DBLoss: Decomposition-based Loss Function for Time Series Forecasting
Authors: Xiangfei Qiu
Xingjian Wu
Hanyin Cheng
Xvyuan Liu
Chenjuan Guo
Jilin Hu
Bin Yang
Venue: NeurIPS 2025 poster ()
Keywords: Time Series Forecasting
Loss Function
Primary area: deep_learning

Abstract: Time series forecasting holds significant value in various domains such as economics, traffic, energy, and AIOps, as accurate predictions facilitate informed decision-making. However, the existing Mean Squared Error (MSE) loss function sometimes fails to accurately capture the seasonality or trend within the forecasting horizon, even when decomposition modules are used in the forward propagation to model the trend and seasonality separately. To address these challenges, we propose a simple yet effective Decomposition-Based Loss function called DBLoss. This method uses exponential moving averages to decompose the time series into seasonal and trend components within the forecasting horizon, and then calculates the loss for each of these components separately, followed by weighting them. As a general loss function, DBLoss can be combined with any deep learning forecasting model. Extensive experiments demonstrate that DBLoss significantly improves the performance of state-of-the-art models across diverse real-world datasets and provides a new perspective on the design of time series loss functions.

DBLoss: Decomposition-based Loss Function for Time Series Forecasting
Xiangfei Qiu', Xingjian Wu', Hanyin Cheng', Xvyuan Liu Chenjuan Guo Jilin Hu Bin Yang!
East China Normal University, -KLATADDS-MOE {xfqiu, xjwu, hycheng, xyliu)āstu. eetresttuuene....mmm edu. en, {cjguo, jihu, byang) ecnu. edu.edu.co cn
Abstract
Time series forecasting holds significant value in various domains such as eco- nomics, traffic, energy, and AIOps, as accurate predictions facilitate informed decision-making. However, the existing Mean Squared Error (MSE) loss function sometimes fails to accurately capture the seasonality or trend within the forecasting horizon, even when decomposition modules are used in the forward propagation model the trend and seasonality separately. To address these challenges, propose simple yet effective Decomposition -Based Based Loss function called DBLoss. This method uses exponential moving averages to decompose the time series into seasonal and trend components within the forecasting horizon, and then calculates the loss for each of these components separately, followed by weighting them. As general loss function, DBLoss can be combined with any deep learning forecasting model. Extensive experiments demonstrate that DBLoss significantly improves the performance of state-of-the-art models across diverse real-world datasets and provides new perspective on the design of time series loss functions.
Resources: http://///wh//oooc///////////n/e
1
Introduction
Time Series Forecasting holds significant value in various domains such as economics [Wang et al. 2025a
Li al.
2025a Ma al.
2025a Liu et al. 2025a traffic [Guo al. 2014 Wu al. 2023a
Lu et al. 2011
Yue et al. [2025a]]]] Ma et al. 2025b c Liu et al. 2025b energy Wang et al.) 2025b
Huang et al.] 2225a Ma al. [222dd]]]] Miao al. 2024), and AlOps Wang et al. [202cc] Yue et al.
2024 Ma et al., 2025e Wu et al. [2024a], as accurate predictions facilitate astute decision-making
To pursue accurate predictions, recent progress Long-term Time Series Forecasting focuses on effectively capturing the inherent seasonality and trend, which reflect the changing laws of the time series, i.e., the inductive bias. Recently, dozens of deep learning models have been designed from light- to multi-scale, such as DLinear [Zeng et al. 2023). OLinear Yuue et al., 2025b). CycleNet [Lin et al. 2024a], TimesNet [Wu al., 2023b). TimeBase [Huang et al. 2025b), PDF [Dai et al. 2024], Mixer [Wang al.) 2024], and DUET [Qiu et al.) 2025a J. which are aiming at capturing such inductive bias consistently within data for more accurate predictions.
Technically, to capture the seasonality and trend within data, decompositionn based techniques are widely applied to disentangle the seasonality and trend parts explicitly. For example, DLinear and DUET apply the moving-averse technigue to obtain the trend part. TimesNet, PDF and TimeMixer apply meticulously-desigeee seasonal decomposition modules to process the seasonal part, and the CycleNet uses a learnable matrix to directly capture the seasonality. All these techniques are
"ooresponning Author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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3.0
-0.1
100
-22
0:0
0.4
025
-08
-025
2301
30
150
40
220
(a) Poor Sensonal Predictions
(b) Poor Trend Predictions
Poor Seasonal and Trend Predictions
Figure Limitations of MSE loss in capturing seasonality or trend within the forecasting horizon.
employed in the forward propagation effectively extract the seasonal and trend components from the contextual time series.
However, if the purpose of extracting seasonality and trend in the contextual time series is to improve predictions, perhaps considering seasonality and trends directly in the forecasting horizon may further enhance prediction performance. As shown in Figure [ we observe that current distance-based loss functions (such as MSE) have the following limitations: 1) they may make poor seasonal predictions; 2) they may make poor trend predictions: 3) they may make both poor seasonal and trend predictions. Even when decomposition techniques are applied in the forward propagation, the seasonality and trend within the forecasting horizon are not effectively modeled, indicating that the inductive bias is not well applied to the predictions.
Inspired by the above motivation, manage explicitly encourage the modeling of the seasonality and trend the forecasting horizon to enhance the performance. Specifically, we propose a simple yet effective Decomposition-BB Based Loss function called DBLoss. This method involves using exponential moving averages [Stitsyuk and Choi 2025) to decompose the time series into seasonal and trend components within the forecasting horizon. It then calculates the loss for each of these components separately and combines them with appropriate weighting. As a general loss function, combining LoLoss with any deep learning forecasting model can lead consistent improvement in performance, which is demonstrated on real-world datasets from multiple domains. The contributions are summarized as follows.
We propose simple yet effective loss function for time series forecasting, called DBLoss, which can refine the characterization and representation of time series through decomposition within the forecasting horizon.
The proposed DBLoss is generally applicable arbitrary deep neural networks with negligible cost. By introducing DBLoss into the baseline, we have achieved performance that generally surpasses the state-of-the-art on eight real-world datasets.
We conduct extensive evaluations DBLoss using quantitative analysis and qualitative visual- izations to verify its effectiveness.
2 Related works
2.1
Time Series Forecasting Methods
Time series forecasting (TSF) predicts future observations based on historical observations. TSF methods are mainly categorized into four distinct approaches: (1) statistical learning- based methods, (2) machine learning-based methods, (3) deep lcarning-basee based methods, and (4) foundation methods.
Early TSF methods primarily rely on statistical learning approaches such ARIMA [Box and Pierce 1970]. ETS [Hyndman et al. 2008), and VAR [Godahewa et al. 2021]. With advancements in machine learning, methods like XGBoost [Chen and Guestrin [016], Random Forests [Breiman 2001), ) and
LightGBM (Ke et al. 2017] gain popularity for handling nonlinear patterns. However, these methods
still require manual feature engineering and model design [Ma al. 2025f Wang et al. 2023 Wu
et al. 2025a]. Leveraging the representation learning of deep neural networks (DNNs) |Huang
al. 2023 Miao al. 20255 Wang et al. [025d], many deep learning- -aased methods emerge.
TimesNet [Wu et al. 2023b and ScgRNN [in et al. 2023 model time series vector sequences,

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Prediction
Seasonal Ys MMMMMMM
Trend Yr.
Input X
MMMMMM
EMA EMA
Backbone
MMMM
Ls C aLs (1 a) Crr
Ground Truth Y
M MMMMMMM
Seasonal Ys
Trend Yr
Figure 2: Overview of the proposed DBLoss.
using CNNs or RNNs to capture temporal dependencies. Additionally, Transformer architectures, including DUET Qiu al. 2025a]. Informer (Zhou al. 2021].
FEDformer (Zhou al. [022]. 2023 capture complex relationships
Triformer [Cirstea et al. 2022]. and PatchTST [Nie et al.
between time points
accurately, significantly improving forecasting performance. MLP-based
methods, including Sparse TSF [Lin et al. 2024bb. CycleNet [Lin et al. 2024a]. SRSNet (Wu et al.,
2025b]. NLinear (Zeng et 2023]. and DLinear [Zeng 2023 adopt simpler architectures with fewer parameters but still achieve highly competitive forecasting accuracy.
However, many of these methods struggle with generalization across domains due to their reliance 2025b]. To address this, foundation methods are proposed, categorized domain-specific data (Li et al.
into LLM-basee methods and time series pre- trained methods. LLM--based methods [Zhou et al.,
2023 Jin et al. 2024 [iu et al., 2024a Pan et al. 2024) leverage the strong representational capacity and sequential modeling capability of LLMSs capture complex patterns for time series modeling. Time series pre-trained methods (Liu et al.) 2024b Gao et al., 2024] Goswami et al. 20244 Das
al. 2024] focus on pre-training over multi-domain time series data, enabling the method to learn domain-agnostic features that are transferable across various applications. This strategy not only enhances performance on specific tasks but also provides greater flexibility when adapting to new datasets or scenarios.
2.2 Loss Functions for Time Series Forecasting
Recently, enhance the training performance of time series forecasting models, researchers have introduced various novel loss functions. These loss functions can be broadly categorized into three types: shape-based losses, dependency-bann
losses, and patch-based structural losses.
Shape-based losses aim to capture structural similarities between true values and predictions by tackling the issue of shape mismatch. For example, techniques based on Dynamic Time Warping (DTW), such as Soft-DTW [Cuturi and Blondel (017) and DILATE [Le Guen and Thome 2019]. achieve alignment performance in improving shape alignment, the high computational complexity of these methods restricts their application in large-scat scenarios. Meanwhile, TILDE-Q Q [Lee al. 2022 introduces transformation invariance, making it robust to amplitude shifts, phase changes, and scale differences, thus focusing more on similarity the shape level. Rependency-based losses are dedicated to characterizing temporal correlations within the forecasting horizon. For instance, FreDF [Wang et al. 2025e] cleverly circumvents complex correlation modeling between labels by performing learning and prediction in the frequency domain. Furthermore, patch-based based structural losses like PSLoss (Kudrat al., 2025) incorporate patch-wise statistical properties into the loss function, enabling a more granular structural measurement of the data. Unlike the aforementioned loss functions, our proposed DBLoss refines the characterization and representation of time series through decomposition within the forecasting horizon, offering novel perspective for the design of time series loss functions.
when time series undergo deformation. However, despite their excellent
3 DBLoss
time series NxT is a time-oriented sequence of N-dimensional time points, where T is the number of timestamps, and N is the number of channels. If N 1, a time series is called univariate, and multivariate if N >>1. Time Series Forecasting aims to predict the next future timestamps, formulated as Y (X,T+1, Xx,+4)) E XF based the historical time
3

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series x= (X,,,'' X.r) E RXT with N channels and T timestamps. For convenience, we separate dimensions with commas. Specifically, we denote X as the i-th channel the j-th timestamp, Xn, I2 as the time series of n-th channel, where =1,- ,
3.1 Overview
As shown in Figure 2 we first generate the prediction Y using an arbitrary backbone method. Next, we input both the prediction Y and the ground truth Y into the EMA Decomposition Module to decompose them into seasonal and trend components. Through this process, we obtain the seasonal component Ys and the trend component Y7 of the prediction, well as the seasonal component Ys and the trend component Y of the ground truth. Subsequently, we compute the errors for both the seasonal and trend components and then combine these errors using weighted sum to form the final loss function. This approach allows for a more accuracy evaluation of the differences between the predicted and ground truth values, leading to more effective optimization and training.
3.2 EMA Decomposition Module
Seasonal-trend decomposition facilitates the learning of complex temporal patterns by breaking down time series signals into trend and seasonal components. Trend components refer to the long-term changes patterns that occur time, intuitively representing the overall direction of the data. In contrast, seasonal components capture the phenomena in the time series that repeat at specific intervals and are typically nonlinear due to the complexity and variability of periodic behavior. This technique is widely applied in time series analysis methods (Wu et al. [202T] Zhou et al.) 2022 Zeng al. 2023 Wang al., 2024. Qiu al., 2025a Unlike above methods, which typically extract the trend and seasonal representations of the time series through decomposition and then combine these two representations obtain more comprehensive time series representation for downstream tasks, our DBLoss computes the loss by separately decomposing the prediction and the ground truth into their trend and seasonal components. We then compute components separately and finally combine these losses using a weighted sum. This process enables the model to better capture the trends and seasonality of the ground truth, resulting in more accurate predictions.
losses for the trend and seasonal
There various methods for seasonal-trend decomposition, such STL decomposition Cleveland
et al.] 1990], Simple Moving Average (SMA) decomposition (Wu et al. 2021 Qiu et al.) 2025a Zeng al. 2023), and Exponential Moving Average (EMA) decomposition Stitsyuk and Choi 2025). In this study, we chose EMA decomposition. Specifically, after obtaining the prediction Y and the ground truth Y, we input them into the EMA decomposition module to decompose them into their trend and seasonal components. We then compute the final loss in the weighted loss function Algorithm I details the calculation process of EMA decomposition module.
described in Section 3.3
Algorithm Calculation Process of EMA Decomposition Module
Input: Time series E IR RxT where B is the batch size, is the time steps, and N is the number channels; Smoothing factor E 10,
Output: Scasonality
Trend of X, denoted as Seasonality E BxTxN Trendd E R BxTXN
I: Get the shape of X: B, T, N X.shape Calculate the weights:
+ [11 - (1) aaaaaa"""''"0""" (1-a) ..,,.,,,1
3: Copy the weights to create divisor: Dai + W.clooe()
4: Update the weights for EMA calculation: W[L :] <- W[1 :] x
5: Reshape the weights and divisor:
W < W.reshape( T,1)
7: DaN + Daw.rreshapeee www.heehhee.,... (1,
8: Compute the cumulative sum of weighted data: C cumsum(X X W,, dim =
Divide the cumulative sum by divisor: Trend <- Prr
10: Calculate the residual: Seasonality X Trend
return Seasonalily, Trend

