TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
Authors: Peiyuan Liu
Beiliang Wu
Yifan Hu
Naiqi Li
Tao Dai
Jigang Bao
Shu-Tao Xia
Venue: ICML 2025 poster (2025)
Keywords: Long-term time series forecasting
Non-stationarity
Dependency Modeling
Primary area: deep_learning->sequential_models_time_series
TL;DR: TimeBridge is a novel framework designed to bridge the gap between non-stationarity and dependency modeling in long-term time series forecasting.

Abstract: Non-stationarity poses significant challenges for multivariate time series forecasting due to the inherent short-term fluctuations and long-term trends that can lead to spurious regressions or obscure essential long-term relationships. Most existing methods either eliminate or retain non-stationarity without adequately addressing its distinct impacts on short-term and long-term modeling. Eliminating non-stationarity is essential for avoiding spurious regressions and capturing local dependencies in short-term modeling, while preserving it is crucial for revealing long-term cointegration across variates. In this paper, we propose TimeBridge, a novel framework designed to bridge the gap between non-stationarity and dependency modeling in long-term time series forecasting. By segmenting input series into smaller patches, TimeBridge applies Integrated Attention to mitigate short-term non-stationarity and capture stable dependencies within each variate, while Cointegrated Attention preserves non-stationarity to model long-term cointegration across variates. Extensive experiments show that TimeBridge consistently achieves state-of-the-art performance in both short-term and long-term forecasting. Additionally, TimeBridge demonstrates exceptional performance in financial forecasting on the CSI 500 and S&P 500 indices, further validating its robustness and effectiveness. Code is available at https://github.com/Hank0626/TimeBridge.

Multivariate time series forecasting aims to predict future changes based on historical observations of time series data, TimeBridge: Non-Stationarity Matters for Long-terd term Time Series Forecasting
Peiyuan Liu Beiliang Wu Yifan Hu
Naiqi Li
Tao Dai 2
Jigang Bao
Shu-tao Xia
Abstract
Y,
Predicted
Predicted Y.
Non-stationarity poses significant challenges for multivariate time series forecasting due to the inherent short-term fluctuations and long-term trends that can lead to spurious regressions or obscure essential long-term relationships. Most existing methods either eliminate or retain non- stationarity without adequately addressing its dis- tinct impacts short-term and long-term mod- eling. Eliminating non- con-stationarity is essential for avoiding spurious regressions and capturing local dependencies in short-term modeling, while preserving it is crucial for revealing long-term cointegration across variates. this paper, we propose TimeBridge, a novel framework designed to bridge the gap between non non-stationarity and dependency modeling in long-term time series forecasting. By segmenting input series into smaller patches, TimeBridge applies Integrated Attention to mitigate short-term non-stationarity and capture stable dependencies within each variate, while Cointegrated Attention preserves non-stationarity to model long-term cointegration across variates. Extensive experiments show that TimeBridge consistently achieves state-of-the-an performance in both short-term and long-term forecasting. Additionally, TimeBridge demon- strates exceptional performance in financial fore- casting on the CSI 500 and S&P 500 indices, fur- ther validating its robustness and effectiveness. Code is available at https:/ /ggttuu......./// com/ Hank 0 62 6/ TimeBridge.
25
Input
Output
Input
Output
20
Speriees
20-
15
15
10-
10-
5-
50
100
150
200
250
30
100
130
200
250
(a) Retain Non-stationarity
(b) Remove Non-stationarity
Figure I. Visualization of the impact of non-stationarity on short- and long-term modeling. The goal forecast two coin- tegrated sequences. Xt and Y.. where X, exhibits two random fluctuations. (a) Retaining -on-stationarity preserves long-tenn cointegration between variates but leads spurious regressions in short-term modeling (orange line). (b) Removing non-stationarity avoids short-term spurious regressions but disrupts long-term simi- lar trends (green line).
which holds significant applications in fields such as finan- cial investment (Sezer et al., 2020), weather forecasting (Karevan & Suykens, 2020), and traffic flow prediction (Shu et al., 2021; Miao et al., 2024b). However, the inherent non- stationarity of time series (Kim et al., 2022), characterized by short-term fluctuations and long-term trends, introduces challenges such as spurious regressions, making time series forecasting a particularly complex task.
Recently, many methods have emerged to utilize comaliizaioonannddennmaaaliiiii
paradigm to address non-stationarity in time series (Kim et al., 2022; Fan et al., 2023; Liu et al., 2023; 2024b). For instance, RevIN (Kim et al., 2022) normalizes the input data and subsequently applies its distributional characteristics to denormalize the output predictions. Building on this approach, other meth- ods have designed distributional prediction networks (Fan et al., 2023) and more refined normalization techniques (Liu et al., 2023) further mitigate non-stationarity. On the other hand, some studies (Liu et al., 2022b; Ma et al., 2024; Fan et al., 2024) argue that over-stabilizing time series may actually reduce the richness of embedded features, leading to a decline in model performance. Existing methods for addressing non- non-stationarity in time series face a dilemma:
1. Introduction
'Equal contribution 'Tsinghua Shenzhen International Grad- uate School "Shenzhen University, Correspondence to: Tao Dai <daitao. edu @ gmail.com. Naiqi Li <linaiqi.thu gmail.com>.
Proceedings of the nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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TimcBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
some prioritize eliminating non-stationary factors reduce overfitting, while others attempt to incorporate these factors but lack comprehensive theoretical frameworks. Further- more, they do not adequately explain the trade-off's between removing non-stationarity and leveraging it for modeling.
sequences effectively capture the long-term cointegration relationships among variates. Experiments across multiple datasets demonstrate that TimeBridge achieves consistent oftt--of-hheart performance in both long-term and short- term forecasting. Furthermore, we validate the effectiveness of TimeBridge on two financial datasets, the CSI 500 and S&P 500, which exhibit significant short-term volatility and strong long-term cointegration relationships among sectors.
is notable that non-stationary characteristics have dis- tinct impacts on modeling short-term and long-term de- pendencies. Non-stationarity can lead to spurious regres- sions in short-term modeling due the high randomness and unpredictability of short-term fluctuations (Noriega & Ventosa- SantoularSaaaaaiiaaa 2007) (see Figure la). For example, a sudden drop in temperature could be caused by a typhoon or cold front, both of which have no intrinsic connection. Retaining non-stationarity can result in false correlations between such unrelated events. However, non-stationarity is crucial for capturing long-term cointegration relation- ships among variates, reflecting their co-movement or syn- chronized changes over time (Fanchon & Wendel, 1992). Removing non-stationarity may also eliminate these essen- tial long-term dependencies (see Figure 1b). Therefore, while non-stationarity can cause spurious regressions in short-term modeling, is essential for modeling long-term dependencies between variates. Conversely, eliminating non-stationarity benefits short-term modeling but erases long-term cointegration.
In nutshell, our contributions are summarized in three folds:
1. Going beyond previous methods, establish novel connection between non-stationarity and dependency modeling, highlighting the importance of climinating non-stationarity in short-term variations while preserv- ing it for long-term cointegration.
2.
We propose TimeBridge, novel framework that em- ploys Integrated Attention to model temporal depen- dencies by mitigating short-term statitaatinn,,ty and Cointegrated Attention to capture long- tnn-termm cointegra- tion across variates while retaining non-stationarity.
3. Comprehensive experiments demonstrate that Time- Bridge achieves state-of-the-art performance in both long-term and short-term forecasting across various datasets. Moreover, further validate its robustness and effectiveness on the CSI 500 and S&P 500 indices, which pose additional challenges due to their complex volatility and cointegration characteristics.
In this paper, to address the dual challenges posed by non- stationarity in short-term and long-term modeling, we pro- pose distinct strategies tailored each scenario. For short- term modeling, we eliminate non- stattioatoontt to capture the strong temporal dependencies within each variate, as short- term causal relationships mainly exist between consecutive time points within single variate rather than across variates. This strategy reduces the risk of spurious regressions from non-stationary fluctuations, enabling the model to better capture the local causal dynamics. For long-term model- ing, we utilize the preserved non-stationarity to uncover long-term cointegration relationships between different vari- ates, thereby enabling more accurate and reliable long-term forecasting.
2. Related Works
As shown in Figure 2, recent advancements in multivariate time series forecasting have predominantly focused on two core directions: Normalization and Dependecy Modeling,
Normalization can be divided into stationary and non- stationary methods. Stationary methods (Kim et al., 2022: Fan et al., 2023; Liu et al., 2024b; 2023) aim to eliminate non-stationarrit through model-agnostic normalization tech- niques, thereby preventing spurious regressions and enhane- ing model performance. For example, RevIN (Kim et al., 2022) applies Z-normalization to the input sequence and then reverses the normalization on the output using the distributional characteristics of the input, assuming that both share similar distributional properties. Dish-TS (Fan et al., 2023) takes this further by predicting the statistical characteristics of the output with a distribution prediction model. Additionally, SAN (Liu et al., 2023) offers more granular letch- prediction method. Conversely, some approaches (Liu et al., 2022b; Ma et al., 2024; Fan et al., 2024) advocate preserving non- non-stationarity. as excessive normalization can eliminate inherent diverse sequence char- acteristics and limit predictive accuracy.
Technically, based on the above motivations, we propose TimeBridge as a novel framework to bridge the gap between non-stationarity and dependency modeling in long-term time series forecasting. TimeBridge first captures short- term fluctuations by partitioning the input sequence into small-length patches, followed by utilizing Integrated At- tention to model these stabilized sub-sequences within each variate. Here, "Integrated" reflects the non-stationary na- ture of the short-term series, also referred to as integrated series (Park& Phillips, 2001). Subsequently, we downsam- ple the patches to reduce their quantity, thereby enriching each patch with more long-term information. Cointegrated Attention retains the non-stationary characteristics of the
2

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
Normalization
Dependency Modeling
and C' represents the number of time variates. It is important to recognize that real-world time series data often exhibit high short-term uncertainty, while long-term dynamics may reveal cointegration relationships among different time vari- ates.
Stationary
Tirwdridge
Channel Independent
Baste DOUEES
Boge (NtISGS)
Trtagrated
Model
3.1. Structure Overview
Non- Stationary
Channel Dependent
Colnttgrated
Evep
NST DUISSE CSSaa EUICAZIN
VWWWy
Mode
Ausibracc (N11533
Fillfomee
As illustrated in Figure 3, our proposed TimeBridge con- sists of four key components: (a) Patch Embedding seg- ments the input sequence into non-overlapping patches and transforms each patch into a patch token; (b) Integrated Attention models the dependencies among all patch tokens of the same variates. By eliminating non-stationarity within each patch token, it mitigates the risk of spurious regres- sions that could arise from abrupt short-term changes; (c) Patch Downsampling aggregates global information and re- duces the number of patches to encapsulate richer long-term features within each patch, while simultaneously lowering computational complexity; (d) Cointegrated Attention pre- serves the non-stationary characteristics of the sequence and models the long-ter term cointegration relationships across different variates within the same temporal window.
Figure 2. Time series forecasting methods categorized by normal- ization and dependency modeling.
Dependency Modeling focuses on designing methods to capture the relationships within multivariate time series, which can be classified into Channel Independent (CI) and Channel Dependent (CD) methods. CI methods (Zeng et al., 2023; Das et al., 2023; Nie et al., 2023; Dai et al., 2024; Lin et al., 2024; Miao et al., 2025) rely exclusively on the historical values of each individual channel for prediction, deliberately avoiding egy not only stabilizes the training process but also excels at capturing rapid temporal dynamics unique to each vari- ate. In contrast, CD methods (Wu al., 2021; Zhou al., 2022; Wu et al., 2023; Zhang& & Yan, 2023; Liu et al., 2024a; Hu et al., 2025b;a) leverage the interrelationships between variates for modeling. While these methods utilize more information, they struggle with spurious regressions when modeling short-term dependencies, failing to capture rapid changes effectively.
interactions. This strat-
3.2. Patch Embedding
In this stage, each variate of the input sequence X. is first divided into non-overlapping patches, and each patch is then mapped to an embedded patch token. Since the process is identical for each variate, we use X to represent a single variate and later restore the dimensionality of the variates. Formally, this process can formulated as follows:
{PI, . PN} Patching(X)) P Embedding(PI. ..pp.......)))
(I)
The challenge with previous methods lies in their isolated treatment of non-stationarity and dependency model- ing, overlooking their intrinsic connection. Due to non- stationarity, time series often exhibit significant short-term fluctuations, leading to severe spurious regressions when modeling short-term dependencies. However, capturing long-term cointegration requires preserving this underlying variability. Therefore, short-term random fluctuations need to be addressed by eliminating non-stationarity and mod- eling intra-variate temporal dependencies, while long-term cointegration demands preserving pon-stationarity for inter- variate modeling. Our proposed TimeBrrdge addresses these issues by employing Integrated Attention and Cointegrated Attention, respectively.
(2)
Here, each patch p. has a length of S, and the number of patches N = [ The Embedding(-)) operation transforms each patch from its original length S to hidden dimension D through a trainable linear layer. This results in embedded patch tokens P E RCXNDD where each of the C' vari- ates contains N patches, capturing local information that is typically subject to rapid short-term fluctuations. For convenience, we denote Pe.; as the set of all patches within single variate and P.,.. as the patches across all variates at the same time position in the following sections.
3.3. Integrated Attention
3. Method
The embedded patch tokens P represent short-term non- stationary sequences, also referred to integrated series of order & (A: > 0) (Park & Phillips, 2001; Mushtaq, 201 1). This non-stationarity makes it challenging to model depen- dencies across different variates, as short-term fluctuations are highly susceptible to external shocks. Furthermore, modeling temporal dependencies within the same variate
In the task of multivariate time series forecasting, the objective is to predict future sequences Y = [x/+1,... xx+0l E Rxx given historical input se- quences X = [x],".. xx] E RCxI Here, I and denote the lengths of the input and output sequences, respectively,
3

