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

Kwangryeol Park, Jaeho Kim, Seulki Lee

Year: 2026Venue: arXiv preprintArea: cs.AIType: PreprintEmbeddings: 168

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%

Last extracted: 10/9/2026, 8:29:21 AM

Summary

The paper introduces AliO (Align Outputs), a novel approach to improve output alignment in Long-Term Time Series Forecasting (LTSF) models. It addresses the issue where state-of-the-art models produce inconsistent predictions for overlapping timestamps across lagged input sequences. AliO reduces discrepancies in both time and frequency domains using a technique called regression pulling with stop-gradient operations. The authors also propose TAM (Time Alignment Metric) to quantify output alignment, distinct from traditional accuracy metrics like MSE. Experiments show AliO improves alignment by up to 58.2% while maintaining or enhancing forecasting performance.

Entities (17)

LTSF · task · 99%AliO · method · 98%TAM · metric · 97%RegPull · technique · 95%DLinear · model · 92%Autoformer · model · 90%CycleNet · model · 90%Electricity · dataset · 90%ETT · dataset · 90%GPT4S · model · 90%ILI · dataset · 90%iTransformer · model · 90%MSE · metric · 90%PatchTST · model · 90%TimesNet · model · 90%Traffic · dataset · 90%Weather · dataset · 90%

Relation Signals (17)

AliO → improves → output alignment

confidence 95% · AliO effectively improves the output alignment, i.e., up to 58.2% in TAM

TAM → measures → output alignment

confidence 95% · TAM quantitatively assesses the model’s output alignment by measuring discrepancies between predictions

AliO → reducesdiscrepanciesin → time and frequency domains

confidence 95% · reducing the discrepancies between prediction outputs for the same timestamps in both the time and frequency domains

AliO → uses → RegPull

confidence 95% · we propose Regression Pulling (RegPull)... To maintain forecasting regression performance while aligning predictions

AliO → appliedto → iTransformer

confidence 90% · We experiment with AliO on ... using various state-of-the-art LTSF models such as ... iTransformer

AliO → appliedto → DLinear

confidence 90% · the right panel, where AliO is applied, shows that consistency is improved while accuracy is maintained

AliO → appliedto → CycleNet

confidence 90% · We experiment with AliO on ... using various state-of-the-art LTSF models such as CycleNet

AliO → appliedto → GPT4S

confidence 90% · We experiment with AliO on ... using various state-of-the-art LTSF models such as ... GPT4S

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Abstract

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.