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3.3
Weighted Loss Function
Based on the EMA decomposition, we obtain the predicted seasonal component Ys and trend component Yr, as well as the corresponding ground truth values Ys and Yr. We then propose weighted loss function, which consists of two parts: the seasonal loss Ls and the trend loss Lr.
Ls Ys Ys LT = Yr Yr)
To prevent the loss of one component from dominating the optimization process due to scale differ- ences, we introduce a scale alignment mechanism. Specifically, the trend loss is adaptively adjusted according to the relative magnitude between Ls and Ly:
(Cf+²)
Ls
Cy aligned = Lr x stopgrad
CT (T)
(2)
where e is a small constant to ensure numerical stability. Here, stopgrad(.) denotes a gradient detachment operation, which prevents the gradient from back- propagatngg through the alignment ratio, thereby avoiding interference between the two loss components.
Finally, we define the total loss L as:
L: 3 Ls + 3) Ly aligned
where 3 is tuning parameter used to balance the contributions of the seasonal loss and the trend loss. By adjusting the value of 3, we can optimize the model's training process according to specific application scenarios.
We provide a theoretical analysis in Appendix C to explain why the proposed DBLoss is more effective than the conventional MSE loss for time series forecasting.
4 Experiments
4.1 Setup
Datasets To conduct comprehensive and fair comparisons for different models, we conduct experi- ments on eight well-known forecasting benchmarks as the target datasets, including ETT (ETThl, ETTh2, ETTml, ETTm2), Solar, Weather, Electricity, and Traffic. For more details on the benchmark datasets, please refer to Table ij Appendix A
Backbones We selected eight state-of-the-ae (SOTA) time series forecasting models to serve as baselines. Specifically, we include four time series specific models: iTransformer (Liu al. [024c] Amplifier [Fei et al. 2025), PatchTST [Nie et al., 2023), and DLinear (Zeng et al.] 2023 as
well four time series foundation models: CALF [Liu al. 2025c). UniTS [Gao et al. 2024
TTM [Eaamarrm et al. 2024], and GPT4TS [Zhou et al. 2023
Implementation Details To keep consistent with previous works, adopt Mean Squared Error (MSE) and Mean Absolute Error (MAE) as evaluation metrics. We consider four forecasting horizon F: (96, 192, 336, 720] for all datasets. We utilize the comprehensive time series forecasting benchmark TFB [Qiu et al., 2024] for unified evaluation, with all baseline results also derived from TFB. Please note that for all the baseline scripts, we directly use the optimal scripts provided by TFB and only replace the training loss function with DBLoss, without making any other modifications. The purpose of this approach is to validate the effectiveness of DBLoss the greatest extent possible. By doing so, we can ensure the accuracy of the experimental results and clearly demonstrate the performance improvements brought by DBLoss. We do not apply the "Drop Last" trick (Qiu et al. 2024 2025blc) to ensure a fair comparison. All experiments of DBLoss are conducted using Py Torch Python 3.8 and executed on an NVIDIA Tesla-A800 GPU. The training process is guided by the MSE loss function and employs the ADAM optimizer. The initial batch size is set to 64, with the flexibility to halve it (down minimum of 8) case of an Out- Out-Of-Memonl Oe-orryry (OOM) issue.
4.2 Main results
We present the MSE and MAE of four state-of-the-art long-term multivariate forecasting models on eight real-worid datasets in Table [ Notably, DBLoss observes performance improvements
5

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Table 1: Long-term multivariate forecasting results. The table reports MSE and MAE for different forecasting horizons F E {96, 192, 336, 720). The parameters for the baselines are kept consistent with those of TFB (Qiu et al. 2024 The better results are highlighted in bold.
Model
iTransformer
Amplifier
PatcTTT
DLincur
Loss
Ori
DBLoss
Ori
DBLoss
Ori
DBLoss
Ori
DBLoss
Metric
MSP
MAF
MSF
MAR
MSF
MAF
MSF:
MAF
MSF
MAF
MSE
MAF
MSF
MAR
MSF
MAE
0.386
0
0.396
0.376
0.393
0.376
0.377
0.397
0.373
0.379
0.403
0.369
0.390
192
0.424
0.440
0.405
0.421
0.414
0.42
0.409
0.415
0.409
0.425
0.395
0.413
0.408
0.419
0.402
0.409
336
0.449
0.466
0,425
0.438
0.442
0.446
0.430
0.432
0.431
0.444
0.414
0.426
01,400
0.440
0.430
0.428
0.495
0.478
0.463
0.479
0.459
0.465
0.457
0.477
0.471
0.193
0.475
Avg
0.449
0.448
0.423
0.430
0.428
0.435
0.419
0.425
0.419
0.436
0.402
0.420
0.425
0.439
0.412
0.425
0.297
0.348
0.288
0.337
0.291
0.342
0.288
0.332
0.274
0.337
0.274
0.334
0.300
0,364
0.284
0.342
0.357
0.389
0.355
0.344
0.348
0.384
0.334
0.387
0.423
0,357
0.390
336
0.388
0.417
0,385
0.416
0.384
0.42
0.377
0.405
0.377
0.416
0.349
0.392
0.490
0.487
01.407
0.430
720
0,424
0.444
0.427
0.443
0.422
0,451
0,400
0.437
0,406
0.441
0,390
0.422
0.704
0.597
0,586
0,533
Avg
0.370
0.364
0.396
0.363
0.403
0.352
0.395
0.337
0.470
0.468
0.424
96
0.300
0.353
0.290
0.341
0.293
0.347
0.287
0.335
0.289
0.343
0.284
0.328
0,300
0.345
0.295
0.337
BT BTTml
192
0.341
0.380
0.328
0.363
0.329
0.367
0.328
0.359
0.329
0.368
0.322
0.355
0.336
0,366
0.331
0.358
0.374
0.368
0.386
0,365
0.387
0.364
0.380
0.362
0.390
0.359
0.367
0.386
0,361
0.378
720
0.429
0.430
0.415
0.415
0,449
0.422
0.424
0.413
0,416
0.425
0.410
0.412
0.419
0.416
0.415
0.409
Avg
0.361
0.390
0,350
0.376
0.354
0.381
0.351
0.372
0.349
0.381
0.344
0.368
0.356
0.378
0.351
0.370
0.175
0.266
0.166
0.254
0.168
0.258
0.163
0.245
0.165
0.255
0.163
0.246
0.164
0.255
0,163
0.247
ET
192
0.242
0.312
0.227
0.295
0.227
0.298
0.222
0.288
0.221
0.293
0.219
0.284
0.224
0.304
0.220
0.290
336
0,282
0.337
0.278
0.330
0.276
0.334
0.271
0.322
0,276
0.327
0.273
0.320
0.277
0.337
0.277
0.329
720
0.375
0.394
0.375
0.388
0.364
0.394
0.350
0.373
0.362
0.381
0.357
0.371
0.401
0,366
0.390
Avg
0.269
0.327
0,262
0.317
0.259
0.321
0.252
0.307
0.256
0.314
0.253
0.306
0.259
0.324
0.257
0.314
0,180
0.215
0.189
0.170
0.234
0.199
0.265
0,202
0.236
Solut
192
0.193
0.257
0,201
0.239
0,202
0,252
0.208
0.239
0.204
0.302
0.182
0.226
0.220
0.282
0,224
0,250
336
0,203
0,266
0,195
0.232
0.232
0.274
0.235
0.251
0,212
0.293
0,187
0,232
0.234
0.295
0.237
0.256
0.223
0.281
0.232
0.265
0.229
0.276
0.242
0.256
0.215
0.307
0.197
0.237
0.243
0.301
0.245
0.260
Avg
0.202
0.262
0.202
0.238
0.212
0.260
0.219
0.243
0,200
0,284
0.183
0.227
0.224
0.286
0.227
0.251
0.157
0.207
0.154
0.196
0.141
0.199
0.145
0.189
0.150
0.200
0.149
0,189
0.170
0.230
0.169
0,221
Weather
0,200
0.197
0.239
0,238
0.186
0.239
0.189
0,229
0.216
0,273
0.262
336
0.252
0.287
0,249
0.278
0.239
0.276
0.239
0.269
0.242
0.279
0.240
0.270
0,258
0.307
0,253
0,293
720
0.320
0.336
0.319
0.335
0.316
0.328
0.316
0.323
0.312
0.330
0.314
0.322
0,323
0.362
0.319
0.346
Avg
0,232
0.230
0.262
0.223
0.260
0,221
0.252
0,262
0.223
0,242
0.293
0.280
Electiicty
96
0.134
0.230
0.131
0.226
0.132
0.227
0.133
0.227
0.143
0.247
0.143
0.244
0.140
0.237
0.140
0.235
192
0.154
0.250
0,149
0.242
0,149
0,241
0.147
0.239
0.158
0,260
0.158
0.257
0.154
0.251
0.154
0.247
0.169
0.163
0.257
0.258
0.163
0.168
0.267
0.165
0.169
0.268
0.264
720
0.194
0.288
0.195
0.284
0.200
0.292
0,203
0.290
0.214
0.307
0.214
0.304
0,204
0.301
0.203
0.295
0,163
0.258
0,160
0.252
0,162
0,255
0.162
0.253
0,171
0.270
0.170
0,266
0.167
0,264
0,167
0,260
0.363
0.265
0.366
0.261
0.396
0.278
0.393
0.270
0.370
0.262
0.369
0.254
0.395
0.275
0.39
0.270
Trflic
192
0.384
0.273
0.587
0.271
0.413
0.285
0.412
0.275
0.386
0. 0.269
0.385
0.260
0.407
0.280
01.407
0.274
336
0.396
0.277
0.397
0.275
0.421
0.291
0:422
0.286
0.396
0.275
0.395
0.266
0.117
0.286
0.415
0,279
0.444
0.306
0.307
0.456
0.295
0.432
0.286
0.454
0.308
0.449
0.298
Avg.
0.397
0.281
0,399
0.278
0.422
0.290
0.421
0.284
0.397
0.275
0.395
0.267
0.418
0.287
0.417
0,280
across all backbone models and significantly outperforms MSE loss in most cases. This validates the robustness and broad applicability of the proposed loss function. Furthermore, DBLoss achieves significant improvements on models that have already adopted trend-seasonal decomposition to
further extract better model representations, such as DLinnarr [Zeng et al. 2023]. This indicates that performing trend-seasonal decomposition during the loss computation does not conflict with the previous trend-seasonal decomposition operations but rather enhances model performance.
4.3 Comparison with Other Loss Functions
To better validate the effectiveness of DBLoss, we compare it with several other loss functions- see Table 2 TILDE-Q emphasizes shape similarity using transfornation-inaariae loss terms. FreDF cleverly circumvents complex correlation modeling between labels by performing learning and prediction in the frequency domain. PSLoss incorporates patch-wie wise statistical properties into the loss function, enabling a more granular structural measurement of the data. The results indicate that DBLoss achieves the lowest MSE and MAE in most cases across various datasets and forecasting horizons. This is due to its ability to refine the characterization and representation of time series through decomposition within the forecasting horizon, thereby achieving more precise alignment between the ground truth and predictions.

