Batch Training for Streaming Time Series: A Transferable Augmentation Framework to Combat Distribution Shifts
Authors: Weiyang Zhang
Xinyang Chen
Yu Sun
Weili Guan
Liqiang Nie
Venue: Accepted by TMLR ()

Abstract: Multivariate time series forecasting, which predicts future dynamics by analyzing historical data, has become an essential tool in modern data analysis. With the development of deep models, batch-training based time series forecasting has made significant progress. However, in real-world applications, time series data is often collected incrementally in a streaming manner, with only a portion of the data available at each time step. As time progresses, distribution shifts in the data can occur, leading to a drastic decline in model performance. To address this challenge, online test-time adaptation and online time series forecasting have emerged as a promising solution. However, for the former, most online test-time adaptation methods are primarily designed for images and do not consider the specific characteristics of time series. As for the latter, online time series forecasting typically relies on updating the model with each newly collected sample individually, which may be problematic when the sample deviates significantly from the historical data distribution and contains noise, which may lead to a worse generalization performance.
In this paper, we propose Batch Training with Transferable Online Augmentation (BTOA), which enhances model performance through three key ideas while enabling batch training. First, to fully leverage historical information, Transferable Historical Sample Selection (THSS) is proposed with theoretical guarantees to select historical samples that are most similar to the test-time distribution. Then, to mitigate the negative impact of distribution shifts through batch training and take advantage of the unique characteristics of time series, Transferable Online Augmentation (TOA) is proposed to augment the selected historical samples from the perspective of amplitude and phase in the frequency domain in a two-stream manner. Finally, a prediction module that utilizes a series decomposition module and a two-stream forecaster is employed to extract the complex patterns in time series, boosting the prediction performance. Moreover, BTOA is a general approach that is readily pluggable into any existing batch-training based deep models.Comprehensive experiments under both ideal and practice experimental settings demonstrate that the proposed method exhibits superior performance across all seven benchmark datasets. Compared to state-of-the-art approaches, our method reduces the Mean Squared Error (MSE) by up to 13.7\%.

Published in Transactions on Machine Learning Research (06/2025)
Batch Training for Streaming Time Series: A Transferable Augmentation Framework to Combat Distribution Shifts
Weiyang Zhang
2hangueiyangg122388@mmaa
School of Computer Science and Technology, Harbin Institute of Technology (Shenzhen)
Xinyang Chen
chencinnyangg55@@ma...cm
School of Computer Science and Technology, Harbón Institute of Technology (Shenzhen)
Yu Sun"
sungu@honkai.edtu. edu.cn
College of Computer Science, DISSec, Nonkai University
Weili Guan
guanwelt@@hi.edt..c edu.cn
School of Information Science and Technology, Harbin Institude of Technology (Shenzhen)
Liqiang Nie
nieliqiang@gmail. com
School of Computer Science and Technology, Harbin Institute of Technology (Shenzhen)
Reviewed on OpenReview: https: // opwnreviee.ee.ee... net/ ftrmm??m? id= UkRCH
Abstract
Multivariate time series forecasting, which predicts future dynamics by analyzing historical data, has become an essential tool in modern data analysis. With the development of deep models, batch-training based time series forecasting has made significant progress. However, in real-world applications, time series data is often collected incrementally in streaming manner, with only a portion of the data available at each time step. As time progresses, distribution shifts the data can occur, leading to a drastic decline in model performance. To address this challenge, online test-time adaptation and online time series forecasting have emerged as promising solutions. However, for the former, most online test- time adaplation methods are primarily designed for images and do not consider the specifie characteristies of time series. As for the latter, online time series forecasting typically relics on updating the model with each newly collected sample individually, which may be problematic when the sample deviates significantly from the historical data distribution and contains noise, which may lead to a worse generalization performance. In this paper, we propose Batch Training with Transferable Online Augmentation (BTOA), which enhances model performance through three key idcas while enabling batch training. First, fully leverage historical information, Transferable Historical Sample Selection (THSS) is proposed with theoretical guarantees to select historical samples that are the most similar the test- time distribution. Then, to mitigate the negative impact of distribution shifts through batch training and take advantage of the unique characteristics of time series, Transferable Online Augmentation (TOA) is proposed to augment the selected historical samples from the perspective amplitude and phase in the frequency domain in a two-stream manner. Finally, a prediction module that utilizes series decomposition module and a two-stream forecaster is employed to extract the complex patterns time series, boosting the prediction performance. Morcover, BTOA is general approach that is readily pluggable into any existing batch-training based deep models. Comprehensive experiments under both ideal and practical experimental settings demonstrate that the proposed method exhibits superior performance across all seven benchmark datasets. Compared to state-of-the-arr approaches, our method reduces the Mean Squared Error (MSE) by up to 13.7%.
"Corresponding authors.

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Published in Transactions on Machine Learning Research (06/2025)
1 Introduction
Time series forecasting is crucial in real-world applications and is widely used across various fields, such as weather forecasting (Zhang et al. 2022a), power demand prediction (Gasparin et al. 2022), traffic flow analysis (Jin et al. 2021), and financial market modeling (Lai et. al.] 2018). In these practical applications, time series forecasting techniques not only help decision-makers better plan and optimize resources but also improve system efficiency and stability, driving the intelligent development of varions industrics. To improve iorecasting accuracy, recent research has proposed advanced forecasting methods (Zhou el al. 2022 Wu el al., 2021 p022p. However, they typically rely on a conventional machine learning assumption that the training and test data follow the same distribution. This assumption often does not hold in real-world applications, where dataset shifts frequently occur (Quionero- Cantelll et al. 2009). Consequently, model performance can be significantly degraded when tested with data that deviates substantially from the training distribution. It is also worth noting that due to the inherent temporal nature, time series often arrive continuoulyy in real- world scenarios, which means that models are typically required to handle streaming data. Recently, online test-time adaptation and online time series forecasting have emerged as promising solutions to address this issue, allowing pre-trained models to adapt to previously unseen data distributions during inference without the need for labeled data (Wang et al. 2023 Liang et al. 2023
Unlike traditional batch training methods, online test-time adaptation and online time series forecasting
adapt models in real-time using streaming data. Current online test-time adaptation methods can be broadly
classified into three categories (Liang et al. 2023]: (1) Data-based methods (Gong et al. 2024 Wang et al.
2022a), which focus on maximizing prediction consistency across different test datasets. (2) Model-based
methods (Jang et al., 2022 Liu et al.) 2023; Shu et al. 2022a), which aim to modify the original model architecture by adapting specific layers or mochanisms. (3) Optimization-basee methods (Wang et al., 2022b: Shhu al. b022b; Mummadi et. al., 2021 which focus on optimizing prediction results using various optimization techniques. However, most existing online test-time adaptation methods are predominantly designed for image-based tasks, with few approaches specifically tailored for the complex patterns inherent in time series data. Current online time series forecasting methods typically utilize traditional Bayesian theory or add additional adapter modules to achieve adaptation (Pham et al.) 2022 Zhang et al., 2023) These methods often rely on updating the model individually with each newly collected sample. When a sample deviates significantly from the historical data distribution and may contain substantial noise, these approaches can lead to reduced generalization performance.
In this paper, Batch training with Transfcrable Online Augmentation (BTOA) framework is proposed for online lest-time adaplation in lime series forecasting, with the challenges of online test-time adaptation being addressed through three key innovations. Firstly, to fully leverage historical distribution informa- tion, we introduce the Transfcrable Historical Sample Selection (THSS) module with theoretical guarantees to precisely select historical samples from the memory bank that are closest to the test-time distribution. This overcomes the inefficiency of traditional online methods relying on random sampling or full-volume updates (Chaudhry et al., 2019 Buzzega et al. 2020), enabling intelligent activation and on-demand utiliza- tion of historical information. Secondly, to address distribution shifts, we propose the Transferable Online Augmentation (TOA) module, which enables batch training while avoiding the frequency-domain distortion (Verma et al. 2021 Demirel & Holz 2024] caused by traditional time-domain data augmentation methods (e.g., ManeerrMMxupp Zhang et al. 2017), Cut-Mixup (Yun et al. 2019)). TOA performs decoupling in the frequency domain and applies two-stream augmentation to the selected samples from both the amplitude and phase dimensions, fully preserving the amplitude-phase coupling information during the augmentation process. This approach uniquely maintains the critical frequeneyddomain characteristics that are essential for time series forecasting. Finally, prediction block consisting of series decomposition module and a two-stream forecaster generates predictions by achieving differential modeling of seasonal components and trend components in non-stationary time series. This design extracts complex patterns in time scries data, significantly enhancing prediction performance by capturing hierarchical temporal dependencies.
Our main contributions are summarized as follows:
The Batch training with Transferable Online Augmentation (BTOA) framework is proposed to ad-
dress the distribution shift in online learning from three key perspectives. First, to fully leverage
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historical distribution information, the Transferable Historical Sample Selection (THSS) module is introduced to select historical samples with minimal distribution discrepancy from the test-time distribution, enabling intelligent adaptation and on-demand activation of historical information. Second, the Transferable Online Augmentation (TOA) module is proposed to perform two-stream augmentation on the selected samples from the perspectives of Trequency-domain amplitude and phase, effectively overcoming the frequency distortion limitations of traditional time-domain aug- mentation. This approach preserves essential frequency-domain features and enables batch training, thereby alleviating distribution shift. Finally, prediction module is employed to achieve differential modeling, capturing complex temporal patterns and enhancing forecasting performance.
BTOA is a general approach that is readily pluggable into any online time series forecasting model. This approach effectively mitigales the negative impact of noise in test-time samples, alleviates distribution shift, and enhances the effectiveness and robustness of the online learning model. We conducted experiments on seven popular real-world datasets in both ideal and practical scenarios. The results show that our method demonstraa rates excellent performance across all benchmark datasets. Compared to state-of-the Methods, our method reduces the Mean Squared Error by up to 13.7%.