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AliO: Output Alignment Matters in Long-Term Time Series Forecasting Kwangryeol Park Affiliation: Artificial Intelligence Graduate School Affiliation: Ulsan National Institute of Science & Technology (UNIST), South Korea Email: pkr7098@unist.ac.kr Jaeho Kim †thanks: Work done at UNIST Affiliation: Artificial Intelligence Affiliation: Korea University, South Korea Email: kjh3690@korea.ac.kr Seulki Lee †thanks: Corresponding author Affiliation: Department of Computer Science and Engineering Affiliation: Ulsan National Institute of Science & Technology (UNIST), South Korea Email: seulki.lee@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 as weather forecasting and planning of electricity consumption. However, state-of-the-art LTSF models often fail to achieve prediction output alignment for the same timestamps across lagged input sequences. Instead, these models exhibit low output alignment, resulting in fluctuation in prediction outputs for the same timestamps, undermining the model’s reliability. To address this, we propose AliO (Align Outputs), a novel approach designed to improve the output alignment of LTSF models by reducing the discrepancies between prediction outputs for the same timestamps in both the time and frequency domains. To measure output alignment, we introduce a new metric, TAM (Time Alignment Metric), which quantifies the alignment between prediction outputs, whereas existing metrics such as MSE only capture the distance between prediction outputs and ground truths. Experimental results show that AliO effectively improves the output alignment, i.e., up to 58.2% in TAM, while maintaining or enhancing the forecasting performance (up to 27.5%). This improved output alignment increases the reliability of the LTSF models, making them more applicable in real-world scenarios. The code implementation is on the GitHub repository11 1 https://github.com/eai-lab/AliO. 1 Introduction The task of long-term time series forecasting (LTSF) is essential in various fields such as prediction of electricity demand Hebrail and Berard (2006), weather forecasting Wu et al. (2021), health data Khadem et al. (2023), and so on. Recently, deep neural network models Zeng et al. (2023); Liu et al. (2023); Nie et al. (2022); Zhou et al. (2023) have shown strong performance in predicting long-term time series based on historical information, which typically aim to minimize the error between the prediction output and the ground-truth sequence as a regression task. However, in real-world applications, merely minimizing the prediction error in LTSF is insufficient. It is equally important to ensure that the forecasting model generates consistent (aligned) prediction outputs for overlapping timestamps across lagged input sequences. For example, Figure 1 illustrates a scenario in which a trained forecasting model predicts future electricity usage in two instances: (1) for the period from April to November (Prediction 1, purple), using input data from January to March, and (2) for the period from May to December (Prediction 2, yellow), using input data from February to April, where the forecasting periods overlap between May and November. A reliable model should provide consistent predictions for these overlapping months, regardless of the partially differing input sequences. If the model produces inconsistent predictions on the electricity usage for the same timestamps (i.e., May to November) between two input sequences, it could result in significant time and financial costs for rescheduling budget allocation and undermine the reliability of the predictions. We refer to this phenomenon as the output alignment problem, which has not been adequately acknowledged and addressed by existing LTSF studies Le Guen and Thome (2019); Cuturi and Blondel (2017); Jhin et al. (2024), despite its significance and substantial impact on real-world applications. To the best of our knowledge, state-of-the-art LTSF models Wu et al. (2021); Nie et al. (2022); Liu et al. (2023) 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. Figure 1: An example of low output alignment between two predictions on future electricity usages: (1) Prediction 1 (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 to Nov) leads to differing budget allocation plannings for electricity power consumption in Aug, resulting in time and financial waste due to rescheduling resource allocation. In this paper, we present AliO (Align Outputs), a novel method designed to enhance output alignment in LTSF models. For the first time, AliO enables LTSF models to produce consistent predictions for overlapping timestamps across lagged input sequences. By aligning predictions for overlapping timestamps through the minimization of discrepancies in both time and frequency domains, AliO improves output consistency even when input sequences lagged. AliO achieves this by directing the model’s predictions toward the ground-truth sequences while simultaneously minimizing discrepancies across multiple predictions obtained from a set of lagged input sequences. It allows AliO to integrate seamlessly with the model’s forecasting objectives, such as regression loss (e.g., MSE), without adding implementation complexity or requiring modifications to the model. As a result, AliO enhances the reliability of forecasts by improving output alignment (consistency), while maintaining or even improving overall forecasting performance. This represents a significant advancement over existing methods Wu et al. (2021); Zeng et al. (2023) that focus solely on minimizing the forecasting objective without considering prediction output alignment. To quantify output alignment, which represents the consistency of a model’s predictions across lagged input sequences, we propose a new metric, Time Alignment Metric (TAM). TAM quantitatively assesses the model’s output alignment by measuring discrepancies between predictions for overlapping timestamps for multiple input sequences. To the best of our knowledge, the proposed TAM is the first metric designed to measure the output alignment. We experiment with AliO on representative LTSF tasks, including ETTh1, h2, m1, m2, Electricity (ECL), Traffic, Weather, and ILI dataset Wu et al. (2021), using various state-of-the-art LTSF models such as CycleNet Lin et al. (2024), GPT4S Zhou et al. (2023), iTransformer Liu et al. (2023), PatchTST Nie et al. (2022), TimesNet Wu et al. (2022), DLinear Zeng et al. (2023) and Autoformer Wu et al. (2021). The evaluation results demonstrate that AliO effectively aligns predictions over overlapping timestamps, i.e., improving TAM up to 58.2%, while maintaining or enhancing forecasting regression accuracy up to 17.4% measured in MSE. The detailed descriptions on additional datasets and experimental configurations are provided in Appendices G, H and I. 2 Motivation A system with low prediction consistency leads to user distrust and increased costs. For instance, studies on the consistency of weather forecasts have reported cases where consistent forecasts lead to greater trust in the system, whereas inconsistent forecasts result in user distrust Burgeno and Joslyn (2023); Losee and Joslyn (2018); Burgeno and Joslyn (2020). Furthermore, accroding to research in non-weather domains, consumers tend to perceive consistency among multiple estimates from the same system as a signal of skill Falk and Zimmermann (2017); Burgeno and Joslyn (2023); Budescu et al. (2003). The inconsistency in predictions necessitates replanning, which consumes unnecessary organizational time. The resulting resource reallocation leads to the investment of more resources, contrary toe th original goal of Rolling forecasting, which aims for resource savings Huhtakangas (2020); Henttu-Aho (2018). This can be described as an additional sunk cost Wang et al. (2022), as frequent fluctuations in predictions can induce irrational decision-making and result in lost opportunity costs Wang et al. (2022); Chung and Cheng (2018). 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 Figure 2 shows the prediction results of DLinear Wu et al. (2021) on the ILI dataset Wu et al. (2021) using only regression loss (MSE), revealing a significant lack of consistency. In contrast, the right panel, where AliO is applied, shows that consistency is improved while accuracy is maintained. As can be seen in Appendix I, this is a phenomenon observed across other models and datasets. Figure 2: (Left) Six prediction outputs from the DLinear Zeng et al. (2023) model trained with regression loss (MSE) only, forecasting a length-24 sequence for ILI data Wu et al. (2021), 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 to the ground truth for all six predictions, which significantly reduces the prediction shifts observed when using regression loss alone. Although both consistency and accuracy are crucial for addressing users’ psychological and economic concerns, our experiments revealed that while modern LTSF models achieve high accuracy, they fall short in terms of consistency. AliO, as a loss function used in conjunction with regression loss, aims to achieve two goals: maintaining the high accruacy of existing LTSF models while improving their prediction consistency. Furthermore, to quantify consistency, we propose the Time Alignment Metric (TAM). Regression metrics such as MSE, MAE, and DTW Sakoe and Chiba (1978) represent accuracy, which is the distance between predictions and ground truth. Therefore, they have limitations in measuring consistency, which is a measure of performance between predictions. Complementing this, TAM measures the distance between predictions, serving as an indicator that represents consistency. Through Sec. 5 and Appendix I, we show that AliO successfully achieves the dual goals of maintaining/improveing the high accuracy of existing models and enhancing consistency. Additionally, in , we experimentally verify the relationship between the inconsistency value (TAM) of existing models and the MSE improvement in models trained with AliO. To the best our knowledge, this is the first study to address the improvemnet of consistency in LTSF. 3 Time alignment metric (TAM) We first define the concept of output alignment and then introduce Time Alignment Metric (TAM), which enables a quantitative assessment of output alignment. Definition 3.1. (Output Alignment) Output alignment refers to the property wherein the prediction outputs of a model, derived from a set of lagged input sequences, exhibit consistent alignment, characterized by uniform patterns across overlapping timestamps in the prediction outputs. For instance, as illustrated in Figure 2, the six prediction outputs for the overlapping timestamps (i.e., from 25 to 43) in the left figure demonstrate a lower degree of output alignment. Conversely, the six prediction outputs in the right figure exhibit similar and consistent patterns, demonstrating a high degree of output alignment. To quantify the output alignment (Definition 3.1) in model predictions, we propose Time Alignment Metric (TAM). We begin by defining the necessary concepts, including lagged input and output sequences, and overlapping output sequences, as provided below. Definition 3.2. (Input and Output Sequence) Given a time-series sequence ∈c×d X∈ ^c× d, where c is the input channel, and d is the length of the sequence, respectively, the n-th input sequence n∈c×d′ X^n∈ ^c× d of the length d′≤d ≤ d is defined as the segment s:s+d′−1 X_s:s+d -1, where s is the starting timestamp, and s+d′−1s+d -1 is the ending timestamp. Then, taking n∈c×d′ X^n∈ ^c× d as input, f(n;)=n∈c′×hf( X^n; θ)= Y^n∈ ^c × h is defined as the prediction output sequence of the length h provided by the forecasting model f, where c′c is the output channel, and θ is the model parameter. Having the input sequence n X^n and the corresponding output sequence n Y^n of the model f, the lagged input and output sequences, n+1 X^n+1 and n+1 Y^n+1, are derived by shifting their timestamps by the lag parameter l, as described below. Definition 3.3. (Lagged Input and Output Sequence) Given the n-th input sequence n∈c×d′ X^n∈ ^c× d of the segment s:s+d′−1 X_s:s+d -1, the sequence n+1∈c×d′ X^n+1∈ ^c× d of the segment s+l:s+d′+l−1 X_s+l:s+d +l-1 is defined as the lagged input sequence of n X^n with l being the lag parameter. Then, given the lagged input sequence n+1∈c×d′ X^n+1∈ ^c× d , the prediction output of the length h, f(n+1;)=n+1∈c′×hf( X^n+1; θ)= Y^n+1∈ ^c × h, is defined as the model f’s lagged output sequence. From n Y^n and n+1 Y^n+1, the overlapping output sequences n P^n and n+1 P^n+1 are derived to compute TAM, as follows. Definition 3.4. (Overlapping Output Sequence) Given two output sequences n∈c′×h Y^n∈ ^c × h and n+1∈c′×h Y^n+1∈ ^c × h of the model f generated from n X^n and n+1 X^n+1 as input, respectively, n∈c′×h′ P^n∈ ^c × h and n+1∈c′×h′ P^n+1∈ ^c × h are defined as the overlapping output sequences between n Y^n and n+1 Y^n+1, whose segments are given by n1+l:h Y^n_1+l:h and n+11:h−l Y^n+1_1:h-l, respectively, where h′=h−lh =h-l is the overlapping length. From these, Time Alignment Metric (TAM) is defined to quantify output alignment by evaluating the consistency between overlapping output sequences n P^n and n+1 P^n+1, as follows. Definition 3.5. (TAM; Time Alignment Metric) Given N overlapping output sequences n∈ℝc′×h′ P^n ^c × h for n=1,2,…,Nn=1,2,…,N, where the input and output sequences, n+1 X^n+1 and n+1 Y^n+1, are offset from n X^n and n Y^n by the lag l, TAMNTAM_N (Time Alignment Metric) is calculated as the average of the MAE values between all pairs of n,m\ P^n, P^m\, which is given by: TAMN≜(N−1)N2∑n=1N−1∑m=n+1N|n−m|1h′TAM_N (N-1)N2 _n=1^N-1 _m=n+1^N P^n- P^m _1h (1) TAM offers a distinct advantage over traditional forecasting regression metrics, such as the MSE or MAE between model outputs and ground truth, which only measure forecasting performance and fail to capture prediction fluctuations across different predictions. Unlike these metrics, TAM evaluates the distances between overlapping predictions, allowing for an assessment of how predictions evolve over time. For instance, regression metrics do not determine whether overlapping predictions remain consistent or exhibit smooth transitions, making it challenging to assess data-model robustness. In contrast, TAM computes the distance between overlapping predictions, providing a robust measure of data-model consistency. A lower TAM value reflects improved alignment and data-model robustness. 