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
Input
(a) Patch Embedding
(b) Integrated Attention
Patch
Operation
Embedding
Patching
Attetmic on
Fetch
Ps
Pg
P5
Integrated
MWC
Attention
Inpet Series
Embedded Patch Tokens
Eliminate Non-stationarity
Patch Downsampl ling
(c)
Patch Downsampling
(d) Cointegrated Attention
Same Operation
P Pg
Pg
Cointegrated Attention
Dowisample
Attetnion on
Attetnion on
Pi
Px
Same Operation
Ps
K
Projection
P Keep Non-statiinaal
Output
Figure Overall architecture of TimeBridge: (a) Patch Embedding divides the imput sequence into non-overlapping patches and embeds each a token; (b) Integrated Attention models temporal dependencies within each variate by mitigating short-term non-stationarity: (c) Patch Downsampling reduce patches to aggregates long-term information and lower complexity; (d) Cointegrated Attention captures long-term relationships across variates while keeping non-stationarity.
can lead to spurious regression due to the inherent non- stationarity of the patches. To address this, we first apply patch-wise normalization to all patches within variate:
3.4. Patch Downsampling
Long-term equilibrium relationships between sequences, cointegration among different variates, often require sequences to contain sufficient long-term information to emerge. Therefore, before modeling the cointegration be- (4) tween variates, is crucial to increase the amount global information represented by each patch. This is achieved by reducing the number of patches and aggregating global information through the attention mechanism:
Trend (Padding pp = =p: Trend
(3)
P = { (pI..... .p'].
where the AvgPool(-) operation is moving average with the Padding(-) operation keep the series length unchanged. We then employ the proposed Integrated Attention mech- anism to capture temporal dependencies within the same variate:
P Downsample (Po..)
(7)
P Attention(P's c.:3 ......PP....
(8)
Here, Downsample()) reduces the patches Pc.: to M patches (M N) using an MLP. By employing the downsampled P E RMxx the Query and the original (5) Pa.: E NDD as the Key and Value the attention mech- anism, we leverage the long-range modeling capability of attention to dynamically aggregate global information. This allows each patch to encapsulate richer long-tern long-term informa- tion, making it possible to capture the intricate cointegration relationships that emerge only over sufficiently extended temporal horizons.
Pa.: = LayerNorm (Pc.: + Attention(PL 0.31
P 6.
Pa.)):
Pes. c; LayerNorm (P MLP(P .L.((P.))
(6)
where MLP(-) represents multi-layer feedforward network, and LayerNorm(.) denotes layer normalization. The atten- tion mechanism uses the normalized P'. as both Query and Key, while the original P.,: serves as the Value. This design generates stationary attention map, which is then directly multiplied by the Value, removing the need for subsequent denormalization. By leveraging Integrated Attention in this way, we effectively model the temporal dependencies with- out being affected by the short-term non-stationary nature of the sequences.
3.5. Cointegrated Attention
Although short-term relationships between integrated series are susceptible to spurious regressions, accurately modeling long-term cointegration between sequences necessitates re- taining their inherent con-statoonary characteristics. Since

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TimcBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
each downsampled patch now encapsulates more extensive long-term information, we leverage Cointegrated Attention to directly model the cointegration relationships among all variates at the same time interval P... E RCXD:
to the Appendix E.
4.1. Long-term Forecasting
Setups. We conduct long-term forecasting experiments on several widely-used real-world datasets, including the Elec- tricity Transformer Temperature (ETT) dataset with its four subsets (ETThI, ETTh2, ETTml, ETTm2) (Wu et al., 2021; Miao et al., 2024a), as well as Weather, Electricity, Traffic, and Solar (Liu et al., 2025a:b). These datasets exhibit strong non-stationary characteristics, detailed in Appendix D. Fol- lowing previous works (Zhou et al., 2021; Wu et al., 2021), we use Mean Square Error (MSE) and Mean Absolute Error (MAE) as evaluation metries. We set the input length I to 720 for our method. For other baselines, we adopt the setting that searches for the optimal input length I and other hyperparameters. Details of the metric and the searching process can be found in Appendix C.I and Appendix F.1.
P :. = LayerNorm P,,, + Attention P.,, P.......... P...)).
(9)
P. s.n LayerNorm AN + MLP(P...))
(10)
This attention mechanism not only captures the global coin- tegration relationships across variates but also adaptively assesses the strength of these relationships: stronger coin- tegration is reflected by higher attention weights, while weaker connections receive lower weights. Finally, the em- bedded patch tokens P E RCMxx are unpatched and projected to the final output Y E RCx.
4. Experiment
Results. As shown Tab. 1, TimeBridge consistently achieves the best overall performance. Notably, the large- scale Traffic dataset, with its 862 channels, presents sub- stantial challenges due to its high dimensionality and in- tricate temporal dependencies. TimeBridge adeptly bal- ances non-stationarity and dependency modeling, achiev- ing consistently strong predictive performance. Quantita- tively, compared to tate-of-theearr based DeformableTST (Luo & Wang, 2024), CNN-base ModernTCN (Donghao & Xue, 2024), and MLP-based TimeMixer (Wang et al., 2024b)- Trinerrgge reduces MSE and MAE by 1.55%/2.99% 5.56%/4.125 and 13.66% (66%/7...%% respectively.
To validate the effectiveness of the proposed TimeBridge, we conduct extensive experiments on variety of time series forecasting tasks, including both long-term and short-term forecasting. Additionally, we evaluate TimeBridge on finan- cial forecasting tasks characterized by significant short-term volatility and strong long-term cointegration relationships among sectors.
Baselines. For long term forecasting, select diverse set of state-of-the-arr baselines representative of recent advance- ments in time series forecasting, including the Transformer- based DeformableTST (Luo & Wang, 2024), the CNN- based ToderaTNN (Donghao & Xue, 2024), the MLP- based TimeMixer (Wang et al., 2024b), as well as other com- petitive methods such iTransformer (Liu al., 2024a), PatchTST (Nic al., 2023), Crossformer (Zhang & Yan, 2023), Leddam (Yu et al., 2024), MICN (Wang et al., 2022),
4.2. Short-term Forecasting
Setups. For short-term forecasting, we conduct experiments on the PeMS datasets (Wang et al., 2024b), which capture complex spatiotemporal correlations among multiple vari- ates across cidy traffic networks. We use mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean squared error (RMSE) as evaluation metrics. The input length I is set to 96 and the output length O to 12 for all baselines. Details of datasets and metrics are in Appendix D and Appendix C.2.
TimesNet (Wu et al., 2023), and DLinear (Zeng et al., 2023). For short-term forecasting, we add two well-performing baselines SCINet (Liu et al., 2022a) and DUET (Qiu et al., 2025). For financial forecasting, we also incorporate the mo- mentum strategy CSM (Jegadeesh & Titman, 1993) and the reversal strategy BLSW (Poterba & Summers, 1988), along with two classic deep learning models, LSTM (Hochreiter & Schmidhuber, 1997) and Transformer (Vaswani et al., 2017a), to provide a comprehensive evaluation.
Results. As shown in Tab. 2, methods that perform well in long-term forecasting with indeeeeddt approaches, such as PatchTST (Nie et al., 2023) and DLinear (Zeng et al., 2023), suffer from significant performance degrada- tion on the PeMS dataset due to its strong inter-variabe dependencies. In contrast, TimeBridge demonstrates ro- bust performance on this challenging task, outperforming even the recent -fate-of-the-art method TimeMixer (Wang et al., 2024b), which highlights its effectiveness in capturing complex spatiotemporal relationships.
Implementation Details. All experiments are implemented in PyTorch (Paszke et al., 2019) and conducted on two NVIDIA RTX 3090 24GB GPUs. We use the Adam opti- mizer (Kingma, 2014) with learning rate selected from (1e-3, 5e-).. The number of patches N is set accord- ingly to different datasets. We adopt hybrid MAE loss that operates in both the time and frequency domains for stable training (Wang et al., 2024a). For additional details on hyperparameter settings and loss function, please refer
5

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
Models
TimeBridge (Ours)
iTransformer (2024a)
DeformhketST (2024)
TimeMixer (2024b)
PatchTST (2023)
Crossformer (2023)
Leddam (2024)
ModenTTCN (2024)
MICN (2022)
TimesNet (2023)
DLincar
(2023)
Metric
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
XSE
MAE
MBE
MAE
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
ETiml
10.344
0.379 10.362
0.391
10,348
0.383
10.355
0.380] 0.353
0.582
[0.420 0.435
|0.354 0.381
00.351
0.381
10,383
0.406 0.400
0.406
|0.357
0.379
ETTm2
|0.246
0.310
[0.26]
0.329
[0,257
0.319
[0.257
0,318 0.255 0.317 [0.518 0.501 10.265 0.320
[0.253
0.314
[02277 0.336 0.291
0.333
[0.267
0.332
ETThi
|0.397
0.424)0.439
0.448
[0.404
0.423
[0.427
0.441 (0:413 0.434
J0.440 0.463 |0.415 0.430
(0.404
0.420
00443
0.462 0.458
0.450
11.5000..72
[0.423
0.437
ETTh2
[0.341
0.382J0.374
0,406
[0.328
0.377
|0.349
0.397|
0.324 0.381 [0.809 0 0.658 |0.345 0.391
[0.322 0. 0.379 0.385
0.430
00.44
0.427
[0.43] 0.447
Weather
|0.219
|0.233
0.271
[0.222
0.262
|0.226
0.264 0.226 0.264 (0.228 0.287 |0.226 0.264
(0.224 0. 0.264
|0224 0.298 0.259 0.287
|0.240 0.300
Eloctricity
[0.149
0.245 0.24510.164 0.261
[0,161 0.261
10.185
0,284| 0.159 0.253 [0.181 0.279 [0.162 0.256 (0.156 0.253 00.182 0,292 0,192 0.295 |0.166 0.264
Traffic
|0.360
0.255 | |0.397 0.282 [0.39] 0.278
[0:409 0.279] (0.991 0.264 [0.523 0.284| 10.452 0.283 (0.396 0.270
0.533
0.312 0.620 0.336|0:434 0.295
Solar
0.188 0.239 12.3910.2000 (0.200 0.260 (0.185 0.254 [0.193 0.252 0.194 0.245 [0.19] 0.242 (0.223 0.264 (0.228 0.282 002233 0.266 0,244 0.334 (0.244 0.309
Table 1. Long-term forecasting hyperparameter search results. All results are averaged across four different prediction lengths:
O E {96, 192, 336, 7203. See Tab. 9 for full results.
TimeBridge (Ours)
TimeMixer (2024b)
SCINet (2022a)
Crossformer (2023)
PatchTST (2023)
TimesNet (2023)
MICN (2022)
DLinear (2023)
DUET (2025)
Stationary (2022b)
Autoformer (2021)
Informer (2021)
Models
MAE
14.52
14.63
15,97
15.64
18.95
16.41
15.71
19.70
15.57
17.64
18.08
19.19
PcMS03
MAPE
11:51
15.74
17.29
15.17
15.27
17.56
18.75
RMSE
23.10
23.28
25.20
25.56
30.15
26.72
24.55
32.35
22.99
28.37
27.82
32,70
MAE
19.24
19.21
20.35
20.38
24.86
21.63
21.62
24.62
20.84
22.
25.00
22,05
PeMS04
MAPE:
12.42
12:53
12.84
12.84
16.65
13:15
13:53
16.12
14.88
14.85
16.70
14.88
RMSE
31,12
30.92
32.31
32.41
40.46
34.90
34,39
39,51
31.41
35.47
38.02
36.20
MAE
20.43
20.57
22:79
22.54
27,87
25.12
22.28
28.65
22:34
26.02
26.92
27.26
PeMS07
MAPE
8.42
8.62
9.11
9.38
12.69
10.60
9.57
12.15
9.92
11.75
11.83
11.63
RMSE
33.59
35.61
42.56
40.71
34.97
42.34
40.60
MAE
14.98
15.22
17.38
17.56
20.35
19.01
17.76
201.2
15.54
19.20
20.47
20.96
PeMS08
MAPE
9.56
9.67
10.92
13:15
11.83
10:76
12.09
9.87
12.21
12:27
13.20
RMSE
23.77
24.26
27.34
27.21
31.04
30.65
27.26
32.38
24.04
38.62
31.52
30.61
Table 2. Short-term forecasting results in the PeMS datasets.
CSI 500
S&P 500
Models
ARR
AVoll
MDD1,
ASRT
CRT
IRT
ARRT
AVoI_
MDDJ
ASRT
CRT
IR1
BLSW (198))1988
0.
0.227
-0.155
0.485
0,710
0.446
0.199
0.318
-0.223
0.626
0.892
0.774
CSM (1993)
0.015
0,229
-0.179
0.066
0.084
0,001
0,099
0.250
-0.139
0.396
0.712
0,584
LSTM (1997)
-0.008
0,159
-0.172
-0.047
-0,044
-0.128
0.142
0.162
-0.178
0.877
0.798
0.929
Transformer (2017a)
0.154
0.156
-0.135
0.986
1.143
0.867
0.135
0.159
-0.140
0.852
0.968
0.908
PatchTST (2023)
0.118
0.152
-0.127
0.776
0,923
0.735
0.146
0.167
-0.140
0.877
1.042
0.949
Crossformer (2023)
-0.039
0,163
.0.217
-0,238
-0.179
-0.350
0.284
0.159
-0.114
1,786
2.491
L646
Transformer (2024a)
0.214
0.168
-0.164
1.276
1.309
1.173
0.159
0.170
-0.139
0.941
1.0
0.955
TimeMixer (2024b)
0.078
0,153
-0.114
0.511
0,685
0,385
0.254
0.162
.0.131
1.568
1.938
1 1,448
TSMixer (2023)
0.086
0.156
-0.143
0.551
0.601
0.456
0.187
0.173
-0.156
1.081
1.199
1.188
Bridgr (Ours)
0.285
0.203
-0.196
1.405
1.453
1.317
0.326
0.169
-0.142
1.927
2.298
1.842
Table 3, Results for financial time series forecasting in CSI 500 and S&P 500 datasets. See Tab. 10 for full results.
4.3. Financial Forecasting
Calmar Ratio (CR), and Information Ratio (IR). Details of datasets and metrics are in Appendix D and Appendix C.3.
Setups. We conduct experiments on both the U.S. and Chi- nese stock markets, including the S&P 500 and CSI 500 indices. Stock price movements are influenced by vari- ous factors such economic indicators, market sentiment, geopolitical events, and company-specifie news, leading to high non-stationarity. We predict next-day returns us- ing historical data and generate investment portfolios with a buy-hold-sell strategy (Sanderson & Sumpkin-Sowent 2018). At day ++1 open, traders sell day stocks and buy tan-ranked ones based on predicted returns. Following previous work (Lin al., 2021), we evaluate performance using Annual Return Ratio (ARR), Annual Volatility (AVol), Maximum Drawdown (MDD), Annual Sharpe Ratio (ASR),
Results. As shown in Tab. 3, the inherent non-stationarity and intricate dependencies within financial markets make it difficult for baseline methods to consistently identify op- timal portfolios across different markets. While financial market fluctuations are notoriously unpredictable, Time- Bridge adapts well to these challenges. By capturing short- term volatility within financial time series and preserving long-term cointegration across sectors, it achieves consis- tently strong performance, outperforming existing methods in overall market efficiency.
6