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Table 2: Comparison between the proposed DBLoss and other loss functions. The model is DLinear and we report the result of three Siiasets-ETThh ETTml, and Traffic. The best results are highlighted in bold, and the second-best results are highlighted in underline.
Dataset
ETTh2
ETTml
Traffic
Forecnst horizon
96
192
720
Avg
966
720
Avg
96
192
336
720
Avg
MSE
0.300
0.387
0.490
0.704
0.470
0.300
0.336
0.367
0.419
0.356
0.395
0.407
0.417
0.454
0.418
Ori
MAE
0.364
0.423
0.487
0.597
0.468
0.345
0.366
0.386
0,416
0.378
0.275
0.250
0.286
0.308
0.287
Q
MSE
0.287
0.362
0.425
0.599
0.418
0.302
0.336
0.371
0.425
0.359
0.416
0.422
0,423
0.461
0.431
3022
MAE
0.345
0.395
0.445
0.551
0.434
0.542
0.362
0.386
0.417
0.577
0.294
0.296
0.293
0,316
0.500
FreDF
MSE
0.284
0.362
0.420
0.587
0.413
0.302
0.333
0.363
0.415
0.353
0.398
0.408
0.416
0.452
0.419
2025e
MAF
0.342
0.396
0.445
0.546
0.344
0.363
0.381
0.411
0.375
0.270
0.275
0,380
0.300
0.282
PSooss
MSE
0.283
0.358
0.111
0.614
0.417
0.296
0.332
0.361
0.116
0.351
0.398
0.408
0.116
0.652
0.119
2025
MAE
0.343
0.393
0.434
0.549
0.430
0.339
0.361
0.380
0.413
0.373
0.270
0.274
0,279
0.299
0.281
MSE
0.284
0.586
0.409
0.361
0.396
0.407
0.417
(Ours)
MAF
0.342
0.390
0.430
0.533
0.424
0.337
0.358
0.378
0.409
0.370
0.270
0.274
0.279
0.298
0.280
Table 3: Zero-shot forecasting results on ETT datasets. The forecasting horizon is 720. The parameters for the baselines are kept consistent with those of TFB QQuu et al. 2024 The better results are highlighted in bold.
Model
Transformer
Ampliderr
PatchTST
DLinew
DBLoss
DBLos
Ori
DBLoss
Metric
MSF
MAF
MSF
MAR
MSE
MAF
MSE
MAR
MSF
MAR
MAE
MSE
MAR
MSF
MAR
ETTh -FTT2
0.46
0.470
0.434
0.446
0.393
0.427
0.401
D.431
0.402
0.43
0.389
0.427
0.647
0.573
0.542
0.520
1.001
0.676
0.771
0.592
0.777
0,571
0.758
0.576
0.753
0.500
0,722
0.570
0.754
0.602
0U735
0.584
ETThI-ETTas
0.454
0.447
0.434
0.420
0.406
0.415
0.412
0,414
0,403
0.414
0.400
0.410
0.640
0,566
0.535
0.510
ETTh2-ETThi
0.672
0.593
0.557
0.521
0.678
0.592
0.530
0.515
0.593
0.490
0.506
0.521
0.450
0.477
ETTm
0.969
0.659
0.802
0.594
0.761
0.585
0.714
0.564
0.762
0.577
0.738
0.551
0.752
0.608
0.738
0.579
ETTM--ETTm2
0.417
0.428
0.436
0.422
0:403
0.417
0.403
0,415
0.393
0.409
0,395
0.404
0.787
0.629
0.550
0,526
ETTul-E[Th]
0,705
0.598
0.528
0,516
0.500
0.494
0.482
0,488
0,710
0.594
0,553
0.534
0.460
0.481
0.445
0.468
ETTml TETTh2
0.433
0.460
0,409
0.444
0.425
0,446
0.421
0.445
0,,418
0.451
0.431
0.427
0.464
0.404
0.444
ETTm)-ETTm2
0.369
0.389
0.370
0.387
0.369
0.384
0.372
0.384
0.370
0.391
0.367
0.389
0:116
0.367
0.394
ETTm2-EEThh
1.001
0.704
0.775
0.613
0.512
0.524
0.479
0.491
0.896
0.695
0.617
0.577
0.488
0.197
0.460
0.481
ETTm2-ETTh2
0.477
0.486
0.456
0.468
0.411
0.464
0.414
0.439
0.412
0.419
0,400
0.431
0.415
0.452
0.410
0.445
FTTm2-<FTT
0.662
0.566
0.551
0.498
0.652
0.547
0.478
0,452
0.484
0.451
0..422
0.429
0.449
0.439
0.436
0.430
4.4 Zero-shot Forecasting Results
To evaluate the effectiveness of DBLoss in enhancing the generalization ability on unseen datasets, we follow the methods outlined in (Chen et al. 2024 Kudrat al., 2025) and conduct zero-shot forecasting experiments. Specifically, we sequentially use ETTh1, ETTh2, ETTm1, and ETTm2 as source datasets, while the remaining datasets serve as target datasets.
TableBshows the results measured on the target datasets when the forecasting horizon is set to 720. These results highlight the consistent advantages of DBLoss In most cases, DBLoss outperforms MSE loss, indicating that it can significantly improve the model's generalization performance across different datasets and sampling frequencies. These improvements stem from DBLoss's 's ability to better capture the intrinsic trends and seasonal patterns within the datasets, thereby enabling the model to more effectively adapt to unseen data patterns.
4.5 Results on Time Series Foundation Models
To further evaluate the effectiveness of the proposed DBLoss, we conducted 5% few-shot experiments on four time series foundation models, using DBLoss only during the fine-tuning stage. These models include two LLM-based time series forecasting models: CALF [Liu et al., 2025c). GPT4TS (Zou et al. 2023). as well as two time series pre-trained models: UniTS [Gao et al. 2024), TTM (Eka baram et al. 2024). The results in show that incorporating DBLoss consistently outperforms the standard MSE loss. These findings highlight that DBLoss not only enhances performance on specific models but also improves the performance of foundation models, further demonstrating its significant role in multivariate time series forecasting.
7

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Table 4: Foundation models results in the 5% few-shot setting. The table reports average MSE and MAE for four forecasting lengths F E {96, 192, 336, 720}. The parameters for the baselines are kept consistent with those of TSFM- Full results are provided in Table of of Appendix G
[Li et al., 2025c The better results are highlighted in bold.
Model
GPT4TS
CALF
TTM
UniTS
Loss
Ori
DBLoss
Ori
DBLoss
Osi
DBLoss
Ori
DBLoss
Metric
MSF
MAR
MAR
MSE
MAE
MAR
MSB
MAR
MAE
MSR
MAF
MAP
ETTa1
0.467
0.470
0.453
0.462
0.44
0.451
0.433
0.446
0.405
0.425
0,395
0.417
0.436
0.434
0.425
0.427
ETT2
0.373
0.414
0.368
0.406
0.373
0.407
0.368
0.404
0.342
0.383
0.332
0.378
0.372
0.405
0.357
0.393
ETTml
0.388
0.404
0.377
0.394
0.372
0.396
0.358
0.382
0.356
0.376
0.354
0.372
0.377
0.402
0.362
0.386
ETTm2
0.278
0.335
0.266
0.320
0.271
0.352
0,259
0,316
0.258
0.313
0.257
0,308
0.292
0.344
0.270
0,320
Solar
0.262
0.335
0.279
00,229
0.246
0.300
0.219
0.269
0.266
0.206
0.261
0.246
Weather
0.253
0.293
0.248
0.284
0.238
0.277
0.236
0.272
0.225
0.260
0,225
0.256
0.230
0.269
0,231
0.260
Electriitty
0.207
0.317
0.207
0.309
0.172
0.268
0.171
0.264
0.179
0.277
0.178
0.274
0.180
0.275
0.181
0.274
Traflie
0:433
0.309
0.428
0.295
0.435
0.316
0.433
0.309
0:484
0.341
0.481
0.339
0.422
0.289
0.420
0.282
4.6 Impact of DBLoss on Generalization
To examine how DBLoss affects training dynamics and generalization capabilities, we use both MSE loss and DBLoss as objective functions and visualize the MSE on the training and testing datasets across all training epochs-soe see Figure We observe a consistent trend across all datasets. During training, models optimized with only MSE loss have lower errors per epoch compared to those optimized with Lossss. However, on the test data, models trained with MSE loss exhibit higher errors per epoch than those trained with Dossss. These observations indicate that while models trained with MSE loss have lower losses during the training phase, they generalize poorly on test data. In contrast, DBLoss enhances the model's generalization and prediction accuracy by encouraging the model to learn the trends and seasonal patterns in the dataset. Additionally, models trained with MSE loss show significant MSE fluctuations in the test loss on some datasets (e.g., Figure 3b and Figure 3d). whereas DBLoss demonstrates greater stability.
Training Error
Testing Error
Training Error
Texting Error
HIBLORs Delom
0,990
0.42
0.77
0.25
0.21
M
11107
33655
0:25
13552
0.19
0.34
33835
0.166
Epochs
Epochs
Epochs
Epochs
(a) Results on ETThl dataset
(b) Results on ETTm2 dataset
Training Error
Testing Errar
Training Error
Testing Error
0.22
0.216
0.00
OBLess
paless
onims
OBLess
0.300
CLUS
6.62
0L150
0:56
0,100
628
0:24
Epochs
Epochs
Epochs
Epochs
(c) Results on Weather dataset
(d) Results on Solar dataset
Figure 3: Training and testing MSE loss curves across all training epochs for the iTransformer model trained with MSE loss and DBLoss on the ETThI, ETTm2, Weather, and Solar datasets. Notably, the model trained with DBLoss exhibits higher training errors but achieves lower testing errors. This highlights the effectiveness of DBLoss in enhancing generalization and mitigating overfitting.
8

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4.7 Hyperparameter Sensitivity
Our method has two hyperparameters: the score weight for weighted loss and the smooth- ing factor a for EMA decomposition. To handle extreme cases, we manually replace a 0 and with approximate values close and I, respectively. Specifically, larger 3 in- creases the proportion of the seasonal component in the loss calculation, while a smaller a results in heavier smoothing, making the trend smoother and the seasonal component more prominent. Conversely, larger results in less smoothing, making the trend less smooth and the seasonal component less noticeable. From Figure 4 we have the following observations: I) When 3 is too large (e.g.. 3 = 1) or too small (e.g., = 0), the model's performance is poor. 2) For datasets with pronounced seasonality, such as traffic, a larger score weight 3 (i.e., considering higher proportion of the seasonal component in the loss cal- culation) yields better performance. A smaller smoothing factor (i.e.. making the seasonal component more prominent) also improves performance. 3) For datasets with less pronounced seasonality, such as ETTh2, moderate 3 value (e.g.. 0.4 0.5) achieves better results, indicating that the proportions of the seasonal and trend components should be balanced. The variation in the smoothing factor a has minimal impact on performance. 4) However, we find that the optimal values of and may vary across different algorithms. At present, there is no definitive method for selecting these parameters. We discuss this limitation Appendix H and leave it as an open problem for future research.
0.41
0.40
0.39
ETTha Trafficc
M5R
32-
0.28
03 Score Weight: BB
0.20
01002030000 08
Smoothing Factor:
Figure 4: The impact the hyperparameter on ETTh2 and Traffic based DLinear (horizon 96).
4.8 Forecasting Visualization
Figure 5 shows the visualization of forecasting results for samples from the ETThl dataset. We can observe that the predictions obtained with DBLoss are closer the ground truth. This is mainly because DBLoss encourages the model to better learn the seasonal and trend patterns in the dataset. More visualization results are provided in Appendix
Trranformer
Amplittie
PaichTST
DLinear
Figure Forecasting visualization comparing DBLoss and MSE loss objective functions.
5 Conclusion
In this study, we propose DBLoss to address the traditional MSE that sometimes fails to accurately capture the seasonality trend within the forecasting horizon, even when decomposition modules are used in the forward propagation to model the trend and seasonality separately. Specifically, our method uses exponential moving averages to decompose the time series into seasonal and trend components within the forecasting horizon, and then calculates the loss for each of these components separately. followed by weighting them. By introducing DBLoss into the baseline model, have achieved performance that surpasses the state-of toatte--f-the-arr the-art art on eight real- world datasets. Additionally, all datasets and code are available at http:://gthbb.ooomddciiommmtmmtt

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Acknowledgments and Disclosure of Funding
This work was partially supported by the National Natural Science Foundation of China (No. 62472174), the Open Research Fund of Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, ECNU, the Fundamental Research Funds for the Central Universities, and the ECNU Multifunctional Platform for Innovation (001).
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A Datasets
To conduct comprehensive and fair comparisons for different models, we conduct experiments on eight well-known forecasting benchmarks as the target datasets, including (I) ETT (Flectricity Transformer Temperature, 4 subsets) data contains seven features of electricity transformer data collected from two separate counties between July 2016 and July 2018. These datasets vary in granularity, with "h" indicating hourly data and "m" indicating 15-minute intervals. The suffixes "I" and "2" refer to two different regions from which the datasets originated. (II) Weather data includes 21 meteorological factors recorded every 10 minutes in 2020 at the Max Planck Biogeochemistry Institute's 's Weather Station. (III) Electricity data contains hourly electricity consumption data of 321 clients from 2012 to 2014. (IV) Solar data records the solar power production of 137 PV plants in 2006, which are sampled every 10 minutes.
(V) Traffic data contains road occupancy rates measured by 862 sensors on freeways in the San Francisco Bay Area from 2015 to 2016, recorded hourly.
Table 5: Statistics of datasets.
Doraset
Domain
Prequency
Lengths
Dim
Split
Description
ETTh1
Electricity
hour
14,400
Power transformer 1, comprising seven indicators such as temnerature useful
ETTh2
Electricity
1 hour
14,400
6:2:2
Power transformer 2. comprising seven indicators such as oil temperature and usefal load
ETTml
Electricity
15 mins
$7,500
7
6:2:2
Power tramsformer 1. comprising seven indicators such as eil temperature and useful load
ETTm2
Electricity
15 mins
57,600
6:2:2
Power transfommer 2, comprising
lemperaiur and usctul load indicatoos such meteveological indicatoos year 2020, which
Weather
Environment
mins
$2,696
7:1:2
Reconedd every for the
Electricity
Electricity
bour
26.904
321
7:1:2
every hour 2012 to 2014 Solar peoduction recurds cellecied from 137 PV plants in Alabuma
Electricity records the electricity consumption
Soha
Energy
mins
$2,560
137
6:2:2
Trallie
Traflic
hour
17,544
862
7:1/2
Roal vecaannc tales micsuuee by 862 sensors on Sun Francisco Bay urea freeways
B Related Works
Time series forecasting (TSF) predicts future observations based on historical observations. TSF methods are mainly categorized into four distinct approaches: (1) statistical learning- bbaen methods, (2) machine learning-based methods. (3) deep learning-based methods, and (4) foundation methods.
Early TSF methods primarily rely on statistical learning approaches such as ARIMA [Bxx and Pierce
1970 ETS (Hyndman al. 2008). J. and VAR [Godahewa et al. 2021]. With advancements in machine
learning, methods like XGBoost [Chen and Guestrin [016]. Random Forests [Breiman 2001]. J, and [017] gain popularity for handling nonlinear patterns. However, these methods
LightGBM (Ke et al.
still require manual feature engineering and model design [Pan el al. |023] |Wu et al. [002bb] Lu et al.
2024 Li et al. 2025d Fu al. 2025 Lu al. 2025 Li et al.
2025e Zhou al. 2025] Leveraging
the representation learning of deep neural networks (DNNs) Cheng et al. 2023 Qiu et al. 2025d
Li ct al. 2025f Huang al., [022cc] Yang et al., 2024| Lu et al. 2003 Wang et al.) 20251 many
deep tearning-basee methods emerge. TimesNet [Wu et al. 2023b) and SegRNN [Lin et al. 2023
model time series as vector sequences, using CNNs or RNNs to capture temporal dependencies.
Additionally, Transformer architectures, including DUET Qiu et al. 2025a], Informer [Zoou et al.
202T]. DAG [Qiu et al. 2025e]. FEDformer [Zhou et al. 2022]. Trifommer [Cirstea et al. 2022], ] and
PatchTST (Nie et al. 2023), J, capture complex relationships between time points more accurately.
significantly improving forecasting performance. MLP-based methods, including Hdmixer (Huang et al. 2024), SPF [Lin et al. 2024b). CycleNer [Lin et al.) 2024a), APN (Liu et al. 2025d].
SRSNet [Wu et al. 2025b). NLinear [Zeng et al. 2023), and Dl.inear (Zeng et al., 2023), J. adopt simpler architectures with fewer parameters but still achieve highly competitive forecasting accuracy.
However, many of these methods struggle with generalization across domains due to their reliance on domain-specific data [Li et al. 2025b). To address this, foundation methods are proposed, categorized into LLM-based methods and time series pre-trained methods. LLM-based methods (Zhou et al.
2023 Jin et al. 2024 Liu et al. 2024a Pan et al. 2024) leverage the strong representational capacity and sequential modeling capability of LLMs to capture complex patterns for time series modeling. Time series pre-trained methods (Liu et al., 2024b Gao et al., 2024| Goswami et al.) 2024, Das jet al. 2024) focus on pre-training over domti- time series data, enabling the method to learn domain-agnostic features that are transferable across various applications. This strategy not only enhances performance on specific tasks but also provides greater flexibility when adapting to new datasets or scenarios.
23