2 Related work
2.1 Online Test-time Adaptation
Online test-time adaptation (OTTA) continuously updates the model in real-time as it encounters new data during inference. This ensures swift adaptation to evolving data distributions without altering the original training procedure (Chen al. 2022: Nguyen et. al., 2023 Zhang et al. 2022b). Notably, TENT (Wang ct al. 2020) addresses distributional shift by dynamically adjusting batch normalization parameters through entropy loss minimization during inference. Similarly, EATA (Niu et al. 2022) introduces selective approach to optimizing unsupervised surrogate losses akin to TENT, focusing solely reliable and informative data points. ViDA [Liu et al. 2023) employs supervision of the student output by leveraging predictions from the teacher with augmented input. Additionally, it introduces high/low low-rank adapters that are updated to accommodate continuous online lest-time adaptation. ECL [Zeng et al., 2024] marks a departure from traditional methods by integrating a memory bank containing output distributions to establish thresholds for complementary labels. This innovative approach ensures the memory bank's continual relevance and effectiveness through periodic updates with the latest model parameters. Although these methods have shown promising results in the fields of computer vision and natural language processing (Wang et all, 2023 Liang et al. 2023), they do not take advantage of the unique characteristics of time series. Dish- TS Fan et al. 2023) proposes the Dual-CONET framework, which learns the distribution discrepancies in input and output spaces respectively, and introduces a coefficient network to mitigate intra- and inter-space distribution differences. SOLID (Chen et al. 2024) employs a residual-based distribution shift detector to quantify the model's vulnerability to distribution shifts by evaluating the mutual information between prediction residuals and their corresponding contexts. TAFAS (Kim et al., 2025) flexibly adjusts the source forecaster to continually adapt to evolving test distributions while preserving the semantic information learned during pre-training.
2.2 Online Time Series Forecasting
Online time forecasting focuses on strcaming data, that is, for cach N variate sample x; received, the model constructs an L-length look-back window X and outputs a H-length prediction window Y, and then the true values used to improve the model's performance in predicting the next sample. Online time series forecasting has a wide range of real-world applications due to the sequential nature of the data (Anava et al.
2013 Gultekin & Paisley 2018
Aydore et al., 2019).
Previous methods have attempted to solve the online time series forecasting problem using Bayesian con- tinuous learning theory, however they are unable to quickly utilize information from historical samples. Inspired by Complementary Learning Systems (CLS) theory, FsNet (Pham et al. (022) achieves great online time series forecasting by quickly adapting to historical data using the adapter module and slowly learning
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Prediction Block
Magnitude Mixup
Series Decomposition
Forecaster
Ma
THSS
rfft
irfft
concat
Input
Phase Mixup
Forecaster
Output
TOA Block
SD Block
sufrea Wiow M4- Keie
mapping
index
Avg Pool
space
THIS Block
Figure 1: Overall architecture of Batch training with Transferable Online Augmentation (BTOA). The THSS block is used to select historical samples, the TOA block implements batch training through data augmentation, and the Prediction block generates the final output.
the newly collected sample with the Temporal Convolutional Network architecture. OneNet Zhang et al. 2023) builds on FsNet by exploring the need for inter-channel dependencies, using the Online Convex Pro- gramming module to balance cross-time dependencies with cross-variate variate dependencies. This allows OneNet to achieve significant performance gains on some datasets with multiple variates such as the ECL dataset. Current models update based on a single received sample when processing streaming data. However, single sample noisy, can disrupt the optimal update path and significantly degrade model performance, To mitigate this issue, our BTOA implements batch training, which enhances the model's robustness against noisy data while maintaining effective online test-time adaptation. Proceed (Zhao & Shen 2024) estimates inter-concept distribution shifts and uses an adaptive generator to effectively translate the estimated shifts into parameter adjustments, proactively adapting the model to test samples.
3 Method
Problem Formulation. Given well-trained time series forecasting model train on the training set and a sequence of unlabeled time series segments. Distribution shift refers to the problem that arises when the distribution of the test data (target) diverges from that of the training data (source) (Liang et al., 2023 This phenomenon presents substantial challenges for machine learning systems deployed in practical scenarios (Saenko et al. 2010: Chen et al. 2017 Online test-time adaptation aims to leverage the labeled knowledge embedded in the prediction model f to infer the future values of samples under distribution shift, in an online manner. In this problem, the learning process takes place over a sequence rounds, where the model receives L-length look-back window X (f1,...,,1]) E N and outputs the forecast. window Y (vI,..., vH) NxH. The true values Y are then revealed to improve the model's performance in the next rounds. Our goal to continuously optimize the prediction model f, which mitigate the negative impact of distribution shifts.
Structure overview. Figure Tillstrates the comprehensive workflow of Batch Training with Transferable Ouline Augmentation (BTOA). BTOA is meticulously structured into three principal modules: a transferable historical sample selection module that fully leverages historical distribution information, transferable online augmentation module that enables batch training to alleviate the negative effects of distribution shift, and a prodiction block that extracts complex temporal patterns and produces predictions.
3.1 Transferable Historical Sample Selection
The distribution of the data stream changes dynamically over time, which can adversely affect time series forecasting accuracy. To mitigate this issue, Transferable Historical Sample Selection is proposed to ef-
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fectively utilize historical data. Firstly, we aim to select historical samples that have smaller distribution discrepancy to the test-time distribution.
Specifically, we establish a memory bank, which is a set and denoted as M. .M stores historical samples and is updated using a Lirst-InFFrrt-Out (FIFO) policy to maintain a fixed size. Upon receiving test-time sample, we use the THSS module to select historical samples that are semantically similar to the test-time sample and most closely align with test-time distribution. This selection is achieved through a mapping model, where intuitively, any model that can preserve semantic consistency between the original and mapped data can be used. Due to the inherent ability of the Variational Autoencoder (VAE) (Higgins et al. 2017) to maintain semantic consistency between input and output, we choose VAE model as our mapping model. Initially, the VAE model is pre-trained in an unsupervised manner on the training set, and during online test-time adaptation, its parameters are frozen to remain unchanged. Once the test-time sample Xtest is introduced, both the test-time sample and the historical samples stored M are projected into a latent space. We then calculate the distances between these samples in the latent space and select those historical samples that. have smaller distribution discrepancy to the test-Lime distribution. These selected samples are semantically similar to the test-time sample, forming the selected historical sample set Xn. The above procedure can be formulated as follows:
Zrest E(test)), 2, E(m:), Vm; E M
Zteat Zz
d. cosine similarity (Ztest: z.)
(1)
||ztat |2:
X,. {miJies, where S, = arg sorss n),
where E(-) represents the encoder of VAE model, and n is a hyperparameter indicating the number of his- torical samples we need to utilize. By effectively utilizing historical samples that have a smaller distribution discrepancy to test-time distribution, we alleviate distribution shift and boost prediction performance.
3.2 Transferable Online Augmentation
We mitigate the negative impact of distribution shift during online test-time adaptation by introducing batch training, which can be achieved through data augmentation techniques. However, existing data augmentation methods, such as Linear-Mixup (Zhang et al. 2017) and Cut-Mixup (Yun et al. 2019), primarily mix time series in the time domain, which can affect the frequency domain information that is crucial for accurate prediction (Demirel & Holz 2024 Ullrich et al. 2020; Zhang et al. 2022ch. Since distribution shift is more pronounced in online test-time adaptation, preserving the frequency domain information of time series becomes particularly important.
To preserve the frequency domain information, we propose a two-stream augmentation approach that focuses both the amplitude and phase in the frequency domain, and we select the aforementioned set of selected historical samples Xn, which are closer to the test-time distribution, the source for augmentation. By doing so, we ensure that the augmented instances' phase and amplitude are properly interpolated based on the test-time sample, avoiding destructive interference in the frequency domain. We first apply the Fast Fourier Transform (FFT) to both the test-time sample and historical samples stored in the set Kn, decomposing them into amplitude and phase components, which can be formulated as:
A ((( = =F(X), X, E (Xeest) U U.K.
(2)
where F denotes the Fast Fourier Transform, P()) means the amplitude and phase. Then, combine the amplitude and phase of the test-time sample with those of the historical samples. This process ensures that the frequency domain information remains intact while enhancing the data with relevant historical pat- terns. To ensure more appropriate historical samples, we perform aggressive data augmentation primarily using historical samples when their distance from the test-time sample the latent space small, indi- cating similar distributions. Conversely, when the distance between the historical and test-time sample is large, implying a significant distributional shift, we prioritize the test-time sample for data augmentation.
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Specifically, the process of mixup is:
,x), X, E A A(Kaug) (X5|AA((X test) + P(XKugg) (XxoA)((X(ex)) (eot)) + (1- (p)P(X,)) ,), X, E &.))
(3)
(4)
where Laug means the augmented sample set. AA, Ap are hyperparameters representing the mixing coeffi- cients for amplitude and phase, respectively. When the distance between latent vectors is below distance threshold, we sample the mixing coefficients for amplitude and phase from a uniform distribution, denoted as Aa Ap ~ (4,,1 1.0), prioritizing data augmentation on the historical samples, with & being a lower value. Conversely, if the distance exceeds the threshold, we focus on augmenting the test-time sample. In this case, the coefficients are drawn from a truncated normal distribution, Ap ~ NN(,, 0), characterized by a high mean and low standard deviation. The process for determining the sampling distribution of da and Ap is follows:
Ap.dp E (1(,,10), if T (N(H, if >T,
(5)
where de represents the distance between latent vectors, and denotes a predefined distance threshold. Finally, the augmented sample set Kang is obtained by applying the inverse FFT to the mixed components. For case of understanding, we use X Xang to represent an element of Xaug hereafter. As shown below:
Xang = =F (A(Xang) )ce )ei
(6)
where F-1 denotes the inverse Fast Fourier Transform. After obtaining the augmented set Xang. we con- catenate these augmented samples with the test-time sample and input them as a batch into the next module for training. Compared to the previous approach, which only used the lest-time sample update the model, this method significantly reduces the negative impact of noise in the test-time sample model optimization.
3.3 Prediction Model and Training Objective
Prediction Model. To effectively learn complex temporal patterns in time series forecasting, we use series dccomposition (RB 1990 Anderson 1976). This technique simplifics complex raw data, allowing the model to make better predictions. Specifically, we extract the trend component of the time series by applying moving average kernel to the input series. The difference between the trend component and the original series is regarded as the seasonal component. These components reflect the long-term trend and cyclical relationship of the time series, respectively. The series decomposition handled as follows:
X, ArgPool(padding(X.o (Xaug)),
(7)
X, Xong Xt,
(8)
where X,, X, denote the extracted long-term trend and seasonal terms, respectively. We use padding to maintain the original scries length, and then apply the AvgPool layer for moving average calculations.