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
5. Ablation Studies
relationships first leads to a loss of important short-term temporal features (Wang al., 2022; Han et al., 2024).
To validate the effectiveness of the proposed TimeBridge, we conduct a comprehensive ablation study on its archi- tectural design. In Tab. 4, Tab. 5. and Tab. 6, the rows highlighted in gray correspond to the original TimeBridge configuration, serving as baseline for comparison with various modified versions.
Ablation on modeling approaches for Integrated Atten- tion and Cointegrated Attention. We conduct the fol- lowing experiments: @ both Integrated and Cointegrated Attention use channel-independent (CI) modeling, @ Inte- grated Attention uses channel-dependent (CD) modeling while Cointegrated Attention uses CI, @ Integrated Atten- tion uses CI while Cointegrated Attention uses CD, and @ both use CD modeling. The results in Tab. 6 show that modeling short-term inter-variates relationships can lead to severe spurious regression. CI modeling generally outper- forms CD in scenarios with fewer channels (e.g.. Weather), while CD excels when the number of channels large (e.g.. Traffic). This aligns with recent findings that inter-channel dependencies become increasingly important as the number of channels grows. We attribute this to the model's abil- ity to extract potential long-term stable relationships from non-stationary sequences when more channels are present, thereby improving both forecasting accuracy and robust- ness.
Ablation on removing or keeping non-stationarity. We conduct the following experiments: @ Non-stationarity re- tained in both Integrated and Cointegrated Attention. @ Retained in Integrated, removed from Cointegrated. @ Re- moved from Integrated, retained in Cointegrated. @ Re-
moved from both. Results in Tab. 4 show that the best per- formance is achieved when non-stationarity is removed in Integrated Attention, which models short-term intra-variate fluctuations, and retained in Cointegrated Attention, which captures long-term inter-variate dependencies. Conversely, retaining non-stationarity in Integrated Attention while re- moving it from Cointegrated Attention yields the worst results.
Casc
Integrated Altuntion
Cointegrated Attaniinn
Weather
Solar
Elecuriiity
Traffic
Case
Integrated Alteniion
Conneerred Altenion
Weather
Swar
Ellctricity
Traffc
Nocms
Norm?
MSE
MAB
MSE
MAE
MSE
MAE
MSB
MAE
Che
Cl CD
MSH
MAR
MSF
MAR
MSF
MAR
MSB
MAR
x
x
0.230
0.260
0.183
0.242
0.153
0.249
0.371
0.260
3
C
CI
1.2188
0.259
0.081
02211
0.157
0255
0,387
0,276
x
0:230
0.200
0.183
0.252
0.150
0.251
9.381
0.883
20
CD
C3
0.222
0.262
0.184
0.347
0.160
0255
0.387
0.280
x
0:319
0.389
6.181
0.239
0.189
0.245
0.360
0.255
2
01
CD)
0.219
0.349
0.081
0.239
(L149
IL245
0,360
0.255
0,219
0.259
0.183
0,232
0.153
0.250
0.574
0.259
8
CD
CD
0.222
0.263
0.183
0.247
0.156
0.254 0.376
0.269
Table 4. Ablation on the effect of removing non-stationarity in Integrated Attention and Cointegrated Attention. / indicates the use of patch normalization to eliminate non-stationarity, while x mcans -on-stationarity is retained.
Table 6. Ablation on modeling approaches for Integrated Attention and Cointegrated Attention. "CI" denotes channel independent and "CD" denotes channel-dependent modeling.
Ablation on Integrated and Cointegrated Attention im- pact and order. We conduct the following experiments: @ Integrated Attention only, @ Cointegrated Attention only, @ Integrated Attention followed by Cointegrated Attention, and @ Cointegrated Attention followed by Integrated Atten- tion, with patch downsampling replaced by upsampling in this case. The results Tab. 5 show that both and @ un- derperform compared to B, indicating that both components are beneficial. Additionally, + shows the weakest perfor- mance, possibly because modeling long- cointegrated
6. Non-stationarity and Dependency Modeling Analysis
Intra-variate Modeling, As shown in Figure 4a, when non- stationarity is retained, the attention map in the temporal dimension diverges, with the model focusing on multiple patches across a broader time span. However, after remov- ing non-stationarity, the attention map becomes more con- centrated adjacent time steps, aligning with the causal nature of time series, where closer time steps are usually more correlated. Non-stationarity may cause the model to mistake distant similarities for causality. By eliminating non-stationariyy the model better captures short-term varia- tions and local dependencies, enhancing its robustness and interpretability in handling complex time series data.
Integnatl Anenion
Cuistagratcd Anemion
Wisther
Solar
Blcarisity
Traffi::
Ocks
Oukar
MSE
MAE
MSE
MAE
MAE
MAE
0:200
0.262
0.184
0.246
0.158
0.252
[.388
0.264
x
0.222
0.264
0.191
0.260
0.165
0,263
0.309
0.655
1
2
0.219
0.249
6.181
0.239
0.149
0,245
0,360
0.255
Inter-variate Modeling. Figure 4b shows that removing non-stationarity narrows the model's ''ss attention to few inter- variate dependencies, while retaining non-stationarity en- ables the capture of more diverse and richer relationships. Non Non-stationary sequences help the model identify cointe- gration, revealing hidden equilibrium mechanisms in multi-
2
| 0.227
0.266
0.190
0.252
0,160
0.255
0.396
0.251
Table Ablation the impact and order of Integrated Attention and Cointegrated Attention. "Order" specifics the sequence, with lower numbers indicating carlier placement. indicates the com- ponent is removed.
7

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
Stationary
Non-stationury
Stationary
Non-stationary
0.33
Qteyy Tlax Pacc
Queyy Tiac Pytt
Qzcn Chandl
Key Patch
Time Purch
Channe
Downssmpled Patch Number (M) (c) Results of Different M
(a) Intra-variate Attention
(b) Inter-variate Attention
Figure (a) Comparison intra- variate attention maps under stationary and non-stationary conditions for different patches in the Flectricity dataset. (h) Comparison of inter-variate attention maps between different variates under stationary and non-stationary conditions in the Solar dataset. (c) Impact of varying the number of downsampled patches on forecasting performance across different datasets. See Tah. 14 for full results.
Integrated Attention: Non-stationary Cointegrated Attention: Non-stationary
Integrated Attention: Stationary Cointegrated Attention Stationary
Integrated Attention: Non-stationary Cointegrated Attention Stationary
Integrated Attention: Stationayy
Cointegrated Attention Non Non-stationary
Teem predicted
Tdee (C))
Teex (C')
Tdew (C') (C))
20
20
20
T(C)
20
(C') predicted
15
TIC' predicte
15
TiC' predeted
15
(C") predicted
predicted
10
10
5
-5
-5
-10
100 200
300
400
500
600
700
-10
0
100
200
300
400
500
600
700
-10
0
100
200
300
400
500
600
700
-10
0
100
200
300
460 500 600 700
(a)
(b)
(d)
Figure 5. Visualization of the effect of retaining removing non-stationarity in Integrated Attention and Cointegrated Attention on the Weather dataset for temperature (I) and dew point temperature (Tazw). (a) Both Integrated and Cointegrated Attention retain
non-stationarity. (b) Both remove non-stationarity. (c) Only Integrated Attention retains non-stationarity. (d) Only Cointegrated Attention retains non-station.rity.
variate time series. Preserving non-stationarity enhances the model's s ability to express complex inter- variata dependen- cies. Additionally, Figure 4c shows the impact of different patch downsampling rates on performance. For datasets with more channels and stronger cointegration (e.g., Solar and Traffic), increasing downsampled patches initially im- proves predictions by preserving long-term features. How- ever, too much downsampling adds computational cost and negatively affects smaller-channel datasets (e.g.. Weather), so we carefully balanced downsampling rates based on dataset characteristics, detailed in Tab. 8.
modeling to capture the underlying cointegration between variables.
7. Conclusion
In this paper, we address the dual challenges of non- stationarity in multivariate time series forecasting, specifi- cally focusing on its distinct impacts on short-term and long- term modeling. To this end, we propose TimeBridge, a novel framework that bridges the gap between non-stationarity and dependency modeling. By employing Integrated Attention to mitigate short-term non-stationarity and Cointegrated At- tention to preserve long-term dependencies, TimeBridge ef- fectively captures both local dynamics and long-term cointe- gration. Comprehensive experiments across diverse datasets demonstrate that TimeBridge consistently achieves state- oh-the-art performance both short-term and long-term forecasting tasks. Moreover, its exceptional performance on the CSI 500 and S&P 500 indices underscores its robustness and adaptability to complex real-world financial scenarios. Our work paves the way for further exploration of models that balance the nuanced effects of non-stationarity, offering a promising direction for advancing multivariate time series forecasting.
Real Case of Weather Forecast. Given the strong interre- lationships between weather variables, we analyze temper- ature 7 and dew point temperature Taew from the Weather dataset. Dew point measures atmospheric moisture and is typically closely linked to temperature. Without external in- fluences, such as water vapor or heat sources, the difference between temperature and dew point is minimal, showing long-term cointegration. However, temperature tends ex- hibit more short-term fluctuations due to external factors like sunlight and weather systems. As shown in Figure 5, the results demonstrate that spurious regressions can only be avoided by eliminating non-stationarity during short-term modeling, while preserving it during long-term dependency
8

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
Acknowledgements
time series forecasting. International Joint Conference on Artificial Intelligence, pp. 3944-3952, 2024.
This work supported part by the National Natural Science Foundation of China, under Grant (62302309, 62171248), Shenzhen Science and Technology Program
Fanchon, P. and Wendel, J. Estimating var models under non-stationarity and cointegration: alternative approaches for forecasting cattle prices. Applied Economics, 24(2): 207-217, 1992.
Impact Statement
Han, L., Ye, H.-J., and Zhan, D.-C. The capacity and ro- bustness trade-off: Revisiting the channel independent strategy for multivariate time series forecasting. IEEE Transactions on Knowledge and Data Engineering, 2024.
Time series forecasting is fundamental in various real-world applications, including finance, weather prediction, and traf- fic management. Our work introduces an innovative ap- proach handling non-stationarity, improving both short- term and long-term dependency modeling. The datasets used in this study are publicly available, ensuring repro- ducibility and transparency. We do not foresee any ethical concerns associated with this research, Our contributions primarily advance the field time series forecasting, with potential benefits across multiple domains.
Hochreiter, S. and Schmidhuber, J. Long short-term memory. Neural Comput., 1997.
Hu, Y. Liu, P., Zhu, P., Cheng, D., and Dai, T. Adap- tive multi-scale decomposition framework for time se- ries forecasting. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pp. 17359-17367, 2025a.
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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
A. Time Series Integration and Cointegration Analysis
A.1. Integration and ADF Test
time series said to be integrated of order k, denoted as [k). if becomes stationary after differencing & times. For instance, a series X2 is I(1) if its first difference AX, = - is stationary. To test for non-stationarity, the Augmented Dickey-Fuller (ADF) test (Mushtaq, 2011) is commonly used. It examines the null hypothesis that a unit root is present, indicating non-stationarity:
AX, = a + 7 - + Σ 6 o
Here, AX, is the differenced series, is the coefficient the lagged series, and 6 is the error term. Rejecting the null hypothesis (7 0) indicates stationarity, while failing to reject it implies non-stationarity. Non-stationary data can lead to spurious regressions, where unrelated temporal intervals appear to be correlated due to common trends. We report the ADF test results in Tab. 7.
A.2. Cointegration and EG Test
Cointegration occurs when two or more non-stationary series move together over time, maintaining stable, long-term relationship. For example, if X, and Y are both 7(1), they are cointegrated if there exists stationary linear combination, Z, = X2 BY This indicates a shared stochastic trend. The Engle-Granger (EG) test (Bilgili, 1998) for cointegration involves two steps:
1. Estimate Long-term Relationship. Regress X, on Y, using Ordinary Least Squares (OLS);
X, = a + BY, + tx.
where are the residuals.
2. ADF Test on Residuals. Apply the ADF test to Ex:
Aa = 74-1 +
8, 8, +
If the residuals are stationary, X, and Y; are cointegrated.
Cointegration is vital for capturing long-term relationships between variables, providing robust foundation for multivariate time series modeling. Ignoring cointegration can result in models that miss significant underlying connections, reducing forecasting accuracy and reliability. We report the EG test results in Tab. 7.
B. Theoretical Analysis of Integrated and Cointegrated Attention
In this section, we provide more detailed theoretical analysis to justify the design choices behind Integrated Attention and Cointegrated Attention in non-stationary time series modeling. Our approach grounded in classical stochastic processes, specifically Brownian motion, to explain the rationale behind these mechanisms.
Proposition 1: Spurious Attention from Non-Stationary Inputs
Consider standard Brownian motion X, I(L), which is commonly used model for non-stationary processes. We define the process as follows:
X, = X + ue, Ue ~ ~((,00
From this, we know that:
12