4 Improving output alignment (AliO) To enhance the output alignment of LTSF models, we propose AliO (Align Outputs), which minimizes the discrepancy between overlapping predictions (Definition 3.4). AliO is designed to simultaneously achieve two key objectives: (1) improving output alignment by reducing discrepancies between the overlapping predictions, and (2) maintaining the model’s forecasting regression performance by aligning the model’s prediction outputs with the ground truths. The two objectives are achieved through the technique that we call regression pulling with the stop-gradient operation, which ensures that the forecasting regression loss remains unaffected while improving output alignment simultaneously. Given two overlapping predictions, n=f(n;)1+l:h P^n=f( X^n; θ)_1+l:h and n+1=f(n+1;)1:h−l P^n+1=f( X^n+1; θ)_1:h-l defined in Definition 3.4, AliO aligns these predictions such that n≃n+1 P^n P^n+1 for n=1,2,…,Nn=1,2,…,N by minimizing the following objective with respect to the model f’s parameter θ, as: minD(f(n;)1+l:h,f(n+1;)1:h−l) _ θD(f( X^n; θ)_1+l:h,f( X^n+1; θ)_1:h-l) (2) where D denotes a distance function and subscripts denote the segment ranges of the model f’s prediction output sequences, i.e., 1+l:h1+l:h and 1:h−l1:h-l correspond to the same timestamps for n X^n and n+1 X^n+1, respectively. Eq. 2 can be easily extended to handle non-consecutive overlapping output sequences, enabling its application to TAM (Definition 3.5). AliO aligns the overlapping predictions n P^n and n+1 P^n+1 in both time and frequency domains, as illustrated in Figure 3 with further details provided in the following subsections. The algorithmic procedure of AliO is summarized in Alg. 1. Figure 3: Output alignments in both time and frequency domains. (Left) In the time domain, AliO minimizes the difference between two predictions in the overlapping timestamps, i.e., Tn P^n_T and Tn+1 P^n+1_T (T 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-gradient operation to 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 Fn P^n_F and Fn+1 P^n+1_F (the transformed frequency domain of Tn P^n_T and Tn+1 P^n+1_T), 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 . 4.1 Time domain alignment To enhance the temporal alignment, AliO aligns the overlapping predictions, Tn P^n_T and Tm P^m_T for n,m=1,2,…,Nn,m=1,2,…,N, in the time domain, where the subscript T denotes the time domain. This alignment can be achieved using a distance function, DTD_T, such as MSE or Dynamic Time Warping (DTW) Sakoe and Chiba (1978). The computed distance is back-propagated to encourage the model f to produce overlapping predictions that align with each other. However, solely aligning the overlapping predictions by minimizing the time domain alignment loss ℒT=DT(Tn,Tm)L_T=D_T( P^n_T, P^m_T) may lead to both prediction outputs deviating from the ground truth, degrading the forecasting regression performance of the model f. Algorithm 1 The procedure of AliO. ⊙ denotes element-wise multiplication. FFT(⋅,⋅,⋅)FFT(·,·,·) returns the frequency domain representation of each signal sequences, sg(⋅)sg(·) is stop-gradient operator. Input: The number of predictions N, consecutive predictions n Y^n with at time lag of l, and their ground truth ^n Y^n, where n∈[1,N]n∈[1,N], and distance function DTD_T and DFD_F for the time and frequency domain, respectively. The subscript T and F denotes the time and frequency domain, respectively. Output: The time domain alignment loss ℒTL_T and frequency domain alignment loss ℒFL_F. Initialize ℒt=0,ℒf=0,count=0L_t=0,L_f=0,count=0 for n=1n=1 to N−1N-1 do for m=n+1m=n+1 to N do gap=|m−n|×lgap= m-n × l (Tn,Tm,T)=(gap:n,:−gapm,^gap:n)( P^n_T, P^m_T, GT_T)=( Y^n_gap:, Y^m_:-gap, Y^n_gap:) idxT=Index(|Tn−T|>|Tm−T|)idx_T=Index( P^n_T- GT_T > P^m_T- GT_T )(1) ℒt=ℒt+DT(T,i∈idxTn,sg(T,i∈idxTm))+DT(sg(T,i∉idxTn),T,i∉idxTm)L_t=L_t+D_T( P^n_T,i∈ idx_T,sg( P^m_T,i∈ idx_T))+D_T(sg( P^n_T,i∉ idx_T), P^m_T,i∉ idx_T)[RegPull] (2) (Fn,Fm,F)=FFT(Tn,Tm,T)( P^n_F, P^m_F, GT_F)=FFT( P^n_T, P^m_T, GT_T) (3) idxF=Index(|Fn−F|>|Fm−F|)idx_F=Index( P^n_F- GT_F > P^m_F- GT_F ) (4) ℒf=ℒf+DF(F,i∈idxFn,sg(F,i∈idxFm))+DF(sg(F,i∉idxFn),F,i∉idxFm)L_f=L_f+D_F( P^n_F,i∈ idx_F,sg( P^m_F,i∈ idx_F))+D_F(sg( P^n_F,i∉ idx_F), P^m_F,i∉ idx_F) [RegPull] (5) count=count+1count=count+1 end for end for (ℒT,ℒF)=(ℒt/count,ℒf/count)(L_T,L_F)=(L_t/count,L_f/count) Return: ℒTL_T and ℒFL_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 Tn P^n_T and Tm P^m_T are relatively further from the ground truth than the other at the same timestamp, indicated by the index variable idxTidx_T, as shown on line (1) in Alg. 1. By applying the stop-gradient operation, denoted as sg(⋅)sg(·), to the prediction output points farther from the ground truth, the time domain alignment loss ℒTL_T pulls these distant points closer to the ground truth, aligning its optimization direction with that of the forecasting regression loss and reinforcing to minimize the regression loss. Consequently, regression pulling effectively reduces the misalignment between prediction outputs with the forecasting regression performance being unaffected, inducing overall predictions closely aligned with the ground truths. For each prediction output point i∈[1,h′]i∈[1,h ] in Alg. 1, DT(T,i∈idxTn,sg(T,i∈idxTm))D_T( P^n_T,i∈ idx_T,sg( P^m_T,i∈ idx_T)) on line (2) encourages the prediction Tn P^n_T to move closer to the prediction Tm P^m_T. On the other hand, DT(sg(T,i∉idxTn),T,i∉idxTm)D_T(sg( P^n_T,i∉ idx_T), P^m_T,i∉ idx_T) on the same line (2) promotes the alignment of Tm P^m_T towards Tn P^n_T. The direction of alignment is determined by the index variable, idxTidx_T. These two distance values are combined into the time domain alignment loss ℒTL_T, which is expressed as: ℒT=DT(T,i∈idxTn,sg(T,i∈idxTm))+DT(sg(T,i∉idxTn),T,i∉idxTm)L_T=D_T( P^n_T,i∈ idx_T,sg( P^m_T,i∈ idx_T))+D_T(sg( P^n_T,i∉ idx_T), P^m_T,i∉ idx_T) (3) Consequently, Eq. 3 aligns the direction of the time domain alignment loss ℒTL_T with the forecasting regression loss while improving output alignment. The regression pulling can also be applied in the same way to the frequency domain; Figure 3 shows a visual illustration of output alignment in the time (left) and frequency domain (right) using regression pulling (red cross marks). 4.3 Frequency domain alignment In addition to the time domain alignment, we propose frequency domain alignment as a complementary approach that supports and enhances time-domain alignment by aligning overlapping predictions in the frequency representation. It promotes the alignment of both the phase and the amplitude components, improving overall consistency in the time domain. As shown in Alg. 1, the time domain overlapping predictions, Tn P^n_T and Tm P^m_T, along with the ground truths T GT_T, are first transformed into frequency domain representations, resulting in Fn P^n_F, Fm P^m_F, and F GT_F on line (3), where the subscript F denotes the frequency domain. Subsequently, the index variable idxFidx_F is determined by identifying output points further from F GT_F between Fn P^n_F and Fm P^m_F on line (4). The distance function DFD_F is then applied with regression pulling to facilitate alignment in the frequency domain on lines (5) in the same manner to the time domain. From this, the frequency domain alignment loss ℒFL_F is obtained as: ℒF=DF(F,i∈idxFn,sg(F,i∈idxFm))+DF(sg(F,i∉idxFn),F,i∉idxFm)L_F=D_F( P^n_F,i∈ idx_F,sg( P^m_F,i∈ idx_F))+D_F(sg( P^n_F,i∉ idx_F), P^m_F,i∉ idx_F) (4) We use MSE (mean squared error) as the main distance function for the frequency domain alignment, i.e., DF=∥Fn−Fm∥22/h′D_F= P^n_F- P^m_F _2^2/h . The following Theorem 4.1 demonstrates that applying MSE in the frequency domain facilitates phase alignment between prediction outputs. Theorem 4.1. Given two frequency domain prediction vectors, Fn p^n_F and Fm p^m_F in ℂh′C^h , where ℂC means the set of complex numbers, minimizing their MSE, ∥Fn−Fm∥22 p^n_F- p^m_F _2^2, results in a reduction of the difference in the phase components, ∠Fn p^n_F and ∠Fm p^m_F, as shown below (the proof is provided in Sec. B.1): 1h′∥Fn−Fm∥22→0⟹|∠Fn−∠Fm|→0 1h p^n_F- p^m_F _2^2→ 0 p^n_F- p^m_F → 0 (5) Figure 3 (Right) shows a polar-coordinate depiction of frequency alignment conducted through Eq. 5 in the frequency domain. Theorem 4.2 shows that minimizing MSE, i.e., DF=∥Fn−Fm∥22D_F= P^n_F- P^m_F _2^2, also reduces the differences in amplitude once the phase components ∠Fn P^n_F and ∠Fm P^m_F have been aligned. Theorem 4.2. Once the phases of two frequency domain prediction vectors, Fn p^n_F and Fm p^m_F in ℂh′C^h , are aligned, i.e., ∠Fn≃∠Fm p^n_F p^m_F, the minimization of MSE between the two prediction outputs, ∥Fn−Fm∥22 p^n_F- p^m_F _2^2, leads to a reduction in the amplitude difference, as shown below (the proof is provided in Sec. B.2): 1h′∥Fn−Fm∥22→0⟹|Fn|−|Fm|→0 1h p^n_F- p^m_F _2^2→ 0 p^n_F - p^m_F → 0 (6) 4.4 AliO loss By combining the time ℒTL_T in Eq. 3 and frequency domain alignment loss ℒFL_F in Eq. 4, we derive the AliO loss, ℒAliOL_AliO, which is incorporated with the forecasting regression loss, ℒregL_reg as shown below, where λT _T and λF _F control the extent of alignment in the time and frequency domain, respectively. ℒAliO=λTℒT+λFℒF∴ℒtotal=ℒreg+ℒAliO _AliO= _TL_T+ _FL_F _total=L_reg+L_AliO (7) 5 Experiment We evaluate AliO on representative LTSF models and datasets. In model training, MSE is employed as the regression loss. The model performance is assessed using both MSE and TAMNTAM_N for output alignment evaluation, with N=2N=2. To reproduce experimental results of baselines and ensure a fair evaluation, we follow the same hyper-parameters presented in each model papers Wu et al. (2021); Lin et al. (2024); Zeng et al. (2023); Zhou et al. (2023); Liu et al. (2023); Nie et al. (2022); Wu et al. (2022). For AliO, we set N=2N=2 and l=1l=1, covering a wide prediction range. The search space for λT _T and λF _F are 1.0,2.0,5.0\1.0,2.0,5.0\ and 0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\, respectively, and we report the best results in the main results. All results are reported as the average performance across all prediction lengths. Detailed descriptions of the experiment and results are provided in Appendices G, H and I. Models. We apply AliO to various LTSF architectures, experimenting it with a diverse range of LTSF approaches, categorized into four groups: (1) Transformer-based models, i.e., Autoformer Wu et al. (2021), PatchTST Nie et al. (2022), and iTransformer Liu et al. (2023), (2) Linear-based models, i.e., DLinear Zeng et al. (2023) and CycleNet Lin et al. (2024), (3) CNN-based models, i.e., TimesNet Wu et al. (2022), and (4) LLM-based models, i.e., GPT4TS Zhou et al. (2023). Datasets. On the main text of the paper, we report results on representative LTSF datasets, i.e., Electricity (ECL) Hebrail and Berard (2006), ETT h1, h2, m1, and m2 Zhou et al. (2021), Traffic, Wu et al. (2021), Weather Wu et al. (2021), and National-Illness (ILI) dataset Wu et al. (2021). The experiments results on full datasets can be found in Appendix I. Context Length Following the official settings for each model, we set the context length to 336336 for PatchTST and DLinear, and 9696 for the other models. For the ILI dataset specifically, the context lengths were set to 104104 for PatchTST and DLinear, and 3636 for TimesNet. Prediction Length Furthermore, as per the official settings, the prediction lengths were set to 96,192,336,720\96,192,336,720\ for most datasets. The exceptions were the PEMS-related datasets with 12,24,48,96\12,24,48,96\, the ILI dataset with 24,36,48,60\24,36,48,60\, and the Autoformer model on the ETT datasets, for which we used 24,48,168,336,720\24,48,168,336,720\. 5.1 Output alignment and forecasting performance Tab. 1 summarizes the performance of LTSF models on seven datasets (excluding ILI) for multivariate LTSF tasks, comparing baselines (without AliO) and AliO-integrated models in terms of MSE and TAM. As shown in the table, integrating AliO with the regression loss substantially improves alignment performance (TAM), achieving gains of up to 70.5% for CycleNet, 45.8% for GPT4TS, 45.8% for iTransformer, 36.8% for PatchTST, 45.6% for TimesNet, 64.0% for DLinear, and 39.0% for Autoformer. Simultaneously, forecasting performance (MSE) also improves by up to 11.5%. As summarized in Tab. 2, for the ILI dataset exhibiting insufficient initial output alignment, AliO achieves up to a 17.4% higher MSE improvement compared to other benchmarks. Table 1: The forecasting and alignment performance, measured by MSE and TAM2, respectively, are compared between the baseline (MSE only) and AliO, with best results indicated in bold. The results are averaged over three random seeds and prediction lengths. The more metrics (MAE, MAPE, and RMSE) with standard deviations are provided in Appendix I. Models CycleNet GPT4TS iTransformer PatchTST TimesNet DLinear Autoformer Metric MSE TAM2 MSE TAM2 MSE TAM2 MSE TAM2 MSE TAM2 MSE TAM2 MSE TAM2 ECL 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 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 ETTh1 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 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 ETTh2 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 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 ETTm1 baseline 0.386 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 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 ETTm2 baseline 0.272 0.027 0.270 0.042 0.292 0.048 0.257 0.059 0.296 0.036 0.289 0.037 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 Traffic baseline 0.487 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 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 Weather 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 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. 2 shows AliO’s effectiveness in improving both forecasting accuracy and alignment performance on the ILI dataset Wu et al. (2021), which exhibits poor output alignment in the absence of AliO. The proposed regression pulling enhances model training by analyzing prediction distances at each timestamp, adaptively strengthening the effect of regression loss to align predictions with each other and ground truth. On ILI, which exhibits poor initial alignment, AliO achieves 17.4% MSE and 58.1% TAM improvement. Conversely, datasets with better initial alignment, presented in Tab. 1, achieve comparatively smaller MSE improvements. Figures 5 and 13 shows this trend; higher initial TAM (poor alignment) correlates with greater MSE gains, demonstrating AliO’s efficacy in suboptimal alignment conditions. Figures 2 and 5 visualize DLinear predictions on the ILI dataset. The left figures (without AliO) exhibit prediction shifts Jhin et al. (2024) in both time (Figure 2) and frequency (Figure 5) domains, reflected in high TAM values. In contrast, the right figures (with AliO) show improved alignment in both domains, which reduces TAM value and enhances regression performance (MSE). Table 2: The forecasting regression and alignment performance on ILI Wu et al. (2021), measured in MSE, and TAM2, are compared between the baseline (without AliO) and AliO. The best results are highlighted in bold. For robust evaluation, we conduct experiments with three random seeds. Results are rounded to three decimal places for simplicity. Models GPT4TS PatchTST TimesNet DLinear Metric MSE TAM2 MSE TAM2 MSE TAM2 MSE TAM2 base 1.898 0.117 1.813 0.110 2.176 0.322 2.247 0.152 AliO 1.755 0.098 1.497 0.046 1.963 0.258 2.187 0.119 Figure 4: The relationship between initial TAM and MSE improvement (%). The correlation coefficient is 0.330.33, indicating a positive trend; when the initial alignment is poor (i.e., higher TAM values), the MSE improvement tends to be greater. For more detailed information, please check Appendix E. Figure 5: (Left) The frequency domain of five predictions in polar coordinates from DLinear Zeng et al. (2023) 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. 