After decomposition, the trend component X, and the seasonal component X, will be fed into two-stream forecaster with identical structures. The outputs from two-stream forecaster are combined to generate the final prediction Y. As shown below:
Y Forecaster, (X.) Forecaster. (X.).
(9)
Training Objective. We use the L2 loss to optimize the parameters of the BTOA model, with the loss function defined as:
N
L =
Y
(10)
N
1:H
1:H
J
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where N represents the number of channels in the time series. During the test-time adaptation process, use the mean MSE and MAE between the ground truth Y and the model's predicted output Y across all samples as the final evaluation metries to compare model performance.
It worth noting that BTOA is a general module designed to mitigate the distributional shift that occur during the learning process. This adaptability makes it applicable to any online time series forecasting model. The inherent flexibility of ВTOA allows to be integrated seamlessly with a variety of models, enhancing their robustness against changes in data distribution. Moreover, BTOA is not limited to specific algorithm framework. This means that more advanced deep models are developed, BTOA can be incorporated into these advanced deep models to further improve performance.
3.4 Theoretical Insights
In the Transferable Historical Sample Selection module, need to select mapping model that can reflect the semantic consistency of samples before and after mapping. Proposition 3.1 theoretically demonstrates the rationale for using the VAE model as mapping model.
Proposition 3.1 (Consistency in Latent Space (Li et al. 2022)) Given well-trained unconditional VAE with the encoder E-) that produces distribution pr(z the decoder D(-) that produces distribution go (x(2)) while the prior for is p(z), let and Z2 be two latent vectors of two different real samples x1 and X2, i.c., E(x1) = 21 and E(x2) = 22- If the distance d(z1, 22) I 6, then D(z1) and D(z2) will have a similar semantic label as in Equation equation 17
I(D(z)):y) I(D((2):))) Sc
(11)
where slands for tolerable semantic difference, 5 is the marimum distance to mainlain semanlic consistency, and d(-) is distance measure such cosine similarity between two vectors.
Let the historical sample P be the set that includes the training set and all samples reccived prior the test-time sample. The Lest-time sample set Q refers the samples being received al present. Due to distribution shift, the data distributions of P and Q may differ. In the Transferable online augmentation module, the augmented set Kaug derived from the selected historical sample set Nr, has data distribution that is closer to the historical sample set P compared to using the test-time sample alone. Proposition 3.2 provides theoretical support for the performance advantages of using Kang as input, suggesting that it can lead to smaller upper bound on the generalization error. Moreover, our choice of L2 loss as the loss function aligns with the requirements of the proposition.
Proposition 3.2 (Generalization Error Upper Bound Mansour el al., 2009) Let Se e arg minjer Lo(f, GQ) and similarly let fp be minimizer of Cr( Note that these minimizers may not be unique. For adaptation to succeed, it is natural to assume that the average loss Lelf ff) between the besl-in-class hypotheses is small. Under that assumplion and for small discrepancy dislance, there is useful bound on the error of hypothesis with respect to the test-time sample set as in Equation equation 12
Lols, G2) S Colle. GQ) Co(ssp) + discr, (2, (2,P)
+ JQ), Lalfs,
(12)
where G represents the ideal prediction model and the loss function C is symmetric and obeys the triangle inequality.
4 Experiments
In this section, we evaluated BTOA across range of online time series forecasting applications, demonstrat- ing its effectiveness in diverse scenarios. In addition the primary evaluation, we conducted comprehensive ablation studies to investigate the contribution of each individual component of BTOA.
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Table 1: Statistics of popular datasets for benchmark.
Datasets
ETTh1
ETTh2
ETTml
ETTm2
Weather
Electricity
Traffic
Features
7
7
7
7
21
321
862
Timestops
17420
17420
69680
69680
52695
26304
17544
ADF
-5.90
-4.13
-14.98
-5.66
-26.66
-8.44
-15.02
Dataset. We evaluate the performance of BTOA on seven real-world datasets, including ETT (with 4 subsets), Weather, ECL, Traffie used in iTransformer (Liu al., (004).. The presence and severity of distributional shifts in these datasets can be measured using the Augmented Dickey-Fuller (ADF) test statistic (Liu et al., 2022). Basic information about these datasets is provided in Table [ where it can be observed that they exhibit varying degrees of distributional shift. Detailed dataset descriptions are available in appendix A.I
Table 2: Full results of the online time-series forecasting task. We compare extensive competitive models under different prediction lengths following the setting of Proccod. The best results are in bold, and the second best are underlined.
Models
BTOA
Proceed
SOLID
OneNet
FaNet
CycleNet
Dish-TS
PatchTST
DER-
ER
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
0.708
0.533
0.729
0.534
0.745
0.552
0.780
0.559
0.993
0.624
0.753
0.556
0.761
0.562
0.756
0.552
0.834
0.604
0.811
0.593
48
0.806
0.576
0.886
0.593
0.848
0.593
0.896
0.600
1.089
0.664
0.857
0.596
0.882
0.603
0.887
0.601
0.921
0.635
0.901
0.627
0.930
0.625
1.003
0.650
0.977
0.645
1.025
0.648
1.350
0.752
0.986
0.644
1.074
0.662
1.086
0.664
1.036
0.675
1.019
0.668
ECL WTH m2' ETTml ETT12
1.688
0.563
1.801
0.603
2.021
0.609
2.606
0.638
2.941
0.696
2.228
0.636
1.931
0.612
1,893
0.608
2.790
0.714
2.492
0.684
48
2.772
0.668
3.291
0.733
3.442
0.718
3.921
0.729
4,090
0.797
3.701
0.752
3.391
0.751
3,28/
0.720
4.090
0.801
3.799
0.778
4.850
0.800
5.79D
0.847
5.802
0.868
6.248
0.927
6.245
0.945
6.674
0.920
6.054
0.911
5.976
0.887
6.363
0.925
6.022
0.905
0.454
0.408
0.422
0.393
0.455
0.400
0.706
0.487
0.627
0.484
0.604
0.481
0.459
0.412
0.470
0.415
0.775
0.579
0.745
0.566
48
0.592
0.477
0.679
0.462
0.578
0.461
0.978
0.582
0.855
0.500
0.808
0.564
0.008
0.481
0.598
0.477
0.847
0.605
0.817
0.592
96
0.657
0.517
0.060
0.519
0.039
0.508
0.882
0.586
1.348
0.606
0.876
0.594
0.697
0.521
0.673
0.510
0.887
0.619
0.850
0.608
0.614
0.406
0.6177
0.409
0.099
0.420
0.7788
0.435
0.877
0.449
0.895
0.442
0.683
0.427
0.658
0.415
1.838
0.619
1.626
0.589
48
1.020
0.515
1.000
0.521
1.113
0.490
1.202
0.5
1.261
0.519
1-431
0.516
1.118
0.490
1.063
0.486
2.223
0.058
1.980
0.629
1.556
0.577
1.979
0.586
1.822
0.563
3.2
0.621
4.224
0.736
2.121
0.585
2.041
0.603
1.867
0.505
2.733
0.700
2.505
0.674
24
0.712
0.352
0.728
0.366
0.735
0.372
00.774
0.385
0.877
0.401
00.916
0.463
0.741
0.372
0.733
0.368
1.502
0.691
1.444
0.668
48
0.976
0.473
0.973
0.477
0.980
0.482
1.017
197
1.328
0.357
0.874
0.138
0.989
0.484
0.981
0. 0.482
1.658
0.739
1.605
0.719
96
1.261
0.584
1.261
0.592
1,263
0.597
1.320
0.603
1.714
0.679
1.188
0.560
1.269
0.601
1.263
0.592
1.793
0.780
1.750
0.761
24
3.941
0.281
3.978
0.285
4.156
0.294
4.112
0.203
6.194
0.381
4.343
0.290
4.432
0.299
4.143
0.294
11.877
0.583
11.304
0.568
48
4.656
0.309
1.661
0.312
1.780
0.315
4.750
0.313
9.380
0.135
5.186
0.322
1.901
0.324
4.762
0.315
12.683
0.600
12.076
0.585
96
5.675
0.348
3.672
0.341
5:835
0.340
5.703
0.336
12.851
0.161
6.350
0.349
6.791
0.371
5.791
0.341
13:221
0.601
12.671
0.585
Traflic
24
0.332
0.231
0.335
0.232
0.376
0.276
[0.351
0.250
0.452
0.316
0.3371
0.264
0.371
0.274
0.377
0.276
0.461
0.322
0.477
0.331
0.356
0.243
0.358
0.245
0.378
0.244
0.374
0.262
0.438
0.337
0.395
0.278
0.393
0.269
0.380
0.267
0.501
0.347
0.519
0.353
96
0.371
0.248
0.375
0.249
0.397
0.249
0.386
0.263
0.565
0.371
00.410
0.284
0.405
0.283
0.398
0.277
0.573
0.372
0.584
0.376
AVG
1.664
0.463
1.771
0.473
1.812
0.476
|.9999
0.501
2.798
0.556
2.025
0.511
1.905
0.491
1.811
0.481|
3.314
0.627
3.143
0.612
Baseline. We evaluate multiple baseline methods in our experiments, covering approaches from continual learning, time series forecasting, and online learning. (1) Experience Replay and its variant DER++ ER stores historical data in a buffer and alternates it with newer samples during training. DER++ [Buzzega
let al. 2020) adds a knowledge distillation module compared to standard ER. (2) Stationary (Liu el al.
