Towards Accurate Time Series Forecasting via Implicit Decoding
Authors: Xinyu Li
Yuchen Luo
Hao Wang
Haoxuan Li
Liuhua Peng
Feng Liu
Yandong Guo
Kun Zhang
Mingming Gong
Venue: NeurIPS 2025 poster ()
Keywords: Time Series Forecasting
Frequency-domain
Primary area: applications

Abstract: Recent booming time series models have demonstrated remarkable forecasting performance. However, these methods often place greater focus on more effectively modelling the historical series, largely neglecting the forecasting phase, which generates long-term forecasts by separately predicting multiple time points. Given that real-world time series typically consist of various long short-term dynamics, independent predictions over individual time points may fail to express complex underlying patterns and can lead to a lack of global views. To address these issues, this work explores new perspectives from the forecasting phase and proposes a novel Implicit Forecaster (IF) as an additional decoding module. Inspired by decomposition forecasting, IF adopts a more nuanced approach by implicitly predicting constituent waves represented by their frequency, amplitude, and phase, thereby accurately forming the time series. Extensive experimental results from multiple real-world datasets show that IF can consistently boost mainstream time series models, achieving state-of-the-art forecasting performance. Code is available at this repository: [https://github.com/rakuyorain/Implicit-Forecaster](https://github.com/rakuyorain/Implicit-Forecaster).

Towards Accurate Time Series Forecasting via Implicit Decoding
Xinyu Li', Yuchen Luo', Hao Wang", Haoxuan Li3,6
Liuhua Peng', Feng Liu², Yandong Guo Kun Zhang Mingming Gongl.6*
'hee University of Melbourne "Zhejiang University PPeking University AAI Robotics Carnegie Mellon University *Mohamed bin Zayed University of Artificial Intelligence x150student, unimelb. edu. au, mingming. goa.liggggoooogmmmmmmmg00 edu. au
Abstract
Recent booming time series models have demonstrated remarkable forecasting performance. However, these methods often place greater focus on more effectively modelling the historical series, largely neglecting the forecasting phase, which generates long-tern forecasts by separately predicting multiple time points. Given that real-world time series typically consist of various long short-term dynamics, independent predictions over individual time points may fail to express complex underlying patterns and can lead to a lack of global views. To address these issues, this work explores new perspectives from the forecasting phase and proposes a novel Implicit Forecaster (IF)
additional decoding module. Inspired by decom- position forecasting, IF adopts a more nuanced approach by implicitly predicting constituent waves represented by their frequency, amplitude, and phase, thereby accurately forming the time series. Extensive experimental results from multiple real-world datasets show that IF can consistently boost mainstream time series models, achieving state-of-the forecasting performance. Code is available at this repository: https:///ittuu: com www//raayorrinnnaaaaaaa wwwpaaiiiiinniiitiiaaeeieeeeiiee
horolliiici-oorestttrrr Forecaster
1 Introduction
Time Series Forecasting (TSF) is the task of predicting future trends and dynamics of time series based on historical observations. is essential for many real-world applications, including weather forecasting 52 28 energy management (21 and traffic flow estimation [ [32 In the past few years, the advancement of deep learning has enabled neural forecasting methods to take significant place in the TSF landscape. Neural networks, especially Recurrent Neural Networks (RNNs) [00 11] are particularly effective on TSF due to their ability to capture sequential information. Convolutional Neural Networks (CNNs) 51, with their strength in recognising intricate temporal structures and local patterns, excel at extracting features from time series. By modelling dependencies between a sequence of representations, Transformers have demonstrated superior ability in learning inter-series 13 and intra-series 20 information, thus becoming ofate-of-the-at art in TSF.
Despite these achievements, existing TSF methods primarily focus on improving the modelling of historical time series, while more precisely translating the modelled features into future series during the forecasting phase is rarely considered. Specifically, previous works typically use elaborately designed architectures to learn and transform the input time series into complex representations, and then leverage MLPs to predict these representations separately into the value of each future time step, thereby generating the output series. We noticed that this scheme independent prediction between consecutive points might not be suitable for long-term forecasting, given that native MLPs lack
"Corresponding author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Constituent Fluctuations
Lemgloe Compoand
Component
Long dorm
Time Series
Complete Series
Time Sertes
wmw
MMMAMAM
Decompose
Short-tomm
Component
Component
Decomposition of a time series into cub-components. Left: illustrates the breakdown of simple time series into its constituent parts, each reflecting fluctuations over different frequencies. Right: provides a complete view of each individual component clearly visualise how the combina- tion of different periodic fluctuations forms the observed time series.
Figure 1:
global view of time series' overall fluctuations 45 and are unable to handle the autocorrelation among predicted points, which may hinder them from forecasting and expressing general trends and long-term patterns. Besides, real-world time series often exhibit a mixture long short-term dynamics, yet point- wise MLPs directly predict the intertwined original series while ignoring this compositional structure. This results in a forecasting process that lacks the understanding of underlying patterns, which not only increases forecasting difficulty but also produce indistinguishable and uninterpretable predictions over various dynamics.
Considering the numerous underlying dynamics, is apparent from Figure that time series be regarded as a combination of multiple different types fluctuations, where long-periee fluctuations reflect the overall trend of the time series and short-period fluctuations can indicate the seasonality. Extensive research [1
36] has proven that forecasting future time series by separately pre- enhance performance. This decomposition forecasting
dicting its trend and seasonal components approach further highlights the potential to forecast future series by predicting more nuanced trends and dynamics. Consequently, we pose the following question: feasible form the future time series by globally predicting constituent fluctuations across different periodicities?
In light of the aforementioned problems and motivations, explored the forecasting phase of TSF, focusing on accurately translating the learned information into actual forecasts. In this paper, we propose Implicit Forecaster (IF), novel decoding module that replaces the output layers of previous TSF methods. IF offers a different forecasting perspective: instead of predicting future time points separately, it globally forms the time series by combining various base fluctuations, where these fluctuations are treated as periodic waves that can be implicitly represented and predicted just by their frequency, amplitude, and phase. Experimentally. IF is compatible with recent mainstream TSF models and can consistently boost their forecasting performance, making them state-of-the-art on 14 real-world datasets widely used in TSF and improving the result by large margin several of them. The contributions of our work lie in three folds:
We refine the forecasting phase of TSF and propose Implicit Forecaster to replace the output layers of existing TSF models. It globally predicts various constituent waves from frequency perspective to achieve accurate long-term time series forecasting.
Experimentally, the proposed Implicit Forecaster can applied to mainstream TSF models, boosting their performance by a large margin and achieving consistent state-of-the-art multiple real-world benchmarks.
We show that better forecasting phase also matters for accurate TSF, an aspect largely overlooked by existing research. We hope our work can draw research attention to this critical point and inspire novel advancements.

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2 Related Work
2.1 Time Series Forecasting with Deep Models
In the rapidly evolving field of time series forecasting, deep learning-based methods have garnered significant attention and attained remarkable success. RNNs have been favoured for their proficiency in handling sequential data. LSTNet [TO] integrates convolutional layers into recurrent structures, effectively learning long short-term patterns. SegRNN [I]] reduces recurrent steps by leveraging Segment-wise Iterations and Parallel Multi-step Forecasting strategies, achieving notable performance and efficiency. CNNs excel in learning local temporal patterns underlaid in time series, where MICN 34 employs multiple convolutional kernels capture local features and global correlations. offering holistic views of time series. TimesNet [40) extends ID time series into 2D space for better time series representation, thereby extracting qulli-periodicity information. MLPs, built upon linear transformations, have demonstrated impressive forecasting accuracy while maintaining high computational efficiency. TimeMixer (36] utilizes a decomposable multiscale mixing, capturing temporal patterns at different granularities. SOFTS 4 introduces the STar Aggregate- Redistii-de (STAR) module, aggregating global information across channels capture inter-series correlations.
Transformer [27] as the most prominent sequence modelling architecture that achieved exceptional success in natural language processing [39] and computer vision (2), has emerged as the most popular approach for solving TSF problems. Autoformer [4] combines traditional time series decomposition with auto-correlation extract better predictive components. PatchTST ([20] adopts a Vision Transformer-lic strategy, segmenting time series into smaller patches to better model intra-series dependencies. Crossformer [50] and CARD 337 explore interdependencies between patches across channels, concurrently learning cross-time and cross- vvrritt information. i'Transformer 13 treats the entire time series as an individual token to model series-wise dependencies, thereby
explicitly learning multivariate correlations.
Given that time series forecasting is an end-to-end process, previous methods predominantly con- centrated on optimizing the "historical series to representation" stage. Our work, however, shifts the focus the "representation to predicted series" stage. bridging the gap in the forecasting phase.
2.2 Time Series Forecasting with Fourier Analysis
Fourier analysis offers methods for effectively modelling the underlying dynamics within time series. Recent advancements demonstrated that integrating rrequencyddomen techniques can enhance the performance of TSF. FEDformer (54) employs frequency-enhancee attention for efficient computation while better capturing the global properties. FiLM (55) applies Legendre Polynomial projections
to approximate historical information and remove noise. FreTS [55 applies MLPs directly to the frequency domain, learning the mappings of time series in the complex plane. FreDF (30) proposes new loss term to penalize forecasting errors within the frequency domain, enhancing the autocorrelation between predicted points. OLinear [46] operates in an orthogonally transformed domain better model time series and utilizes normalized weight matrix efficiently capture multivariate correlations. FreEformer [47] further leverages frequeney -nhaannceeeeeeee Transformer to model cross-variate dependencies in the complex domain, introducing an enhanced attention mechanism that improves feature diversity and gradient flow. In contrast to these methods, which primarily focus on leveraging Fourier analysis to model the historical time series, our approach explores a novel strategy that incorporates frequency techniques in the forecasting phase, aiming to achieve more accurate pattern predictions and address the lack of global views of output series.
3 Methodology
Problem Definition Consider a length multivariate time series with N number of variates,
represented X {xx, X2, x7 E RN',, where each x, R corresponds to N concurrent variates observed at time I. Also, consider the subsequent time series of X over a future horizon L, denoted as Y (xT+1, -.......... xT+ 4xN. Given a forecasting model fo parameterized by 0 that maps the historical time series X to its future forecast as Y fo (X), where Y
(X141, 8 & E R the objective of time series forecasting is to optimize 0 such that the forecast Y closely approximates the true future Y.
3