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
Mean (X,) 0, Var(X,) = to?, Cov(X xt,) min(t1, 12)
Let two input patches of length S be:
P: = [x+++t1..... X(t+ts), Pj = (Xxtstt...,,,4-.1
The attention score between these patches pt and pj, based on the dot product between the two vectors, can be approximated as:
S score(pi.P) Pj) x P.P & Σ X & Σ
s
(X4+44 - () 0) x _ (oXXxtttss
s=1
Substituting the covariance expression, get:
S score(ps, Pj) α ((XXttttxxX min(t s,t s)o² x o² (smin(i,))+ S + + S
This score grows quadratically with both the time index and the patch length S. As increases, the score is dominated by long-term trends, leading spurious attention due to the influence of global trends, rather than capturing genuine short-term dependencies between patches. Figure 8 demonstrates this phenomenon, where many patches show high attention scores due to the accumulation of long-term variance, even though they do not represent meaningful short-term dependencies.
To mitigate this effect, we propose patch- wise detrending strategy:
p (ptrend(pp) [AX AXH++t I(0), AX, Xx Xt-1
This operation removes the non-stationary trend from the input series, leading stable variance:
Var (X,, = o²
Now, the attention score between the detrended patches yi and p/ becomes:
score(y/, x So²
This ensures that the attention mechanism focuses on the genuine short-term dependencies between the patches, unaffected by long-term drift.
Proposition Importance of Non-Stationarity in Capturing Cointegration
Cointegration is statistical property of a collection of time series variables, where linear combination of non- -ta-ssationary series can result stationary process. Let us consider two non-stationary time series X, and Y., both following a unit root process (1):
~ I(1,,
Y ~ I(1)
these series are cointegrated, there exists a linear combination of X, and Y, that is stationary:
Z, X, 3Y T(0)
13

---

TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
where 3 is constant coefficient that defines the relationship between the two series.
However, if we remove the non-stationarity by detrending the series, get:
AX; (Xtr)
AY, Detrend(Y;)
This leads to the following change in Ze:
== = AX, 3AY, Ft
where é, represents random noise sequence, which destroys the cointegration signal. Removing non-stationary components eliminates the vast majority of cointegration information, as shown in Figure 9.
Thus, to capture long-term dependencies accurately, we must preserve the non-stationarity between the variables. In our framework, Cointegrated Attention (CD) is designed to maintain these long-term dependencies, ensuring that the model can recognize and capture the cointegration between variables across time.
Summary of Propositions
These two propositions explain the necessity of Integrated Attention and Cointegrated Attention:
For short-term modeling, we eliminate non-stationarity to avoid spurious regressions and focus on stable, local dependencies within the data.
For long-term modeling, preserve non-stationarity to capture meaningful cointegration relationships between variables, which are crucial for modeling long-term equilibrium dynamics.
C. Metrics
C.1. Long-term Forecasting
We use Mean Squared Error (MSE) and Mean Absolute Error (MAE) as evaluation metrics. Given the ground truth values X, and the predicted values X,, these metrics are defined as follows:
MSE
E X,
x,)3
MAE
X
X
N
N 1=1
where N is the total number of predictions.
C.2. Short-term Forecasting
We use MAE (the same as defined above), Mean Absolute Percentage Error (MAPE), and Root Mean Squared Error (RMSE) to evaluate the performance. These metrics are defined as follows:
1 N
X;
X;
MAPE
100,
RMSE
Σ XX X,)
X,
C.3. Financial Forecasting
We use six widely recognized metrics to assess the overall performance of each method: Annual Return Ratio (ARR). Annual Volatility (AVol), Maximum Drawdown (MDD), Annual Sharpe Ratio (ASR), Calmar Ratio (CR), and Information Ratio (IR). Lower absolute values of AVol and MDD, coupled with higher values ARR, ASR, CR, and IR, indicate better performance.
14

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
A RR quantifies the percentage increase decrease in the value of an investment a year.
ARR (1 + Total Return) 1.
AVol measures the volatility of investment's returns over the course of year. Rp denotes the daily return of the portfolio.
AVol
V ())
MDD indicates the maximum decline from a peak to trough in the value of an investment.
Ppeea Ptrough
MDD max
Ppeak
ASR reflects the risk-adjusted return of an investment a year.
ARR
ASR
AVol
CR compares the average annual return of an investment to its maximum drawdown.
ARR
CR
MDD
IR evaluates the excess return of an investment relative to a benchmark, adjusted for its volatility. Rs is the daily return of the market index.
mean R2 Rs) std Rp -R))
IR
D. Datasets
We conduct extensive experiments on several widely-used time series datasets for long-term forecasting. Additionally, we use the PeMS datasets for short-term forecasting and the CSI 500 and S&P 500 indices for financial forecasting. We report the statistics in Tab. 7. Detailed descriptions these datasets are as follows:
(I) ETT (Electricity Transformer Temperature) dataset (Zhou al., 2021) encompasses temperature and power load data from electricity transformers in two regions of China, spanning from 2016 2018. This dataset has two granularity levels: ETTh (hourly) and ETTm (15 minutes).
(2) Weather dataset (Wu et al., 2023) captures 21 distinct meteorological indicators in Germany, meticulously recorded at 10-minute intervals throughout 2020. Key indicators in this dataset include air temperature, visibility, among others, offering comprehensive view of the weather dynamics.
(3) Electricity dataset (Wu et al., 2023) features hourly electricity consumption records in kilowatt-hours (kWh) for 321 clients. Sourced from the UCL Machine Learning Repository, this dataset covers the period from 2012 to 2014, providing valuable insights into consumer electricity usage patterns.
(4) Traffic dataset (Wu al., 2023) includes data hourly road occupancy rates, gathered by 862 detectors across the freeways of the San Francisco Bay area. This dataset, covering the years 2015 to 2016, offers detailed snapshot of traffic flow and congestion.
(5) Solar-Energy dataset (Liu al., 2024a) contains solar power production data recorded every 10 minutes throughout 2006 from 137 photovoltaic (PV) plants in Alabama.
(6) PeMS dataset (Liu et al., 2022a) comprises four public traffic network datasets (PeMS03, PeMS04, PeMS07, and PeMS08), constructed from the Caltrans Performance Measurement System (PeMS) across four districts in California. The data is aggregated into 5-minute intervals, resulting in data points per hour and 288 data points per day.
15

---

TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
CSI 500' contains 502 stocks listed on the Shanghai and Shenzhen stock exchanges in China from 2018 2023, including close, open, high, low, volume and turnover data.
(8) S&P 5002 contains 487 stocks representing diverse sectors within the U.S. economy from 2018 2023, including close, open, high, low and volume data.
Tasks
Dataset
Dim
Prediction Length
Dataset Size
Frequency
ADFi
EG
ETTm1
7
(96, 192, 336, 720}
(34465, 11521, 11521)
15 min
14.98
20
ETTm2
7
(96, 192, 336, 720
(34465, 11521, 11521)
15 min
5.66
17
ETTh1
7
(96, 192, 336, 720)
(8545, 2881, 2881)
1 hour 55.91
11
Long-term
ETTh2
7
(96, 192, 336, 720
(8545, 2881, 2881)
1 hour 4.13
10
Forecasting
Electricity
321
(96, 192, 336, 720}
(18317, 2633, 5261)
hour 8.44
39567
Traffic
862
(96, 192, 336,777,7 720)
(12185, 1757, 3509)
1 hour 15.02
354627
Weather
21
(96, 192, 336, 720)
(36792, 5271, 10540)
10 min 266.88
77
Solar-Energy
137
(96, 192, 336, 720}
(36601, 5100,,,161,,1111
10 min
37.23
8373
PeMS03
358
12
(15617, 5135, 5135)
5 min
19.05
Short-term
PeMS04
307
12
(17172,3375, 3375)
5 min min -20.60
- 15..66
Forccasting
PeMSO7
883
12
(16911, 5622, 5622)
PeMS08
170
12
(10690, 3548, 265)
5 min 16.04
Financial
CSI 500
502
1
(943, 242, 242)
day 3.06
Forecasting
S&P 500
487
(1008, 251, 249)
day 2.80
Augmented Dickey-Fuller (ADF) Test: smaller ADF test result indicates more stationary time series
data.
Engle-Granger (EG) Test: A bigger EG test result indicates the data contains more cointegration relationships.
Table Dataset detailed descriptions. "Dataset Size" denotes the total number of time points (Train, Validation, Test) split respectively. "Prediction Length" denotes the future time points to be predicted. "Frequency" denotes the sampling interval of time points.
To further illustrate the degree of soatiootion the datasets, we conduct additional experiments using a Random Walk series (representing maximum non-stationarity) and Gaussian white noise (representing near-stationarity), The Random Walk series is generated using the formula X, X,-1 + with E, ~O(0,, 1), where we set 10, 000 and simulate 100 iterations. The average ADF value for the Random Walk series is -1.53, indicating a high degree of non-stationarity In contrast, for the Gaussian white noise series, generated as X, N , 1) with the same settings, the average ADF value is -97.54, indicating strong stationarity. Comparing these results with those Tab. 7, we can see that most datasets exhibit significant non-stationarity, especially the ETT, CSI 500, and S&P 500 datasets.
To analyze cointegration, we conducted the Engle-Granger (EG) test on all eight datasets of long-term forecasting. The results indicate that datasets with more channels tend to exhibit more extensive cointegration relationships. This is particularly evident in datasets like Electricity and Traffic, which show significantly higher EG test values, reflecting a greater abundance of long-term equilibrium relationships among variates. For these high-dimensional datasets, effectively modeling the intricate cointegration structures is crucial, as neglecting these long-term dependencies can result in suboptimal predictions.
'https:// invest i ing. com/ ndi seee/chiie na na-naenee securities-500 180 Pttpps: //hk, finance, yahoo. cww.qucte///cccccctootcccccccc.t htthho/.cmmmgottt///sss///// /tt///////sss/////////
16

---

TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
E. Implementation Details
All experiments are implemented in PyTorch (Paszke et al., 2019) and conducted on two NVIDIA RTX 3090 24GB GPUs. We use the Adam optimizer (Kingma, 2014). All models are trained for 100 epochs. Following the protocol outlined in the comprehensive benchmark TFB (Qiu et al., 2024), the drop-last trick is disabled during the test phase. Tab. 8 provides detailed hyperparameter settings for each dataset. For the four ETT datasets, the relatively small number of channels results in less pronounced long-term cointegration relationships, as evidenced by the low EG test results in Tab. 7. Therefore, we focus on modeling short-term intra-variate variations only. To improve training stability, we adopt a hybrid MAE loss that operates in both the time and frequency domains. Compared to MSE, which amplifies differences, MAE provides a more stable optimization process. Additionally, the frequency-domait loss mitigates label autocorrelation, making training more effective. The loss function defined as:
L
X
X
b FFT(X;) FFT(X,)),
N
L L 0) I a T.j,
where FFT denotes the Fast Fourier Transform. a the hyperparameter. We conduct additional experiments where TimeBridge was trained using the MSE loss and compare the results with the best baseline, DeformableTSTT in Tab. 15. The results show that the hybrid MAE loss improves TimeBridge's performance
four ETT datasets. The ETT datasets have relatively few channels (C 7), which limits TimeBridge 's ability to utilize the Cointegrated Attention module to capture cointegration information. We only use the Integrated Attention to model ETT datasets (see Tab. 8). Additionally, the ETT data exhibits high degree of random fluctuation, which is better modeled using loss function that weighs both time and frequency components. Hence, the hybrid MAE loss strengthens the modeling of short-term dependencies in such datasets. On datasets with more channels (c.g., Electricity, Traffic, Solar), the choice of loss function has minimal impact, as both losses yield similar results.
Num. of Integrated
Num. of Cointegrated
N
M
d_model
d_ff
ETTh1 ETTh2
3
0
30
30
2e-4
128
128
0.35
3
0
15
15
lc-4
128
128
0.35
ETTml
3
0
15
15
2e-4
64
128
0.35
ETTm2
3
0
15
15
2e-4
64
64
0.35
Weather
1
1
30
12
le-4
128
128
0.1
Solar
1
1
30
12
5e-4
128
128
0.05
Electricity
1
2
30
4
5e-4
512
512
0.2
Traffic
1
3
30
5e-4
512
512
0.35
Table Hyperparameter settings for different datasets. "N" denotes the number of patches. "M" denotes the number of patches after the patch downsampling block. "Ir" denotes the learning rate. "d model" and "d_ff" denote the model dimension of attention layers and feed-forward layers, respectively.
F. Full Results
F.1. Main Experiments
Tab. 9 present the full results for long-term forecasting through hyperparameter search. The hyperparameter search process involved exploring input lengths I E {96, 192, 336, 512, 720}, learning rates from 10-5 to 0.05, encoder layers from to 5, dmadel values from 16 to 512, and training epochs from 10 to 100. In both settings, TimeBridge consistently achieved the best performance, demonstrating its effectiveness and robustness.
Additionally, for financial forecasting, we included three additional strong baselines: ALSTM (Qin et al., 2017), GRU (Chung et al., 2014), and TRA (Lin et al., 2021). The results in Tab. 10 show that TimeBridge continues to outperform these
17