2022] focuses on modeling non-stationary time series through a De-stationary Attention module. (3) Revin
[Kim et al. ) dynamically normalizes time series to mitigate the negative impact of non-stationarity, where we use PST as a high-quality backbone. (4) Dish-TS (Fan et al. (203)) proposes a Dual-
CONET framework that learns distribution discrepancies in input, and output spaces respectively, using
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coefficient network to alleviate intra- and inter-space distribution differences. We also use PatchTST as high-quality backbone here. (5) i'Transformer (Liu et al., 2024) is traditional time series forecasting method that models time series by transforming the roles of attention mechanisms and feed-forward networks. (6) CycleNet (Lin et al.) 2024) uses a residual periodic prediction technique, leveraging Icarnable recursive periods to model inherent periodic patterns sequences and making predictions on the modeled periodic residual components. (7) FsNet (Pham et al., 2022) avoids catastrophic forgetting of historical samples through TCN structures and adapter modules, enabling fast adaptation to new samples.
(8) One (Zhag et al. 2023) combines various strategies for time series forecasting. (9) SOLID (Chen et al., 2024
uses pre-trained parameters for linear probing at each update and does not inherit fine-tuned parameters from the previous update. We use the SOLID++ variant, continuously fine-tuning all model parameters on the online data. (10) Proceed (Zhao & Shen. 2024) estimates inter-concepl distribution shifts and uses an adaptive generator to effectively translate the estimated shifts into parameter adjustments, proactively adapting the model to test samples.
Table 3: Full results of the online timo-scrics forecasting task. We compare extensive competitive models under different prediction lengths following the setting of F&Net. The best results are in bold, and the second best are underlined.
Models
BTOA
OneNet
PaNet
Transformer
CycleNet
Dish-TT
Revin
Stationazy
DER++
DER
Metric
MSE
MAE
MSE
MAR
MSE
MAR
MSE
MAE
MSE
MAE
MSE
MAF
MSE
MAR
MSE
MAR
MSF
MAF
MSE
MAE
0.223
0.301
0.235
0.303
0,286
0.343
0.223
0.294
0.389
0.385
0.257
0.318
0,238
0.301
0.383
0.395
0,239
0.305
0.240
0.316
24
0.271
0.325
0.400
0.442
0.411
0.436
0.703
0.524
0.959
0.621
0.692
0.517
0.672
0.510
0.759
0.565
0.648
0.531
0.673
0.547
48
0.265
0.356
0.447
0.451
0.402
0.432
0.828
0.570
1,080
0.661
0.622 0.941
0.792
0.537
0.747
0.570
0.606
0.525
0.634
0.538
ECL WTH CTTm2 ETTml ETThh
0.390
0.362
0.383
0.351
0.467
0.371
0.418
0.352
0.559
0.384
0.514
0.302
0.383
0.344
0.770
0.383
0.508
0.375
0.508
0.378
24
0.505
0.397
0.538
0.414
0.603
0.473
1.716
0.587
2.206
0.602
1.584
0.794
1.741
0.581
2.090
0.659
0.828
0.540
0.808
0.543
48
0.587
0.436
0.604
0.445
0.867
0.516
2.781
0.676
3.254
0.892
2.119
0.677
2.762
0.664
2.938
0.722
1.157
0.577
1.136
0.571
0.106
0.187
0.117
0.202
0.101
0.187
9.106
0.192
0.125
0.210
0.120
0.201
0.122
0.208
00.11
0.197
0.110
0.192
0.114
0.197
24
0.114
0.222
0.134
0.243
0.137
0.249
1.663
0.692
0.816
0.535
0.427
0.471
1.531
0.704
0.536
0.449
0.196
0.326
0.202
0.333
0.118
0.227
0.118
0.228
0.124
0.240
1.648
0.722
1.322
0.692
0.553
0.549
1.018
0.614
1.433
0.721
0.208
0.340
0.220
0.351
0.174
0.226
0.191
0.233
0.179
0.229
0.1.8
0.221
0.173
0.224
0.321
0.271
0.173
0.226
0..194
0.228
0.190
0.231
0.191
0.233
24
0.206
0.259
0.267
0.261
0.233
0.276
0.639
0,430
0.659
0.411
0.611
0.450
0.652
0.435
0.954
0.679
0.307
0.3445
0.310
0.347
48
0.204
0.267
0.273
0.284
0.299
0.313
0.
0.987
0.502
1.067
0.480
0.906
0.507
1.083
0.502
1:209
0.592
0.329
0.359
0.331
0.363
0.156
0.197
0.138
0.201
0.161
0.215
0.160
0.205
0.169
0.210
0.156
0.195
0.165
0.211
0.152
0,196
0.208
0.233
0,180
0.244
24
0.161
0.241
0.189
0.273
0,189
0.276
0.375
0.399
0.388
0. 104
0.340
0.382
0.37U
01399
0.428
0,446
0,270
0.351
0,283
0.356
48
0.173
0.255
0.197
0.278
0.223
0.303
0.472
0.467
0.478
0.468
0.412
0.440
0.453
0.432
0.488
0.481
0.294
0.339
0,297
0.363
2.430
0.266
2.500
0.258
3.317
0.542
1.897
0.218
2.274
0.229
2.966
0.288
3.873
0.331
2.611
0.508
2.657
0.421
2.579
0.508
2.193
0.346
2,700
0.366
6.071
1.024
1.009
0.313
6.585
0.368
3,236
0.523
3.469
0.579
8.996
1.035
9.265
1.066
9.327
1.057
48
2.423
0.162
3.261
0
1000
7.231
1.089
1.787
0.316
7,276
0.393
5.911
0.355
6.583
0.379
1.987
0.789
9.009
1.048
9.685
1.074
Trafii
0.232
0.205
0.233
0.215
0.295
0.253
0.298
0.321
0.313
0.318
0.410
0.315
0.257
0.296
0..418
0.325
0.280
0.241
0.286
0.247
0.310
0.261
0.348
0.269
0.360
0.287
1.187
0.498
1.006
0.508
0.913
0.508
1.097
0.502
1.275
0.575
0.384
0.289
0.383
0.299
48
0.351
0.293
0.384
01.002
0.378
0.297
1.615
0.568
1.786
0.570
1.479
0.583
1.765
0.617
0.399
0.295
0.391
0.310
0.394
0.307
AVG
|0.677
0.285
0.656
0.306
1.068
0.396
1.183
0.418
1.630
0.454
|0.00
0.456
1.452
0.434
1.320
0.504
1.325
0.425
1.371 0.437
Implementation details. To ensure a more fair comparison of the model's performance, we followed the experimental setups of FsNet (Pham et al. 2022 and Proceed (Zhao & Shen 2024), testing the model under two different scenarios: one is the ideal experimental setup from F&Net, and the other is the practical experimental setup from Proceed. For the ideal experimental sctup, we set the prediction lengths to 1, 24, and 48. For the practical experimental setup, we set the prediction lengths to 24, 48, and 96 to better compare the model's performance with longer prediction lengths. Since learning is conducted in i sequential rounds, the model receives a look-back window in cach round and predicts the forecast window. All models are evaluated using cumulative Mean Squared Error (MSE) and Mean Absolute Error (MAE), which assess the models' performance over the entire learning process. We use the AdamW (Loshchilov & Hutter 2017)
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optimizer to minimize the L2 loss. To simulate streaming data, set the batch size to 1, reflecting the arrival of data in streaming fashion. Our method is general-purpose module. In the ideal setup, we instantiate it using OneNet as the forecaster, while in the practical setup, we consistently use PatchTST as the forecaster. More details about the implementation, architectures, and hyperparametens with the trained VAE model are given in Appendix B.4
4.1 Online Forecasting Results
Cumulative performance. To validate the effectiveness of the BTOA framework, we compared its fore- casting performance with other baselines under two experimental scttings across seven popular datasets. Each experiment was repeated three times, and we report the average results. In Appendix A.3 and B.3 we provide the standard deviations of the methods and additional experimental results, offering a more compre- hensive evaluation of the model performance and the robustness of our approach across multiple trials. As shown in Tables 2 and 3 BTOA achieved the best performance in most cases. Notably, BTOA significantly improved forecasting accuracy on datasets with strong non-stationarity, such as ETTh2 and ETTm2. This highlights BTOA's 's critical role in addressing distribution shift issues during online test-time adaptation. By effectively leveraging historical samples, BTOA mitigates distribution shifts, reduces the impact of potential noise in test-time samples on the model, and significantly enhances model robustness.
(a) ETTh1
(b) ETTm2
(c) WTH
Loss
Instances
Instances
Instances
Miar Dud buutoon snat
Distribution
Denaty
Densty
Feature Value
Feature Value
Feature Value
Figure Evolution of the cumulative MSF loss and visualization of distribution shifts.
Convergence of different deep models. As shown in Figure 2 we analyze the convergence behavior of BTOA compared with baseline models across three datasets and select 200 time-point samples before and after loss peaks to observe their distribution differences. It is evident that loss curve peaks tend to occur in the early and middle stages of training, and samples before and after these peaks exhibit distribution shifts, fully indicating that changes in data distribution significantly impair model performance. Such distribution shifts pose particular challenges for time-series models, as they disrupt the patterns learned by the model, leading to temporary performance degradation. Traditional batch learning settings typically evaluate performance only on small subset of data at the end of training, often neglecting the distribution shifts that occur during training, which results in suboptimal adaptation to practical scenarios. From Figure 2 we observe that when distribution shift occurs, the increase in MSE for BTOA is significantly smaller than that of all baseline models. This demonstrates BTOA's superior capability to detect and rapidly adapt to distributional changes, thereby maintaining more stable performance. The key to this adaptability lies in the integration of its data augmentation module, which leverages historical data and a rwo-stream augmentation technique lo capture shifts in both the amplitude and phase of the data, enhancing the model's robustness. This ability enables BTOA to make effective adjustments without compromising critical frequency domain information, sharply distinguishing from other models that struggle with such changes. These results highlight BTOA's effectiveness in quickly adapting to distribution shifts, ensuring more stable and reliable performance. Its
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capacity to mitigate the impact of such shifts underscores BTOA's potential as versatile and robust solution for real-world time-series forecasting challenges.
Table 4: Performance metrics for different batch sizes.