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3.1 Implicit Forecaster
Discrete Fourler Transform
Time Series Encoders
amplitude
Implicit Forecaster
MLP- based
CNN- CNN-based
PHead
w
prediction
DFT
iDFT
www.mo/www0.
window
RNN- boN-based
Trm-based
phass
AHead
mM
tegean;: STARt..
Mimeter: sPacchrs..........
predicton
Crop
Output
Input
Output
Figure 2: The overall architecture of Implicit Forecaster. The input time series will first be learned and converted into hidden representations via the encoder part of other time series models. The proposed Implicit Forecaster then decodes such representations into amplitudes and phases of various periodic waves, composing them into output series.
As depicted in Figure 2 our Implicit Forecaster mainly consists of an Amplitude Head (AHead), Phase Head (PHead), Discrete Fourier Transform (DFT) module, and inverse DFT (iDFT) module. We will dissect its architecture around how it generates the forecasts in this section.
Encoder Integration IF receives two inputs for forecasting: 1) hidden encoder representation that can maximally capture the inherent information and temporal features of the input time series, and 2) the original input time series provided by skip-connection. Since IF mainly serves decoding module for improving the forecasting phase, we take existing forecasting models (e.g.. the encoder part of TSF Transformers) the time series encoder for learning the encoder representation, learning the
additional
and replace their output layers with IF. Given a multivariate time series X E RR 7 encoder representation be easily formulated follows:
(1) where EnnceSerissEncoderr : R TxN + IR D is the encoder module, Xene NxD is the encoder representation of the input time series and D stands for the model dimension. Note that the encoder representation is seepare-eeparated over N, as IF independently forecasts each variate of time series.
X
TimeSeriesEncoder/
Implicit Decoding In the forecasting phase, rather than directly predicting the learned information into each output time point like conventional methods, IF implicitly predicts the constituent fluctua- tions of the future time series. Concretely, for desired forecast, IF presupposes frequency pool (i.e., fixed-size spectrum) containing waves with different long short-term frequencies and separately predicts the amplitude and phase of each wave. These implicitly represented waves will serve as the constituent components and will be combined as the future time series, where the part within the forecasting length will be cropped as the final output. Additionally, IF uses a skip-connection to extract and incorporate the amplitude and phase information of the original input series, given that such information is likely to persist throughout future observations, which will be concatenated the encoder representation as a supplementary feature map to assist the wave prediction.
Taking the encoder representation Xenc E . N. D and the historical time series X E RRTX as its
input, IF forecasts the future time series Y E IR Lx as follows:
S = F(X )
Å AHead Concat (Enne (S)),
(2)
b = PHead( ConcaoC (x ennc arg
Y CFop.y(F=11 (

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where F denotes the Discrete Fourier Transform and F-1 is its inverse operation, and j is the imaginary unit defined as the square root of -1. S E CNT is a complex matrix denoting the spectrum of the original input X. |S] is the amplitude of each constituent wave from X and arg is the argument operation that computes the phase of each constituent wave. AHead is the amplituee predictor and PHead is the phase predictor. Suppose the frequency pool of IF contains P waves with different frequencies to form the future time series, then Å E B the predicted amplitudes and phases of these waves, respectively. We provide detailed discussion of IF in the next section.
xPP and Ø E [- x denote
3.2 Overall Workflow
In this section, we provide a comprehensive introduction to the internal components of Implicit Forecaster, explaining the motivations behind various designs.
Discrete Fourier Transform and its Inverse IF aims at forming time series with various periodic waves, while the DFT [26) and its inverse provide effective tools for representing and processing these waves. By transforming signal from the time domain into the frequency domain, DFT enables the decomposition of a time series into its constituent frequency components, each represented by a complex number that encodes both the amplitude and phase of the corresponding sinusoidal wave at a certain frequency. Consider a univariate time series X {01, it to its spectral representation Z = {21, .2., 2N E C with N waves by DFT as follows:
.N } E R.W, we can convert
|xx| arg((*)) As ci00 k =1=1 N, =2
2k
Fn
where 2g E C corresponds to the complex of k-th wave components with a frequency of k-1 By considering each wave component =ą in polar coordinate, |zx| = A Re(ez)= Im( (z. )a
N
R20 represents its amplitude and arg(zs) = db = alan2(Imı (zx), Re(=x)) E [-x,, π| represents its phase, where Re(=)), [m((=)) E RR are the real part and the imaginary part of zk. respectively.
On the other hand, the inverse DFT allows for the reconstruction of time series from provided frequency components with known amplitudes and phases. This feature is particularly critical for our approach, as IF essentially seeks to predict amplitudes and phases of various waves with different frequencies to implicitly represent underlying fluctuations of future time series, where the inverse DFT is used to combine these waves to obtain the final forecast. Reversely, given amplitudes A = {Ai, A AN) E Ryy, and phases & = {0r, $2, ...,,.. ΦN) E [--,,]] T] N of N waves, we can construct a time series signal (21, ENN RR through inverse DFT as follows:
Σ 1=1
zk jan
1 N
Σ An edoe d2an n n= I,..,,.,AA N,
In
=
where In E is the n-th value x and 2 E C is the complex of the k-th wave.
Frequency Pool Based on DFT properties, any length-L time series can be completely represented as a linear combination of L wave components, where each wave corresponds a normalized frequency of f& k/L. L. for k = 0, 0,..., L 1. Therefore, it is possible for IF to forecast such time series just by predicting the amplitudes and phases of these waves, while we can define P,= = (So. f f as the frequency pool of IF. However, we observed that there may exist very long- long-period fluctuations in time series, such that the input time series or the forecast can only encompass small portion of their complete cycle. Given that these fluctuations operate at lower frequencies than those that can be directly predicted by IF (i.e., beyond the IF's frequency pool), they can only be approximated and represented by blending the pool's existing frequencies (a phenomenon known as Spectral Leakage), which complicates the forecasting process.
To better capture these autra-low-frequenc components, we extend IF's frequency pool. Specifically, when forecasting a time series of length L, we consider increasing the size of the frequency pool to P (P > L).. such that it contains frequencies f k/ P for k O.... P 1. Consequently, IF predicts the amplitudes and phases of waves rather than L., allowing to directly access and predict lower-fregge frequeeyy components. This extension enhances the representational capacity of IF, enabling
5