5.3 Hyper-parameter Analysis Time and Frequency Domain Coefficients. AliO utilizes the two coefficients, λT _T and λF _F in Eq. 7, to balance the regression performance and output alignment by adjusting the alignment strengths over the time and frequency domains, respectively. To evaluate AliO with respect to these two coefficients, we provide Figure 7, which presents heatmaps of the normalized MSE and TAM for various coefficient values. The x-axis represents λF _F in 0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\, and the y-axis corresponds to λT _T in 1.0,2.0,5.0\1.0,2.0,5.0\. The left heatmap shows normalized MSE, where the maximum MSE in each dataset is scaled to 1.0 and the minimum to 0, followed by averaging across all models. The right heatmap presents normalized TAM in the same manner. Additional results are in Appendix F. As AliO is designed to enhance alignment, a clear trend is observed in TAM where increasing the time domain alignment coefficient λT _T from 1.0 to 5.0 leads to a consistent improvement. Interestingly, the normalized MSE also shows a general improvement as λT _T increases, suggesting that the regression pulling effectively aligns the directions of the AliO loss with those of the regression loss. The influence of the frequency domain alignment coefficient λF _F is less pronounced compared to the time coefficient λT _T. However, in cases where λT _T alone cannot improve performance, λF _F shows its impact on both the MSE and TAM metrics, as exemplified by values of 0.280.28 and 0.500.50, 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. Figure 6: Comparison of the normalized MSE (left) and TAM (right) over different settings of λT _T and λF _F. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (1.0,2.0,5.0\1.0,2.0,5.0\). Increasing λT _T consistently improves TAM performance, while λF _F shows its impact depending on the value of λT _T. Blue represents better performance. Figure 7: Comparison of the normalized MSE (left) and TAM (right) over different settings of the overlapping predictions N and the lag l. The x-axis represents the lag l, and the y-axis represents the number of overlapping predictions N. Increasing N improves TAM, while increasing l degrades the performance. No distinct tendencies are observed regarding MSE. Blue represents better performance. 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. Figure 7 shows the normalized MSE and TAM with a range between 0 and 1.0. TAM is improved as the lag decreases, which can be attributed to an increased length of overlapping timestamps at smaller lag value. This allows AliO to align longer overlapping sequence effectively. Additionally, the model performance improves as N increases. As shown in Alg. 1, a higher N enables AliO to align predictions from distant timestamps, significantly enhancing TAM. In contrast, no clear trend is observed for MSE, since the window size increases as both N and l grow. Prior studies Freitas et al. (2023); Azlan et al. (2019); Iqbal et al. (2025); Ma et al. (2024) suggest that large window sizes can cause overfitting, whereas an optimal window size leads to better performance. As shown in Figure 7, when both N and l are large, MSE tends to increase, indicating degraded performance. Conversely, lower MSE values (0.29,0.37,0.200.29,0.37,0.20) are observed along the diagonal elements, representing proper window sizes, such as (l,N)=(1,8),(2,4),(8,2)(l,N)=(1,8),(2,4),(8,2), which indicates that balanced combinations of l and N yield better performance. 6 Related works To the best of our knowledge, no prior work has adequately identified the output alignment problem and provided a solution for it, as AliO proposed in this paper. Data-Model Robustness. While existing research Liu et al. (2023) primarily focuses on model initialization robustness, i.e., the consistency of model performance across randomly initialized weights Liu et al. (2023), the output alignment emphasizes data-model robustness. This aspect underscores the model’s ability to produce consistent prediction outputs for lagged input sequences. The data-model robustness is equally critical for practical applications as discussed in this paper and can be quantified using the proposed TAM, which can be enhanced through the proposed time and frequency domain alignment. We anticipate this aspect of LTSF study will inspire further active research in the field. Contrastive Learning. Contrastive learning Zhang et al. (2022); Eldele et al. (2021); Akerkar and Sajja (2009); Eldele et al. (2023); Yue et al. (2022); Kim et al. () employ pretraining strategies that encourage augmented positive pairs to be closely aligned in the representation space. However, they differ from AliO in several key aspects. First, they operate in the representation space rather than in the output space where AliO functions. Second, they minimize the distance between whole vectors rather than focusing on overlapping timestamps. Third, they do not incorporate regression-aware algorithms like regression pulling. These differences highlight the distinctive approach of AliO in output alignment, operating independently of existing methodes and can thus be integrated with them. 7 Limitations and discussions On Volatile Datasets. TAM is a metric designed to assess the output alignment of LTSF models. However, its application should be approached with caution on highly volatile datasets that fall outside the typical scope of LTSF tasks, as model predictions may experience considerable volatility. Nevertheless, as demonstrated in Appendix I, the results on the Exchange Rate dataset Wu et al. (2021), which is influenced by abrupt external shocks, suggest that TAM retains potential applicability even in volatile settings. Designing more robust metrics for inherently volatile data is our next research objective. Distance Functions. AliO improves regression performance under high initial TAM conditions (a low degree of output alignment). This is likely attributable to its use of MSE as the distance function, which is consistent with the regression loss employed during model training. To further enhance performance, alternative distance functions such as Le Guen and Thome (2019); Liu et al. (2021); Huber (1992); Cuturi and Blondel (2017), which capture different characteristics of time series data (e.g., shape), may serve as promising directions for future exploration. 8 Conclusion In this paper, we investigate the output alignment problem, which can commonly arise in LTSF tasks and propose Time Alignment Metric (TAM) as a quantitative measure for this problem. To solve the output alignment problem, we propose AliO (Output Alignment) applied in both time and frequency domains. AliO achieves up to 27.5% improvement in MSE across various LTSF models and datasets, and up to 58.2% improvement in TAM, effectively addressing the output alignment problem. Acknowledgment This work was supported by the Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT)(No.RS-202400508465) and Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (No.RS-2020-I201336, Artificial Intelligence Graduate School Program(UNIST)). References Akerkar and Sajja (2009) R. Akerkar and P. Sajja Knowledge-based systems. Jones & Bartlett Publishers. Cited by: §6. Azlan et al. (2019) A. Azlan, Y. Yusof, and M. F. M. Mohsin Determining the impact of window length on time series forecasting using deep learning. International Journal of Advanced Computer Research 9 (44), p. 260–267. Cited by: §5.3. Budescu et al. (2003) D. V. Budescu, A. K. Rantilla, H. 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), p. 178–194. 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One fits all: power general time series analysis by pretrained lm. Advances in neural information processing systems 36, p. 43322–43355. Cited by: Appendix G, Appendix H, Appendix H, Table 23, Table 24, Appendix I, §1, §1, §5, §5. Appendix A Exemplar illustrations on importance of output alignment 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. Appendix B Proof B.1 Proof of Theorem 4.1 Proof. Rewriting the equation on the left side of the theorem, we can express it form the perspective of individual elements: |F1−F2|22h′=1h′∑ih′|F,i1−F,i2|2 p^1_F- p^2_F ^2_2h = 1h Σ^h _i p^1_F,i- p^2_F,i ^2 (8) The squared magnitude of the difference between two complex numbers F,i1 p^1_F,i and F,i2 p^2_F,i are given as follows: |F,i1−F,i2|2 p^1_F,i- p^2_F,i ^2 =(F,i1−F,i2)(1¯F,i−2¯F,i) =( p^1_F,i- p^2_F,i)( p^1_F,i- p^2_F,i) (9) Where 1¯F,i p^1_F,i and 2¯F,i p^2_F,i are the complex conjugate of F,i1 p^1_F,i and F,i2 p^2_F,i. Now, expressing F,i1 p^1_F,i and F,i2 p^2_F,i including the complex conjugate in polar form: F,i1=|F,i1|ejθF,i1,F,i2=|F,i2|ejθF,i2 p^1_F,i= p^1_F,i e^j _ p^1_F,i, p^2_F,i= p^2_F,i e^j _ p^2_F,i (10) 1¯F,i=|F,i1|e−jθF,i1,2¯F,i=|F,i2|e−jθF,i2 p^1_F,i= p^1_F,i e^-j _ p^1_F,i, p^2_F,i= p^2_F,i e^-j _ p^2_F,i (11) where j=−1j= -1 and θ is the angle (phase) for the corresponding complex number. We substitute into the expression and simplify the expression: |F,i1−F,i2|2 p^1_F,i- p^2_F,i ^2 =(|F,i1|ejθF,i1−|F,i2|ejθF,i2)(|F,i1|e−jθF,i1−|F,i2|e−jθF,i2) =( p^1_F,i e^j _ p^1_F,i- p^2_F,i e^j _ p^2_F,i)( p^1_F,i e^-j _ p^1_F,i- p^2_F,i e^-j _ p^2_F,i) (12) =|F,i1|2+|F,i2|2−|F,i1||F,i2|(ej⁡(θF,i1−θF,i2)+e−j⁡(θF,i1−θF,i2)) = p^1_F,i ^2+ p^2_F,i ^2- p^1_F,i p^2_F,i (e^j( _ p^1_F,i- _ p^2_F,i)+e^-j( _ p^1_F,i- _ p^2_F,i)) (13) Using Euler’s formula ejθ=cos(θ)+jsin(θ)e^jθ=cos(θ)+jsin(θ) [10], we can convert the exponential expressions to trigonometric expressions: |F,i1−F,i2|2 p^1_F,i- p^2_F,i ^2 =|F,i1|2+|F,i2|2−|F,i1||F,i2|cos(θF,i1−θF,i2)+jsin(θF,i1−θF,i2) = p^1_F,i ^2+ p^2_F,i ^2- p^1_F,i p^2_F,i \cos( _ p^1_F,i- _ p^2_F,i)+jsin( _ p^1_F,i- _ p^2_F,i) (14) +cos(−(θF,i1−θF,i2))+jsin(−(θF,i1−θF,i2)) +cos(-( _ p^1_F,i- _ p^2_F,i))+jsin(-( _ p^1_F,i- _ p^2_F,i))\ (15) =|F,i1|2+|F,i2|2−2|F,i1||F,i2|cos(θF,i1−θF,i2) = p^1_F,i ^2+ p^2_F,i ^2-2 p^1_F,i p^2_F,i cos( _ p^1_F,i- _ p^2_F,i) (16) Utilizing the angle (phase) notation ∠ , we can express the equation as: |F,i1−F,i2|2=|F,i1|2+|F,i2|2−2|F,i1||F,i2|cos(∠F,i1−∠F,i2) p^1_F,i- p^2_F,i ^2= p^1_F,i ^2+ p^2_F,i ^2-2 p^1_F,i p^2_F,i cos( p^1_F,i- p^2_F,i) (17) Reducing the final equation implies increasing the cosine term, which in turn signifies aligning the two phases, since one of predictions is constant by stop-gradient sg(⋅)sg(·) operator in Alg. 1. |F,i1−F,i2|2 p^1_F,i- p^2_F,i ^2 →0⟹cos(∠F,i1−∠F,i2)→1 → 0 cos( p^1_F,i- p^2_F,i)→ 1 (18) cos(∠F,i1−∠F,i2) cos( p^1_F,i- p^2_F,i) →1⟹|∠F1−∠F2|→0 → 1 p^1_F- p^2_F → 0 (19) ∴|F,i1−F,i2|2 p^1_F,i- p^2_F,i ^2 →0⟹|∠F1−∠F2|→0 → 0 p^1_F- p^2_F → 0 (20) ∎ B.2 Proof of Theorem 4.2 Proof. Since the two phases, i.e., ∠F,i1 p^1_F,i and ∠F,i2 p^2_F,i, are aligned, the Eq. 17 in Sec. B.1 is modified as follows: |F,i1−F,i2|2=|F,i1|2+|F,i2|2−2|F,i1||F,i2| p^1_F,i- p^2_F,i ^2= p^1_F,i ^2+ p^2_F,i ^2-2 p^1_F,i p^2_F,i (21) By factoring, it can be expressed in the following perfect square form. |F,i1−F,i2|2=(|F,i1|2−|F,i2|2)2 p^1_F,i- p^2_F,i ^2=( p^1_F,i ^2- p^2_F,i ^2)^2 (22) Since one of predictions is constant by stop-gradient sg(⋅)sg(·) operator in Alg. 1, minimizing |F,i1−F,i2|2 p^1_F,i- p^2_F,i ^2 leads to a reduction in the amplitude difference ∎ Appendix C Visualization of alignment In this section, we visualize the model’s prediction outputs and alignment differences when trained solely with regression loss versus with both regression loss and AliO. Each figure is arranged in a 2×2 grid: the top row shows results using regression loss alone, while the bottom row presents results incorporating AliO; the left column illustrates the time domain, and the right column displays the frequency domain visualized in a polar coordinate system, where the azimuth angle represents the phase and the radial distance from the origin corresponds to the amplitude. Overlapping timestamps of prediction outputs for five lagged inputs are shown, with the ground truth indicated by black dashed lines. Figures 9, 10, 11 and 12 show the results from DLinear [40], PatchTST [33], TimesNet [36], and iTransformer [30]. Figure 9: Visualization: Upper shows the prediction of DLinear [40] 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. Figure 10: Visualization: Upper shows the prediction of PatchTST [33] using only the regression loss on the ETTm1 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. 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. 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. Appendix D On sudden event This section expands on the sudden events mentioned in Sec. 7. Definition D.1. The sudden event refers to an anomalous data point in the input sequence that can be interpreted as out-of-distribution (OOD), causing perturbations in the output sequence. Such event occur when input-level perturbations lead to unpredictable output patterns. These scenarios deviate from the core objective of Long-term Time Series Forecasting (LTSF), which focuses on predictable sequences. Instead, they align more closely with: • Domain Adaptation [19] • Anomaly Detection [39] • Test-Time Adaptation [21] The TAM metric is specifically designed for LTSF tasks with predictable, stable output sequences (i.e., non-perturbed scenarios). In contexts involving sudden events or OOD (out-of-distribution) data, the inherent assumptions of TAM-particularly its reliance on sequence stability-may not hold, necessitating cautious application. However, if a sudden event includes precursor signals (i.e., detectable input patterns) and the model can sufficiently predict the output sequence using this information, TAM remains valid even under such conditions. Appendix E Initial TAM vs. MSE improvement Figure 5 may be difficult to interpret as there is no explicit distinction between models and datasets. In this section [Figure 13], to facilitate easier interpretation, models are represented by different shapes and datasets are represented by different colors. Figure 13: The relationship between initial TAM and MSE improvement (%). Appendix F Zero coefficient of time domain alignment (λT _T) By default, AliO operates in the time domain, so scenarios where the time-scaling coefficient is zero were excluded from the main context. However, we conducted experiments on CycleNet [27], iTransformer [30], TimesNet [36] under conditions with a zero time-scaling coefficient. The results, shown in Figures 15, 15, 17, 17, 19, 19, 19, 21, 21, 23, 23, 25 and 25, are normalized between 0 and 1. Since the condition (λT=0 _T=0, λF=0 _F=0) implies that AliO is not used, so it is empty. As demonstrated in Figure 7 increasing λT _T concsistently improves TAM performance. When it comes to MSE performance, there’s a general trend for it to improve as the coefficient increases, similar to what’s seen with TAM (I’m assuming this refers to a specific model or method you’re using). However, in some environments, the coefficient can become excessively large, dominating the regression loss and causing performance to degrade. Despite this, our experiments confirmed that an appropriate, non-zero coefficient leads to improved performance. The tables show the results where the lag is 1 and the number of sequences is 2 (default). Figure 14: Comparison of the normalized MSE (left) and TAM (right) of CycleNet trained on ETTm1 dataset. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\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 λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\0.0,1.0,2.0,5.0\). Figure 16: Comparison of the normalized MSE (left) and TAM (right) of CycleNet trained on Traffic dataset. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\0.0,1.0,2.0,5.0\). Figure 17: Comparison of the normalized MSE (left) and TAM (right) of iTransformer trained on ETTh1 dataset. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\0.0,1.0,2.0,5.0\). Figure 18: Comparison of the normalized MSE (left) and TAM (right) of iTransformer trained on ETTh2 dataset. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\0.0,1.0,2.0,5.0\). Figure 19: Comparison of the normalized MSE (left) and TAM (right) of iTransformer trained on ETTm1 dataset. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\0.0,1.0,2.0,5.0\). Figure 20: Comparison of the normalized MSE (left) and TAM (right) of iTransformer trained on ETTm2 dataset. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\0.0,1.0,2.0,5.0\). Figure 21: Comparison of the normalized MSE (left) and TAM (right) of iTransformer trained on Traffic dataset. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\0.0,1.0,2.0,5.0\). Figure 22: Comparison of the normalized MSE (left) and TAM (right) of iTransformer trained on Weather dataset. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\0.0,1.0,2.0,5.0\). Figure 23: Comparison of the normalized MSE (left) and TAM (right) of TimesNet trained on ETTh1 dataset. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\0.0,1.0,2.0,5.0\). Figure 24: Comparison of the normalized MSE (left) and TAM (right) of TimesNet trained on ETTm1 dataset. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\0.0,1.0,2.0,5.0\). Figure 25: Comparison of the normalized MSE (left) and TAM (right) of TimesNet trained on ETTm2 dataset. The x-axis represents λF _F (0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\), and the y-axis represents λT _T (0.0,1.0,2.0,5.0\0.0,1.0,2.0,5.0\). Appendix G Dataset explanation We provide a brief description of each dataset used in our experiments, as referenced in Secs. 5 and I. • Electricity transformer temperature (ETT) [42]: This dataset consists of two years of data collected from two counties in China. It is divided into four subsets: ETTh1 and ETTh2 (sampled hourly), and ETTm1 and ETTm2 (sampled every 15 minutes). Each record contains six power load features and one target variable representing the oil temperature. • Exchange [37]: Contains daily exchange rates from eight countries, spanning from 1990 to 2016. • Electricity (ECL) [13]: Comprises electric power consumption data sampled every minute over four years for a single household. • ILI (https: / / github.com / thuml / Autoformer): Weekly records from 2002 to 2021, provided by the US Centers for Disease Control and Prevention, representing the number of influenza-like illness patients. • PEMS: Contains traffic information measured at 5-minute intervals on California highways. The subsets PEMS03, PEMS04, PEMS07, and PEMS08 correspond to different regions and time spans, with varying numbers of sensors [28]. • Solar-Energy [24]: Includes solar energy production data from 137 plants in 2006, measured at 10-minute intervals. • Traffic [37]: Consists of hourly traffic congestion data collected by 862 sensors on San Francisco freeways from January 2015 to December 2016. To ensure fair performance evaluation, we adopted the sequence length, prediction length, and label length settings used in recent LTSF models, such as Autoformer [37], DLinear [40], PatchTST [33], TimesNet [36], iTransformer [30], GPT4TS [43], CycleNet [27]. Following previous works [37, 30, 28]. Tab. 3 shows the descriptions of datasets including the number of variate in each dataset, prediction length, dataset size, sampling frequency, and domain. Since the official Autoformer implementation provides the results in different prediction length (24, 48, 168, 336, 720), we follow their implementation and show the results on Appendix I. Table 3: Descriptions of datasets. # of vars means the number of variate in each dataset. Dataset # of vars Prediction Length Dataset size (Train / Validation / Test) Frequency Domain ETTh1 7 24, 48, 168, 336, 720 (Autoformer) 96, 192, 336, 720 (other models) 8545 / 2881 / 2881 Hourly Temperature ETTh2 7 24, 48, 168, 336, 720 (Autoformer) 96, 192, 336, 720 (other models) 8545 / 2881 / 2881 Hourly Temperature ETTm1 7 24, 48, 168, 336, 720 (Autoformer) 96, 192, 336, 720 (other models) 34465 / 11521 / 11521 15 min Temperature ETTm2 7 24, 48, 168, 336, 720 (Autoformer) 96, 192, 336, 720 (other models) 34465 / 11521 / 11521 15 min Temperature Weather 21 96, 192, 336, 720 36792 / 5271 / 10540 10 min Weather Electricity (ECL) 321 96, 192, 336, 720 18317 / 2633 / 5261 Hourly Electricity Traffic 862 96, 192, 336, 720 12185 / 1757 / 3509 Hourly Transportation ILI 7 24, 36, 48, 60 617 / 74 / 170 Weekly Health Exchange Rate (Exchange) 8 96, 192, 336, 720 5120 / 665 / 1422 Daily Economy Solar 137 96, 192, 336, 720 36601 / 5161 / 10417 10 min Energy PEMS03 358 12, 24, 48, 96 15617 / 5135 / 5135 5 min Transportation PEMS04 307 12, 24, 48, 96 10172 / 3375 / 3375 5 min Transportation PEMS07 883 12, 24, 48, 96 16911 / 5622 / 5622 5 min Transportation PEMS08 170 12, 24, 48, 96 10690 / 3548 / 3548 5 min Transportation Appendix H Experiment configurations To ensure fair model evaluation, we utilized the official GitHub codes provided by six benchmark models: Autoformer [37], DLinear [40], PatchTST [33], TimesNet [36], iTransformer [30], GPT4TS [43], CycleNet [27]. For all experiments, we adopted the same prediction length, label length, and sequence length as the official implementations, and maintained the original architecture of each model. Context lengths. According to the papers [37, 40, 33, 36, 30, 43, 27], the used context lenghts are followed by: • CycleNet, Autoformer, iTransformer: 96 • TimesNet: 96 (36 for the ILI dataset) • PatchTST, DLinear: 336 (104 for the ILI dataset) Prediction lengths. We primarily used prediction lengths of 96,192,336,720\96,192,336,720\. For the ILI dataset (https://github.com/thuml/Autoformer), we followed prior works and used 24,36,48,60\24,36,48,60\. As an exception, for the ETTh1, h2, m1, m2 datasets, we followed the Autoformer paper and used 24,48,168,336,720\24,48,168,336,720\ (for Autoformer) and 96,192,336,720\96,192,336,720\ (for other models). Optimization. The optimizer and scheduler were used under the same conditions as specified in the official codes for each model. Our method. For our method (AliO), the hyperparameters—the number of samples N in Alg. 1 and lag l in Alg. 1—were set to their default values of 22 and 11, 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 (λT _T) and frequency domain (λF _F) were selected from the ranges 1.0,2.0,5.0\1.0,2.0,5.0\ and 0.0,0.5,1.0,2.0\0.0,0.5,1.0,2.0\, respectively (default values are λT=1 _T=1 and λF=0 _F=0 since the frequency domain is optional). Additionally, the AWL [26] technique was employed to automatically tune these coefficients and report the best performance. Evaluation. We used Mean Squared Error (MSE) as our baseline loss function and reported forecasting performance using MSE, Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), and TAM. For all these metrics, lower values indicate better performance. Robustness. To ensure robust results, we used three random seeds (2023,2024,20252023,2024,2025) for initialization and report the standard deviation using ±. All experiments were conducted on NVIDIA RTX 3090 and A6000 GPUs. We conducted all experiments using the same GPU when comparing the baseline and AliO under the same conditions. Training hyper-parameters. We used the same learning rate, batch size, and epoch as the official implementation of each model for reproducibility and fair comparison. The implementation code for each model is as follows. • Autoformer: https://github.com/thuml/Autoformer • DLinear: https://github.com/vivva/DLinear • PatchTST: https://github.com/yuqinie98/PatchTST • TimesNet: https://github.com/thuml/TimesNet • iTransformer: https://github.com/thuml/iTransformer • GPT4TS: https://github.com/DAMO-DI-ML/NeurIPS2023-One-Fits-All • CycleNet: https://github.com/ACAT-SCUT/CycleNet We followed the implementation code listed above (GitHub) and used the same learning rate, batch size, and number of epochs as shown in Tabs. 4, 5, 6, 7, 8, 9, 10 and 11. The optimizer we used is Adam [23]. Table 4: Learning rate and batch size of Autoformer used in each datasets. Dataset ECL ETTh1 ETTh2 ETTm1 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 3 10 Table 5: Learning rate and batch size of DLinear used in each datasets. Dataset ECL ETTh1 ETTh2 ETTm1 ETTm2 Exchange ILI Traffic Weather Learning rate 0.001 0.0001 0.05 0.0001 0.001 0.01 0.0005 0.005 0.05 0.05 0.0001 batch size 16 8 32 8 32 8 32 32 16 16 Epoch 10 10 10 10 10 10 10 10 10 Table 6: Learning rate and batch size of PatchTST used in each datasets. Dataset ECL ETTh1 ETTh2 ETTm1 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 each datasets. Dataset ECL ETTh1 ETTh2 ETTm1 ETTm2 ILI 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 10 Table 8: Learning rate and batch size of iTransformer used in each datasets (excluding PEMS and Solar). Dataset ECL ETTh1 ETTh2 ETTm1 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 16 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 Learning rate 0.001 0.0005 0.001 0.0001 0.001 0.0005 batch size 32 32 32 16 32 16 32 Epoch 10 10 10 10 10 Table 10: Learning rate and batch size of GPT4TS used in each datasets. Dataset ECL ETTh1 ETTh2 ETTm1 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 Table 11: Learning rate and batch size of CycleNet used in each datasets. Dataset ECL ETTh1 ETTh2 ETTm1 ETTm2 Traffic Weather Solar Learning rate 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 batch size 128 128 128 128 128 128 128 128 Epoch 30 30 30 30 30 30 30 30 Appendix I Full experimental results This section presents the comprehensive results of each Long-Term Series Forecasting (LTSF) models, Autoformer [37] (Tabs. 12 and 13), DLinear [40] (Tabs. 14 and 15), PatchTST [33] (Tabs. 16 and 17), TimesNet [36] (Tabs. 18 and 19), iTransformer [30] (Tabs. 20, 21 and 22), GPT4TS [43] (Tabs. 23 and 24), and CycleNet [27] (Tabs. 25 and 26). The evaluation metrics include Mean Squared Error (MSE), Temporal Alignment Metric (TAM), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Root Mean Squared Error (RMSE). The ± symbol denotes the standard deviation across multiple seeds (we used three random seeds for robust experiment). AVG represents the average value across all prediction lengths, with the best performance highlighted in bold. Table 12: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, 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/Autoformer) 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 is MSE: 27.90%, TAM: 81.18%, MAE: 15.17%, MAPE: 15.42%, and RMSE: 8.58%. Exchange, Traffix, and Weather are in Tab. 13. Models Autoformer Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ ECL 96 0.204±0.008± 0.008 0.035±0.002± 0.002 0.319±0.008± 0.008 3.398±0.230± 0.230 0.452±0.009± 0.009 0.193± 0.003 0.029± 0.001 0.308± 0.003 3.296± 0.132 0.440± 0.004 192 0.219±0.005± 0.005 0.042±0.002± 0.002 0.329±0.003± 0.003 3.416±0.185± 0.185 0.468±0.005± 0.005 0.215± 0.003 0.038± 0.004 0.324± 0.004 3.402± 0.091 0.464± 0.004 336 0.232±0.004± 0.004 0.040±0.004± 0.004 0.340±0.004± 0.004 3.466±0.224± 0.224 0.481±0.004± 0.004 0.223± 0.007 0.034± 0.004 0.331± 0.005 3.350± 0.136 0.472± 0.007 720 0.260±0.013± 0.013 0.046±0.007± 0.007 0.362±0.009± 0.009 3.501±0.118± 0.118 0.509±0.013± 0.013 0.253± 0.008 0.042± 0.002 0.356± 0.006 3.404± 0.035 0.503± 0.008 AVG 0.229±0.018± 0.018 0.041±0.003± 0.003 0.337±0.015± 0.015 3.445±0.036± 0.036 0.478±0.019± 0.019 0.221± 0.019 0.036± 0.004 0.330± 0.015 3.363± 0.040 0.469± 0.020 ETTh1 24 0.408±0.008± 0.008 0.063±0.005± 0.005 0.427±0.002± 0.002 10.853±0.413± 0.413 0.639±0.006± 0.006 0.373± 0.008 0.051± 0.002 0.416± 0.007 10.747± 0.477 0.611± 0.007 48 0.408±0.014± 0.014 0.048±0.004± 0.004 0.431±0.008± 0.008 11.632±0.477± 0.477 0.639±0.011± 0.011 0.395± 0.024 0.037± 0.002 0.420± 0.012 10.889± 0.175 0.628± 0.019 168 0.502±0.028± 0.028 0.051±0.012± 0.012 0.482±0.016± 0.016 11.810±0.142± 0.142 0.708±0.020± 0.020 0.445± 0.014 0.036± 0.005 0.451± 0.010 11.573± 0.593 0.667± 0.011 336 0.517±0.013± 0.013 0.039± 0.003 0.491±0.006± 0.006 11.786± 0.326 0.719±0.009± 0.009 0.484± 0.013 0.048±0.002± 0.002 0.479± 0.008 11.957±0.223± 0.223 0.696± 0.009 720 0.549±0.016± 0.016 0.047± 0.002 0.527±0.008± 0.008 12.533± 0.659 0.741±0.011± 0.011 0.521± 0.009 0.057±0.009± 0.009 0.516± 0.010 13.217±1.015± 1.015 0.722± 0.007 AVG 0.477±0.053± 0.053 0.050±0.007± 0.007 0.472±0.034± 0.034 11.723±0.488± 0.488 0.689±0.039± 0.039 0.444± 0.050 0.046± 0.008 0.456± 0.034 11.677± 0.811 0.665± 0.038 ETTh2 24 0.280±0.009± 0.009 0.054±0.004± 0.004 0.356±0.006± 0.006 1.524±0.038± 0.038 0.529±0.008± 0.008 0.265± 0.003 0.034± 0.003 0.340± 0.003 1.392± 0.028 0.515± 0.003 48 0.335±0.029± 0.029 0.051±0.009± 0.009 0.390±0.021± 0.021 1.624±0.067± 0.067 0.578±0.025± 0.025 0.301± 0.009 0.025± 0.002 0.360± 0.006 1.422± 0.054 0.549± 0.008 168 0.442±0.008± 0.008 0.044±0.009± 0.009 0.448±0.009± 0.009 1.868±0.150± 0.150 0.665±0.006± 0.006 0.421± 0.008 0.027± 0.002 0.427± 0.005 1.580± 0.084 0.649± 0.006 336 0.479±0.012± 0.012 0.031±0.012± 0.012 0.482±0.012± 0.012 2.003±0.194± 0.194 0.692±0.009± 0.009 0.448± 0.002 0.018± 0.004 0.455± 0.002 1.749± 0.020 0.669± 0.001 720 0.470±0.022± 0.022 0.025±0.005± 0.005 0.483±0.013± 0.013 2.273±0.166± 0.166 0.685±0.016± 0.016 0.450± 0.005 0.020± 0.000 0.466± 0.003 2.044± 0.024 0.670± 0.004 AVG 0.401±0.072± 0.072 0.041±0.010± 0.010 0.432±0.046± 0.046 1.858±0.245± 0.245 0.630±0.059± 0.059 0.377± 0.071 0.025± 0.005 0.410± 0.046 1.637± 0.219 0.610± 0.060 ETTm1 24 0.377±0.015± 0.015 0.092±0.003± 0.003 0.415±0.005± 0.005 2.563±0.005± 0.005 0.614±0.012± 0.012 0.351± 0.023 0.084± 0.001 0.394± 0.011 2.455± 0.019 0.592± 0.019 48 0.447±0.048± 0.048 0.079±0.014± 0.014 0.452±0.017± 0.017 2.789±0.130± 0.130 0.668±0.035± 0.035 0.381± 0.002 0.053± 0.001 0.412± 0.003 2.458± 0.074 0.617± 0.001 96 0.493±0.054± 0.054 0.074±0.010± 0.010 0.478±0.019± 0.019 2.834±0.081± 0.081 0.701±0.038± 0.038 0.440± 0.029 0.055± 0.006 0.445± 0.012 2.579± 0.078 0.663± 0.022 288 0.579±0.030± 0.030 0.070±0.013± 0.013 0.513±0.012± 0.012 2.851±0.060± 0.060 0.760±0.020± 0.020 0.515± 0.003 0.041± 0.002 0.479± 0.004 2.604± 0.049 0.718± 0.002 672 0.544± 0.013 0.036±0.001± 0.001 0.501±0.005± 0.005 2.788±0.062± 0.062 0.737± 0.009 0.554±0.013± 0.013 0.031± 0.002 0.501± 0.009 2.728± 0.044 0.744±0.009± 0.009 AVG 0.488±0.065± 0.065 0.070±0.017± 0.017 0.472±0.032± 0.032 2.765±0.095± 0.095 0.696±0.047± 0.047 0.448± 0.070 0.053± 0.016 0.446± 0.036 2.565± 0.093 0.667± 0.053 ETTm2 24 0.161±0.013± 0.013 0.078±0.010± 0.010 0.268±0.010± 0.010 1.081±0.039± 0.039 0.401±0.016± 0.016 0.148± 0.005 0.057± 0.005 0.259± 0.005 1.054± 0.041 0.385± 0.007 48 0.217±0.012± 0.012 0.080±0.005± 0.005 0.308±0.009± 0.009 1.196±0.046± 0.046 0.466±0.013± 0.013 0.182± 0.006 0.041± 0.009 0.279± 0.007 1.094± 0.033 0.426± 0.007 96 0.252±0.029± 0.029 0.058±0.008± 0.008 0.324±0.015± 0.015 1.302±0.024± 0.024 0.502±0.028± 0.028 0.217± 0.003 0.030± 0.006 0.298± 0.004 1.158± 0.019 0.466± 0.003 288 0.319±0.005± 0.005 0.027±0.004± 0.004 0.362±0.003± 0.003 1.407±0.029± 0.029 0.565±0.005± 0.005 0.313± 0.001 0.019± 0.005 0.354± 0.001 1.307± 0.013 0.559± 0.001 672 0.408± 0.002 0.017±0.003± 0.003 0.410±0.001± 0.001 1.577±0.014± 0.014 0.639± 0.001 0.409±0.003± 0.003 0.012± 0.004 0.409± 0.002 1.538± 0.047 0.640±0.002± 0.002 AVG 0.271±0.078± 0.078 0.052±0.024± 0.024 0.334±0.044± 0.044 1.313±0.156± 0.156 0.514±0.075± 0.075 0.254± 0.087 0.032± 0.015 0.320± 0.050 1.230± 0.161 0.495± 0.084 Table 13: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, 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/Autoformer) 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 is MSE: 27.90%, TAM: 81.18%, MAE: 15.17%, MAPE: 15.42%, and RMSE: 8.58%. ECL, ETTh1, ETTh2, ETTm1, and ETTm2 are in Tab. 12. Models Autoformer Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ Exchange 96 0.155±0.012± 0.012 0.031±0.016± 0.016 0.286±0.011± 0.011 1.700±0.053± 0.053 0.394±0.015± 0.015 0.139± 0.005 0.018± 0.007 0.270± 0.005 1.648± 0.010 0.373± 0.007 192 0.284±0.017± 0.017 0.014± 0.004 0.389±0.009± 0.009 2.345±0.025± 0.025 0.532±0.016± 0.016 0.275± 0.012 0.015±0.008± 0.008 0.379± 0.009 