AliO: Output Alignment Matters in Long-Term Time Series Forecasting
Authors: Kwangryeol Park
Jaeho Kim
Seulki Lee
Venue: NeurIPS 2025 poster ()
Keywords: Forecasting
Novel challenge
Primary area: applications
TL;DR: In the field of predictable time series forecasting task, the newly identified issue of output alignment, the metrics to evaluate it, and potential solutions.

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.

Artificial Intelligence Graduate School AliO: Output Alignment Matters in Long-Term Time Series Forecasting
Kwangryeol Park
Ulsan National Institute of Science & Technology (UNIST), South Korea pkr7098 @ unist.ac.kr
Jaeho Kim Artificial Intelligence
Korea University, South Korea kjh3690@koreeal korea..ckrr.cc
Seulki Leef
Department of Computer Science and Engineering Ulsan National Institute of Science & Technology (UNIST), South Korea sculki. @oneeuuniti unist.ac.kr
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 weather forecasting and planning of electricity consumption. However, state-of-the-att LTSF models often fail to achieve pre- diction 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 im- prove the output alignment of LTSF models by reducing the discrepancies between prediction outputs for the same timestamps in both the time and frequency domains. measure output alignment, we introduce new metric, TAM (Time Alignment Metric), which quantifies the alignment between prediction outputs, whereas ex- isting metrics such as MSE only capture the distance between prediction outputs and ground truths. Experimental results show that AliO effectively improves the output aligmment, i.e., up to 58.2% 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 on the GitHub repository"
Introduction
The task of long-term time series forecasting (LTSF) is essential in various fields such as prediction of
electricity demand [3 weather forecasting (37] health data 200, and on. Recently, deep neural
network models 40 30 33 43
have shown strong performance predicting long-term time series
"Work done UNIST 1 "oonesponding author
https://gsthub.com lab/Al10
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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based on historical information, which typically aim to minimize the error between the prediction output and the ground-trute -rutth 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. Iilustrrates scenario in which 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 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.
the electricity usage for the same timestamps (i.e., May to
We refer to this phenomenon as the output alignment problem, which has not been adequately acknowledged and addressed by existing LTSF studies [25] 7 [8]. despite its significance and substantial impact on real-world applications. To the best of our knowledge, state-of-the-art LTSF models [3 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 Out- puts), novel method designed to enhance
output alignment in LTSF models. For the first time, AliO enables LTSF mod-
Eleetricity Power Demand
Prediction Piediction Ieputs
Waste budget
els to produce consistent predictions for overlapping timestamps across lagged in- put sequences. By aligning predictions for overlapping timestamps through the min-
Input 1
Prediction
Lag
Input
Prediction
Overlapping oirpur sequence
imization of discrepancies in both time and frequency domains, AliO improves out- put consistency even when imput sequences lagged. AliO achieves this by directing the model's predictions toward the ground-truth sequences while simultaneously minimiz- ing discrepancies across multiple predic- tions obtained from a set of lagged input sequences. It allows AliO to integrate seam- lessly with the model's forecasting objec- tives, 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 im- proving overall forecasting performance.
Jan. Feb. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. Nov. Dec.
First buget decision for Aug
Second buget decision for Aug
Figure I: An example of low output alignment between two predictions on future electricity usages: (1) Predic- tion I (purple) forecast spanning from Apr to Nov and (2) Prediction 2 (yellow) forecast spanning from May to Dec. These two predictions are generated from two partially overlapping input sequences (1) one from Jan to Mar (Input 1), and (2) the other from Feb to Apr (Input 2). The inconsistency between these two prediction outputs over the same timestamps (i.e., May Nov) leads to differing budget allocation plannings for electricity power consumption in Aug, resulting in time and financial waste due to rescheduling resource allocation.
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 model's predictions across lagged input sequences, we propose a new metric, Time Aligmment Metric (TAM). TAM quantitatively as- sesses 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(hI, h2, m1, m2), Electricity
(ECL), Traffic, Weather, and ILI dataset (37). using various -ate-off-the-at LTSF models such as CycleNet (27) GPT4S 43), iTransformer (30, PatchTST [K]] TimesNet (36), DLinear [40) and Autoformer [37). The evaluation results demonstrate that AliO effectively aligns predictions over overlapping timestamps, i.c., improving TAM up to 58.2%, while maintaining or enhancing

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Japut 11 19
Preduction 0 200
Ground Truth
Prediction 2
Lipux [1:20
Inputs
Psediction 5
Prediction Piediction
Prediction
"Espur (5:
Pocdiction
Without AlIO
With AlIo
-05
Timeslamp
Timestamp 30
Figure 2: (Left) Six prediction outputs from the DLinear (MSE) only, forecasting a length-24 sequence for ILI data 37 where six input sequences (green dotted line) are lagged one timestamp apart. (Right) Six predicted sequences from DLinear trained with AliO, under the same conditions. The predicted sequences exhibit improved output alignment, showing a closer match the ground truth for all six predictions, which significantly reduces the prediction shifts observed when using regression loss alone.
40 model trained with regression loss
forecasting regression accuracy up to 17.4% measured in MSE. The detailed descriptions on additional datasets and experimental configurations are provided in Secs. G
2 Motivation
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 31
Furthermore, accroding research in non-weather domains, consumers tend to perceive consistency among multiple estimates from the same system as a signal skill [1 predictions necessitates replanning, which consumes unnecessary organizational time. The resulting resource reallocation leads to the investment more resources, contrary toe th original goal of Rolling forecasting, which aims for resource savings [16 14]. This can be described as additional sunk cost [35], as frequent fluctuations in predictions can induce irrational decision decision-making and result in lost opportunity costs [35 6].
5 3 The inconsistency in
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. [ shows the prediction results of DLinear (37 on the ILI dataset (37] using only regression loss (MSE), revealing a significant lack of consistency.
contrast, the right panel, where AliO is applied, shows that consistency is improved while accuracy is maintained. As can be seen in Sec. this is 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 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 Kd represent accuracy, which the distance between predictions and ground truth. Therefore, they have limitations in measuring consistency, which a measure of performance between predictions. Complementing this, TAM measures the distance between predictions, serving an indicator that represents consistency.
Through Sec. D and Sec. we show that AliO successfully achieves the dual goals of maintain- ing/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 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 quantitative assessment of output alignment.
3

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Definition 3.1. (Output Alignment) Output alignment refers to the property wherein the prediction oulputs 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 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 imput and output sequences, and overlapping output sequences, as provided below.
Definition 3.2. (Input and Output Sequence) Given a time-series sequence X E Rexd, where è is the input channel, and is the length of the sequence, respectively, the n-th input sequence X" E Rexd' of the length d' & d is defined as the segment X ss: where is the starting timestamp, and + d' - 1 is the ending timestamp.
Then, taking X" E _ge input, f(x"; Y ge'xh is defined the prediction output sequence of the length h provided by the forecasting model f. where c' is the output channel, and is the model parameter. Having the input sequence X" and the corresponding output sequence Y" of the model f. the lagged input and output sequences, x and y are derived by shifting their timestamps by the lag parameter I, as described below.
Definition 3.3. (Lagged Input and Output Sequence) Given the n-th input sequence X" cRd of the segment X 11, the sequence X Rexd of the segment X x is defined as the lagged inpul sequence of X" with being the lag parameter.
Then, given the lagged input sequence X E Rexd", the prediction output of the length h, f(x"+1; Q) = y E Ted' xh is defined as the model f's lagged output sequence. From Y and y the overlapping output sequences p and p=+1 are derived to compute TAM, as follows.
Definition 3.4. (Overlapping Output Sequence) Given two output sequences Y Re'xh and y ge'xh of the model generated from X" and x=+1 input, respectively, pd Eg'' and p=+1 Re' are defined as the overlapping output sequences between Y" and y whose segments are given by 1+tsh and Y respectively, where N' = h - 7 is the overlapping length. 1:h-1-
From these, Time Alignment Metric (TAM) is defined to quantify output alignment by evaluating the consistency between overlapping output sequences pd and p as follows.
Definition 3.5. (TAM; Time Alignment Metric) Given N overlapping output sequences P E IR xh' for n = 1, 2,2,.... N. where the input and output sequences, X x*+ and y are offset from X and Y by the lag 1, TAMx (Time Alignment Mefric) is calculated the average of the MAE values between all pairs of {P", p'm,, which is given by:
oulputsequencss
TAM~ A -N- - 1)N
P Pm
Σ
Σ
2
B= m
TAM offers a distinct advantage over traditional forecasting regression metrics, such 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 metries 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 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 simultancously

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Overlapping Ouiput Sequence (P? )
80"
135
15"
Regrossion Loss Dovction
AliO Leea Direction
Ground
1807
Stop gradicnt
Prediction
Regressim Loss Direction
AIIO Loss Dirction
315
Prediction
Stop grathent
270"
Figure 3: Output alignments in both time and frequency domains. (Left) In the time domain, AliO minimizes the difference between two predictions the overlapping timestamps, i... P7 and p7+1 (I means time domain). To preserve the regression loss direction while improving the output alignment, the regression pulling (Sec. 4.2 is applied through the stop-guu gradient operation each time point, ensuring that the overall loss is calculated in the direction towards the ground truth. (Right) In the frequency domain, AliO aligns both the phase and amplitude components of P and P (the transformed frequency domain of PT and Ptt applied with the stop-gradient operation. The figure is represented in polar coordinates, where the angle indicates the phase, and the distance from the center point corresponds to the amplitude
achieve two key objectives: (I) improving output alignment by reducing discrepancies between the overlapping predictions, and (2) maintaining the model's forecasting regression performance by align- ing the model's s prediction outputs with the ground truths. The two objectives are achieved through the technique that we call regression pulling with the stop-gradien operation, which ensures that the forecasting regression loss remains unaffected while improving output alignment simultaneously.
Given two overlapping predictions, P" = f(X"; 0)14h and pa+1 f(x*** -h defined in Theorem 3.4 AliO aligns these predictions such that pa p=+1 for n = 1,2,... ..., ..., by minimizing the following objective with respect to the model f's parameter 0, as:
([x*+I
min D(f(X"; ( f(x*+1, 1)
(2)
where D denotes distance function and subscripts denote the segment ranges of the model f's prediction output sequences, i.e., 1 1+
and h - correspond to the same timestamps for X" and x respectively. Eq. 2 can be casily extended to handle non-consecutive overlapping output sequences, enabling its application to TAM (Theorem 3.5). AliO aligns the overlapping predictions pa and pa+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.
4.1 Time domain alignment
To enhance the temporal alignment, AliO aligns the overlapping predictions, P and P for n, m 1, 2..... N, in the time domain, where the subscript T denotes the time domain. This alignment can be achieved using distance function, Dr, such as MSE or Dynamic Time Warping (DTW) [34]. The computed distance is back-propagated encourage the model f to produce overlapping predictions that align with each other. However, solely aligning the overlapping predictions by minimizing time domain alignment loss LT Dr(Pi. PT)) may lead to both prediction outputs deviating from 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 and Pi are relatively further from the ground truth than the other at the same timestamp. indicated by the index variable idrT, as shown on line (1) in Alg. [
By applying the stop-gradient operation, denoted as sg(-), ) the prediction output points farther from the ground truth, the time domain alignment loss Lr pulls these distant points closer the ground truth, aligning its optimization direction with that of the forecasting regression loss and reinforcing to minimize the regression loss. Consequently, regression pulling
5

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Algorithm 1 The procedure of AliO. @ denotes element-wis multiplication. FFT(; -. returns the frequency domain representation of each signal sequences, sg(-) is stop-gradiem operator.
The number of predictions N, consecutive predictions Y" with at time lag of I, and their ground truth Y where n E [1, .V, and distance function Dr and D 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 Cr and frequency domain alignment loss Lr.
Input:
Initialize L = 0, L = 0, count for n to N 1 do
for
1 to N do
gap |m - n x
PT. GT (Y sapa
Y
gaps gap:
Y
ider Index([
GT P GT
(1)
C. = L. +
Tigddny sg( Ta }) + Dr(sg Pr P Tiigider)
[RegPull]
(2)
(Pr, PT, GT FFT P GT
(3)
i = Index( P GTrl > PF GT
(4)
Cj = C, + Dr (PP (PP P,jeidos 5g( P
Fjtiday) )) + Dr(8g Peagddd P Fagidan)
[RegPull] (5)
count count
end for
end for
(LT, LF) = (Lr Coonnt, f.y /count))
Return: LiT and CF
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 [1, in Alg. | Dr Pr(Pr PPr idizy sg( ( (eidddzy))) )) line (2) encourages the prediction PT to move closer to the prediction PT. On the other hand, [sg PT sigdder)) ), Pm Tiigidzy on the same line (2) promotes the alignment of PT towards PT The direction of alignment determined by the index variable, idær. These two distance values combined into the time domain alignment loss Lr, which expressed as:
Lr = (r Tiididyy sg( ((Tricidey)) + (r (T. Tigghta)) P Pr.sgizz))
(3)
Consequently, Fq. (3) aligns the direction of the time domain alignment loss Cr 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. 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 complemen- tary 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 the time domain.
As shown in Alg. I the time domain overlapping predictions, P and PT, along with the ground truths GT T. are first transformed into frequency domain representations, resulting in P P and GT F on line (3), where the subscript F denotes the frequency domain. Subsequently. the index variable idæ r is determined by identifying output points further from GT r between P and PT on line (4). The distance function Dr 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 LF is obtained as:
LF = Dr (P Fiieddy ag( P (Ficide))) )) + Dr(s P Figgider ) Pm Figgddey)
We use MSE (mean squared error) as the main distance function for the frequency domain alignment, i..., Dr = P P ///hh' hh. The following Theorem 4. demonstrates that applying MSE in the frequency domain facilitates phase alignment between prediction outputs.
Theorem 4.1. Given two frequency domain prediction vectors, ph and p in CK,, where C means the set of complex numbers, minimizing their MSE, l|p] P 12.. results in reduction of the difference

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in the phase components, Lpp and Lpp, as shown below (the proof is provided in Sec.
B.1
h p PF =
1 LpF 0
(5)
Fig. 3 (Right) shows a polar-coordings depiction of frequency alignment conducted through Eq. ( in the frequency domain. Theorem 4.2 shows that minimizing MSE, i.c., Dr = || PF 12 also reduces the differences in amplitude once the phase components <P's and / <P have been aligned.
Theorem 4.2. Once the phases of two frequency domain prediction vectors, p and p in C are aligned, ie.. Lp's Lp", f, the minimization of MSE between the two prediction outputs, lp? p leads to a reduction in the amplitude difference, as shown below (the proof is provided in Sec. B.2
1 N llp
2
+ = 1pr
1F
00
4.4 Alio loss
By combining the time Lr in Eq. and frequency domain alignment loss Lr in Eq. 4. we derive the AliO loss, LASO. which incorporated with the forecasting regression loss, freg as shown below, where \T and Xp control the extent of alignment in the time and frequency domain, respectively.
CAHO = XT + AFCF
Ltoral = -reg + LANO
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 T' AM for output alignment evaluation, with N = 2. To reproduce experimental results of baselines and ensure fair
evaluation, we follow the same hyper-parameterr presented in each model papers [37 27 40 43 30
33 [6].
For AliO, set N 2 and = covering wide prediction range. The search space for ÀT and XØ 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 reported as the average performance across all prediction lengths. Detailed descriptions of the experiment and results are provided in Secs. to
Models. We apply AliO to various LTSF architectures, experimenting it with diverse range of LTSF approaches, categorized into four groups: (1) Transformer-bawe models, i.e., Autoformer [37]
PatchTST [33, and iTransformer 330, (2) Linear-based models, i.e., DLinear [40] and CycleNet 27
(3) CNN-based models, i.e., TimesNet [36], and (4) LLMe-ased models, i.e., GPT4TS
43
Datasets. On the main text of the paper, we report results on representative LTSF datasets, i.e.,
Electricity (ECL) 133 ETT (h1, h2, ml, and m2) 42 Traffic, [37]. Weather (37), and National-
Illness (ILI) dataset 33) The experiments results on full datasets can be found in Sec. [|]
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 IIII dataset specifically, the context lengths were to 104 for PStchTST 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 used (24, 48, 168, 336, 720}.
5.1 Output alignment and forecasting performance
Tab. summarizes the performance of LTSF models on seven datasets (excluding ILI) for multivariate LTSF tasks, comparing baselines (without AliO) and AliO-integraal 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 PStchTST, 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 II.I dataset exhibiting insufficient initial output alignment, AliO achieves up 17.4% higher MSE improvement compared to other benchmarks.
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Table 1: The forecasting and alignment performance, measured by MSE and TAM2, respectively, compared between the baseline (MSE only) and AliO, with best results indicated in bold. The results averaged over three random seeds and prediction lengths. The more metries (MAE, MAPE, and
RMSE) with standard deviations are provided in Sec.
Models
CycleNet
GPT4TS
iTransformer
PatcTTST
TimesNet
DLincar
Autoforner
Metric
MSB
TAM2
MSE
TAM2
MSE TAM2
MSE
TAMa
MSE
TAM2
MSE
TAM2
MSE
TAM2
baseline
0.171
0.016 0.167 0.036
0.176
0.050
0.162 0.037
0.196
0.040
0.166
0.013
0.229
0.041
ECL
Alio
0.170
0.014
0.167
0.025
0.172
0.031
0.161 0.024
0.191
0.028
0.166
0.010
0.221
0.036
baseline
0.432
0.044
0,424
0.056
0.455
0.088
0.418
0.076
0.476
0.05
0.422
0.025
0.477
0.05
ETTh1
AliO
0.429
0.013
0.417
0.039
0.438
0.052
0.415
0.048
0.468
0.032
0.419
0.014
0.444
0.046
baseline
0,385
0.047
0.363
0.048
0.382
0.074
0.341
0.061
0,416
0.079
0,449
0.054
0,401
0,041
ET
AliO
0.381
0.025
0.354
0.033
0.377
0.052
0.341
0.050
0,402
0.043
0.446
0.033
0.377
0.025
baseline
0.3366
0.032
0.351
0.044
0.407
0.060
0.352
0.050
0.412
0.045
0.358
0.024
0,488
0.070
ETTml
AliO
0.383
0.014
0.346
0.031
0.396
0.033
0.349
0.032
0.398
0.026
0.354
0.010
0.448
0,053
FTTm2
bascline
0.272
0.027
0,270
0.042
0.292
0.048
0.257
0.059
0.296
0.036
0.289
0.271
0.052
AliO
0.271
0.019
0.265
0.028
0.289
0.026
0.255
0.039
0.295
0.022
0.268
0.020
0.254
0,032
baseline
0..877
0.021
0.411
0.059
0,422
0.061
0.389
0.041
0.626
0.040
0,436
0.025
0.634
0.048
Traffic
AliO
0.481
0.008
0.405
0.032
0.423
0.044
0388
0.027
0.554
0.030
0.434
0.009
0.624
0.045
baseline
0.255
0.010
0.227
0.024
0.260
0.024
0.229
0.018
0.262
0.018
0.245
0.007
0.342
0.056
Weather
AliO
0.255
0.008
0.225
0.011
0.259
0.017
0.228
0.012
0.262
0.014
0.244
0.005
0.311
0.042
5.2
How Alio maintains or improves regression performance
Tab. 0 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 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. [ achieve com- paratively smaller MSE improvements. Figs. 4 and shows this trend; higher initial TAM (poor alignment) correlates with greater MSE gains, demonstrating AliO's s efficacy in suboptimal alignment conditions.
Figs. 2 andd 5 visualize DLinear predic- tions on the ILI dataset. The left fig- ures (without AliO) exhibit prediction shifts [18] in both time (Fig. 2 and fre- quency (Fig. 5) domains, reflected in high TAM values. In contrast, the right figures (with AliO) show improved align- ment in both domains, which reduces TAM value and enhances regression per- formance (MSE).
The forecasting regression and alignment per- formance on ILI (37], measured in MSE, and TAM2, are compared between the baseline (without AliO) and AliO. The best results are highlighted in bold. For robust eval- uation, we conduct experiments with three random seeds. Results are rounded to three decimal places for simplicity.
Table 2:
Models
GPT4TS
PaccTTST
TimesNet
Dl
Metric
MSE
TAM2
MSE TAM2 MSE TAM2 MSB
TAMg
base
1898
0.117
1813 0.110
2.16 0.322 2,247
0.132
AliO
1.755
0.098
1.497 0,046
1963
0,258
2.187 0.119
5.3 Hyper-parameter Analysis
Time and Frequency Domain Coefficients. AliO utilizes the two coefficients, ÀT and X in Eq. 77. 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 X in {0.0. 0.5, 1.0, 2.0), and the y-axis corresponds to \T in {1.0, 5.0]. The left heatmap shows normalized MSE, where the maximum MSE in each dataset is scaled to 1.0 and the minimum to O, 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 aligmment, clear trend is observed in TAM where increasing the time domain alignment coefficient \r from 1.0 to 5.0 leads to a consistent improvement. Interestingly, the normalized MSE also shows a general improvement as XT 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 Xp is less pronounced compared to the time
8