---

TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
methods, further validating its superiority.
F.2. Ablation Studies
We present the full results of the ablation studies discussed in the main text. Tab. 11 provides the complete results of the ablation on removing non-stationarity in both Integrated and Cointegrated Attention. Tab. 12 reports the full results on
TimeBridge
iTransformer DeformaleeST
TimeMixer
PatchTST
Crossformer
Leddam (2024)
ModemIIN
MICN
TimesNet (2023)
DLinear (2023)
Models
(Ours)
(20244)
(2024)
(20245)
(2023)
(2023)
(2024)
(2022)
Metric
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAB
MSE
MAE
MSE
MAE
96
0.284
0.337
0.300
0.353
0.291
0,347
0,293
0.345
0,293
0.346
0.310
0.361
0.294
0.347
0.292
0,346
0,314
0.360
0.338
0.375
0.299
0.343
192
0.317
0.367
0.345
0.382
0.325
0.372
0.335
0.372
0.333
0.370
0.363
0.402
0.334
0.370
0.332
0,368
0.359
0.387
0.371
0.387
0.335
0.365
0.361
0.394
0.374
0.398
0.359
0.390
0.368
0.386
0.369
0.369
0.389
0.430
0.392
0.425
0.365
0.391
0.398
0.413
0.410
0.411
0.369
0.386
720
0.413
0:418
0.429
0.430
0.418
0.423
0.426
0.417
0.416
0.420
0.600
0.547
0.421
0.419
0.416
0.417
0.459
0.464
0.478
0.450
0.425
0.421
Aug.
00.344
0.379
00.362
0.391
D.348
0.383
0.355
0.380
0.353
0.382
0.420
0.435
0.354
0.381
|0:351
0.381
0.383
0.406
(0.400
0.406
00.357
0.379
96
00.157
0.243
0.175
0.266
0.169
0.258
0.165
0.256
0.166
0.256
0.263
0.359
0.174
0.260
0.166
0.256
0.178
0.273
0.187
0.267
0.167
0.260
157
192
0.217
0.285
0.242
0.312
0.229
0.299
0.225
0.298 0.223
0.296 0.345
0.400
0.231
0.301
0.222
0.293
0.245
0.316
0.249
0.309
0.224
0.303
336
0.269
0.321
0.282
0.340
0.280
0,333
0.277
0.332
0.274
0.329
0.469
0.496
0.288
0.336
0.272
0.324
0.295
0.350
0.321
0.351
0,281
0.342
720
0.348
0.378
0.378
0.398
0.349
0.384
0,360
0,387 0,362
0.385
0.996
0.750
0.368
0.386
0.351
0.381
0,389
0.406
0.497
0.403
0.397
0.421
Arg.
(0.246
0.310
002699
0.329
10,257
0.319
0,257
0.318 0,256
0.317 0.518
0.501
0.265
0.320
0.255
0.314
0.277
0.336
0.291
0.333
0.266
0.332
96
0.350
0,389
0.386
0.405
0.369
0.396
0,372
0.401
0.370
0.400
0.386
0.426
0.377
0.394
0.368
0.394
0.396
0.427
0.384
0.402
0.375
0.399
192
0,388
0.414
0.424
0.440
0.410
0.417
0,413
0.430
0.413
0.429
0.413
0.442
0.408
0.427
0:405
0.413
0.430
0.453
0.557
0.436
0.405
0.416
336
0.408
0.430
0.449
0.460
0.391
0.414
0.438
0.450
0.422
0.440
0.440
0.461
0.424
0.437
0.391
0.412
0.433
0.458
0.491
0.469
0.439
0.443
720
0.443
0.463
0.495
0.487
0.447
0.464
0.486
0.484
0.447
0.468
0.519
0.524
0.451
0.465
0.450
0.461
0.474
0.508
0.521
0.500
0.472
0.490
Aug.
(0.997
0.424|
00.439
0.448
10.404
0.423
0.427
01.41
0.413
0.434
01.40
0.463
0..415
0.430
0.404
0.420
0..43
0.462
0..558
0.450
[0.423 0..37
96
0.271
0.331
0.2997
0.348
00.272
0.334
0,281
0.351
0.274
0.337
0.611
0.557
0.283
0,345
0.263
0.332
0,289
0.357
0.340
0.374
0,289
0.353
192
0.335
0.370
0.371
0.403
0.325
0.369
0.349
0.387
0.314
0.382
0.703
0.624
0.339
0,381
0.320
0.374
0,409
0.438
0.402
0.414
0,383
0.418
336
0.371
0.402
0.404
0.428
0.319
0.373
0.366
0.413
0.329
0.384
0.827
0.675
0.366
0.405
0,313
0.376
0.417
0.452
0.452
0.452
0.448
0.465
720
0.387
0.425
0.424
0.444
0.395
0.433
0.401
0.436
0.379
0.422 1.094
0.775
0.395
0.436
0.392
0.433
0.426
0.473
0.462
0.468
0.605
0.551
Argg
|0.341
0.382
0..374
0.406
10.328
0.377
0.349
0.397
0.324
0.381
0.809
0.658
0.345
0.391
|o,322
0.379
0.385
0.430
0.414
0.427
00.431
0.447
96
00.144
0.184
0.159
0.208
0.146
0.198
0.147
0.198
0.149
0.198
0. 0.1
0.212
0.149
0.200
0.149
0,200
0.166
0.226
0,112
0.220
0..15
0.237
Weatter
192
0.186
0.225
0.200
0.248
0.191
0.239
0.192
0.243
0.194
0.241
0.195
0.261
0.193
0.240
0.196
0.245
0.220
0.283
0.219
0.261
0.220 0.282
0.237
0.267
0.253
0.289
0.241
0,280
0,247
0,284
0.245
0.282
0,252
0.241
0.279
0.238
0.277
0.275
0.328
0,280
0.306
0.265 0.319
720
0.307
0.320
0.321
0.338
0.310
0.331
0,318
0.330
0.314
0.334
0.318
0.363
0.324
0.338
0.314
0.334
0.311
0.356
0.366
0.359
0.323 0.362
Augg
(0.219
0.249
0.2233
0.271
0.222
0.262
0.226
0,264
0.226
0.264
0.228
0.287
0.226
0.264
00.224
0.264
0.242
0.298
0.259
0.287
0.2240 0.300
96
00.120
0.214
0.138
0.237
0.132
0.234
0.153
0.256
0.129 0.222
0.135 0.237
0.134
0.228
0.129
0.226
0.159
0.267
0.168
0.272
0.140 0.237
Blectricity
0.142
0.237
0.157
0.256
0.148
0.248
0.168
0.269
0.147 0.240
0. 06160 0.262
0.155
0.248
0.143
0.239
0.168
0.279
0.184
0.289
0.152 0.249
336
0.156
0.252
0.167
0.264
0.165
0.266
0.189
0.291
0.163
0.259
0. 182 0.282
0.173
0. 0.268
0.161
0.259
0.196
0.308
0.198
0.300
0. 069 0.267
720
0.179
0.278
0.194
0.286
0.197
0.296
0,228
0.320
0.197
0.290
0,246 0.337
0.186
0,282
0.191
0.286
0,203
0.312
0.220
0.320
0.203 0.301
Aug.
[0.149
0.245
00.164
0.261
[0.161
0.261
0,185
0.284
0.159
0.253
0. 081
0.279
0.162
0.256
0.156
0.253
0.182
0.292
0.119
0.295
0,1666 0.264
96
00.340
0.240
0.363
0.265
00.355
0.261
0.369
0.257
0.360
0.249
0.512
0.282
0.415
0.264
0.368
0.253
0.508
0.301
0.593
0.321
00.400
0.282
Traffic
192
0.343
0.250
0.385
0.273
0,380
0.271
0,400
0.272
0.379
0.256
0.501
0.273
0.445
0.277
0.379
0.261
0.536
0.315
0.617
0.336
0.423
0.287
336
0.363
0.257
0.396
0.277
0.393
0.281
0.407
0.272
0.392
0.264
0.507
0.279
0.461
0.286
0.397
0.270
0.525
0,310
0.629
0.336
0..46
0.296
720
0.393
0.271
0.445
0.312
0.434
0.300
0.461
0.316
0.432
0.286
0.571
0.301
0.489
0.305
0.440
0.296
0.571
0.323
0.640
0.350
0.466
0.315
Aug.
(0.360
0.255
0..997
0.282
[0.391
0.278
0.409
0.279
0.391
0.264
0.523
0.284
0.452
0.283
00.99
0.270
0.535
0.312
0.620
0.336
(0.434 0.295
96
00.161
0.224
0.188
0.242
0.165
0.238
0.179
0.232
0.178
0.229
0.166
0.230
0.197
0.241
0.202
0.263
0.188
0.252
0.219
0.314
0.216
0.287
192
0.177
0.237
0.193
0.258
0.184
0.254
0
0.259
0.189
0.246
0.
186
0.237
0.231
0.264
0.223
0.279
0.215
0.280
0.231
0.322
0.244
0.305
Solar
336
0.188
0.244
0.195
0.259
0.191
0.263
0.190
0.256
0.198
0.249
0.203
0.243
0.216
0.272
0.241
0.292
0.222
0.267
0.246
0.337
0,263
0.319
720
0.197
0.252
0.223
0.281
0.199
0.262
0.203
0.261
0.209
0.256
0.210
0.256
0.250
0.281
0.247
0.292
0.226
0.264
0.280
0.363
0.264
0.324
Avg.
[0.181
0.239
0.200
0.260
0.185
0.254
0,193 0.252 0.194 0.245 0.191
0.242
0.223
0.264
0.228
0.282
0.213
0,266
0.244 0.334
00.247 0.309
Table 9. Full results of long-term forecasting of hyperparameter searching. All results are averaged across four different prediction lengths: O E [96, 192, 336, 720). The best and second-best results are highlighted in bold and underlined, respectively.
18

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TimcBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
the impact and order of Integrated and Cointegrated Attention, with illustrative visualization in Figure 6. Additionally, Tab. 13 shows the results of ablation on different modeling approaches for these attention mechanisms. Finally, Tab. 14 presents the results of varying the number of downsampled patches M and its effect on forecasting performance.
G. Statistical Analysis
We repeat all experiments three times and report the standard deviations for both our model and the second-best baseline, along with the results of statistical significance tests. Tab. 16, Tab. 17, and Tab. 18 present the results for long-term tnggteemm forecasting, short-term forecasting, and financial forecasting, respectively.
CSI 500
S&P 500
Models
ARRI
AVol
MDD1
ASR
CR$
IRT
ARRI
AVoll
MDD1
ASR
CR
IR1
BLSW (1988)
0.110
0.227
-0.155
0.485
0.710
0.446
0.199
0.318
-0.223
0.626
0.892
0.774
CSM (1993)
0.015
0.229
-0.179
0.066
0.084
0.001
0.099
0.250
-0.139
0.396
0.712
0.584
LSTM (1997)
-0.008
0.159
-0.172
-0.047
-0.044
-0.128
0.142
0.162
-0.178
0.877
0.798
0.929
ALSTM (2017)
0.016
0.162
-0,192
0.101
0.086
0.014
0.191
0.161
-0.150
1.186
1,273
1,115
GRU (2014)
-0,004
0.159
-0.193
-0.028
-0.023
-0.118
0.124
0.169
-0.139
0.734
0.829
1.023
Transformer (2017b)
0.154
0.156
-0.135
0.986
1.143
0.867
0.135
0.159
-0.140
0.852
0.968
0.908
TRA (2021)
0.125
0.162
-0.145
0.776
0.866
0.657
0.184
0.166
-0.158
1.114
1.172
1.106
PatcTTST (2023)
0.118
0.152
-0.127
0.776
0.923
0.735
0.146
0.167
-0.140
0.877
1.042
0.949
iTransformer (2024a)
0.214
0.168
-0.164
1.276
1.309
1.173
0.159
0.170
-0.139
0.941
1.150
0.955
TimeMixer (2024h)
0.078
0.153
-0.114
0.511
0.685
0.385
0.254
0.162
-0.131
1,568
1.938
1.448
Crossformer (2023)
00.39
0.163
-0,217
-0.238
-0.179
-0,350
0.284
0.159
-0.114
1.786
2.491
1.646
TSMixer (2023)
0.086
0.156
-0.143
0.551
0.601
0.456
0.187
0.173
-0.156
1.081
1.199
1.188
TimeBridge
0.285
0.203
-0.196
1.405
1.453
1.317
0.326
0.169
-0.142
1.927
2.298
1.842
Table 10. Full results for financial time series forecasting in CSI 500 and S&P 500 datasets.
Input
Input
Patch Embedding
Patch Embedding
Integrated
Cointegrated Attention
Attention
Input
Input
Patch Embedding
Patch Embedding
Patch Downsampling
Patch Upsampl ing
Integrated Attention
Cointegrated Attention
Cointegrated Attention
Integrated Attention
Projection
Projection
Projection
Projection
Output
Output
Output
Output
Figure 6. Illustration of the impact and order of Integrated Attention and Cointegrated Attention in Tab. 12: ① Integrated Attention only, @ Cointegrated Attention only, @ Integrated Attention followed by Cointegrated Attention, and @ Cointegrated Attention followed by Integrated Attention, with patch downsampling replaced by upsampling.
19

