD then applies a channel-adaptive routing mechanism that dynamically adjusts the degree of inter-channel interaction for each patch, enabling selective activation of frequency-specific experts. This facilitates fine-grained input-aware modeling of smooth trends, local fluctuations, and abrupt transitions. xCPD can be seamlessly integrated on top of existing CI and CD forecasting models, consistently enhancing both accuracy and generalization across benchmarks. The code is available https://github.com/Clearloveyuan/xCPD.

Figures & tables

Figure 1: The proposed xCPD plugin consists of three modules: (A) spectral channel-patch embedding, (B) channel-patch grouping, and (C) channel-patch routing with Mixture-of-Experts (MoE).

Model		TSMixer		+ xCPD		DLinear		+ xCPD		PatchTST		+ xCPD		TimesNet		+ xCPD
Metric		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.361	0.392	0.357	0.388	0.384	0.397	0.381	0.392	0.414	0.419	0.402	0.404	0.384	0.402	0.378	0.398
	192	0.404	0.418	0.389	0.405	0.433	0.427	0.430	0.421	0.460	0.445	0.443	0.435	0.436	0.429	0.422	0.423
	336	0.420	0.431	0.402	0.411	0.479	0.456	0.472	0.444	0.501	0.466	0.482	0.451	0.491	0.469	0.478	0.456
	720	0.463	0.472	0.458	0.461	0.506	0.503	0.497	0.485	0.500	0.488	0.494	0.482	0.521	0.500	0.509	0.491
ETTh2	96	0.274	0.341	0.270	0.336	0.314	0.369	0.302	0.357	0.302	0.348	0.274	0.335	0.340	0.374	0.330	0.368

Table 1: Long-term forecasting results on 9 real-world datasets in terms of MSE and MAE. The lookback horizon is set 96 (512 for TSMixer) and forecasting horizons are {96, 192, 336, 720}. The better performance in each setting is shown in bold . The best results for each row are underlined .

Model	TSMixer		+ LIFT		+ CCM		+ xCPD		DLinear		+ LIFT		+ CCM		+ xCPD
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	0.412	0.428	0.407	0.424	0.413	0.428	0.401	0.416	0.456	0.452	0.451	0.448	0.451	0.451	0.445	0.435
ETTh2	0.355	0.401	0.351	0.397	0.351	0.399	0.345	0.392	0.559	0.515	0.553	0.511	0.552	0.511	0.507	0.479
ETTm1	0.347	0.375	0.342	0.372	0.351	0.380	0.343	0.371	0.403	0.407	0.399	0.404	0.397	0.403	0.389	0.395
ETTm2	0.267	0.322	0.264	0.319	0.258	0.319	0.251	0.312	0.350	0.401	0.346	0.398	0.306	0.369	0.340	0.388
Exchange	0.381	0.414	0.378	0.411	0.355	0.402	0.349	0.396	0.354	0.414	0.351	0.411	0.348	0.409	0.312	0.380

Table 2: Averaged long-term forecasting results with horizons

\{96,192,336,720\}

and look-back horizon are 512 for TSMixer and 96 for DLinear on 9 datasets in terms of MSE and MAE. The better performance is shown in bold .

		CD	CI	+PRReg	+LIFT	+PCD	+CCM	+xCPD
ETTh1	Linear	0.402	0.345	0.342	0.345	None	0.342	0.338
ETTh1	Transformer	0.861	0.655	0.539	0.528	0.621	0.518	0.512
ETTm1	Linear	0.404	0.354	0.311	0.316	None	0.310	0.305
ETTm1	Transformer	0.458	0.379	0.349	0.356	0.404	0.300	0.289
Exchange	Linear	0.051	0.119	0.048	0.050	None	0.045	0.042
Exchange	Transformer	0.101	0.511	0.100	0.099	0.233	0.097	0.094

Table 3: Comparison of baselines under a general setting.

