Title: AliO: Output Alignment Matters in Long-Term Time Series Forecasting
Venue: NeurIPS 2025
Accepted: yes
Keywords: Forecasting

Novel challenge
Abstract: Long-term Time Series Forecasting (LTSF) tasks, which leverage the current data sequence as input to predict the future sequence, have become increasingly crucial in real-world applications such as weather forecasting and planning of electricity consumption. However, state-of-the-art LTSF models often fail to achieve prediction output alignment for the same timestamps across lagged input sequences. Instead, these models exhibit low output alignment, resulting in fluctuation in prediction outputs for the same timestamps, undermining the model's reliability. To address this, we propose AliO (Align Outputs), a novel approach designed to improve the output alignment of LTSF models by reducing the discrepancies between prediction outputs for the same timestamps in both the time and frequency domains. To measure output alignment, we introduce a new metric, TAM (Time Alignment Metric), which quantifies the alignment between prediction outputs, whereas existing metrics such as MSE only capture the distance between prediction outputs and ground truths. Experimental results show that AliO effectively improves the output alignment, i.e., up to 58.2\% in TAM, while maintaining or enhancing the forecasting performance (up to 27.5\%). This improved output alignment increases the reliability of the LTSF models, making them more applicable in real-world scenarios. The code implementation is on an anonymous GitHub repository.

AliO: Output Alignment Matters in Long-Term Time Series Forecasting

Abstract

Long-term Time Series Forecasting (LTSF) tasks, which leverage the current data sequence as input to predict the future sequence, have become increasingly crucial in real-world applications such as weather forecasting and planning of electricity consumption. However, state-of-the-art LTSF models often fail to achieve prediction output alignment for the same timestamps across lagged input sequences. Instead, these models exhibit low output alignment, resulting in fluctuation in prediction outputs for the same timestamps, undermining the model's reliability. To address this, we propose AliO (Align Outputs), a novel approach designed to improve the output alignment of LTSF models by reducing the discrepancies between prediction outputs for the same timestamps in both the time and frequency domains. To measure output alignment, we introduce a new metric, TAM (Time Alignment Metric), which quantifies the alignment between prediction outputs, whereas existing metrics such as MSE only capture the distance between prediction outputs and ground truths. Experimental results show that AliO effectively improves the output alignment, i.e., up to 58.2% in TAM, while maintaining or enhancing the forecasting performance (up to 27.5%). This improved output alignment increases the reliability of the LTSF models, making them more applicable in real-world scenarios. The code implementation is on the GitHub repository 3 .

1 Introduction

The task of long-term time series forecasting (LTSF) is essential in various fields such as prediction of electricity demand [13], weather forecasting [37], health data [20], and so on. Recently, deep neural network models [40, 30, 33, 43] have shown strong performance in predicting long-term time series

based on historical information, which typically aim to minimize the error between the prediction output and the ground-truth sequence as a regression task. However, in real-world applications, merely minimizing the prediction error in LTSF is insufficient. It is equally important to ensure that the forecasting model generates consistent (aligned) prediction outputs for overlapping timestamps across lagged input sequences. For example, Fig. 1 illustrates a scenario in which a trained forecasting model predicts future electricity usage in two instances: (1) for the period from April to November (Prediction 1, purple), using input data from January to March, and (2) for the period from May to December (Prediction 2, yellow), using input data from February to April, where the forecasting periods overlap between May and November. A reliable model should provide consistent predictions for these overlapping months, regardless of the partially differing input sequences. If the model produces inconsistent predictions on the electricity usage for the same timestamps (i.e., May to November) between two input sequences, it could result in significant time and financial costs for rescheduling budget allocation and undermine the reliability of the predictions.

We refer to this phenomenon as the output alignment problem, which has not been adequately acknowledged and addressed by existing LTSF studies [25, 7, 18], despite its significance and substantial impact on real-world applications. To the best of our knowledge, state-of-the-art LTSF models [37, 33, 30] often fail to maintain the prediction output consistency, and none of existing works has explicitly recognized or attempted to address this inconsistency in LTSF tasks.

In this paper, we present AliO (Align Outputs) , a novel method designed to enhance output alignment in LTSF models. For the first time, AliO enables LTSF models to produce consistent predictions for overlapping timestamps across lagged input sequences. By aligning predictions for overlapping timestamps through the minimization of discrepancies in both time and frequency domains, AliO improves output consistency even when input sequences lagged. AliO achieves this by directing the model's predictions toward the ground-truth sequences while simultaneously minimizing discrepancies across multiple predictions obtained from a set of lagged input sequences. It allows AliO to integrate seamlessly with the model's forecasting objectives, such as regression loss (e.g., MSE), without adding implementation complexity or requiring modifications to the model. As a result, AliO enhances the reliability of forecasts by improving output alignment (consistency), while maintaining or even improving overall forecasting performance.

This represents a significant advancement over existing methods [37, 40] that focus solely on minimizing the forecasting objective without considering prediction output alignment.

To quantify output alignment, which represents the consistency of a model's predictions across lagged input sequences, we propose a new metric, Time Alignment Metric (TAM). TAM quantitatively assesses the model's output alignment by measuring discrepancies between predictions for overlapping timestamps for multiple input sequences. To the best of our knowledge, the proposed TAM is the first metric designed to measure the output alignment.

We experiment with AliO on representative LTSF tasks, including ETT{h1, h2, m1, m2}, Electricity (ECL), Traffic, Weather, and ILI dataset [37], using various state-of-the-art LTSF models such as CycleNet [27], GPT4S [43], iTransformer [30], PatchTST [33], TimesNet [36], DLinear [40] and Autoformer [37]. The evaluation results demonstrate that AliO effectively aligns predictions over overlapping timestamps, i.e., improving TAM up to 58.2%, while maintaining or enhancing

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forecasting regression accuracy up to 17.4% measured in MSE. The detailed descriptions on additional datasets and experimental configurations are provided in Secs. G to I.

2 Motivation

A system with low prediction consistency leads to user distrust and increased costs. For instance, studies on the consistency of weather forecasts have reported cases where consistent forecasts lead to greater trust in the system, whereas inconsistent forecasts result in user distrust [5, 31, 4]. Furthermore, accroding to research in non-weather domains, consumers tend to perceive consistency among multiple estimates from the same system as a signal of skill [11, 5, 3]. The inconsistency in predictions necessitates replanning, which consumes unnecessary organizational time. The resulting resource reallocation leads to the investment of more resources, contrary toe th original goal of Rolling forecasting, which aims for resource savings [16, 14]. This can be described as an additional sunk cost [35], as frequent fluctuations in predictions can induce irrational decision-making and result in lost opportunity costs [35, 6].

While modern Long-Term Times Series Forecasting (LTSF) models demonstrate high accuracy, research on the aforementioned consistency has been insufficient. Consequently, we examined the consistency of existing models through rolling forecasting on representative LTSF datasets. The left panel of Fig. 2 shows the prediction results of DLinear [37] on the ILI dataset [37] using only regression loss (MSE), revealing a significant lack of consistency. In contrast, the right panel, where AliO is applied, shows that consistency is improved while accuracy is maintained. As can be seen in Sec. I, this is a phenomenon observed across other models and datasets.

Although both consistency and accuracy are crucial for addressing users' psychological and economic concerns, our experiments revealed that while modern LTSF models achieve high accuracy, they fall short in terms of consistency. AliO, as a loss function used in conjunction with regression loss, aims to achieve two goals: maintaining the high accruacy of existing LTSF models while improving their prediction consistency. Furthermore, to quantify consistency, we propose the Time Alignment Metric (TAM). Regression metrics such as MSE, MAE, and DTW [34] represent accuracy, which is the distance between predictions and ground truth. Therefore, they have limitations in measuring consistency, which is a measure of performance between predictions. Complementing this, TAM measures the distance between predictions, serving as an indicator that represents consistency.

Through Sec. 5 and Sec. I, we show that AliO successfully achieves the dual goals of maintaining/improveing the high accuracy of existing models and enhancing consistency. Additionally, in ?? , we experimentally verify the relationship between the inconsistency value (TAM) of existing models and the MSE improvement in models trained with AliO. To the best our knowledge, this is the first study to address the improvemnet of consistency in LTSF.

3 Time alignment metric (TAM)

We first define the concept of output alignment and then introduce Time Alignment Metric (TAM), which enables a quantitative assessment of output alignment.

Definition 3.1. ( Output Alignment ) Output alignment refers to the property wherein the prediction outputs of a model, derived from a set of lagged input sequences, exhibit consistent alignment, characterized by uniform patterns across overlapping timestamps in the prediction outputs.

For instance, as illustrated in Fig. 2, the six prediction outputs for the overlapping timestamps (i.e., from 25 to 43) in the left figure demonstrate a lower degree of output alignment. Conversely, the six prediction outputs in the right figure exhibit similar and consistent patterns, demonstrating a high degree of output alignment. To quantify the output alignment (Theorem 3.1) in model predictions, we propose Time Alignment Metric (TAM). We begin by defining the necessary concepts, including lagged input and output sequences, and overlapping output sequences, as provided below.