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90"
90"
MSE Improvement Initaa TAM
45
135'
45"
Trend line
TAM
0.08
180"
Ampliiade
00
190"
Gronnd Irum Predkcijon Prodiction
0.04
Phose
0.00
225
225
Production
Prediction
MSE Improvement (%)
270"
270
Figure 4: The relationship between initial TAM and MSE improvement (%). The correlation coefficient is 0.33, indicating positive trend; when the initial alignment poor (i.e., higher TAM values), the MSE improvement tends be greater. For more detailed information, please check Sec. B
Figure 5: (Left) The frequency domain of five predic- tions in polar coordinates from DLinear [40] trained with regression loss, forecasting the ILI dataset. (Right) Five phases from the same setup, with AliO applied. AliO effectively aligns both the phase and amplitude of the model's predictions in the frequency domain.
Time Coofisiint (A))
MSE (Normalized)
Time Coefiictint (A,)
TAM (Normalized)
Number of onetlappingg
ouppu sequenees (N)
MSE (Normalized)
Number of twerfapping
TAM (Normalized)
0.34
0.33
0.36
0.33
2.0 50
0.07 0.03 0.05 0.01
se.
00.99
0.61
0.57
0.50
outpur sequences
0.00
0.17 0.32
0.65
86
2.0
0.37
0.28
0.38
0. 0.36
0.55 0.50
0.58
0.62
A5
-00..1
0.27
0.52 0.64
84
0.64
0.69
0.56
0.56
0.92
0.95
0.93
0.95
0.52
0.54
0.44
0.20
-12
Froquancy
0.00
Coefficient (3,)
Trequency Coefficient (..)
Lag ch
Lag ch
Figure 6: Comparison of the normalized MSE (left) and TAM (right) over different settings of XT and Xp. The x-axis represents Xp (00.0. 0.5, 1.0, 2.0)), and the y-axis represents ÀT ([1.0, 2.0, 5.0)). Increasing \r consis- tently improves TAM performance, while X shows its impact depending the value of \y- Blue represents better performance.
Figure 7: Comparison of the normalized MSE (left) and TAM (right) over different settings the over- lapping predictions N and the lag L. The x-axis represents the lag 1, and the y-axis represents the number of overlapping predictions N. Increasing N improves TAM, while increasing I degrades the performance. No distinet tendencies observed regarding MSE. Blue represents better performance.
coefficient År. However, in cases where \T alone cannot improve performance. AF shows its impact on both the MSE and TAM metrics, as exemplified by values of (..88 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 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 align longer overlapping sequence effectively. Additionally, the model performance improves as N increases, As shown in Alg, [ 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 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 I 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 NI, (1,8), (2, 4), (8,2), 2), which indicates that balanced combinations of and N yield better performance.
Related works
To the best of our knowledge, 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

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to produce consistent prediction outputs for lagged input sequences. The data-moder 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 L.TSF study will inspire further active research in the field.
Contrastive Learning. Contrastive learning [41 g 38 [22] employ pretraining strategies that encourage augmented positive pairs be closely aligned 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-awer -waree 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 metric designed assess the output alignment of L.TSF models. However, its application should be approached with caution on highly volatile datasets that fall outside the typical scope of LTSF tasks, model predictions may experience considerable volatility, Nevertheless, as demonstrated in Sec. !1 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 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 per- formance, alternative distance functions such as [25] 29 7 which capture different characteristics of time series data (e.g.. shape), may serve 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 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. 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 (Sw.eeeeMMMTT/NNNRR222222222 Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea governmen((MSIT) (No.RS-2020-1 100200112013300 336,, Artificial Intelligence Graduate School Program(UNIST).
and
References
[1] R. Akerkar and P. Sajja. Knowledge-based systems. Jones & Bartlett Publishers, 2009.
[2] A. Azlan, Y. Yusof, and M. M. Mohsin. Determining the impact of window length on time series forecasting using deep learning. International Journal of Advanced Computer Research, 9444):200-26) 2019.
[3] 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): (0(1):10 178 00:177-1944 194, 2003.
[4] Burgeno and S. Joslyn. The impact of weather forecast inconsistency on user trust. wea. climate soc., 12, 67966696669 690996944400 2020.
10

---

[5] J. N. Burgeno and S. L. Joslyn. The impact of forecast inconsisteney and probabilistic forecasts
on users' trust and decision-making making. Weather, Climate, and Society, 15(3):693-709, 20233
[6] 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.
[7] M. Cuturi and M. Blondel. Soft-dtw: a differentiable loss function for time-series. In International conference machine learning, pages 894-903. PMLR, 2017.
[8] E. Eldele, M. Ragab, Z. Chen, M. Wu, C. K. Kwoh, X. Li, and C. Guan. Time-series repre- sentation learning via temporal and contextual contrasting. arXiv preprint 1rXiiv2106.14111 2021.
[9] 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,
2023.
[10] L. Euler. Introductio in analysin infinitorum, volume MM Bousquet, 1748.
[11] A. Falk and F. Zimmermann. Consistency as signal of skills. Management Science, 63(7):2197- 2210, 2017.
[12] 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.
[13] G. Hebrail and A. Berard. Individual Household Electric Power Consumption. UCI Machine Learning Repository, 2006. DOI: https://doi.orr// wtt.:/oorrrr..44322//. 10.24432//C8K544
[14] 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.
[15] P. J. Huber. Robust estimation of location parameter. In Breakthroughs in statisties: Methodology and distribution, pages 492-518. Springer, 1992.
[16] O. Huhtakangas. Rolling forecasting in budgetary systems: case study. 2020.
[17] A. Iqbal, Shahid, M. Roman, M. 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)xc0009862, 2025.
[18) S. Y. Jhin, S. Kim, and N. Park. Addressing prediction delays in time series forecasting: continuous gru approach with derivative regularization. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pages 1234 1245, 2024.
[19] X. Jin, Park, D. Maddix, H. Wang, and Y. Wang. Domain adaptation for time series forecasting via attention sharing. In International Conference on Machine Learning, pages 10280-10299 PMLR, 2022.
[20] 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.
[21] H. Kim, S. Kim, J. Mok, and S. Yoon. Battlig the non-stationarity in time series forecasting via test-time adaptation. arXiv preprint arXiv:2501. .44900, 2025.
[22] J. Kim, K. Park, S. Yun, and S. Lee. Ppt: Patch order do matters in time series pretext task, In The Thirteenth International Conference Learning Representations.
[23] D. Kingma and J. Ba. Adam: A method for stochastic optimization, 2017.
[24] 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- 2018.

---

[25] V. Le Guen and N. Thome. Shape and time distortion loss for training deep time series forecasting models. Advances neural information processing systems, 32, 2019.
[26] L. Liebel and M. Körner. Auxiliary tasks in multi-task learning. trXiv:1805.06333 2018.
arXiv preprint
[27] 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, 112006615-106644 2024.
[28] M. Liu, A. Zeng, M. Chen, 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.
[29] 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.
[30] 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:23 10.00625,, 2023.
[31] E. Losee and S. Joslyn. The need trust: How features of the forecasted weather influence forecast trust. International journal of disaster risk reduction, 30:95-104, 2018.
[32] C. Ma, Y. Hou, X. Li, Y. Sun, and H. Yu. Long input sequence network for long time series forecasting, arviv preprint EXXv:2407.15869, 2024.
[33] Y. Nie, H. Nguyen, P. Sinthong, and J. Kalagnanam. A time series worth 64 words: Long-term forecasting with transformers. arXiv preprint arXiv:2211. 2022.
[34] H. Sakoe and S. Chiba. Dynamic programming algorithm optimization for spoken word recognition. IEEE transactions on acoustics, speech, and signal processing. 26(1):43-4)) 49, 1978.
[35] 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.
[36] 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:2210.021166 2022.
[37] H. Wu, J. Xu, J. Wang, and M. Long, Autoformer: Decomposition transformers with auto- correlation for long-term series forecasting. Advances in neural information processing systems, 34:22419-22430, 2021.
[38] Z. Yue, Y. Wang. J. Duan, T. Yang, C. Huang. Y. Tong, and B. Xu. Ts2vec: Towards universal representation of time series, 2022.
[39] 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.
[40] 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 11121- 11 128. 2023.
[41] X. Zhang, Z. Zhao, Т. Tsiligkaridis, and M. Zitnik. Self-supervised contrastive pre-training for time series via time-frequency consistency. Advances in neural information processing systems, 55:3988-4003 2022.
[42] 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 1110--1115,, 2021.
[43] T. Zhou, P. Niu, L. Sun, R. Jin, et al. One fits all: Power general time series analysis by pretrained Im. Advances in neural information processing systems, 36:43322-43355, 2023.
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A Exemplar illustrations on importance of output alignment
Budget allecation in March
Budget allocation April
Wast of budget
Figure 8: A simple example illustrating the importance of output alignment in electricity power demand forecasting. The two predictions differ, causing confusion among people, and they realize that the budget allocated in March was a waste.
B Proof
B.1 Proof of Theorem 4.1
Proof. Rewriting the equation on the left side of the theorem, we can express form the perspective of individual elements:
pf-1 1 h' W 1DP Pr. 1
(8)
The squared magnitude of the difference between two complex numbers pk, and pi,, are given as follows:
1/ - 1
(p), -P:3) F. - F3))
(9)
Where pf F and p² F, are the complex conjugate of p\z, and p¹ Now, expressing Pky and pd including the complex conjugate polar form:
10
j0 rf PF. 1p², e
P P
(10)
18 p pd F. = :
p F. = pr
(11)
where j V and is the angle (phase) for the corresponding complex number. We substitute into the expression and simplify the expression:
lpk, p (Ipk:le 10.
lp X(pbale 110t.. lpp.le -P00...) 110 rh.,
(12)
= |Ppp,11 12 + 1p..1 2 |p
-8 -0,3,) + -100 -1(0,b, - (o2
(13)
Using Euler's formula e0 co) + (s) trigonometric expressions:
10
we can convert the exponential expressions to
1p: 12
|p. 12 p 2 1lppp 1p fcos((0 -pp. + jisin(p -6p3
(14)
+ cos( c -0p:.1 )) + jsin( - (0ph. - ppz.)) ))}
(15)
= 1p/.12 + 1 2 pppp 1p: |cos(0 0pz.)
(16)
Utilizing the angle (phase) notation Z, we can express the equation as:
|p: 2 1p!,1 + 1p 2 ph /cos <<pp: (
(17)
13

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Reducing the final equation implies increasing the cosine term, which in turn signifies aligning the Iwo phases, since one predictions is constant by stop-get gradient sg(.) operator Alg.
PF. p + 0 = cos(<pp; - LPF,))
(18)
cos <pp; Lpr,)) + = Lp) - Lppr -0 0
(19)
PF. pr::l +0 1/pp /p/ +
(20)
B.2 Proof of Theorem 4.2
Proof. Since the two phases, i.c., <p),, and <p; are aligned, the Eq. (17 Sec. B.1 modified
as follows:
PF. pf = 1p/,11 + p 2/p p
(21)
By factoring, it can be expressed in the following perfect square form.
P pi: (1p 12 1))
(22)
Since one predictions is constant by stop- graddrradent sg(-) operator Alg. minimizing (p), -PP; leads to 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 2x2 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 the amplitude. Overlapping timestamps of prediction outputs for five lagged inputs are shown, with the ground truth indicated by black
dashed lines. Figs. to 12 show the results from DLinear 40 PatchTST 33, TimesNet [36], and iTTansformer
30
Value
Timestamp
Tedk
Groand bark
Pradkcioa 2
Prodiction
Productios
Production
Pock
Dedictiss
Timestampp
Figure 9: Visualization: Upper shows the prediction of DLinear 44 using only the regression loss on the ECL dataset (13), while Bottom shows the result under the same conditions when AliO is also applied. It is observed that AliO improves alignment performance for overlapping timestamps.
14

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6
8
Timesiamp
Goozad Buch
Prodeton
Prilisiia
Timessamp
Figure 10: Visualization: Upper shows the prediction of PSt 33 using only the regression loss on the ETTml dataset [42). while Bottom shows the result under the same conditions when AliO is also applied. It is observed that Alio improves alignment performance for overlapping timestamps.
Timestamp
Greund Trals
Prodetion
Pediiiin
Prodecion
Timestamp
Figure 11: Visualization: Upper shows the prediction of TimesNet (36] using only the regression loss on the ETTm2 dataset [42]. while Bottom shows the result under the same conditions when AliO is also applied. It is observed that AliO improves alignment performance for overlapping timestamps.
15

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Timestamp
Ground Truth
Prediction
Prediction
Prediction
Pockttss
Proketen
Figure 12:
Visualization: Upper shows the prediction of iTransformer (30 using only the regression loss on the Solar dataset (24), while Bottom shows the result under the same conditions when AliO is also applied. It is observed that AliO improves alignment performance for overlapping timestamps.
16

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D
On sudden event
This section expands the sudden events mentioned in Sec.
Definition D.I. The sudden event refers to an anomalous data point input sequence that can be interpreted
(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 predictable sequences. Instead, they align closely with:
Domain Adaptation
19
Anomaly Detection
39
Test-Time Adaptation
The TAM metric is specifically designed for LTSF tasks with predictable, stable output sequences (i.e., non-perturbed scenarios). In contexts involving sudden events OOD (out-of-distribution) data, the inherent assumptions of TAM-particularly its reliance on sequence stability-may not hold, necessitating cautious application. However, if 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. may difficult to interpret as there is no explicit distinction between models and datasets. In this section [Fig. 3 facilitate easier interpretation, models are represented by different shapes and datasets are represented by different colors.
MSE Improvement vs TAM
0.12
TAM
0.04
+
0.00
10
20
MSE Improvement (%)
Dataset
ETTh1
ETTm1 ETTm2
Exchange
Traffic Weather
ETTh2
PEMS08
Model Linear
Autoformer
TimesNet
GPT2
PatchTST
iTransformer
Figure 13: The relationship between initial TAM and MSE improvement (%).
17