Model		TSMixer	+ LIFT	+ CCM	+ xCPD	DLinear	+ LIFT	+ CCM	+ xCPD	PatchTST	+ LIFT	+ CCM	+ xCPD	TimesNet	+ LIFT	+ CCM	+ xCPD
M4 (Yearly)	SMAPE	14.70	14.88	14.67	14.56	16.96	15.93	14.33	14.12	13.47	13.35	13.30	13.28	15.37	15.12	14.42	14.36
	MASE	3.343	3.341	3.370	3.221	4.283	4.038	3.144	3.121	3.019	3.006	2.997	2.989	3.554	3.512	3.448	3.434
	OWA	0.875	0.875	0.873	0.870	1.058	0.937	0.834	0.829	0.792	0.788	0.781	0.778	0.918	0.864	0.802	0.795
M4 (Quarterly)	SMAPE	11.18	11.02	10.98	10.88	12.14	11.48	10.51	10.49	10.38	10.36	10.35	10.35	10.46	10.28	10.12	10.10
	MASE	1.346	1.338	1.332	1.326	1.520	1.482	1.243	1.224	1.233	1.228	1.224	1.218	1.227	1.221	1.183	1.176
	OWA	0.998	0.992	0.984	0.976	1.106	1.098	0.931	0.928	0.921	0.918	0.915	0.912	0.923	0.913	0.897	0.889

Table 4: Short-term forecasting results on M4 dataset in terms of SMAPE, MASE, and OWA, and Stock dataset in terms of MSE and MAE. The forecasting horizon is

\{7,24\}

for the Stock.

Model	TSMixer	+ CCM	+ xCPD	DLinear	+ CCM	+ xCPD	PatchTST	+ CCM	+ xCPD	TimesNet	+ CCM	+ xCPD	IMP(%)
h1 $\rightarrow$ h2	0.288	0.283	0.280	0.358	0.344	0.335	0.442	0.438	0.418	0.391	0.388	0.384	2.318
h1 $\rightarrow$ h2	0.374	0.370	0.366	0.855	0.843	0.822	0.582	0.571	0.544	0.540	0.516	0.510	2.366
h1 $\rightarrow$ m1	0.763	0.710	0.702	0.782	0.723	0.702	0.754	0.692	0.678	0.887	0.827	0.801	2.300
h1 $\rightarrow$ m1	1.252	1.215	1.198	1.243	1.224	1.212	1.892	1.135	1.128	1.623	1.601	1.589	0.937
h1 $\rightarrow$ m2	0.959	0.937	0.928	1.213	0.912	0.890	1.132	0.903	0.892	1.199	1.122	1.086	1.950
h1 $\rightarrow$ m2	1.765	1.758	1.712	2.132	1.732	1.680	2.015	1.732	1.712	2.204	1.874	1.835	2.214

Table 5: Zero-shot forecasting MSE results on ETT datasets. The forecasting horizon is

\{96,720\}

Figure 2: Model efficiency comparison on Traffic.

Variants		ETTh1		ETTm1		Exchange		Solar-Eng.		Weather		Electricity		Traffic
Variants		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
DLinear+xCPD		0.445	0.435	0.389	0.395	0.312	0.380	0.322	0.393	0.253	0.301	0.197	0.287	0.606	0.372
Remove	w/o Shared Basis	0.462	0.456	0.404	0.415	0.331	0.395	0.341	0.399	0.267	0.310	0.205	0.295	0.618	0.379
	w/o Freq. Partition	0.448	0.394	0.399	0.403	0.318	0.388	0.329	0.398	0.264	0.307	0.203	0.292	0.610	0.376
	w/o Node Group	0.461	0.455	0.406	0.415	0.329	0.394	0.344	0.402	0.266	0.310	0.204	0.293	0.616	0.377
	w/o Filters	0.465	0.456	0.408	0.416	0.333	0.397	0.344	0.402	0.272	0.313	0.209	0.298	0.623	0.383

Table 6: Ablation study of xCPD. Look-back horizon is 96 and results are averaged over forecasting horizons

\{96,192,336,720\}

. Best and second-best results are marked in bold and underline .

Figure 3: Filters visualization of HULL on ETTh1.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Type	Dim	Lookback Length	Prediction Length	Total Length	Frequency	Domain
Long-term Forecasting Datasets
ETTh1	Multivariate	7	96	{96, 192, 336, 720}	17,420	Hourly	Electricity
ETTh2	Multivariate	7	96	{96, 192, 336, 720}	17,420	Hourly	Electricity
ETTm1	Multivariate	7	96	{96, 192, 336, 720}	69,680	15-min	Electricity
ETTm2	Multivariate	7	96	{96, 192, 336, 720}	69,680	15-min	Electricity
Weather	Multivariate	21	96	{96, 192, 336, 720}	52,695	10-min	Meteorology

Appendix

Table 7: Statistics of all datasets used in our experiments. “Frequency” denotes the sampling interval of time points.