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more accurate forecasting of time series with large-scale or slow fluctuations. Moreover, according to the conjugate symmetry property of the DFT. the spectrum of real-valued time series symmetric, where the second half is the complex conjugate of the first half. Therefore, in practical implementation, IF only needs to predict the first half of its requency-doman components to reconstruct the complete time signal, substantially reducing computational cost.
The Amplitude Head is essentially an MLP that maps the encoder representations to the amplitudes of the waves in the frequency pool. To enhance information utilization, it also incorporates the amplitude information derived from the original input time series, thereby obtaining global view of the input's s energy distribution and improving the accuracy of future amplitude prediction. Given the n-th variate's s encoder representation X Xene,n E TR. and corresponding input amplitude information An E 20 the AHead can be formulated as follows:
AHead
AHead(X (a Ken,n; A MLP amp (Concat(Xene.)) A. ))1,
where MLP amp: IR D+T + IR. is dense network with two layers for predicting the amplitude.
PHead The Phase Head is responsible for predicting the relative position of the waves in the frequency pool, ensuring they are properly aligned within the forecasting window. However, directly regressing raw phase values challenging due to the inherent numerical discontinuity of phases at their boundaries, where abrupt changes may occur from π to -T. Such discontinuities make difficult for IF to produce stable and consistent phase predictions. To mitigate this issue, we adopt a continuous representation of phases by predicting their sine and cosine components instead. Predicting these components ensures phases have a smooth transition across their periodic boundaries, thereby facilitating more accurate phase prediction. Therefore, taking the encoder representation Xene, R D and input phase Φ., E |- T of the n-th variate as its input, the PHead consists of two MLPs and can be formulated as follows:
å Tanh(MLP ein (Concat(Xeno.n)) pn))),
B Fanh(MLP....Coonnnc (Concatt (XeaattXeeemm... @n)))
(6)
S alan2 ;,))
where Φ. E I- T| is the predicted phases of pool waves. á 3, E i-1,, are the sine and cosine components, and MLP sin, MLP cos :R D+T + Rº are their corresponding predictors.
4 Experiments
We conduct extensive experiments to evaluate the performance of mainstream TSF models equipped with IF across various datasets and forecasting scenarios, comparing them with their original results to analyze the effectiveness of IF.
Datasets We comprehensively include 14 benchmark datasets commonly used in TSF for our experiments, covering various real-life domains such as energy, traffic, weather, economics, and disease. Specifically, these datasets are ETT (Electricity Transformer Temperature) with 4 subsets, ECL (Electricity Consuming Load), Traffic, Weather (53), Exchange Rate, ILI (Influenza-Like Illness) 41] PeMS (Traffic data of Caltrans Performance Measurement System) with 4 subsets [12, and Solar Energy [1O]. We split each dataset into training, validation, and test sets in respective ratios of 70%, 15%, and 15%, with all datasets divided strictly in chronological order to prevent data leakage issues. The validation set is used for monitoring the training loss of models, and the test set evaluates the model's 's final performance quantitatively. Lastly, all datasets are preprocessed with standardisation based on the training set. Further descriptions for each dataset are provided in Appendix A.
well well-acknowledged TSF models as our benchmark methods for comparison. These methods are from a diverse range of model types (we choose the most competitive method of each type). including: 1) one MLP- based method: SOFTS 4, 2) four Transformer-basee
Baselines We carefully select
methods: TimeXer (38] iTransformer [3], PatchTST 200 and Crossformer (50) 3) one RNN-based method: SegRNN [1I, and 4) one CNN-based method: TimesNet 44) These models will be paired with to compare against their original performance, with all models being evaluated under the
same framework and environment. We provide a detailed introduction to basclines in Appendix A.2

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Implementations All models are implemented entirely in Python and built upon Py PyTorch 2.0, with baseline methods directly adopted from their official implementations. The experiments reported in this paper were conducted on I 16-core AMD EPYC 9654 CPU and a single NVIDIA RT'X 4090 GPU. We choose Adam optimizer [9] and 1.2 loss to learn the model parameters and take MSE (Mean Squared Error) and MAE (Mean Absolute Error) as metrics to evaluate the models. More implementation and experimental details are provided in Appendix A.3
4.1 Main Results
Table I presents the forecasting performance of all models across 14 benchmarks, including their results after being equipped with the proposed Implicit Forecaster (w/ IF), where better values are bolded. Since MSE/MAE measures the discrepancy between the forecast and the ground truth, lower value indicates better model performance.
Table I: Multivariate time series forecasting results, averaged over respective prediction lengths I. E {24,30 36, 48, 60) for the IIII dataset, I. E {12, 24, 48, 96] for the PEMS datasets, and I. e (96, 192, 192,336, 720} for the remaining datasets. The lookback window length is set to T = 36 for the II and 7 = 96 for the others.
Madel
500TS
(MP0
(Ownd)
W/FF
Therker
(304))
(Duro
IF
Tra Bormerr
2000
(Own))
wr
SegRWN (700))
(0wmn)
W/FF
(NPS))
(Owed)
WrFFF
Cheeffrrer (303))
10wm
Tiieeder (2037)
(Dero
W/IF
Mennc
MAB MAI
MIE
MAE NSE MAE
MAR
MSE MAR
MAE
MAE
MII
MAR
MAE
MAR
MAE
ASE
MAE
MSEE MAZ
MSE MAA
MBB MAB
8.986
633
0.59
8.96
BAR
(1.46
0.414
0,47
9.690
0.66
0.87
0.086
(0.97
0.0
0.478
0.14
0.41
6,800
11396
0.56
0653 BA11
0.595
0.511
6.06 0.497
PITW
0151
8.167
0.366
8.361
10155
0:33
6.36
(6.12
6.59
0180
0
0.66
0.111
6.362
(150
0.319
(8:8)
6:771
6.18
01361
0354
8,57
0:34
(.39]
0.33
0.39
6,396
(6T9) 8,30
0.207
(0.490
0.317
Weater
0.66
8:324
(4333
0.354
(4321
1979)
9.36
(1335
(0736)
0,37
(0055
0336
0.324
16362
(1312
(175) 0/515 (121 6.770)
(1369
Tackege
0.451
8.402
(1757
0/334
[6/07
0.155
0,412
(6699
1.294
0.743
08432
14766
1-314
(V/25
(.454
11
3380
8364
1.163
3,754
(003
11112
8:167
(1113 0:31 IUPM
9:387
0331 04134
(135)
61191
0.28%
(139)
0.386
1.200
11313
(1332 6:36
(1322
0/555
B/11T (1333
PRM97
1000
9.146
9880 W.86 100
0:355
8,116
(0:556
0.535 10.10 6.399
(0.311
0.00
9.37
(.35500351)
(.11)
6:307
(1196
0,50 (0:555 6,550
(1167
6:550
6.100 0.315
FFM199
01118-0714
0718
8.107
0:535
011
8,557
(0.70
Prieedsi)
(8715
BARIG]
1.95% 3.89
18.4% 11.85)
18.85 K12
14.56 8836)
KAA2% 4.7%)
[4.75 7.20%
Compared with the initial results, all baselines equipped with IF exhibit consistently improved performance, demonstrating obviously lower MSE/MAE across most of the datasets. This clearly validates the effecctivness and generality of IF, indicating that decoding time series by implicitly predicting and combining constituent components indeed leads to more accurate forecast. Notably, the performance gains are particularly pronounced on PEMS datasets, in which all models outperform their original results with average decreased MSE and MAE of 30.4% and 18.2%. We attribute these substantial improvements to IF's ability to directly access low-frequency waves and to perform separated predictions over long short-term patterns, which facilitates forecasting time series that are mixed with extreme fluctuations and rapid changes, such as PEMS. Furthermore, it is evident from the table that the average improvement of IF in MSE is more significant than that in MAE, revealing that IF produces globally more precise forecasts, and predicting frequency components does offer more holistic view of the future series. Finally, the experimental results also imply that the encoders of these models are actually capable of extracting useful information from the data, while their original point-wise decoding schemes are insufficient to fully leverage the learned information.
4.2 Method Analysis
In this section, we further analyze the proposed Implicit Forecaster. For fair comparison, all models in the following experiments will adopt the same time series encoder: a standard Transformer encoder with inverted time series embedding 13)
Ablation study We gradually replace/adjust the proposed module/mechanism, thereby investigating their contribution to the method's overall performance. We consider two ablations: 1) IF: we replace IF with other decoders, including Linear forecaster that is common in recent works 4 13 20 31] MI. forecaster with the same nonlinear process and parameter amount as IF, and the traditional Transformer decoder, 2) Skip-connection: to ensure that the strong performance of IF is not attributed
7