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F Zero coefficient of time domain alignment (AT)
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 (Ar = 0. Ap = 0) implies that AliO is not used, so it is empty. As demonstrated in Fig. 6 increasing Xr concsistently improves TAM performance. When it comes to MSE performance, there's '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 specific model or method you're using). However, in some environments, the coefficient can become excessively large, dominating the regression loss and causing performance 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 I and the number of sequences is 2 (default).
10/07
0.000
61011
0000
The OltReee
0.364
0.364
0.375
0375
0.77
0.74
8:006
0.199
3.19
0.305
0.202
0:331
0.25
0.300
6410
0.00
6.91
0A41
0,500
0.27
0.007
0.055
0.00
0.105
1,00
0.594
Fappery Crrlle and
Trogerng Cocthoe By
Fing Confloae (o)
Figure 14: Comparison of the normalized MSE (left) and TAM (right) of CycleNet trained on ETTml dataset. The x-axis represents Ap ((0.0, 0.5, 1.0, 2.0)), and the y-axis represents ÀT ((0.0, 1.0, 2.0, 5.0)).
Figure 15: Comparison of the normalized MSE (left) and TAM (right) of CycleNet trained on ETTm2 dataset. The x-axis represents Xp ((0.0, 0.5, 1.0, 2.0}), and the y-axis represents Xr (f0.0, 1.0, 2.0, 5.0)).
0.444
0.44
04444
0-4-5
1.00
E.00
0.00
0141
01107
0:54
0.55
3153
0111
1.74
677
0.7%
01995
000
0.522
0.95
1.000
1:50
01993
1:00
0085
1.
L0K0
0000
Prepanay Coethioon1))
Pragerecy Coetferm Ou1
Pregenacy Uoerhoor DJ
Faeparay Cuffissen
Figure 16: Comparison of the normalized MSE (left) and TAM (right) of CycleNet trained on Traffic dataset. The x-axis represents X ((0.0. 0.5, 1.0, 2.0)), and the y-axis represents \T ((0.0, 1.0, 2.0, 5.0)).
Figure 17: Comparison of the normalized MSE (left) and TAM (right) of i'Transformer trained ETTh1 dataset. The x-axis represents Xf ((0.0, 1.0, 2.0 and the y-axis represents Ar (f0.0, 1.0, 2.0, 5.03).
0609
3.4%
2-111
0161
(1:00
101
FIGH
0100
04
2.344
0490
2.09
The
0464
2571
0.411
0223
1.00
03773
120
D45%
0-55
5.59
0.453
60
Prousday Costioies
Proguacy Costicer
12
d0
Figure 18: Comparison of the normalized MSE (left) and TAM (right) of iTransformer trained on ETTh2 dataset. The x-axis represents Xr ((0.0. 0.5, 1.0, 2.0)), and the y-axis represents X7 (f0.0, 1.0, 2.0, 5.03).
Figure 19: Comparison of the normalized MSE (left) and TAM (right) of iTransformer trained on ETTml dataset. The x-axis represents X (f0.0. 0.5, 1.0, 2.03), and the y-axis represents Xx (f0.0, 1.0, 2.0, 5.0)).
18

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3:90
11401
1.90
8-911
0161
01000
0140
100
F-0H1
0140
01/01
1.50
1009
Tox Cww.iaieeeeo
0235
01236
A2212
1.09
1:30
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0211
HABS
5.00
B404
0432
04444
1:9
033
1:0
6416
0-15
CSw
0095
1000
2.32
0281
222
1M00
0000
21
Figure 20: Comparison of the normalized MSE (left) and TAM (right) of iraansfformrr trained on ETTm2 dataset. The x-axis represents X ((0.0, 0.5, 1.0, 2.0 and the y-axis represents Xr (f0.0, 1.0, 2.0, 5.0)).
Figure 21: Comparison of the normalized MSE (Ieft) and TAM (right) of iTransformer trained Traffic dataset. The x-axis represents Xp ((0.0, 0.5, 1.0, 2.0}), and the y-axis represents Xy (f0.0, 1.0, 2.0, 5.0)).
00/00
3:433
00064
B.30
01077
L.K0
0.0.0
2.00
0:34
0.33
3
4
6122
2.94
(U0)
2013
B.41
0441
2.50
0462
0:20
6222
0203
6225
1440
04566
0.411
0.41)
60
Pmngerec Oed
Conficien
Properry Conthoen
Figure 22: Comparison of the normalized MSE (left) and TAM (right) of iTransformer trained on Weather dataset. The x-axis represents Xf ([0.0, 0.5, 1.0, 2.03), and the y-axis represents AT ((0.0, 1.0, 2.0, 5.03).
Figure 23: Comparison of the normalized MSE (left) and TAM (right) of TimesNet trained on ETTh dataset. The x-axis represents Xp ({0.0, 0.5, 1.0, 2.03). and the y-axis represents Ar (10.0, 1.0, 2.0, 5.0}).
0:00
64M
0:50
0.181
0.555
0.944
199
0.594
0119
0.518
0,581
1.522
B4S44
44
0.335
1000
0.682
1.000
1.771
0072
Emperiy Codllied
Trogitng Cocthoe DJ
Progeciy Decticar
Fregel Coffiiett
Figure 24: Comparison of the normalized MSE (left) and TAM (right) of TimesNet trained on ETTml dataset. The x-axis represents Ay ((0.0, 0.5, 1.0, 2.0)), and the y-axis represents \r ((0.0, 1.0, 2.0, 5.03).
Figure 25: Comparison the normalized MSE (left) and TAM (right) of TimesNet trained on ETTm2 dataset. The x-axis represents Xp ((0.0, 0.5, 1.0, 2.0}), and the y-axis represents Xr (f0.0, 1.0, 2.0, 5.03).
G Dataset explanation
We provide brief description of each dataset used in our experiments, referenced in Secs. 5 and
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: ETThl and ETTh2 (sampled hourly), and ETTml 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 single houschold.
19

---

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 influenzaliike illness patients.
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).
PEMS:
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 [3
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 Sec. [
Table 3: Descriptions of datasets. vars means the number variate each dataset.
Dataset size (Train / Validation Test)
Dataset
vars
Prediction Length
Frequency
Domain
24, 48. 168, 336, 720 (Autoformer)
ETTh1
7
96, 192, 336, 720 (other models)
8545 / 2881 2881
Hourly
Temperature
24, 48, 168, 336, 720 (Autoformer)
ETTh2
7
96, 192, 336, 720 (other models)
8545 28811 2881
Hourly
Temperature
24, 48, 168, 336, 720 (Autolormer)
ETTml
7
34465 / 11521 / 11521
15 min
Temperature
96, 192, 336, 720 (other models)
24, 48. 168, 336, 720 (Autoformer)
ETTm2
7
34465 11521 / 11521
15 min
Temperature
96, 192. 336, 720 (other models)
Weather
21
192, 336, 720
36792 5271 10540
10 min
Weather
Electricity
321
96, 192, 336, 720
18317 / 2633 5261
Hourly
Electricity
(FCL.)
Traffic
862
96, 192, 336, 720
12185 71577 3509
Hourly
Transportation
ILI
7
24, 36, 48, 60
617 74 170
Weekly
Health
Exchange Rate
(Fxchange)
8
96. 192, 336, 720
5120 665 1422
Daily
Economy
Solar
137
96, 192, 336, 720
36601 / 5161 10417
10 min
Energy
PEMSO3
358
12, 24, 48, 96
15617/3133
5135
5 min
Transportation
PEMS04
307
12, 24, 48, 96
10172 3375 / 3375
5 min
Transportation
PEMS07
883
12, 24, 48, 96
169117 56227 5622
min
Transporiation
PEMS08
170
12, 24, 48, 96
10690/ 3548 / 3548
5 min
Transportation
H Experiment configurations
To ensure fair model evaluation, we utilized the official GitHub codes provided by six benchmark mod- els: Autoformer (37), DLinear (40), PatchTST (33). TimesNet (36), iTransformer (30), GPT4TS [33), CycleNet (27]. For all experiments, 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
36
30
43
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)
20

---

Prediction lengths. We primarily used prediction lengths of {96, 192, 336, 720). For the ILI dataset http://gghuub.ooomu/mm//aae/e
followed prior works and used {24, 36, 48, 60). As an exception, for the [TT h2, ml, m2) datasets, we followed the Autoformer paper and used (24, 48, 168, 336, 720) (for Autoformer) and (96, 192, 336, ]20} (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--tii number of samples N in Alg. and lag in Alg. [] -wwere
their default values of and I, 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 (Ar) and frequency domain (Ap) were selected from the ranges (1.0, 5.0} and {0.0, 0.5, 1.0, 2.0}. respectively (default values are Xr and A 0 since the frequency domain is optional). Additionally, the AWL [26] technique was employed 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 initializa- tion 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:
DLinear:
PatchTST:
TimesNet: httee///..ttppp//g/hhbboco/////t
iTransformer: htaps///tn/argggeooooooo///o//aa
CycleNet: httppr//gttulocm/////////////100
We followed the implementation code listed above (GitHub) and used the same learning rate, batch size, and number of epochs shown in Tabs. 4 to II The optimizer used is Adam 23
Table 4: Learning rate and batch size of Autoformer used in each datasets.
Dataset
ECL
ETTh1
ETTh2
ETTml
ETTm2
Exchange
Traffic
Weather
Learning rate
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
batch size
32
32
32
32
32
32
32
32
Epoch
10
10
10
10
10
10
10
21

---

Table 5: Learning rate and batch size of DLinear used in each datasets.
Dataset
ECL
ETTh1
ETTh2
ETTml
ETTm2
Exchange
ILI
Traffic
Weather
0.001
0.0005
Learning rate
0.001
0.0001
0.05
0.0001
0.05
0.05
0.0001
0.01
0.005
batch size
16
8
32
32
16
32
Epoch
10
10
10
10
10
10
10
10
10
Table 6: Learning rate and batch size of PatchTST used in each datasets.
Datasct
ECL
ETTh1
ETTh2
ETTml
ETTm2
ILI
Traffic
Weather
Learning rate
0.0001
0.0001
0.0001
0.0001
0,0001
0.0025
0.0001
0.0001
batch size
32
128
128
128
128
16
6
128
Epoch
100
100
100
100
100
100
100
100
Table 7: Learning rate and batch size of TimesNet used in cach datasets.
Dataset
ECL
ETTh1
ETTh2
ETTml
ETTm2
ILII
Exchange
Traffic
Weather
Learning rate
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
batch size
32
32
32
32
32
32
32
16
32
Epoch
10
10
10
10
10
10
10
10
Table 8: Learning rate and batch size of iTrrnsformer used in each datasets (excluding PEMS and Solar).
Dataset
ECL
ETTh1
ETTh2
ETTml
ETTm2
ILI
Exchange
Traffic
Weather
Learning rate
0.0005
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.001
0.0001
batch size
32
32
32
32
32
32
16
32
Epoch
10
10
10
10
10
10
10
10
10
Table 9: Learning rate and batch size of iTransformer used in each datasets (including PEMS and Solar.
Dataset
PEMS03
PEMS04
PEMS07
PEMS08
Solar
0.0001
Learning rate
0.001
0.0005
0.001
0.0005
0.001
32
batch size
32
32
32
16
16
Epoch
10
10
10
10
10
Table 10: Learning rate and batch size of GPT4TS used in each datasets.
Datasct
ECL.
ETTh1
ETTh2
ETTml
ETTm2
ILI
Traffic
Weather
Learning rate
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.001
0.0001
batch size
512
256
256
256
256
16
256
512
Epoch
10
10
10
10
10
10
10
10
22

---

Table 11: Learning rate and batch size of CycleNet used in each datasets.
Dataset
ECL
ETTh1
ETTh2
ETTml
ETTm2
Traffic
Weather
Solar
Learning rate
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
batch size
128
128
128
128
128
Epoch
30
30
30
30
30
30
I Full experimental results
This section presents the comprehensive results of each Long- Term Series Forecasting (LTSF) models,
Autoformer (37] (Tabs, [2 and 13). DLinear [40] (Tabs. 14 and 13 PatchTST
33
(Tabs. 16 and
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 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.
23

---

Table 12: Full results of the proposed AliO method on the ECL. ETThI. ETTH2, ETTml, ETTm2, Exchange, Traffic, and Weather datasets and Autoformer [37] The results are reported in terms of MSE. TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https. //github. com/ thuml of ormer) ormer) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 6.98%, TAM: 24.24%, MAE: 3.38% 3.38%, MAPE: 3.76%, and RMSE: 3.12%. The maximum improvement of AliO over the baseline MSE: 27. 7.9%, TAM: 11.88% MAE: 15.17%, MAPE: 15. 4,44%, and RMSE: 8.58%. Exchange, Traffix, and Weather are in Tab. [3
RMSEL
0.400
0-464--00000
00000
0.472--0.000
0.503
10000
100000
1,610 11611=10000
0.628--0.01
1101
1067-00111
13.2 13 11.11 0.665110000
0,549 100000
100000
10
0.004 10
10000
10053
10000
100013 10000000000
0.464
0,,409
00.175 0,628
0,05 0.451=0.0100 11.573++0.5 5959 0,667
11.957 0995=00222 .22 (16900000000
0,665
0.515
0.008 6649
0.000 0.669
01.50
100009 06667
10004 1-220
1.094 10.03 01.40
18.10000 00660
1.158 01.550
0.360=10,006 1.422 0.000
0.
MAPEL
3.296:10.133
3.402 0.099
3.350
0.253 0.008 0.042 0,0 0.006 1.004
3.363+-0.040 0.000
0.47
10.74710.44
0,002 0.420=-0.012 10.889::.113
11.677-10.81
1.392==0.023
0.054
1.580
1.749-10.02
2.044
1.637
1437-102210
0.410 -0.01 2.455-
1.402 2.458-1
2.458-1..075
100078 00078
2.579 1.579110.05
0.005
2.60
1:00000000
2.565
1.307
1.538-0.004
161
1.230-0.100
1.054
AliO
MAEL
0.029 0.001 0.308-10.000
0.038110.008 004 0.324
0.223 0.034 004 1.331-00005
0.330==0.015
0.05 1000000000 0002 0.416
0022 0.479=00.00
0.516
0.456 1.0034
E0, 00,003 0.340=0.003
100005
0.427
0.455
0.466
10.003
1000-000-0
0,394-0.011
0.445.0.01
10000
10,000
0.44-10.080
0.036
0.25910.000
0.259 10.05
0.279 0.007
2998
0.354
000001
0.01 0.00 4.00-00.02
3020000000
0.479
0.501
TAML
0.036-10. 0,004
0.037=0.000
00000
0,036
100000
0.057
100008
0,046
0.034
0,025
0,003
0. 0057-00000
0.02 0400
1000-00.004
LIO1
0.05
0.05 S000O
1019
N032-=0.015
100008 0.000.00000000
0.4521: .009 0.193
0.033
005 0.215 0.003
3.501 118 0.509+0.00 013 0.221
0073=0.000
0.395-0.022
11 11.63220.44 477 0.639+0.011 .011 0.445-0.0114
000024
0.521 009
12111000
0.444 1.000 0000000000
10.0000000
0099
200
2
10000
5
+514+00073 0544100000
MSEJ
0.484
1444100000
0.265
0.301
0,421
448
0.450
0
0.38
2.83410.081 0.701 0.STE
0.440
118
5655 1
0.313
Autoformer
0.
0.
0000
3.445 0.478+0.015 .010
000
020
12.533-0.652 0.689 10
022
000
0000
0.668 10.000
(00
10001110
009
0.39--0.001
RMSEL
0.46810
0.481
10. 10.85320 413 0.63910
0,708
0,7194
0.3 0,741
12. 123.0.06 0,529
06692
0.685
(1614
766010
7737
0.69
0.500
968 0.00000000000
0.
0
.002 0311 00S 3.398 230
3.41610.188 185
000 3.46610.224 224
150
191
191
245
10.005
2.789 130
180
0000
000
100100000
0229
35
MAPE]
11 11.810
11.786=
12.533
1,624-00
2
1
2,5633
8.85110
78811
1.8366
1.966
1.5
1.08
1
0239 000
1000
2011
002
0.411 1000
0100
1000
030
0. COCDEEEAE
1000
009
(006
0.501 100000
R60
1000
SSS 0330230 1003
100
Baseline
MAEL
0.340
0.8622H
0.433
0,5272
0472
0415
0-200 FOTTE
0.49
228
0.
0.42
(.
0.
0.
96 0044 .008 0.03540 000
000 0.04010.0001
S0US
005
T0U
T SNN
TAM
0.219 01042
0 0.20 013 0.04000
410 1
0.280 000 T
0.377-0101 10994000
0,579 00864
0.041
0.05
0.517=0 1A
0.549 000000000000000000000
0.442 100
336 0.470 TEDD
40101100000 01000
0.493 0.00
0000
0
0. 0.502 19% 0
1..0000000
000
0.232+00000
AVG 0,229
0.40 4088 000
0010116
720 0.477
129
0
10.000
0.479
072
0.447
0.544
0,48 000
111611100011
012
220
0006
0.408 0
AVG 0,77
MSE
0.408
0.10
0.3
192
336
720
24
48
168
336
AVG
24
48
168
720
AVG
24
48
96
288
672
AVG
24
44
96
288
672
Models Method
Metric
ECL
ETThl
m
m2
24