---

C Theoretical Proofs
In this section, provide theoretical analysis to explain why the proposed DBLoss more effective than the conventional MSE loss in time series forecasting.
Motivated by the success of recent methods such DLinear Zeng et al. 2023 DUET Qiu et al. [225a] TimeMixer Wang et al. [2024], and xPatch SSttsyuk and Choi 2025]. which model time
series by decomposing them into trend and seasonal components, achieving excellent performance, we assume that the trend and seasonal components are highly independent.
C.1 Problem Formulation
Let the original time series be denoted as Vs, which can be decomposed into a trend component Ti and seasonal component Sy:
T. + S.. Similarly, the model prediction de can be expressed as:
(4)
Út = T. S.-
(5)
C.2 Analysis of MSE Loss
Under this setting, the Mean Squared Error (MSE) can be expanded as:
LASE 11ye 0.llă [I(T, St) (Ý, + 5
|I(T; f;)) + (S; - 8;)11
||T f21 + ||S, S.112 + (Tt -Te--))(SS -.)-
(6)
The key part from MSE lies in the cross term 2 (T; (i)(S, S.). Our assumption is that the trend and seasonal components are highly independent. However, this cross term introduces an interaction between them, potentially making it difficult for the model optimize the two components independently, which can degrade the overall prediction performance. For instance, the trend component is poorly predicted while the seasonal component is well captured, the interaction term still yield large negative value of (Te (S) S,), disproportionnaely affecting the total loss.
C.3 Gradient Analysis of MSE
We further analyze the MSE loss from the perspective of gradient propagation, and reveal MSE loss being unable to independently consider these two components during the optimization process. According to the chain rule:
Vel. = ((T((TT)))))) -T) (St 9:)] ve(-f,
= [TT -;) + (St S.)1 (Voli + VoS.).
Let the Jacobians of the trend and seasonal components be:
Jr = VoTi. Js VoS.
(8)
Then the gradient can be expressed as:
V = = (T, T.JJ + (Te 4:JJs + (S, - -S.JJ + (S, 5,JJs
(9)
We decompose this gradient into two parts:
(1) Ideal Decoupled Term
G = 2. [ (77-f.JJ-- + (St 5.JJs
(10)
which represents the desired independent optimization of the two components.
24

---

(2) Coupled Term
coupling -2 [ - -T)J + (S, - S.)J,
(11)
The coupled term Coopling causes mutual interference: the trend error influences the seasonal optimization and vice versa. As long as T₃ * Fh or St * S, (i.e., the model has not converged), Geoupling O, meaning that the optimization of one component will inevitably affect the other.
C.4 Analysis of DBLoss
Los 3 3-11, 11,8,-,.12++ (1 1)- -Till:
(12)
where 3 is hyperparameters controlling the relative weights of the trend and seasonal components.
Unlike MSE, DBLoss explicitly separates the optimization of the two components, thus removing the coupling term Geoupling from the gradient computation. Furthermore, by adjusting the coefficients or using different distance norms, targeted learning and better modeling of both parts.
precisely control the loss scale for each component, enabling
D Visualization
iTransformer
Amplitier
PatchTST
DLinear
ETTh
(a) Forecasting visualization ETThl dataset
Amplißeer
022
92
02
100
(b) Forecasting visualization on ETTm2 dataset
(Transformer
Amplifier
PatchTST
DI invar
0000
(uB
8:00
Woither
-L.b
0.50
0.5
23
150
250
30
100
100
124
200
100
120
200
(c) Forecasting visualization on Weather dataset
Amplitier
Jnnrr
Siar
(d) Forecasting visualization Solar dataset
Figure 6: Forecasting visualization comparing DBLoss and MSE loss as objective functions.
25

---

E
Comparison with Other Loss Functions
Table Comparison between the proposed DBLoss and other loss functions. The model is DLinear and we report the result of ETTh2. The best results are highlighted in bold, and the second-best results are highlighted in underline. The standard deviation of methods calculated through 5 random seeds are also reported.
Dataset
ETTh2
Forecust horizon
96
192
336
720
Avg
MSE
0.300 002
0.387
2.400
0.704
0.470 001
MAE
0.361s0.002
0.123
0,487 28.020
0.597+00001
0.1168e0.00
TILDE Q
MSE
a.287
0.062
0.425 20.002
0.418 0.509
2022
MAF
0.34520.001 0.39510..00
0.44520.002
0.001
0.434 001
DF
MSE
0 000000000 0.362:0.001
12020.002
0.587+0.000
0
2025c
MAE
0.342
0.396
0. 1.55500.004
0.5:66i0.000
0. 432:0.0008
PSLoss
MSE
0.283 0.358+0.002
0 0.000000002
0.61/1+0.008
0.117=0.002
2025
MAE
0.343
0.303
4..0000002
0.549
0.430=6.002
DBI Joss
MSF
0.2842..00z 0.357 000
0.407
0.586 o01
0.409
(Ours)
MAE
0.34220.001
0.390
0.430
0.533 004
0.424
Table 7: Comparison between the proposed DBLoss and other loss functions. The model is DLinear and we report the result of ETTml. The best results are highlighted in bold, and the second-best results are highlighted in underline. The standard deviation of methods calculated through 5 random seeds are also reported.
Dataset
ETTml
Forecast harizon
96
192
336
720
Avg
MSE
0.300 0.336
0.367
0.419#0.000
0.356=s0.0m
Ori
MAE
0.34520.002 0.36619..00
0.38620.005
0.41620.000
0.378.2.005
TILDE-Q
MSE
0.302=0.002 0.336
0.371 20.002
0.425a0.00
0.359=0.00 o03
2022
MAE
0.312.0.005
0.362+0.008
0,386
0.117+0.002
0.377=6.002
DF
MSE
0. 0.302e0.003 0.333+0.004
0.363
0,415 001
0.353
2025e
MAE
0.344=0.001
0.3634:0.001
0,381 20.00s
D.411+0.002
0.375
0,296 1.832
0.361
0.41610.001
0.351
2025
MAE
0.33920.002
0.36 13661:0.000
0,380
0.413a0.00 not
0.373
DBLoss (Ours)
MSE
0.205
0.331 s00000
0.361=0.002
0.415
0.351 0.351-0.002
MAE
0.337
0.358 10.001
0.378
0.009:00001
0.370
Table 8: Comparison between the proposed DBLoss and other loss functions. The model is DLinear and we report the result of Traffic. The best results are highlighted in bold, and the second- results highlighted in underline. The standard deviation of methods calculated through 5 random seeds are also reported.
Dataset
Traflic
Forecast horizon
192
336
Avg
MSE
0.395 0.407
0.417a0.00
0.45440.cm
0.418
Ori
MAE
0.275=0.002
0.250+0.000
0.286
0.308=0.00l
0.287=0.002
TILDE
MSE
0.41620.00s 0.422
2..0000000
0.461
0.431 :0:00000000
2022
MAF
0.294.0.002
0.296
0.293
0.31610.000
0.300a..002
DF
MSE
0 2..00
0.408 100000
0.416:0.002
0.419 0.452
2025e
MAE
0.270
0.275100000
0. 0.200200002
0.3022
0.282.10.0m 001
PSLos
MSE
0. 0.398=0.001
0. 0.408000001
0.416=0.003
0.152+0.005
0.119
2025
MAE
0.270
0.274
0.279:0.00.
0.299
0.281 =0.002
DBLoss
MSE
0.396 0.407:0.000
0.415
0.449
0.417=0.0m2
(Ours)
MAE
0.270
0.274:0000
0.279
0.298
0.280.0.002
26

---

Efficiency
Table 9 presents the epoch-level training times (in seconds) of PatchTST when using DBLoss and MSE across different datasets. The results show the average values for the four forecasting horizons of each dataset, with the same parameters, where only MSE is replaced by DBLoss. Based on the experimental results in the table, we can observe that DBLoss does lead to an increase in training time compared to MSE, this increase is not significant. As dataset size grows, the time difference becomes even more negligible.
Table Comparison of average epoch-level training times (in seconds) between DBLoss and MSE using PatchTST across four forecasting horizons on different datasets.
Train Time
ETTh
ETTh2
ETTml
ETTm2
Solar
Weather
Electricity
Traffic
MSE
2.36
2.37
14.45
14.39
183.11
36.07 37.23
258.47 260.85
1035.77 1039.67
DBLoss
3.11
3.37
15.93
15.73
186.31
G Full Results on Time Series Foundation Models
Table T0 presents the results of foundation models under the 5% few-shot setting. reports both
MSE and MAE across different forecasting horizons F E 96, 192, 336, 720. The baseline parameters
are kept consistent with those used in TSFM-Bench [Li al. 2025c]. The best results are highlighted in bold.
H Limitations
Potential limitations The DBLoss demonstrates efficacy in TSF scenarios. However, there are several potential limitations of DBLoss that warrant discussion here:
The proposed method involves two critical hyperparameters: the score weight for the weighted loss and the smoothing factor for Exponential Moving Average (EMA) decomposition. Specifically, larger B increases the proportion the seasonal component the loss calculation, while smaller results in stronger smoothing, making the trend smoother and the seasonal component more prominent. However, the current research has not yet proposed an automated strategy to optimize the selection of these two parameters. The absence of systematic tuning approach may still limit the model's 's performance improvement and generalization capability. There- fore, developing an efficient automated hyperparameter tuning mechanism to adaptively determine the optimal parameter combination is an important direction for future research.
Lack of Automated Hyperparameter Tuning Strategy:
27