Dataset
ETTh1
ETTml
WTH
ECL
Traffic
batch-size
96
24
48
96
24
48
96
96
24
48
96
8
0.710
0.809
0,931
0.437
0.600
0.663
0.714
0.981
1.262
3,971
4.724
5.795
0.325
0.359
0.376
16
0.708
0.806
0.930
0.454
0.592
0.657
0.712
0.976
1.261
3.941
4.656
5.675
0.320
0.356
0.371
32
0.711
0.803
0.933
0.437
0.593
0.660
0.715
0.979
1.261
3.949
4.659
5.678
0.319
0.356
0.379
64
0.725
0.814
0.940
0.471
0.613
0.672
0.732
0.991
1.290
3.978
4.711
5.801
0.334
0.369
0.380
The batch size at the online test-time adaptation. In Table we also investigated the impact of the batch size hyperparameter, which determines the number of historical samples to be auginented during the online batch training phase. Our experiments revealed that optimal performance is achieved with batch sizes of either 16 or 32. Considering both performance and computational efficiency, we selected a batch size of 16 as the optimal choice. This decision strikes a balance between ensuring high model performance and reducing training time, thus improving the model's practicality for real-world applications, especially when handling large streaming data. By optimizing this hyperparameter, we can significantly accelerate the training process without sacrificing accuracy, making our approach better suited for time-sensitive tasks.
(a) PatchTST
(b) PatchIST++Proeed
(c) PatchTST+BBOA
ETTh
chec
volus
1//
N
S0
16
ETTm2
Time
Serkes
613
Series
01-
rouu Sertes Ground Truh Reviittinn
Ground Tust
Ground Trush
WTH
medi
505
Tirme
Time
Time
Figure 3: Forecast results under distribution shift with forecasting window 96.
Visualization. To validate the effectiveness of our method, we compare the prediction results of three models PStchTST, Proceed, and our BTOA on three datasets. Specifically, we selected samples from the post-peak loss curve period, these samples better reffect the models' adaptability after distribution shifts occur. As shown in Figure 3 BTOA exhibits faster adaptation to distribution shifts and closer alignment with ground truth compared to basclines. Specifically, BTOA's predictions remain tightly aligned with true values, whereas PatchTST and Proceed show prolonged deviations. As training progresses, BTOA not only adapts faster but also captures complex temporal patterns more effectively, leading to superior prediction performance. This highlights BTOA's advantage mitigating the negative impacts of distribution shifts and underscores its ability to maintain and even improve-predictivve accuracy over time, making more reliable solution for online scenarios.
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4.2 Ablation Study
In this section, we will examine the contribution of each module in BTOA individually. The first module is Transferable Historical Sample Selection module, which selects historical samples that are close to the information of the test-time sample, a critical factor in fully leveraging historical information. The second module is the Transferable Online Augmentation module, which performs batch training to help mitigate noise present the test-time sample and reduce distribution shifts. More ablation results are provided Appendix B.2
Table 5: Comparison of Historical Sample Selection Methods.
Dataset
ETTh1
ETTml
WTH
ECL
Traffic
Methods
24
48
96
24
48
96
24
48
96
24
48
96
24
48
96
closest-16
0.715
0.809
0.944
0.458
0.600
0.672
0.719
0.991
1.283
3.986
4.678
5.732
0.334
0.372
0.391
L2-distance
0.711
0.812
0.935
0.459
0.601
0.659
0.717
0.982
1.273
3.956
4.671
5.723
0.324
0.362
0.374
THSS (ouns)
0.708
0.806
0.930
0.454
0.592
0.657
0.712
0.976
1.261
3.941
4.656
5.675
0.320
0.356
0.371
The Components of BTOA. First, we examine the THSS module. We conducted a comparison between three methods: (1) directly selecting the 16 most recent samples in time closest to the test-time sample for data augmentation, without choosing from the memory bank; (2) using L2 distance as representative naive distance metric for the selection criterion; and (3) using the THSS module as the selection standard. As shown in Table 5 the THSS module outperforms the other two methods in selecting historical samples. This result demonstrates that the VAE successfully captures the semantic information of time series in the latent space, confirming the effectiveness of the latent space.
Table 6: Comparison of Data Augmentation Methods.
Dataset
ETThh
ETTmm
WITH
ECL
Trafffc
Methods
24
96
24
48
96
48
96
24
48
24
48
96
Non-Aug
0.756
0.882
1.086
0.470
0.598
0.673
0.732
0.981
1.263
4.143
4.762
5.791
0.376
0.380
0.398
Mixur
0.762
0.890
1.121
0.471
0.602
0.681
0.744
0.992
1.283
4.152
4.801
3.822
0.379
0.384
0.406
Cut-Mixup
0.753
0.869
0.941
0.467
0.608
0.669
0.729
0.983
1.271
4.097
4.722
5.731
0.355
0.374
0.380
TOA (ours)
0.708
0.806
0.930
0.454
0.592
0.657
0.712
0.976
1.261
3.941
4.656
5.675
0.320
0.356
0.371
Second, we evaluate the effectiveness of the TOA module. We compare four methods on five datascts: (1) without data augmentation; (2) using Linear-Mixup (Zhang et al. 2017;; (3) using Cut-Mixup (Yun) jtt al.) 2019); and (4) using our proposed TOA module. Table 6 demonstrates the superior performance of the TOA method across all five datasets. This indicates that, for time series, simple time domain data augmentation can casily lead to interference between different scrics. In contrast, the TOA method performs data augmentation in the frequency domain, focusing on amplitude and phase, reducing the interference between time series and achieving positive data augmentation, thereby improving prediction performance.
Table 7: The Generality of BTOA.
Dataset
ETTTh
ETTm2
WTH
ECL
Traffic
Methods
24
48
96
24
48
96
24
48
96
24
48
96
24
48
96
TCN
1.006
1.138
1.360
0.899
1.233
1.823
0.872
1.418
1.805
6.964
8.155
10:001
0.478
0.530
0.573
TCN+BTOA
0.912
1.021
1.244
0.781
0.942
1.184 0.720
0.994
1.298
6.031
7.152 9.077 0.409
0.447
0.523
(Tranformmer
0.758
0.900
1.103
0.874 1.313
1.948 0.845 1.124 1.411 3.940 4.667 5.844 0.340 0.361 0.379
f'ransformer +BTOA 0.702 0.835
0.991 0.802 1.302 1.766 0.3 0.998 1.280 3.818 4.596 5.705 0.329 0.351 0.376
Generalisability. As plug- and-play module, BTOA has demonstrated significant effectiveness in improv- ing model performance. In Table 7 we compare the results of adding BTOA to the TCN and iTransformer
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models. The experimental results show that, in all models, the performance after adding BTOA outperforms the original versions of these models. These results fully validates the effectiveness of BTOA in enhancing the models' generalization ability and overall performance. Specifically, BTOA improves the model's robust- ness against performance degradation caused by distribution shifts online time series prediction through batch-training. Additionally, the inclusion of BTOA also positively impacts the stability of model training. Therefore, BTOA, as a general-purpose module, not only improves the performance of various models but also enhances their feasibility and stability in real-world applications.
5 Conclusion
In this paper, a general online lest-lime adaplation method, Batch training with Transferable Online Aug- mentation (BTOA) for time series, is designed to effectively mitigate the performance degradation caused by distribution shifts during the test process. Our approach incorporates transferable historical sample selection module, transferable online augmentation module, and prediction block. The THSS module fully leverages the distributional information from historical samples. Through the TOA module, we rein- troduce batch training in the online setting to alleviate the negative impact of distribution shifts. Finally, the prediction block extracts complex patterns in time series, enabling more accurate outputs. Extensive experiments demonstrate that our approach can be integrated into any online time series forecasting model and achieves superior performance.
Broader Impact Statement
Our work proposes a plug-aud-play module from the perspective of online test-time adaptation to mitigate the negative impact of distribution shift. This crucial for practical time series forccasting tasks. This research can serve as a reference for future studies in machine learning and data science, promoting the development of more complex and accurate online time series forecasting techniques. Therefore, our paper primarily focuses scientific research and does not have any obvious negative social impact.
Acknowledgements
This work was supported by the Shenzhen Science and Technology Program ZDDSYSS222366260110000000
National Natural Science Foundation of China (62306085, 62302241, 62476071, 62236003, U23B2055, U24A20328), Shenzhen College Stability Support Plan (GXWD2001133011:2200)
GXWD2002061744428000
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A Additional experimental details
A.1 Datasets
We conduct experiments on 7 real-world datasets to evaluate the performance of our method including (1) ETT ETT(eetttiiciy Transformer Temperature) (Wu et al., 2021) are collected from two electricity transformers with factors. There are four subsets where ETTh1 and ETTh2 are recorded every hour, and ETTml and ETTm2 are recorded every 15 minutes. (2) WTH (Wu et al., 2021 includes 21 meteorological factors collected every 10 minutes from the Weather Station of the Max Planck Biogeochemistry Institute in 2020. (3) FCC. (Wu et al. 2021) collects 321 customers' hourly electricity consumption. (4) Traffic (Wu et al. colllecs the road occupancy rates from different sensors San Francisco freeways.
A.2 Implementation Details
All experiments are implemented in PyTorch and conducted on a single NVIDIA RTX4090 24GB GPU. The learning rate for the experiments is set between 1e-3 and 7e-3.
A.3 Details about the standard deviation
We report the standard deviations of BTOA for three backbones on seven datasets across three random soods Table
Table 8: Standard deviation of three backbones.
Backbones
Metric
ETTh1
ETTh2
ETTml
ETTm2
Weather
Electricity
Traffic
MSE
0.007
0.008
0.005
0.003
0.001
0.04
0.0007
TCN
MAE
0.002
0.003
0.001
0.0005
0.0007
0.0002
0.001
MSE
0.001
0.01
0.01
0.001
0.009
0.04
0.0006
iTransformer
MAE
0.0005
0.002
0.001
0.0001
0.004
0.0005
0.0005
MSE
0.001
0.03
0.008
0.003
0.001
0.04
0.0007
PSt
MAE
0.0002
0.003
0.002
0.0003
0.0001
0.001
0.001
B Additional experimental Results
B.1 Analysis of Inference Time and Memory Consumption
Onobit
3.5
Tahtt
DCRsp
Onetut
17
MSE
3.3
2.59
3.4
host 0,190
SOLD
Hile:
33
1.7
SOLE: SG
OLB 0450
153
Dan 6.6G
0.00
634
822
2008
Inference Time (s/sample)
Inference Time (s/sample)
Figure 4: Comparison of inference time and memory consumption on ETTh2 and ETTm2.