Definition 3.2. (Input and Output Sequence) Given a time-series sequence X ∈ R c × d , where c is the input channel, and d is the length of the sequence, respectively, the n -th input sequence X n ∈ R c × d ′ of the length d ′ ≤ d is defined as the segment X s : s + d ′ -1 , where s is the starting timestamp, and s + d ′ -1 is the ending timestamp.

Then, taking X n ∈ R c × d ′ as input, f ( X n ; θ ) = Y n ∈ R c ′ × h is defined as the prediction output sequence of the length h provided by the forecasting model f , where c ′ is the output channel, and θ is the model parameter. Having the input sequence X n and the corresponding output sequence Y n of the model f , the lagged input and output sequences, X n +1 and Y n +1 , are derived by shifting their timestamps by the lag parameter l , as described below.

Definition 3.3. (Lagged Input and Output Sequence) Given the n -th input sequence X n ∈ R c × d ′ of the segment X s : s + d ′ -1 , the sequence X n +1 ∈ R c × d ′ of the segment X s + l : s + d ′ + l -1 is defined as the lagged input sequence of X n with l being the lag parameter.

Then, given the lagged input sequence X n +1 ∈ R c × d ′ , the prediction output of the length h , f ( X n +1 ; θ ) = Y n +1 ∈ R c ′ × h , is defined as the model f 's lagged output sequence . From Y n and Y n +1 , the overlapping output sequences P n and P n +1 are derived to compute TAM, as follows.

Definition 3.4. (Overlapping Output Sequence) Given two output sequences Y n ∈ R c ′ × h and Y n +1 ∈ R c ′ × h of the model f generated from X n and X n +1 as input, respectively, P n ∈ R c ′ × h ′ and P n +1 ∈ R c ′ × h ′ are defined as the overlapping output sequences between Y n and Y n +1 , whose segments are given by Y n 1+ l : h and Y n +1 1: h -l , respectively, where h ′ = h -l is the overlapping length.

From these, Time Alignment Metric (TAM) is defined to quantify output alignment by evaluating the consistency between overlapping output sequences P n and P n +1 , as follows.

Definition 3.5. (TAM; Time Alignment Metric) Given N overlapping output sequences P n ∈ R c ′ × h ′ for n = 1 , 2 , . . . , N , where the input and output sequences, X n +1 and Y n +1 , are offset from X n and Y n by the lag l , TAM N ( Time Alignment Metric ) is calculated as the average of the MAE values between all pairs of { P n , P m } , which is given by:

TAM offers a distinct advantage over traditional forecasting regression metrics, such as the MSE or MAE between model outputs and ground truth, which only measure forecasting performance and fail to capture prediction fluctuations across different predictions. Unlike these metrics, TAM evaluates the distances between overlapping predictions, allowing for an assessment of how predictions evolve over time. For instance, regression metrics do not determine whether overlapping predictions remain consistent or exhibit smooth transitions, making it challenging to assess data-model robustness. In contrast, TAM computes the distance between overlapping predictions, providing a robust measure of data-model consistency. A lower TAM value reflects improved alignment and data-model robustness.

4 Improving output alignment (AliO)

To enhance the output alignment of LTSF models, we propose AliO (Align Outputs), which minimizes the discrepancy between overlapping predictions (Theorem 3.4). AliO is designed to simultaneously

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achieve two key objectives: (1) improving output alignment by reducing discrepancies between the overlapping predictions, and (2) maintaining the model's forecasting regression performance by aligning the model's prediction outputs with the ground truths. The two objectives are achieved through the technique that we call regression pulling with the stop-gradient operation, which ensures that the forecasting regression loss remains unaffected while improving output alignment simultaneously.

Given two overlapping predictions, P n = f ( X n ; θ ) 1+ l : h and P n +1 = f ( X n +1 ; θ ) 1: h -l defined in Theorem 3.4, AliO aligns these predictions such that P n ≃ P n +1 for n = 1 , 2 , . . . , N by minimizing the following objective with respect to the model f 's parameter θ , as:

where D denotes a distance function and subscripts denote the segment ranges of the model f 's prediction output sequences, i.e., 1 + l : h and 1 : h -l correspond to the same timestamps for X n and X n +1 , respectively. Eq. (2) can be easily extended to handle non-consecutive overlapping output sequences, enabling its application to TAM (Theorem 3.5). AliO aligns the overlapping predictions P n and P n +1 in both time and frequency domains, as illustrated in Fig. 3 with further details provided in the following subsections. The algorithmic procedure of AliO is summarized in Alg. 1.

4.1 Time domain alignment

To enhance the temporal alignment, AliO aligns the overlapping predictions, P n T and P m T for n, m = 1 , 2 , . . . , N , in the time domain, where the subscript T denotes the time domain. This alignment can be achieved using a distance function, D T , such as MSE or Dynamic Time Warping (DTW) [34]. The computed distance is back-propagated to encourage the model f to produce overlapping predictions that align with each other. However, solely aligning the overlapping predictions by minimizing the time domain alignment loss L T = D T ( P n T , P m T ) may lead to both prediction outputs deviating from the ground truth, degrading the forecasting regression performance of the model f .

4.2 Regression pulling (RegPull)

To maintain forecasting regression performance while aligning predictions, we propose Regression Pulling (RegPull) , which identifies which prediction output points of P n T and P m T are relatively further from the ground truth than the other at the same timestamp, indicated by the index variable idx T , as shown on line (1) in Alg. 1. By applying the stop-gradient operation, denoted as sg ( · ) , to the prediction output points farther from the ground truth, the time domain alignment loss L T pulls these distant points closer to the ground truth, aligning its optimization direction with that of the forecasting regression loss and reinforcing to minimize the regression loss. Consequently, regression pulling

Algorithm 1 The procedure of AliO. ⊙ denotes element-wise multiplication. FFT ( · , · , · ) returns the frequency domain representation of each signal sequences, sg ( · ) is stop-gradient operator.

Input: The number of predictions N , consecutive predictions Y n with at time lag of l , and their ground truth ˆ Y n , where n ∈ [1 , N ] , and distance function D T and D F for the time and frequency domain, respectively. The subscript T and F denotes the time and frequency domain, respectively. Output: The time domain alignment loss L T and frequency domain alignment loss L F . Initialize L t = 0 , L f = 0 , count = 0 for n = 1 to N -1 do for m = n +1 to N do gap = | m -n | × l ( P n T , P m T , GT T ) = ( Y n gap : , Y m : -gap , ˆ Y n gap : ) idx T = Index ( | P n T -GT T | > | P m T -GT T | ) (1) L t = L t + D T ( P n T,i ∈ idx T , sg ( P m T,i ∈ idx T )) + D T ( sg ( P n T,i/ ∈ idx T ) , P m T,i/ ∈ idx T ) [RegPull] (2) ( P n F , P m F , GT F ) = FFT ( P n T , P m T , GT T ) (3) idx F = Index ( | P n F -GT F | > | P m F -GT F | ) (4) L f = L f + D F ( P n F,i ∈ idx F , sg ( P m F,i ∈ idx F )) + D F ( sg ( P n F,i/ ∈ idx F ) , P m F,i/ ∈ idx F ) [RegPull] (5) count = count +1 end for end for ( L T , L F ) = ( L t /count, L f /count ) Return: L T and L F

effectively reduces the misalignment between prediction outputs with the forecasting regression performance being unaffected, inducing overall predictions closely aligned with the ground truths.

For each prediction output point i ∈ [1 , h ′ ] in Alg. 1, D T ( P n T,i ∈ idx T , sg ( P m T,i ∈ idx T )) on line (2) encourages the prediction P n T to move closer to the prediction P m T . On the other hand, D T ( sg ( P n T,i/ ∈ idx T ) , P m T,i/ ∈ idx T ) on the same line (2) promotes the alignment of P m T towards P n T . The direction of alignment is determined by the index variable, idx T . These two distance values are combined into the time domain alignment loss L T , which is expressed as:

Consequently, Eq. (3) aligns the direction of the time domain alignment loss L T with the forecasting regression loss while improving output alignment. The regression pulling can also be applied in the same way to the frequency domain; Fig. 3 shows a visual illustration of output alignment in the time (left) and frequency domain (right) using regression pulling (red cross marks).

4.3 Frequency domain alignment

In addition to the time domain alignment, we propose frequency domain alignment as a complementary approach that supports and enhances time-domain alignment by aligning overlapping predictions in the frequency representation. It promotes the alignment of both the phase and the amplitude components, improving overall consistency in the time domain.