---

Table 10: Foundation models results in the 5% few-shot setting, The table reports MSE and MAE for different forecasting lengths F' E {96, 192, 336, 720}. The parameters for the baselines are kept consistent with those of TSFM-Bench [Li et al. 2025c The better results are highlighted in bold.
Model
GPT4TS
CALF
TTM
UniTS
Loss
Ori
18s
Ori
DBI
LBss
Ori
DBLLos
Ori
DBLOs
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
0.438
0.445
0.459
0.446
0,405
0.426
0,401
0.422
0.363
0.392
0.361
0.390
0.381
0.394
0,381
0.393
192
0.460
0.458
0.455
0.456
0.128
0.442
0.423
0.436
0,391
0.409
0.387
0.405
0.421
0.428
0.405
0.424
336
0.462
0.467
0.449
0.460
0.443
0.454
0.437
0.447
0.411
0.429
0.404
0.422
0.443
0.437
0.429
0.429
0.509
0.511
0.470
0.486
0.495
0.494
0.472
0.479
0.453
0.471
0.429
0.448
0.498
0.475
0.486
0.463
Avg
0.467
0.470
0.453
0.462
0.413
0.454
0.433
0.446
0.405
0.425
0.395
0.417
0.436
0.434
0.425
0.427
0.329
0.380
0.323
0.371
0.302
0.362
0.308
0.363
0.271
0.329
0.270
0.328
0.305
0.353
0.299
0.346
192
0.368
0.406
0.364
0.399
0.385
0,400
0.383
0.397
0.339
0.373
0.329
0.374
0.369
0,403
0,357
0.390
336
0.378
0.421
0,374
0.412
0.387
0.418
0,375
0.412
0.372
0.401
0.346
0.392
0.388
0.412
0.361
0.399
720
0.418
0.450
0.412
0.443
0.416
0.449
0.407
0.445
0.385
0,428
0.382
0.419
0,425
0.451
0.410
0.439
Avg
0.373
0.414
0.368
0,406
0.373
0.407
0.368
0.404
0.342
0.383
0.332
0.378
0.372
0,405
0.357
0.393
96
0.343
0.379
0,330
0.168
0.317
0.366
0.299
0.347
0,299
0.343
0.294
0.337
0,313
0.363
0.360
0.349
ETTal
0.361
0.387
0,380
0.337
0.369
0.367
0.340
0.357
0,390
0.373
336
0.394
0.106
0.384
0.398
0.385
0,405
0.371
0
0.365
0.381
0.363
0.379
0,381
0.105
0,370
0.392
720
0.440
0.434
0.432
0.425
0.439
0.433
0.427
0.420
0.420
0.412
0.419
0.409
0.457
0.448
0.438
0.428
Avg
0,388
0.377
0.394
0.396
0.358
0.376
0.354
0.377
0.402
0.386
96
0.190
0.279
0.181
0.266
0.180
0.272
0.170
0.258
0.164
0,250
0.162
0.244
0.188
0.278
0.173
0.255
ETTal
192
0.241
0.312
0.231
0.300
0,237
0.310
0.228
0.295
0.222
0,299)
0.224
0.287
01255
0.317
0.247
0.302
0.281
0.330
0.348
0.277
0.282
0.330
0.321
0.366
720
0.385
0.401
0.371
0.384
0.372
0.397
0.363
0.382
0.364
0.381
0.363
0.376
0.404
0.415
0.374
0.392
Avg.
0.278
0,335
0,266
0.329
0.271
0.332
0.259
0.316
0,258
0.313
0.257
0.348
0.292
0.344
0,270
0,320
0.253
0.243
0.267
0,274
0.238
0.154
0.186
0.226
Solar
192
0.2096
0.336
0.252
0.281
0.224
0.290
0.243
0.294
0.225
0.270
0.235
0.267
0.198
0.255
0.200
0.241
336
0,262
0.341
0,260
0.284
0.243
0.308
0.252
0,303
0,222
0.274
0.251
0,271
0.208
0,259
0,222
0,254
0.265
0.261
0.285
0.247
0.314
0.251
0.306
0.226
0.277
0.231
0.284
0.243
0.265
Avg.
0.262
0.335
0,254
0.279
0.229
0.297
0. 0.346
0.300
0.219
0.269
0.224
0.266
0.206
0.261
0.214
0.246
0.187
0.244
0,181
0.234
0,163
0.217
0.162
0.210
0.147
0.195
0.148
0.191
0.154
0,206
0.156
0,198
Weather
0.225
0.220
0.265
0.206
0.253
0.205
0.194
0.238
0.199
0.248
0.237
336
0,268
0.304
0.265
0.297
0.266
0.297
0,257
0.292
0,244
0.277
0.243
0.272
0.248
0.285
0.248
0.275
0.330
0.348
0,327
0.340
0,322
0,339
0.322
0.338
0,314
0.329
0.316
0,327
0.320
0.337
0,320
0,329
Avg
0.25%
0.291
0.248
0.284
0.238
0.277
0.236
0,260
0.230
0.269
0,231
0.260
Eleeriikty
96
0.178
0.294
0.179
0.286
0.141
0,240
0,140
0.236
0.146
0,246
0.146
0.245
0.150
0.249
0.152
0.248
192
0.192
0.306
0.192
0.300
0.156
0.254
0.154
0.250
0.165
0.264
0.165
0.262
0.167
0.264
0,168
0,263
0.208
0.318
0.208
0.310
0.174
0.271
0.174
0.268
0,281
0.180
0.278
0.181
0.277
0,182
0.276
720
0,248
0,348
0,248
0.339
0,216
0,306
0,214
0.302
0.223
0.315
0.223
0,313
0.220
0.309
0,224
0,310
0.207
0.317
0,207
0.309
0.172
0.268
0.171
0.264
0.179
0.277
0.178
0.274
0.180
0.275
0,181
0.274
0.411
0.300
0.408
0.286
0.406
0.298
0.405
0.290
0.448
0.324
0.445
0.322
0.101
0.278
0,400
0.272
Thilic
192
0.422
0.304
0.416
0.289
0.423
0.309
0.420
0.301
0.466
0.330
0.463
0.329
0.414
0.284
0.412
0.278
0.432
0.308
0.426
0.293
0.436
0.317
0.432
0,491
0.345
0.487
0.34.
0,421
0.250
0,280
720
0.468
0.325
0.463
0.311
0.477
0.340
0.476
0.335
0.533
0.365
0.527
0,361
0.452
0.305
0,451
0.298
Avg
0.433
0.309
0.428
0.295
0.435
0.316
0.433
0.309
0.484
0.341
0.481
0.339
0.422
0.289
0.420
0.282
28