2.338± 0.025 0.525± 0.011 336 0.447±0.041± 0.041 0.020±0.007± 0.007 0.496±0.023± 0.023 3.403±0.130± 0.130 0.668±0.031± 0.031 0.436± 0.012 0.011± 0.000 0.488± 0.007 3.369± 0.002 0.660± 0.009 720 1.110±0.035± 0.035 0.037±0.044± 0.044 0.814± 0.022 6.640±0.123± 0.123 1.053±0.017± 0.017 0.800± 0.011 0.007± 0.002 1.079±0.022± 0.022 6.627± 0.094 1.039± 0.011 AVG 0.499±0.329± 0.329 0.025±0.008± 0.008 0.496± 0.177 3.522±1.699± 1.699 0.662±0.220± 0.220 0.413± 0.221 0.013± 0.004 0.554±0.280± 0.280 3.495± 1.707 0.649± 0.221 Traffic 96 0.637±0.018± 0.018 0.051±0.005± 0.005 0.400±0.014± 0.014 4.222±0.261± 0.261 0.798±0.011± 0.011 0.613± 0.009 0.041± 0.003 0.375± 0.002 3.782± 0.014 0.783± 0.006 192 0.626±0.009± 0.009 0.055±0.006± 0.006 0.389±0.006± 0.006 4.104± 0.090 0.791±0.006± 0.006 0.621± 0.008 0.051± 0.005 0.386± 0.006 4.130±0.079± 0.079 0.788± 0.005 336 0.623±0.006± 0.006 0.042± 0.002 0.386±0.005± 0.005 4.207± 0.091 0.789±0.004± 0.004 0.614± 0.003 0.048±0.006± 0.006 0.379± 0.003 4.313±0.167± 0.167 0.784± 0.002 720 0.651±0.010± 0.010 0.045±0.004± 0.004 0.398±0.006± 0.006 4.326±0.033± 0.033 0.807±0.006± 0.006 0.648± 0.005 0.039± 0.001 0.386± 0.003 3.970± 0.063 0.805± 0.003 AVG 0.634±0.010± 0.010 0.048±0.005± 0.005 0.393±0.005± 0.005 4.215±0.070± 0.070 0.796±0.006± 0.006 0.624± 0.013 0.045± 0.004 0.382± 0.004 4.049± 0.175 0.790± 0.008 Weather 96 0.269±0.032± 0.032 0.069±0.011± 0.011 0.340±0.023± 0.023 13.016± 1.557 0.518±0.031± 0.031 0.224± 0.009 0.051± 0.003 0.288± 0.011 13.076±0.455± 0.455 0.474± 0.010 192 0.298±0.013± 0.013 0.055±0.003± 0.003 0.356±0.008± 0.008 14.001± 0.960 0.546±0.012± 0.012 0.282± 0.007 0.043± 0.005 0.332± 0.006 15.517±0.589± 0.589 0.531± 0.007 336 0.367±0.026± 0.026 0.046±0.006± 0.006 0.398±0.020± 0.020 13.587± 2.218 0.606±0.021± 0.021 0.338± 0.011 0.045± 0.014 0.368± 0.008 14.002±0.378± 0.378 0.581± 0.010 720 0.434±0.014± 0.014 0.056±0.004± 0.004 0.441±0.013± 0.013 11.666± 0.645 0.659±0.011± 0.011 0.401± 0.005 0.027± 0.007 0.410± 0.004 12.468±0.723± 0.723 0.633± 0.004 AVG 0.342±0.057± 0.057 0.056±0.007± 0.007 0.384±0.035± 0.035 13.068± 0.788 0.582±0.049± 0.049 0.311± 0.058 0.042± 0.008 0.350± 0.040 13.766±1.028± 1.028 0.555± 0.053 Table 14: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, ETTm2, Exchange, ILI, Traffix, and Weather datasets and 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/vivva/DLinear) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 5.62%, TAM: 36.21%, MAE: -3.82%, MAPE: 2.25%, and RMSE: 0.79%. The maximum improvement of AliO over the baseline is MSE: 36.07%, TAM: 77.73%, MAE: 8.21%, MAPE: 15.19%, and RMSE: 5.61%. Exchange ILI Traffic Weather are in Tab. 15. Models DLinear Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ ECL 96 0.140± 0.000 0.017± 0.000 0.237± 0.000 2.149± 0.005 0.374± 0.000 0.140± 0.000 0.013± 0.000 0.237± 0.000 2.149± 0.003 0.374± 0.000 192 0.153± 0.000 0.013± 0.000 0.250± 0.000 2.300± 0.003 0.391± 0.000 0.153± 0.000 0.010± 0.000 0.250± 0.000 2.301± 0.001 0.391± 0.000 336 0.169± 0.000 0.011± 0.000 0.268± 0.000 2.283± 0.004 0.411± 0.000 0.169± 0.000 0.008± 0.000 0.267± 0.000 2.283± 0.001 0.411± 0.000 720 0.204± 0.000 0.011± 0.001 0.301± 0.000 2.457± 0.002 0.451± 0.000 0.203± 0.000 0.008± 0.001 0.300± 0.000 2.451± 0.004 0.451± 0.000 AVG 0.166± 0.021 0.013± 0.002 0.264± 0.021 2.297± 0.098 0.407± 0.026 0.166± 0.021 0.010± 0.002 0.264± 0.021 2.296± 0.096 0.407± 0.026 ETTh1 96 0.371± 0.000 0.028± 0.001 0.395± 0.001 8.691± 0.011 0.609± 0.000 0.369± 0.000 0.019± 0.000 0.392± 0.000 8.743± 0.003 0.608± 0.000 192 0.408± 0.002 0.028± 0.002 0.419± 0.002 8.481± 0.081 0.638± 0.002 0.404± 0.000 0.015± 0.000 0.413± 0.000 8.496± 0.006 0.635± 0.000 336 0.435± 0.002 0.024± 0.005 0.439± 0.003 8.545± 0.053 0.660± 0.001 0.432± 0.000 0.013± 0.000 0.435± 0.000 8.489± 0.001 0.658± 0.000 720 0.474± 0.001 0.022± 0.003 0.493± 0.001 9.459± 0.056 0.689± 0.001 0.469± 0.000 0.010± 0.000 0.488± 0.000 9.429± 0.001 0.685± 0.000 AVG 0.422± 0.034 0.025± 0.003 0.436± 0.032 8.794± 0.350 0.649± 0.026 0.419± 0.033 0.014± 0.003 0.432± 0.032 8.789± 0.343 0.646± 0.025 ETTh2 96 0.291± 0.003 0.069± 0.016 0.354± 0.002 1.359± 0.048 0.539± 0.003 0.290± 0.006 0.046± 0.010 0.353± 0.006 1.321± 0.028 0.538± 0.006 192 0.381± 0.008 0.049± 0.005 0.416± 0.006 1.369± 0.007 0.617± 0.007 0.369± 0.007 0.023± 0.004 0.405± 0.006 1.376± 0.022 0.607± 0.006 336 0.438± 0.011 0.039± 0.001 0.456± 0.007 1.494± 0.008 0.662± 0.008 0.454± 0.009 0.050± 0.027 0.465± 0.004 1.497± 0.041 0.674± 0.007 720 0.687± 0.044 0.057± 0.020 0.586± 0.018 1.605± 0.033 0.828± 0.026 0.672± 0.025 0.013± 0.002 0.577± 0.013 1.544± 0.019 0.820± 0.015 AVG 0.449± 0.131 0.054± 0.010 0.453± 0.076 1.457± 0.090 0.662± 0.095 0.446± 0.128 0.033± 0.014 0.450± 0.075 1.434± 0.080 0.660± 0.093 ETTm1 96 0.300± 0.001 0.031± 0.002 0.344± 0.001 2.021± 0.011 0.548± 0.001 0.296± 0.000 0.014± 0.000 0.338± 0.000 1.983± 0.001 0.544± 0.000 192 0.338± 0.000 0.026± 0.001 0.369± 0.001 2.079± 0.008 0.581± 0.000 0.333± 0.000 0.010± 0.000 0.361± 0.000 2.068± 0.001 0.577± 0.000 336 0.369± 0.000 0.020± 0.001 0.386± 0.001 2.131± 0.008 0.608± 0.000 0.367± 0.000 0.008± 0.000 0.382± 0.000 2.124± 0.001 0.606± 0.000 720 0.427± 0.000 0.019± 0.001 0.422± 0.001 2.240± 0.018 0.653± 0.000 0.422± 0.000 0.006± 0.000 0.416± 0.000 2.237± 0.001 0.650± 0.000 AVG 0.358± 0.041 0.024± 0.004 0.380± 0.025 2.118± 0.072 0.597± 0.034 0.354± 0.042 0.010± 0.002 0.374± 0.025 2.103± 0.082 0.594± 0.035 ETTm2 96 0.171± 0.002 0.034± 0.002 0.265± 0.004 1.047± 0.009 0.413± 0.002 0.166± 0.001 0.018± 0.001 0.258± 0.001 1.056± 0.003 0.407± 0.001 192 0.233± 0.004 0.031± 0.004 0.312± 0.007 1.153± 0.024 0.483± 0.004 0.229± 0.002 0.015± 0.001 0.307± 0.002 1.162± 0.005 0.478± 0.002 336 0.311± 0.022 0.053± 0.023 0.366± 0.017 1.325± 0.085 0.557± 0.019 0.284± 0.005 0.026± 0.007 0.344± 0.004 1.274± 0.018 0.533± 0.005 720 0.443± 0.012 0.030± 0.003 0.453± 0.008 1.315± 0.019 0.665± 0.009 0.394± 0.004 0.020± 0.005 0.415± 0.003 1.384± 0.007 0.628± 0.003 AVG 0.289± 0.091 0.037± 0.009 0.349± 0.062 1.210± 0.104 0.530± 0.083 0.268± 0.075 0.020± 0.004 0.331± 0.051 1.219± 0.110 0.512± 0.072 Table 15: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, ETTm2, Exchange, ILI, Traffix, and Weather datasets and 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/vivva/DLinear) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 5.62%, TAM: 36.21%, MAE: -3.82%, MAPE: 2.25%, and RMSE: 0.79%. The maximum improvement of AliO over the baseline is MSE: 36.07%, TAM: 77.73%, MAE: 8.21%, MAPE: 15.19%, and RMSE: 5.61%. ECL, ETTh1, ETTh2, ETTm1, and ETTm2 are in Tab. 14. Models DLinear Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ Exchange 96 0.086± 0.006 0.017± 0.000 0.208± 0.008 1.198± 0.036 0.293± 0.010 0.078± 0.000 0.015± 0.000 0.201± 0.000 1.120± 0.001 0.280± 0.000 192 0.160± 0.002 0.015± 0.001 0.295± 0.002 1.584± 0.033 0.400± 0.003 0.157± 0.002 0.014± 0.001 0.291± 0.002 1.594± 0.019 0.396± 0.002 336 0.330± 0.032 0.012± 0.005 0.434± 0.016 2.120± 0.139 0.574± 0.028 0.320± 0.034 0.012± 0.003 0.425± 0.022 2.222± 0.207 0.565± 0.030 720 0.833± 0.129 0.033± 0.017 0.689± 0.050 2.298± 0.441 0.910± 0.072 0.822± 0.135 0.054± 0.033 0.679± 0.051 2.186± 0.258 0.904± 0.074 AVG 0.352± 0.260 0.019± 0.007 0.407± 0.163 1.800± 0.390 0.544± 0.209 0.344± 0.259 0.024± 0.016 0.399± 0.161 1.781± 0.407 0.536± 0.210 ILI 24 2.335± 0.149 0.196± 0.039 1.078± 0.060 4.052± 0.445 1.527± 0.049 2.229± 0.019 0.131± 0.003 1.036± 0.005 3.689± 0.059 1.493± 0.006 36 2.072± 0.026 0.144± 0.009 1.018± 0.012 2.497± 0.075 1.439± 0.009 2.076± 0.036 0.153± 0.063 1.024± 0.005 2.507± 0.190 1.441± 0.013 48 2.269± 0.077 0.136± 0.020 1.091± 0.032 2.685± 0.249 1.506± 0.026 2.161± 0.016 0.078± 0.003 1.049± 0.003 2.278± 0.016 1.470± 0.005 60 2.315± 0.022 0.132± 0.028 1.088± 0.006 2.542± 0.176 1.521± 0.007 2.280± 0.080 0.112± 0.021 1.078± 0.019 2.412± 0.050 1.510± 0.027 AVG 2.247± 0.093 0.152± 0.023 1.069± 0.027 2.944± 0.576 1.498± 0.031 2.187± 0.069 0.119± 0.025 1.047± 0.018 2.721± 0.505 1.478± 0.023 Traffic 96 0.412± 0.001 0.024± 0.001 0.286± 0.001 3.123± 0.030 0.642± 0.001 0.269± 0.000 0.007± 0.000 0.426± 0.000 2.749± 0.000 0.653± 0.000 192 0.424± 0.000 0.026± 0.003 0.291± 0.000 3.073± 0.018 0.651± 0.000 0.276± 0.000 0.007± 0.002 0.430± 0.000 2.718± 0.006 0.656± 0.000 336 0.438± 0.001 0.023± 0.000 0.299± 0.001 3.047± 0.008 0.662± 0.000 0.282± 0.000 0.006± 0.002 0.440± 0.000 2.690± 0.006 0.664± 0.000 720 0.468± 0.001 0.026± 0.004 0.319± 0.001 3.139± 0.022 0.684± 0.001 0.299± 0.000 0.009± 0.002 0.466± 0.000 2.769± 0.005 0.683± 0.000 AVG 0.436± 0.019 0.025± 0.001 0.299± 0.011 3.095± 0.033 0.660± 0.014 0.282± 0.010 0.007± 0.001 0.441± 0.014 2.731± 0.027 0.664± 0.010 Weather 96 0.175± 0.001 0.009± 0.001 0.237± 0.003 10.585± 0.247 0.419± 0.001 0.175± 0.001 0.007± 0.000 0.236± 0.002 10.679± 0.122 0.419± 0.001 192 0.216± 0.001 0.007± 0.000 0.275± 0.001 10.767± 0.080 0.465± 0.001 0.217± 0.000 0.005± 0.000 0.273± 0.000 10.893± 0.009 0.466± 0.000 336 0.264± 0.002 0.007± 0.000 0.316± 0.003 11.698± 0.232 0.513± 0.002 0.262± 0.000 0.004± 0.000 0.310± 0.000 11.930± 0.005 0.511± 0.000 720 0.326± 0.001 0.006± 0.000 0.367± 0.002 10.920± 0.188 0.571± 0.001 0.324± 0.000 0.003± 0.000 0.362± 0.000 11.236± 0.009 0.569± 0.000 AVG 0.245± 0.050 0.007± 0.001 0.299± 0.043 10.993± 0.380 0.492± 0.051 0.244± 0.049 0.005± 0.001 0.295± 0.041 11.184± 0.424 0.491± 0.050 Table 16: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, ETTm2, ILI, Traffix, and Weather datasets and PatchTST [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/yuqinie98/PatchTST) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 2.58%, TAM: 35.69%, 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: 14.44%, MAPE: 29.81%, and RMSE: 11.01%. ILI, Traffic, and Weather are in Tab. 17. Models PatchTST Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ ECL 96 0.130± 0.000 0.030± 0.000 0.223± 0.000 2.332± 0.034 0.360± 0.000 0.130± 0.000 0.016± 0.000 0.221± 0.000 2.276± 0.006 0.361± 0.000 192 0.149± 0.001 0.032± 0.002 0.241± 0.001 2.488± 0.040 0.386± 0.001 0.148± 0.001 0.021± 0.000 0.238± 0.000 2.462± 0.004 0.385± 0.001 336 0.166± 0.000 0.037± 0.001 0.260± 0.001 2.493± 0.008 0.408± 0.001 0.165± 0.001 0.028± 0.001 0.257± 0.000 2.468± 0.008 0.406± 0.001 720 0.203± 0.001 0.049± 0.002 0.293± 0.001 2.583± 0.027 0.451± 0.001 0.202± 0.001 0.030± 0.000 0.289± 0.001 2.554± 0.012 0.449± 0.001 AVG 0.162± 0.024 0.037± 0.007 0.254± 0.023 2.474± 0.081 0.401± 0.030 0.161± 0.024 0.024± 0.005 0.251± 0.023 2.440± 0.091 0.400± 0.029 ETTh1 96 0.379± 0.000 0.065± 0.002 0.401± 0.000 9.413± 0.032 0.616± 0.000 0.378± 0.001 0.053± 0.002 0.401± 0.001 9.358± 0.026 0.615± 0.001 192 0.412± 0.000 0.063± 0.002 0.420± 0.000 9.490± 0.037 0.642± 0.000 0.411± 0.000 0.050± 0.002 0.419± 0.000 9.454± 0.027 0.641± 0.000 336 0.435± 0.002 0.072± 0.007 0.436± 0.001 9.524± 0.066 0.659± 0.001 0.432± 0.001 0.045± 0.002 0.434± 0.000 9.425± 0.038 0.657± 0.000 720 0.448± 0.003 0.103± 0.013 0.465± 0.002 10.184± 0.163 0.669± 0.002 0.439± 0.003 0.042± 0.005 0.461± 0.002 9.896± 0.092 0.663± 0.002 AVG 0.418± 0.023 0.076± 0.014 0.431± 0.021 9.653± 0.277 0.646± 0.018 0.415± 0.021 0.048± 0.004 0.429± 0.020 9.533± 0.190 0.644± 0.017 ETTh2 96 0.276± 0.000 0.067± 0.001 0.338± 0.000 1.378± 0.002 0.525± 0.000 0.275± 0.000 0.055± 0.001 0.337± 0.000 1.368± 0.003 0.525± 0.000 192 0.336± 0.000 0.062± 0.001 0.378± 0.000 1.510± 0.003 0.580± 0.000 0.337± 0.000 0.051± 0.001 0.377± 0.001 1.495± 0.024 0.580± 0.000 336 0.361± 0.000 0.060± 0.002 0.401± 0.001 1.700± 0.011 0.601± 0.000 0.360± 0.000 0.048± 0.002 0.399± 0.002 1.672± 0.027 0.600± 0.000 720 0.391± 0.000 0.055± 0.002 0.429± 0.000 2.013± 0.003 0.625± 0.000 0.391± 0.001 0.044± 0.002 0.429± 0.001 2.010± 0.003 0.625± 0.001 AVG 0.341± 0.038 0.061± 0.004 0.387± 0.030 1.650± 0.214 0.583± 0.033 0.341± 0.038 0.050± 0.004 0.386± 0.030 1.636± 0.216 0.583± 0.033 ETTm1 96 0.290± 0.001 0.052± 0.001 0.342± 0.001 2.175± 0.015 0.538± 0.001 0.285± 0.001 0.029± 0.000 0.334± 0.001 2.102± 0.015 0.534± 0.001 192 0.334± 0.002 0.049± 0.001 0.370± 0.002 2.302± 0.014 0.578± 0.002 0.329± 0.002 0.035± 0.001 0.364± 0.001 2.241± 0.001 0.574± 0.001 336 0.366± 0.001 0.049± 0.001 0.391± 0.001 2.350± 0.014 0.605± 0.001 0.367± 0.002 0.026± 0.000 0.386± 0.001 2.284± 0.008 0.606± 0.002 720 0.418± 0.005 0.052± 0.002 0.423± 0.004 2.452± 0.026 0.646± 0.004 0.415± 0.002 0.037± 0.001 0.419± 0.001 2.459± 0.016 0.644± 0.001 AVG 0.352± 0.042 0.050± 0.001 0.382± 0.026 2.320± 0.089 0.592± 0.035 0.349± 0.043 0.032± 0.004 0.376± 0.028 2.271± 0.114 0.589± 0.036 ETTm2 96 0.164± 0.001 0.051± 0.001 0.252± 0.000 1.057± 0.004 0.405± 0.001 0.162± 0.000 0.036± 0.001 0.251± 0.000 1.053± 0.001 0.403± 0.000 192 0.220± 0.000 0.059± 0.002 0.292± 0.001 1.194± 0.004 0.469± 0.000 0.218± 0.001 0.033± 0.001 0.289± 0.001 1.194± 0.006 0.467± 0.001 336 0.275± 0.001 0.063± 0.000 0.329± 0.000 1.318± 0.001 0.525± 0.001 0.274± 0.000 0.045± 0.001 0.327± 0.000 1.316± 0.003 0.523± 0.000 720 0.368± 0.002 0.063± 0.002 0.384± 0.001 1.478± 0.003 0.606± 0.001 0.366± 0.001 0.044± 0.000 0.383± 0.001 1.480± 0.002 0.605± 0.001 AVG 0.257± 0.067 0.059± 0.004 0.314± 0.043 1.262± 0.139 0.501± 0.066 0.255± 0.067 0.039± 0.005 0.312± 0.044 1.261± 0.141 0.500± 0.067 Table 17: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, ETTm2, ILI, Traffix, and Weather datasets and PatchTST [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/yuqinie98/PatchTST) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 2.58%, TAM: 35.69%, 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: 14.44%, MAPE: 29.81%, and RMSE: 11.01%. ECL, ETTh1, ETTh2, ETTm1, and ETTm2 are in Tab. 16. Models PatchTST Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ ILI 24 2.132± 0.179 0.128± 0.015 0.917± 0.064 4.121± 0.100 1.459± 0.062 1.686± 0.053 0.056± 0.003 0.811± 0.011 3.004± 0.048 1.298± 0.020 36 1.559± 0.045 0.106± 0.017 0.830± 0.013 2.305± 0.216 1.248± 0.018 1.391± 0.047 0.044± 0.002 0.767± 0.004 1.751± 0.077 1.179± 0.020 48 1.736± 0.078 0.099± 0.009 0.900± 0.021 2.306± 0.099 1.317± 0.030 1.431± 0.109 0.042± 0.001 0.774± 0.034 1.682± 0.049 1.195± 0.045 60 1.824± 0.140 0.107± 0.005 0.923± 0.046 2.476± 0.098 1.349± 0.053 1.479± 0.097 0.042± 0.002 0.790± 0.034 1.738± 0.096 1.216± 0.040 AVG 1.813± 0.186 0.110± 0.010 0.893± 0.033 2.802± 0.684 1.343± 0.068 1.497± 0.101 0.046± 0.005 0.785± 0.015 2.044± 0.497 1.222± 0.041 Traffic 96 0.358± 0.000 0.040± 0.000 0.245± 0.000 2.553± 0.018 0.598± 0.000 0.357± 0.000 0.029± 0.001 0.242± 0.000 2.472± 0.005 0.598± 0.000 192 0.378± 0.001 0.039± 0.000 0.254± 0.000 2.570± 0.022 0.615± 0.001 0.378± 0.000 0.022± 0.000 0.248± 0.000 2.482± 0.001 0.615± 0.000 336 0.391± 0.001 0.040± 0.001 0.262± 0.001 2.620± 0.004 0.626± 0.001 0.390± 0.001 0.027± 0.001 0.257± 0.001 2.542± 0.003 0.625± 0.000 720 0.431± 0.000 0.044± 0.001 0.285± 0.000 2.845± 0.018 0.657± 0.000 0.427± 0.001 0.029± 0.000 0.278± 0.000 2.707± 0.017 0.653± 0.001 AVG 0.390± 0.024 0.041± 0.002 0.262± 0.013 2.647± 0.105 0.624± 0.019 0.388± 0.023 0.027± 0.003 0.256± 0.012 2.551± 0.084 0.623± 0.018 Weather 96 0.152± 0.002 0.020± 0.000 0.201± 0.003 11.697± 0.303 0.390± 0.003 0.151± 0.000 0.015± 0.000 0.197± 0.000 11.913± 0.090 0.388± 0.000 192 0.196± 0.001 0.018± 0.001 0.242± 0.001 13.084± 0.650 0.443± 0.002 0.196± 0.001 0.013± 0.000 0.240± 0.001 13.325± 0.405 0.442± 0.001 336 0.247± 0.000 0.017± 0.000 0.282± 0.000 14.374± 0.254 0.497± 0.000 0.246± 0.000 0.012± 0.000 0.279± 0.001 14.242± 0.366 0.496± 0.000 720 0.320± 0.001 0.016± 0.000 0.334± 0.001 13.976± 0.119 0.566± 0.000 0.321± 0.000 0.007± 0.000 0.329± 0.000 14.243± 0.026 0.566± 0.000 AVG 0.229± 0.056 0.018± 0.001 0.265± 0.044 13.283± 0.919 0.474± 0.058 0.228± 0.056 0.012± 0.003 0.261± 0.044 13.431± 0.853 0.473± 