As shown in Alg. 1, the time domain overlapping predictions, P n T and P m T , along with the ground truths GT T , are first transformed into frequency domain representations, resulting in P n F , P m F , and GT F on line (3), where the subscript F denotes the frequency domain. Subsequently, the index variable idx F is determined by identifying output points further from GT F between P n F and P m F on line (4). The distance function D F is then applied with regression pulling to facilitate alignment in the frequency domain on lines (5) in the same manner to the time domain. From this, the frequency domain alignment loss L F is obtained as:

We use MSE (mean squared error) as the main distance function for the frequency domain alignment, i.e., D F = ∥ P n F -P m F ∥ 2 2 /h ′ . The following Theorem 4.1 demonstrates that applying MSE in the frequency domain facilitates phase alignment between prediction outputs.

Theorem 4.1. Given two frequency domain prediction vectors, p n F and p m F in C h ′ , where C means the set of complex numbers, minimizing their MSE, ∥ p n F -p m F ∥ 2 2 , results in a reduction of the difference

in the phase components, ∠ p n F and ∠ p m F , as shown below (the proof is provided in Sec. B.1):

Fig. 3 (Right) shows a polar-coordinate depiction of frequency alignment conducted through Eq. (5) in the frequency domain. Theorem 4.2 shows that minimizing MSE, i.e., D F = ∥ P n F -P m F ∥ 2 2 , also reduces the differences in amplitude once the phase components ∠ P n F and ∠ P m F have been aligned.

Theorem 4.2. Once the phases of two frequency domain prediction vectors, p n F and p m F in C h ′ , are aligned, i.e., ∠ p n F ≃ ∠ p m F , the minimization of MSE between the two prediction outputs, ∥ p n F -p m F ∥ 2 2 , leads to a reduction in the amplitude difference, as shown below (the proof is provided in Sec. B.2):

4.4 AliO loss

By combining the time L T in Eq. (3) and frequency domain alignment loss L F in Eq. (4), we derive the AliO loss, L AliO , which is incorporated with the forecasting regression loss, L reg as shown below, where λ T and λ F control the extent of alignment in the time and frequency domain, respectively.

5 Experiment

We evaluate AliO on representative LTSF models and datasets. In model training, MSE is employed as the regression loss. The model performance is assessed using both MSE and TAM N for output alignment evaluation, with N = 2 . To reproduce experimental results of baselines and ensure a fair evaluation, we follow the same hyper-parameters presented in each model papers [37, 27, 40, 43, 30, 33, 36]. For AliO, we set N = 2 and l = 1 , covering a wide prediction range. The search space for λ T and λ F are { 1 . 0 , 2 . 0 , 5 . 0 } and { 0 . 0 , 0 . 5 , 1 . 0 , 2 . 0 } , respectively, and we report the best results in the main results. All results are reported as the average performance across all prediction lengths. Detailed descriptions of the experiment and results are provided in Secs. G to I.

Models. We apply AliO to various LTSF architectures, experimenting it with a diverse range of LTSF approaches, categorized into four groups: (1) Transformer-based models, i.e., Autoformer [37], PatchTST [33], and iTransformer [30], (2) Linear-based models, i.e., DLinear [40] and CycleNet [27], (3) CNN-based models, i.e., TimesNet [36], and (4) LLM-based models, i.e., GPT4TS [43].

Datasets. On the main text of the paper, we report results on representative LTSF datasets, i.e., Electricity (ECL) [13], ETT {h1, h2, m1, and m2} [42], Traffic, [37], Weather [37], and NationalIllness (ILI) dataset [37]. The experiments results on full datasets can be found in Sec. I.

Context Length Following the official settings for each model, we set the context length to 336 for PatchTST and DLinear, and 96 for the other models. For the ILI dataset specifically, the context lengths were set to 104 for PatchTST and DLinear, and 36 for TimesNet.

Prediction Length Furthermore, as per the official settings, the prediction lengths were set to { 96 , 192 , 336 , 720 } for most datasets. The exceptions were the PEMS-related datasets with { 12 , 24 , 48 , 96 } , the ILI dataset with { 24 , 36 , 48 , 60 } , and the Autoformer model on the ETT datasets, for which we used { 24 , 48 , 168 , 336 , 720 } .

5.1 Output alignment and forecasting performance

Tab. 1 summarizes the performance of LTSF models on seven datasets (excluding ILI) for multivariate LTSF tasks, comparing baselines (without AliO) and AliO-integrated models in terms of MSE and TAM. As shown in the table, integrating AliO with the regression loss substantially improves alignment performance (TAM), achieving gains of up to 70.5% for CycleNet, 45.8% for GPT4TS, 45.8% for iTransformer, 36.8% for PatchTST, 45.6% for TimesNet, 64.0% for DLinear, and 39.0% for Autoformer. Simultaneously, forecasting performance (MSE) also improves by up to 11.5%. As summarized in Tab. 2, for the ILI dataset exhibiting insufficient initial output alignment, AliO achieves up to a 17.4% higher MSE improvement compared to other benchmarks.

5.2 How AliO maintains or improves regression performance

Tab. 2 shows AliO's effectiveness in improving both forecasting accuracy and alignment performance on the ILI dataset [37], which exhibits poor output alignment in the absence of AliO. The proposed regression pulling enhances model training by analyzing prediction distances at each timestamp, adaptively strengthening the effect of regression loss to align predictions with each other and ground truth. On ILI, which exhibits poor initial alignment, AliO achieves 17.4% MSE and 58.1% TAM improvement. Conversely, datasets with better initial alignment, presented in Tab. 1, achieve comparatively smaller MSE improvements. Figs. 4 and 13 shows this trend; higher initial TAM (poor alignment) correlates with greater MSE gains, demonstrating AliO's efficacy in suboptimal alignment conditions.

Figs. 2 and 5 visualize DLinear predictions on the ILI dataset. The left figures (without AliO) exhibit prediction shifts [18] in both time (Fig. 2) and frequency (Fig. 5) domains, reflected in high TAM values. In contrast, the right figures (with AliO) show improved alignment in both domains, which reduces TAM value and enhances regression performance (MSE).

5.3 Hyper-parameter Analysis

Time and Frequency Domain Coefficients. AliO utilizes the two coefficients, λ T and λ F in Eq. (7), to balance the regression performance and output alignment by adjusting the alignment strengths over the time and frequency domains, respectively. To evaluate AliO with respect to these two coefficients, we provide Fig. 6, which presents heatmaps of the normalized MSE and TAM for various coefficient values. The x-axis represents λ F in { 0 . 0 , 0 . 5 , 1 . 0 , 2 . 0 } , and the y-axis corresponds to λ T in { 1 . 0 , 2 . 0 , 5 . 0 } . The left heatmap shows normalized MSE, where the maximum MSE in each dataset is scaled to 1.0 and the minimum to 0, followed by averaging across all models. The right heatmap presents normalized TAM in the same manner. Additional results are in Sec. F.

As AliO is designed to enhance alignment, a clear trend is observed in TAM where increasing the time domain alignment coefficient λ T from 1.0 to 5.0 leads to a consistent improvement. Interestingly, the normalized MSE also shows a general improvement as λ T increases, suggesting that the regression pulling effectively aligns the directions of the AliO loss with those of the regression loss. The influence of the frequency domain alignment coefficient λ F is less pronounced compared to the time

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coefficient λ T . However, in cases where λ T alone cannot improve performance, λ F shows its impact on both the MSE and TAM metrics, as exemplified by values of 0 . 28 and 0 . 50 , respectively. This demonstrates the significance of the frequency domain alignment to overcome limitations in the time domain alignment, indicating that both phase and amplitude contribute to performance enhancements.

Number of Overlapping Output Sequences and Lag Size. We investigate how the model's MSE and TAM vary with different numbers of overlapping output sequences N and the lag l . Fig. 7 shows the normalized MSE and TAM with a range between 0 and 1.0. TAM is improved as the lag decreases, which can be attributed to an increased length of overlapping timestamps at smaller lag value. This allows AliO to align longer overlapping sequence effectively. Additionally, the model performance improves as N increases. As shown in Alg. 1, a higher N enables AliO to align predictions from distant timestamps, significantly enhancing TAM. In contrast, no clear trend is observed for MSE, since the window size increases as both N and l grow. Prior studies [12, 2, 17, 32] suggest that large window sizes can cause overfitting, whereas an optimal window size leads to better performance. As shown in Fig. 7, when both N and l are large, MSE tends to increase, indicating degraded performance. Conversely, lower MSE values ( 0 . 29 , 0 . 37 , 0 . 20 ) are observed along the diagonal elements, representing proper window sizes, such as ( l, N ) = (1 , 8) , (2 , 4) , (8 , 2) , which indicates that balanced combinations of l and N yield better performance.

6 Related works

To the best of our knowledge, no prior work has adequately identified the output alignment problem and provided a solution for it, as AliO proposed in this paper.

Data-Model Robustness. While existing research [30] primarily focuses on model initialization robustness, i.e., the consistency of model performance across randomly initialized weights [30], the output alignment emphasizes data-model robustness. This aspect underscores the model's ability

to produce consistent prediction outputs for lagged input sequences. The data-model robustness is equally critical for practical applications as discussed in this paper and can be quantified using the proposed TAM, which can be enhanced through the proposed time and frequency domain alignment. We anticipate this aspect of LTSF study will inspire further active research in the field.