---

Table 13; Full results of the proposed Alio method on the ECL. ETTh ETTH2, ETTml, ETTm2, Exchange, Traffic, and Weather datasets and Autoformer [37] The results are reported in terms of MSE. TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We usc official GitHub code (https. //github. com thuml/ out of ormer) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 6.98%, TAM: 24.24%, MAE: 3.38%, MAPE: 3.76%. and RMSE: 3.12%. The maximum improvement of AliO over the baseline MSE: 27 900%, TAM: 11.88% MAE: 15.17%, 3.38%
0.000
0,000
0,011
0.788-0.000
0,,00
0784::0002
10,000
1000000
0010
1,07
RMSE.
0.373
0.525
3.369 1.039
0.600
3.782 0.004 0.783
0,7844
101 03800--00
0663 00900-10000000000000000000000000
0.788
0.787
0
0 0.554 000000002 0130 0.700 0.78
0..5
455
MAPEL
0.005 1,648
2.338
0.,00
00.04
6.627-00.99
3,495--11 1111770
17707
3.782-0.01
.006 1313+D
388
14 SOOTEE
766611 1024
0.386 1386-00000 4.313
4,130+
170000005 3,970
10.004 13.076
13.7661
12.
0.009
0.488=0.000 00.007
1.079 022
280
0000
0,368 000
(350 10,00
0.270
0.375
0. 1.000000
ATiO
MAEL
0.379
0.379
0.386-0
0.332
0.333
04410
0.38
0.41
0.350
0.018 0,007
0.275 0.001 0.01 0.011=0.0000
008
0.0000
0 0,007 D0000
104003
0,041 10,000
S005
TAM
0.015
5 SOSSEE
0,010
0.05
0.139-10.000 00005
111
00000000000000
000
300
10000000
0.4361-0.012
1021-00000
MSEL
0.6
0.621
0.613
01222
0.:300
04401
MAPE: 15. 4,44%, and RMSE: 8.58%. ECL, ETTh1, ETTh2, ETTml, and m2 are in Tab. [2
Autoformer
0.53
00.00
0
220
0010000000
0000
1000
122
100011
RMSEL
0.394:10.015
0.532+0.010
-0003 1..0 130 0.66811.0031
6.640 123 1,0534
0.662--0
079810.000
0659
0.7880
0.3
0.582
1.700
10.0200002
3.522 699
201
4,207 MAMD
218
11666 0.655
0.78
MAPEL
1222
4.14
4,320
1215
13,01
1 SESTES
11
14
Baseline
MAEL
0286-10001
0009
049610.0
18844-00022
1.496--4177
0400
000
0ER00 DDS
023
UND 0 SELLOOI
120
0110
1004
0.447 0,0 100
FZD
ON0
S00
AN
0,031 1031
0014
TAM1
8 0.5T
0,631 -0 00099
336 0.367 022 01966
01$510000
0.284 100400101
1.110 185 100
0.499 16.00000
529
0.626 600
1634000000
20 013
0434
AVG 0.342
MSEL
06377
0.634
0.634
96
192
336
720
AVG
96
192
336
720
AVG
96
192
720
Models
Method
Metric
Exchange
Traffic
Wcather
25

---

Table 14: Full results of the proposed AlIO method on the ECL, ETThl ETTH2 ETTml, ETTm2, Exchange, ILI, Traffix, and Weather datasets and DL [40]. The results are reported in terms of MSE. TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https //github. com vi vvav /DLineDrn to train the model with the official configuration The average improvement of AliO over the bascline is MSE: 5.62%, TAM: 36.21% 16, MAE: -3.8 5.82%, MAPE: 2.25% and RMSE: 0. 0.79%. The maximum im improvement of AliO over the bascline is MSE: 36.07%. %,, TAM: 77.1 77.73%, MAE; 8.2 %. 2.25%
RMSEL
000
1000
1411:000000
000
0.451 1000000
0.4002
0,608
0.000
1435-0.0000
0.635 0.000
0.000
0.685
10,000
0,646
0.538
0.000
0.607=0.000
0.607 0.006
0.674
0.007
$20-100
0.099
0.660
0.00
0.544
0.577 10,000
0,000
0,606
0.650
0.650.1.0000
0.000
0.594 1.594.0.035
0.400
0,478
0.002
10.00S
10.000
0.628
0.51210.772
0
0.478
1
1L59T 0000
100
111
2,30 10000
2.2 10000
MAPE!
0.30 1ED MN 1983
,000 2.14
0.267 0.267=-0.000 2,451
2,
8.766
1.344
2
11:-1----00 0000 8.
8.
00000
0000
0000
1.000000.000
000
AliO
MAE
0,237
0.010 0.000 2550--0
0.300-
0,2644
0.55
0.013 0.0000
1.008=0.000
0.008 0.008=10.000 10.001 100
10.000
(00
TONTE OMN
TAML
0,040
0.030
0,010
0.0
0010
0.00
0.00
0.00
0.00
0. 1.1400 000
1.3: 000
1.169 000
1.203:00.000 0WO
121
0. 1000
M00
2AN
NON
OON
100
0.40 10000010000
20N
0.284 1S0400
50000
0.394 MA075
MSE,
0.330
0,404
4600
0.440
0,419
0.290
0,300
0.677
0.446
0.296
0.354
0,160
0.225
DL DLinear
0.
0.
100000000
000
4000
=: MO0000
0000
10.00 VM 0A17400
IM 1001400000
0000
1004
$37100.00
083
RMSE
0.374-3
0.250= 000 2.300 1
0.264 021 2,297-1-000 00009
04662
0.3 1ET
& 0663
0.312 10.017 1325 1315 0.530-100000
0.
0.6651::06.0
MAPE: 15. 19%, and RMSE: 5.61%. Exchange ILI Traffic Weather are in Tab. 15
0,391
0:41 1
0. 0.301 0000 2.57 0.55
8.691 =0.011 0
0.44
0.5
0.55
0.
MAPET
000 0.237 000 2.149
1.0005
100005
000000
0.268 000 2,283+00.04
200
8.794
1.359
1.494
1005
1
2 TOLT-OO
10000
21 3131
2 2240
11
2
1047
1151000
0.34910.00 1.210
0,395 848
1.0
8
2
Baseline
MAE,
001
100
1419
0V3
0.4330
0.493
0 2200
0.436
0.436
0. 0.354
0.413
000
0.416
1077
0.456
0.5 0.586.10.000
0.453
0.300
100
0'886
101
10.000
1100
0,422
S0000
0007
0,366
10.000
0.411
0.380
0.
0.
00171-0.000
0013ED
00000
-0000
0023 SUU
I
20
101
ION
0.443 0037
TAML
M028
0023
1 MO57
10.00 0653
720 0.204 1.01
166600002 1.00
0.687 0000
0.00
0.0
0.28910.09 100000000000000
MSEL
1110-10.000
0,153 00000
0. 0.169-0.000 00000
0000
0:37==0000
000
0.455
0.474
4.740.0000
42210
0.42210000000
0.291
+0,00
1011
0.44
00:000 1
+00.0000
160
100000
0000
10.0000000
0.358 10004 1
0111
2231
0.:00
0.427
0,408
0,481
0.442
0.330
0.
0.
0
96
192
336
AVG
96
192
192
330
720
AVG
96
192
1
3.00
720
720
AVG
960
192
336
720
AVG
AV0
96
192
336
720
AVG
Models Method Metric
BTThl
ETTh2
l
ETTm2
ECL
26

---

Table 15: Full results of the proposed Ali method on the ECL, ETThl ETTh2, ETTml, ETTm2, Exchange, ILI, Traffix, and Weather datasets and D DLinear [40] The results are reported in terms of MSE. TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https. //github. com vi vvaa SaivaaD/innee /Diineer to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 5. 5.62%, TAM: 2.25PE22.55. 2.25% and RMSE: 0.79%. The maximum improvement of AliO over the baseline is MSE: 36.07%. TAM: 77.73%, MAE; 8.2 %. MAPE: 15.19% 19%, and RMSE: 5.61%. ECL., ETTh1, ETTh2, ETTml, and ETTm2 are in Tab. [4]
RMSEJ
0.280 0.000
0.396 0.002 0.0
5.03=00.030
-9044:0.074
0.536=
00.200
1.493 000
14.1100011
1.441
1.470==0.000
0,005
1412--0.050 1.510=
0.002
1.478 0.003
0.65.
00000
2 0.664401000000
0.656
(0000
10,000
0.683:00
-000000
00.000
4411
1110000001
10,000
0.511 1-1.000.000
0,000
0.000
0,000
141111002 0-491
224 0.49 0
0.000
0.505 114877788
0.664
0.569
0027
MAPEL
1.120
1,594
2,222
207
0.051 0.186
0,258
1.78 1,781 -
3.689
0.059
2.507 190
0.000000000
2,278=00.016 0,016
2,721
2,749
0,000
10,000
2.600
7769
TOLEEEET MO 10,800 COLEEE
11.9
11.236
2.76%
11.9
0.0000
0.002
0.425
0.002
0.679
1161
1,030 1O0S
500
AliO
MAEL
0.015 0.000 0.201
000 0.291
0.3992
EE MOR9E
500000
SS
TAM
0014
10.02 0010
6 SIIZE CAD
NN
UUN
CAN
19 1111
D.
MSEJ
0
1.000000
0.3201
135
00.000
1000000000
ON
OM
0.823
5220
2.100
2
D DLinear
0
22
20009
02933
RMSEL
0.160 0,015 001 0.295 2,12 000000000000000000000000
00,03 0.
1.
1521
0.577
1766
1198
MAPET
1584
2,98
163 1052-
0.833 129 0.033-0.017 0.689 0,0 1.800
10993
200
1 1100
11000
10.
0.29 10.
00.008
00.00
0.434 10,016
36 2.072 0.144 00009 1
1,091 200000
1191=0,05
20,027
1000
101
II
20000000
167--00002 2002
Baseline
MAEL
0.208
2..2+-00.60 0.019- 0.407-0
1,08
1,088
2.247 0.15210.02 1,69
0.412=00.01 0.026 0,29
1999
2299
00000 0.316 0.367
1,078
0.25
0.27
0.330 0.055
24 2.335 149 0.196E0 039
120
2.269=0.077 0.136 020
60 2.315 0.132 422
01521000023
0.436 1000
I
0.264 0.006-0.00 0.000
0.0171100000
0,024
720 0.468 100001 0,026--000
1.007-00000
0.007
TAML
0.023
0.175 100 (
0.216 0.001 0
36.21 % MAE: -3.8 8.82%, MAPE
0.086
192 0.404=00.000
0.43::0.00)
720 00.00
AVG 10,000
MSEJ
96
192
336
720
AVG
48
AVG
96
336
AVG
96
192
336
Models
Method Metric
Exchange
Weather
Traffic
ILI
27

---

Table 16: Full results of the proposed AliO method on the BCL, ETTh1, ETTh2, ETTml, ETIm2, ILI, Traffix, and Weather datasets and Patch [33]. The results are reported in terms of MSE. TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https:
//github. com/ wwg.nieeg//e.mm PataccttTTT to train the model with the official configuration The average improvement of AliO over the bascline is MSE: 2.58%
TAM: 35. %6, MAE: 2.42%, MAPE: 4.40%, and RMSE: 1.34%. The maximum improvement of AliO over the bascline is MSE: 20.93%. TAM: 60.37%. MAE:
0000
1002
000
10000
1001
10,000
0000
0000
RMSE,
0.361 1551100000
0.4065
0,61 06115
0,641
0.000 0966--1110 0,525
0.385:FV
0,40 1401=00000 9,454 0,641
0.550
SA0O 0..0
0.000 1:38 1000001000
9,533 025.00.52..
0,000
00000
-.0000
0,000 0.289 0001 2.554410.00 20000
.
199
1000
0.337 1-955--0000
1T1
THE T
MAPE]
0.221 0.000 2.276
0.238= 0.000 2.462-
0.257=0.000 0.000 2.468=-0.000
0.251-00.02 2.440--000000
9,358--00000
9.454
9.425
2459
1,630
2.34
2.271
0.477 1
10.0000000 2.
0.434=0.000 0.000
0.461 0.002
1.0000
900
0,00
10000
1.000000
100
107700000
AliO
MAE]
0.419=0.000
0.429-0.000
0.3990
0.429
0,434
0.34--00
0.386
1419100000
0,289
031271
0.386
0.376
0.25%
TAM1
0000 0,016:10.000
00011
0.165--0.000 0.,01 0.028 00,00
0.202 1000 0.0.30==0.00
0,024
10,005
0.616-0.000 0.378 0.050
00,02 2
MOO
0055
0,055 10,00
10.00
0,032 1000
1000
0.045 1..00000
0.033
0,033
-00000 0,048
0,042
0,050
0,62
0,044
0.000
0.00
0.00
0.130
0.148=00000
0.451=00.00 0.01 11611:00002
0-111=0,000
0.452-0.00
SES
.02 12---11
100002
0.367
0.415 1
0.349 00000004
00000
0.366-00000
MSEJ
0.642-0.000 -00000 0 0.411
0.669 +6.0000000 -0.002 0.439
0.341
0,28.::1
0.30
PatchTST
0.
14. 44,,4% MAPE: 9.. and RMSE: 11.01%. ILI, Traffic, and Weather are in Tab, [7]
RMSE,
0.360 0.000
0.3 0.386=:0.001
0. 0.408=0.001
0.401=:0.030
00000 1
9.653 10.008
100000
0.525
(000
-0000000000 00000
0000
000
1.0000
1.65010.214 1.5033 1
1
0.538=0.000
0.578
1.00500.001
0.6 0.64610.0044
0.59210.035
0.405=00001
0.46910.000
100000
10.00 0.005
0.606=10.01
0.501
006
0.580
0.601
0.625
MAPET
0.030 000 0.223+:0.000 2.3321:0.034
0. 0.149 0.032 002 0.241 2.488
2.493+00000
0.260=00.01 2.583
0.254+:0.23 2.74410.081
9.413
0.420=00000 9.490=000037
9.524100066
10. 10.184
1.378+0002
0.378 1.510
1.700+00.01
2.01310.003
0.342+0.001 2.175
0.370+0.002 2.350
2.302+00.04
10.02
2.320
1.057
100004
1.194
0.003
1.478
1.262
Baselint
MAE
0.037 10.001 0.203
00 ZN 1.40
0.436100001
0.465
0.431
0.338
0.401
1.200000000
0,38710030
0.391
0.23
0.382
0.252
0,291100011
0.32910.000
0..8410.001
0.314
0.203=00.00 0.011 0002
0022
DA 100000
MANOOOTTEN WI
2A
D'0S 2M
2
200
10000
II
M22
100000
10.002 200000000
TAM1
720 0.162::0.02 0.037
-00 0,03
0,065
192 0.435 0.072
0,448 :0.003 0,103
0,076
0,276=0.000 0,077
1
10:01 0149
0002
0,051
0,09
10001 0,063
720 0.368-0..000 0.063
AVG 0.557=0.66 0,059
0,06
=:0.00 DOWO
10000000000 0.
0.341 1341 0.00
0,390 11
0.
0.00
0.130=0.000
0.166=00.00
96 1412-=0.000
0:418-00023
0.36
0,000000000000000000000000000000
0.306-10:011
S000
0.418 100000
0.352-10.422
0,16:::000 1000000111
0:200=:000
336 0.275
MSE,
0.31
0,391
0.366
0.2
0110
96
192
336
AVG
336
720
AVG
96
192
336
720
AVG
96
192
336
720
96
192
Models Method
Metric
BTThl
BTTh2
ETTml
ETTm2
ECL
28

---

Table 17: Full results of the proposed AliO method on the BCL, ETThl, ETTH2. ETTml, ETTm2, ILI, Traflix, and Weather datasets and Patch [33] The results are reported in terms of MSE, TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https. httpaniieg/PPacccSS///e Pat to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 2.58% //github. com/ maginie98//a
TAM: 35.6 %.,6 MAE: 2.42% MAPE: 4.40%, and RMSE: 1.34%. The maximum improvement of AliO over the baseline is MSE: 20.93%, TAM: 60.37%, MAE: 14. 4.44, MAPE: 29.81 9. and RMSE: 11.0 11.01%. ECL. ETTh1, ETTh2, ETTml, and ET'm are in Tab. T6]
RMSEL
1.298
1.19
1.195:10.045
0.005
+0003 1.738 099 1.216
0,,00
0.598 1.222.11222020
-0,041
0.598
0,0000
0.615 0,00
1000000000
0.6.5 0,0000
1000004 0.
0000000000
142200.001
0,001 100
100
0,248 0000 2.542 0.653:11100100
0.442:3
0.4000
0.56000
13451 0.473
1.682 00,09
0,977
00,05
142000,003
0003
405
2166
MAPEJ
3,004
1,751
0000000022200000000000000 0.790 2,044:1111
1022=000000 2.482
2,472
2,551
14
10000000111
AliO
MAET
0.811:1.0111
1.391:10.00 0,042--0.000 100000 0,774
0.044 0.000 0,767
=0,004
0,785
1000
CEEEEEE NN ON 119770M00
100
100
10004
00000 0.811
0,25
S90 0.272
0.256
0.35 A3335
0.056.10.000
0,0422-0.002
TAML
0.097 0,042
0.357 0000
1.686
1228 1.431-0.100
1.317 0.11
1.349 1.479=00.09
0.39 1.98 11000000001
000000000000
1
007000000
11
1.0.000
11
MSEL
1.343 1.497
0...0.33 0337-
0.615 0.390
0.422
0,388
0.2.0
PatchTST
0.37
000000
0.566 1000
1,59
RMSEL
0.25 0000 2,570 0163
2,647 0
0
0.55
4.121
2.30 2.30510:216
2. 2.306-10.099
2.76++0.098
2.802+00688
2,553 1000
110008
8455-000
11.697 -0
14.372 233
0,282 100 1776-111
MAPEJ
2,620
13.
13.24
0,262 11..111166
Baseline
MAEL
0997110000
O. 0.900
0.92330.066
0.803
0,245 0
100 0.262
0. 1000
0,20
0,
0.33410.000 10000
0.265
40004
23122 179 0.128-000015
1.559 0.106=10017
1.736=00.78 0.09
005
1.82440.11 40 0.107
1.813+00. 86 0.1 0010000
0.110-:0.000
000
0.35810:000 000000 00
000
1040=0000
2002
D00
96 0. 0.220000022 0000=000000
0.018
101
1000000000000
336 0,247-10.000 0.117
000
000
0,010 10.00000000000000000000000000000
0. 0.229-00.00 000000000000000
TAM,
0.00 0,000
0.018-000
0.390 0,01
192 1.378=0.000 0.0
0.391 #0,001 0.
0.431 =0.00000000 1.00
0.0
0. 196:
720 1.00-00000
MSE
24
36
48
60
AVG
96
336
720
AVG
192
AVG
Models
Method Metric
Weather
Traffic
ILI
29