Since online test-time adaptation deals with streaming data that arrives in continuous and sequential manner, Figure compares the inference time for single sample and overall memory usage between
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method and the baselines. The results show that BTOA introduces minimal increases in inference time and memory usage, which are acceptable in online setting, while outperforming the baseline methods. Notably, for the VAE module during inference, BTOA only requires simple forward propagation without the need for backpropagation. On the ETTh2 dataset, compared to Procced, BTOA improves prediction performance by 18% with only a 9% increase in inference time; compared OneNet, BTOA enhances prediction performance by 28% while reducing inference time by 35%. This comparison further highlights the superiority of BTOA.
B.2 Ablation Study
Table 9: Performance Impact of Series Decomposition Block.
Dataset
ETThh
ETTmm
WTH
ECL
Traffic
Methods
24
48
96
24
48
96
24
48
96
24
48
96
24
48
96
Non-SD
0.712
0.820
0.955
0.461
0.596
0.668
0.721
0.982
1.270
4.021
4.712
5.721
0.342
0.367
0.380
SD
0.708
0.806
0.930
0.454
0.592
0.657
0.712
0.976
1.261
3.941
4.656
5.675
0.320
0.356
0.371
we evaluate the effect of using a series decomposition module in the prediction block by comparing the performance of models with and without series decomposition. Although OneNet suggests that series de- composition is not universally effective method for improving performance, Table g shows that for the BTOA model, series decomposition can effectively extract complex temporal patterns and enhance the model's predictive ability.
Table 10: Performance Impact of different distance threshold.
Dataset
ETThl
WTH
Traffic
distance
24
48
96
24
48
0.7
0.241
0.295
0.259
0.157
0.169
0.171
0.249
0.306
0.362
0.8
0.223
0.271
0.265
0.156
0.161
0.174
0.232
0.310
0.351
0.9
0.233
0.272
0.266
0.159
0.170
0.182
0.262
0.316
0.355
We also conduct ablation experiments regarding the distance threshold hyperparameter as shown in Table I0, which determines from which distribution aa and AP are sampled. the distance between the historical sample and the newly received sample below a predefined distance threshold (set to 0.8 our experiment), Xa, Xp are sampled from a uniform distribution with lower mcan. If the distance between the historical sample and the newly received sample exceeds the predefined distance threshold, Aa, Ap sampled from truncated Gaussian distribution with higher mean and lower standard deviation.
B.3 More experimental results
Due to space limitations in the main text, we added comparisons with more methods in the appendix. Tables 11 and 12 show comparisons with additional Test-time adaptation methods, including EVO-MSN (Qin et al., 2024h and TAFSA (Kim el al. 2025), under practical scenario settings. Tables 13 and 14 present comparisons
with more series decomposition and Test-time adaptation methods under ideal scenario settings.
B.4 VAE Models Architecture
We use the Total Correlation Variational Autoencoder (TC-VAE) (Chen et al. 2018 to compute the distance in latent space between the current incoming sample and the historical samples stored in the memory bank. The model is trained for 100 epochs with a learning rate of 3e-3, using the Evidence Lower Bound (ELBO) as the loss function, and a batch size of 128. The latent dimensions and the parameter are set to 10 and 5, respectively. Below, we provide detailed information about the encoder and decoder architectures, which differ across datasets due variations in input channels.
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Table 11: The MSE results of the online time-series forecasting task. We compare extensive competitive models under different prediction lengths following the sctting of Proccod. The best results are bold, and the second best are underlined.
Method
BTOA
Proceed
SOLID++
OneNet
F&Net
CycleNet
EVO-MSN
TAFAS
Dish-TS
PatchTST
DER++
ER
21
0.708
0.729
0.745
0.780
0.993
0.753
0.769
0.743
0.761
0.756
0.831
0.811
0.806
0.886
0.818
0.896
0.857
0.881
0.851
0.882
0.887
0.901
0.930
1,003
0.977
1.025
1.359
0.986
1,010
0.979
1.074
1.086
1.036
1.019
ECL WTH ETTm2 ETTml ET
24
1.688
1.011
2.021
2.606
2.941
2.228
2.435
1.990
1,931
1.893
2.790
2.492
48
2.772
3.291
3.442
3.921
4.090
3.701
3.777
3.426
3.391
3.283
4.090
3.799
96
4.880
5,790
6,802
6.248
6.245
6.674
6.112
5.808
6.05-1
6.976
6.363
6.022
24
0.454
0.422
0.455
0.766
0.627
0.604
0.677
0.451
0.459
0.470
0.775
0.745
48
0.592
0.579
0.578
0.978
0.855
0.808
0.858
0.679
0.008
0.598
0.847
0.817
0.657
0.660
0.659
0.882
1.318
0.876
0.811
0. 0.659
0.697
0.673
0.887
0.859
0.614
0.617
0.699
0.768
0.877
0.895
0.747
0.690
0.693
0.658
1.626
1.020
1.090
1.113
1.202
1.261
1.431
1.173
1.117
1.118
1.063
2.223
1.989
96
1.556
1.979
1.822
3.102
4.224
2.121
2.718
1.837
2.041
1.867
2.733
2.505
21
0.712
0.728
0,735
0.774
0.877
0.874
0,762
0.734
0.711
0,732
1.502
1.4414
48
0.976
0.973
0,980
1.017
1.328
1.19
1.029
0.979
0.989
0,981
1.658
1.605
96
1.261
1.264
1,263
1.320
1.714
1.471
1.309
1.261
1.269
1.263
1.793
1.750
24
3.941
3.078
4.156
4.112
6.194
4.343
4.122
4.182
4.432
4.143
11.877
11.304
4.656
4.664
4.780
4.750
9.380
5.186
4.759
4.764
4.901
4.762
12.683
12.076
5.675
5.672
5.835
5.703
12.851
6.350
5.726
5.887
6.791
5.791
13.221
12.671
Taflle
0.332
0.3335
U.376
0.351
0.452
0.371
0.355
0.379
0.371
U.376
0.461
0.477
0.356
0.358
0.378
0.374
0.498
0.395
0.372
0.376
0.393
0.380
0.501
0.519
96
0.371
0.375
0.397
0.386
0.565
0.41
0.383
0.398
0.405
0.398
0.573
0.584
Table 12: The MAE results of the online timo-scrics forecasting task. We compare extensive competitive models under different prediction lengths following the setting of Proceed. The best results are in bold, and the second best are underlined.
Method
BTOA
Proceed
SOLID++
OneNet
PoNett
CycloNet
EVO.MSN
TAPAS
Dish-TS
PatchTST
DER++
ER
BCL WTH ETTmm2 ETTml ETTh2 ETTh1
24
0.533
0.534
0.552
0.559
0.624
0.556
0.557
0.550
0.761
0.552
0.604
0.593
48
0.576
0.593
0.593
0.600
0.664
0.596
0.598
0.593
0.882
0.601
0.635
0.627
96
0.625
0.650
0.645
0.648
0.752
0.644
0.647
0.646
1.074
0.664
0.675
0.668
24
0.563
0.603
0.609
0.638
0.696
0.636
0.629
0.608
1.931
0.608
0.714
0.684
48
0.686
0.733
0.718
0.729
0.797
0.752
0.726
0.720
3.391
0.720
0.801
0.778
0.800
0.847
0.868
0.927
0.945
0.920
0.909
0.883
6.054
0.887
0.925
0.905
24
0.408
0.393
0.406
0.487
0.484
0.481
0.463
0.413
0.459
0.415
0.579
0.566
0.477
0.462
0.46.
0.582
0.560
0.564
0.546
0.473
0.608
0.474
0.605
0.592
96
0.517
0.519
0.503
0.586
0.006
0.594
0.561
0.511
0.697
0.510
0.619
0.608
0.406
0.409
0.420
0.435
0.449
0.442
0.431
0.414
0.693
0.415
0.619
0.589
0.515
0.521
0.490
0.511
0.519
0.516
0.505
0.490
1.118
0.486
0.658
0.629
96
0.577
0.586
0.563
0.621
0.636
0.585
0.604
0.567
2.041
0.565
0.700
0.674
24
U.473
0.477
0.482
0.497
0.537
0.560
0.493
0.482
0.969
0.4882
0.739
0.719
48
0.473
0.477
0.482
0.497
0.557
0.560
0.493
0.482
0.989
0.482
0.739
0.719
96
0.584
0.592
0.597
0.603
0.679
0.602
0.601
0.592
1.269
0.592
0.780
0.761
24
0.281
0.285
0.294
0.293
0.381
0.299
0.293
0.293
4.432
0.2994
0.583
0.566
48
0.309
0.312
0.315
0.313
0.435
0.322
0.314
0.315
4.901
0.315
0.600
0.583
96
0.3-18
0.340
0.340
0.336
0. 0.4664
0.349
0.337
0.341
6.791
0.341
0.601
0.585
Tralic
24
0.231
0.233
0.276
0.250
0.316
0.264
0.258
0.272
0.371
0.276
0.322
0.331
0.244
0.262
0.337
0.278
0.257
0.265
0.393
0.267
0.347
0.353
0.248
0.249
0.249
0.263
0.371
0.281
0.259
0.275
0.405
0.278
0.372
0.376
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Published in Transactions on Machine Learning Research (06/2025)
Table 13: The MSE results of the online timo-scries forccasting task. We compare extensive competitive models under different prediction lengths following the setting of FsNet. * means 'Former'. The best results are in bold, and the second best are underlined.