Contrastive Learning. Contrastive learning [41, 8, 1, 9, 38, 22] employ pretraining strategies that encourage augmented positive pairs to be closely aligned in the representation space. However, they differ from AliO in several key aspects. First, they operate in the representation space rather than in the output space where AliO functions. Second, they minimize the distance between whole vectors rather than focusing on overlapping timestamps. Third, they do not incorporate regression-aware algorithms like regression pulling. These differences highlight the distinctive approach of AliO in output alignment, operating independently of existing methodes and can thus be integrated with them.

7 Limitations and discussions

On Volatile Datasets. TAM is a metric designed to assess the output alignment of LTSF models. However, its application should be approached with caution on highly volatile datasets that fall outside the typical scope of LTSF tasks, as model predictions may experience considerable volatility. Nevertheless, as demonstrated in Sec. I, the results on the Exchange Rate dataset [37], which is influenced by abrupt external shocks, suggest that TAM retains potential applicability even in volatile settings. Designing more robust metrics for inherently volatile data is our next research objective.

Distance Functions. AliO improves regression performance under high initial TAM conditions (a low degree of output alignment). This is likely attributable to its use of MSE as the distance function, which is consistent with the regression loss employed during model training. To further enhance performance, alternative distance functions such as [25, 29, 15, 7], which capture different characteristics of time series data (e.g., shape), may serve as promising directions for future exploration.

8 Conclusion

In this paper, we investigate the output alignment problem, which can commonly arise in LTSF tasks and propose Time Alignment Metric (TAM) as a quantitative measure for this problem. To solve the output alignment problem, we propose AliO (Output Alignment) applied in both time and frequency domains. AliO achieves up to 27.5% improvement in MSE across various LTSF models and datasets, and up to 58.2% improvement in TAM, effectively addressing the output alignment problem.

Acknowledgment

This work was supported by the Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT)(No.RS-202400508465) and Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (No.RS-2020-II201336, Artificial Intelligence Graduate School Program(UNIST)).

References

R. Akerkar and P. Sajja. Knowledge-based systems. Jones & Bartlett Publishers, 2009.

A. Azlan, Y. Yusof, and M. F. M. Mohsin. Determining the impact of window length on time series forecasting using deep learning. International Journal of Advanced Computer Research, 9(44):260-267, 2019.

D. V. Budescu, A. K. Rantilla, H.-T. Yu, and T. M. Karelitz. The effects of asymmetry among advisors on the aggregation of their opinions. Organizational Behavior and Human Decision Processes, 90(1):178-194, 2003.

J. Burgeno and S. Joslyn. The impact of weather forecast inconsistency on user trust. wea. climate soc., 12, 679-694, 2020.

J. N. Burgeno and S. L. Joslyn. The impact of forecast inconsistency and probabilistic forecasts on users' trust and decision-making. Weather, Climate, and Society, 15(3):693-709, 2023.

S.-H. Chung and K.-C. Cheng. How does cognitive dissonance influence the sunk cost effect? Psychology research and behavior management, pages 37-45, 2018.

M. Cuturi and M. Blondel. Soft-dtw: a differentiable loss function for time-series. In International conference on machine learning, pages 894-903. PMLR, 2017.

E. Eldele, M. Ragab, Z. Chen, M. Wu, C. K. Kwoh, X. Li, and C. Guan. Time-series representation learning via temporal and contextual contrasting. arXiv preprint arXiv:2106.14112, 2021.

E. Eldele, M. Ragab, Z. Chen, M. Wu, C.-K. Kwoh, X. Li, and C. Guan. Self-supervised contrastive representation learning for semi-supervised time-series classification. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(12):15604-15618, 2023.

L. Euler. Introductio in analysin infinitorum, volume 2. MM Bousquet, 1748.

A. Falk and F. Zimmermann. Consistency as a signal of skills. Management Science, 63(7):21972210, 2017.

J. D. Freitas, C. Ponte, R. Bomfim, and C. Caminha. The impact of window size on univariate time series forecasting using machine learning. In Symposium on Knowledge Discovery, Mining and Learning (KDMiLe), pages 65-72. SBC, 2023.

G. Hebrail and A. Berard. Individual Household Electric Power Consumption. UCI Machine Learning Repository, 2006. DOI: https://doi.org/10.24432/C58K54.

T. Henttu-Aho. The role of rolling forecasting in budgetary control systems: reactive and proactive types of planning. Journal of management control, 29(3):327-360, 2018.

P. J. Huber. Robust estimation of a location parameter. In Breakthroughs in statistics: Methodology and distribution, pages 492-518. Springer, 1992.

O. Huhtakangas. Rolling forecasting in budgetary systems: case study. 2020.

A. Iqbal, A. Shahid, M. Roman, M. T. Afzal, and U. u. Hassan. Optimising window size of semantic of classification model for identification of in-text citations based on context and intent. PloS one, 20(3):e0309862, 2025.

S. Y. Jhin, S. Kim, and N. Park. Addressing prediction delays in time series forecasting: A continuous gru approach with derivative regularization. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pages 1234-1245, 2024.

X. Jin, Y . Park, D. Maddix, H. Wang, and Y . Wang. Domain adaptation for time series forecasting via attention sharing. In International Conference on Machine Learning, pages 10280-10297. PMLR, 2022.

H. Khadem, H. Nemat, J. Elliott, and M. Benaissa. Blood glucose level time series forecasting: nested deep ensemble learning lag fusion. Bioengineering, 10(4):487, 2023.

H. Kim, S. Kim, J. Mok, and S. Yoon. Battling the non-stationarity in time series forecasting via test-time adaptation. arXiv preprint arXiv:2501.04970, 2025.

J. Kim, K. Park, S. Yun, and S. Lee. Ppt: Patch order do matters in time series pretext task. In The Thirteenth International Conference on Learning Representations.

D. P. Kingma and J. Ba. Adam: A method for stochastic optimization, 2017.

G. Lai, W.-C. Chang, Y. Yang, and H. Liu. Modeling long-and short-term temporal patterns with deep neural networks. In The 41st international ACM SIGIR conference on research & development in information retrieval, pages 95-104, 2018.

V. Le Guen and N. Thome. Shape and time distortion loss for training deep time series forecasting models. Advances in neural information processing systems, 32, 2019.

L. Liebel and M. Körner. Auxiliary tasks in multi-task learning. arXiv preprint arXiv:1805.06334, 2018.

S. Lin, W. Lin, X. Hu, W. Wu, R. Mo, and H. Zhong. Cyclenet: enhancing time series forecasting through modeling periodic patterns. Advances in Neural Information Processing Systems, 37:106315-106345, 2024.

M. Liu, A. Zeng, M. Chen, Z. Xu, Q. Lai, L. Ma, and Q. Xu. Scinet: Time series modeling and forecasting with sample convolution and interaction. Advances in Neural Information Processing Systems, 35:5816-5828, 2022.

X. Liu, N. Li, and S. Xia. Gdtw: A novel differentiable dtw loss for time series tasks. ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 2860-2864, 2021.

Y. Liu, T. Hu, H. Zhang, H. Wu, S. Wang, L. Ma, and M. Long. itransformer: Inverted transformers are effective for time series forecasting. arXiv preprint arXiv:2310.06625, 2023.

J. E. Losee and S. Joslyn. The need to trust: How features of the forecasted weather influence forecast trust. International journal of disaster risk reduction, 30:95-104, 2018.

C. Ma, Y. Hou, X. Li, Y. Sun, and H. Yu. Long input sequence network for long time series forecasting. arXiv preprint arXiv:2407.15869, 2024.

Y. Nie, N. H. Nguyen, P. Sinthong, and J. Kalagnanam. A time series is worth 64 words: Long-term forecasting with transformers. arXiv preprint arXiv:2211.14730, 2022.

H. Sakoe and S. Chiba. Dynamic programming algorithm optimization for spoken word recognition. IEEE transactions on acoustics, speech, and signal processing, 26(1):43-49, 1978.

J. Wang, B. Zhang, S. Liang, and J. Li. Sunk cost effects hinge on the neural recalibration of reference points in mental accounting. Progress in Neurobiology, 208:102178, 2022.

H. Wu, T. Hu, Y. Liu, H. Zhou, J. Wang, and M. Long. Timesnet: Temporal 2d-variation modeling for general time series analysis. arXiv preprint arXiv:2210.02186, 2022.

H. Wu, J. Xu, J. Wang, and M. Long. Autoformer: Decomposition transformers with autocorrelation for long-term series forecasting. Advances in neural information processing systems, 34:22419-22430, 2021.

Z. Yue, Y. Wang, J. Duan, T. Yang, C. Huang, Y. Tong, and B. Xu. Ts2vec: Towards universal representation of time series, 2022.