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to its skip-connection, we test the effectiveness of the skip-connection, including disabling it (w/o) and only using it (only, i.e. without using the time series encoder).
Table 2: Ablations of IF. This table presents the standard Transformer's ' average performance across various forecasting lengths when equipped with different decoders.
Table 3: Ablations of skip-connection. This
table details the impact of adjusting the skip- connection in IF.
IF
Skip-connection
Transformer
Docuder
Replace
w/ Skip.
w/o Skip.
only Skip.
Replace
Linear
Transformer
Metric
MSE
MAE
MSE
MAE|
MSE
MAE
Metric
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAE
ETThl
0,503
0.516
0.537
0.521
0.617
0.577
ETTh2
0.188
0.301
0.201
0.313
0.199
0,311
0.202
0.316
ECL
0.157
0.256
0.100
0.254
0,210
0.300
Exchange
0.286
0.384
0.111
0.442
0.428
0.448
0.412
0.454
11.1
2.387
1,018
2,689
1.107
2.486
1.040
3,504
1.282
Weather
0.221
0.259
0.227
0.257
0.250
0.291
PEMSOS
0.103
0.209
0.184
0.272
0.198
0.278
0.166
0.272
Solar
0.189
0.251
0.190
0.251
0.284
0.345
Performance
100%
100%
17.00%
88.0%
77.35
88.9%
(73.1%
84.0%
Performance
100%
99 99.
97.2%%
100%
77.8%
84.0%
From Table 2. we can clearly witness that changing IF to other decoders results in significant degradation the model performance. This supports that when processing equivalent input from the encoder, IF is considerably better at transforming high-level time series representations into more precise forecasts, which underscores its substantial effectiveness. As a comparison, particularly on the PEMS dataset, Linear forecaster performs much poorer than IF, revealing that point-wise prediction can struggle handling long-term patterns. Besides, MLP forecaster is still inferior to IF, which implies that the superiority of IF is not own to nonlinear transformation, but rather to the success of the
forecasting method. Notably, the Transformer decoder performs the worst among these forecasters, indicating conventional heavy decoder designs may be overly complicated and thus susceptible to noise interference.
From the results in 3] the skip-connection in IF contributes effectively to some of the datasets, in which disabling it only leads to a performance degradation of 2.8% in MSE, indicating its influential but not decisive role in our method's 's strong performance. Besides, using only skip-connection also shows good forecasting performance, proving that the frequency-domain information in the input time series can associated with future observations, thus can provide effective features for forecasting.
Computational efficiency We use the standard big-O asymptotic notation for an- alyzing the time complexity of IF. Accord- ing to the symmetric properties in the fre- quency domain described in Section 3.2 for a frequency pool size P. there will be output neurons in IF used for mak- ing predictions ( +1 1 neurons for predict- ing amplitudes, sine of phases and cosine of phases, respectively). Therefore, the forward pass complexity of IF is simply bounded by (PP). However, after predicting frequency- domain features, an iDFT is applied to re- construct the time signal. The iDFT can be implemented efficiently with a complexity of ((P log P) using FFT-based algorithms. Combining both terms, the total time complexity of IF is: prediction complexity + iDFT complexity O(P P P) O(P log P). Conventional point-wise forecasters, in contrast, predict future values independently at each time step, with distinct output neurons corresponding to different time points. Hence, their overall computational complexity is OLL. In experiments, we measured the training speed (ms/iter) and memory footprint (GB) of IF, comparing its computational cost with other forecasters. The results are provided in Table 4 showing that IF has higher computational cost compared to simpler forecasters such as Linear or MLP, yet it remains more efficient than the Transformer decoder, thus this additional cost does not represent a significant disadvantage.
Table 4: The computational cost of different forecast- across various datasets. The results are measured over the prediction length of L = 96 for the PEMS08 dataset and L 720 for the rest of the datasets.
+ P
Decoder
IF
Linear
MLP
Transfomer
ETTb]
Speed (mofiter)
14.12
12.11
11.45
14:44
Memory (GB)
0.241
0,190
0,217
0,251
BCL
Speed (ma/itrr)
65.43
59.73
61.05
68.57
Menory (CB)
1.119
0.791
0,888
1.164
Weather
Spced (insiiter)
19.23
17.51
17.76
20
Memory (GB)
0.388
0.328
0.352
0.406
PEMS08
Speed (mn/iterr)
34.48
31.25
Memory (GB)
0.899
0.538
0.637
0.940
8

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Varying lookback length The previous methods [13] have verified that treating the entire time series token and learning from the variate dimension can benefit the model from longer lookback window, achieving more accurate forecasting. Considering this can be attributed to the encoder, we believe that the distinct advantages of decoders only emerge with a shorter input. Therefore, we take another perspective that varies the lookback window size from long to short, further validating IF's effectiveness compared to point-wise decoding methods. Figure shows performance changes of Transformer when shortening the input length, where the results surprisingly exhibit that using IF as the decoder not only outperforms the Linear forecaster across input lengths but that the gap widens as the input length reduces. This reveals that adopting IF as the decoder could also empower the model's robustness under limited input data, illustrating its unique strength in extracting maximum details from minimal information. DTTm
PENSH
PENSAE
82-
4.17
1.0
0.
3.4
822
Liou
Figure 3: Forecasting performance of Transformer when equipped with Implicit Forecaster (green) and Linear forecaster (yellow) after shortening the lookback window to T' E (96, 72, 48, 24). Results are averaged over all prediction lengths, where Implicit Forecaster consistently outperforms Linear forecaster, especially for shorter input lengths.
Visualization analysis Implicitly predicting constituent fluctuations with IF also offers better interpretability, Figure shows case visualization from the ECL dataset, where the left heat map illustrates the absolute weights learned by AHead for the waves in IF's frequency pool. These higher- weighted frequencies
be regarded the most influential ones identified across the entire training set. The right heat map displays the corresponding predicted energy distribution (i.e., amplitudes) for these frequencies. We can clearly observe that frequencies with larger weights tend concentrate higher energy, indicating that IF not only learns which frequencies are important but also actively allocates more representational capacity them during forecasting. This pattern suggests that IF implicitly captures the dominant periodic components driving the whole temporal dynamics, offering an interpretable decomposition for the forecasting process.
1.0
0.8
0.8
300
100
250
150
04
200
04
250
0.2
250
02
300
300
50
100
150
200
250
300
350
50
100
150
200
250
300
Figure 4: Weights and Prediction Visualisation. The x-axis corresponds IF's frequency pool.
How does IF solve the problem of lacking global views in previous methods?
Frequency components can capture global characteristics time series, as they are computed from the entire series. This global property has been validated by many forecasting methods and utilized to enhance the model's global views when learning the input [54 45). Accordingly, we believe that they can also be used to globally express the output when making predictions. Therefore, the proposed module predicts amplitudes phases of frequency components, and these components computed as global fluctuations of future series, improving global views of the forecasting phase.

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4.3 Limitation Analysis
Despite its strong forecasting capabilities, our Implicit Forecaster still exhibits opportunities for further enhancement in terms of efficiency and overall performance. For example, predicting all spectral components may introduce some inefficiencies, as selectively focusing on the more contributive frequencies could potentially reduce computational overhead without significantly compromising the forecasts' accuracy. Besides, employing fixed-size frequency pool might occasionally be suboptimal, as a manually defined spectrum might not precisely align with the optimal frequencies that best fit the dataset. Moreover, while the Implicit Forecaster is primarily tailored for forecasting tasks, exploring its adaptations across different scenarios could provide additional values. By highlighting these limitations, we aim to encourage further research to refine our method.
Conclusion
In this paper, improve the forecasting phase of recently booming TSF methods, which have lacked attention in previous works. We propose Implicit Forecaster, a novel decoding module to replace the output layers of existing forecasting models. It predicts learned historical information into various implicitly represented constiteent waves and effectively combines them to form the forecast. Experimental results demonstrate that our Implicit Forecaster enhance the performance of current mainstream forecasting models, achieving consistent state-of-the-ar on multiple real-works datasets. We discover that the forecasting phase of TSF can be an interesting and compelling point research. Future improvements may include: 1) efficient decoding by predicting key frequencies only, and 2) accurate prediction for localized patterns by incorporating other frequency methods (e.g., Wavelets).
Acknowledgments and Disclosure of Funding
We would like to acknowledge the support from ARC DP240102088, WIS-MBZUAI 142571, NSFC 623B2002, ARC DE240101089, LP240100101, DP230101540, NSF & CSIRO Responsible AI Program 2303037, the NSF under Award No. 2229881, the AI Institute for Societal Decision Making (AI-SDM), the National Institutes of Health (NIH) under Contract RO1HL159805, and grants from Quris AI, Florin Court Capital, MBZUAI-WET Joint Program, and Al Deira Causal Education Project.
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A Experimental Details
A.1 Dataset Descriptions
We use 14 real-world datasets to comprehensively evaluate the performance of baseline methods on the long-term time series forecasting task. Detailed descriptions of these datasets are follows:
ETT The Electricity Transformer Temperature dataset records oil temperatures and multiple power load situations collected from two counties in China between July 2016 and July 2018. It consists of four subsets with different time granularities: ETThl and ETTh2 are sampled every hour, while ETTml and ETTm2 are sampled every 15 minutes.
ECI 53 : The Electricity Consuming Load dataset records the hourly electricity consumption (in kWh) of 321 clients over the period from 2012 to 2014.
Traffi 4
The Traffic dataset records road occupancy rates San Francisco Bay area freeways, measured hourly by 862 sensors from January 2015 to December 2016, sourced from the California Department of Transportation.
Weather [33: The Weather dataset from the Max Planck Biogeochemistry Institute in Germany records 21 metcorological factors, such as air temperature and humidity. This climatological time series is sampled every 10 minutes throughout the entire year of 2020.
Exchange Rate 4 The Exchange dataset compiles daily exchange rate panel data for eight countries, including Australia, Britain, Canada, Switzerland, China, Japan, New Zealand, and Singapore, covering the period from 1990 to 2016.
IL 255
The Influenza-Likk Illness dataset captures the weekly ratio of influenza-like illness patients to the total number of patients, reported by the Centers for Disease Control and Prevention of the United States from 2002 to 2021.
This transportation dataset provided by the Caltrans Performance Measurement System records California traffic features like flow, occupancy, and speed. We use four of its public subsets: PEMS03, PEMS04, PEMS07, and PEMS08, each with the same sampling rate of 5 minutes.
PeMS [12]
Solar Energy [10]: The Solar- Enlery dataset records solar power production from 137 photovoltaic (PV) plants in Alabama State for the year 2006, with data sampled every 10 minutes.
Table 5: Dataset details. Dim refers to the number of variates in the dataset. Input Length refers to the number of historical timesteps used for making forecasts and Prediction Length denotes to the number of future timesteps to be predicted. Dataset Size denotes the respective timepoints in training, validation and test set. Frequency refers to the sampling interval between each timepoint.
Dataset
Dim Inpus Lngghh Prediction Lnggth
Dataset Size
Frequency
Domain
ETTHL BTTh2
7
[96, 192, 336, 720}
(12191,2613 2613, 2613)
Hourly
Temperature
ETTml, ETTm2
7
[96,, 192, 336, 720)
(18776, 10152, 10152)
15min
Temperature
Electricity
321
96
[996, 192, 336, 720]
(18412, 39H1, 39H1,,34477
Hourly
Electricity
Traffie
862
96
[906, 192, 336, 720]
(121290, 2631, 2633)
Hourly
Transportation
Weather
[96, 192, 192,336, 720]
[36887, 7004, 7905)
10min
Weather
Exchange Rate
96
[966, 192, 3366, 720
(5311, 1138, 11:39)
Daily
Finance
Ilness
7
36
[24, 36, 48, 60)
(676, 144, 146)
Weekly
Health
PEMSU3
358
(12, 24, 48, 96)
(18345,3991 3,313,9111330 39113)
5min
Transpoctation
PEMSIH
307
(12, 48, 96)
1,18991,,3488. 23500)
Smin
Trunspoctation
PEMS07
883
96
(12, 21, 18, 96)
(19756, 1233, 1235)
Smin
Transportation
PEMS08
170
96
{12, 24, 18, 96)
(12199,
2679)
Smin
Transportation
Solar-Bnegyy
137
[96, 192, 338, 720)
(36792, 78844, 7884)
10min
Energy
sttps://archive. iic.shiveeiiss..o uci
httcc/////.aeeeeccccccccccaaaaaa htap://pems.dot.ca.ggg
https: https://www.gcc-eene dewpgg/.ee///////e
22