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
Integrated Attention
Cointegrated Attention
Weather
Solar
Electricity
Traffic
Norm?
Norm?
Length
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.144
0.193
0.163
0.227
0.124
0.221
0.342
0.241
192
0.186
0.235
0.180
0.240
0.144
0.240
0.351
0.255
x
X
336
0.239
0.279
0.191
0.248
0.158
0.254
0.374
0.261
720
0.311
0.333
0.197
0.253
0.184
0.282
0.418
0.284
Avg.
0.220
0.260
0.183
0.242
0.153
0.249
0.371
0.260
96
0.144
0.193
0.164
0.227
0.124
0.220
0.348
0.247
192
0.188
0.237
0.180
0.240
0.146
0.240
0.370
0.257
336
0.239
0.279
0.191
0.248
0.161
0.258
0.382
0.264
720
0.308
0.331
0.197
0.293
0.189
0.285
0.422
0.283
Avg.
0.220
0.260
0.183
0.252
0.155
0.251
0.381
0.263
96
0.144
0.184
0.161
0.224
0.120
0.214
0.340
0.240
192
0.186
0.225
0.177
0.237
0.142
0.237
0.343
0.250
x
336
0.237
0.267
0.188
0.244
0.156
0.252
0.363
0.257
720
0.307
0.320
0.197
0.252
0.179
0.278
0.393
0.271
Avg.
0.219
0.249
0.181
0.239
0.149
0.245
0.360
0.255
96
0.143
0.193
0.163
0.227
0.123
0.219
0.343
0.241
192
0.186
0.236
0.180
0.239
0.144
0.239
0.367
0.254
336
0.238
0.278
0.191
0.247
0.159
0.256
0.379
0.262
720
0.307
0.330
0.197
0.253
0.185
0.284
0.405
0.277
Avg.
0.219
0.259
0.183
0.242
0.153
0.250
0.374
0.289
Table 1. Full results of ablation on the effect of removing non-stationarity in Integrated Attention and Cointegrated Attention. indicates the use patch normalization eliminate non- stationarity, while means -onsstationartty retained.
Integrated Attention
Cointegrated Attention
Weather
Solar
Electricity
Traffic
Order
Order
Length
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.144
0.196
0.163
0.227
0.127
0.221
0.356
0.245
192
0.186
0.238
0.182
0.242
0.145
0.239
0.377
0.259
1
x
336
0.241
0.283
0.192
0.246
0.162
0.257
0.390
0.265
720
0.310
0.332
0.199
0.260
0.197
0.289
0.427
0.286
Avg.
0.220
0.262
0.144
0.244
0.158
0.252
0.388
0.264
96
0.147
0.200
0.161
0.240
0.127
0.227
0.348
0.252
192
0.191
0.242
0.195
0.259
0.155
0.254
0.358
0.262
x
1
336
0.242
0.283
0.198
0.268
0.173
0.273
0.367
0.267
720
0.308
0.331
0.209
0.273
0.203
0.299
0.401
0.278
Avg.
0.222
0.264
0.191
0.260
0.165
0.263
0.369
0.265
96
0.144
0.184
0.161
0.224
0.120
0.214
0.340
0.240
192
0.186
0.225
0.177
0.237
0.142
0.237
0.343
0.250
2
336
0.237
0.267
0.188
0.244
0.156
0.252
0.363
0.257
720
0.307
0.320
0.197
0.252
0.179
0.278
0.393
0.271
Avg.
0.219
0.249
0.181
0.239
0.149
0.245
0.360
0.255
96
0.148
0.199
0.174
0.237
0.130
0.225
0.370
0.274
192
0.193
0.243
0.187
0.251
0.147
0.240
0.386
0.279
2
336
0.245
0.284
0.195
0.258
0.165
0.262
0.394
0.276
720
0.320
0.336
0.203
0.262
0.199
0.291
0.432
0.295
Avg.
0.227
0.266
0.190
0.252
0. 160
0.255
0.396
0.281
Table 12. Full results of ablation on the impact and order of Integrated Attention and Cointegrated Attention. "Order" specifies the sequence, with lower numbers indicating carlier placement. indicates the component is removed.
20

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
Integrated Atlention
Cointegrated Attention
Weather
Solar
Electricity
Traffic
Cl CD
CI CD
Length
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.144
0.192
0.163
0.227
0.125
0.221
0.358
0.260
192
0.185
0.236
0.180
0.240
0.144
0.242
0.373
0.271
CI
CI
336
0.237
0.278
0.191
0.247
0.161
0.255
0.391
0.282
720
0.307
0.330
0.197
0.258
0.196
0.288
0.425
0.292
Avg.
0.218
0.259
0.183
0.243
0.157
0.252
0.387
0.276
96
0.146
0.195
0.165
0.229
0.125
0.222
0.362
0.266
192
0.188
0.239
0.178
0.245
0.145
0.241
0.374
0.274
CD
C1
336
0.242
0.284
0.191
0.252
0.166
0.263
0.388
0.282
720
0.310
0.331
0.201
0.261
0.205
0.293
0.423
0.296
Avg.
0.222
0.262
0.184
0.247
0.160
0.255
0.387
0.280
96
0.144
0.184
0.161
0.224
0.120
0.214
0.340
0.240
192
0.186
0.225
0.177
0.237
0.142
0.237
0.343
0.250
C1
CD
336
0.237
0.267
0.188
0.244
0.156
0.252
0.363
0.257
720
0.307
0.320
0.197
0.252
0.179
0.278
0.393
0.271
Avg.
0.219
0.249
0.181
0.239
0.149
0.245
0.360
0.255
96
0.146
0.197
0.162
0,229
0.125
0.221
0.352
0.254
192
0.188
0,238
0.178
0,245
0.148
0,246
0.361
0.266
CD
CD
336
0.241
0.282
0.191
0.254
0.161
0.261
0.377
0.267
720
0.313
0.333
0.199
0.260
0. 189
0.289
0.412
0.288
Avg.
0.222
0.263
0.183
0.247
0.156
0.254
0.376
0.269
Table 13. Full results of ablation on modeling approaches for Integrated Attention and Cointegrated Attention. "CI" denotes channel independent and "CD" denotes channcl-dependent modeling.
Downsampled
Weather
Solar
Electricity
Traffic
Patch Number M
Length
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.171
0.231
0.172
0.234
0.135
0.231
0.349
0.252
192
0.207
0.261
0.187
0.252
0.158
0.255
0.358
0.259
336
0.257
0.298
0.196
0.257
0.182
0.278
0.382
0.269
720
0.323
0.344
0.203
0.258
0.191
0.290
0.414
0.282
Avg.
0.239
0.284
0.189
0.250
0.166
0.264
0.375
0.266
96
0.147
0.201
0.168
0.231
0.120
0.214
0.340
0.240
192
0.189
0.241
0.183
0.245
0.142
0.237
0.343
0.250
4
336
0.240
0.282
0.195
0.251
0.156
0.252
0.363
0.257
720
0.309
0.332
0.200
0.255
0.179
0.278
0.393
0.271
Avg.
0.221
0.264
0.186
0.246
0.149
0.245
0.360
0.255
96
0.144
0.195
0.163
0.225
0.119
0.219
0.338
0.240
192
0.186
0.237
0.183
0.243
0.146
0.244
0.341
0.249
336
0.238
0.280
0.195
0.251
0.161
0.260
0.379
0.264
720
0.308
0.330
0.196
0.250
0.177
0.277
0.400
0.280
Avg.
0.219
0.261
0.184
0.242
0.151
0.250
0.365
0.258
96
0.143
0.184
0.161
0.224
0.120
0.220
0.334
0.238
192
0.186
0.225
0.177
0.237
0.148
0.247
0.337
0.250
12
336
0.237
0.267
0.188
0.244
0.163
0.264
0.363
0.256
720
0.307
0.320
0.197
0.252
0.176
0.286
0.387
0.268
Avg.
0.219
0.249
0.181
0.239
0.151
0.254
0.355
0.253
96
0.145
0.195
0.149
0.223
0.122
0.223
0.333
0.235
192
0.187
0.238
0.175
0.236
0.149
0.249
0.343
0.254
16
336
0.241
0.282
0.187
0.244
0.165
0.265
0.354
0.256
720
0.312
0.328
0.196
0.250
0.179
0.276
0.386
0.269
Avg.
0.222
0.261
0.177
0.238
0.152
0.253
0.354
0.254
Table 14. Full results of varying the number of downsampled patches M on forecasting performance.
21

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
Model
TimeBridge (Hybrid MAE)
TimeBridge (MSE)
DeformableTST (2024)
Dataset
MSE
MAE
MSE
MAE
MSE
MAE
ETTml
0.344
0.379
0.353
0.388
0.348
0.383
ETTm2
0.246
0.310
0.246
0.310
0.257
0.319
ETThl
0.397
0.424
0.399
0.420
0.404
0.423
ETTh2 0.341
0.382
0.346
0.394
0.328
0.377
Weather
0.219
0.249
0.218
0.250
0.222
0.262
Electricity 0.149
0.245
0.149
0.246
0.161
0.261
Traffic 0.360
0.255
0.360
0.252
0.391
0.278
Solar
0.181
0.239
0.181
0.238
0.185
0.254
Table 15. Average results of different prediction length O E {96, 192, 336, 720} with Hybrid MAE loss and MSE loss.
Model
TideBridge
DeformableTST (2024)
Confidence
Dataset
MSE
MAE
MSE
MAE
Interval
ETTml
0.344 + 0.014
0.379 + 0.010
0.348 + 0.008
0.383 + 0.006
99%
ETTm2
0.246 + 0.004
0.310 + 0.012
0.257 + 0.003
0.319 + 0.003
99%
ETTh1
0.397 + 0.010
0.424 + 0.008
0.404 + 0.015
0.423 + 0.006
99%
ETTh2
0.341 + 0.018
0.382 + 0.015
0.328 0.009
0.377 + 0.010
99%
Weather
0.219 + 0.006
0.249 + 0.004
0.222 + 0.009
0.262 + 0.006
99%
Electricity
0.149 + 0.011
= 0 0.007
0.161 + 0.010
0.261 + 0.015
99%
Traffic
0.360 + 0.008
0.255
+ 0.013
0.391 + 0.016
0.278 + 0.010
99%
Solar
0.181 + 0.002
0.239 + 0.003
0.185 + 0.008
0.254 + 0.005
99%
Table Standard deviation and statistical tests for TimeBridge and second-best method (DeformablcTSTT on ETT, Weather, Electricity, Traffic and Solar datasets.
Model
TimeBridge
TimeMixer (2024b)
Confidence
Dataset
MAE
MAPE
RMSE
MAE
MAPE
RMSE
Interval
PeMS03
14.63 0.164
14.21 + 0.133
23.10 + 0.186
14.63 + 0.112
14.54 0.105
23.28 + 0.128
99%
PeMS04
19.24 0.131
12.42 0.108
0.112
19.21 + 0.217
12.53 0.154
30.92 + 0.143
99%
PeMSO7
20.43 0.173
8.42 0.155
33.44
0.118
8.62 0.112
33.59 0.273
99%
PeMS08
14.98 0.278
9.56 + 0.126
23.77
15.22 + 0.311
9.67 0.101
24.26 + 0.212
99%
Table 17. Standard deviation and statistical tests for TimeBridge and second-hest method (TimeMixer) the PeMS dataset.
Model
TimcBridge
Crossformer (2023)
Confidence
Dataset
ARR
AVol
MDD
ARR
AVol
MDD
Interval
CSI 500
0.285 1: 0.033
0.203 1 0.012
0.196 1 0.010
0.039 1 0.027
0.163 0.009
0.217 1 0.011
95%
S&P 500
0.326 0.022
0.009
0.142 1 0.010
0.284 1
0.159 0.011
0.114 4 0.014
95%
Table 18. Standard deviation and statistical tests for TimeBridge and second-bes method (Crossformer) the CSI 500 and S&P 500 datasct.
22

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
H. Visualization
Figure visualizes short-term fluctuations and long-term cointegration across stock sectors. Figure 8 provides additional examples of intra-variate attention maps comparing stationary and non-stationary conditions for different patches in the Electricity dataset. Figure 9 shows further examples of inter-variate attention maps in the Solar dataset under both stationary and non-stationary conditions. Figure 10, Figure 11, Figure 12, and Figure 13 present long-ter forecasting visualizations for Weather, Solar, Electricity, and Traffic datasets, respectively. We display the last 96 input steps based on each model's s optimal input length, along with the corresponding 96 predicted steps. Finally, Figure 14 illustrates short-term forecasting for the PeMS03 dataset, where each model predicts 12 steps from 96-step input.
Banks
23
Long-term Cointegration
NBFIs
Retails
Short-term
Fluctuations
Price
35-
2012
2019
2023
2021
2034
Time
Figure 7. Visualization of short-term term fluctuations and long-ten cointegration across stock sectors. "NBFIs" represents Non-Bank Financial Institutions. The figure highlights how sectors experience short-term price volatility while maintaining long-term cointegration.
Stationary
stationary
Stationary
Non-stationary
Quay Tineaas
Dwer Time
Quer timetaach
Key
Figure 8. Additional examples comparing intra-variate attention maps under stationary and non-stationary conditions for different patches the Electricity dataset.
Stationary
Non-stationary
Stationary
Non-stationary
Qvey Cland
Quary Chumil
Clannel
Channel
Chamna
Figure 9. Additional examples comparing inter-variate attention maps between different variates under stationary and non-stationary conditions the Solar dataset.
23

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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
Production
Poodition
0.08
0.08
0.08
0.08
0.00
0.00
0.06
Muw
0.06
Wap
0.00
0.00
0.00
0.00
50
1.00
1550
200
50
3,00
150
200
50
2.00
130
200
50
3.00
150
200
(a) TimeBridge
(b) TTransformer
(c) PatchTST
(d) TimeMixer
0.10
Groandhhutt
0.10
Groundiruth
0.10
Grounditum
11.00
0.08
0.00
0.08
0.00
0.04
0.64
0.04
0.02
0.02
0.02
0.02
(e) TimesNet
(f) ModernTCN
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Figure 11. Visualization predictions from different models the Solar dataset.
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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
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TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting
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26