Reviews:
[{"review_id": "4c3g0f5Pfu", "reviewer": "Reviewer_oJJG", "summary": "This study proposes a simple yet effective loss function for time series forecasting, called DBLoss, which can refine the characterization and representation of time series through decomposition within the forecasting horizon. By decomposing the future sequence into different trend and seasonal components, DBLoss enhances the loss function's ability to improve prediction performance for different items.", "questions": "Q1: The end-to-end model in the paper was comprehensively evaluated on 8 benchmark datasets, while the foundation model was only validated on 4 datasets. Could you please explain the rationale behind this inconsistency in experimental design?\n\nQ2: In Algorithm 1, the range of the EMA smoothing coefficient α is defined as (0, 1), suggesting that 0 and 1 are excluded. However, Figure 4 appears to include these boundary values. Could you clarify this discrepancy?", "limitations": "The paper includes a dedicated section addressing the inherent limitations of the proposed approach.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "S1. This paper is well motivated as it faces the challenge of effectively capturing the seasonality or trend within the forecasting horizon.. The notations are clear, the experimental results show that the proposed method is effective.\n\nS2. This paper points out that the traditional MSE Loss sometimes fails to accurately capture the seasonality or trend within the forecasting horizon and designs a new Loss Function accordingly, providing a new perspective.\n\n\nW1. The end-to-end model in the paper was comprehensively evaluated on 8 benchmark datasets, while the foundation model was only validated on 4 datasets. Could you please explain the rationale behind this inconsistency in experimental design?\n\nW2. The paper employs EMA (Exponential Moving Average) decomposition. Why was STL (Seasonal-Trend decomposition using Loess) not adopted to decompose the series into trend, seasonal, and residual components, with separate loss calculations and weighted aggregation?\n\nW3. In Algorithm 1, the range of the EMA smoothing coefficient α is defined as (0, 1), suggesting that 0 and 1 are excluded. However, Figure 4 appears to include these boundary values. Could you clarify this discrepancy?\n\nW4. The proposed method is intuitively feasible, but a more fundamental theoretical explanation would help enhance the significance of the study. The study relies too heavily on empirical experimental analysis and lacks in-depth experimental justification.", "was_revised": true, "final_justification": "I appreciate the author's response and have accordingly increased my score.", "rebuttal": "Dear  **Reviewer oJJG**, thank you for providing your detailed and constructive feedback.\n\n**W1&Q1:** The end-to-end model in the paper was comprehensively evaluated on 8 benchmark datasets, while the foundation model was only validated on 4 datasets. \n\nThank you for your valuable feedback. We recognize the importance of consistency in experimental design. To address this, we have added additional experiments and extended the evaluation of the foundation model on more datasets to provide a more comprehensive comparison with the end-to-end model. The results of the additional experiments are as follows, and it is evident that DBLoss performs well on these datasets. We are reporting the average values for four prediction horizons for each dataset.\n\n|    Model    |      | GPT4TS |       |           |           |   CALF    |       |           |           |  Moment   |       |           |           |    TTM    |       |           |           |\n| :---------: | :--: | :----: | :---: | :-------: | :-------: | :-------: | :---: | :-------: | :-------: | :-------: | :---: | :-------: | :-------: | :-------: | :---: | :-------: | :-------: |\n|             |      |  Ori   |       |  DBLoss   |           |    Ori    |       |  DBLoss   |           |    Ori    |       |  DBLoss   |           |    Ori    |       |  DBLoss   |           |\n|             |      |  MSE   |  MAE  |    MSE    |    MAE    |    MSE    |  MAE  |    MSE    |    MAE    |    MSE    |  MAE  |    MSE    |    MAE    |    MSE    |  MAE  |    MSE    |    MAE    |\n|    Solar    | Avg  | 0.262  | 0.334 | **0.254** | **0.278** | **0.229** | 0.297 |   0.232   | **0.291** |   0.243   | 0.277 | **0.241** | **0.274** |   0.218   | 0.269 | **0.215** | **0.257** |\n|   Weather   | Avg  | 0.252  | 0.293 | **0.249** | **0.285** |   0.237   | 0.275 | **0.234** | **0.268** |   0.230   | 0.269 | **0.229** | **0.257** |   0.225   | 0.260 | **0.224** | **0.254** |\n| Electricity | Avg  | 0.207  | 0.316 | **0.205** | **0.311** |   0.172   | 0.268 | **0.170** | **0.264** | **0.193** | 0.296 | **0.193** | **0.294** | **0.179** | 0.276 | **0.179** | **0.274** |\n|   Traffic   | Avg  | 0.433  | 0.309 | **0.429** | **0.299** |   0.435   | 0.316 | **0.434** | **0.313** |   0.434   | 0.304 | **0.430** | **0.300** |   0.484   | 0.341 | **0.478** | **0.335** |\n\n**W2:** Why was STL (Seasonal-Trend decomposition using Loess) not adopted to decompose the series into trend, seasonal, and residual components, with separate loss calculations and weighted aggregation?\n\nThank you for your valuable feedback. While STL is indeed a popular for time series decomposition, we chose to use EMA decomposition for the following reasons:\n\n- EMA provides a smooth and efficient way to model the trend without the computational complexity of Loess, which we found to be particularly slow. For example, when using the DLinear algorithm to predict 96 steps on the ETTh1 dataset, it required over 11 hours to run, severely limiting the practical usability of STL. \n- Our main contribution lies in proposing a new loss paradigm that, through a reasonable decomposition approach, enhances model performance. In fact, any decomposition method could be applied, but the quality of the decomposition directly affects the final performance. We adopted the currently advanced EMA decomposition method [1] with the aim of improving performance.\n\n[1] xPatch: Dual-Stream Time Series Forecasting with Exponential Seasonal-Trend Decomposition, AAAI, 2025\n\n\n\n**W3&Q2:** In Algorithm 1, the range of the EMA smoothing coefficient α is defined as (0, 1), suggesting that 0 and 1 are excluded. However, Figure 4 appears to include these boundary values. Could you clarify this discrepancy?\n\nThank you for your valuable feedback. The range of α in the algorithm's definition is constrained to (0, 1), excluding 0 and 1, primarily to avoid computational errors such as division by zero when α = 1, and also to prevent the model from degenerating into a non-typical or meaningless smoothing method when α = 0. However, in Figure 4, to account for extreme cases, we manually replaced the value of 0 with an approximate value close to 0 and the value of 1 with an approximate value close to 1. We apologize for not clarifying this in the explanation of Figure 4. We will revise this in the corrected version of the paper.\n\n\n\n**W4:** The proposed method is intuitively feasible, but a more fundamental theoretical explanation would help enhance the significance of the study. The study relies too heavily on empirical experimental analysis and lacks in-depth experimental justification.\n\nThank you for your insightful feedback and constructive observations regarding our work. In order to better explain the differences between DBLoss and MSE. We provide a theoretical analysis explaining why DBLoss is more effective than MSE in this context. Motivated by the success of methods such as Dlinear [1], DUET [2], TimeMixer [3], and xPatch [4], which model time series by decomposing them into trend and seasonal components, achieving excellent performance, we assume that the trend and seasonal components are highly independent. \n\nWe represent the time series as $y_t$, which can be decomposed into a trend component $T_t$ and a seasonal component $S_t$, such that:\n\n$y_t = T_t + S_t$\n\nFor the model’s prediction $\\hat{y}_t$, we similarly have:\n\n$\\hat{y}_t = \\hat{T}_t + \\hat{S}_t$\n\nUnder this setting, the Mean Squared Error (MSE) can be expanded as:\n\n$$L _{MSE} = ||y _t - \\hat{y} _t|| _2 ^2 = ||(T _t + S _t) - (\\hat{T} _t + \\hat{S} _t)||_2 ^2$$\n$$  ~~~~~~~~~~ = ||y _t - \\hat{y} _t|| _2^2 = ||(T _t - \\hat{T}_t) + (S _t - \\hat{S} _t)||_2^2$$\n$$ ~~~~~~~~~~ = ||T _t - \\hat{T}_t|| _2^2 + ||S _t - \\hat{S}_t||_2 ^2 + 2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$$\n\nThe key part from MSE lies in the cross term $2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$. Our assumption is that the trend and seasonal components are highly independent.However, this cross term introduces an interaction between them, potentially making it difficult for the model to optimize the two components independently, which can degrade the overall prediction performance. For instance, if the trend component is poorly predicted while the seasonal component is well captured, the interaction term can still yield a large negative value of $2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$, disproportionately affecting the total loss.  \n\nFurthermore, we analyze $L_{MSE}$ from the perspective of differentiation, and reveal MSE loss function being unable to independently consider these two components during the optimization process:\n\nAccording to the chain rule:\n\n$\\nabla _{\\boldsymbol{\\Theta}} L _t = 2 \\cdot [(T _t - \\hat{T}_t) + (S _t - \\hat{S}_t)] \\cdot \\nabla _{\\boldsymbol{\\Theta}} (-\\hat{T}_t - \\hat{S}_t) = -2 \\cdot [(T _t - \\hat{T}_t) + (S _t - \\hat{S}_t)] \\cdot \\left( \\nabla _{\\boldsymbol{\\Theta}} \\hat{T}_t + \\nabla _{\\boldsymbol{\\Theta}} \\hat{S}_t \\right)$\n\nUsing Jacobian matrix notation:\n\n- Jacobian of Trend: $$ \\mathbf{J} _T := \\nabla _{\\boldsymbol{\\Theta}} \\hat{T} _t  $$\n- Jacobian of Seasonality: $$ \\mathbf{J}_S := \\nabla _{\\boldsymbol{\\Theta}} \\hat{S}_t $$\n\nThen:\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot [(T_t - \\hat{T}_t) + (S_t - \\hat{S}_t)] \\cdot (\\mathbf{J}_T + \\mathbf{J}_S) $$\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot \\left[ (T_t - \\hat{T}_t)(\\mathbf{J}_T + \\mathbf{J}_S) + (S_t - \\hat{S}_t)(\\mathbf{J}_T + \\mathbf{J}_S) \\right] $$\n\nExpanding the terms:\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_T + (T_t - \\hat{T}_t)\\mathbf{J}_S + (S_t - \\hat{S}_t)\\mathbf{J}_T + (S_t - \\hat{S}_t)\\mathbf{J}_S \\right] $$\n\nWe split this into two parts:\n\n1、Ideal Decoupled Term\n\n$$ \\mathbf{G}_{\\text{ideal}} = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_T + (S_t - \\hat{S}_t)\\mathbf{J}_S \\right] $$\n\n2、Coupled Term\n\n$$ \\mathbf{G}_{\\text{coupling}} = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_S + (S_t - \\hat{S}_t)\\mathbf{J}_T \\right] $$\n\nThis term represents the problematic coupling:\n\n- The trend error influences the seasonality‑guided optimization.  \n- The seasonal error influences the trend‑guided optimization.  \n\nThe full gradient is expressed as the sum of the ideal and coupled terms: $$ \\nabla _{\\boldsymbol{\\Theta}} L _t = \\mathbf{G} _{\\text{ideal}} + \\mathbf{G} _{\\text{coupling}} $$\n\nAs long as $T _t \\neq \\hat{T}_t $ or $S _t  \\neq \\hat{S} _t $ (i.e., the model has not fully converged), the coupled term $  \\mathbf{G} _{\\text{coupling}}  \\neq \\mathbf{0} $. Due to the presence of $ \\mathbf{G} _{\\text{coupling}} $, the MSE loss function being unable to independently consider these two components during the optimization process.\n\n\n\nCompared with MSE, DBLoss ($L _{DB} = \\lambda_1 ||T _t - \\hat{T} _t||_1 + \\lambda_2 ||S _t - \\hat{S}_t||_2^2$ , where $\\lambda_1$ and $\\lambda_2$ are hyperparameters controlling the weights of the trend and seasonal losses, respectively.) removes this cross term (or $  \\mathbf{G} _{\\text{coupling}} $), enabling the model to treat these two components independently throughout the optimization process.   Furthermore, by adjusting the coefficients or using different distance norms, one can precisely control the loss scale for each component, enabling targeted learning and better modeling of both parts.\n\n[1] Are Transformers Effective for Time Series Forecasting? AAAI, 2023\n\n[2] DUET: Dual Clustering Enhanced Multivariate Time Series Forecasting, SIGKDD, 2025\n\n[3] TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting, ICLR, 2024\n\n[4] xPatch: Dual-Stream Time Series Forecasting with Exponential Seasonal-Trend Decomposition, AAAI, 2025"}, {"review_id": "MsnKWteVEp", "reviewer": "Reviewer_GNuv", "summary": "This study proposes a simple yet effective Decomposition-Based Loss function called DBLoss. This method uses exponential moving averages to decompose the time series into seasonal and trend components within the forecasting horizon, and then calculates the loss for each of these components separately, followed by weighting them.", "questions": "no.", "limitations": "YES", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "S1. Multivariate time series forecasting is important to time-series analysis.\n\nS2. This work focuses on an important problem that could have real-world applications.\n\nS3. The tables and figures used in this work are clear and easy to read.\n\nS4. Experimental results demonstrate that DBLoss consistently outperforms state-of-the-art baselines across multiple real-world datasets.\n\n---\n\nW1. Why was EMA (Exponential Moving Average) decomposition chosen over other similar decomposition mechanisms? Providing corresponding explanations would make the paper more comprehensive.\n\nW2. This infamous \"ETT long term forecasting benchmark\" is often criticized for its flaws such as limited domain coverage and the practice of forecasting at unreasonable horizons (e.g., 720 days into the future for exchange rate or oil temperature of a transformer at a specific hour months into the future). Every new model somehow beats this benchmark; however, there is still barely any absolute progress, only an illusion of it. Please refer to the talk (and paper) from Christoph Bergmeir [1, 2] where he discusses the limitation of this benchmark and current evaluation practices. A very recent position paper [3] also conducted a comprehensive evaluation of models on this benchmark showing that there's no obvious winner.\n\nOne (not so difficult) way to improve the quality of evaluation is to include results on better benchmarks.\n\n---\n\nMinor. Some citations are nonstandard by referencing arXiv preprints. Please ensure all citations point to formally published versions.\n\n[1] https://neurips.cc/virtual/2024/workshop/84712#collapse108471\n\n[2] Hewamalage, Hansika, Klaus Ackermann, and Christoph Bergmeir. \"Forecast evaluation for data scientists: common pitfalls and best practices.\" Data Mining and Knowledge Discovery 37.2 (2023): 788-832.\n\n[3] Brigato, Lorenzo, et al. \"Position: There are no Champions in Long-Term Time Series Forecasting.\" arXiv preprint arXiv:2502.14045 (2025).", "was_revised": true, "final_justification": "My concerns have been addressed. Regarding the additional theoretical analysis, I agree with the hypothesis and the corresponding theoretical insights.", "rebuttal": "Dear  **Reviewer GNuv**, thank you for providing your detailed and constructive feedback.\n\n**W1:** Why was EMA (Exponential Moving Average) decomposition chosen over other similar decomposition mechanisms? Providing corresponding explanations would make the paper more comprehensive.\n\nThank you for your valuable feedback.  Our main contribution lies in proposing a new loss paradigm that, through a reasonable decomposition approach, enhances model performance. In fact, any decomposition method could be applied, but the quality of the decomposition directly affects the final performance. We adopted the currently advanced EMA decomposition method [1] with the aim of improving performance.\n\n[1] xPatch: Dual-Stream Time Series Forecasting with Exponential Seasonal-Trend Decomposition, AAAI, 2025\n\n**W2:** This infamous \"ETT long term forecasting benchmark\" is often criticized for its flaws such as limited domain coverage and the practice of forecasting at unreasonable horizons.Every new model somehow beats this benchmark; however, there is still barely any absolute progress, only an illusion of it. \n\nThank you for your valuable feedback. To demonstrate the effectiveness of DBLoss, we conducted experiments on eight datasets from the widely recognized and authoritative TFB benchmark [2]. The results indicate that DBLoss significantly improves performance. We are reporting the average values for four prediction horizons for each dataset.\n\n[2] TFB: Towards Comprehensive and Fair Benchmarking of Time Series Forecasting Methods, PVLDB, 2024, Best Paper Award Nomination.