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A.2 Baseline Descriptions
We carefully select 7 mainstream TSF models baselines for comparison, which are representative methods from diverse model types. Detailed descriptions of these baselines are as follows:
SOFTS 4
SOFTS an MLP-based method that leverages a STar Engregate-Rediitrrbuee (STAR) module capture the core representation of time series for efficient and accurate forecasting. The core representation is shared across channels and carries global information on all variates. Official implementation of SOFTS available at this repository: SOFTS
TimeXer [8] TimeXer is a Transformer- based method that can simultaneously leverage endogenous and exogenous variate information. It captures internal information of time series via patch-wise self-attention while integrating external time series information via variate-wise cross-attentionn thereby achieving accurate forecasting. Official implementation of TimeXer is available this repository: TimeXer
iTransformer [3] TTransformer is Transformer-basee method that embeds the entire time series of each variate as a token and inverts the Transformer's role to apply the attention and forward network the variate dimension. It models series-wise dependencies via multivariate correlating, making it proficient in capturing and learning cross-variate information. Official implementation of iTransformer is available this repository: Transf
SegRNN I
SegRNN an RNN-based method that introduces Segment- wise Iterations and Parallel Multi-step Forecasting (PMF) to significantly reduce the required recurrent iterations for forecasting, achieving remarkable performance and inference speed. Official implementation of SegRNN is available at this repository: SegRNN
Transformer-bast based method that considers channel independence between variates and adopts a Vision Transformer-like strategy [2]. segmenting time series into smaller, semantically richer patches. models patch-wise dependencies, thereby enhancing local information processing for each variate. Official implementation of PatchTST is available at this repository: PatchTST
PatchTST [20 PatchTST
Crossformer is a serefoommrr-base method that introduces the Dimension- Segment-Wise (DSW) embedding and Two-Stage Attention (TSA) layer. models patch-wise dependencies across multiple variates, allowing it to effectively learn cross-time and cross-variate information. Official implementation of Crossformer available at this repository: Crossformer
Crossformer [50]:
TemesNet [40]: TimesNet is a Frequency-domain integrated CNN-based method that extends ID time series into 2D space with Fast Fourier Transform (FFT) for better time series representation. This improves its capability for learning multi celti-ppriodicity and extracting complex long short-term temporal patterns. Official implementation of TimesNet is available this repository: TimesNet
The Time- Mories--ibbaan provided by TimesNet 40 offers fair implementation of baseline models. It is built on the source code and configurations provided by each model's 's original paper.
A.3
Implementations
All the models and experimental frameworks are implemented entirely in Python and built upon PyTorch 2.0 [22]. All the experiments reported this paper are conducted on a 16-core AMD EPYC 9654 CPU and single NVIDIA RTX 4090 GPU. We select Adam [9] as the optimizer and MSE loss to learn the model parameters. The learning rate is scheduled to follow an exponential decay pattern during training, which is halved at the end of each epoch. The number of training epochs is determined using an early stopping strategy, where the training is stopped when the model's performance (i.e., loss) ceases to improve the validation set for maximum of times.
For experimental fairness, throughout the experiments, we did not perform any hyperparameter tuning for any of the baseline models or models equipped with IF. Instead, to ensure consistency and accurately assess our approach, all models used the same hyperparameters before and after applying IF. All baseline methods and their hyperparameters, scripts, and experimental frameworks are strictly followed by the Lime-Series: -eries-Library repository provided by TimesNet 44). which comprehensive and fair platform for time series analysis.
23

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Loss Function In our experiments, all models are trained by minimizing the MSE loss to optimize their model parameters, Given the predicted future time series Y E IR L and the corresponding ground truth Y E RLXN, where I. denotes the forecasting horizon and N represents the number of variates, the MSE loss between Y and Y defined as follows:
L
N
Y - -Y,,)
LMSE Y) =
LN
==
where Yig, Yij E IR are the value of the time series at the i-th time step for the j-th variate.
To align with previous works, we evaluate the performance of models with two metries: the Mean Squared Error (MSH) and the Mean Absolute Error (MAB). In multivariate time series forecasting, given the predicted series Y and the true series Y.
Evaluation Metries
Mean Squared Error (MSE):
L
N
1
MSE Y Y)
Yog - Yo.s)
(8)
LN
Mean Absolute Error (MAE):
L
1 LN
MAE(Y, Y)
Y
Yi.
(9)
j=1
B Algorithmic Details
We provide pseudocode of IF's algorithmic details, clearly demonstrating its step-by-step process for time series forecasting. Note that: 1) MLP, amp, sin, cos) D+T + R are nonlinear neural networks with dense layers, which are applied to the last dimension of the input, 2) Concatenate is the concatenation operation that concatenates tensors at their last dimension.
Algorithm Implicit Forecaster
Define: input length T: output length L; variates number N; model dimension D; spectrum size Require: encoder representation Xeno E Nx, input time series X RTxN
Ensure: output time series Y R
1: Discrete Fourier Transform the input time series, taking amplitudes and phases.
2: S F(XT)
b S E CNT
3: A- [S]
> E R
4: Φ arg(S)
Φ n, n NxT
5: Run Amplitude Head, predict amplitudes.
6: Å JMLPamm amp
(Concatenate MMLPamp(Connatena encc A))l
b Å E R NxP
Run Phase Head, predict phases.
8: & Tanh(MLP sin (Concatenatel X Xenc: +)))
Dâ E [-1,1]
9: 3 Tanh(ML.P.... (Concaaaatee Xenc: Φ)))
b b E [1,1 NxP
10: - atan2(@, 33
b -,, NxP
11: & Combine predicted waves through inverse Discrete Fourier Transform.
Y- FA قهي
a r e NxP
Take the part within the output length.
14: & = -rrp,, (T
Y Y
15: return &
24