Method
BTOA
OneNet
PoNet
TAFAS
iTras"
CycleNet
Dish-TS
Revin
Stationary
Auto"
NLinnar
Pori-mid"
PutchTST
DER++
WTTI ETTo2 ET ETTh2 ETTai
0.228
0.235
0.286
0.229
0.223
0.380
0.257
0.237
0.383
0.501
0.287
0.445
0.246
0.239
0.240
0.271
0,400
0.411
0.703
0.999
0,082
0.671
0.708
1.496
0.759
1.228
0.810
0.648
0.673
0.265
0.447
0.402
0.828
1.090
0.941
0.702
0.747
0.891
0.846
0.986
0.831
0.906
0.634
0.350
0.383
0.467
0.387
0.418
0.550
0.514
0.382
0.770
1,890
0.820
1.225
0.362
0.508
0.508
0.683
0.638
0.633
0.522
1.716
2:303
1.584
1.741
2,088
2,030
2.332
2,008
1-022
0,828
0.808
48
0.587
0.604
0.867
0. 0.506
2,781
3.254
2.119
2.762
2,988
3.946
3.405
3,600
2.716
1.157
1.136
0.100
0.104
0.120
0.111
0.385
0.109
0.255
0.116
0.110
0.114
0.114
0.134
0.137
0.124
0.777
1.083
0.816
1.531
0.536
1.885
0.922
1.774
0.427
0.196
0.2002
0.118
0.118
0.124
0.118
0.783
1.648
1.522
1.018
1.433
1.991
0.504
1,800
0.553
0.208
0.200
0.174
0.181
0.179
0.183
0.168
0.173
0.321
0.173
0.194
0.611
0.178
0.342
0.184
0.190
0.191
24
0.2006
0.267
0.223
0.237
0.639
0,020
0
0.672
0.354
1.076
0.605
0.807
0.647
0.307
0.310
0.200
0.906
1.083
1.406
0.968
0.608
0.329
0.331
0.1.5
0.158
0.161
0.157
0.160
0.160
0.156
0.165
0.152
0.208
0.160
0.180
0.162
0.208
0.180
0.189
0.376
0.340
0.370
0.428
0.445
0.350
0.372
0.270
0.253
0.178
0.107
0.223
0.185
0.472
0.478
0.412
0:453
0.487
0.624
0:430
0.501
0.465
0.294
0.297
Thaifcc BCL
2.430
2,590
3:317
2,510
1.897
2:274
3.000
3.873
2.613
3.847
3:532
3.061
2.022
2.657
2.679
2.493
2,700
6.071
2.507
4.009
6,085
4.008
4.862
3.463
7,289
7.148
6.587
4.325
8,996
9.327
48
2.423
3.261
7:231
2.842
4.787
7:276
4.479
6.583
4.987
8.004
8:658
8.140
5.030
9.000
0.685
0.232
0.295
0.410
0.418
1.062
0.904
0.533
0:250
0.286
0.510
0.348
0.360
0.322
1.097
1.187
1.006
1.097
1.275
1.795
1.641
1.471
0.913
0.384
0.383
0.351
0.364
0.378
0.368
1,615
1.786
1.479
1,678
1.765
2,281
2.182
2.031
1,519
0.508
0.394
Table 14: The MAE results of the online time-series forecasting task. models under different prediction lengths following the setting of FsNet. * meaas 'Former. The best results are in bold, and the second best are underlined.
We compare extensive competitive
Method
BTOA
OueNett
PoNet
TAFAS
iTrans"
CyeleNet
Disb-TS Revin
Stationazy
Auto"
NLinow
Per-mid*
PatchTST
DER+++
ER
ETTh2 ETTh:
0.343
0.347
0.294
0.385
0:318
0.304
0.385
0.474
0.341
0.430
0.311
0.305
24
0.325
0.442
0.496
0.384
0.524
B.621
0.517
0.510
0.565
0.801
0.550
0.711
0.570
0.534
0.547
0.356
0.454
0.452
0.400
0,570
0.661
0.022
0.557
0.370
0.617
0.580
0.639
0.575
0.325
0.538
0.362
0.351
0.371
0.367
0,352
0.384
0.302
0.344
0.363
0.574
0.426
0.479
0.351
0.375
0.376
24
0.397
0.414
0.473
0.406
0.587
0.602
0.504
0.581
0.659
0.706
0.685
0.654
0.577
0.540
0.643
0.434
0.445
0.516
0.441
0.678
0.002
0.077
0.684
0.722
0.772
0.706
0.732
0.672
0.577
0.671
ETTa2 E2 Raa
0.18T
0.202
0.187
0.195
0.192
0.210
0.204
0.208
0.157
0.406
0.153
0.308
0.186
0.192
0.197
0.222
0.249
0.233
0.529
0.602
0.535
0.704
0.449
0.842
0.767
0.471
0.525
0.118
0.118
0.128
0.118
0.783
1.648
1.322
1.018
1.433
1.991
0.804
1,820
0.558
0.208
0.220
0.229
0.230
0,221
0.224
0:271
0.226
0.228
0.373
0.221
0.259
0.228
0.233
0.206
0.267
0.203
0.237
0,639
0.039
0.611
0.852
0.954
1.075
0.695
0.867
0,547
0,307
0.310
0.204
0.273
0.299
0.230
0.987
1.067
0.506
1,083
1.209
1.405
0.0G8
1.236
0.608
0.329
0.331
WT11
0.197
0.201
0.215
0.199
0.205
0.210
0.195
0.211
0.156
0.297
0.203
0.249
0.200
0.235
0.244
0.241
0.273
0.276
0.257
0.399
0.406
0,382
0.394
0.446
0.471
0.388
0.439
0,393
0
01356
0.303
0.207
0.467
0.468
0.440
0.452
0.484
0.659
0
DCL
0.266
0.258
0.542
0.262
0.218
0,229
0.315
0.331
0.508
0.335
0.334
0.282
0.311
0.421
0.506
0.313
1.024
0.355
0.366
0.368
0.308
0.332
0.329
0.455
01.412
0.375
1,0035
1.067
0.462
0.400
1.089
0.031
0.318
0.383
0.563
0.379
0.789
0.835
0.587
0.514
0.399
1.048
1.074
Ttaffec
0,205
0.210
0.253
0.215
0:521
0.318
0.315
0.295
0.325
0.066
0.436
0.392
0.307
0:241
0.217
0.261
0,205
0.297
0,519
0.498
0..008
0.502
0.375
0.584
0.507
0.541
0.508
0.289
0.299
48
0.293
0.302
0.297
0.288
0.568
0.570
0.563
0.573
0.617
0.6:0
0.626
0.605
0.571
0.295
0.307
Table 15: Encoder Network for ETT dataset
Layer Name
Output size
# of kernels
Kernel size
Stride
Activation
Input
N 1 7
Convolution
N x 32 x 26 5
32
9 3
2 x
ReLU
Convolution
N x 32 x 10 x
32
7 x 3
2 x
ReLU
Convolution
N x 64 x 2 x 1
64
5 5 3
3 x
ReLU
Convolution
N x 128 1 X
128
2 1
1
ReLU
Convolution
N x 128 1 X
128
1x1 1
1x1
20

---

Published in Transactions on Machine Learning Research (06/2025)
Table 16: Decoder Network for ETT dataset
Layer Name
Output size
# kernels
Kernel size
Stride
Activation
Input
N 10 x
Transposed
Convolution
N
512 512
x
512
2 x 7
1 x
ReLU
Transposed Convolution
N x 128 2 x
128
x 1
6
ReLU
Transposed Convolution
x 64
2
ReLU
Transposed Convolution
N x 32 32
32
1
2
ReL.U
Transposed Convolution
32
x 1
2
Table 17: Encoder Network for WTH dataset
Layer Name
Output size
# of kernels
Kernel size
Stride
Activation
Input
N Nx1x 60 12
Convolution
N x 32 18 11
32
9 x
3 1
ReLU
Convolution
N x 32 x 5
32
3 x
3 x
ReLU
Convolution
N
64
3
ReLU
Convolution
N x 128 1 x
128
2 x
x
ReLU
Convolution
N x 128 1 x
128
1 x
1 x
Table 18: Decoder Network for WTH dataset
Layer Name
Output size
# of kernels
Kernel
Stride
Activation
Input
N 10 x
Transposed Convolution
N x 512 x x 12
512
2 x 7
x
ReLU
Transposed Convolution
x 128 x 12
128
4
x
ReLU
Transposed Convolution
x
ReLU
Transposed Convolution
N x 32 32 12
32
4
2 x
ReLU
Transposed Convolution
N x 32 x x 12
4 x
2 x
Table 19: Encoder Network for ECL datasct
Layer Name
Output size
# of kernels
Kernel size
Stride
Activation
Input
N x x 60 321
Convolation
N x 32 18 64
32
9 x
3 x 5
ReLU
Convolation
N x 32 x x 13
32
3 x
x
ReLU
Convolution
N
64
3 x
3
ReLU
Convolution
N x 128 1 x
128
2 x
x 3
ReLU
Convolution
N x 128 1 x
128
1 x
x
21

---

Published in Transactions on Machine Learning Research (06/2025)
Table 20: Decoder Network for ECL dalaset
Layer Name
Output size
# of kernels
Kernel size
Stride
Activation
Input
N x 10 x
Transposed Convolution
N x 512 x 2 x 321
512
2 x 321
1x 1
ReLU
Transposed Convolution
N x
x
128
x
1
ReLU
Transposed Convolution
N 64 16 x 321
64
X
ReLU
Transposed Convolution
N x 32 x 32 x 321
32
4 x
2 x
ReLU
Transposed Convolution
32 321
1
x
Table 21: Encoder Network for traffic dataset
Layer Name
Output size
# of kernels
Kernel size
Stride
Activation
Input
X x 60 x 862
Convolution
32 x 18 x
32
x
5
ReLU
Convolution
N x. x x 35
32
x 3
3 5
ReLU
Convolution
N x 64 x
64
x 3
3 5
ReLU
Convolution
x 128
128
x
5
ReLU
Convolution
N 128
128
Table 22: Decoder Network for traffic dataset
Layer Name
Output size
# kernels
Kernel size
Stride
Activation
Input
N 10
Transposed Convolution
N x 512 x x 862
512
2 862
1x
ReLU
Transposed Convolution
N x 128 x x 862
128
4 x
x 1
ReLU
Transposed Convolution
x 64 x 16 x 862
64
x
1
ReLU
Transposed Convolution
N 32 32 862
32
1
ReLU
Transposed Convolution
N 32 1 x 862
1
4 x
2 x
22


Reviews:
[{"review_id": "hYbuVhRjBI", "reviewer": "Reviewer_wLse", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. This paper addresses distribution shifts in time series via frequency-domain augmentation, which is innovative and rarely explored.\n2. Extensive experiments on real-world datasets with ablation studies validating each module’s contribution.\n\nWeaknesses:\n1. The manuscript emphasizes the online nature of the proposed time series forecasting method. However, it lacks a comprehensive analysis of computational costs and inference latency, which are critical considerations for real-time applications.\n2. A systematic comparison with state-of-the-art test-time adaptation techniques for time series forecasting is notably absent, which would help substantiate the claimed advancements of the proposed method (e.g., [1][2][3]).\n3. The paper lacks ablation experiments examining the two-stream paradigm, which is a core design element in both the TOA module (amplitude/phase separation) and the forecasting architecture.\n\n[1] Online test-time adaptation of spatial-temporal traffic flow forecasting.\n[2] Augmented Contrastive Clustering with Uncertainty-Aware Prototyping for Time Series Test Time Adaptation.\n[3] Battling the Non-stationarity in Time Series Forecasting via Test-time Adaptation.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a novel framework called Batch Training with Transferable Online Augmentation (BTOA) for addressing online test-time adaptation (TTA) in multivariate time series forecasting. Three key innovations are introduced: (1) a Transferable Historical Sample Selection (THSS) module leveraging a VAE to identify historically similar samples, (2) a Transferable Online Augmentation (TOA) module performing frequency-domain augmentation to preserve critical time-series features, and (3) a prediction block combining time series decomposition and a two-stream forecaster for complex pattern extraction.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see weaknesses.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "vwtJOGZfZg", "reviewer": "Reviewer_Xfxf", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The main **strong points** of the work are the following: \n- Prominent online time series forecasting methods update the model with each new sample. However, this can be problematic if the incoming data significantly deviates from the historical distribution or is noisy. In contrast, in BTOA, authors enable *batch-based training*.\n- The authors in the current benchmark present improved performances with respect to **both offline and online methods** in terms of cumulative point-wise metrics. Several *ablation studies* accompany the experimental evaluation of the method’s main components along with *sensitivity analysis* on the batch size selection and the models' convergence.\n\nThe following **weak points** can be identified for the submission:\n- **[W1 - Positioning of the Work \\& Missing Related Works]:** The main contribution of this work over existing online methods remains somewhat unclear, focusing primarily on batch-based adaptation and alternative design choices in key components (e.g., augmentation strategy, prediction module). Additionally, the title is very general and does not adequately refer to the methods-specific characteristics.