0.059 Table 18: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, 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: 1.81%, MAPE: 4.01%, and RMSE: 1.76%. The maximum improvement of AliO over the baseline is MSE: 29.03%, TAM: 51.96%, MAE: 9.34%, MAPE: 23.09%, and RMSE: 16.56%. Exchange, Traffic, ILI, and Weather are in Tab. 19. Models TimesNet Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ ECL 96 0.168± 0.001 0.038± 0.001 0.271± 0.001 2.666± 0.048 0.409± 0.001 0.166± 0.001 0.025± 0.000 0.269± 0.001 2.563± 0.067 0.408± 0.001 192 0.186± 0.001 0.038± 0.002 0.288± 0.002 2.787± 0.097 0.431± 0.001 0.183± 0.001 0.023± 0.000 0.284± 0.001 2.712± 0.013 0.428± 0.002 336 0.202± 0.005 0.041± 0.002 0.302± 0.003 2.944± 0.075 0.449± 0.006 0.196± 0.001 0.025± 0.001 0.297± 0.002 2.944± 0.091 0.443± 0.002 720 0.228± 0.012 0.044± 0.005 0.322± 0.011 2.883± 0.092 0.477± 0.013 0.219± 0.002 0.039± 0.003 0.313± 0.001 2.933± 0.027 0.468± 0.002 AVG 0.196± 0.020 0.040± 0.002 0.296± 0.017 2.820± 0.094 0.442± 0.022 0.191± 0.017 0.028± 0.006 0.291± 0.014 2.788± 0.143 0.436± 0.020 ETTh1 96 0.409± 0.010 0.055± 0.001 0.425± 0.006 10.015± 0.311 0.640± 0.008 0.410± 0.010 0.045± 0.001 0.426± 0.006 10.036± 0.280 0.641± 0.008 192 0.469± 0.006 0.054± 0.003 0.460± 0.004 10.999± 0.070 0.685± 0.004 0.461± 0.005 0.030± 0.002 0.455± 0.001 10.429± 0.286 0.679± 0.003 336 0.507± 0.013 0.046± 0.003 0.478± 0.008 10.926± 0.266 0.712± 0.009 0.497± 0.003 0.027± 0.001 0.472± 0.002 10.342± 0.156 0.705± 0.002 720 0.521± 0.006 0.046± 0.003 0.497± 0.003 10.981± 0.954 0.722± 0.004 0.502± 0.007 0.027± 0.001 0.487± 0.003 10.451± 0.332 0.708± 0.005 AVG 0.476± 0.039 0.050± 0.004 0.465± 0.024 10.730± 0.370 0.689± 0.029 0.468± 0.033 0.032± 0.007 0.460± 0.020 10.314± 0.148 0.683± 0.024 ETTh2 96 0.328± 0.014 0.067± 0.006 0.370± 0.010 1.607± 0.053 0.573± 0.012 0.318± 0.003 0.041± 0.006 0.362± 0.002 1.500± 0.021 0.564± 0.003 192 0.429± 0.030 0.089± 0.034 0.424± 0.019 1.646± 0.069 0.655± 0.023 0.398± 0.011 0.044± 0.002 0.408± 0.005 1.626± 0.040 0.631± 0.008 336 0.452± 0.013 0.085± 0.023 0.453± 0.009 1.826± 0.034 0.672± 0.010 0.448± 0.002 0.051± 0.003 0.447± 0.002 1.839± 0.052 0.669± 0.001 720 0.454± 0.006 0.075± 0.025 0.463± 0.006 2.177± 0.158 0.673± 0.005 0.446± 0.016 0.036± 0.005 0.459± 0.010 2.064± 0.132 0.668± 0.012 AVG 0.416± 0.046 0.079± 0.008 0.428± 0.032 1.814± 0.201 0.643± 0.037 0.402± 0.047 0.043± 0.005 0.419± 0.034 1.757± 0.192 0.633± 0.038 ETTm1 96 0.334± 0.005 0.052± 0.001 0.374± 0.003 2.366± 0.011 0.578± 0.004 0.328± 0.005 0.029± 0.001 0.368± 0.002 2.308± 0.045 0.572± 0.004 192 0.386± 0.000 0.045± 0.001 0.399± 0.001 2.478± 0.070 0.622± 0.000 0.381± 0.001 0.033± 0.002 0.397± 0.001 2.458± 0.045 0.618± 0.001 336 0.429± 0.007 0.044± 0.002 0.427± 0.003 2.593± 0.091 0.655± 0.005 0.409± 0.003 0.022± 0.001 0.417± 0.001 2.483± 0.014 0.640± 0.003 720 0.499± 0.004 0.042± 0.002 0.465± 0.003 2.801± 0.051 0.706± 0.003 0.475± 0.001 0.020± 0.001 0.452± 0.002 2.686± 0.023 0.689± 0.001 AVG 0.412± 0.054 0.045± 0.003 0.416± 0.030 2.560± 0.144 0.640± 0.042 0.398± 0.048 0.026± 0.005 0.409± 0.027 2.484± 0.121 0.630± 0.038 ETTm2 96 0.186± 0.000 0.040± 0.002 0.266± 0.001 1.181± 0.013 0.431± 0.000 0.186± 0.002 0.030± 0.002 0.265± 0.001 1.155± 0.026 0.431± 0.002 192 0.255± 0.002 0.037± 0.001 0.308± 0.002 1.290± 0.020 0.505± 0.002 0.253± 0.001 0.020± 0.001 0.307± 0.002 1.292± 0.036 0.503± 0.001 336 0.315± 0.003 0.033± 0.003 0.345± 0.001 1.351± 0.017 0.561± 0.003 0.317± 0.004 0.020± 0.002 0.348± 0.002 1.335± 0.015 0.563± 0.003 720 0.430± 0.001 0.034± 0.004 0.410± 0.001 1.584± 0.039 0.656± 0.001 0.423± 0.001 0.017± 0.001 0.406± 0.000 1.545± 0.021 0.650± 0.000 AVG 0.296± 0.080 0.036± 0.003 0.332± 0.047 1.352± 0.132 0.538± 0.073 0.295± 0.078 0.022± 0.005 0.331± 0.047 1.332± 0.125 0.537± 0.072 Table 19: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, 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: 1.81%, MAPE: 4.01%, and RMSE: 1.76%. The maximum improvement of AliO over the baseline is MSE: 29.03%, TAM: 51.96%, MAE: 9.34%, MAPE: 23.09%, and RMSE: 16.56%. ECL, ETTh1, ETTh2, ETTm1, and ETTm2 are in Tab. 18. Models TimesNet Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ Exchange 96 0.112± 0.001 0.038± 0.004 0.241± 0.001 1.405± 0.100 0.334± 0.002 0.105± 0.002 0.023± 0.001 0.232± 0.001 1.300± 0.041 0.324± 0.003 192 0.217± 0.008 0.036± 0.004 0.337± 0.005 1.951± 0.061 0.466± 0.009 0.207± 0.002 0.020± 0.000 0.330± 0.001 2.002± 0.054 0.455± 0.003 336 0.366± 0.008 0.031± 0.003 0.440± 0.006 3.197± 0.076 0.605± 0.007 0.364± 0.005 0.029± 0.004 0.439± 0.004 3.249± 0.070 0.603± 0.004 720 0.964± 0.025 0.033± 0.005 0.746± 0.012 6.519± 0.188 0.982± 0.013 0.944± 0.034 0.024± 0.004 0.740± 0.015 6.473± 0.189 0.972± 0.017 AVG 0.415± 0.295 0.035± 0.003 0.441± 0.170 3.268± 1.777 0.597± 0.217 0.405± 0.290 0.024± 0.003 0.435± 0.171 3.256± 1.775 0.588± 0.217 Traffic 96 0.593± 0.007 0.042± 0.001 0.313± 0.002 2.957± 0.073 0.770± 0.004 0.589± 0.003 0.028± 0.001 0.310± 0.001 2.798± 0.051 0.767± 0.002 192 0.618± 0.005 0.041± 0.004 0.325± 0.004 3.115± 0.107 0.786± 0.003 0.614± 0.003 0.025± 0.001 0.320± 0.001 2.882± 0.053 0.783± 0.002 336 0.631± 0.008 0.041± 0.005 0.334± 0.006 3.420± 0.219 0.794± 0.005 0.448± 0.128 0.026± 0.001 0.303± 0.018 2.712± 0.320 0.663± 0.092 720 0.661± 0.002 0.035± 0.002 0.347± 0.001 3.477± 0.016 0.813± 0.001 0.653± 0.002 0.032± 0.005 0.344± 0.002 3.456± 0.094 0.808± 0.001 AVG 0.626± 0.022 0.040± 0.003 0.330± 0.011 3.242± 0.192 0.791± 0.014 0.576± 0.069 0.028± 0.003 0.319± 0.014 2.962± 0.261 0.755± 0.050 ILI 24 2.759± 0.839 0.371± 0.026 0.972± 0.077 4.452± 0.551 1.643± 0.246 2.239± 0.347 0.256± 0.011 0.924± 0.045 3.424± 0.129 1.492± 0.114 36 1.949± 0.033 0.330± 0.031 0.919± 0.006 2.976± 0.224 1.396± 0.012 1.734± 0.025 0.262± 0.027 0.846± 0.007 2.508± 0.167 1.317± 0.010 48 1.976± 0.112 0.305± 0.030 0.896± 0.029 2.906± 0.050 1.405± 0.040 1.911± 0.033 0.263± 0.016 0.862± 0.015 2.686± 0.047 1.382± 0.012 60 2.002± 0.098 0.289± 0.006 0.919± 0.033 3.060± 0.179 1.414± 0.035 1.970± 0.054 0.251± 0.021 0.899± 0.002 2.779± 0.123 1.403± 0.019 AVG 2.171± 0.304 0.324± 0.028 0.926± 0.025 3.348± 0.572 1.465± 0.092 1.963± 0.162 0.258± 0.004 0.883± 0.027 2.849± 0.309 1.399± 0.056 Weather 96 0.174± 0.003 0.021± 0.001 0.223± 0.002 11.767± 0.301 0.417± 0.004 0.174± 0.003 0.017± 0.000 0.223± 0.001 11.659± 0.718 0.417± 0.003 192 0.231± 0.005 0.017± 0.002 0.271± 0.004 14.158± 0.953 0.481± 0.006 0.237± 0.007 0.014± 0.001 0.276± 0.006 13.831± 0.810 0.487± 0.007 336 0.285± 0.002 0.020± 0.002 0.307± 0.001 13.782± 0.351 0.534± 0.002 0.280± 0.001 0.011± 0.001 0.302± 0.000 13.819± 0.139 0.529± 0.001 720 0.359± 0.001 0.015± 0.001 0.353± 0.002 14.045± 0.438 0.599± 0.001 0.358± 0.001 0.012± 0.001 0.353± 0.001 14.367± 0.365 0.599± 0.001 AVG 0.262± 0.061 0.018± 0.002 0.289± 0.043 13.438± 0.871 0.508± 0.060 0.262± 0.060 0.014± 0.002 0.289± 0.042 13.419± 0.930 0.508± 0.059 Table 20: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, ETTm2, Solar, PEMS03, PEMS04, PEMS07, PEMS08, Exchange, Traffix, and Weather datasets and iTransformer [30]. The results are reported in terms of MSE, TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https://github.com/thuml/iTransformer) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 6.59%, TAM: 28.04%, MAE: 5.41%, MAPE: 0.46%, and RMSE: 3.85%. The maximum improvement of AliO over the baseline is MSE: 88.52%, 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, and Weather are in Tab. 22. Models iTransformer Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ ECL 96 0.148± 0.000 0.046± 0.000 0.240± 0.000 2.525± 0.030 0.385± 0.000 0.147± 0.000 0.034± 0.000 0.238± 0.000 2.512± 0.018 0.383± 0.000 192 0.165± 0.001 0.042± 0.000 0.256± 0.001 2.732± 0.033 0.406± 0.001 0.160± 0.000 0.031± 0.000 0.250± 0.000 2.671± 0.023 0.400± 0.000 336 0.180± 0.001 0.049± 0.001 0.273± 0.001 2.736± 0.005 0.424± 0.001 0.175± 0.001 0.036± 0.001 0.267± 0.000 2.802± 0.025 0.418± 0.001 720 0.211± 0.000 0.063± 0.000 0.301± 0.000 3.068± 0.024 0.459± 0.000 0.207± 0.001 0.025± 0.001 0.294± 0.001 3.073± 0.032 0.455± 0.001 AVG 0.176± 0.021 0.050± 0.007 0.267± 0.020 2.765± 0.174 0.418± 0.024 0.172± 0.020 0.031± 0.004 0.262± 0.019 2.764± 0.184 0.414± 0.024 ETTh1 96 0.386± 0.001 0.083± 0.001 0.404± 0.001 10.066± 0.052 0.621± 0.001 0.380± 0.000 0.057± 0.000 0.397± 0.000 9.673± 0.026 0.616± 0.000 192 0.443± 0.001 0.083± 0.001 0.437± 0.001 9.867± 0.093 0.666± 0.001 0.431± 0.000 0.064± 0.001 0.428± 0.000 9.624± 0.060 0.657± 0.000 336 0.487± 0.003 0.089± 0.002 0.459± 0.002 10.520± 0.334 0.698± 0.002 0.471± 0.001 0.038± 0.000 0.445± 0.001 10.034± 0.076 0.686± 0.001 720 0.506± 0.007 0.098± 0.001 0.492± 0.003 11.807± 0.649 0.711± 0.005 0.471± 0.008 0.048± 0.002 0.469± 0.004 11.258± 0.051 0.687± 0.006 AVG 0.455± 0.041 0.088± 0.005 0.448± 0.029 10.565± 0.676 0.674± 0.031 0.438± 0.034 0.052± 0.009 0.435± 0.023 10.147± 0.591 0.661± 0.026 ETTh2 96 0.300± 0.001 0.080± 0.002 0.350± 0.001 1.469± 0.010 0.548± 0.001 0.300± 0.000 0.045± 0.001 0.348± 0.000 1.422± 0.004 0.548± 0.000 192 0.378± 0.000 0.076± 0.001 0.398± 0.001 1.596± 0.014 0.615± 0.000 0.375± 0.001 0.058± 0.002 0.396± 0.000 1.560± 0.006 0.612± 0.001 336 0.421± 0.002 0.073± 0.002 0.431± 0.001 1.775± 0.008 0.649± 0.001 0.413± 0.001 0.055± 0.003 0.427± 0.001 1.712± 0.002 0.643± 0.001 720 0.429± 0.001 0.069± 0.002 0.447± 0.000 2.018± 0.012 0.655± 0.001 0.419± 0.002 0.050± 0.002 0.441± 0.001 1.981± 0.024 0.647± 0.001 AVG 0.382± 0.046 0.074± 0.003 0.406± 0.033 1.714± 0.184 0.617± 0.038 0.377± 0.042 0.052± 0.005 0.403± 0.032 1.669± 0.186 0.613± 0.036 ETTm1 96 0.343± 0.003 0.067± 0.001 0.377± 0.001 2.354± 0.012 0.586± 0.002 0.330± 0.001 0.037± 0.000 0.361± 0.000 2.142± 0.005 0.575± 0.001 192 0.380± 0.002 0.060± 0.001 0.394± 0.001 2.383± 0.013 0.617± 0.001 0.375± 0.001 0.031± 0.000 0.383± 0.001 2.203± 0.011 0.612± 0.001 336 0.418± 0.000 0.058± 0.001 0.418± 0.000 2.443± 0.006 0.647± 0.000 0.408± 0.001 0.037± 0.000 0.406± 0.000 2.302± 0.005 0.639± 0.001 720 0.488± 0.001 0.056± 0.001 0.457± 0.001 2.670± 0.012 0.699± 0.001 0.472± 0.001 0.026± 0.000 0.441± 0.001 2.451± 0.005 0.687± 0.001 AVG 0.407± 0.048 0.060± 0.004 0.412± 0.027 2.462± 0.111 0.637± 0.037 0.396± 0.046 0.033± 0.004 0.398± 0.026 2.275± 0.104 0.628± 0.037 ETTm2 96 0.185± 0.001 0.050± 0.001 0.270± 0.001 1.169± 0.009 0.430± 0.001 0.184± 0.000 0.038± 0.000 0.267± 0.000 1.160± 0.002 0.428± 0.000 192 0.254± 0.000 0.050± 0.001 0.314± 0.001 1.302± 0.018 0.504± 0.000 0.250± 0.001 0.025± 0.000 0.310± 0.001 1.287± 0.005 0.500± 0.001 336 0.315± 0.004 0.049± 0.006 0.352± 0.003 1.411± 0.005 0.562± 0.004 0.312± 0.001 0.022± 0.000 0.348± 0.001 1.399± 0.003 0.559± 0.001 720 0.413± 0.001 0.044± 0.003 0.407± 0.001 1.615± 0.006 0.643± 0.001 0.411± 0.001 0.019± 0.000 0.404± 0.001 1.594± 0.009 0.641± 0.001 AVG 0.292± 0.075 0.048± 0.002 0.336± 0.045 1.374± 0.146 0.534± 0.070 0.289± 0.075 0.026± 0.007 0.332± 0.045 1.360± 0.142 0.532± 0.070 Table 21: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, ETTm2, Solar, PEMS03, PEMS04, PEMS07, PEMS08, Exchange, Traffix, and Weather datasets and iTransformer [30]. The results are reported in terms of MSE, TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https://github.com/thuml/iTransformer) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 6.59%, TAM: 28.04%, MAE: 5.41%, MAPE: 0.46%, and RMSE: 3.85%. The maximum improvement of AliO over the baseline is MSE: 88.52%, TAM: 60.37%, MAE: 73.75%, MAPE: 18.04%, and RMSE: 66.10%. ECL, ETTh1, ETTh2, ETTm1, and ETTm2 are in Tab. 20. Exchange, Traffix, and Weather are in Tab. 22 Models iTransformer Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ Solar 96 0.206± 0.002 0.038± 0.002 0.237± 0.002 1.846± 0.033 0.454± 0.002 0.192± 0.002 0.021± 0.001 0.217± 0.002 1.843± 0.035 0.438± 0.002 192 0.237± 0.001 0.040± 0.000 0.263± 0.001 1.976± 0.014 0.487± 0.001 0.225± 0.001 0.020± 0.000 0.241± 0.001 1.980± 0.008 0.474± 0.001 336 0.250± 0.001 0.038± 0.001 0.275± 0.001 2.011± 0.011 0.500± 0.001 0.242± 0.001 0.019± 0.000 0.257± 0.000 2.017± 0.011 0.492± 0.001 720 0.251± 0.000 0.035± 0.000 0.275± 0.000 2.066± 0.025 0.501± 0.000 0.247± 0.001 0.016± 0.000 0.261± 0.000 2.092± 0.012 0.497± 0.001 AVG 0.236± 0.016 0.038± 0.002 0.263± 0.014 1.975± 0.072 0.485± 0.017 0.227± 0.019 0.019± 0.002 0.244± 0.015 1.983± 0.081 0.475± 0.021 PEMS03 12 0.069± 0.000 0.048± 0.000 0.175± 0.001 1.417± 0.006 0.263± 0.001 0.069± 0.000 0.028± 0.000 0.175± 0.000 1.424± 0.006 0.263± 0.000 24 0.098± 0.001 0.052± 0.000 0.209± 0.001 1.671± 0.011 0.313± 0.001 0.098± 0.000 0.041± 0.000 0.209± 0.000 1.663± 0.006 0.313± 0.001 48 0.163± 0.001 0.061± 0.001 0.274± 0.001 2.022± 0.009 0.404± 0.001 0.162± 0.001 0.047± 0.000 0.273± 0.001 2.000± 0.015 0.402± 0.001 96 0.918± 0.310 0.117± 0.026 0.720± 0.127 3.747± 0.531 0.944± 0.162 0.576± 0.044 0.080± 0.009 0.575± 0.028 3.383± 0.266 0.759± 0.029 AVG 0.312± 0.314 0.069± 0.025 0.345± 0.196 2.214± 0.814 0.481± 0.243 0.226± 0.183 0.049± 0.017 0.308± 0.141 2.117± 0.679 0.434± 0.173 PEMS04 12 0.081± 0.000 0.042± 0.000 0.188± 0.000 1.283± 0.007 0.284± 0.000 0.082± 0.000 0.033± 0.000 0.190± 0.001 1.294± 0.013 0.286± 0.000 24 0.100± 0.000 0.041± 0.000 0.212± 0.000 1.512± 0.019 0.316± 0.000 0.101± 0.000 0.031± 0.000 0.213± 0.000 1.521± 0.009 0.317± 0.001 48 0.134± 0.002 0.041± 0.000 0.248± 0.002 1.782± 0.029 0.366± 0.002 0.131± 0.001 0.029± 0.000 0.245± 0.001 1.771± 0.017 0.362± 0.002 96 0.169± 0.001 0.041± 0.000 0.280± 0.000 2.090± 0.015 0.411± 0.001 0.166± 0.002 0.028± 0.000 0.277± 0.002 2.081± 0.008 0.407± 0.003 AVG 0.121± 0.030 0.041± 0.001 0.232± 0.031 1.667± 0.270 0.344± 0.043 0.120± 0.028 0.030± 0.002 0.231± 0.030 1.667± 0.262 0.343± 0.041 PEMS07 12 0.067± 0.000 0.044± 0.000 0.164± 0.001 1.670± 0.003 0.258± 0.000 0.070± 0.001 0.036± 0.001 0.168± 0.001 1.706± 0.017 0.264± 0.001 24 0.087± 0.000 0.042± 0.000 0.190± 0.001 1.898± 0.019 0.295± 0.001 0.090± 0.001 0.035± 0.000 0.193± 0.001 1.943± 0.028 0.300± 0.001 48 0.995± 0.054 0.020± 0.019 0.832± 0.032 1.540± 0.702 0.997± 0.027 0.114± 0.001 0.031± 0.001 0.218± 0.002 2.158± 0.017 0.338± 0.002 96 1.172± 0.189 0.021± 0.023 0.895± 0.050 2.412± 1.881 1.079± 0.085 0.402± 0.161 0.052± 0.017 0.435± 0.089 4.015± 0.667 0.623± 0.121 AVG 0.580± 0.454 0.031± 0.010 0.520± 0.308 1.880± 0.298 0.657± 0.342 0.169± 0.121 0.038± 0.007 0.254± 0.095 2.455± 0.818 0.381± 0.127 PEMS08 12 0.088± 0.001 0.053± 0.000 0.193± 0.001 1.637± 0.004 0.297± 0.001 0.089± 0.000 0.046± 0.000 0.193± 0.000 1.643± 0.003 0.298± 0.001 24 0.138± 0.000 0.059± 0.000 0.243± 0.001 2.074± 0.005 0.371± 0.001 0.138± 0.000 0.051± 0.000 0.242± 0.000 2.076± 0.011 0.371± 0.001 48 0.247± 0.010 0.052± 0.001 0.287± 0.006 2.242± 0.070 0.497± 0.010 0.232± 0.003 0.037± 0.001 0.271± 0.002 2.085± 0.038 0.481± 0.003 96 0.452± 0.020 0.065± 0.005 0.431± 0.013 2.921± 0.217 0.672± 0.015 0.282± 0.003 0.030± 0.001 0.312± 0.002 2.394± 0.052 0.531± 0.003 AVG 0.231± 0.125 0.057± 0.005 0.288± 0.079 2.219± 0.413 0.459± 0.127 0.185± 0.068 0.041± 0.007 0.255± 0.039 2.050± 0.239 0.420± 0.082 Table 22: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, ETTm2, Solar, PEMS03, PEMS04, PEMS07, PEMS08, Exchange, Traffix, and Weather datasets and iTransformer [30]. The results are reported in terms of MSE, TAM, MAE, MAPE, and RMSE. The best results are highlighted in bold. We use official GitHub code (https://github.com/thuml/iTransformer) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 6.59%, TAM: 28.04%, MAE: 5.41%, MAPE: 0.46%, and RMSE: 3.85%. The maximum improvement of AliO over the baseline is MSE: 88.52%, TAM: 60.37%, MAE: 73.75%, MAPE: 18.04%, and RMSE: 66.10%. ECL, ETTh1, ETTh2, ETTm1, and ETTm2 are in Tab. 20. Solar, PEMS03, PEMS04, PEMS07, and PEMS08 are in Tab. 21. Models