Z. Zamanzadeh Darban, G. I. Webb, S. Pan, C. Aggarwal, and M. Salehi. Deep learning for time series anomaly detection: A survey. ACM Computing Surveys, 57(1):1-42, 2024.

A. Zeng, M. Chen, L. Zhang, and Q. Xu. Are transformers effective for time series forecasting? In Proceedings of the AAAI conference on artificial intelligence, volume 37, pages 1112111128, 2023.

X. Zhang, Z. Zhao, T. Tsiligkaridis, and M. Zitnik. Self-supervised contrastive pre-training for time series via time-frequency consistency. Advances in neural information processing systems, 35:3988-4003, 2022.

H. Zhou, S. Zhang, J. Peng, S. Zhang, J. Li, H. Xiong, and W. Zhang. Informer: Beyond efficient transformer for long sequence time-series forecasting. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pages 11106-11115, 2021.

T. Zhou, P. Niu, L. Sun, R. Jin, et al. One fits all: Power general time series analysis by pretrained lm. Advances in neural information processing systems, 36:43322-43355, 2023.

A Exemplar illustrations on importance of output alignment

Budget

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B Proof

B.1 Proof of Theorem 4.1

Proof. Rewriting the equation on the left side of the theorem, we can express it form the perspective of individual elements:

The squared magnitude of the difference between two complex numbers p 1 F,i and p 2 F,i are given as follows:

Where p 1 F,i and p 2 F,i are the complex conjugate of p 1 F,i and p 2 F,i . Now, expressing p 1 F,i and p 2 F,i including the complex conjugate in polar form:

where j = √ -1 and θ is the angle (phase) for the corresponding complex number. We substitute into the expression and simplify the expression:

Using Euler's formula e jθ = cos ( θ ) + jsin ( θ ) [10], we can convert the exponential expressions to trigonometric expressions:

Utilizing the angle (phase) notation ∠ , we can express the equation as:

Reducing the final equation implies increasing the cosine term, which in turn signifies aligning the two phases, since one of predictions is constant by stop-gradient sg ( · ) operator in Alg. 1.

B.2 Proof of Theorem 4.2

Proof. Since the two phases, i.e., ∠ p 1 F,i and ∠ p 2 F,i , are aligned, the Eq. (17) in Sec. B.1 is modified as follows:

By factoring, it can be expressed in the following perfect square form.

Since one of predictions is constant by stop-gradient sg ( · ) operator in Alg. 1, minimizing | p 1 F,i -p 2 F,i | 2 leads to a reduction in the amplitude difference

C Visualization of alignment

In this section, we visualize the model's prediction outputs and alignment differences when trained solely with regression loss versus with both regression loss and AliO. Each figure is arranged in a 2×2 grid: the top row shows results using regression loss alone, while the bottom row presents results incorporating AliO; the left column illustrates the time domain, and the right column displays the frequency domain visualized in a polar coordinate system, where the azimuth angle represents the phase and the radial distance from the origin corresponds to the amplitude. Overlapping timestamps of prediction outputs for five lagged inputs are shown, with the ground truth indicated by black dashed lines. Figs. 9 to 12 show the results from DLinear [40], PatchTST [33], TimesNet [36], and iTransformer [30].

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D On sudden event

This section expands on the sudden events mentioned in Sec. 7.

Definition D.1. The sudden event refers to an anomalous data point in the input sequence that can be interpreted as out-of-distribution (OOD), causing perturbations in the output sequence. Such event occur when input-level perturbations lead to unpredictable output patterns.

These scenarios deviate from the core objective of Long-term Time Series Forecasting (LTSF), which focuses on predictable sequences. Instead, they align more closely with:

Domain Adaptation [19]

Anomaly Detection [39]

Test-Time Adaptation [21]

The TAM metric is specifically designed for LTSF tasks with predictable, stable output sequences (i.e., non-perturbed scenarios). In contexts involving sudden events or OOD (out-of-distribution) data, the inherent assumptions of TAM-particularly its reliance on sequence stability-may not hold, necessitating cautious application. However, if a sudden event includes precursor signals (i.e., detectable input patterns) and the model can sufficiently predict the output sequence using this information, TAM remains valid even under such conditions.

E Initial TAM vs. MSE improvement

Fig. 4 may be difficult to interpret as there is no explicit distinction between models and datasets. In this section [Fig. 13], to facilitate easier interpretation, models are represented by different shapes and datasets are represented by different colors.

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F Zero coefficient of time domain alignment ( λ T )

By default, AliO operates in the time domain, so scenarios where the time-scaling coefficient is zero were excluded from the main context. However, we conducted experiments on CycleNet [27], iTransformer [30], TimesNet [36] under conditions with a zero time-scaling coefficient. The results, shown in Figs. 14 to 25, are normalized between 0 and 1. Since the condition ( λ T = 0 , λ F = 0 ) implies that AliO is not used, so it is empty. As demonstrated in Fig. 6 increasing λ T concsistently improves TAM performance. When it comes to MSE performance, there's a general trend for it to improve as the coefficient increases, similar to what's seen with TAM (I'm assuming this refers to a specific model or method you're using). However, in some environments, the coefficient can become excessively large, dominating the regression loss and causing performance to degrade. Despite this, our experiments confirmed that an appropriate, non-zero coefficient leads to improved performance. The tables show the results where the lag is 1 and the number of sequences is 2 (default).

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0.267

0.232

0.285

5

0.001

0.000

0.001

0.008

0.0

0.2

0.4

0.6

0.8

1.0

F

F

G Dataset explanation

We provide a brief description of each dataset used in our experiments, as referenced in Secs. 5 and I.

Electricity transformer temperature (ETT) [42]: This dataset consists of two years of data collected from two counties in China. It is divided into four subsets: ETTh1 and ETTh2 (sampled hourly), and ETTm1 and ETTm2 (sampled every 15 minutes). Each record contains six power load features and one target variable representing the oil temperature.

Exchange [37]: Contains daily exchange rates from eight countries, spanning from 1990 to 2016.

Electricity (ECL) [13]: Comprises electric power consumption data sampled every minute over four years for a single household.

0.0

0.5

1.0

2.0

Frequency Coefficient (λ

F

)

0

0.332

0.351

0.329

1

0.028

0.006

0.351

0.000

2

0.212

0.199

0.207

0.205

5

0.985

0.988

0.988

1.000

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.5

1.0

2.0

Frequency Coefficient (λ

F

)

1.000

0.994

0.977

0

0.416

0.415

0.509

0.411

1

0.213

0.211

0.211

0.210

2

0.001

0.000

0.000

0.000

5

Time Coefficient (λ

T

)

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.5

1.0

2.0

Frequency Coefficient (λ

F

)

0.210

0.218

0.270

0

0.298

0.222

0.213

0.225

1

0.000

0.184

0.291

0.136

2

1.000

0.915

0.965

0.939

5

Time Coefficient (λ

T

)

0.0

0.5

1.0

2.0

Frequency Coefficient (λ

F

)

0

1.000

0.986

0.983

1

0.460

0.458

0.457

0.470

2

0.250

0.251

0.255

0.264

5

0.000

0.003

0.004

0.001

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.5

1.0

2.0

Frequency Coefficient (λ

)

0

1.000

0.836

0.775

1

0.487

0.402

0.411

0.368

2

0.355

0.542

0.268

0.293

5

0.103

0.154

0.202

0.000

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.5

1.0

2.0

Frequency Coefficient (λ

)

0.998

1.000

0.972

0

0.454

0.444

0.450

0.431

1

0.245

0.259

0.257

0.252

2

0.000

0.005

0.022

0.012

5

Time Coefficient (λ

T

)

F

F

ILI (https: / / github.com / thuml / Autoformer): Weekly records from 2002 to 2021, provided by the US Centers for Disease Control and Prevention, representing the number of influenza-like illness patients.

PEMS : Contains traffic information measured at 5-minute intervals on California highways. The subsets PEMS03, PEMS04, PEMS07, and PEMS08 correspond to different regions and time spans, with varying numbers of sensors [28].

Solar-Energy [24]: Includes solar energy production data from 137 plants in 2006, measured at 10-minute intervals.

Traffic [37]: Consists of hourly traffic congestion data collected by 862 sensors on San Francisco freeways from January 2015 to December 2016.

To ensure fair performance evaluation, we adopted the sequence length, prediction length, and label length settings used in recent LTSF models, such as Autoformer [37], DLinear [40], PatchTST [33], TimesNet [36], iTransformer [30], GPT4TS [43], CycleNet [27]. Following previous works [37, 30, 28]. Tab. 3 shows the descriptions of datasets including the number of variate in each dataset, prediction length, dataset size, sampling frequency, and domain. Since the official Autoformer implementation provides the results in different prediction length (24, 48, 168, 336, 720), we follow their implementation and show the results on Sec. I.