\n- **[W2 - Missing Related Works and Baselines]:** The prediction module is based on a simple decomposition framework to extract patterns in both the frequency and time domains. However, a substantial body of related work employs similar components, which the authors do not adequately discuss [1,2,3,4]. Furthermore, since the paper compares to offline methods (e.g., iTransformer), it would be valuable to extend these comparisons to recent state-of-the-art offline forecasting models, particularly those leveraging decomposition-based neural architectures. Additionally, several offline methods have been particularly designed for alleviating distribution shifts in forecasting (see baselines in RevIN and [5]), but the authors only compare against Reversible Instance Normalization.\n- **[W3 - Missing Definitions\\& Presentation of Distribution Shift Examples]:** The paper lacks a clear presentation of distribution shift definitions and specific examples/cases (e.g., concept drift is presented in FSNet, OneNet), which would help readers better understand the specific challenges in time series forecasting. Additionally, the analysis of distribution shifts in the proposed datasets is limited to ADF tests. In contrast, related methods such as RevIN provide more comprehensive insights using, for instance, density plots to visualize train-test distribution discrepancies and demonstrate how their approach mitigates these shifts. \n- **[W4 - Incremental Performances and Missing Details in Experimental Evaluation]:** The performance gains in cumulative MSE/MAE are generally modest for the proposed method, and their statistical significance is unclear due to the absence of standard deviations. It is also not specified whether the authors conduct multiple runs to assess variance. For instance, in Table 6, the authors fail to clearly show the adaptability of BTOA since combining it with other common methods results in minor improvements for several horizons and datasets.\n\n\n[1] Wu, H., Xu, J., Wang, J., & Long, M. (2021). Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting. Advances in neural information processing systems, 34, 22419-22430.\n\n[2] Lin, S., Lin, W., Hu, X., Wu, W., Mo, R., & Zhong, H. (2024). Cyclenet: enhancing time series forecasting through modeling periodic patterns. Advances in Neural Information Processing Systems, 37, 106315-106345.\n\n[3] Zeng, A., Chen, M., Zhang, L., & Xu, Q. (2023, June). Are transformers effective for time series forecasting?. In Proceedings of the AAAI conference on artificial intelligence (Vol. 37, No. 9, pp. 11121-11128).\n\n[4] Wu, Q., Yao, G., Feng, Z., & Shuyuan, Y. (2024). Peri-midformer: Periodic pyramid transformer for time series analysis. Advances in Neural Information Processing Systems, 37, 13035-13073.\n\n[5] Fan, W., Wang, P., Wang, D., Wang, D., Zhou, Y., & Fu, Y. (2023, June). Dish-ts: a general paradigm for alleviating distribution shift in time series forecasting. In Proceedings of the AAAI conference on artificial intelligence (Vol. 37, No. 6, pp. 7522-7529).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors present an online test-time adaptation method for time series forecasting to alleviate prominent distribution shift challenges in related offline neural network methods. More specifically, they propose a **batch training-based method with transformable online augmentations** (so-called BTOA), which first leverages the historical information by selecting historical samples that are similar to the test-time distribution (named Transferable Historical Sample Selection - THSS component), followed by an augmentation technique that enhances the selected samples by phase and amplitude in the frequency domain (called the Transferable Online Augmentation - TOA component). Finally, a prediction module based on decomposition and a two-stream forecaster is used to encode the temporal data dependencies. The proposed BTOA framework can be incorporated into any online time series (batch-based) forecasting framework to boost robustness in terms of shifts between train and test distributions.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Based on **[W1]**, could the authors provide intuitive examples or scenarios where their method clearly outperforms conventional test-time baselines? This would help clarify the practical advantages of the approach and better highlight the challenges addressed by the work. The introduction section could be further adapted to intuitively motivate the core contribution of the presented method.\n\n2. Based on **[W2]**, the related works section could be improved to contain relevant modeling techniques in terms of architectural design. Additionally, important basic methods in terms of mitigating distribution shifts are missing. Could the authors include these babies in discussions and experiments? Are performance comparisons against the proposed baselines still favorable for BTOA?\n\n3. Concerning **[W3]**, could the authors improve the presentation and connection of their methodological contribution to well-defined cases of time series distribution shifts?\n\n4. To address **[W4]**, it is advised that the authors justify the significance of performance improvements to support their main claims in terms of experimental evaluation.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "FcDflsDoPN", "reviewer": "Reviewer_fMvx", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n1. The paper is clearly written and well-structured, making it easy to read and understand.\n2. The problem of test-time adaptation is a timely and important direction to explore.\n3. The proposed method is simple, general, and effective.\n4. The experimental results of the proposed method are competitive in their setup.\n\n### Weakness\n1. The paper does not follow the latest developments in this area. In fact, several recent works have explored test-time adaptation for online time series forecasting [1-4]. However, this paper fails to include these recent advances and only compares with baselines that were published several years ago.\n2. Due to the issue above, the paper still adopts the outdated online learning setup introduced by FSNet. This setup has been shown to be problematic in [4], as it overlooks the lagging nature of future horizons, which can result in label leakage during online training.\n3. As a result of the concerns above, the experimental results and conclusions are difficult to evaluate fairly. While such a paper might have been acceptable two years ago, it does not meet the expectations for timeliness and relevance in 2025.\n4. Some claims in the paper are made without sufficient evidence and come across as intuitive speculation rather than being supported by empirical or theoretical justification.\n\n[1] Calibration of Time-Series Forecasting: Detecting and Adapting Context-Driven Distribution Shift (KDD 2024)\n\n[2] Evolving Multi-Scale Normalization for Time Series Forecasting under Distribution Shifts (Openreview 2024)\n\n[3] Battling the Non-stationarity in Time Series Forecasting via Test-time Adaptation (AAAI 2025)\n\n[4] Proactive Model Adaptation Against Concept Drift for Online Time Series Forecasting (KDD 2025)", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper investigates the problem of test-time adaptation for time series forecasting. A simple and general method is proposed to select and augment similar historical samples for online batch learning, effectively reducing the noise introduced by distribution shift in the single test sample. Enhanced with a two-stream forecaster architecture, the proposed framework achieves competitive performance in the experimental results.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "No", "audience_explanation": "", "requested_changes": "1. **Please carefully review the related literature listed above**. Several recent works have addressed test-time or online adaptation in time series forecasting. A more comprehensive **literature review** is necessary, and **these papers should be included as baselines** in the experimental comparison to ensure fairness and relevance.\n\n2. **Revise the experimental setup to follow the protocol discussed in [4].** While an exact replication is not required, the current setup suffers from overlapping prediction horizons during training, which makes the evaluation unreliable and unrealistic. This issue has been pointed out in [4]. As it stands, only the results for prediction length = 1 are valid under this setting.\n\n3. Provide supporting evidence for key claims:\n\n   * **Naive online learning vs. batched online learning**:\n    The current claim that batch-based online learning improves performance over using a single sample per step is intuitive but lacks empirical support. Experiment results in [4] show that naive online learning with a single sample can sometimes outperform batch-based methods (Solid++). It would strengthen the paper to include a comparative analysis to verify this assumption.\n\n    * **Explanation of sudden loss increases in Figure 2:**\n    The paper attributes these spikes to significant distribution shifts in the data. However, this claim is currently speculative. Please include visualizations of the corresponding time series to substantiate this observation.\n\n4. Minor issues: \n\n     * The notations for ground truth and prediction in Section 3.3 appear to be swapped and are inconsistent with the definitions provided earlier in Section 3 \n    * Are the augmented samples use the original label for training? Additionally, how many augmented samples are generated per mini-batch?\n    * The proposed framework incorporates two forecasters.  But the impact of this design is not investigated in ablation study.\n     * The paper does not discuss the computational cost or training/inference time, which is an important consideration for online scenarios.", "broader_impact_concerns": "The paper does not include a Broader Impact Statement. However, given that the proposed method is general-purpose and does not directly raise ethical or societal concerns, I believe this omission is acceptable.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)