\n\n|     Model     |      | iTransformer |       |            |           | TimeMixer |       |            |           | PatchTST  |       |            |           |  DLinear  |       |            |           |\n| :-----------: | :--: | :----------: | :---: | :--------: | :-------: | :-------: | :---: | :--------: | :-------: | :-------: | :---: | :--------: | :-------: | :-------: | :---: | :--------: | :-------: |\n| Loss Function |      |     Ori      |       |   DBLoss   |           |    Ori    |       |   DBLoss   |           |    Ori    |       |   DBLoss   |           |    Ori    |       |   DBLoss   |           |\n|    Metric     |      |     MSE      |  MAE  |    MSE     |    MAE    |    MSE    |  MAE  |    MSE     |    MAE    |    MSE    |  MAE  |    MSE     |    MAE    |    MSE    |  MAE  |    MSE     |    MAE    |\n|      ILI      | Avg  |    1.857     | 0.892 | **1.765**  | **0.838** |   1.820   | 0.886 | **1.651**  | **0.798** |   1.902   | 0.879 | **1.900**  | **0.878** |   2.185   | 1.040 | **2.004**  | **0.979** |\n|   AQShunyi    | Avg  |    0.706     | 0.506 | **0.701**  | **0.500** | **0.706** | 0.506 |   0.707    | **0.499** |   0.703   | 0.507 | **0.700**  | **0.502** |   0.706   | 0.522 | **0.704**  | **0.512** |\n|     AQWan     | Avg  |    0.809     | 0.496 | **0.804**  | **0.487** |   0.810   | 0.495 | **0.807**  | **0.486** |   0.812   | 0.498 | **0.810**  | **0.486** |   0.818   | 0.512 | **0.817**  | **0.502** |\n|    ZafNoo     | Avg  |    0.523     | 0.456 | **0.519**  | **0.453** |   0.518   | 0.451 | **0.515**  | **0.440** | **0.509** | 0.454 | **0.509**  | **0.441** |   0.496   | 0.452 | **0.492**  | **0.446** |\n|    CzeLan     | Avg  |  **0.218**   | 0.272 | **0.218**  | **0.259** |   0.218   | 0.268 | **0.208**  | **0.244** |   0.222   | 0.274 | **0.219**  | **0.249** |   0.285   | 0.343 | **0.252**  | **0.288** |\n|    Covid19    | Avg  |    1.488     | 0.050 | **1.431**  | **0.047** |  12.941   | 0.133 | **9.755**  | **0.083** |   1.607   | 0.056 | **1.528**  | **0.051** |   8.074   | 0.239 | **5.721**  | **0.135** |\n|    FRED-MD    | Avg  |    75.609    | 1.437 | **63.029** | **1.335** |  76.156   | 1.439 | **65.032** | **1.355** |  78.174   | 1.478 | **66.261** | **1.350** |  96.092   | 1.693 | **82.883** | **1.489** |\n|      NN5      | Avg  |    0.660     | 0.550 | **0.657**  | **0.549** |   0.656   | 0.556 | **0.654**  | **0.554** |   0.689   | 0.590 | **0.687**  | **0.589** | **0.692** | 0.580 |   0.693    | **0.578** |\n\n**W3:** Minor. Some citations are nonstandard by referencing arXiv preprints. Please ensure all citations point to formally published versions.\n\nThank you for your feedback. We appreciate your attention to detail. We understand the importance of citing formally published versions, and we will make sure to replace all references to arXiv preprints with their corresponding formally published versions.\n\n\n\n**Additional theoretical analysis:**\n\nConsidering that we had not previously provided a theoretical analysis of DBLoss, we have now included this additional analysis to make our paper more complete.\n\n Motivated by the success of methods such as Dlinear [1], DUET [2], TimeMixer [3], and xPatch [4], which model time series by decomposing them into trend and seasonal components, achieving excellent performance, we assume that the trend and seasonal components are highly independent. \n\nWe represent the time series as $y_t$, which can be decomposed into a trend component $T_t$ and a seasonal component $S_t$, such that:\n\n$y_t = T_t + S_t$\n\nFor the model’s prediction $\\hat{y}_t$, we similarly have:\n\n$\\hat{y}_t = \\hat{T}_t + \\hat{S}_t$\n\nUnder this setting, the Mean Squared Error (MSE) can be expanded as:\n\n$$L _{MSE} = ||y _t - \\hat{y} _t|| _2 ^2 = ||(T _t + S _t) - (\\hat{T} _t + \\hat{S} _t)||_2 ^2$$\n$$  ~~~~~~~~~~ = ||y _t - \\hat{y} _t|| _2^2 = ||(T _t - \\hat{T}_t) + (S _t - \\hat{S} _t)||_2^2$$\n$$ ~~~~~~~~~~ = ||T _t - \\hat{T}_t|| _2^2 + ||S _t - \\hat{S}_t||_2 ^2 + 2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$$\n\nThe key part from MSE lies in the cross term $2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$. Our assumption is that the trend and seasonal components are highly independent.However, this cross term introduces an interaction between them, potentially making it difficult for the model to optimize the two components independently, which can degrade the overall prediction performance. For instance, if the trend component is poorly predicted while the seasonal component is well captured, the interaction term can still yield a large negative value of $2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$, disproportionately affecting the total loss.  \n\nFurthermore, we analyze $L_{MSE}$ from the perspective of differentiation, and reveal MSE loss function being unable to independently consider these two components during the optimization process:\n\nAccording to the chain rule:\n\n$\\nabla _{\\boldsymbol{\\Theta}} L _t = 2 \\cdot [(T _t - \\hat{T}_t) + (S _t - \\hat{S}_t)] \\cdot \\nabla _{\\boldsymbol{\\Theta}} (-\\hat{T}_t - \\hat{S}_t) = -2 \\cdot [(T _t - \\hat{T}_t) + (S _t - \\hat{S}_t)] \\cdot \\left( \\nabla _{\\boldsymbol{\\Theta}} \\hat{T}_t + \\nabla _{\\boldsymbol{\\Theta}} \\hat{S}_t \\right)$\n\nUsing Jacobian matrix notation:\n\n- Jacobian of Trend: $$ \\mathbf{J} _T := \\nabla _{\\boldsymbol{\\Theta}} \\hat{T} _t  $$\n- Jacobian of Seasonality: $$ \\mathbf{J}_S := \\nabla _{\\boldsymbol{\\Theta}} \\hat{S}_t $$\n\nThen:\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot [(T_t - \\hat{T}_t) + (S_t - \\hat{S}_t)] \\cdot (\\mathbf{J}_T + \\mathbf{J}_S) $$\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot \\left[ (T_t - \\hat{T}_t)(\\mathbf{J}_T + \\mathbf{J}_S) + (S_t - \\hat{S}_t)(\\mathbf{J}_T + \\mathbf{J}_S) \\right] $$\n\nExpanding the terms:\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_T + (T_t - \\hat{T}_t)\\mathbf{J}_S + (S_t - \\hat{S}_t)\\mathbf{J}_T + (S_t - \\hat{S}_t)\\mathbf{J}_S \\right] $$\n\nWe split this into two parts:\n\n1、Ideal Decoupled Term\n\n$$ \\mathbf{G}_{\\text{ideal}} = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_T + (S_t - \\hat{S}_t)\\mathbf{J}_S \\right] $$\n\n2、Coupled Term\n\n$$ \\mathbf{G}_{\\text{coupling}} = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_S + (S_t - \\hat{S}_t)\\mathbf{J}_T \\right] $$\n\nThis term represents the problematic coupling:\n\n- The trend error influences the seasonality‑guided optimization.  \n- The seasonal error influences the trend‑guided optimization.  \n\nThe full gradient is expressed as the sum of the ideal and coupled terms: $$ \\nabla _{\\boldsymbol{\\Theta}} L _t = \\mathbf{G} _{\\text{ideal}} + \\mathbf{G} _{\\text{coupling}} $$\n\nAs long as $T _t \\neq \\hat{T}_t $ or $S _t  \\neq \\hat{S} _t $ (i.e., the model has not fully converged), the coupled term $  \\mathbf{G} _{\\text{coupling}}  \\neq \\mathbf{0} $. Due to the presence of $ \\mathbf{G} _{\\text{coupling}} $, the MSE loss function being unable to independently consider these two components during the optimization process.\n\n\n\nCompared with MSE, DBLoss ($L _{DB} = \\lambda_1 ||T _t - \\hat{T} _t||_1 + \\lambda_2 ||S _t - \\hat{S}_t||_2^2$ , where $\\lambda_1$ and $\\lambda_2$ are hyperparameters controlling the weights of the trend and seasonal losses, respectively.) removes this cross term (or $  \\mathbf{G} _{\\text{coupling}} $), enabling the model to treat these two components independently throughout the optimization process.   Furthermore, by adjusting the coefficients or using different distance norms, one can precisely control the loss scale for each component, enabling targeted learning and better modeling of both parts.\n\n[1] Are Transformers Effective for Time Series Forecasting? AAAI, 2023\n\n[2] DUET: Dual Clustering Enhanced Multivariate Time Series Forecasting, SIGKDD, 2025\n\n[3] TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting, ICLR, 2024\n\n[4] xPatch: Dual-Stream Time Series Forecasting with Exponential Seasonal-Trend Decomposition, AAAI, 2025"}, {"review_id": "eJIBQPMTjK", "reviewer": "Reviewer_PehU", "summary": "This paper identifies a key limitation of the commonly used MSE loss in time-series forecasting. To address this limation, the authors propose DBLoss, a decomposition-based loss function. DBLoss works by applying EMA decomposition to both the model's prediction and the ground truth, extracting their seasonal and trend components. It then computes the losses for each component separately, L2 loss for seasonal component and L1 loss for trend component, and combined via a weighted sum, controlled by a tunable hyperparameter $\\beta$. The method is model-agnostic and can be seamlessly applied to various time series forecasting models.", "questions": "1. Time series data inherently follows certain statistical principles and often comes with well-estabilished decomposition theories. However,  the authors do not discuss which theoretical assumptions or statistical interpretations justify the use of DBLoss. While the empirical results are strong, I believe that a theoretical foundation is essential --- especially when introducing a new loss function --- so that we can better understand when and why DBLoss should work. The paper would be strengthened by a deeper discussion of such theoretical grounding. \n2. Follow question from Weakness 3 : Could the limited impact of the score weight $\\beta$ be due to  the residual noise component remaining in both the trend and seasonal parts after decomposition?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths : \n1. The paper address a timely and important issue in time-series forecasting: the limitations of MSE loss in modeling trend and seasonality. The motivation is clearly articulated and well-grounded.\n2. DBLoss is model-agnostic and can be easily incorporated into various deep learning forecasting models without requiring architectural changes.\n3. Extensive experiments on diverse datasets and models demonstrate consistent performance improvements, including in zero-shot and few-shot scenarios.\n\nWeakness : \n1. In Figure 4, the sensitivity analysis of the score weight $\\beta$ shows relatively small performance difference across a wide range of values. Although the authors acknowledge in Appendix D that the score weight $\\beta$ has limited impact within a certain range, I believe this observation points to a more fundamental  issue: DBLoss may lack explicitly control over which component dominates the learning process. For instance, if a dataset has dominant seasonal pattern, the model may still overly focus on that, similar to implicit bias in MSE training. \n2. While DBLoss decomposes the prediction and ground truth into seasonal and trend components, it ultimately sums the two losses and backpropagates through both. This raises the question: does it truly overcome the limitations of MSE, or is it just a reweighted version of the same distance based loss concept? \n3. The method applies EMA for decomposition, which is a smoothing technique. However, it does not explicitly account for residuals or noise components after decomposition. It remains unclear how well DBLoss handles unpredictable or irregular patterns that are not capturd by trend or seasonality.", "was_revised": true, "final_justification": "The paper addresses a timely and important problem in time-series forecasting: the limitations of MSE loss in modeling trend and seasonality. The motivation is well-grounded, and the proposed DBLoss is model-agnostic, requiring no architectural changes for adoption. Extensive experiments across diverse datasets and models demonstrate consistent and meaningful performance improvements, including in zero-shot and few-shot scenarios, highlighting its broad applicability.\n\nAll initial concerns raised during the first review round have been adequately addressed in the rebuttal and discussion. In particular, the authors clarified the role of the score weight $\\beta$ and its impact on component dominance, explained how DBLoss differs conceptually from simple reweighting of MSE, and discussed its robustness to residual/noise patterns. These clarifications, combined with the thorough empirical evidence, resolve my earlier reservations.\n\nGiven the clarity of the motivation, the strength of the experimental results, and the successful resolution of prior concerns, I now consider the contribution solid and impactful. I therefore lean toward a borderline accept recommendation.", "rebuttal": "We would like to sincerely thank **Reviewer PehU** for providing a detailed review and insightful comments. We have revised our paper accordingly.\n\n**W1&Q2:** The limited impact of the score weight.\n\nDear Reviewer **PehU,** Thank you for your insightful feedback and constructive observations regarding our work. We truly appreciate your engagement with our study. Your comments prompted us to further reflect on our approach, and upon deeper investigation, we found that, as you mentioned, the parameter β did not adequately control the balance between the trend and seasonality components in the learning process.\n\nWe believe this issue arose because we did not account for the differing value scales of the trend loss and season loss in our original approach. To address this, we introduced a modification in the loss function:\n\n$$\\text{trend\\\\_loss} = \\text{trend\\\\_loss} \\times \\frac{\\text{season\\\\_loss}}{(\\text{trend\\\\_loss} + 1e^{-8})}.\\text{detach()}$$\n\nBy introducing this equation, we normalize the trend loss by the relative scale of the season loss, ensuring that both components are numerically consistent, treated as a good initialization strategy. In particular, the use of `.detach()` ensures that the gradient of the season loss does not influence the computation of the trend loss during backpropagation. This means that the season loss is used solely for scaling purposes, without affecting the gradient updates for the trend loss, thus maintaining the balance and stability between the two components. Then we can use the β to further control the influence of trend and seasonality on the training process, which makes the importance of β more explict. We subsequently conducted a reanalysis of the sensitivity of β on the ETTh2 and Traffic datasets, and retested the results in Tables 1 using the modified version. We are pleased to report that this adjustment led to an improvement in DBLoss performance. **Due to character limitations, we will present the results (Table 1) during the rebuttal discussion.** Thank you for your valuable feedback.\n\n| |       |       |       |       |       | $\\beta$ |       |       |       |       |       |\n| ---------- | ----- | ----- | ----- | ----- | ----- | ------- | ----- | ----- | ----- | ----- | ----- |\n|            | 0     | 0.1   | 0.2   | 0.3   | 0.4   | 0.5     | 0.6   | 0.7   | 0.8   | 0.9   | 1     |\n| ETTh2      | 0.332 | 0.281 | 0.281 | 0.282 | 0.283 | 0.284   | 0.286 | 0.291 | 0.300 | 0.314 | 0.340 |\n| Traffic    | 0.418 | 0.410 | 0.408 | 0.405 | 0.402 | 0.399   | 0.398 | 0.398 | 0.396 | 0.392 | 0.421 |\n\n The impact of the hyperparameter ($\\beta$) on ETTh2 and Traffic based DLinear (horizon 96), the metric is MSE.\n\n- For datasets with pronounced seasonality, such as traffic, a larger score weight *β* ($\\beta$=0.8 or  0.9) (i.e., considering a higher proportion of the seasonal component in the loss calculation) yields better performance.\n\n- For datasets with pronounced trend, such as ETTh2, a small score weight *β* ($\\beta$=0.1or 0.2) yields better performance.\n\n \n**W2&Q1:** The paper would be strengthened by a deeper discussion of such theoretical grounding.\n\nThank you for your insightful feedback and constructive observations regarding our work. In order to better explain the differences between DBLoss and MSE. We provide a theoretical analysis explaining why DBLoss is more effective than MSE in this context. Motivated by the success of methods such as Dlinear [1], DUET [2], TimeMixer [3], and xPatch [4], which model time series by decomposing them into trend and seasonal components, achieving excellent performance, we assume that the trend and seasonal components are highly independent. \n\nWe represent the time series as $y_t$, which can be decomposed into a trend component $T_t$ and a seasonal component $S_t$, such that:\n\n$y_t = T_t + S_t$\n\nFor the model’s prediction $\\hat{y}_t$, we similarly have:\n\n$\\hat{y}_t = \\hat{T}_t + \\hat{S}_t$\n\nUnder this setting, the Mean Squared Error (MSE) can be expanded as:\n\n$$L _{MSE} = ||y _t - \\hat{y} _t|| _2 ^2 = ||(T _t + S _t) - (\\hat{T} _t + \\hat{S} _t)||_2 ^2$$\n$$  ~~~~~~~~~~ = ||y _t - \\hat{y} _t|| _2^2 = ||(T _t - \\hat{T}_t) + (S _t - \\hat{S} _t)||_2^2$$\n$$ ~~~~~~~~~~ = ||T _t - \\hat{T}_t|| _2^2 + ||S _t - \\hat{S}_t||_2 ^2 + 2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$$\n\nThe key part from MSE lies in the cross term $2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$. Our assumption is that the trend and seasonal components are highly independent.However, this cross term introduces an interaction between them, potentially making it difficult for the model to optimize the two components independently, which can degrade the overall prediction performance. For instance, if the trend component is poorly predicted while the seasonal component is well captured, the interaction term can still yield a large negative value of $2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$, disproportionately affecting the total loss.  \n\nFurthermore, we analyze $L_{MSE}$ from the perspective of differentiation, and reveal MSE loss function being unable to independently consider these two components during the optimization process:\n\nAccording to the chain rule:\n\n$\\nabla _{\\boldsymbol{\\Theta}} L _t = 2 \\cdot [(T _t - \\hat{T}_t) + (S _t - \\hat{S}_t)] \\cdot \\nabla _{\\boldsymbol{\\Theta}} (-\\hat{T}_t - \\hat{S}_t) = -2 \\cdot [(T _t - \\hat{T}_t) + (S _t - \\hat{S}_t)] \\cdot \\left( \\nabla _{\\boldsymbol{\\Theta}} \\hat{T}_t + \\nabla _{\\boldsymbol{\\Theta}} \\hat{S}_t \\right)$\n\nUsing Jacobian matrix notation:\n\n- Jacobian of Trend: $$ \\mathbf{J} _T := \\nabla _{\\boldsymbol{\\Theta}} \\hat{T} _t  $$\n- Jacobian of Seasonality: $$ \\mathbf{J}_S := \\nabla _{\\boldsymbol{\\Theta}} \\hat{S}_t $$\n\nThen:\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot [(T_t - \\hat{T}_t) + (S_t - \\hat{S}_t)] \\cdot (\\mathbf{J}_T + \\mathbf{J}_S) $$\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot \\left[ (T_t - \\hat{T}_t)(\\mathbf{J}_T + \\mathbf{J}_S) + (S_t - \\hat{S}_t)(\\mathbf{J}_T + \\mathbf{J}_S) \\right] $$\n\nExpanding the terms:\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_T + (T_t - \\hat{T}_t)\\mathbf{J}_S + (S_t - \\hat{S}_t)\\mathbf{J}_T + (S_t - \\hat{S}_t)\\mathbf{J}_S \\right] $$\n\nWe split this into two parts:\n\n1、Ideal Decoupled Term\n\n$$ \\mathbf{G}_{\\text{ideal}} = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_T + (S_t - \\hat{S}_t)\\mathbf{J}_S \\right] $$\n\n2、Coupled Term\n\n$$ \\mathbf{G}_{\\text{coupling}} = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_S + (S_t - \\hat{S}_t)\\mathbf{J}_T \\right] $$\n\nThis term represents the problematic coupling:\n\n- The trend error influences the seasonality‑guided optimization.  \n- The seasonal error influences the trend‑guided optimization.  \n\nThe full gradient is expressed as the sum of the ideal and coupled terms: $$ \\nabla _{\\boldsymbol{\\Theta}} L _t = \\mathbf{G} _{\\text{ideal}} + \\mathbf{G} _{\\text{coupling}} $$\n\nAs long as $T _t \\neq \\hat{T}_t $ or $S _t  \\neq \\hat{S} _t $ (i.e., the model has not fully converged), the coupled term $  \\mathbf{G} _{\\text{coupling}}  \\neq \\mathbf{0} $. Due to the presence of $ \\mathbf{G} _{\\text{coupling}} $, the MSE loss function being unable to independently consider these two components during the optimization process.\n\n\n\nCompared with MSE, DBLoss ($L _{DB} = \\lambda_1 ||T _t - \\hat{T} _t||_1 + \\lambda_2 ||S _t - \\hat{S}_t||_2^2$ , where $\\lambda_1$ and $\\lambda_2$ are hyperparameters controlling the weights of the trend and seasonal losses, respectively.) removes this cross term (or $  \\mathbf{G} _{\\text{coupling}} $), enabling the model to treat these two components independently throughout the optimization process.   Furthermore, by adjusting the coefficients or using different distance norms, one can precisely control the loss scale for each component, enabling targeted learning and better modeling of both parts.\n\n[1] Are Transformers Effective for Time Series Forecasting? AAAI, 2023\n\n[2] DUET: Dual Clustering Enhanced Multivariate Time Series Forecasting, SIGKDD, 2025\n\n[3] TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting, ICLR, 2024\n\n[4] xPatch: Dual-Stream Time Series Forecasting with Exponential Seasonal-Trend Decomposition, AAAI, 2025\n\n**W3:** DBLoss does not explicitly account for residuals or noise components after decomposition.\n\nThank you for your valuable feedback. We deeply appreciate your thoughtful observations. The reason we did not explicitly consider the residual term in our methodology is that we believe the residual is typically an unpredictable random component. Since residuals often contain noise and irregular patterns, we think that explicitly modeling or penalizing the residual during the decomposition process wouldn't be very meaningful and may even have negative consequences.\n\nInstead, we believe that the foundation of time series forecasting lies in the relative stability and predictability of the data. Therefore, we focus on the trend and seasonal components, which are the primary driving forces behind the predictable patterns in time series data. By using the Exponential Moving Average (EMA) for decomposition, we aim to effectively capture these components."}, {"review_id": "vKU7vbFxdK", "reviewer": "Reviewer_4VN9", "summary": "The paper introduces DBLoss, decomposition-based loss function designed to improve time series forecasting performance by modeling seasonality and trend components within the forecasting horizon.  DBLoss employs EMA, exponential moving average, to seperate into each components. Experiments demonstrate that the DBLoss consistently outperforms traditional MSE loss by comparing SOTA models trained by each loss.", "questions": "As weaknesses.", "limitations": "As weaknesses.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\nIdentifies a limitation of MSE loss function for TSF, making the problem clearly. Proposed DBLoss is easily integrable into various forecasting models without significant increasing computational costs, which can lead to high potential for practical adoption. Experiments are comprehensive, covering diverse datasets and various type of SOTA models. Several robustness experiments are shown, too.\n\nWeaknesses:\n1. Contrasting to strong empirical results, theoretical justification or analysis of why DBLoss improves forecasting performance is lacking. A clear theoretical explanation or analysis of empirical results is needed.\n2. There should be a empirical or theoretical justification of why 2-norm is used for seasonality loss and 1-norm is used for trend loss.\n3. The method relies on existing, well-known exponential moving average decomposition. Novelty of the methodological contribution is quite limited.\n4. Computational cost analysis must be shown for adapting this loss. Empirical timing experiments would make it clearer.", "was_revised": true, "final_justification": "They gave the detailed response and the additional experimental results provided for clarification. Incorporating this information into your paper would enhance the clarity of your explanations and strengthen the experimental support for your contributions. I will incorporate these response into the evaluation score under the assumption that these revisions will be included in your final version.", "rebuttal": "Dear Reviewer **4VN9**, thank you for providing your detailed and constructive feedback.\n\n**W1:** A clear theoretical explanation or analysis of empirical results is needed.\n\nMotivated by the success of methods such as Dlinear, DUET, TimeMixer, and xPatch, which model time series by decomposing them into trend and seasonal components, achieving excellent performance, we assume that the trend and seasonal components are highly independent. \n\nWe represent the time series as $y_t$, which can be decomposed into a trend component $T_t$ and a seasonal component $S_t$, such that:\n\n$y_t = T_t + S_t$\n\nFor the model’s prediction $\\hat{y}_t$, we similarly have:\n\n$\\hat{y}_t = \\hat{T}_t + \\hat{S}_t$\n\nUnder this setting, the Mean Squared Error (MSE) can be expanded as:\n\n$$L _{MSE} = ||y _t - \\hat{y} _t|| _2 ^2 = ||(T _t + S _t) - (\\hat{T} _t + \\hat{S} _t)||_2 ^2 = ||(T _t - \\hat{T}_t) + (S _t - \\hat{S} _t)||_2^2= ||T _t - \\hat{T}_t|| _2^2 + ||S _t - \\hat{S}_t||_2 ^2 + 2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$$\n\nThe key part from MSE lies in the cross term $2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$. Our assumption is that the trend and seasonal components are highly independent.However, this cross term introduces an interaction between them, potentially making it difficult for the model to optimize the two components independently, which can degrade the overall prediction performance. For instance, if the trend component is poorly predicted while the seasonal component is well captured, the interaction term can still yield a large negative value of $2 \\cdot (T_t - \\hat{T}_t) \\cdot (S_t - \\hat{S}_t)$, disproportionately affecting the total loss.  \n\nFurthermore, we analyze $L_{MSE}$ from the perspective of differentiation, and reveal MSE loss function being unable to independently consider these two components during the optimization process:\n\nAccording to the chain rule:\n\n$\\nabla _{\\boldsymbol{\\Theta}} L _t = 2 \\cdot [(T _t - \\hat{T}_t) + (S _t - \\hat{S}_t)] \\cdot \\nabla _{\\boldsymbol{\\Theta}} (-\\hat{T}_t - \\hat{S}_t) = -2 \\cdot [(T _t - \\hat{T}_t) + (S _t - \\hat{S}_t)] \\cdot \\left( \\nabla _{\\boldsymbol{\\Theta}} \\hat{T}_t + \\nabla _{\\boldsymbol{\\Theta}} \\hat{S}_t \\right)$\n\nUsing Jacobian matrix notation:\n\n- Jacobian of Trend: $$ \\mathbf{J} _T := \\nabla _{\\boldsymbol{\\Theta}} \\hat{T} _t  $$\n- Jacobian of Seasonality: $$ \\mathbf{J}_S := \\nabla _{\\boldsymbol{\\Theta}} \\hat{S}_t $$\n\nThen:\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot [(T_t - \\hat{T}_t) + (S_t - \\hat{S}_t)] \\cdot (\\mathbf{J}_T + \\mathbf{J}_S) $$\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot \\left[ (T_t - \\hat{T}_t)(\\mathbf{J}_T + \\mathbf{J}_S) + (S_t - \\hat{S}_t)(\\mathbf{J}_T + \\mathbf{J}_S) \\right] $$\n\nExpanding the terms:\n\n$$ \\nabla_{\\boldsymbol{\\Theta}} L_t = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_T + (T_t - \\hat{T}_t)\\mathbf{J}_S + (S_t - \\hat{S}_t)\\mathbf{J}_T + (S_t - \\hat{S}_t)\\mathbf{J}_S \\right] $$\n\nWe split this into two parts:\n\n1、Ideal Decoupled Term\n\n$$ \\mathbf{G}_{\\text{ideal}} = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_T + (S_t - \\hat{S}_t)\\mathbf{J}_S \\right] $$\n\n2、Coupled Term\n\n$$ \\mathbf{G}_{\\text{coupling}} = -2 \\cdot \\left[ (T_t - \\hat{T}_t)\\mathbf{J}_S + (S_t - \\hat{S}_t)\\mathbf{J}_T \\right] $$\n\nThis term represents the problematic coupling:\n\n- The trend error influences the seasonality‑guided optimization.  \n- The seasonal error influences the trend‑guided optimization.  \n\nThe full gradient is expressed as the sum of the ideal and coupled terms: $$ \\nabla _{\\boldsymbol{\\Theta}} L _t = \\mathbf{G} _{\\text{ideal}} + \\mathbf{G} _{\\text{coupling}} $$\n\nAs long as $T _t \\neq \\hat{T}_t $ or $S _t  \\neq \\hat{S} _t $ (i.e., the model has not fully converged), the coupled term $  \\mathbf{G} _{\\text{coupling}}  \\neq \\mathbf{0} $. Due to the presence of $ \\mathbf{G} _{\\text{coupling}} $, the MSE loss function being unable to independently consider these two components during the optimization process.\n\n\nCompared with MSE, DBLoss ($L _{DB} = \\lambda_1 ||T _t - \\hat{T} _t||_1 + \\lambda_2 ||S _t - \\hat{S}_t||_2^2$ , where $\\lambda_1$ and $\\lambda_2$ are hyperparameters controlling the weights of the trend and seasonal losses, respectively.) removes this cross term (or $  \\mathbf{G} _{\\text{coupling}} $), enabling the model to treat these two components independently throughout the optimization process.   Furthermore, by adjusting the coefficients or using different distance norms, one can precisely control the loss scale for each component, enabling targeted learning and better modeling of both parts.\n\n\n**W2:** There should be a empirical or theoretical justification of why 2-norm is used for seasonality loss and 1-norm is used for trend loss.\n\nYour suggestion is crucial for enhancing the clarity of our paper. Based on the experimental results, we chose to use the 2-norm for seasonality loss and the 1-norm for trend loss. Specifically, we tested the following combinations: 2-norm for both seasonality and trend losses (L2-L2), 1-norm for seasonality and 2-norm for trend loss (L1-L2), and 2-norm for seasonality and 1-norm for trend loss (DBLoss (L2-L1)). We found that the best performance was often achieved when using the 2-norm for seasonality loss and the 1-norm for trend loss. \n\n\n\n| Model         |      | iTransformer |       |      |       |             |       |     |      | TimeMixer |       |       |       |    |       |     |       |\n| ------------- | ---- | ------------ | ----- | ----------- | ----- | ----------- | ----- | --------------------- | ----- | --------- | ----- | ----------- | ----- | ----------- | ----- | --------------------- | ----- |\n| Loss Function |      | MSE          |       |L2-L2 |       | L2-L1 |       | DBLoss |       | MSE       |       | L2-L2 |       | L2-L1|       | DBLoss |       |\n| Metric        |      | MSE          | MAE   | MSE         | MAE   | MSE         | MAE   | MSE                   | MAE   | MSE       | MAE   | MSE         | MAE   | MSE         | MAE   | MSE                   | MAE   |\n| ETTh2         | Avg  | 0.370        | 0.403 | 0.368       | 0.401 | **0.362**       | 0.397 | 0.363                 | **0.395** | 0.349     | 0.397 | **0.344**       | **0.386** | 0.352       | 0.397 | 0.346                 | 0.387 |\n| ETTm2         | Avg  | 0.269        | 0.327 | 0.264       | 0.324 | 0.265       | 0.324 | **0.262**                 | **0.317** | 0.257     | 0.318 | 0.253       | 0.312 | 0.254       | 0.312 | **0.251**                 | **0.307** |\n| Weather       | Avg  | 0.232        | 0.270 | 0.231       | 0.268 | 0.234       | 0.270 | **0.229**                 | **0.261** | 0.226     | 0.264 | 0.225       | 0.259 | 0.224       | 0.259 | **0.222**                 | **0.254** |\n\n\n\n| Model         |      | PatchTST |       |    |       |     |       |    |       | DLinear |       |    |       |    |       |   |     |\n| ------------- | ---- | -------- | ----- | ----------- | ----- | ----------- | ----- | --------------------- | ----- | ------- | ----- | ----------- | ----- | ----------- | ----- | --------------------- | ----------- |\n| Loss Function |      | MSE      |       | L2-L2 |       | L2-L1 |       | DBLoss  |       | MSE     |       | L2-L2 |       | L2-L1 |       | DBLoss  |             |\n| Metric        |      | MSE      | MAE   | MSE         | MAE   | MSE         | MAE   | MSE                   | MAE   | MSE     | MAE   | MSE         | MAE   | MSE         | MAE   | MSE                   | MAE         |\n| ETTh2         | Avg  | 0.351    | 0.395 | **0.337**       | 0.386 | 0.339       | 0.385 | 0.339                 | **0.383** | 0.470   | 0.468 | 0.409       | 0.435 | 0.411       | 0.436 | **0.391**                 | **0.414**       |\n| ETTm2         | Avg  | 0.256    | 0.314 | 0.256       | 0.309 | 0.256       | 0.310 | **0.252**                 | **0.307** | 0.259   | 0.324 | 0.257       | 0.320 | **0.254**       | 0.319 | 0.256                 | **0.313**       |\n| Weather       | Avg  | 0.224    | 0.262 | **0.222**       | 0.252 | **0.222**       | **0.250** | 0.223                 | 0.252 | 0.242   | 0.293 | 0.241       | 0.288 | **0.238**       | **0.279** | 0.239                 | 0.280 |\n\n**W3:** The method relies on existing, well-known exponential moving average decomposition.\n\nThank you for your valuable feedback. Indeed, we did not make an innovative contribution in terms of the decomposition method itself. Our main contribution lies in proposing a new loss paradigm that, through a reasonable decomposition approach, can enhance model performance. In fact, any decomposition method could be applied, but the quality of the decomposition directly affects the final performance. We adopted the currently advanced EMA decomposition method [1] with the aim of improving performance.\n\n[1] xPatch: Dual-Stream Time Series Forecasting with Exponential Seasonal-Trend Decomposition, AAAI, 2025\n\n**W4:** Empirical timing experiments would make it clearer.\n\n| Train Time  | ETTh1 | ETTh2 | ETTm1 | ETTm2 | Solar  | Weather | Electricity | Traffic |\n| -------------------------- | ----- | ----- | ----- | ----- | ------ | ------- | ----------- | ------- |\n| MSE                        | 2.36  | 2.37  | 14.45 | 14.39 | 183.11 | 36.07   | 258.47      | 1035.77 |\n| DBLoss                     | 3.11  | 3.37  | 15.93 | 15.73 | 186.31 | 37.23   | 260.85      | 1039.67 |\n\nThe table presents the epoch-level training times (in seconds) when using DBLoss compared to MSE on different datasets. The results show the average values for the four prediction steps of each dataset, with the same parameters, where only MSE is replaced by DBLoss. Based on the experimental results in the table, we can observe that DBLoss does lead to an increase in training time compared to MSE, but this increase is not significant. As the dataset size grows, the time difference becomes even more negligible."}]

Author rebuttal:

Meta-review: The paper introduces a new loss, DBLoss, for time-series forecasting, which aims to properly capture seasonality and trend in forecasts. The paper has gone through several discussions between the authors and reviewers, and eventually, the final scores of A, A, BA, BA have been received. The quality of the paper has been increased throughout the rebuttal phase, and I support accepting the paper. I ask the authors to ensure that all the changes and additions requested by the authors be carefully addressed in the final version of the paper.
Decision: Accept (poster)