H Experiment configurations

To ensure fair model evaluation, we utilized the official GitHub codes provided by six benchmark models: Autoformer [37], DLinear [40], PatchTST [33], TimesNet [36], iTransformer [30], GPT4TS [43], CycleNet [27]. For all experiments, we adopted the same prediction length, label length, and sequence length as the official implementations, and maintained the original architecture of each model.

Context lengths. According to the papers [37, 40, 33, 36, 30, 43, 27], the used context lenghts are followed by:

CycleNet, Autoformer, iTransformer: 96

TimesNet: 96 (36 for the ILI dataset)

PatchTST, DLinear: 336 (104 for the ILI dataset)

Prediction lengths. We primarily used prediction lengths of { 96 , 192 , 336 , 720 } . For the ILI dataset (https://github.com/thuml/Autoformer), we followed prior works and used { 24 , 36 , 48 , 60 } . As an exception, for the ETT{h1, h2, m1, m2} datasets, we followed the Autoformer paper and used { 24 , 48 , 168 , 336 , 720 } (for Autoformer) and { 96 , 192 , 336 , 720 } (for other models).

Optimization. The optimizer and scheduler were used under the same conditions as specified in the official codes for each model.

Our method. For our method (AliO), the hyperparameters-the number of samples N in Alg. 1 and lag l in Alg. 1-were set to their default values of 2 and 1 , respectively. The Mean Squared Error (MSE) function was used as the distance function for both the time and frequency domains. The coefficients for the time domain ( λ T ) and frequency domain ( λ F ) were selected from the ranges { 1 . 0 , 2 . 0 , 5 . 0 } and { 0 . 0 , 0 . 5 , 1 . 0 , 2 . 0 } , respectively (default values are λ T = 1 and λ F = 0 since the frequency domain is optional). Additionally, the AWL [26] technique was employed to automatically tune these coefficients and report the best performance.

Evaluation. We used Mean Squared Error (MSE) as our baseline loss function and reported forecasting performance using MSE, Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), and TAM. For all these metrics, lower values indicate better performance.

Robustness. To ensure robust results, we used three random seeds ( 2023 , 2024 , 2025 ) for initialization and report the standard deviation using ± . All experiments were conducted on NVIDIA RTX 3090 and A6000 GPUs. We conducted all experiments using the same GPU when comparing the baseline and AliO under the same conditions.

Training hyper-parameters. We used the same learning rate, batch size, and epoch as the official implementation of each model for reproducibility and fair comparison. The implementation code for each model is as follows.

Autoformer: https://github.com/thuml/Autoformer

DLinear: https://github.com/vivva/DLinear

PatchTST: https://github.com/yuqinie98/PatchTST

TimesNet: https://github.com/thuml/TimesNet

iTransformer: https://github.com/thuml/iTransformer

GPT4TS: https://github.com/DAMO-DI-ML/NeurIPS2023-One-Fits-All

CycleNet: https://github.com/ACAT-SCUT/CycleNet

We followed the implementation code listed above (GitHub) and used the same learning rate, batch size, and number of epochs as shown in Tabs. 4 to 11. The optimizer we used is Adam [23].

I Full experimental results

This section presents the comprehensive results of each Long-Term Series Forecasting (LTSF) models, Autoformer [37] (Tabs. 12 and 13), DLinear [40] (Tabs. 14 and 15), PatchTST [33] (Tabs. 16 and 17), TimesNet [36] (Tabs. 18 and 19), iTransformer [30] (Tabs. 20 to 22), GPT4TS [43] (Tabs. 23 and 24), and CycleNet [27] (Tabs. 25 and 26). The evaluation metrics include Mean Squared Error (MSE), Temporal Alignment Metric (TAM), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Root Mean Squared Error (RMSE). The ± symbol denotes the standard deviation across multiple seeds (we used three random seeds for robust experiment). AVG represents the average value across all prediction lengths, with the best performance highlighted in bold .

Models

Autoformer

Method

Baseline

AliO

Metric

MSE

↓

TAM

↓

MAE

↓

MAPE

↓

RMSE

↓

MSE

↓

TAM

↓

MAE

↓

MAPE

↓

RMSE

↓

ECL

96

0.204

±

0

.

008

0.035

±

0

.

002

0.319

±

0

.

008

3.398

±

0

.

230

0.452

±

0

.

009

0.193

±

0.003

0.029

±

0.001

0.308

±

0.003

3.296

±

0.132

0.440

±

0.004

192

0.219

±

0

.

005

0.042

±

0

.

002

0.329

±

0

.

003

3.416

±

0

.

185

0.468

±

0

.

005

0.215

±

0.003

0.038

±

0.004

0.324

±

0.004

3.402

±

0.091

0.464

±

0.004

336

0.232

±

0

.

004

0.040

±

0

.

004

0.340

±

0

.

004

3.466

±

0

.

224

0.481

±

0

.

004

0.223

±

0.007

0.034

±

0.004

0.331

±

0.005

3.350

±

0.136

0.472

±

0.007

720

0.260

±

0

.

013

0.046

±

0

.

007

0.362

±

0

.

009

3.501

±

0

.

118

0.509

±

0

.

013

0.253

±

0.008

0.042

±

0.002

0.356

±

0.006

3.404

±

0.035

0.503

±

0.008

AVG

0.229

±

0

.

018

0.041

±

0

.

003

0.337

±

0

.

015

3.445

±

0

.

036

0.478

±

0

.

019

0.221

±

0.019

0.036

±

0.004

0.330

±

0.015

3.363

±

0.040

0.469

±

0.020

ETTh1

24

0.408

±

0

.

008

0.063

±

0

.

005

0.427

±

0

.

002

10.853

±

0

.

413

0.639

±

0

.

006

0.373

±

0.008

0.051

±

0.002

0.416

±

0.007

10.747

±

0.477

0.611

±

0.007

48

0.408

±

0

.

014

0.048

±

0

.

004

0.431

±

0

.

008

11.632

±

0

.

477

0.639

±

0

.

011

0.395

±

0.024

0.037

±

0.002

0.420

±

0.012

10.889

±

0.175

0.628

±

0.019

168

0.502

±

0

.

028

0.051

±

0

.

012

0.482

±

0

.

016

11.810

±

0

.

142

0.708

±

0

.

020

0.445

±

0.014

0.036

±

0.005

0.451

±

0.010

11.573

±

0.593

0.667

±

0.011

336

0.517

±

0

.

013

0.039

±

0.003

0.491

±

0

.

006

11.786

±

0.326

0.719

±

0

.

009

0.484

±

0.013

0.048

±

0

.

002

0.479

±

0.008

11.957

±

0

.

223

0.696

±

0.009

720

0.549

±

0

.

016

0.047

±

0.002

0.527

±

0

.

008

12.533

±

0.659

0.741

±

0

.

011

0.521

±

0.009

0.057

±

0

.

009

0.516

±

0.010

13.217

±

1

.

015

0.722

±

0.007

AVG

0.477

±

0

.

053

0.050

±

0

.

007

0.472

±

0

.

034

11.723

±

0

.

488

0.689

±

0

.

039

0.444

±

0.050

0.046

±

0.008

0.456

±

0.034

11.677

±

0.811

0.665

±

0.038

ETTh2

24

0.280

±

0

.

009

0.054

±

0

.

004

0.356

±

0

.

006

1.524

±

0

.

038

0.529

±

0

.

008

0.265

±

0.003

0.034

±

0.003

0.340

±

0.003

1.392

±

0.028

0.515

±

0.003

48

0.335

±

0

.

029

0.051

±

0

.

009

0.390

±

0

.

021

1.624

±

0

.

067

0.578

±

0

.

025

0.301

±

0.009

0.025

±

0.002

0.360

±

0.006

1.422

±

0.054

0.549

±

0.008

168

0.442

±

0

.

008

0.044

±

0

.

009

0.448

±

0

.

009

1.868

±

0

.

150

0.665

±

0

.

006

0.421

±

0.008

0.027

±

0.002

0.427

±

0.005

1.580

±

0.084

0.649

±

0.006

336

0.479

±

0

.

012

0.031

±

0

.

012

0.482

±

0

.

012

2.003

±

0

.

194

0.692

±

0

.

009

0.448

±

0.002

0.018

±

0.004

0.455

±

0.002

1.749

±

0.020

0.669

±

0.001

720

0.470

±

0

.

022

0.025

±

0

.

005

0.483

±

0

.

013

2.273

±

0

.

166

0.685

±

0

.

016

0.450

±

0.005

0.020

±

0.000

0.466

±

0.003

2.044

±

0.024

0.670

±

0.004

AVG

0.401

±

0

.

072

0.041

±

0

.

010

0.432

±

0

.

046

1.858

±

0

.

245

0.630

±

0

.

059

0.377

±

0.071

0.025

±

0.005

0.410

±

0.046

1.637

±

0.219

0.610

±

0.060

ETTm1

24

0.377

±

0

.

015

0.092

±

0

.

003

0.415

±

0

.

005

2.563

±

0

.

005

0.614

±

0

.