---

ETTh1 C Statistical Significance Test
Error Bar We evaluate the robustness of IF across different random seeds using six datasets. Table reports the standard deviations of the Transformer's performance when equipped with IF over five random runs, demonstrating that IF yields consistently stable results.
Table 6: This table presents the performance robustness when the standard Transformer is equipped with IF. The results are obtained from 5 random runs.
Dataset
ETTh2
ECL
Weather
Horizon
MSE
MAE
MSE
MAE
MSE
MAE
96
0.15241 0.000
0.272.1 0.002
0.132:1 0.000
0.229 0.000
0.151
0.002
0.19911 0.001
192
0.181 0.002 0.296:: 0.002
0.147 0.001
0.244 0.000
0.197
0.001
0.244 0.002
336
0.195 0.002
0.307 0.002
0.161
0.001
0.262 0.001
0.237
0.004
0.274+ 0.003
720
0.223++ 0.005
0.328++ 0.003
0.188+
0.003
0.289 0.004
0.301
0.001
0.319+ 0.003
Dataset
PEMS03
PEMS07
PEMS08
Horizon
MSE
MAE
MSE
MAE
MSE
MAE
12
0.058:: 0.001
0.160:: 0.001
10.052+ 0.001
0.144 0,001
0.062:: 0.000
0.16341 0.000
24
0.074: 0.002 0.181 0.002
0.063 0.001
0.157 0.001
0.077 0.001
0.1831 0.001
48
100:
0.002
0.209+ 0.002
0.084
0.002
0.185 0.002
0. 107 0.002
0.218 0.002
96
0.1454 0.004
0.2591 0.003
0.120 0.002
0.221 0.002
0.165 0.002
0.2734 0.002
Paired t-test To verify whether the improvement the model by IF is statistically significant, we have also repeated the experiments across random runs and conducted paired t-test to further validate the significance of the performance gain. Specifically, we applied to the Transformer backbone while keeping all hyperparameters identical (including the random seed) before and after applying IF to ensure strictly fair comparison. For the paired t-test, we test the hypothesis that the Transformer equipped with IF achieves significantly lower MSE than without IF. The experimental results averaged over 4 prediction lengths are as follows:
Table 7: Performance comparison of the standard Transformer without and with IF, Significance
levels: p 0.05, ** p < 0.01, **8 p< 0.001, n.s. not significant.
Transformer (MSE)
Original
w/ IF
p-value
Significance
0.539 + 0.001
0.504 + 0.002
1.98 X 10
ETTh2
0.200 + 0.001
0.188 + 0.001
4.68
10
ETTml
0.508 + 0.001
0.493 0.003
1.04
10-3
ETTm2
0.155 = 0.001
0.148 0.001
1.23
10-5
ECL
0.162 + 0.001
0.159
0.000
5.71
10-3
Traffic
0.620 = 0.001
0.550 + 0.002
7.85 x 10
88
Weather
0.228 + 0.000
0.220 + 0.000
2.15 x 10-5
Exchange
0.406 + 0.004
0.297
0.003
1.31 10-5
ILI
2.985 + 0.013
2.770
0.053
1.03 10-2
PEMS03
0.123 0.001
0.095
0.000
9.71
10-6
PEMS04
0.152 = 0.022
0.099
0.006
3.67 X 10-2
PEMS07
0.103 + 0.001
0.081 + 0.001
6.70 X 10-6
#*
PEMS08
0.176 + 0.011
0.104 + 0.002
1.67 X 10
Solar
0.188 + 0.001
0.189 - 0.001
5.44
10-1
N.S.
From Table 7 it is evident that the performance of our method is considerably robust, and the proposed IF module stably and statistically significantly outperforms the Transformer's original MLP-based point-wise prediction method (in 13 out of 14 datasets).
25

---

D Showcases
D.1 Model Prediction Visualisations
We provide visualisations of model forecasts supplementary showcases, offering a clearer parison between baseline models and their enhanced versions with the decoder. The left column shows the predictions from the baseline models, while the right column shows their predictions after incorporating with IF. The forecasts from each model on the ECL and PEMS03 datasets are illustrated in Figure 6 where the visual comparisons clearly show that incorporating IF leads to more accurate and aligned predictions.
SOFTS
w/ IF
3.9
3.9
-2.5
-372
-199
-1.93
iTransformer
w/ IF
1.9
3.5
0.35
3:25
-222
-350
-359
-125
-125
106
359
389
25
75
125
335
SegRNN
w/ IF
3.5
3.53
2700
0.
3.9
45
-199
PatchTST
w/ IF
022
3.20
-352
-35
-126
-1.0
1384
ide
200
Crossformer
w/ IF
0.5
0.-
-325
-330
-100
-1a
100
350
200
125
201
Figure 5: Visualization of input- 1ol------ouuut-- results on the ECL dataset
26

---

TimeXer
w/ IF
33
03
-13
M M
-13
126
369
203
125
390
159
293
iTransformer
w/ IF
yonry mypple
179t
1 M M
133
1.
SegRNN
w/ IF
4.
0.5
-1.
PStchTST
w/ IF
3.3
0.3
-15
-1.
TimesNet
w/ IF
mardy pod
3.3
0.3
Figure 6: Visualization of input-96-outpann results the PEMS03 dataset
27

---

D.2 Decoder Prediction Visualisations
We further provide forecast visualisations of the Transformer equipped with different decoders (i.e., Implicit Forecaster, Linear Forecaster, MLP Forecaster, and Transformer Decoder). The results on the ETTh2 and ILI datasets are shown in Figure 7 where IF produces visibly more accurate and smoother predictions compared to the other decoders.
IF
Linear
2.3
1.
3.
0.0
0.3
200
MLP
Transformer
2.0
1.5
3.5
0.3
Figure 7: Visualization of 9 results on the ETTh2 dataset
IF
Linear
Grandiuth
Greandrudt
MIPP
Transformer
GreasdTudh
M
Figure 8: Visualization of input- 16 36-prccc results on the ILI dataset
E
Full Results
Table provides the complete multivariate time series forecasting results of all baseline models across 14 datasets, along with their enhanced versions equipped with the proposed Implicit Forecaster (w/ The results show that IF brings consistent and substantial performance improvements in most
IF).
28

---

cases. The more pronounced gains in MSE compared to MAE also indicate that IF enhances the model's global views of the future time series, leading to more accurate forecasts.
Table 8: The comprehensive results of multivariate time series forecasting. The lookback window length is set to T 36 for the ILI and T = 96 for the others. Avg means the average result from all four prediction lengths.
Modets
SOFTS (2000)
IF
TaneNa (2004)
IF
ScaRNN (3005)
PachTST
w
Cuaaamee
w/FF
w/
Merri:
MSS: MAR:
VSS MAT
MS: MAR
MST
MAT
MSE MAP
MSS MARE] MSR MAR MSS: MAR
MSTE
MAI
MSI
MAR
MAR
MATE
MSR
MAT:
NSSI MAA
0.519
0,510
0.528
0.541
0:557
0.533
0.517
0.552
0.640
0.641
0:606
0.553
700
0/007
9.618
9,412
0.007
0.582
0.750
0.519
8:506
8.516
0.525
8:510
1.555
0,525
0.535
0.538 0.523 0:523 0,228 (U555
01522
0.215
0:511
0.500
8:587
0:564
[1556
11:500
11.055 01.54
(0.159
0,25310.153
8.171
192
0.354
0.154
0.252
0.187
0.153
(1200)
576 (0.303 0:130
0.195
0.215
0.117
8.209
0.322
0.311
0.212
0:200
0.112
1965
(1256
0.251
0.171
01716
0.352
0.322
0.355
9.240
0.:42
0.261
0.967
8:219
0.341
0.257
0.244
0.346
(1242
0.250
(1467
0.386
9.413
(1257
(136)
11.2% 01549
[Awg/0.202
0.315 0,195
0.365
0.214 0,317 0,154
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29