---

Table 18: Full results of the proposed AlIO method on the ECL. ETTh ETTh2, ETTml, ETTm2, Exchange, Traffic, ILI. and Weather datasets and TimesNet [36] The results are reported in terms of MSE. TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https //github. com/ thuml TimesNet to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 3.42%. TAM: 33,36%, MAE: 1.8 81%, MAPE: 4.01 %, and RMSE: 1.76%. The maximum improvement of AliO over the baselinc MSE; 29.0 99.03%. TAM: 51. 1..66% MAE: 9.34%.
0,408 10.0000001
0.428 0.428=00.002
+0.002
0.443
-000000
0.436-1-.0220
100020
0.641 10000000000
-0,003
0,705 10000
1000000
50,00
156:::0000
-0000
10000011
110000000
5000 1000000000000
RMSE,
0.468
0,436
0,679
0,705
0,708
0,566
1.626 0,63
(668
0,640
1660
0.630
0..00
10.00 0.65
05563
20.20000
10.33
10.15
199
-
022
121
100000
MAPEJ
2.563
2.712-+0003
2. 2.944+
0.9
2. 2.933+-0.227
0.07
1-00001144 2,788--0.144
10.036:0
10 10.429=
10. 10.342.000000
10.551-000
1.500
10.00
2248410.1221
10.3141111111
2
2
2,68
2.484
1,299
1.35
0.348- 1
0,487 10.000033300
2
0.269 00001
0.284=00.000
0.025-10.001 00.001 0,297=00.002
0.219 .002 0.039 0.003 0.313--0.001
0.426 0066
0.455 0.001
0.472-10.000 200
10000
2000
100000
100000
2012
00307- 0000
200
1331400047
AliO
MAE
0.29 0.291
(1.62-
(449--
397
0.460
409
0.265
0.31710.000 0.017 0
0.406
0
0.
1..000
100
200
S
20
TAML
0.025
0.023
1028--0.000
0.442-:0.022 0.191 0.028
0.410 000000 0.045
1030000000
0022
11610.111 0,043
0.020 1O.0OOO
0.030
10,00 1000000000
0,020
0.00
1,040
1 04533-000000 2.17710.155 158 0.64310.007 0.402 0.047 1.0.0.0453
10.00
0.026
14111000000 0.000.0.00000
0.253 0.0000000000
1.0000000
MSE
0.166
000
0. 0.183 000
0.196 .001
10. 10.999100070 0.712 0.497
0.461 0.00
00000 0.685 0004 00
0.497
502.5
1000001 0488--0000
0.468
0.318
0.318 00.00 1
0.655 0.448 10.0000
0.398
200
0.672--0.000 0.446
0..288
181 10 10000
TIW
10000
0. 4099-0000
103
1.398110
0.398 10.00
0.186 0.000
20
0.423 +0.00
110
10.0000000
0.381
TimesNet
MAPE: 23, 09.. and RMSE: 66.56%. Exchange, Traffic, ILI, and Weather are in Tab, [9]
RMSE
0.409 001
0.431 001
0. 0.449= ,006
0.477+:0003
0.640=0.008
10.981 1.0.24 0.722
0.6894:0.002
10.029
1.607 0.573
0,211100119 1.646 0.655
5000005
0.673
0.5783:0.003
1 00220000000
0.622
0.655
00.00000003
0.706
(540110004)
0.640
1100000
0.431
0.505
00000 E003
1.351-0.011 10007 1.561-0.000
0.656110.001
000039 0538100003
0.561
0.706
0.503
MAPE
0.271 100001 2.666
2.787=0.099
0.302+00000 0.945
2.883--0.092
2.820-00.0%
10.015
10.9261102266 20.26 2
10.730
1.826- -0004
0.42810033 1.81410.20
2.366
2.478 070
2.593
2,801 10.0000051
10.14
1.181 1010001111110
00000 0.020
1.290
1.584-10.03
1.32-10.112
2.560
Baseline MAE
0.288 00002
005 0.322 10.011
0.196:10.022 0.040 0.296 -0.001
0.425
-000 04000-0:0044
00000000000000000000 1080000000
000 1465--0025
1497--00000
0,3 01370=01100
336 0.452+:0013 0.085-.0.022 0.463
0.334 0.052+0.000 0.374
10001
100000
192 0.386 000 0.045 0.399
2000 0,42710.00
2000 0.465 146$100.003
10.003 0.416 10,000
266-=0
0.308
+0.002
0.345 0.0001
00.00 0.410110.001
0.352-00047
TAM
0.038 001
0. 186 001 0.038
0.202 005 0.041=00002
720 0.2222 0.0444:0.000
0.409=0.0000 0,055
192 0.469 000 0.054
0.507 0.046
0.521 0.046
00000
0.476:10.035 0.050
0.328 0.007
192 0,429 0.089
0.454 0.075
0.416.10.046 0.079
00000000000000
001
0.429 077 0.044
0.499 0,04210:000
0.412.0.0044 0.045
000
0. 1.18::10.00 0.040
0.255 0.037
0.315 0.003 0.033
720 0.430-1.0001 0.03410.004
0.296:10.0m 080 0.03611.000 003
01180000001
MSEJ
96
192
336
AVG
96
336
720
AVG
96
720
AVG
96
336
720
AVG
96
192
336
AVG
Models
Method Metric
ETThl
ETTh2
ETTml
ETTm2
ECL
30

---

Table 19: Full results of the proposed AliO method on the ECL. ETTh1 ETTh2, ETTml, ETTm2, Exchange, Traffi,, ILI. and Weather datasets and TimesNer [36 The results are reported in terms of MSE. TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https. //gitubb. com/ thuml. TimesNet to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 3.42%. TAM: 33,36%, MAE: 1.8 16, MAPE: 4.01% %, and RMSE: 1.76%. The maximum improvement of AliO over the baseline MSE; 29, 99.03%. TAM: 51. 1.966% MAE: 9.34% MAPE: 23. 0.%, and RMSE: 16.56%. ECL. ETThl, ETTh2, ETTml, and mm are in Tab. [8]
RMSEL
0.324 0.003
0.455 0.003
0.004
0.603
1.972=0.017
1.88--0.217
0.767=0.000
0.783 0.002
0.663 0,092
0.808=0.000 0.001
0.755
0.045 2.508=0.167 1.317=:0.010
1.492=0.114
1.382
1.403
0.001
2,849 0.309 1.399-
0.055
0.417=0.003
10.718 0:487=00.00
3000
100007
10 139 0.529--0.00
-0.001
0.599-00.000
0.001
000059
0.508
10.365 0.550.9999
MAPEL,
0.001 1.300
2.002
3.249=00.070
0.740 0.015 6.473
0.99
3.256-1-1 1.775
2.798=0..08
0.001 0.303-00005 0.01 2.712-
0,001 2.882=0.055
0.320
3.45-=0.094
1011 3,424=0.129
2,962
2015 2.779-0.123
2.686
2.849-0.306
T 11,699 11659-0077
STO
1- 1000 10.36
13 1E819E
2.779
13831
0011
0.439-00.000 0.004
0,840 0,07
1000000000
4002
100
AliO
MAEL
00011 0.232.10.000
0.0000 0.330
1..00 0,024 0.435-00133
0.310
0.320
0.344
0,028 - 0.319
09924-0.06
0.862
0.2700
0,890
0.55
0.289
0.004
1001
100000
DIG
2T MOND
TAM
10000002 0,023
0.020-000000
0.020
0.029
0.025
0,02
0,024
SSS
0.00
0.00
0.00
100000 0. 2.002
128
1..00.0
II
0.9.4
0.105
0200
MSEL
0.364
0.500
0.614
0.200
1
TimesNet
117
100000
0.480 COMMU
00,0000000
100 0354
RMSEL
0
0,508
111501100 0.1000.0000
1.
0.4
4405 1960
1888
192
1
179
122
135
0 0,307 200002 18888-0877
MAPET
1951
14, 1188888
1336=10.000 00000 0,746H0.012 6.5108
6.5
3
11..
500000 0.353 14,044
1000
500000
0. TO0OOTTT
1011
Baseline
MAEL
004 0.241 0:41-00.00
0.03610 0K4 0,9700
0.410H0.0W
I METEEE
0.
0.300
60 2.131 22241000028 022666000000000000
0.277
0.23
HN
IS
2NI
TAML
0.1121:0.00 0.03810.000
0,00 11
1.976=0.112 112 0. 2889
100304 03241
0011-00
192 0.285 0.015
0020--0000
0.018:0.000
0.115-0.295 000.000005500
10.18 1101
(402
00.02 00
2.759=0.888 0
0.174=0.005 0
10.00 0
MSEJ
0.217 000000
0.964
1593
192 00005
0.661 0.661-00000
0.626 00.020
36 1.949=0.033
2.002
0000
0.359100000
0.262:10.6
96
192
336
720
AVG
96
336
720
AVG
24
48
AVG
96
336
720
AVG
Models
Method Metric
Exchange
Traffic
Weather
ILI
31

---

Full results of the proposed AliO method on the ECL. ETThl, ETTh2, ETTml ETTm2, Solar, PEMS03 PEMS04, PEMSOT, PEMS08. Exchange, Trafli,, 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 bold. We use official GitHub code (https:/ Pettp:://gthhan ggithub. com/ tttmtthuual///mmm / Transformes to train the model with the official configuration. The average improvement of AliO over the bascline line is MSE: 6.59%, TAM: 28.02 28. 04.04% %. MAE: 5.41%. MAPE: 0.46%. and RMSE: 3.8 3.85%. The maximum improve of AliO over the bascline is MSE:
88.5 5.52%. TAM: 60. 37%. MAE; 73.75%, MAPE: 18.04%, and RMSE: 66. 10%. Solar, PEMS03, PEMS04, PEMS07, and PEMS08 are in Tab, 21 Exchange, Traffic,
RMSEL
1.383-1-0.000
1:0,0000
0.250 0.000 2.67 0.05 11551100000
0,400 1000000
140001000000
1O000
1661 TO000
1000001
0.036-00000 0.00 0.267==0.000 0.000 2,8024 0.418
0,383
0.001 3.073 (145
0.680
01.50
1688
0.5
0.55
1453
4441
10,000 2.512-10.018
100023
188
0226
0.00000000 21288.0000
(60
10,0%
1
186
1
10
MAPEL
00000000 02244-0000
2262-00019 2,764
0.428 00.001 10.03
0.445--0.001 11
11
24
0.46--00.004
10000000000000
0000
1000000
196 1000
AliO
MAE
1..000 0.238
0.294
0.266
01397=
0.469
0.455
0.340
0.33
0.396
01.40
444
(1.50
0.257
10.000
0.160 10000 102--00001
100
1031-00000
100000
TO
OOD
TAML
0,0000 0.034
0.031
1045
0.0.8
0.050
10,05 0.026
0,030
1.000
0000
000
0. 10.00
0.172: 0200
100
10000000
0. 0.666- 0.431 000
0.471 100
11 11.807 0.711 0 00.
000
00000
-0.5000000
100000
000000
1000
100
1000000
10.000
TO
T0V
0.112 1OT
0411 10,00
iTransformer
MSE.
0.147
0.175
0.207
0.471
0.438
500000
01548 0.375
0.3000
0.413
0.4433
0.377
0.375
00 0.400
0.472-
0.2500
0.
RMSEL
0.385110 000
0.406 000
0.424 000
0.459 000
0.418+00002 024
0.621 =:0.000
1
0.69==0
00000 2000
10 10.565 10.666 07774440000
1100000
3 0114-00000
10000000
1000000
069940000
00551100000
000
03304400
100004
0.562.1:000
000001
00000 0.643100.00)
000
0.534
46699 0.
1,596 0
1.411 10.1000005 0..62
2000 7 146210.111 0
MAPET
2.52510.030
0.042 000 0,256 0.256-10.001 10.001 2.732++0.33
2.736=00.008
3.068 =0004
2.765=0.114
10.066
9,867
093
10.520=
06666
1.775
0018--0000
101181
1,71410
1
2333--0000
200000
2,443
2012
2,670
1.169::0.000
1,302-::.013
1.374
1.77
10000
100000
Baseline
000 0,240110.000
0.233+0.011
=0000
200000
100000
04
20000000
100000
01:11+-0000
100000001
014122000
1457100011
0.407 0.04 0.06010.000 004 0.270=000001
0.314-10.000
0.352 10.00000003
0.407 1.0001
00.05
MAEL
0.300 000 0,08010.000 05:0
0:412
0000 0.257
0.404
0.45
0.492
0.448
0 01406
01.77
0130
0.41
0.301
0.083 00
0.33
TAM
0.04610.
0.049 100001
0. 0.063-00000 00,000
0.050
[00
10.001
0,08
0.487=0.003 0000 0.08 100000000
200000
0.506:10.077 0.098
0.088-00000
192 0.378 0.0000 10000000
002
0.421 0.073
720 0.429- 0.060
000
2
10,000
AVG 0.382 046 0,07410100
(00000
0.343 0.067-0.000
192 0.380==0.002 0000000000002200 0.060 001
0.418:10.00 000 0,0.8
001
720 0.488 001 0.056-0.00
100
0. 0..000000001 0.050
192 0.254 0,0 0,050
100
0.315 004 0.04910.000
720 0.413.1.0001 0.04410.000
0.048=00000
002
192 0.443 001 0.083
0. 1.1488:0.000 000
0.165:0:001
0. 180==00001 001
0.211 000
0. 0.102200002
0.386:10:001
0.455
0.292-10.075
MSEJ
and Weather are in Tab. [2
96 00.
96
192
336
720
AVG
96
336
720
AVG
96
336
336
AVG
96
336
AVG
Models
Method Metric
E
ETTh2
El
ET Tm
ECL
Table 20:
32

---

Full results of the proposed AliO method on the ECL. ETThl, ETTh2, ETTml ETTm2, Solar, PEMS03, PEMS04, PEMSO7, PEMSO8. Exchange, Trafii,
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 bold. We use official GitHub code (https:/ Pettp:://gihhen /github. com/ ttuml / Sransffrmeee to train the model with the official configuration. The average improvement of AliO over
the baseline line is MSE: 6.59%, TAM: 28.04 04.04% %. MAE: 5.41%. MAPE: 0.46%. and RMSE: 3.85 3.85%. The maximum improvement of AliO over the bascline is MSE:
Exchange, Traffix, and
101
1011
1100
127
RMSEL
0.4
0.2000
18 38
0.
0.
0..
0.3
0.40
0.S.
1000000
5000
11
180
17
MAPET
88. 9... TAM: 60. 37%, MAE: 73.75%, MAPE: 18. 8.04%, and RMSE: 66. 10%. ECL, ETTh1, ETTh2, ETTml, and ETTm2 are in Tab, 20
00000 1,845
20000
2.
1881
2.
40
2200
0000000
10000000
1IT75E 55
2
HHT
IN
11
00.09
2
2
MAE]
2357
0.61
0.2440000000
0. 1OEEE0E
AliO
308--
0.552
0.2.
0.
0,021 1,,000
000
TAMI
000 1U22S
0.00
0.00
0.00
AAN 0,00
0.00
0,00
0000
1000
II
LIN
MON
MSEJ
002 0.1922200
1225
1242--
iTransformer
0.485 0.
0.
2. 2.006 0. 0.00 1000000
0.487 001
1000
000
COD 100
RMSEL
0.45410.000
2.011=00011 0.500
1811
D.
D.
20,072
108433
1.976:1.0044
1.975
1000000000
MAPET
74777
2.99 2221972
1
2
I.
2
I
2
0000
10000000000 0.275 0000
U. AANAAN
0.263
96
Baseline
MAE.
0237230
0.263-00000
0.275
0,173
0.43
100000
ON
ON
ON
103810000
0040-00000
1060 DO00
THO MOUNO
T
20061100000 20 0.038 0038-
UN D048
0.069-10000 00522
1000 T
OIN MOOO
ONN 1
100000
1000000
0.251 W
0.25 0000
10.00
II
UII
T
I
MSEI
0.237
0,500
0.2363
0918
1001110.00
0
00000000000000000000000000000000
0.
0.
100000
0.00
00000
0.23
0
0,
Weather are in Tab. 22
96
192
336
720
AVO
12
24
48
96
AVG
12
24
48
96
XV
2-
24
48
96
DVC
12
24
49
96
AVG
Models
Method
Metric
PEMS03
PEMS04
PEMS07
PEMS08
Solar
Table 21:
33