Reviews:
[{"review_id": "Qxiird3vSs", "reviewer": "Reviewer_1BTh", "summary": "This paper introduces a new framework to tackle the challenges posed by non-stationarity in multivariate time series forecasting. It addresses both short-term fluctuations and long-term trends by employing two specialized attention mechanisms, i.e., Integrated Attention and Cointegrated Attention. The framework is validated with extensive experiments on real-world datasets, demonstrating superior performance in both short-term and long-term forecasting.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your insightful comments on our work. Below, we address each of your questions:\n\n**Q1:** A more detailed theoretical exploration could strengthen the framework's validity.\n\n**A1:** Thank you for your valuable comments. Below we provide a concise theoretical explanation grounded in classical stochastic processes, specifically Brownian motion, to clarify the rationale behind our design.\n\n**Proposition 1: Spurious Attention from Non-Stationary Inputs**\n\nConsider a standard Brownian motion $X_t \\sim I(1)$, where: \n\n$X_t = X_{t-1} + u_t,\\quad u_t \\sim \\mathcal{N}(0, \\sigma^2)$.\n\nThen we have:\n\n$\\text{Mean}(X_t)=0, \\quad \\text{Var}(X_t) = t\\sigma^2,\\quad \\text{Cov}(X_{t_1}, X_{t_2}) = \\min(t_1, t_2)\\sigma^2$\n\nLet two input patches of length $S$ be:\n\n$p_i = [X_{t+i+1}, \\dots, X_{t+i+S}],\\quad p_j = [X_{t+j+1}, \\dots, X_{t+j+S}]$\n\nTheir attention score $\\text{score}(p_i, p_j)$ is approximated as:\n\n$\\text{score}(p_i, p_j) \\propto p_i p_j^T \\propto \\sum_{s=1}^{S} (X_{t+i+s} X_{t+j+s}) \\propto \\sum_{s=1}^{S} \\text{Cov}(X_{t+i+s}, X_{t+j+s})$\n\n$\n\\propto \\sigma^2 \\left(S\\min(i, j) + \\frac{S^2 + 2St + S}{2}\\right)\n$\n\nThis score grows with both the time index $t$ and the square of the patch length $S$, leading to spurious attention caused by global trends rather than genuine short-term dependencies. **As shown in Figure 4(a), many patches tend to exhibit high attention scores.**\n\nTo mitigate this, we perform patch-wise detrending:\n\n$p_i' = \\text{Detrend}(p_i) = [\\Delta X_{t+i}, \\dots, \\Delta X_{t+i+S}] \\sim I(0), \\quad \\Delta X_t = X_t - X_{t-1} $\n\nIn this case, the variance becomes stable:\n\n$\\text{Var}(\\Delta X_t) = \\sigma^2,\\quad \\text{score}(p_i', p_j') \\propto S\\sigma^2$\n\nThis makes the attention mechanism focus on true short-term patterns, unaffected by long-term drift.\n\n---\n**Proposition 2: Importance of Non-Stationarity in Capturing Cointegration**\n\nLet $X_t, Y_t \\sim I(1)$ be two non-stationary time series with a cointegration relationship:\n\n$Z_t = X_t - \\beta Y_t \\sim I(0)$\n\nwhere $\\beta$ is a constant coefficient. If we remove non-stationarity via detrending:\n\n$\\Delta X_t = \\text{Detrend}(X_t),\\quad \\Delta Y_t = \\text{Detrend}(Y_t)$\n\nThen:\n\n$\nZ_t = \\Delta X_t - \\beta \\Delta Y_t = \\epsilon_t\n$\n\nwhich becomes a **random noise sequence**. This destroys the cointegration signal, making it impossible for attention mechanisms to capture long-term equilibrium relationships. **Figure 4(b) illustrates that removing non-stationary components eliminates the vast majority of cointegration information.**\n\n---\nThese two propositions together explain our architectural choice:\n- For **short-term modeling**, we eliminate non-stationarity to avoid spurious regressions.\n- For **long-term modeling**, we preserve non-stationarity to retain meaningful cointegration.\n\nWe will incorporate this theoretical analysis in greater detail in the revision of our paper.\n\n**Q2:** Highlighting distinct advantages or improvements over these methods would clarify the contribution of this work.\n\n**A2:** **We design our model by focusing on the distinct characteristics of non-stationary data, specifically addressing short-term stability and long-term dependencies.** This insight leads to the adoption of simple yet effective strategies that align with the nature of non-stationary data. Unlike other attention methods that directly model short-term dependencies and are prone to spurious regressions, Integrated Attention stabilizes short-term modeling by removing non-stationary features from the Query-Key pairs. In contrast, Cointegrated Attention emphasizes the necessity of retaining non-stationary features for capturing long-term cointegration relationships, setting it apart from recent channel-dependent Transformer models. These models either fail to consider long-term cointegration [1] or overlook it entirely [2].\n\n[1] Crossformer: Transformer Utilizing Cross-Dimension Dependency for Multivariate Time Series Forecasting\n\n[2] Revitalizing Multivariate Time Series Forecasting: Learnable Decomposition with Inter-Series Dependencies and Intra-Series Variations Modeling\n\n**Q3:** ARIMA is a strong baseline for short-term forecasting; the authors should compare its performance to highlight the advantages of their framework.\n\n**A3:** Since ARIMA is for univariate forecasting and our PeMS datasets are multivariate, we compared TimeBridge with VARIMA, which is more appropriate for multivariate forecasting. The table below shows that TimeBridge outperforms VARIMA in most cases. This comparison will be updated in the revised paper.\n||TimeBridge|VARIMA|\n|-|-|-|\n||MAE/MAPE/RMSE|MAE/MAPE/RMSE|\n|PeMS03|14.52/**14.21**/23.10|**13.78**/17.95/**19.95**|\n|PeMS04|**19.24**/**12.42**/**31.12**|24.87/15.61/36.26|\n|PeMS07|**20.43**/**8.42**/**33.44**|26.00/11.21/37.67|\n|PeMS08|**14.98**/**9.56**/**23.77**|19.38/12.49/28.02|\n\nWe sincerely appreciate your constructive feedback. If you have any further questions or need clarification, please feel free to reach out.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "Not applicable", "experimental_designs_or_analyses": "The overall experimental designs look great.", "relation_to_broader_scientific_literature": "The authors have discussed the related work in Normalization and Dependecy Modeling in time series forecasting.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths:\n1. Addressing the dual challenge of non-stationarity in short-term stability and long-term dependencies is interesting, as traditional methods either neglect or insufficiently manage these aspects.\n2. The framework has been rigorously tested across multiple datasets, demonstrating consistent state-of-the-art performance. The inclusion of financial forecasting datasets enhances the credibility of the results.\n\n\nWeaknesses:\n1. While the authors criticize some existing methods for incorporating non-stationary factors without a robust theoretical framework, the proposed method in this paper also lacks a rigorous theoretical foundation. A more detailed theoretical exploration could strengthen the framework's validity.\n2. The novelty of the proposed work could be further justified. The use of attention mechanisms to manage correlations between patches or among variates is not entirely novel, as similar approaches have been explored in channel-dependent transformers. Highlighting distinct advantages or improvements over these methods would clarify the contribution of this work.\n3. In short-term forecasting, where ARIMA is recognized as a strong baseline method, the authors might benefit from discussing its relative performance. This comparison could provide a clearer benchmark for evaluating the advantages of the proposed framework.", "other_comments_or_suggestions": "See above", "questions_for_authors": "See above", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FT1jVhzNB8", "reviewer": "Reviewer_CeE3", "summary": "This paper introduces TimeBridge, a methodological framework addressing non-stationarity in long-term time series forecasting. The proposed approach is structured around two core mechanisms: Integrated Attention, which seeks to mitigate short-term non-stationarity by stabilizing localized variations, and Cointegrated Attention, designed to retain and model long-term non-stationary dependencies. Empirical evaluations across multiple benchmark datasets suggest that TimeBridge performs competitively relative to state-of-the-art (SOTA) forecasting models.\n\n## update after rebuttal\n\nAfter carefully reading through all the reviewers' comments, I believe the strong performance of this work across multiple dimensions is truly commendable. Accordingly, I have raised my score, with the expectation that the authors will revise and improve the identified concerns in the camera-ready version.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your feedback and insightful comments on our work. Below, we address your questions:\n\n**Q1:** It omits direct comparisons with seasonal-trend decomposition methods.\n\n**A1:** We add comparison with seasonal-trend decomposition methods (Autoformer [1] and TDformer [2]). The table below shows the average prediction results across all datasets. Full results will be included in the revised paper.\n\n||TimeBridge|Autoformer|TDformer|\n|-|-|-|-|\n||MSE/MAE|MSE/MAE|MSE/MAE|\n|ETTm1|**0.344**/**0.379**|0.588/0.517|0.380/0.406|\n|ETTm2|**0.246**/**0.310**|0.324/0.368|0.267/0.325|\n|ETTh1|**0.397**/**0.424**|0.496/0.487|0.486/0.479|\n|ETTh2|**0.341**/**0.382**|0.453/0.462|0.378/0.411|\n|Weather|**0.219**/**0.249**|0.338/0.382|0.236/0.271|\n|Electricity|**0.149**/**0.245**|0.227/0.338|0.177/0.282|\n|Traffic|**0.360**/**0.255**|0.628/0.379|0.570/0.322|\n|Solar|**0.181**/**0.239**|0.340/0.380|0.208/0.251|\n|Climate|**1.057**/**0.494**|1.456/0.584|1.364/0.626|\n\n**Q2:** The definitions of stationary and non-stationary in the paper seem to be mere rebranding of trend and seasonal components.\n\n**A2:** While trend and seasonal components can be viewed as corresponding to non-stationary and stationary elements, previous works have mainly focused on modeling them within individual channels. They haven’t fully addressed the need to preserve non-stationarity across channels to capture long-term cointegration. **Our paper’s core insight is that short-term dependencies can be modeled within stationary channels, while long-term cointegration requires preserving the non-stationary relationships between variables.** Thus, we analyze non-stationarity’s distinct impacts on both intra- and inter-variable modeling, rather than just decomposing the sequence within channels as previous methods [1,2] have done.\n\n**Q3:** The improvements are modest and don't definitively prove TimeBridge's superiority over existing methods.\n\n**A3:** As shown in **Table 1**, TimeBridge consistently achieves superior performance in long-term forecasting, reducing MSE and MAE by 1.85%/2.49%, 5.56%/4.12%, and 13.66%/7.58% compared to DeformableTST, ModernTCN, and TimeMixer, respectively. Additionally, our model demonstrates exceptional performance in financial forecasting (**Table 3**), highlighting its ability to capture complex cointegration relationships in financial markets. These results confirm the effectiveness of our approach.\n\n**Q4:** The methodology and structure of TimeBridge bear strong resemblance to several prior works, including [3,4,5,6].\n\n**A4:** We design our model with a unique focus on non-stationarity in both short-term stability and long-term dependencies. **Integrated Attention stabilizes short-term dependencies by removing non-stationary features from the Query-Key pairs, while Cointegrated Attention emphasizes the necessity of preserving non-stationarity for modeling long-term cointegration relationships between channels.**\nIn contrast, PatchTST [5] does not address non-stationarity and focuses solely on the temporal dimension, missing the rich cointegration relationships between variables. Crossformer [3] and UniTST [6] model short-term dependencies across and within channels but are theoretically susceptible to spurious regressions and lack explicit consideration of long-term cointegration. Fredformer [4], though utilizing frequency-domain attention to capture long-term signals, struggles with fine-grained temporal trends in the time domain, limiting its ability to model short-term dependencies effectively.\n\n**Q5:** A more detailed ablation study is needed to clarify the individual contributions of Integrated and Cointegrated Attention in mitigating non-stationarity.\n\n**A5:** We have conducted an ablation study on the impact and order of Integrated Attention and Cointegrated Attention in **Table 5** in our paper. The results indicate that both attention mechanisms must be used together, with Integrated Attention applied first, followed by Cointegrated Attention.\n\n**Q6:** The theoretical grounding of Integrated and Cointegrated Attention should be more explicitly justified within the context of existing time series modeling literature.\n\n**A6:** We have provided a more explicit theoretical justification for Integrated and Cointegrated Attention in Reviewer 1BTh’s Answer 1 (**A1**).\n\n**Q7:** Would a decomposition-based or frequency-domain approach offer comparable performance with potentially greater computational efficiency?\n\n**A7:** We compared TimeBridge with methods such as TimeMixer (decomposition-based), TimesNet (frequency-domain), and the newly added Autoformer and TDformer in **A1**, all of which demonstrate that TimeBridge consistently outperforms them. While decomposition-based or frequency-domain methods may offer comparable performance, we currently do not have conclusive evidence to support this.\n\nWe appreciate your insightful feedback. If you have any further questions or need clarification, feel free to follow up.", "claims_and_evidence": "The manuscript presents claims that are only partially substantiated by empirical results, and several key limitations emerge:\n- The proposed framework purports to be an effective mechanism for handling non-stationarity; however, it omits direct comparisons with seasonal-trend decomposition methods [1][2], which are widely recognized as robust techniques for handling non-stationary components in time series data. Moreover, the definitions of stationary and non-stationary in the paper seem to be mere rebranding of trend and seasonal components.\n- The methodological novelty of the work appears to be constrained. The proposed architecture largely constitutes a reconfiguration of existing methodologies [3][4][5][6], rather than introducing a fundamentally novel paradigm.\n- While the reported experimental results indicate performance improvements, the magnitude of these improvements is relatively modest and does not conclusively establish the superiority of TimeBridge over existing approaches.\n\n[1]  Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting.\n\n[2] First De-Trend then Attend: Rethinking Attention for Time-Series Forecasting.\n\n[3] Crossformer: Transformer Utilizing Cross-Dimension Dependency for Multivariate Time Series Forecasting\n\n[4] Fredformer: Frequency Debiased Transformer for Time Series Forecasting\n\n[5] A Time Series is Worth 64 Words: Long-term Forecasting with Transformers\n\n[6] UniTST: Effectively Modeling Inter-Series and Intra-Series Dependencies for Multivariate Time Series Forecasting", "methods_and_evaluation_criteria": "The selection of benchmark datasets and evaluation metrics (MSE, MAE) aligns with standard practices in long-term time series forecasting. However, the study would benefit from:\n- **Explicit comparisons with seasonal-trend decomposition methods**, given their proven efficacy in handling non-stationarity.\n- A more rigorous ablation study to disentangle the individual contributions of Integrated and Cointegrated Attention, ensuring a precise understanding of their respective roles in mitigating non-stationarity.", "theoretical_claims": "The manuscript does not provide formal mathematical proofs. \n\nAlthough it references cointegration principles, it lacks a rigorous theoretical justification for why the proposed attention-based mechanisms offer an optimal means of capturing non-stationary dependencies. The absence of a formal derivation raises concerns regarding the theoretical soundness of the proposed approach.", "experimental_designs_or_analyses": "The experimental design and results were examined, revealing the following limitations:\n- The reported performance gains are insufficiently substantial to position TimeBridge as a transformative advancement over existing methodologies.", "relation_to_broader_scientific_literature": "The study contributes to time series forecasting by integrating non-stationarity mitigation techniques into an attention-based framework. However, several key concerns remain unaddressed:\n- Seasonal-trend decomposition methods have long been established as effective solutions for handling non-stationarity, yet the paper does not explicitly contrast its approach with these techniques.