Method	ETTh1	ETTh2	ETTm1	ETTm2	Exchange	Weather	Electricity	Traffic
PatchTST+PCD	0.468/0.456	0.379/0.399	0.382/0.393	0.272/0.325	0.366/0.403	0.255/0.281	0.198/0.288	0.482/0.300
PatchTST+xCPD	0.455/0.443	0.367/0.393	0.371/0.388	0.265/0.317	0.358/0.397	0.248/0.272	0.191/0.280	0.466/0.292

Appendix

Table 8: Long-term forecasting results with lookback window 96 and prediction horizons {96, 192, 336, 720}. Results show MSE/MAE averaged over three runs with best in bold .

Model	ETTh1	ETTh2	ETTm1	ETTm2	Solar-Energy	Weather
iTransformer	0.455/0.447	0.383/0.407	0.407/0.411	0.288/0.332	0.233/0.262	0.258/0.278
iTransformer+xCPD	0.438/0.440	0.375/0.400	0.394/0.401	0.277/0.325	0.226/0.256	0.250/0.269
TimeMixer	0.447/0.440	0.365/0.396	0.381/0.395	0.275/0.323	0.216/0.280	0.240/0.271
TimeMixer+xCPD	0.435/0.432	0.358/0.389	0.372/0.388	0.268/0.317	0.210/0.275	0.233/0.266
DUET	0.443/0.437	0.373/0.398	0.390/0.394	0.280/0.325	0.238/0.235	0.252/0.273
DUET +xCPD	0.436/0.433	0.364/0.392	0.381/0.390	0.275/0.318	0.231/0.234	0.246/0.270

Appendix

Table 9: Performance on iTransformer and TimeMixer. Results averaged over forecasting horizons {96, 192, 336, 720} with lookback window 96. Values show MSE/MAE with best results in bold .

Method	Forecasting Horizon
Method	96	192	336	720
	MSE/MAE	MSE/MAE	MSE/MAE	MSE/MAE
DLinear	0.650/0.396	0.598/0.370	0.605/0.373	0.645/0.394
DLinear+xCPD	0.636 / 0.386	0.574 / 0.356	0.586 / 0.362	0.630 / 0.387

Appendix

Table 10: Performance on Traffic dataset with look-back horizon 96.

Method	Prediction Horizon
Method	96	192	336	720
	MSE/MAE	MSE/MAE	MSE/MAE	MSE/MAE
DLinear	0.412/0.282	0.423/0.287	0.437/0.296	0.468/0.316
DLinear+xCPD	0.406 / 0.280	0.411 / 0.283	0.428 / 0.295	0.457 / 0.311

Appendix

Table 11: Performance on Traffic dataset with look-back horizon 336.

Model	Configuration	MSE $\downarrow$	MAE $\downarrow$
DLinear+xCPD	w/ Shared Basis	0.389	0.395
DLinear+xCPD	w/o Shared Basis	0.413	0.425
TimesNet+xCPD	w/ Shared Basis	0.373	0.384
TimesNet+xCPD	w/o Shared Basis	0.395	0.403

Appendix

Table 12: Ablation study under synthetic non-stationarity on ETTm1 (lookback window: 96, forecasting horizon: 96). The shared spectral basis maintains performance despite temporal perturbations.