---

Full results of the proposed AliO method on the ECL. ETThl, ETTh2, ETTml, ETTm2 Solar, PEMS03, PEMS04, PEMS07, PEMSO8. Exchange, Traflix, and Weather datasets and Trrnnfoommr (30] The results are reported in terms of MSE, TAM, MAE, MAPE, and RMSE, The best results are highlighted in bold. Wo use official GitHub code (https: : github. com/ thuml / Trransf ormer to train the model with the official configuration. The average impro improvement of AliO over the baseline line is MSE: 6. 6,59% TAM: 28.04%, MAE: 5.41%. MAPE: 0.46%, and RMSE: 3 3.85%. The maximum improvement of AliO over the baseline is MSE:: 88.52% TAM: 60.3 37%, MAE: 73.75%. MAPE: 18. 8.04% and RMSE: 66. 10%. ECL, ETThl, ETTH2, ETTml, and Em'm2 are in Tab 20 Solar, PEMS03, PEMS04,
0.293 0.293310..0 0,001
0.421=:0.00 0,001
0.576 0.001
0.926 0003
0.554
(662710.00
0.642 0.000
0.654 =00000
0.678 000000
0.650=00.017
0.418 0.000
0.473 0,000
0.530= 0,001
0.000 0,000
15.5500.000
RMSE.
0.597
1,280
0,0006
1.895 0,003
2.914
6.150=1..02
3.060 1.679
2.723 0.,010
0.000 2.757
2.788 0.009
0.296 0.000 2.889=00000 0.008
2,789 0.005
13.4100.300
0.257 0.000 15.938-0.133
15.376-00000
15.819-000522
10.565
MAPEL
14,334
15,367
0.205:10.000 100001
00000 0.300 0,001
0,417-0.000 0000.00
0.002
0,405 0,166
0,0000
0.277-000011
000 0.298=00000 0.000
0.347
0.000
AliO
MAEL
0.699-
0,271
0.278
0.215
0,279
0,264
1.:0-0.0000
(4032 - .002
1001
MNN
1.000.00 0016-00000
000
1MM011- 00000 100
1017100000
0003
TAML
I 1
100000 0,030
0,031
LEOT
0,04
(,04
OMW 10110
LE000
10.0000
IM
110000000
000
100000000
00
OM
02
102 0.250025550 1 15.7155777 00.11 1:32 0.531 10,000 0.259 1
0.086
03331110000
iTrrastormer
MSEL
0.178100
0.857-000
0
0.300
0.412
4222
0.400
D.175-1-
0225
0
0.281
0.382
0.204
1
4076-00000
10000000
10000001
01.500000000000000 1000000001
RMSEL
0,423
0,649
0
100 00.0900.111 151 0.59921
0.506.:0.000
0.579
603 0000001--0000000000
0,67
110111
15. 2
MAPET
1.287
0.037 0000 0.302 1,9266
0,042 100 0.420 002 2952-0000
20016
6,170
11
1,0000
1000000
1897
0,28 110011 1108
-
22975
000
182--00000
00
4.40 166 3.08
16, 16.1887711
0.257 10 1600
15.409
2
2.
3.
Baseline
MAEL
0.206
10000
10.000 0,038E 1002 7,00 0003
10,001
0
0.800
0,282
0.216 000
000
1
000000 0.299
0.350=0..00
0,281
0.408
0.000
0.035:10.001
0.366-10.270 00227 0.0E
0O0
AND
0.422 0.020 1
0402410000
336 0.282=0.001 0,024
0.023+0.001
720 0.359 100300000000000
0.260-0.06 0.0244
TAML
0.457 0.00000022220 01000
192 0.225 0X024
192 0.113100000 100000 0,061
0,061
PEMS07, and PEMS08 are in Tab. 21
0.394 10,001
0.425-00.00 1.000111111000
0.086-10.001
0.17910.001
0.335
00.00200002
MSEL
96
192
336
720
AVG
96
336
720
AVG
96
AVG
Models
Method Metric
Exchange
Traffic
Weather
Table 22:
34

---

Table 23: Full results of the proposed AliO method on the ECL. ETTh1 ETTH2, ETTml, ETTm2, ILI, Traffix, and Weather datasets and GPT4TS [43] The results are reported in terms of MSE. TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https //iitub.. com/ DAMO DI- ML/ Neur One Fits- All to train the model with the official configuration The average improvement of AliO over the bascline is MSE: 2.27% TAM: 33.13%, MAE: 2.43%, MAPE: 1.66% and RMSE: 1.14%. The maximum improvement of AliO over the baseline is MSE: 11 11.4%.
RMSE,
0.612-10.002
9.532 27.6 N.642=00.001
6656=00.003
10. 10.804 153 0.672
0.646 11446-10.0000
-00000
3337=00003
0.592--00000
0000100001
10,001
1.611=00.00
0035:-000000
10,002
10.212 0555:000001
100000
5700000001
11410=0.001
100000
10000000
13533-00000
L00000
10000
0.656
2412--0000 13397
0.537
222-00025 16335--1
0,611
1100010017 0.635
01594
1.606--0-233 11..5
0.500
(660
(0,64
01587
0,474
1,323 0.553
0,533
1,511 10.001 0.618
0.509 1.5091111
9.675
9.519 180
479
1522=0.02
0.001 1.470--0.022
0.388--0.001 1,622-10.025
2,180:00011
12200:0.0064
0008
1079::0MM
012
1.28310.180 0
4665-00022
MAPEJ
9.883--0
1.00002 1,981
0.340 1,001 2.2.202000000000
1385--0.000 2.
1.07
1.283
2,665
0.206 1,,000 0.028 10.003 1.296-10.00 2.811
0.396- -0.000 1.60
-0.06 2.282.2.333
0.262-10.020 22022663
AliO
MAE
0.393 000
100001 0.416=0.000
0.431 0.004 0.031--0.000 0.000 0.427=
0,009 0.053 0000 0.456--0.004
0.423-10.022
0.348
00000
100000
0.409--0.000
10.000 0.362-10.001 1001
10..000.000 0.362-
(1418--0.000
0.376-
(...0..333 0,257
(257=00.00)
1001
0,297--1
10000
0.337
1.00100.000
0.323 12.3100047
0.025-0.000 0.237-:0.00
L.025-0.000 1.,000 1.2511:0.000
0.266110.00 100001
0.388
0.41
0.3855
0.41
0.400
TAM,
0000 0.034
.020 0.0.37
00000
0.417 0.000 1.039-100000
0.039
10,0000
0,04
0022 0.288 00.003 0033-00000
200
0000 0.351 100000 0,022
000000 0.033
1000000
1029
10,000
0,027
SUUU
0,033
0,03
1029
0.927
00033
10.0000000
0,031
100:0000
0.033
1.00000
0.025
10.000000 1000
0.026
0.0000
10.000
0.028
0.265 10.071 1.02810.000
0.025
0. 0.169 0.001 0.021 10.000
0.025:10.002
TAM: 62.66%..6.. 22.06% MAE: 10. 98% MAPE: 8.16%. and RMSE: 5.93%. ILI, Traffix, and Weather are in Tab. 224
0.354 0.0000000000000
0.00
0.042 0.000..00000
MSE,
0.374
0.412:
0.452:
10.00000
0.373 2M
1.674=00.000 - 0.644-10.000 0.404
1000003 404.000000
0.286 0.286
1000
0.00
0.325 0,000 (I
0.361 1.361-00.00
101
0..00
0.414
0.346
16168
0,000
1.225.00.00 000
0.003
0.284
1.382.1..00
100
0. 1.13 000
0.154 0.000
0.167=0.022
GPT4TS
0.613 2200000000
RMSE
0.616 0.000 .02
0.646 001
0.664= 0006
10.152+0.119 0.677 000
9.679 0.651 (6511:0.02) +20000
0.540=0.000
0.606
0.601
0.540=0.000
2-241=000111 0.575
0.605
0.646-10.000
2.32010.110 0.592
0.412
003 0.305 100000 1.224 -0.000 0.483--0
4004
10.008
10.00 1.363 0.545
0.618100.00
10.005
0.51410.066
10.006
000
0.372-10.000
0.3931100000
0.41110.000
0.454-10.000
0.407-1..027
1.532100.034 0.618
9.556-10.065
9.502==0.101
9.507-0.157
1,377=00031
1.484 000
+0.056
2.184=0.033
2.350-00054
1.096
10047 1.304-0..14
2.42--00003
2.679-:0.020
0.266 0.266110.001 100001 2.707
2.915-10033
2.68611.155 150
MAPE]
1.646
2.506
Baseline
MAE
0.401 1:0001
0411 10003
0.441 0.054 - 045550000
0.458 006 0.058 005 00460000000
50000 0.
1353500111
0000
1.441-0.000
5,0,005
0.347 10000
100001
000 0.169
03392--00000
0422110000 1100001110111000
0.351 0.044 200 0.383
0.25
300 0.401 10,005
0.328 0.328
00.000000000
0.252-10.001
0.297 10.001-
000020
0,.49
1H44
720 0.418.10.11 0.045 0
1.299-10.003 0.033 0.
900 0.
TAM
0.380 003 0.060
0.001
0.418 001 0.051
0.42410.025 0.086..0.000
192 0.368 008 00000000000
0.376:000003 0.04
002
0.414 0,42
100
0.048
1000100001
192 0.3 0.03+0.000
0.365. 0.02400000
002
000
0.1 70 0.03910
192 0.233 04 0.042...000
0.297 0.048
0.382.1 0.006 0,041
0.270LL00070 0.042
100000
96 0.138 0.500.000 0.000 0.034
192 0.155 000 0.034
0.169L100000 0.03610.000
0.200 0.042+00001
0.0033 0.03610.000
003
0.363 1.00000000
0.292
MSEJ
0.
96
192
336
720
AVG
96
336
720
AVG
96
336
AVG
96
336
720
AVG
336
720
AVG
Models
Method
Metric
ETTh
ETTh2
ETTml
ETTm2
ECL
35

---

Table 24: Full results of the proposed AliO method on the ECL. ETTh1, ETTH2, ETTml, ETTm2, ILI, Traflix, and Weather datasets and GPT4TS [43 The results are reported in terms of MSE, TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https: //gitub.. com/ DAMO DI- ML/ Neur One Fits- All to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 2.27% TAM: 13% MAE: 2.43%, MAPE: 1.56%, and RMSE: 1.14%. The maximum improvement of AliO over the baseline is MSE: 11 11.5%.
S1001S
1.771000013
1.312-10.015
1.306-10.012
3.376 155 +0.022
0.614 20002
0,632-0.000
1000
1000
RMSEL
137733
1.324-1
0.662
0.636
0.633
0.6
00,002
0117
3007
3,967
3.182.10.000
3.268 04.
MAPEJ
2.854::000
000000
1111
10.000
1000
1000
TAM: 2.006% MAE: 10. 98%. MAPE: 8. 8.16%, and and RMSE: 5.93% 5.93%. ECL., ETThI, ETTh2, ETTml, and and ET'mm are in Tab 23
AliO
MAET
0,839
08812
0.852-
0.8.40
0.200
100000
0.000 10000
IO
COREEE OMD
TAML
1036
1
11
1722--004
100 100
MSEL
1300 1.701
1
GPT4TS
1.300 1
1402-HO00
10000
0661 CO000
000000
RMSEL
1,225 133777
0.S.
1211
101103
3.03
801188
00000
400
II
MАPET
3. 1994
3,423
S000 3.00
3.1
3
IS
II
Baseline
MAEL
0. 0157700000 0010 0822-1
C REREERR
220 099133
02800
0.
TO0 0.3
0.
1000
1. 866 1.861 10.050 0. 10000
I
100
TAM,
1898 10500
0. 1.15 20224100
336 0.319 00241000
0. 0.411 1000
0.1.0
0.00
00.02 0.00
10000000000
MSE
1,965
191415000
1.85
10:00)
0.36--1000
0.40 10.00
100000000000000
0.444
0.411
0,148
0.148=0.00
0000
0.246
1000
0. +027
0.411
24
36
48
60
AVG
96
192
336
720
AVG
96
192
720
AVG
Models
Method
Metric
Weather
Traffic
ILI
36

---

Table 25: Full results of the proposed AlIO method on the ECL, ETTh1, ETTh2, ETTml. ETTm2, Solar, traffic, and Weather datasets and CycleNet [27] The results are reported in terms of MSE. TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https. //github com ACAT SCUT/ Cyclellet to train the model with the official configuration. The average improvement of AliO over the bascline is MSE: 0.52% TAM: 37.91 %. MAE: 0.91%, I%, MAPE: 1.84%, and RMSE: 0.26%. The maximum improvement of AliO over the basclinc is MSE; 2.73%, TAM: 74. 74.35%. MAE: 6.79%. MAPE: 13.27%, and RMSE: 1.38%. Solar, Traffix, and Weather are in Tab, 26
(010
RMSE.
1.00 000 2.304 0.376 0.000
0.394
0.000
2,439=0000 0.459
0.415
0.0000
0,000
0.411
1111000022
0.613-10 10,000
0.000
0.678
0.0000
466610.000
0,000
0.024
1532200.000
0.0000
10.000000
000
1000
0.650
0.532
0.010=00.000 0.297 2,584 0.41
0,676
0.566
0.61
0.001
189
1002
MAPEJ
2,444
2.443
OON 0.122
0.2.66
2
9.
100 2220
0.015 00000 0.247- 0.000
0.265 10,0000000
0000
2
OON
0ON
0.261 1611--0000
AliO
MAE
0.234.10
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-.0000
0.0000
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0. 1.14 000 0.019
0.00 .00 0.010
00161-000
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TAM,
0.00
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0.00
0. 1.156 000
0.173 .00
0.170 0.023
0.000
14000
1000
TO
10000
NOW
MM
MSE
0.211
0,376-
0.42-
0.4530
0.429
28233
0.577
0.5..
0.277
CycleNet
0.
0..
0.
0.
0.
0.376 0000000000 000
0.395 000
0.265= 000 2.439=0.005 0.415 ,000
0.459 ,,000
+00002
0.41 0411-100020
0002
000
000
0620=:0.034
10000000000
(000
447000000 1088-00000
0315::00073
RMSE
9.71 S 016800
0.438 9,559 01657
0.460 000000000 9.463 01334
22 1111
06673
0.57
0,629
4022 001 2.76 0.620
0.676
0) 4088200
1.041 10.000000 4882244
0
MAPET
0.000 0.234 000 2.305 1:0002
0.247= 000 2.450
O. 7.297=00000 000 2.583++0.000
0.26 0.261 10111:00000 0.00 2,444-0000
2022
0.393 002 9.380
32
4443
0.427 022 1.43 20000
2 1
2 1ON61000
2 067210000
10811000
1000
2263
2316-0000
10000
2002
J.1 1881100000
0.389 0,330 000 H1.0000
1.233+0.144
1.770
2.208
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MAE,
0.418 001
0. 10.001
100
0.335
10,002
0.392
0.4 0.43500 (000
0. 0.459
2002
042
0.405
O. 16.000000
0.382 000
000
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0. 0.395 023
0.248 000
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0.29
0.330
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00000000
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ION
WESSED 20N
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0.272 0.075 M027-0.00 0000000044
TAML
0.019
0017
0,014
720 0.211 O,01
96 0.380- 0.002 1O43DD
0 0.462-0.000 0.000 0.01
MOSTE
0.32610.000 0039
0.000-0.0002 00000000000
0.000 1.00
0.432-003 100000000 MO000000000
0.
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0. 0.394 M
0.150:0.0000 0,000
0.73000000 0000000000000
1000
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0 223544 1000 1
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0.385 10008
1000000000000
1000
0000
0.000
MSEL
0.141=10.000
00100-0002
0.455
000000
0.373
0.424
0.456-0008
0,360
45
0 0.386-00066
06 10.000
0555
0,386
0.461
0.
AVG 0.
0
0
0.
0.53
0.
96
192
336
192
336
720
AVG
96
36
330
720
AVG
96
192
1
336
720
AVG
96
192
336
720
AVG
Models Method Metric
E
ETTh2
ETTml
ETTm2
ECL
37

---

Table 26: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTml. ETTm2, Solar, traffic, and Weather datasets and CycleNel [27] The results are reported in terms of MSE, TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https. //github. com ACAT SCUT/ HUUT/Cyc Cyclelett) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 0.52% TAM: 37.91 %. MAE: 0.91% 11, MAPE: 1.84%, and RMSE: 0.26%. The maximum improvement of AliO over the bascline is MSE: 2.73%, TAM: 74.35 74.35%. MAE: 6.79 6.79%. MAPE: 13.27%. and RMSE: 1. ..8%% ECI., ETThI, ETTh2, ETTml, and ET'm2 are in Tab. 25
0,50010 100000000
0000
10,000
000
104000
00:00000000
0,0000
RMSEL
039
0.581
000000
0...
2.328 10,003
1082+00.000
MAPEJ
0.277 1777:00000 000 1.982-100
2,110
3.188
3.
00000000000000000000000000000000
100000000
18
&
000
1000
AliO
MAE]
0.299
0.33
0.333
0.
0.33
0.
1.026-10.000
0.026 2000000
0.538 10,000 29000 00261
CON
SOODA WNN
20
NNN
TAML
0,500:1 10,00 0.250 000
0000
0.000
0.352 MM
55 UNS
222-4 MM
ON
MSEJ
1:33==0.000
00 TOTTEEE
0.338
0.485
0.33
CycleNet
0.
0.
0
1 1000000
0.500 SS00
RMSEL
0.327 00,000 2,478 0.59
01. 1
0.50
0.500
14 1000
100
1984
0.289 0.289-10.000 10.000 0.027 0.298 001 2.335110001
2.101
100000
MАPEJ
3.5.
3.
0.278-0.0011
0.322-10.000 0.000
1006-00015
0000
0000
0. 1000000
280 00002
Baseline
MAEL
0.345
0,024 0.000
0AREE S00
TO
I
1000
0.030
720 1:.51:00000 1021
TAM,
+0.00 0021
0.507 0021
0. 0.480 10000 SODT
0.307 0.307--0000 +0.00% 1.000
0.00
10.00 0.00
0.276 0.00
0.250 100001
336 0.338=000000
4.001000000
0.170 000000000000
0. 7733000
0.350-00000
0.255 10,059
MSE
0.223
0
96
192
AVG
96
192
336
720
AVG
96
192
336
720
AVG
Models Method Metric
Weather
Solar
Traffic
38