012

0.351

±

0.023

0.084

±

0.001

0.394

±

0.011

2.455

±

0.019

0.592

±

0.019

48

0.447

±

0

.

048

0.079

±

0

.

014

0.452

±

0

.

017

2.789

±

0

.

130

0.668

±

0

.

035

0.381

±

0.002

0.053

±

0.001

0.412

±

0.003

2.458

±

0.074

0.617

±

0.001

96

0.493

±

0

.

054

0.074

±

0

.

010

0.478

±

0

.

019

2.834

±

0

.

081

0.701

±

0

.

038

0.440

±

0.029

0.055

±

0.006

0.445

±

0.012

2.579

±

0.078

0.663

±

0.022

288

0.579

±

0

.

030

0.070

±

0

.

013

0.513

±

0

.

012

2.851

±

0

.

060

0.760

±

0

.

020

0.515

±

0.003

0.041

±

0.002

0.479

±

0.004

2.604

±

0.049

0.718

±

0.002

672

0.544

±

0.013

0.036

±

0

.

001

0.501

±

0

.

005

2.788

±

0

.

062

0.737

±

0.009

0.554

±

0

.

013

0.031

±

0.002

0.501

±

0.009

2.728

±

0.044

0.744

±

0

.

009

AVG

0.488

±

0

.

065

0.070

±

0

.

017

0.472

±

0

.

032

2.765

±

0

.

095

0.696

±

0

.

047

0.448

±

0.070

0.053

±

0.016

0.446

±

0.036

2.565

±

0.093

0.667

±

0.053

ETTm2

24

0.161

±

0

.

013

0.078

±

0

.

010

0.268

±

0

.

010

1.081

±

0

.

039

0.401

±

0

.

016

0.148

±

0.005

0.057

±

0.005

0.259

±

0.005

1.054

±

0.041

0.385

±

0.007

48

0.217

±

0

.

012

0.080

±

0

.

005

0.308

±

0

.

009

1.196

±

0

.

046

0.466

±

0

.

013

0.182

±

0.006

0.041

±

0.009

0.279

±

0.007

1.094

±

0.033

0.426

±

0.007

96

0.252

±

0

.

029

0.058

±

0

.

008

0.324

±

0

.

015

1.302

±

0

.

024

0.502

±

0

.

028

0.217

±

0.003

0.030

±

0.006

0.298

±

0.004

1.158

±

0.019

0.466

±

0.003

288

0.319

±

0

.

005

0.027

±

0

.

004

0.362

±

0

.

003

1.407

±

0

.

029

0.565

±

0

.

005

0.313

±

0.001

0.019

±

0.005

0.354

±

0.001

1.307

±

0.013

0.559

±

0.001

672

0.408

±

0.002

0.017

±

0

.

003

0.410

±

0

.

001

1.577

±

0

.

014

0.639

±

0.001

0.409

±

0

.

003

0.012

±

0.004

0.409

±

0.002

1.538

±

0.047

0.640

±

0

.

002

AVG

0.271

±

0

.

078

0.052

±

0

.

024

0.334

±

0

.

044

1.313

±

0

.

156

0.514

±

0

.

075

0.254

±

0.087

0.032

±

0.015

0.320

±

0.050

1.230

±

0.161

0.495

±

0.084

and Weather are in Tab. 22.

Weather are in Tab. 22

Table 22: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, ETTm2, Solar, PEMS03, PEMS04, PEMS07, PEMS08, Exchange, Traffix, and Weather datasets and iTransformer [30]. The results are reported in terms of MSE, TAM, MAE, MAPE, and RMSE. The best results are highlighted in

J Pronunciation of a.li.o

To prevent confusion regarding the pronunciation, this section explicitly describes how to pronounce a.li.o .

[a.li.o]

The pronunciation consists of three syllables:

a : Pronounced like the 'a' in father (ah).

li : Pronounced with a clear 'L' sound, as in Lee (lee).

o : Pronounced like the 'o' in go (oh).

Consequently, it is pronounced as 'Ah-Lee-Oh' , ensuring the 'L' is clearly articulated.