Model

TSMixer

+ CCM

+ xCPD

DLinear

+ CCM

+ xCPD

PatchTST

+ CCM

+ xCPD

TimesNet

+ CCM

+ xCPD

MSE / MAE

\rightarrow

0.288 / 0.357

0.283 / 0.353

0.280 / 0.348

0.358 / 0.405

0.344 / 0.392

0.335 / 0.382

0.442 / 0.458

0.438 / 0.449

0.418 / 0.426

0.391 / 0.412

0.388 / 0.410

0.384 / 0.404

0.374 / 0.414

0.370 / 0.413

0.366 / 0.406

0.855 / 0.682

0.843 / 0.677

0.822 / 0.654

0.582 / 0.565

0.571 / 0.562

0.544 / 0.551

0.540 / 0.508

0.516 / 0.491

0.510 / 0.488

\rightarrow

0.763 / 0.677

0.710 / 0.652

0.702 / 0.645

0.782 / 0.679

0.723 / 0.658

0.702 / 0.632

0.754 / 0.688

0.692 / 0.645

0.678 / 0.630

0.887 / 0.718

0.827 / 0.700

0.801 / 0.682

1.252 / 0.815

1.215 / 0.803

1.198 / 0.792

1.243 / 0.835

1.224 / 0.812

1.212 / 0.798

1.892 / 0.954

1.135 / 0.832

1.128 / 0.821

1.623 / 0.981

1.601 / 0.964

1.589 / 0.952

\rightarrow

0.959 / 0.694

0.937 / 0.689

0.928 / 0.680

1.213 / 0.808

0.912 / 0.682

0.890 / 0.673

1.132 / 0.732

0.903 / 0.682

0.892 / 0.671

1.199 / 0.794

1.122 / 0.731

1.086 / 0.719

Appendix

Table 13: Zero-shot forecasting results on ETT datasets. Look-back horizon is 96 and the forecasting horizon is

\{96,720\}

. Best performance on each setting is in bold .

Method	Memory (MB)	Time (ms/iter)
Baseline Models
PatchTST	12,458	29
TimesNet	13,005	70
TSMixer	13,088	89
With CCM Plugin
PatchTST + CCM	25,034	161

Appendix

Table 14: Computational efficiency comparison on Traffic dataset. Memory usage and training time measured on single NVIDIA A100 GPU.

Tasks	Dataset	Length of Patch	e_layers	lr	d_ff	Num. of Epochs
	ETTh1	$T^{\prime}/16$	$1$	$1e$ - $4$	$256$	$10$
	ETTh2	$T^{\prime}/16$	$1$	$1e$ - $4$	$256$	$10$
	ETTm1	$T^{\prime}/12$	$2$	$1e$ - $4$	$256$	$10$
Long-term	ETTm2	$T^{\prime}/6$	$2$	$1e$ - $4$	$128$	$10$
Forecasting	Exchange	$T^{\prime}/24$	$2$	$1e$ - $4$	$256$	$10$
	Weather	$T^{\prime}/2$	$2$	$1e$ - $4$	$256$	$10$

Appendix

Table 15: Hyperparameter settings for different datasets, where

T^{\prime}

is the forecasting horizon, e_layers is the number of graph blocks, lr is the learning rate, and d_ff is the dimension of feed-forward layers.

Variant	ETTh1	ETTh2	ETTm1	ETTm2	Exchange	Solar
k-NN (top-30%)	0.449 / 0.438	0.511 / 0.482	0.393 / 0.399	0.346 / 0.394	0.315 / 0.381	0.326 / 0.396
k-NN (top-40%)	0.447 / 0.437	0.509 / 0.480	0.391 / 0.398	0.342 / 0.388	0.314 / 0.381	0.325 / 0.395
k-NN (top-50%)	0.445 / 0.435	0.507 / 0.479	0.389 / 0.395	0.340 / 0.388	0.312 / 0.380	0.322 / 0.393
k-NN (top-60%)	0.446 / 0.435	0.507 / 0.479	0.389 / 0.396	0.341 / 0.388	0.311 / 0.379	0.321 / 0.393
k-NN (top-70%)	0.450 / 0.438	0.510 / 0.482	0.394 / 0.400	0.346 / 0.393	0.317 / 0.382	0.325 / 0.395

Appendix

Table 16: Sensitivity to sparsification ratio

\alpha

(top-

\alpha

edges retained). Reported metrics are averaged MSE/MAE over forecasting horizons {96, 192, 336, 720} with input length 96 on DLinear+xCPD.