---

Pronunciation of a. a.li.o o
To prevent confusion regarding the pronunciation, this section explicitly describes how to pronounce a. 4..
[a.li.o]
The pronunciation consists of three syllables:
Pronounced like the 'a' in father (ah).
li: Pronounced with clear 'L' sound, as in Lee (lee).
o: Pronounced like the 'o" in go (oh).
Consequently, it is pronounced as "Ah-Lee-Ok", ensuring the 'L' is clearly articulated.
39

---

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Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, formatting purposes and does not impact the core methodology. scientific rigorousness, or originality of the research, declaration is not required.
Question:
Answer: [NA].
Justification:
We did use LLMs for core development.
Guidelines:
The answer NA means that the method development in this research does not involve LLMs as any important, original, or non-standard components.
Please refer to our LLM policy pettts://neurre http:///eerriiscc///bo/eeeeeee/r cc//oonferences//225//LL. for what should or should be described.
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Reviews:
[{"review_id": "lgQRPpWbag", "reviewer": "Reviewer_J5tA", "summary": "This 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.", "questions": "### 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.", "limitations": "### 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.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "### 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", "was_revised": true, "final_justification": "Considering a proposed revisions including an enhanced narrative, improved figures, additional results and visualizations, deeper insights into the key issues, and expanded discussions of the results in light of new findings and rebuttal points, I believe the submission has strong potential for NeurIPS. My primary remaining concern is whether the authors can effectively incorporate all these improvements and clearly present them in the camera-ready version.\nHowever, I'm willing to reconsider and increase my evaluation score to borderline accept, in recognition of the progress made.", "rebuttal": "We thank the reviewers for their thorough and insightful feedback. We have carefully considered all comments and will revise the paper to reflect these valuable suggestions (including typo). Below, we address each point in detail.\n\n---\nWe apologize for the typo in Table 1. Due to the character limits, the corrected results have been organized in a clear table in our first response to reviewer Q8Hn. We would appreciate it if you could refer to that section for the accurate figures.\n\n---\n[W5,21] [Improvements] Regarding the suggestion to delve deeper into performance variations, our analysis shows that the primary factor determining AliO’s performance gain is the baseline model’s initial instability on a given dataset, which we quantify as the initial TAM [Section 5.2].\n* **Empirical Evidence** As shown in Figure 4, there is a positive correlation between a high initial TAM (poor initial alignment) and greater MSE improvement. This is confirmed by the results on the high-TAM ILI dataset, where AliO achieved the largest MSE gain of up to 17.4%\n* **Theoretical Mechanism (Iterative Dynamic Self-Distillation)** This performance gain is driven by AliO's mechanism, which functions as an iterative, dynamic self-distillation. At each overlapping timestamp, the more accurate prediction (closer to the ground truth) acts as a dynamic Teacher for the less accurate one (the Student) . When the initial gap between them (a high Initial TAM) is large, the alignment loss provides a strong and informative gradient, guiding the model toward a more accurate solution. This principle, where a larger initial gap allows for greater gains through repeated distillation, is supported by recent literature [Understanding the Gains from Repeated Self-Distillation]. Also, for well-aligned data at the beginning (Low initial TAM), it is natural that there is little room for performance improvement since it is already well-aligned.\n* **Root Cause and Future Analysis** We hypothesize that the data characteristics (especially high noise) you mentioned are the root causes that result in a high initial TAM since high noise makes the model increase the prediction variation. Crucially, this means the initial TAM is a characteristic of the baseline model’s performance on the data, measured before our method is applied; it is not an artifact of AliO. Acknowledging this valuable feedback, we will add a dedicated analysis of this relationship in the final paper to further strengthen our contribution.\n\n---\n[W1,2,7] Regarding the implementation, we acknowledge this section could be clearer and will revise the paper as follows:\n* **Clarification of No Model Modification** We will clarify in the paper that our claim refers to the model’s architecture (e.g., Transformer blocks, forward function), which remains entirely unchanged. This highlights that AliO is a model-agnostic method.\n* **Explanation of Data Pipeline** We agree that the data loading pipeline requires adjustment. We will add a detailed explanation to the appendix demonstrating how we access consecutive samples by pairing them in the Dataset class. This is a standard preprocessing technique in time-series [Towards Universal Representation of Time Series, Unsupervised Representation Learning for Time Series] and does not alter the core model.\nThis modified pipeline is used only during the training phase and remains the same as baseline during the inference phase.\n* **Ensuring Reproducibility** To address reproducibility concerns, we provide the main GitHub link in Abstract.\n\n---\n[W10,11,12] In response to the request for Table 1: All results presented in Table 1 are the average of three runs with different random seeds [Table 1 caption].\n* **Input and Prediction Lengths** The specific prediction lengths used for each dataset are detailed in Appendix F. We acknowledge that the context lengths were not explicitly listed. They followed the official SOTA settings, and we will add this to the appendix F for full transparency:\n* * CycleNet, Autoformer, iTransformer: 96\n* * TimesNet: 96, 36(for ili dataset)\n* * PatchTST, DLinear: 336, 104 (for ili dataset)\n* **lambda values** The values for lambdas in Table 1 represent the best results achieved from a grid search over the hyper-parameter space defined in Appendix G. Reporting the best performance found via a search is a standard practice in research [Learning to Forecast in the Frequency Domain, Shape and Time Distortion Loss, etc.].\n\n---\n[W3,8] Regarding the important question about the relationship between sampling frequency and planning cycles, this is critical because it directly impacts operational stability and user trust. We argue that high-frequency forecast consistency remains crucial even for coarser planning cycles for the following reasons:\n* **Impact on Trust and Operational Monitoring** Even with daily planning, volatile hourly forecasts for the same future period erode planners' confidence and can trigger unnecessary monitoring alarms.\n* **Evidence of Real-World Costs** This instability is not theoretical; it leads to significant, real-world costs. This is a well-documented industrial and economical phenomenon known as MRP Nervousness, where frequent re-planning ultimately incurs substantial economic losses (e.g., sunk and opportunity costs). AliO directly mitigates this costly issue.\n* **Importance of Mid-Term Alignment** Planning considers the entire forecast period, not just the final endpoint. The high short- and mid-term misalignment shown in Figure 2 is precisely the problem that erodes trust and forces costly re-evaluations, even if the predictions eventually converge.\n\nAs you mentioned, not all methodologies may be important in every situation, and AliO can be appropriately used in situations like ILI mentioned by the reviewer.\n\n---\n[W4,19] Regarding the important questions about our proposed metric, TAM is necessary because it is specifically designed to measure prediction consistency, a dimension distinct from the accuracy measure by regression metrics. We chose MAE over DTW as the default distance function to directly address the practical need for point-wise stability.\n* **Necessity of TAM (vs. regression metrics)** While regression metrics like MSE measure accuracy (prediction vs. ground truth), TAM is essential for measuring consistency (prediction vs. prediction). This allows us to distinguish between a model that is accurate but unreliable (volatile) and a model that is both accurate and robust. (Table 1).\n* **MAE vs. DTW** We use MAE as the default because our goal is to ensure point-wise consistency — the stability of the predicted value for a specific future timestamp, which is critical for planning. DTW, which measures shape similarity by warping time, is unsuited for this specific, time-aligned problem and could distort the direct value difference.\n* **Practicality and Future Work** MAE is also far more computationally efficient than DTW. However, we agree that a DTW-based TAM could be a valuable tool for the different goal of measuring shape consistency, which we consider a promising future research direction.\n\n---\n[W9] We agree that regression loss provides some implicit alignment but argue this is often insufficient. RegPull mechanism is designed to enforce stronger consistency while preventing the error propagation the reviewer rightly pointed out.\n* **Limits of Implicit Alignment** While regression loss provides some alignment, it doesn’t guarantee consistency. The non-zero baseline TAM values in Table are the evidence. We posit that AliO provides the proper training the reviewer alluded to by adding the necessary explicit loss for consistency while maintaining or enhancing accuracy.\n* **Preventing Error Propagation with RegPull** RegPull mechanism is specifically designed to prevent error propagation. At each timestamp, it identifies the more accurate prediction (teacher) and only pulls the less accurate one (student) towards it. This one-way correction is enforced by applying a stop-gradient to the teacher (point-wise), which blocks any gradient flow from the student back to the teacher. Therefore, a poor prediction cannot corrupt a good one, ensuring the alignment process constructively reinforces the main accuracy objective.\n\n---\n[W13] We thank the reviewer for these valuable suggestions, which allow us to both address concerns about overgeneralization from our main paper's heatmaps and clarify the indicidual and synergistic effects. To address the valid concern about Figures 6 and 7, we will add a comprehensive set of individual heatmaps for each model and dataset to the appendix. This detailed view also allows us to clarify the role of each component. We provide additional result as below (X-axis: $\\lambda_F$, Y-axis: $\\lambda_T$):\n\n|TimesNet-ETTm1 (MSE)|0|0.5|1|2|\n|-|-|-|-|-|\n|5|0.01|0.01|0|0.01|\n|2|0.06|0.06|0.05|0.06|\n|1|0.11|0.11|1|0.12|\n|0|-|0.11|0.12|0.14|\n\n|TimesNet-ETTm1 (TAM)|0|0.5|1|2|\n|-|-|-|-|-|\n|5|0|0|0.01|0|\n|2|0.33|0.33|0.33|0.33|\n|1|0.55|0.54|0.46|0.55|\n|0|-|1|1|1|\n\n* **Individual Component Analysis** As the reviewer proposed, the impact of each component can be seen by setting the other to zero. The effect of the time domain alone is visible in Figure 6. Similarly, the effect of the frequency domain alone is demonstrated in the heatmaps in Appendix E (single model).\n* **Synergistic Effect** The optimal accuracy in Figure 6 (i.e., lowest MSE) is achieved when both time and frequency coefficients are non-zero. This experimentally demonstrates that the two domains contain complementary information and create a synergistic effect to maximize prediction accuracy when used together. This finding aligns with empirical support for the ideas presented in prior works such as [Learning to Forecast in the Frequency Domain].\n\nTo make this point clearer for all readers, we will add a sentence about Figure 6 in the main paper and present the individual effect heatmaps as in Appendix E."}, {"review_id": "PflDbwCp7Q", "reviewer": "Reviewer_5d8j", "summary": "This 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.", "questions": "- **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.", "limitations": "- 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.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### **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.", "was_revised": true, "final_justification": "Thank you for the rebuttal. While it partially addressed my concerns, I agree with another reviewer that, although the problem is interesting, its practical significance and theoretical soundness remain in question. Given the overall status of the paper, I tend to maintain my current score.", "rebuttal": "We are grateful to all the reviewers for their comprehensive and constructive feedback. We have carefully considered every comment, which has helped us to clarify and strengthen our paper's contributions.\n\nWe apologize for the typo in Table 1. Due to the character limits of the rebuttal platform, the corrected results have been organized in a clear table in our first response to reviewer Q8Hn. We would appreciate it if you could refer to that section for the accurate figures.\n\n---\n[W1] We agree that a good model must align with the Ground Truth, and we argue that this fundamental premise necessitates the temporal consistency that AliO is designed to enforce.\n* **Purpose of AliO** The purpose of AliO is maintaining or enhancing the regression performance while enhancing the consistency. This means the purpose of AliO aims to match the ground truth while the prediction is stable.\n\n* **Ideal Model’s Requirement** An ideal model predicting a single, unique Ground Truth for a future point T must produce the same output, regardless of minor input variations (e.g., from $X_n$ to $X_{n+1}$). If the model's output changes for the same single time point, it either contradicts its ideal nature (by failing to pinpoint the single Ground Truth) or implies a physical impossibility (multiple Ground Truths for one time point). Therefore, consistency is an essential property of an ideal model. AliO is the mechanism designed to explicitly enforces this property, which standard regression loss only addresses implicitly.\n\nRegarding the comparison to Weighted Quantile Loss (WQL), we posit that AliO and WQL are not competing baselines but are complementary tools that address different, essential aspects of a reliable forecast.\n* **WQL** is designed to quantify uncertainty by providing a prediction interval. It answers the question: “What is the probable range where the true value might lie?”. It cannot guarantee both accuracy and temporal consistency.\n* **AliO** is designed to ensure consistency. It answers the question: “Is the single most probable prediction path stable and reliable over time? It can guarantee consistency while maintaining or enhancing performance (Table 1 and Appendix H)”\n\nAn ideal system could use both WQL to define the uncertainty bounds and AliO to ensure the central forecast within those bounds is stable. A direct comparison is thus less a baseline and more a future direction for a combined approach.\n\n---\n[W2,6,9,11] We thank the reviewer for these important questions regarding the hyperparameter l and generalization. Our focus on l=1 is a deliberate choice to address the most fundamental case of temporal alignment, and our results indicate that this approach confers a more general alignment to the model. \n* **Rationale for Focusing on l=1** We centered our analysis on l=1 as it represents the most fundamental scenario. A model that is inconsistent for immediately adjacent inputs exhibits a core instability. Furthermore, this setting creates a chain of local alignments: the prediction at time t (Y_t) is aligned with Y_{t+1}, which is then aligned with Y_{t+2}, and so on. This series of constraints encourages the model to learn a global alignment over larger lags as an emergent property.\n* **Interpreting Performance at Larger l (Figure 7)** The performance trend for larger l in Figure 7 reflects the task’s inherent difficulty, not a limitation of AliO. As l increases, the overlapping window for alignment shrinks, making the task harder. The MSE results show a trade-off between l and N, indicating a need to find an appropriate context window size, as discussed in Section 5.3.\n* **Evidence of Generalization** Our results suggest that a model trained on a specific $l$ does generalize. The visualizations in Figure 2 and Appendix C show that a model trained with a fixed l exhibits improved alignment across a wider range of effective lags (e.g., Prediction 0 through 5) compared to the baseline.\n* **Commitment to Further Analysis** To address this, we will add more detailed tabular comparisons for different (N, l) settings to the appendix in the camera-ready version.\n\nThe table below shows the extended results normalized for each lag value (l = 16, 32, 48, 64).\n\n|iTransformer-ETTh1 (MSE)|16|32|48|64|\n|-|-|-|-|-|\n|N=2|0.59|0.59|0.60|0.61|\n\n|iTransformer-ETTh1 (TAM)|16|32|48|64|\n|-|-|-|-|-|\n|N=2|0.16|0.38|0.54|0.70|\n\n---\n[W3,W12] We thank the reviewer for this important suggestion regarding recent foundation models. Our experiments were designed to rigorously validate AliO's primary strength as a model-agnostic method, and the results suggest its applicability to these newer architectures.\n* **Proven Model-Agnostic Performance** Our main goal was to demonstrate that AliO is a model-agnostic technique. We validated this by applying it to a diverse range of architectures — including Transformer, Linear, CNN, and even a Pre-trained Model (GPT4TS) — and showed the performance of AliO across all of them.\n* **Expected Efficacy on PFMs** Since recent foundation models like Chronos or Morai are built upon these fundamental architectural principles (e.g., Transformer, MLPs), we anticipate that AliO, as a model-agnostic loss function, would similarly enhance their output consistency.\n* **Commitment to Further Validation** We agree that an explicit experiment on a recent PFM would further strengthen our paper. While this is not feasible during the short rebuttal period, we will add these results to the appendix of the camera-ready version to enhance our contribution.\n\n---\n[W10,13] The inconsistency problem does appear in short-term settings; however, AliO is most effective in LTSF due to its reliance on a long overlapping sequence for alignment.\n* **Problem Presence in Short-Term** The inconsistency issue is not exclusive to long-term horizons. As shown in Figure 2, significant variance between predictions exists even in the short- and mid-term intervals of the forecast.\n* **Methodological Suitability for LTSF** AliO’s effectiveness relies on having a sufficiently long overlapping sequence to perform alignment. LTSF provides these long overlapping windows, making it the ideal setting. In short-term forecasting, the overlapping period is too brief for the alignment mechanism to be as impactful, a trend suggested by the results in Figure 7 and Appendix H.\n* **Commitment to Broader Scope** We agree that a broader evaluation is a valuable direction. We will add a discussion to the camera-ready paper clarifying this scope and explicitly mentioning the application to short-term forecasting as an area for future work.\n\n---\n[W5] We thank the reviewer for this suggestion, which allows us to highlight a key strength of our experimental design. Our results demonstrate that AliO is highly effective in the high-TAM scenarios you suggested, and also provides significant benefits for datasets with already low TAM.\n* **Proven Efficacy in High-TAM Scenarios** We included the ILI dataset specifically to test our method in a high-TAM environment. As shown in Table 2 and Figure 4, on this dataset, AliO improved TAM by up to 58.1% and, critically, improved the regression metric (MSE) by up to 17.4%, demonstrating its strength in the exact scenario you described.\n* **General Applicability in Low-TAM Scenarios** AliO is also effective on standard benchmarks with lower initial TAM. For instance, on models like iTransformer and Autoformer, AliO still improved TAM by an average of 31.08% and MSE by 6.79% (Appendix H), showcasing its broad utility.\n* **Theoretical Rationale** This performance characteristic is explained by viewing AliO as a form of iterative self-distillation. The large initial gap (high TAM) between the teacher (better prediction) and student (worse prediction) provides a strong learning signal, leading to greater performance gains, a principle supported by recent distillation research [Understanding the Gains from Repeated Self-Distillation]. [This point is also discussed in our response to reviewer HfaX.]\n\n---\n[W8] We thank the reviewer for this insightful question. As you mentioned TAM performance in Figure 6 improves as the $\\lambda_T$ increases. This allows us to clarify the trade-off in our methodology. Our goal is not just to maximize consistency at all costs, but to balance between improving consistency and maintaining or improving forecasting performance (MSE).\n* **Trade-Off Between Consistency and Accuracy** While it is true that TAM performance improves with a larger coefficient, as you rightly observed, the model’s core performance (MSE) does not follow the same trend. The optimal average MSE was already achieved at a coefficient ($\\lambda_T$= 2), after which performance began to degrade, as shown in Figure 6.\n* **Risk of Overfitting to the Alignment Objective** This indicates that exploring even larger coefficient values would likely continue to improve TAM but at the cost of overfitting to the consistency objective. This would diminish the influence of the primary regression loss, which is counterproductive. (While RegPull exists, its purpose is to ensure the direction of the alignment loss matches that of the regression loss; it is not a direct regression loss itself.)\n\nTo further validate these findings, we will add experiments with a broader coefficient range to the appendix.\n\n---\n[W7] We thank the reviewer for pointing out this omission and for the valuable suggestion.\n\nOur experiments used the official context lengths from each SOTA paper. These specifics, listed below, will be added to the appendix for full transparency:\n* CycleNet, Autoformer, iTransformer: 96\n* TimesNet: 96 (36 for the ILI dataset)\n* PatchTST, DLinear: 336 (104 for the ILI dataset)\n\nWe agree that analyzing AliO's effectiveness across various context lengths is a valuable direction for future work. However, our study's primary goal was to isolate the performance improvement solely from AliO under conditions identical to the SOTA baseline settings."}, {"review_id": "GcjIxXLloK", "reviewer": "Reviewer_Q8Hn", "summary": "This 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.", "questions": "1. 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.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths: \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.", "was_revised": true, "final_justification": "I think authors have resolved all my concerns. So I keep my score to support this paper.", "rebuttal": "We are grateful to all the reviewers for their comprehensive and constructive feedback. We have carefully considered every comment, which has helped us to clarify and strengthen our paper's contributions.\n\n---\nThank you for correcting the typo. The results for 'Traffic' in Table 1 were indeed incorrect (they were shifted). The actual results are as follows.\n\n|Models|CycleNet||GPT4TS||iTransformer||PatchTST||TimesNet||DLinear||Autoformer||\n|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|Metric|MSE|TAM2|MSE|TAM2|MSE|TAM2|MSE|TAM2|MSE|TAM2|MSE|TAM2|MSE|TAM2|\n|baseline|0.487|0.021|0.411|0.059|0.422|0.061|0.389|0.041|0.626|0.04|0.436|0.025|0.634|0.048|\n|AliO|0.481|0.008|0.405|0.032|0.423|0.044|0.388|0.027|0.554|0.03|0.434|0.009|0.624|0.045|\n\n---\n\n**Core Motivation: The Economic Costs and Foundational Questions of Inconsistency**\n\nBefore addressing the specific points, we wish to reiterate the core motivation for this work. Our research was driven by a set of fundamental questions regarding the practical reliability of forecasting models:\n* Can a model whose predictions for the same future point fluctuate erratically with minimal new data truly be considered ideal?\n* Can a user trust a system whose predictions are constantly volatile?\n* Given two systems with equal accuracy, will a user not always prefer the stable system over the unstable one?\n\nWe believe the answers to these questions highlight the critical need for output alignment. Prediction inconsistencies are not a theoretical issue; they cause  sunk costs when plans are modified and opportunity costs when they are maintained, leading to significant real-world losses. These losses can amplify across an entire organization through phenomena like MRP Nervousness and the Bullwhip Effect. Therefore, our methodology is critical in fields where value chains are vital, such as energy, public health, manufacturing, and supply chain management.\n\n---\n[W1] We thank the reviewer for this valuable suggestion on validating our metric. Our paper demonstrates TAM's effectiveness through visual showcases and a practical, quantitative design.\n* **Visual Showcases**: The paper provides several visual showcases of TAM's effectiveness. In Figure 2 and the figures in Appendix C, it is visually clear that scenarios with high TAM values exhibit poor alignment, while scenarios with low TAM values show strong, consistent alignment .\n\n* **Quantitative & Practical Design**: TAM is designed to be a direct and interpretable metric for planning. By calculating the point-wise MAE between predictions for the same timestamp, it directly quantifies the value difference that is critical for resource allocation. This differs from shape-based metrics like DTW. However, the TAM framework is flexible, and for applications where shape consistency is the priority, DTW could certainly be adopted as the distance function within TAM.\n\n* **User Study as Future Work**: We agree that a user study is a valuable validation approach. Given that this is a technical paper proposing a new quantitative method, we prioritized establishing an objective and reproducible metric first. We take this suggestion seriously and are considering a user-focused study as a separate, independent paper to properly explore the perceptual aspects of alignment.\n\n---\n[W2] We thank the reviewer for this valuable suggestion. We agree that a more comprehensive related work section will better situate our contribution. Taking advantage of the increased page limit for the camera-ready version, we will expand this section to more clearly differentiate AliO from two key areas: regression losses and time-series contrastive learning.\n* **vs. Regression Losses (e.g., MSE, Soft-DTW)**: While regression losses focus on improving regression accuracy (the relationship between prediction and ground truth), AliO is distinct in that it improves reliability/consistency (the relationship between predictions) while maintaining or enhancing accuracy via the RegPull mechanism. Our metric, TAM, is similarly designed to quantify this reliability, whereas standard metrics quantify accuracy.\n* **vs. Contrastive Learning (e.g., TS2Vec)**: Time-series contrastive learning methods that design inputs to have overlapping timestamps [Towards Universal Representation of Time Series, Unsupervised Representation Learning for Time Series with Temporal Neighborhood Coding] align the overlapping inputs in the latent space. In contrast, AliO performs alignment directly in the output space. Critically, our RegPull mechanism ensures this alignment process is constructive, simultaneously aligning predictions with each other and with the ground truth, which is a unique contribution.\n\n---\n[W4] We thank the reviewer for this critical question on experimental fairness. We ensured a fair comparison through two key principles: 1) faithfully reproducing strong, well-trained baselines, and 2) designing our RegPull mechanism to be a constructive, rather than merely additive, loss.\n\n* **Faithful Baseline Reproduction**: To ensure our baselines were sufficient and strong, we used the official code and hyperparameters (e.g., learning rate, epochs) from each SOTA paper . Our reproduced baseline performance closely matches the results reported in the original papers, confirming they are well-trained.\n* **Principled Loss Design (RegPull)**: To ensure AliO's loss term does not unfairly enhance performance, the RegPull mechanism is designed to be constructive. It ensures alignment occurs only in the direction of the better prediction (the one closer to the ground truth) . This prevents the alignment loss from conflicting with the primary accuracy objective.\nIn fact, our internal experiments confirmed that removing RegPull worsened MSE performance by 1-3%. This demonstrates that the MSE improvement is a direct result of the intentionally designed RegPull algorithm, not merely an artifact of adding a [regression loss + new loss]."}, {"review_id": "zXJOYBb3bf", "reviewer": "Reviewer_HfaX", "summary": "This 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.", "questions": "Can 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?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \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.", "was_revised": true, "final_justification": "I have already read all the replies from the authors, I would like to to change my rate to \"Borderline accept\".", "rebuttal": "We sincerely thank all the reviewers for their thorough and insightful feedback. We have carefully considered all the comments, which have helped us to clarify and strengthen our paper's contributions. We hope that our detailed responses below successfully address the main concerns raised, and we are committed to incorporating these valuable suggestions into the final camera-ready version of our paper.\n\n---\n[W1] We thank the reviewer for this question on practical applications. A key feature of AliO is that it is a training-time-only methodology, meaning the inference process is identical to the baseline with no additional overhead. The training scenario, which uses lagged inputs, is a direct reflection of the rolling forecast method used in real-world deployments.\n* **Zero Inference Overhead** AliO is applied as a loss function only during the training phase. Therefore, at inference time, the model operates as a standard single model, making its deployment pipeline and computational cost completely identical to the baseline.\n* **Training for Real-World Rolling Forecasts** The lagged input scenario used during training is not artificial; it is a direct situation of the rolling forecast method. Real-world systems constantly update predictions using a sliding window of the most recent data (as shown in Figure 1). Our training method prepares the model to be consistent and accurate in exactly this type of real-world, sequential deployment.\n\n---\n[W2] We thank the reviewer for this question. The Regression Pulling (RegPull) mechanism is not a heuristic but a principled design rooted in the well-established methodology of knowledge distillation. It functions as a form of dynamic self-distillation, and the dual-domain alignment is a well-justified choice for recent forecasting.\n\n* **Principled Design via Self-Distillation** RegPull functions as a dynamic self-distillation. At each step, the more accurate prediction acts as a ‘teacher’ for the less accurate ‘student’. The stop-gradient is a standard, principled tool in this framework to ensure a one-way correction, preventing the student’s error propagation from corrupting the teacher and ensuring our alignment loss constructively reinforces the primary regression objective.\n* **Justified Dual-Domain Approach** We align in both time and frequency domains to capture complementary data characteristics, a practice supported by recent theoretical research like [Learning to Forecast in the Frequency Domain], which highlights the importance of using both domains for robust forecasting. The computational overhead of the frequency alignment (FFT) is an efficient O(NlogN).\n* **Training-Time-Only Mechanism** It is important to emphasize that the entire AliO mechanism, including RegPull and the dual-domain loss calculation, is applied only during the training phase.\n\n---\n[W3] We thank the reviewer for their insightful point. We agree that adaptability is a key virtue of a good forecaster. We argue that AliO enhances this adaptability by enabling the model to adapt with long-term consistency, which is crucial for the specific challenges of LTSF.\n\n* **Distinguishing Adaptability and Consistency** We differentiate between adaptability and consistency. Adaptability is the ability to rationally update predictions based on new information, quantified by accuracy. Consistency is the stability of predictions for the same future point despite minor input changes.\n* **Adaptation in LTSF** For LTSF, which focuses on long-term trends, a good model should not change its forecast drastically based on new short-term information. Instead, it should demonstrate a gradual analysis, reflecting a continuous analysis of long-term trends from the past to ensure a consistent evolution of its forecast (Figure 2).\n* **How AliO Achieves Consistent Adaptation** AliO creates a dependency between past and future data points during the backward pass. The RegPull mechanism guides this process towards the ground truth, ensuring the consistent adaptation is also an accurate adaptation. Figure 2 demonstrates the result, showing how the model learns to adapt consistently from past time steps.\n\nFrom a more fundamental perspective, this notion of consistent adaptation aligns with the definition of an ideal model, which must ultimately follow the singular Ground Truth.\n* **The Goal of Adaptation** The ultimate goal of an adaptive model is to follow the ground truth. As the reviewer noted, predictions can naturally fluctuate, but the Ground Truth itself for a given future point does not fluctuate.\n* **Low Variance as an Ideal Trait** Therefore, a model that truly follows the Ground Truth should exhibit low prediction variance for that specific future point. Reducing this variance is a step toward becoming a more ideal model.\n* **AliO’s Guiding Principle** Following this principle, AliO guides the model to produce adaptively consistent predictions, enhancing both its accuracy and reliability.\n\n---\n[W4] We thank the reviewer for this critical question, which prompts us to provide more concrete evidence by connecting our work to established principles in economics and management science.\n\n**Evidence from Economic and Management Science Principles**\n* **The Economic Dilemma of the Planner**\n* * Real-world planners operate under Bounded Rationality, making decisions with limited information and cognitive capacity [A Behavioral Model of Rational Choice].\n* * Output inconsistency forces a costly dilemma: either pay tangible re-planning costs to change course or incur intangible opportunity costs by sticking to a now-outdated plan.\n* * This constant re-evaluation can lead to suboptimal decisions, a pattern often explained by the Sunk Cost Fallacy [The Sunk Cost and Concorde Effects].\n\n* **Systemic Cost Amplification (The Bullwhip Effect)**\n* * This individual planning dilemma often amplifies into massive systemic costs across the supply chain, a phenomenon famously known as the Bullwhip Effect.\n* * Seminal research identifies demand forecast updating as a primary cause of this effect, where small forecast instabilities are magnified upstream [The Bullwhip Effect in Supply Chains].\n* * AliO's goal of improving output alignment is a direct technical approach to mitigate this specific root cause. AliO achieves this through its core mechanisms:\n* * * AliO is designed to achieve two simultaneous goals: it regularizes forecast instabilities via an explicit loss term ($L_{AliO}$), while also ensuring that the model's primary forecasting performance is maintained or enhanced.\n* * * Crucially, the Regression Pulling (RegPull) mechanism ensures this stabilization process is constructive. It forces the alignment to occur only in the direction of the prediction that is closer to the ground truth. \n* * * This principled correction prevents the model from converging to a stable but inaccurate forecast, ensuring the resulting consistency is also reliable while maintaining or enhancing overall forecasting performance.\n\n**Most Benefiting Industries and Applications**\n\nBased on these principles, AliO is most beneficial in industries where such planning dilemmas and systemic effects lead to significant costs. This includes any sector with a critical value chain:\n\n* Manufacturing and Supply Chain Management: Where unstable forecasts directly fuel MRP Nervousness and the Bullwhip Effect.\n* Retail and Consumer Goods: Where inventory and stockout costs are directly tied to forecast accuracy and stability.\n* Energy: Where inconsistent load forecasts lead to inefficient and costly power grid dispatch decisions.\n* Public Health: Where stable forecasts are critical for reliable resource allocation during health crises.\n\n**Commitment to Paper Enhancement**\n\nWe thank the reviewer for prompting this clarification. We will enhance this economic justification to the final camera-ready version, taking advantage of the increased page limit to better motivate our work.\n\n---\n[W5] We thank the reviewer for this important comparison. We argue that AliO is not a direct competitor but a complementary tool to Ensemble and Bayesian methods. While these methods provide crucial model robustness, they do not inherently guarantee the temporal consistency that AliO is specifically designed to enforce.\n* **Different Types of Consistency** The 'consistency' from Ensemble/Bayesian methods refers to model robustness — stability against variations in parameters or initializations, as discussed in works like iTransformer. AliO, however, targets temporal consistency — the stability of predictions for the same future point across slightly shifted time windows.\n* **Lack of Inherent Alignment Mechanism** Ensemble and Bayesian methods lack an explicit temporal alignment algorithm. Averaging a set of temporally inconsistent models will still produce an inconsistent average. Furthermore, without a directional correction mechanism like our RegPull , simple averaging could even pull the final forecast away from the ground truth, whereas AliO always aligns towards the more accurate prediction.\n* **Practical Advantages and Synergy** From a practical standpoint, AliO adds zero inference overhead, unlike ensembles which have a K-fold cost. It is also a computationally efficient loss function. In contrast, Bayesian methods are often significantly more computationally expensive than standard training. AliO can be used with these powerful methods to ensure the central forecast path is stable, while they quantify its uncertainty."}]