Reviews:
[{"review_id": "sFwjTY6k0u", "reviewer": "Reviewer_Cqck", "summary": "This paper studies the problem of time series forecasting, and proposes an implicit forecaster to predict decomposed constituent waves. The proposed method addresses the overlooked forecasting phase, and the empirical performance is demonstrated on various real-world datasets.", "questions": "Is there any other intuition why the proposed decoding strategy is better beyond DFT properties?\n\nMaybe I missed it, how is the frequency pool determined? Is the performance sensitive to the its size?\n\nMinor: It would be better to repeat the experiments with different runs to exam the robustness of the results and report \"error bars\".", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is generally well-written and easy to follow. The idea is intuitive and focuses on the decoding phase of time series forecasting, which is less explored than the encoder. \n\nWhile the method is intuitive and demonstrated empirically, the paper lack of theoretical justification, which limits the applicability of this method. More discussion would enhance the significance of this work, and guide the application more broadly.\n\nThe proposed implicit forecaster is plug-and-play, and leads to an inherent dependence and potential sensitivity to the quality of the encoder. It would be better to have a thorough investigation on the robustness with mediocre encoders.", "was_revised": true, "final_justification": "Most of my concerns are addressed, I therefore increased my rating to 4.", "rebuttal": "Dear Reviewer Cqck. Thank you for your insightful and valuable comments. We would like to address each of your concerns in detail below.\n\n---\n\n> **W1. While the method ... application more broadly.**\n\nWe appreciate your feedback regarding the theoretical justification of our work. Below, we discuss a theoretical advantage of the proposed **Implicit Forecaster (IF)** in the long-term forecasting setting:\n\nLet $X \\in \\mathbb{R}^{T \\times N}$ denote the historical input time series with $N$ variates and length $T$, and $Y \\in \\mathbb{R}^{L \\times N}$ denote the ground-truth future series over horizon $L$. We denote the DFT of each variate $n \\in \\\\{1, \\dots, N\\\\}$ by $z^{(n)} = \\mathcal{F}(Y\\_{:,n}) \\in \\mathbb{C}^{L}$. IF reconstructs predictions as:\n$$\n\\hat{Y}\\_{:,n} = \\mathcal{F}^{-1}(\\hat{A}^{(n)} \\cdot e^{j\\hat{\\phi}^{(n)}})\\_{:L},\\quad\\forall n \\in \\\\{1, \\dots, N\\\\},\n$$\nwhere $\\hat{A}^{(n)} \\in \\mathbb{R}\\_{\\geq 0}^{P}$ and $\\hat{\\phi}^{(n)} \\in [-\\pi, \\pi]^P$ are the predicted amplitudes and phases for variate $n$, with frequency pool size $P \\geq L$. We formalize IF’s improved stability and conditioning compared to point-wise decoders.\n\n**Theorem 1** (Decoder Conditioning Advantage)**.** *Let $Y\\_{:,n} \\in \\mathbb{R}^L$ be the ground-truth future series for variate $n$, and let $h \\in \\mathbb{R}^D$ be a shared latent representation. Suppose:*\n\n- *A point-wise decoder (pw) predicts $\\hat{Y}\\_{\\mathrm{pw},:,n} = \\Phi^{(n)}h$ with decoder matrix $\\Phi^{(n)} \\in \\mathbb{R}^{L \\times D}$;*\n- *An implicit decoder (IF) predicts $\\hat{Y}\\_{\\mathrm{IF},:,n} = G \\hat{z}^{(n)}$ using a **unitary** inverse DFT matrix $G \\in \\mathbb{C}^{L \\times L}$ (i.e.,* $G^\\*G = I\\_L$*) and learned complex coefficients $\\hat{z}^{(n)} \\in \\mathbb{C}^L$;*\n- *The latent representation is perturbed by noise:* $h = h^* + \\eta$*, where $\\eta \\sim \\mathcal{N}(0,\\sigma^2 I\\_D)$;*\n\n*Assume the decoders are unbiased:* $\\Phi^{(n)} h^* = Y\\_{:,n}$ *and $G z^{(n)} = Y\\_{:,n}$. Then, the expected prediction error satisfies:*\n$$\n\\mathbb{E} \\left[ \\\\| \\hat{Y}\\_{\\mathrm{pw},:,n} - Y\\_{:,n} \\\\|^2 \\right]\n\\geq \\sigma\\_{\\min}^2(\\Phi^{(n)}) \\cdot \\mathbb{E} \\left[ \\\\| \\hat{Y}\\_{\\mathrm{IF},:,n} - Y\\_{:,n} \\\\|^2 \\right],\n$$\n*where $\\sigma\\_{\\min}(\\Phi^{(n)})$ is the smallest singular value of $\\Phi^{(n)}$.*\n\nThe theorem shows that the error of the point-wise decoder is lower bounded by a factor of our IF decoder’s error, where the factor is the smallest singular value of the point-wise decoder’s weight matrix. If this value is small (i.e., the decoder is ill-conditioned), the bound becomes loose, so the theorem does not guarantee that IF always performs better. However, the key advantage of IF lies in its prediction stability: since it uses a unitary inverse DFT matrix, the decoder does not amplify latent noise. In contrast, point-wise decoders can suffer significant error amplification if not well-conditioned. Thus, while **Theorem 1** alone doesn’t prove IF is always superior, it highlights the robustness of IF in noisy or ill-conditioned settings.\n\n*Proof.* We analyze the variance of prediction error under noise. For the point-wise decoder:\n$$\n\\hat{Y}\\_{\\mathrm{pw},:,n} = \\Phi^{(n)} (h^* + \\eta) = Y\\_{:,n} + \\Phi^{(n)} \\eta,\n$$\nso the expected squared error is:\n$$\n\\mathbb{E} \\left[ \\\\| \\hat{Y}\\_{\\mathrm{pw},:,n} - Y\\_{:,n} \\\\|^2 \\right]\n= \\mathbb{E} \\left[ \\\\| \\Phi^{(n)} \\eta \\\\|^2 \\right]\n= \\sigma^2 \\cdot \\mathrm{Tr} \\left( \\Phi^{(n)} (\\Phi^{(n)})^\\top \\right).\n$$\nFor the IF decoder, let $\\hat{z}^{(n)} = z^{(n)} + \\tilde{\\eta}$ where $\\tilde{\\eta} \\in \\mathbb{C}^L$ has zero-mean independent components with $\\mathbb{E}[\\tilde{\\eta}\\tilde{\\eta}^\\*] = \\sigma^2 I\\_L$. Then:\n$$\n\\hat{Y}\\_{\\mathrm{IF},:,n} = G (\\hat{z}^{(n)}) = G (z^{(n)} + \\tilde{\\eta}) = Y\\_{:,n} + G \\tilde{\\eta},\n$$\nand since $G$ is unitary:\n$$\n\\mathbb{E} \\left[ \\\\| \\hat{Y}\\_{\\mathrm{IF},:,n} - Y\\_{:,n} \\\\|^2 \\right]\n= \\mathbb{E} \\left[ \\\\| G \\tilde{\\eta} \\\\|^2 \\right]\n= \\mathbb{E} \\left[ \\tilde{\\eta}^* G^* G \\tilde{\\eta} \\right]\n= \\mathbb{E} \\left[ \\tilde{\\eta}^* \\tilde{\\eta} \\right]\n= \\sigma^2 L.\n$$\nNow for the point-wise decoder, we have:\n$$\n\\mathrm{Tr} \\left( \\Phi^{(n)} (\\Phi^{(n)})^\\top \\right)\n= \\sum\\_{i=1}^L \\sigma\\_i^2 \\geq L \\cdot \\sigma\\_{\\min}^2(\\Phi^{(n)}),\n$$\nwhere $\\sigma\\_i$ are the singular values of $\\Phi^{(n)}$. This implies:\n$$\n\\mathbb{E} \\left[ \\\\| \\hat{Y}\\_{\\mathrm{pw},:,n} - Y\\_{:,n} \\\\|^2 \\right]\n\\geq \\sigma^2 L \\cdot \\sigma\\_{\\min}^2(\\Phi^{(n)})\n= \\sigma\\_{\\min}^2(\\Phi^{(n)}) \\cdot \\mathbb{E} \\left[ \\\\| \\hat{Y}\\_{\\mathrm{IF},:,n} - Y\\_{:,n} \\\\|^2 \\right].\n$$\nThis completes the proof.\n\nWe have provided more theoretical analysis of our work. Please kindly refer to the response: **Reviewer P8Px, W2, W3P1 and W3P3**.\n\n> **W2. The proposed implicit ... with mediocre encoders.**\n\nThank you for this valuable suggestion. As part of our robustness study, we have applied the proposed IF module to a previous time series encoder, **Informer** [1]. This encoder can be reasonably considered as **mediocre** compared to recent SOTA encoders. The corresponding results are presented below:\n\n| Informer (MSE) | Original | w/ IF |\n| - | - | - |\n| ETTh1 | 1.165 | **0.697** |\n| ETTh2 | 0.321 | **0.290** |\n| ETTm1 | 0.717 | **0.618** |\n| ETTm2 | 0.242 | **0.191** |\n| ECL | 0.353 | **0.262** |\n| Weather | **0.589** | 0.620 |\n| Exchange | **1.523** | 1.544 |\n| ILI | 6.075 | **5.106** |\n| PEMS03 | 0.162 | **0.122** |\n| PEMS04 | 0.121 | **0.106** |\n| PEMS07 | 0.185 | **0.126** |\n| PEMS08 | 0.150 | **0.115** |\n| Solar | 0.224 | **0.206** |\n| Promotion | | **16.9%** |\n\nThe above results demonstrate that our IF module can consistently improve Informer's performance across different datasets, confirming its robustness and general applicability in mediocre time series encoders.\n\n[1] Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting. Proceedings of the AAAI Conference on Artificial Intelligence. 2021.\n\n> **Q1. Is there any ... beyond DFT properties?**\n\nIndeed, there is another intuitive explanation that supports the effectiveness of the proposed decoding strategy. Our IF decoder employs a **pattern-separated** prediction strategy, enabling the model to focus on simpler, more predictable underlying dynamics rather than the complex, intertwined original series. In our approach, the key point is that these dynamics are represented as frequency components, such that **each output neuron is specialized in predicting one particular wave at a certain frequency**. Unlike previous point-wise decomposition forecasting methods, which cannot reliably guarantee that their predictions align accurately with trend-seasonal components, IF has **component distinctness**. This is because our frequency-based prediction strategy inherently ensures that the predicted components have well-defined, distinct periodic patterns (i.e., low frequencies correspond to trends, and high frequencies represent seasonalities), offering better performance and interpretability.\n\n> **Q2P1. Maybe I missed ... frequency pool determined?