Model	Patch Length	Running Time (ms/iter)	MSE	MAE	Memory (GB)
DLinear+xCPD	16	78	0.183	0.270	$\sim$ 7.0
DLinear+xCPD	24	55	0.181	0.269	$\sim$ 5.5
DLinear+xCPD	32	30	0.180	0.268	$\sim$ 4.5
DLinear+xCPD	64	6	0.181	0.269	$\sim$ 3.2
DLinear+xCPD	96	5.5	0.184	0.271	$\sim$ 2.8

Appendix

Table 17: Running time, forecasting accuracy, and GPU memory usage of DLinear+xCPD on the Electricity dataset with forecasting horizon

T^{\prime}=192

Setting	Horizon=96	Horizon=192	Horizon=336	Horizon=720
Overlap = 1	0.180 / 0.267	0.187 / 0.275	0.199 / 0.296	0.245 / 0.328
Overlap = 2	0.178 / 0.267	0.189 / 0.275	0.201 / 0.297	0.248 / 0.329
No Overlap	0.174 / 0.262	0.180 / 0.268	0.195 / 0.293	0.240 / 0.326

Appendix

Table 18: Forecasting accuracy (MSE/MAE) of DLinear+xCPD on the Electricity dataset under different patch-overlap settings and forecasting horizons.

Multivariate Time Series Forecasting

Multivariate time series forecasting predicts future values across related variables by learning their temporal and cross-variable dependencies.

See all 302 papers

XCTFormer: Leveraging Cross-Channel and Cross-Time Dependencies for Enhanced Time-Series Analysis
May 18, 2026Israel Zexer, Omri AzencotMultivariate Time Series Forecasting Time Series Imputation
The Stein Faculty of Computer and Information Science Ben-Gurion University of the Negev
Learning What to Share: Expert Specialization for Time-Series Forecasting
Oct 24, 2026Bashar Khoury, Shadi Omari, Assaf SchusterMultivariate Time Series Forecasting Mixture-of-Experts Expert Specialization
Department of Computer Science, Technion – Israel Institute of Technology
TiWeaver: Unified Temporal Dynamics Modeling via Contextual Patching
Jun 2, 2026Zhe Li, Jindong Tian, Hao Miao +3Irregular Time-Series Modeling Multivariate Time Series Forecasting
East China Normal University Shanghai, China · Hong Kong Polytechnic University Hong Kong, China · Aalborg University Aalborg, Denmark

May 18, 2026J/K move · Enter open · S save

XCTFormer: Leveraging Cross-Channel and Cross-Time Dependencies for Enhanced Time-Series Analysis

Israel Zexer, Omri Azencot

The Stein Faculty of Computer and Information Science Ben-Gurion University of the Negev

Multivariate Time Series Forecasting Time Series Imputation Cross-Attention Time Series Transformers

Open paper AskPDF arXiv

Multivariate time-series analysis involves extracting informative representations from sequences of multiple interdependent variables, supporting tasks such as forecasting, imputation, and anomaly detection. In real-world scenarios, these variables are typically collected from a shared context or underlying phenomenon, suggesting the presence of latent dependencies across time and channels that can be leveraged to improve performance. However, recent findings show that channel-independent (CI) models, which assume no inter-variable dependencies, often outperform channel-dependent (CD) models that explicitly model such relationships. This surprising result indicates that current CD models may not fully exploit their potential due to limitations in how dependencies are captured. Recent studies have revisited channel dependence modeling with various approaches; however, these methods often employ indirect modeling strategies, which can lead to meaningful dependencies being overlooked. To address this issue, we introduce XCTFormer, a transformer-based channel-dependent (CD) model that explicitly captures cross-temporal and cross-channel dependencies via an enhanced attention mechanism. The model operates in a token-to-token fashion, modeling pairwise dependencies between every pair of tokens across time and channels. The architecture comprises (i) a data processing module, (ii) a novel Cross-Relational Attention Block (CRAB) that increases capacity and expressiveness, and (iii) an optional Dependency Compression Plugin (DeCoP) that improves scalability. Through extensive experiments on three time-series benchmarks, we show that XCTFormer achieves strong results compared to widely recognized baselines; in particular, it attains state-of-the-art performance on the imputation task, outperforming the second-best method by an average of 20.8% in MSE and 15.3% in MAE.

Abstract

Multivariate Time Series Forecasting

Explore similar work

XCTFormer: Leveraging Cross-Channel and Cross-Time Dependencies for Enhanced Time-Series Analysis

Learning What to Share: Expert Specialization for Time-Series Forecasting

TiWeaver: Unified Temporal Dynamics Modeling via Contextual Patching