Author rebuttal:

Meta-review: The paper identifies the problem of inconsistent predictions across overlapping forecasting windows in long-term time series forecasting (LTSF), arguing that such inconsistencies reduce reliability in real-world deployment. To address this, this paper proposes AliO, a method that enforces consistency through “regression pulling” and joint time–frequency alignment. It also introduces a new evaluation metric, TAM, to measure alignment quality. Experiments across several architectures (e.g., Autoformer, PatchTST, TimesNet, GPT4TS) and datasets are presented, with results suggesting AliO improves both predictive accuracy and alignment.

Strengths: This paper addresses a novel and underexplored issue in LTSF, by introducing a new evaluation metric (TAM) that complements existing error-based measures. Experimental results have demonstrated broad applicability via evaluating across multiple models and datasets. This paper provides mathematical formulations and reasonably clear writing with illustrative figures.

Weaknesses: There are concerns about the practical necessity of strict prediction consistency, which is not convincingly justified; reviewers question whether variability may instead reflect appropriate adaptation to new data. The “regression pulling” and stop-gradient mechanisms appear heuristic, with limited theoretical grounding. There are also concerns about the missing evaluation against recent pretrained foundation models, and unclear generalization across different lag values. The TAM metric’s interpretability and necessity relative to established metrics remain underexplored, with limited validation (e.g., user studies).

Rebuttal: The rebuttal provided clarifications to these concerns but did not fully resolve concerns ---- the real-world necessity of strict consistency was not substantiated, the design choices remained ad hoc, and the scope of baselines remained limited. As a result, reviewers maintained reservations, and after weighing these points, I conclude that the weaknesses outweigh the strengths.

Summary:  Despite the above limitations, this paper addresses an interesting and less studied issue in time series forecasting and offers a practical method with encouraging empirical results. While some aspects of the methodology are heuristic and the broader impact is not yet fully proven, the novelty of the problem formulation, the introduction of a new evaluation metric, and the experimental evidence across diverse models make this work a valuable contribution to the community. These strengths outweigh the shortcomings.
Decision: Accept (poster)