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: We briefly described our methodology in the Abstract and stated its effects accordingly.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We provide the limitations in Sec. 7.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: We provide the proofs of our theorems in Secs. B.1 and B.2.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses output alignment issues in long-term time series forecasting (LTSF) by proposing AliO (Align Outputs), a novel training approach that minimizes prediction discrepancies for overlapping timestamps across lagged input sequences.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \nThe paper identifies an interesting and previously unaddressed issue in LTSF - inconsistent predictions for the same timestamps when using lagged input sequences, which could impact practical deployment reliability.\n\nExtensive experiments across 7 state-of-the-art models (Autoformer, DLinear, PatchTST, TimesNet, iTransformer, GPT4TS, CycleNet) and multiple datasets demonstrate broad applicability.\n\nWeaknesses:\n\nThe core scenario of using lagged input sequences for prediction is artificially constructed and rarely encountered in real forecasting applications, where models typically use the most recent available data. The claimed financial and operational costs from output misalignment lack empirical validation.\n\nThe \"regression pulling\" mechanism appears heuristic without strong theoretical foundations. The stop-gradient operation seems like an ad-hoc solution to prevent performance degradation rather than a principled design choice. The necessity of simultaneous time and frequency domain alignment is insufficiently justified given computational overhead.\n\nThe paper assumes that forecasts for the same timestamp should remain consistent, but this ignores the dynamic nature of time series. Forecasts will naturally change as new information is added, which may actually reflect the model's adaptability to new data rather than a flaw.\n\nQUESTIONS:\nCan you provide concrete evidence or case studies demonstrating that output alignment issues cause significant problems in actual forecasting deployments? What specific industries or applications would benefit most from this approach?\n\nHow does AliO compare against ensemble methods or Bayesian approaches that naturally provide prediction uncertainty and consistency? Have you considered comparing against methods that use prediction intervals or confidence bounds?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper identifies a crucial but overlooked issue in long-term time series forecasting (LTSF): the lack of consistency when models generate predictions for overlapping futrue timestamps using lagged input windows. To address this, the authors introduce Alio, a simple yet effective framework that explicitly aligns prediction outputs by minimizing discrepancies across overlapping segments in both the time and frequency domains. \n\nTo quantify this aspect, the authors propose Time Alignment Metric (TAM) to measure the consistency of overlapping outputs. To enhance the consistency, the authors design a novel loss function by adding an alignment loss alongside traditional forecasting objective, using a regression pulling mechanism with a stop-gradient to ensure that the improving alignment does not degrade predictive accuracy.\n\nThe authors conduct extensive experiments on multiple benchmark datasets and popular LTSF models to demonstrate the effectiveness of the method.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n\n* This paper identifies a crucial yet previously overlooked problem in long-term time series forecasting: existing models often fail to produce consistent predictions for overlapping forecasting windows. To address this, the authors introduce a new metric, TAM, which effectively captures prediction alignment and complements traditional error-based metrics such as MSE and MAE.\n\n* The paper proposes a novel method that successfully mitigates these inconsistencies and demonstrates its effectiveness through comprehensive experiments.\n* This paper also provides solid mathematical proofs, which strengthen the theoretical foundation of the work.\n\n* Overall, the paper is clearly written and easy to follow, with well-designed and readable figures that help illustrate the key ideas.\n\nWeaknesses: \n\n* For metric design, authors should propose more showcases or human study results to demonstrate the metric could effectively measure the degree of alignment.\n* The related work section could be expanded. A more comprehensive review and clearer positioning with respect to recent advances would help highlight the unique aspects of this work and situate its contributions more effectively.\n* There are some typos/errors. For example, in Traffic results of Table 1, AliO achieve 0.280 in GPT4TS in TAM, which is higher than baseline (0.059). But the authors use bold to indicate it is the best result. Moreover, also in this table, some results are significantly higher than baseline. The authors should also check the correctness.\n\nQUESTIONS:\n1. How do authors get the results of baselines? As Alio framework includes forecasting objective, it will inherently enhance the performance on MSE metric. For fair comparison, I think the authors should ensure that the training of baselines are sufficient.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of improving output consistency in long-term time series forecasting. The authors propose a method called OliO, which jointly considers alignment in both the time and frequency domains to reduce inconsistencies in forecast outputs. They also introduce a new evaluation metric—Time Alignment Metric (TAM)—to quantify consistency. To demonstrate the effectiveness of their approach, the authors conduct experiments using multiple model architectures across seven standard benchmark datasets and show that OliO can reduce TAM, indicating improved forecasting consistency.\n\nSTRENGTHS AND WEAKNESSES:\n### **Strengths**\n- The paper tackles an interesting and underexplored problem: improving output consistency in long-term time series forecasting (LTSF).\n- The proposed approach, OliO, incorporates both time-domain and frequency-domain alignment, which is a thoughtful and well-motivated design. The final loss function is carefully crafted to encourage consistent  as well as accurate predictions.\n- The method is evaluated across various LTSF architectures, demonstrating its potential generality.\n- A new evaluation metric—Time Alignment Metric (TAM)—is introduced to quantify consistency, contributing a novel perspective to the field.\n\n### **Weaknesses**\n- While consistency is an important property, a good forecaster should ultimately aim to match the ground truth. In many cases, lower consistency may indicate appropriate variability or uncertainty. The paper does not compare against models trained with **Weighted Quantile Loss**, which explicitly models uncertainty and could offer a more principled baseline than OliO. \n- The study only evaluates a very limited range of lag values (\\(l\\)), with Table 1 focusing solely on \\(l=1\\). This setting is not representative of practical long-term forecasting, and limits insight into the generalization capabilities of the proposed method. Results in Figure 7 further suggest that OliO underperforms for larger \\(l\\).\n- The paper omits discussion and evaluation of recent **pretrained time series foundation models** (e.g., Morai, Morai-MoE, Time-MoE, Chronos). A comparison against these strong baselines is essential to assess the true effectiveness of OliO.\n- The results on the **Traffic** dataset raise concerns (not sure if this a typo) : OliO appears to degrade alignment significantly\n- TAM is already low on these datasets. The authors should consider identifying datasets with higher TAM values to better evaluate the method’s strengths and limitations.\n\nQUESTIONS:\n- **Figure 7** makes it difficult to clearly assess the gains over baselines. More comparisons—similar to table 1—across different values of \\(N\\) and \\(l\\) would better highlight the effectiveness of OliO.\n- The paper does not specify the **context length** used during training. It would be valuable to understand how varying context lengths impact the effectiveness of OliO.\n- **Figure 6** suggests that increasing the time alignment coefficient consistently improves performance. Why not explore even larger values to assess the full potential of the method?\n- It remains unclear whether a model trained on a specific lag value \\(l\\) can **generalize** to other lag settings. This is an important aspect of the method’s robustness.\n- Is the **inconsistency issue** targeted by OliO specific to long-term forecasting, or does it also appear in short- and mid-term settings? A broader evaluation would clarify the scope of the problem.\n\nLIMITATIONS:\n- A more extensive exploration of hyperparameter settings (e.g., N, l) is needed to fully assess the robustness of the proposed method.\n\n- The paper would benefit from a more thorough discussion and comparison with existing approaches, particularly recent pretrained time series foundation models.\n\n- A broader range of datasets should be evaluated to better understand the generalizability of the method across diverse forecasting scenarios.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a method for aligning predictions of time series samples that share overlapping prediction periods. The proposed alignment technique ensures consistency across both temporal and spectral domains. To quantify the alignment of time series samples that share overlapping periods, the authors propose a new evaluation metric called the Time Alignment Metric. \n\nThe approach is evaluated through integration with several state-of-the-art models and tested on multiple standard multivariate time series datasets, demonstrating its potential to improve prediction consistency in overlapping time periods.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n * Proposes an interesting method for aligning predictions across overlapping time periods in both temporal and spectral domains\n * Integrates the method with multiple state-of-the-art models and tests on common multivariate time series datasets\n * Introduces a Time Alignment Metric (TAM) specifically designed to evaluate alignment accuracy\n\n### Weaknesses\n * Lacks clear explanation of how consecutive samples are accessed in batch processing\n * Provides only AliO implementation without showing integration with existing models or TAM implementation\n * Needs better justification for strict alignment requirements in real-world applications\n * Lack demonstration of the necessity of TAM (why not using regular metrics on the overlapping period)\n * Lacks analysis of improvement consistency across datasets and explanations for varying performance\n * Contains formatting inconsistencies and unclear result highlighting\n\nQUESTIONS:\n### Implementation Concerns\nThe paper raises significant questions about the practical implementation of the proposed AliO method. I would disagree with the statement \"not [...] requiring modifications to the model\". A crucial implementation detail involves how batches of samples are created and processed. For AliO to function effectively, it needs access to both the current sample ($x_k$) and previous samples ($x_{k-t}$) with their corresponding predictions ($y_k$ and $y_{k-t}$), where $t \\in ${$1$, ..., number_previous_samples}. However, most time series models process samples in batches that typically don't maintain chronological ordering, making it unclear how the authors ensure access to consecutive samples.\n\nThe code provided only includes the AliO implementation itself, not its integration with existing prediction models. This omission makes it difficult to assess how modifications were made to models like DLinear, CycleNet, or iTransformer, where timestamps are typically separated from the data. Additionally, the implementation code for the Time Alignment Metric (TAM) is missing, which is essential for reproducing the results and understanding the evaluation methodology.\n\n### Motivation and Practical Relevance\nWhile the concept of aligning predictions for overlapping time periods is theoretically interesting, its practical motivation requires further justification. The assumption that close samples should provide similar forecasts isn't necessarily problematic, but strict alignment may not always be necessary or beneficial in real-world applications. In resource planning scenarios, the temporal resolution of forecasts is crucial. For hourly or half-hourly data, aligning outputs for consecutive timesteps may not provide significant practical benefits, as planning and allocation decisions are typically made at coarser temporal resolutions (e.g., daily).\n\nThe paper's motivation should address how often planning and allocation processes are updated in practice. Many real-world systems run models at regular intervals (e.g., daily for electricity consumption forecasting) rather than at the highest available resolution. For weekly datasets like ILI, alignment makes sense as consecutive timesteps represent different weeks. However, for hourly datasets like ETT or Electricity, the benefits are less clear. In addition, Figure 2 shows that predictions diverge significantly at the beginning of overlapping intervals but converge toward the end, suggesting that alignment might not be critical for all applications, depending on the planning horizon.\n\n### Theoretical Considerations\nThe theoretical justification for alignment needs clarification. If a model minimizes error such that $\\hat{y}$ is close to $y$ based on $x$, and $\\hat{y_l}$ is close to $y_l$ based on $x_l$, then $\\hat{y}[1:]$ should naturally be close to $\\hat{y_l}[:-1]$ because they're compared to the same ground truth $y[1:] = y_l[:-1]$. This suggests that some alignment is inherently ensured through proper training. The authors argue that differences in input lead to misalignment, implying that the method effectively makes the model \"remember\" previous predictions. This raises concerns about error propagation—if an initial prediction is poor, subsequent aligned predictions might compound this error.\n\n### Reproducibility concerns include:\n\n * Missing values for λ_T and λ_F in Table 1\n * Unspecified input and prediction lengths for Table 1 results\n * Unclear whether results represent averages across multiple runs\n * Undisclosed datasets used for Figures 6 and 7, with a need for similar plots across all datasets to avoid overgeneralization\n\n### Clarity \nSeveral clarity issues should be addressed:\n\n * The phrase \"the same timestamps\" is misleading—\"overlapping timestamps across different predictions\" would be more accurate\n * Table 1 should consistently highlight improvements (e.g., GPT4TS and ECL results should match the formatting used for CycleNet and Weather)\n * Results for DLinear and Traffic with AliO appear reversed, showing worse TAM performance—such performance needs further discussion why TAM is worsened where it should be better as it is specifically designed to align\n * The claim of \"achieving gains of 70.5% for CycleNet\" should specify that this is the maximum gain for ETTh1, not an average (the average gain is 50%, with minimum 12.5% and maximum 70.45%)\n * The notation for lag samples should use n - l to clearly indicate historical samples\n\n### Conclusion\nGiven these concerns, particularly regarding implementation details and practical motivation, I currently score this submission as borderline reject. However, my assessment could change during the rebuttal phase depending on how well the authors address these points, especially the implementation questions, and based on discussions with other reviewers. A strong rebuttal that clarifies the implementation details and better justifies the practical relevance of strict alignment could potentially improve my evaluation of this work.\n\nLIMITATIONS:\n### Comparison and Evaluation Needs\nA comparison between the proposed Time Alignment Metric (TAM) and Dynamic Time Warping (DTW) would provide valuable context for understanding TAM's advantages and limitations. Why using TAM instead of DTW for overlapping period? This comparison should address how TAM differs from established alignment metrics and what specific benefits it offers for time series prediction tasks.\n\n### Ablation Study Limitations\nThe current ablation study lacks sufficient clarity in the main paper regarding the individual contributions of $\\lambda_F$ and $\\lambda_T$ components. Specifically, the analysis should more clearly demonstrate the impact of each component when the other is set to zero ($\\lambda_F=0$ or $\\lambda_T=0$). \n\n### Results Discussion \nThe paper would benefit from a more comprehensive discussion of the results. Currently, there's insufficient analysis of whether the observed improvements are consistent across all datasets or vary significantly between them. A detailed examination of the homogeneity of improvements would help readers understand the method's general applicability. Furthermore, the paper should provide insights into why certain datasets show greater benefits from the proposed alignment method. Are there specific characteristics (such as temporal resolution, seasonality patterns, or noise levels) that make some datasets more responsive to the alignment technique? Offering hypotheses or intuitions about these variations would significantly enhance the paper's contribution.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}