iTransformer Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ Exchange 96 0.086± 0.001 0.035± 0.001 0.206± 0.001 1.287± 0.007 0.294± 0.001 0.086± 0.000 0.030± 0.000 0.205± 0.001 1.280± 0.006 0.293± 0.001 192 0.179± 0.001 0.037± 0.000 0.302± 0.001 1.926± 0.011 0.423± 0.001 0.178± 0.001 0.031± 0.000 0.300± 0.001 1.895± 0.003 0.421± 0.001 336 0.335± 0.003 0.042± 0.000 0.420± 0.002 2.952± 0.014 0.579± 0.003 0.331± 0.002 0.032± 0.001 0.417± 0.001 2.914± 0.013 0.576± 0.001 720 0.865± 0.006 0.038± 0.002 0.702± 0.003 6.170± 0.019 0.930± 0.003 0.857± 0.006 0.031± 0.002 0.699± 0.002 6.150± 0.021 0.926± 0.003 AVG 0.366± 0.270 0.038± 0.002 0.408± 0.166 3.084± 1.680 0.557± 0.213 0.363± 0.267 0.031± 0.001 0.405± 0.166 3.060± 1.679 0.554± 0.212 Traffic 96 0.394± 0.001 0.066± 0.001 0.269± 0.001 2.897± 0.005 0.628± 0.001 0.393± 0.001 0.049± 0.000 0.264± 0.000 2.723± 0.010 0.627± 0.001 192 0.413± 0.001 0.061± 0.000 0.277± 0.000 2.948± 0.011 0.643± 0.001 0.412± 0.000 0.043± 0.000 0.271± 0.000 2.757± 0.011 0.642± 0.000 336 0.425± 0.001 0.059± 0.000 0.283± 0.001 2.947± 0.021 0.652± 0.001 0.428± 0.000 0.041± 0.000 0.278± 0.000 2.788± 0.009 0.654± 0.000 720 0.457± 0.002 0.060± 0.000 0.300± 0.001 3.108± 0.034 0.676± 0.001 0.460± 0.000 0.041± 0.000 0.296± 0.000 2.889± 0.008 0.678± 0.000 AVG 0.422± 0.020 0.061± 0.002 0.282± 0.010 2.975± 0.071 0.649± 0.016 0.423± 0.022 0.044± 0.003 0.277± 0.011 2.789± 0.055 0.650± 0.017 Weather 96 0.176± 0.002 0.024± 0.001 0.216± 0.002 16.182± 0.603 0.420± 0.002 0.175± 0.000 0.021± 0.000 0.215± 0.000 14.334± 0.300 0.418± 0.000 192 0.225± 0.001 0.024± 0.002 0.257± 0.001 16.029± 0.036 0.474± 0.001 0.223± 0.000 0.019± 0.000 0.257± 0.000 15.938± 0.131 0.473± 0.000 336 0.282± 0.001 0.024± 0.000 0.299± 0.001 15.771± 0.132 0.531± 0.001 0.281± 0.001 0.016± 0.000 0.298± 0.000 15.376± 0.069 0.530± 0.001 720 0.359± 0.001 0.023± 0.001 0.350± 0.001 15.409± 0.151 0.599± 0.001 0.357± 0.000 0.013± 0.000 0.347± 0.000 15.819± 0.052 0.597± 0.000 AVG 0.260± 0.061 0.024± 0.000 0.281± 0.045 15.848± 0.262 0.506± 0.059 0.259± 0.061 0.017± 0.003 0.279± 0.044 15.367± 0.565 0.505± 0.060 Table 23: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, 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://github.com/DAMO-DI-ML/NeurIPS2023-One-Fits-All) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 2.27%, TAM: 32.13%, MAE: 2.43%, MAPE: 1.56%, and RMSE: 1.14%. The maximum improvement of AliO over the baseline is MSE: 11.54%, TAM: 62.06%, MAE: 10.98%, MAPE: 8.16%, and RMSE: 5.93%. ILI, Traffix, and Weather are in Tab. 24. Models GPT4TS Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ ETTh1 96 0.380± 0.003 0.060± 0.001 0.401± 0.001 9.556± 0.065 0.616± 0.002 0.374± 0.003 0.034± 0.001 0.393± 0.001 9.675± 0.079 0.612± 0.002 192 0.418± 0.001 0.051± 0.001 0.419± 0.003 9.502± 0.101 0.646± 0.001 0.412± 0.002 0.037± 0.001 0.416± 0.001 9.532± 0.276 0.642± 0.001 336 0.441± 0.007 0.054± 0.004 0.435± 0.004 9.507± 0.157 0.664± 0.006 0.431± 0.004 0.031± 0.000 0.427± 0.003 9.519± 0.180 0.656± 0.003 720 0.458± 0.006 0.058± 0.005 0.465± 0.003 10.152± 0.159 0.677± 0.004 0.452± 0.009 0.053± 0.002 0.456± 0.004 10.804± 0.153 0.672± 0.007 AVG 0.424± 0.026 0.056± 0.003 0.430± 0.021 9.679± 0.245 0.651± 0.020 0.417± 0.026 0.039± 0.008 0.423± 0.020 9.883± 0.479 0.646± 0.020 ETTh2 96 0.292± 0.003 0.053± 0.003 0.353± 0.001 1.377± 0.031 0.540± 0.002 0.288± 0.003 0.041± 0.001 0.348± 0.004 1.352± 0.023 0.537± 0.003 192 0.368± 0.008 0.054± 0.006 0.400± 0.004 1.484± 0.017 0.606± 0.006 0.351± 0.001 0.033± 0.002 0.388± 0.001 1.470± 0.024 0.592± 0.001 336 0.376± 0.003 0.041± 0.002 0.414± 0.002 1.674± 0.030 0.613± 0.002 0.373± 0.002 0.029± 0.001 0.409± 0.002 1.622± 0.025 0.611± 0.001 720 0.414± 0.004 0.042± 0.003 0.449± 0.005 2.051± 0.056 0.644± 0.003 0.404± 0.002 0.027± 0.002 0.441± 0.002 1.981± 0.017 0.635± 0.002 AVG 0.363± 0.040 0.048± 0.005 0.404± 0.031 1.646± 0.230 0.601± 0.034 0.354± 0.038 0.033± 0.005 0.396± 0.030 1.606± 0.212 0.594± 0.032 ETTm1 96 0.292± 0.002 0.047± 0.001 0.347± 0.002 2.184± 0.033 0.540± 0.002 0.286± 0.001 0.036± 0.000 0.340± 0.001 2.136± 0.011 0.535± 0.001 192 0.330± 0.001 0.043± 0.002 0.369± 0.001 2.241± 0.011 0.575± 0.001 0.325± 0.001 0.029± 0.000 0.362± 0.001 2.220± 0.004 0.570± 0.001 336 0.365± 0.002 0.042± 0.002 0.392± 0.003 2.350± 0.054 0.605± 0.002 0.361± 0.001 0.027± 0.001 0.385± 0.000 2.305± 0.008 0.601± 0.000 720 0.418± 0.001 0.043± 0.001 0.423± 0.001 2.506± 0.031 0.646± 0.001 0.414± 0.001 0.033± 0.001 0.418± 0.001 2.469± 0.009 0.643± 0.001 AVG 0.351± 0.041 0.044± 0.002 0.383± 0.025 2.320± 0.110 0.592± 0.035 0.346± 0.042 0.031± 0.003 0.376± 0.026 2.283± 0.110 0.587± 0.036 ETTm2 96 0.170± 0.002 0.039± 0.002 0.259± 0.002 1.096± 0.010 0.412± 0.002 0.168± 0.000 0.033± 0.001 0.257± 0.000 1.079± 0.003 0.410± 0.001 192 0.233± 0.004 0.042± 0.003 0.305± 0.003 1.224± 0.020 0.483± 0.004 0.225± 0.000 0.025± 0.001 0.297± 0.001 1.199± 0.012 0.474± 0.001 336 0.297± 0.009 0.048± 0.007 0.348± 0.005 1.363± 0.011 0.545± 0.008 0.284± 0.003 0.026± 0.001 0.337± 0.002 1.323± 0.016 0.533± 0.002 720 0.382± 0.006 0.041± 0.003 0.401± 0.005 1.532± 0.034 0.618± 0.005 0.382± 0.001 0.028± 0.000 0.400± 0.000 1.531± 0.015 0.618± 0.001 AVG 0.270± 0.070 0.042± 0.003 0.328± 0.047 1.304± 0.145 0.514± 0.068 0.265± 0.071 0.028± 0.003 0.323± 0.047 1.283± 0.150 0.509± 0.068 ECL 96 0.138± 0.000 0.034± 0.000 0.237± 0.000 2.442± 0.003 0.372± 0.000 0.138± 0.000 0.025± 0.000 0.237± 0.001 2.412± 0.003 0.372± 0.000 192 0.155± 0.000 0.034± 0.000 0.252± 0.001 2.679± 0.026 0.393± 0.000 0.154± 0.000 0.025± 0.000 0.251± 0.000 2.652± 0.033 0.392± 0.000 336 0.169± 0.000 0.036± 0.000 0.266± 0.001 2.707± 0.030 0.411± 0.001 0.169± 0.001 0.021± 0.000 0.266± 0.001 2.665± 0.023 0.411± 0.001 720 0.206± 0.000 0.042± 0.001 0.297± 0.001 2.915± 0.033 0.454± 0.000 0.206± 0.000 0.028± 0.003 0.296± 0.000 2.818± 0.063 0.454± 0.000 AVG 0.167± 0.023 0.036± 0.003 0.263± 0.020 2.686± 0.150 0.407± 0.027 0.167± 0.022 0.025± 0.002 0.262± 0.020 2.637± 0.130 0.407± 0.027 Table 24: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, 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://github.com/DAMO-DI-ML/NeurIPS2023-One-Fits-All) to train the model with the official configuration. The average improvement of AliO over the baseline is MSE: 2.27%, TAM: 32.13%, MAE: 2.43%, MAPE: 1.56%, and RMSE: 1.14%. The maximum improvement of AliO over the baseline is MSE: 11.54%, TAM: 62.06%, MAE: 10.98%, MAPE: 8.16%, and RMSE: 5.93%. ECL, ETTh1, ETTh2, ETTm1, and ETTm2 are in Tab. 23. Models GPT4TS Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ ILI 24 1.965± 0.046 0.137± 0.010 0.862± 0.009 3.994± 0.129 1.402± 0.016 1.897± 0.036 0.115± 0.007 0.839± 0.007 3.967± 0.117 1.377± 0.013 36 1.914± 0.090 0.104± 0.026 0.913± 0.037 3.408± 0.334 1.383± 0.033 1.693± 0.044 0.090± 0.007 0.812± 0.020 3.182± 0.090 1.301± 0.017 48 1.851± 0.096 0.112± 0.014 0.915± 0.032 3.073± 0.103 1.360± 0.036 1.722± 0.041 0.091± 0.008 0.852± 0.006 3.268± 0.148 1.312± 0.015 60 1.861± 0.056 0.114± 0.011 0.927± 0.031 3.225± 0.188 1.364± 0.020 1.707± 0.032 0.097± 0.005 0.880± 0.011 3.376± 0.155 1.306± 0.012 AVG 1.898± 0.041 0.117± 0.011 0.904± 0.023 3.425± 0.313 1.377± 0.015 1.755± 0.074 0.098± 0.009 0.846± 0.022 3.448± 0.275 1.324± 0.028 Traffic 96 0.386± 0.001 0.056± 0.000 0.279± 0.000 2.996± 0.001 0.621± 0.001 0.377± 0.002 0.039± 0.000 0.267± 0.002 2.819± 0.007 0.614± 0.002 192 0.404± 0.001 0.058± 0.002 0.286± 0.003 3.081± 0.043 0.636± 0.001 0.399± 0.001 0.032± 0.001 0.276± 0.001 2.854± 0.002 0.632± 0.001 336 0.411± 0.000 0.061± 0.001 0.290± 0.000 3.106± 0.016 0.641± 0.000 0.406± 0.002 0.032± 0.000 0.282± 0.002 2.888± 0.011 0.637± 0.002 720 0.444± 0.000 0.063± 0.001 0.304± 0.001 3.212± 0.029 0.667± 0.000 0.438± 0.000 0.024± 0.000 0.295± 0.000 2.950± 0.007 0.662± 0.000 AVG 0.411± 0.019 0.059± 0.002 0.289± 0.008 3.099± 0.069 0.641± 0.015 0.405± 0.020 0.032± 0.005 0.280± 0.009 2.878± 0.043 0.636± 0.015 Weather 96 0.148± 0.002 0.026± 0.001 0.199± 0.003 12.027± 0.525 0.385± 0.002 0.147± 0.001 0.018± 0.000 0.194± 0.002 11.946± 0.167 0.384± 0.002 192 0.195± 0.000 0.024± 0.001 0.244± 0.000 14.653± 0.401 0.442± 0.000 0.192± 0.001 0.017± 0.000 0.237± 0.001 14.616± 0.251 0.439± 0.001 336 0.246± 0.002 0.024± 0.000 0.284± 0.001 15.431± 0.356 0.496± 0.002 0.244± 0.001 0.013± 0.000 0.276± 0.001 14.887± 0.367 0.494± 0.001 720 0.319± 0.001 0.022± 0.001 0.336± 0.002 14.770± 0.506 0.565± 0.001 0.316± 0.001 0.011± 0.000 0.328± 0.001 14.877± 0.092 0.562± 0.000 AVG 0.227± 0.057 0.024± 0.001 0.266± 0.045 14.220± 1.163 0.472± 0.060 0.225± 0.056 0.015± 0.002 0.259± 0.044 14.081± 1.107 0.469± 0.059 Table 25: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, 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/CycleNet) 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%, MAPE: 1.84%, and RMSE: 0.26%. The maximum improvement of AliO over the baseline is MSE: 2.73%, TAM: 74.35%, MAE: 6.79%, MAPE: 13.27%, and RMSE: 1.38%. Solar, Traffix, and Weather are in Tab. 26. Models CycleNet Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ ECL 96 0.141± 0.000 0.019± 0.000 0.234± 0.000 2.305± 0.002 0.376± 0.000 0.142± 0.000 0.019± 0.000 0.234± 0.000 2.304± 0.004 0.376± 0.000 192 0.156± 0.000 0.017± 0.002 0.247± 0.000 2.450± 0.007 0.395± 0.000 0.156± 0.000 0.015± 0.000 0.247± 0.000 2.444± 0.003 0.394± 0.000 336 0.173± 0.000 0.014± 0.000 0.265± 0.000 2.439± 0.003 0.415± 0.000 0.173± 0.000 0.010± 0.000 0.265± 0.000 2.439± 0.001 0.415± 0.000 720 0.211± 0.000 0.013± 0.000 0.297± 0.000 2.583± 0.001 0.459± 0.000 0.211± 0.000 0.010± 0.000 0.297± 0.000 2.584± 0.002 0.459± 0.000 AVG 0.170± 0.023 0.016± 0.002 0.261± 0.021 2.444± 0.088 0.411± 0.028 0.170± 0.023 0.014± 0.003 0.261± 0.021 2.443± 0.089 0.411± 0.028 ETTh1 96 0.380± 0.002 0.054± 0.011 0.393± 0.002 9.380± 0.023 0.616± 0.002 0.376± 0.000 0.016± 0.001 0.387± 0.000 9.142± 0.027 0.613± 0.000 192 0.425± 0.001 0.043± 0.005 0.418± 0.001 9.471± 0.032 0.652± 0.001 0.422± 0.000 0.013± 0.001 0.414± 0.000 9.185± 0.033 0.650± 0.000 336 0.462± 0.001 0.041± 0.005 0.438± 0.001 9.443± 0.011 0.680± 0.001 0.459± 0.001 0.012± 0.001 0.434± 0.000 9.210± 0.029 0.678± 0.000 720 0.461± 0.000 0.038± 0.002 0.460± 0.000 9.559± 0.015 0.679± 0.000 0.458± 0.001 0.011± 0.000 0.454± 0.000 9.408± 0.029 0.676± 0.000 AVG 0.432± 0.030 0.044± 0.005 0.427± 0.022 9.463± 0.058 0.657± 0.023 0.429± 0.030 0.013± 0.002 0.422± 0.022 9.236± 0.091 0.654± 0.024 ETTh2 96 0.285± 0.001 0.047± 0.002 0.335± 0.001 1.343± 0.005 0.534± 0.001 0.283± 0.000 0.032± 0.000 0.334± 0.000 1.324± 0.001 0.532± 0.000 192 0.373± 0.002 0.042± 0.002 0.392± 0.002 1.512± 0.011 0.611± 0.001 0.371± 0.000 0.029± 0.001 0.390± 0.000 1.504± 0.004 0.609± 0.000 336 0.424± 0.004 0.048± 0.013 0.435± 0.002 1.770± 0.012 0.651± 0.003 0.418± 0.000 0.015± 0.000 0.431± 0.000 1.735± 0.005 0.647± 0.000 720 0.456± 0.003 0.053± 0.013 0.459± 0.002 2.066± 0.009 0.675± 0.003 0.449± 0.000 0.023± 0.005 0.456± 0.000 2.049± 0.001 0.670± 0.000 AVG 0.385± 0.058 0.047± 0.003 0.405± 0.042 1.672± 0.244 0.618± 0.048 0.381± 0.056 0.025± 0.006 0.402± 0.041 1.653± 0.243 0.615± 0.047 ETTm1 96 0.326± 0.001 0.039± 0.003 0.364± 0.001 2.208± 0.001 0.571± 0.001 0.320± 0.000 0.017± 0.000 0.355± 0.000 2.112± 0.002 0.566± 0.000 192 0.366± 0.000 0.032± 0.003 0.382± 0.000 2.263± 0.005 0.605± 0.000 0.364± 0.000 0.013± 0.000 0.376± 0.000 2.204± 0.000 0.603± 0.000 336 0.396± 0.001 0.031± 0.003 0.402± 0.001 2.316± 0.002 0.629± 0.000 0.394± 0.000 0.012± 0.000 0.396± 0.000 2.261± 0.000 0.627± 0.000 720 0.457± 0.000 0.026± 0.001 0.434± 0.000 2.476± 0.003 0.676± 0.000 0.455± 0.000 0.016± 0.001 0.431± 0.000 2.449± 0.007 0.675± 0.000 AVG 0.386± 0.043 0.032± 0.004 0.395± 0.023 2.316± 0.089 0.620± 0.034 0.383± 0.044 0.014± 0.002 0.390± 0.025 2.256± 0.110 0.618± 0.035 ETTm2 96 0.167± 0.001 0.033± 0.002 0.248± 0.001 1.041± 0.004 0.408± 0.001 0.166± 0.000 0.022± 0.000 0.247± 0.000 1.032± 0.000 0.407± 0.000 192 0.233± 0.000 0.029± 0.002 0.291± 0.000 1.158± 0.002 0.482± 0.001 0.232± 0.000 0.020± 0.000 0.290± 0.000 1.151± 0.000 0.481± 0.000 336 0.294± 0.001 0.025± 0.001 0.330± 0.000 1.265± 0.005 0.542± 0.001 0.293± 0.000 0.018± 0.000 0.330± 0.000 1.264± 0.000 0.541± 0.000 720 0.394± 0.000 0.022± 0.000 0.389± 0.000 1.470± 0.004 0.628± 0.000 0.394± 0.000 0.017± 0.000 0.389± 0.000 1.469± 0.000 0.628± 0.000 AVG 0.272± 0.075 0.027± 0.004 0.315± 0.046 1.233± 0.141 0.515± 0.072 0.271± 0.075 0.019± 0.002 0.314± 0.047 1.229± 0.144 0.514± 0.072 Table 26: Full results of the proposed AliO method on the ECL, ETTh1, ETTh2, ETTm1, 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/CycleNet) 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%, MAPE: 1.84%, and RMSE: 0.26%. The maximum improvement of AliO over the baseline is MSE: 2.73%, TAM: 74.35%, MAE: 6.79%, MAPE: 13.27%, and RMSE: 1.38%. ECL, ETTh1, ETTh2, ETTm1, and ETTm2 are in Tab. 25. Models CycleNet Method Baseline AliO Metric MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ MSE↓ TAM↓ MAE↓ MAPE↓ RMSE↓ Solar 96 0.250± 0.001 0.030± 0.002 0.278± 0.001 1.984± 0.005 0.500± 0.001 0.250± 0.000 0.026± 0.000 0.277± 0.000 1.982± 0.000 0.500± 0.000 192 0.289± 0.000 0.027± 0.000 0.298± 0.001 2.101± 0.005 0.538± 0.000 0.290± 0.000 0.026± 0.000 0.299± 0.000 2.110± 0.001 0.539± 0.000 336 0.338± 0.000 0.024± 0.000 0.322± 0.000 2.335± 0.001 0.582± 0.000 0.338± 0.000 0.026± 0.002 0.323± 0.001 2.328± 0.003 0.581± 0.000 720 0.351± 0.000 0.021± 0.001 0.327± 0.000 2.478± 0.005 0.593± 0.000 0.352± 0.000 0.020± 0.000 0.327± 0.000 2.480± 0.001 0.593± 0.000 AVG 0.307± 0.036 0.026± 0.003 0.306± 0.018 2.224± 0.173 0.553± 0.033 0.308± 0.036 0.024± 0.002 0.307± 0.018 2.225± 0.172 0.553± 0.033 Traffic 96 0.481± 0.000 0.028± 0.003 0.314± 0.001 3.675± 0.005 0.693± 0.000 0.468± 0.000 0.007± 0.000 0.293± 0.001 3.187± 0.028 0.684± 0.000 192 0.482± 0.001 0.021± 0.000 0.313± 0.000 3.619± 0.004 0.694± 0.000 0.469± 0.000 0.007± 0.000 0.295± 0.000 3.181± 0.007 0.685± 0.000 336 0.480± 0.005 0.015± 0.004 0.308± 0.006 3.428± 0.058 0.693± 0.004 0.481± 0.000 0.006± 0.000 0.301± 0.000 3.185± 0.009 0.694± 0.000 720 0.507± 0.005 0.022± 0.001 0.324± 0.006 3.490± 0.038 0.712± 0.003 0.506± 0.005 0.012± 0.004 0.323± 0.006 3.465± 0.050 0.711± 0.004 AVG 0.487± 0.010 0.021± 0.004 0.315± 0.005 3.553± 0.088 0.698± 0.007 0.481± 0.014 0.008± 0.002 0.303± 0.011 3.255± 0.109 0.693± 0.010 Weather 96 0.170± 0.000 0.010± 0.000 0.216± 0.000 12.814± 0.055 0.413± 0.000 0.171± 0.000 0.010± 0.000 0.217± 0.001 12.942± 0.059 0.413± 0.000 192 0.223± 0.001 0.013± 0.003 0.260± 0.000 13.923± 0.100 0.472± 0.001 0.222± 0.000 0.010± 0.002 0.259± 0.000 13.935± 0.035 0.472± 0.000 336 0.276± 0.000 0.009± 0.001 0.297± 0.000 14.299± 0.037 0.525± 0.000 0.276± 0.000 0.007± 0.000 0.297± 0.000 14.302± 0.006 0.525± 0.000 720 0.350± 0.000 0.009± 0.002 0.345± 0.000 14.876± 0.083 0.591± 0.000 0.350± 0.000 0.006± 0.000 0.345± 0.000 14.851± 0.003 0.591± 0.000 AVG 0.255± 0.059 0.010± 0.001 0.280± 0.042 13.978± 0.674 0.500± 0.059 0.255± 0.059 0.008± 0.002 0.280± 0.042 14.007± 0.623 0.500± 0.059 Appendix J Pronunciation of a.li.o To prevent confusion regarding the pronunciation, this section explicitly describes how to pronounce a.li.o. [a.li.o] The pronunciation consists of three syllables: • a: Pronounced like the ‘a’ in father (ah). • li: Pronounced with a clear ‘L’ sound, as in Lee (lee). • o: Pronounced like the ‘o’ in go (oh). Consequently, it is pronounced as “Ah-Lee-Oh”, ensuring the ‘L’ is clearly articulated. NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? Answer: [Yes] Justification: We briefly described our methodology in the Abstract and stated its effects accordingly. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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