\n- The methodology and structure of TimeBridge bear strong resemblance to several prior works, including [3][4][5][6]. The extent to which TimeBridge provides a substantive departure from these works remains unclear.\n\n[3] Crossformer: Transformer Utilizing Cross-Dimension Dependency for Multivariate Time Series Forecasting\n\n[4] Fredformer: Frequency Debiased Transformer for Time Series Forecasting\n\n[5] A Time Series is Worth 64 Words: Long-term Forecasting with Transformers\n\n[6] UniTST: Effectively Modeling Inter-Series and Intra-Series Dependencies for Multivariate Time Series Forecasting", "essential_references_not_discussed": "Yes, the study should provide an in-depth discussion of:\n\n- **Seasonal-trend decomposition techniques** to contextualize the proposed approach within established non-stationarity handling frameworks.\n- **Recent advancements in hybrid modeling approaches**, which also address long-term dependencies in time series forecasting.", "other_strengths_and_weaknesses": "- **Strengths**: The paper correctly identifies non-stationarity as a critical issue and attempts to address both short-term and long-term dependencies within a unified forecasting framework.\n- **Weaknesses**: The degree of methodological novelty is limited, as the proposed framework appears to primarily constitute a recombination of existing techniques rather than a fundamentally new paradigm. Furthermore, the lack of direct comparisons with seasonal-trend decomposition models significantly undermines the empirical validation of the proposed approach.", "other_comments_or_suggestions": "- The theoretical grounding of Integrated and Cointegrated Attention should be more explicitly justified within the context of existing time series modeling literature.\n- The ablation study should provide greater granularity in assessing the distinct contributions of each proposed component.", "questions_for_authors": "1. How does TimeBridge substantively differentiate itself from existing attention-based time series forecasting models?\n2. Would a decomposition-based or frequency-domain approach offer comparable performance with potentially greater computational efficiency?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "waaRUukhUZ", "reviewer": "Reviewer_2KKn", "summary": "This study focuses on the problem of time series forecasting (TSF) by addressing the dual challenge of integrating both long-term and short-term relationships. The proposed TimeBridge employs a dual-attention framework: initial patch-level integrated attention for short-term pattern analysis and cointegrated attention to model long-term temporal correlations. The practical performance of TimeBridge has been validated across various real-world datasets.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your feedback and insightful comments on our work. Below, we address each of your concerns and questions:\n\n**Q1:** As indicated in Table 6, the sequential selection strategy of CI and CD significantly impacts prediction performance on Electricity and Traffic datasets, yet this phenomenon remains insufficiently explained in the text.\n\n**A1:** We believe that modeling short-term dependencies first is crucial because these local temporal features are essential for accurate short-term forecasting. **By using Integrated Attention (CI) to capture short-term dependencies before Cointegrated Attention (CD) models long-term relationships, we ensure that important immediate patterns are not lost, which significantly improves performance.** A similar \"local first, global later\" modeling strategy is also used in other works [1].\n\n**Q2:** The manuscript lacks computational complexity analysis, making it difficult to assess the model's efficiency and memory requirements.\n\n**A2:** \nWe provide a detailed comparison of the theoretical complexity of our method with other mainstream Transformer-based models. The table below summarizes the theoretical computational complexity, where $C$ is the number of channels, $I$ and $O$ are the input and output lengths, and $S$ is the patch length:\n|TimeBridge|iTransformer|PatchTST|Crossformer|FEDformer|\n|-|-|-|-|-|\n|$O(\\text{max}(C·(\\frac{I}{S})^2，C^{3/2}))$|$O(C^2)$|$O(C·(\\frac{I}{S})^2)$|$O(\\frac{C}{S^2}O(I+O))$|$O(I/2+O)$|\n\nMoreover, theoretical complexity alone cannot fully capture real-world performance due to implementation differences. We tested on 2 NVIDIA 3090 GPUs, measuring training (1 epoch) and inference times for three datasets of increasing size, with results averaged over 5 runs. The results in the table below indicate that TimeBridge outperforms most Transformer-based models.\n\n|||TimeBridge|iTransformer|PatchTST|Crossformer|FEDformer|\n|-|-|-|-|-|-|-|\n|ETTm1 ($C=7$)|Training Time|72s|65s|73s|280s|449s|\n||Inference Time|33s|31s|39s|48s|62s|\n|Electricity ($C=321$)|Training Time|252s|89s|450s|328s|510s|\n||Inference Time|125s|78s|141s|116s|130s|\n|Traffic ($C=862$)|Training Time|409s|175s|649s|360s|485s|\n||Inference Time|175s|154s|207s|171s|196s|\n\n\n**Q3:** While the study represents an effort in modeling cointegration through deep learning, the architectural design of the proposed modules appears relatively simplistic and could benefit from further sophistication.\n\n**A3:** \nWe would like to clarify that our method was **intentionally designed with a simple, intuitive structure to address the challenges of non-stationarity and dependency modeling effectively**:\n- **Integrated Attention** focuses on capturing short-term dependencies while mitigating spurious regressions, using a streamlined approach that normalizes the attention map to remove non-stationary components.\n- **Cointegrated Attention** preserves non-stationarity to model long-term cointegration relationships between variables.\n- **Patch Downsampling** bridges the two blocks, allowing the Integrated Attention block to process aggregated short-term information, which feeds into the Cointegrated Attention block for long-term modeling.\n\nWe further provide a theoretical analysis of the modeling rationale behind these design choices, which can be found in Reviewer 1BTh’s Answer 1 (**A1**).\n\n\n**Q4:** I am particularly interested in whether the financial experiments have been implemented in real-world trading scenarios? Could the authors provide some insights into the practical deployment?\n\n**A4:** Thank you for your interest in the practical deployment of our model. Our financial experiments were conducted using the **Qlib** [2] platform for historical backtesting, with simulated trading costs to approximate real-world scenarios. While real-world deployment would need to address challenges such as latency and liquidity, we have not yet integrated with live trading systems due to data privacy and regulatory constraints.\n\n\n[1] MICN: Multi-scale Local and Global Context Modeling for Long-term Series Forecasting  (ICLR 2023)\n\n[2] https://github.com/microsoft/qlib\n\n\nThank you for your valuable feedback. If you have any further questions or need clarification, feel free to follow up.", "claims_and_evidence": "The claim aligns with previous work in TSF.", "methods_and_evaluation_criteria": "The experimental datasets and evaluation metrics employed in this study strictly follow the most prevalent standards established in the TSF field.", "theoretical_claims": "No theoretical claims.", "experimental_designs_or_analyses": "The experimental design adheres to the most widely adopted baselines in TSF, ensuring the fairness.", "relation_to_broader_scientific_literature": "Non-stationarity represents a crucial aspect in series modeling, and this article possesses the potential to exert a broader impact on related scientific endeavors.", "essential_references_not_discussed": "The most relevant works have already been encompassed.", "other_strengths_and_weaknesses": "Strengths:\n\n1. The paper is well-motivated. It introduces a novel approach and a new perspective from classical time series analysis for addressing non-stationary time series data.\n\n2. This paper is well written, with well-defined mathematical notations.\n\n3. The experiment is comprehensive, featuring extensive ablation studies that systematically validate the contribution of each component.\n\nWeaknesses:\n\n1. As indicated in Table 6, the sequential selection strategy of CI and CD significantly impacts prediction performance on Electricity and Traffic datasets, yet this phenomenon remains insufficiently explained in the text.\n\n2. The manuscript lacks computational complexity analysis, making it difficult to assess the model's efficiency and memory requirements.", "other_comments_or_suggestions": "While the study represents an effort in modeling cointegration through deep learning, the architectural design of the proposed modules appears relatively simplistic and could benefit from further sophistication.", "questions_for_authors": "I am particularly interested in whether the financial experiments have been implemented in real-world trading scenarios? Could the authors provide some insights into the practical deployment?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vL6rHMDQaY", "reviewer": "Reviewer_yRZB", "summary": "This paper proposes a novel long-short-term representation modeling approach for handling non-stationarity in multivariate time series. For short-term representation, the integrated attention mechanism is employed to model temporal dependencies. For long-term representation, the cointegrated attention mechanism captures inter-channel dependencies while preserving cointegration relationships. Overall, it provides a novel perspective on modeling non-stationary multivariate time series.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your constructive feedback and valuable insights into our work. Below, we address your concerns:\n\n**Q1:** The loss function lacks sufficient discussion on its contribution to performance improvements.\n\n**A1:** \nWe conducted additional experiments where TimeBridge was trained using the **MSE loss**, denoted as \"TimeBridge (MSE loss)\", and compared the results with the best baseline, DeformableTST. Regardless of the loss function, TimeBridge outperforms DeformableTST in most datasets (See Table A), highlighting the effectiveness of our framework. We will incorporate the full results in the revision of our paper.\n\n**Table A: Average results of different prediction length {$96, 192, 336, 720$} with MSE Loss and Hybrid MAE Loss**\n||TimeBridge|TimeBridge (MSE loss)|DeformableTST|\n|-|-|-|-|\n|ETTm1|**0.344**/**0.379**|0.353/0.388|0.348/0.383|\n|ETTm2|**0.246**/**0.310**|**0.246**/**0.310**|0.257/0.319|\n|ETTh1|**0.397**/0.424|0.399/**0.420**|0.404/0.423|\n|ETTh2|0.341/0.382|0.346/0.394|**0.328**/**0.377**|\n|Weather|0.219/**0.249**|**0.218**/0.250|0.222/0.262|\n|Electricity|**0.149**/**0.245**|**0.149**/0.246|0.161/0.261|\n|Traffic|**0.360**/0.255|**0.360**/**0.252**|0.391/0.278|\n|Solar|**0.181**/0.239|**0.181**/**0.238**|0.185/0.254|\n|Climate|1.057/0.494|**1.052**/**0.486**|1.060/0.496|\n\n\n**Q2:** The font size in Table 17 could be increased for better readability.\n\n**A2:** \nThank you for the suggestion. We will adjust the font size of the large tables in the revised version to enhance readability.\n\n**Q3:** Could the authors provide further insights into the rationale behind their loss function choice and its comparative effectiveness? If all methods were evaluated under the same loss function, would the proposed approach still achieve significant improvements? Clarifying this point is crucial to distinguishing whether the performance gains stem from handling non-stationarity or from the loss function itself.\n\n**A3:** \nFrom Table A above, it is clear that the hybrid MAE loss improves TimeBridge's performance on four ETT datasets. The ETT datasets have relatively few channels ($C=7$), which limits TimeBridge’s ability to utilize the Cointegrated Attention module to capture cointegration information. In fact, we only use the Integrated Attention to model ETT datasets (see Table 8 in our paper). Additionally, the **ETT data exhibits a high degree of random fluctuation, which is better modeled using a loss function that weighs both time and frequency components.** Hence, the hybrid MAE loss strengthens the modeling of short-term dependencies in such datasets. On datasets with more channels (e.g., Electricity, Traffic, Solar), the choice of loss function has minimal impact, as both losses yield similar results. We will incorporate this analysis in the revised version of our manuscript.\n\n\nWe are deeply grateful for your recognition of our work and your constructive feedback. If you have any further questions or suggestions, we would be more than happy to address them.", "claims_and_evidence": "This claim aligns well with classical time series analysis and is supported by existing literature.", "methods_and_evaluation_criteria": "The experimental setup is comprehensive and fair, covering mainstream datasets and models in multivariate time series forecasting.", "theoretical_claims": "No relevant theoretical claims are made in this paper.", "experimental_designs_or_analyses": "The experiments are well-designed, covering standard datasets and baselines. However, the choice of loss function (L1 loss) warrants further analysis, as it may involve unfair comparisons.", "relation_to_broader_scientific_literature": "The paper's motivation aligns well with classical time series theories. These concepts have been well-established in prior research.", "essential_references_not_discussed": "The literature review is thorough.", "other_strengths_and_weaknesses": "**Strengths**: While traditional theories on non-stationarity are well-established, many recent deep learning-based approaches either overlook or only partially incorporate them, limiting their effectiveness. This paper successfully integrates these classical principles into deep learning models, demonstrating strong empirical performance, which provides valuable insights for the field.\n\n**Weaknesses**: As mentioned, the loss function design (L1 loss) lacks sufficient discussion on its contribution to performance improvements.", "other_comments_or_suggestions": "The font size in Table 17 could be increased for better readability.", "questions_for_authors": "Could the authors provide further insights into the rationale behind their loss function choice and its comparative effectiveness? If all methods were evaluated under the same loss function, would the proposed approach still achieve significant improvements? Clarifying this point is crucial to distinguishing whether the performance gains stem from handling non-stationarity or from the loss function itself.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes a methodological framework addressing nonstationarity in long-term time series forecasting. The proposed approach is structured around two core mechanisms: Integrated Attention, which seeks to mitigate short-term non-stationarity by stabilizing localized variations, and Cointegrated Attention, designed to retain and model long-term non-stationary dependencies. As pointed out by the reviewers, the studied problem, addressing the dual challenge of nonstationarity in short-term stability and long-term dependencies, is interesting, and the empirical performance of the proposed approaches is impressive. A more detailed theoretical exploration of why the proposed approach works would strengthen the work.
Decision: Accept (poster)