**\n\nWe apologize for not clearly describing how the frequency pool size is determined, as we set it as a **hyperparameter**. As discussed in Paragraph **Frequency Pool**, the pool size $P$ must be greater than or equal to the forecasting length $L$ to ensure signal reconstruction completeness. Therefore, in our experiments, we set the pool size to a default value of $P=720$ for almost all cases, as this size sufficiently covers the length of major fluctuations observed across all datasets and is greater than most forecasting lengths (Note that the maximum forecasting length is 720.). We have clarified this point in our revised version of the paper.\n\n> **Q2P2. Is the performance ... the its size?**\n\nThe forecasting performance is sensitive to IF's pool size. As explained in Paragraph **Frequency Pool**, we believe that a larger pool size can benefit the model from finer frequency granularity. This allows the predicted frequencies to align closer with the frequencies that best fit the dataset, thereby alleviating the **Spectral Leakage** issue (We have clearly demonstrated and discussed this point in the appendix of the revised paper.). Moreover, a larger pool size enables IF to have access to ultra-low-frequency components, enhancing predictions of long-term patterns. Below, we present the MSE results of shrinking the pool size from its default $P=720$ to the prediction length $L$:\n\n| Pool Size (MSE) | P | L |\n| - | - | - |\n| ETTh2 | **0.188** | 0.191 |\n| Exchange | **0.286** | 0.339 |\n| ILI | **2.387** | 2.941 |\n| PEMS-08 | **0.103** | 0.145 |\n| Performance | **100%**  | 83.7% |\n\nThe table shows that reducing the pool size leads to a noticeable decline in performance, indicating that a broader frequency pool can enhance IF’s ability to access lower frequency components and represent diverse waveforms, facilitating better performance.\n\n> **Q3. Minor: It would ... report \"error bars\".**\n\nThank you for pointing this out. In fact, we have already tested the robustness of our model across five runs with different random seeds and reported the corresponding error bars in the appendix of our paper (see **Supplementary Material**). We kindly direct your attention to this supplementary information.\n\n---\n\nWe have revised our paper by adding the above theoretical results and experimental analysis. We hope our responses and clarifications addressed your concerns, and we are happy to deal with any further questions."}, {"review_id": "f35wVryjmG", "reviewer": "Reviewer_P8Px", "summary": "This paper proposes a novel decoding module, Implicit Forecaster (IF), which decomposes time series into frequency components (amplitude, phase, frequency) for implicit prediction and reconstructs future series via inverse DFT. Experiments show IF consistently boosts mainstream models (e.g., 23.9% MSE reduction on PEMS) and achieves SOTA on 13 real-world datasets, especially for long-term and mixed-frequency patterns.", "questions": "Seen in the weak part", "limitations": "Seen in the weak part", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "strong\n(1) It introduces a novel innovative decoding: Implicit Forecaster (IF) replaces point-wise prediction with frequency-based implicit modeling, addressing the lack of global views. And it could seamlessly integrates with Transformers, RNNs, CNNs, etc., with significant improvements (e.g., 16.2% avg. MSE drop on Crossformer).\n(2) By setting P≥L, IF directly models ultra-low-frequency waves, enhancing long-term forecasting. It also gives the interpretability analysis to reveal key frequencies learned by the model.\n(3) It covers 13 datasets and 7 baselines, validating generality (e.g., 10.9% MAE improvement on SegRNN). And rigorous ablations (decoder replacements, skip-connection, window length) isolate each design choice.\n(4) The O(P) time and less memory than a Transformer decoder while improving accuracy.\n\nweak:\n(1) Lacks theoretical comparison between frequency vs. point-wise prediction.\n(2) Theoretical analysis is shallow, the convergence or approximation error of implicit reconstruction is not explored. \n(3) There is no adaptive frequency selection because it needs to predict all frequencies, which may be inefficient. And manual P setting may not match optimal data specific frequencies. While sin/cos decomposition stabilises phase prediction, it fails to model abrupt phase shifts (e.g., traffic congestion spikes in PEMS), causing residual errors in high frequency components (Figure 4).\n(4) Channel-independent forecasting overlooks inter-variable frequency correlations (e.g., synchronized power grid fluctuations), potentially losing multivariate dynamics.", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer P8Px. We sincerely thank you for recognizing our work and for giving valuable, thorough reviews. We provide detailed responses to each of your points as follows:\n\n---\n\n> **W1. Lacks theoretical comparison ... vs. point-wise prediction.**\n\nMany thanks for the insightful feedback on the theoretical aspects of our work. We have provided theoretical results demonstrating that, compared to conventional point-wise decoders, the proposed Implicit Forecaster offers better decoder conditioning and stability under noise. Please kindly refer to **Theorem 1** in response: **Reviewer Cqck, W1**, for details.\n\n> **W2, W3P1. Theoretical analysis is ... may be inefficient.**\n\nWe appreciate your concerns regarding these valuable points. We agree that our current framework lacks mechanisms for adaptive frequency selection, which may limit the forecasting efficiency. Nevertheless, it is possible to directly discard less contributive frequencies (e.g., ultra-high-frequency waves) from IF's frequency pool for improving efficiency. To support this, we have established a theoretical foundation demonstrating that our IF decoder offers precise control over approximation and prediction errors from frequency truncation. Below, we quantify reconstruction errors of IF:\n\n**Theorem 2** (Approximation Error from Frequency Truncation)**.** *Let $Y\\_{:,n} \\in \\mathbb{R}^L$ be a real-valued time series and let $z^{(n)} = \\mathcal{F}(Y\\_{:,n}) \\in \\mathbb{C}^L$ be its Discrete Fourier Transform (DFT). Let $S \\subseteq \\\\{0, 1, \\dots, L-1\\\\}$ denote the frequency indices retained by the model, and define the truncated spectrum:*\n$$\n\\hat{z}\\_k^{(n)} = \n\\begin{cases}\nz\\_k^{(n)} & \\text{if } k \\in S, \\\\\\\\\n0 & \\text{otherwise}.\n\\end{cases}\n$$\n*Let $\\hat{Y}\\_{:,n} = \\mathcal{F}^{-1}(\\hat{z}^{(n)})$ be the reconstructed signal from the truncated spectrum. Then the squared reconstruction error satisfies:*\n$$\n\\\\| Y\\_{:,n} - \\hat{Y}\\_{:,n} \\\\|\\^2 = \\sum\\_{k \\notin S} |z\\_k^{(n)}|^2.\n$$\n\nThis result quantifies the exact loss incurred when we truncate the frequency pool in IF. The reconstruction error is just the sum of the squared magnitudes of the discarded frequency coefficients. IF provides transparent and controllable approximation: you can directly trade off forecasting efficiency and accuracy by selecting which frequencies to keep.\n\n*Proof.* Let $e := Y\\_{:,n} - \\hat{Y}\\_{:,n}$ denote the reconstruction error in the time domain. Since the inverse DFT is a linear transformation, and $\\mathcal{F}^{-1}$ is unitary up to a scaling constant, we can analyze the error in the frequency domain.\n\nDefine the error spectrum:\n$$\ne = \\mathcal{F}^{-1}(z^{(n)} - \\hat{z}^{(n)}),\n\\quad \\text{so} \\quad\n\\\\|e\\\\|\\_2^2 = \\left\\\\| \\mathcal{F}^{-1}(z^{(n)} - \\hat{z}^{(n)}) \\right\\\\|\\_2^2.\n$$\nWe use the unitary DFT, i.e., both $\\mathcal{F}$ and $\\mathcal{F}^{-1}$ include $1/\\sqrt{L}$, now apply **Parseval's identity** for the DFT:\n$$\n\\\\| \\mathcal{F}^{-1}(w) \\\\|\\_2^2 =  \\\\| w \\\\|\\_2^2\n\\quad \\text{for any } w \\in \\mathbb{C}^L.\n$$\nApplying this identity to the difference vector $z^{(n)} - \\hat{z}^{(n)}$, we obtain:\n$$\n\\\\|e\\\\|\\_2^2 = \\left\\\\| \\mathcal{F}^{-1}(z^{(n)} - \\hat{z}^{(n)}) \\right\\\\|\\_2^2\n= \\\\| z^{(n)} - \\hat{z}^{(n)} \\\\|\\_2^2.\n$$\nBy the definition of $\\hat{z}^{(n)}$, we have:\n$$\nz\\_k^{(n)} - \\hat{z}\\_k^{(n)} =\n\\begin{cases}\n0 & \\text{if } k \\in S, \\\\\\\\\nz\\_k^{(n)} & \\text{if } k \\notin S,\n\\end{cases}\n$$\nso:\n$$\n\\\\| z^{(n)} - \\hat{z}^{(n)} \\\\|\\_2^2 = \\sum\\_{k \\notin S} |z\\_k^{(n)}|^2.\n$$\nPutting it all together:\n$$\n\\\\| Y\\_{:,n} - \\hat{Y}\\_{:,n} \\\\|\\_2^2 = \\sum\\_{k \\notin S} |z\\_k^{(n)}|^2.\n$$\nThis completes the proof.\n\n> **W3P2. And manual P ... data specific frequencies.**\n\nIndeed, a manual $P$ can not precisely align with the optimal data specific frequencies, as stated in Section 4.3 **Limitation Analysis**. However, a larger $P$ can provide our IF decoder with a **finer frequency granularity**, which makes the predictable frequencies have the chance to align closer with the frequencies that best fit the dataset, thereby alleviating the **Spectral Leakage** issue (We have explicitly demonstrated and discussed this issue in the appendix of the revised paper.).\n\n> **W3P3. While sin/cos decomposition ... components (Figure 4).**\n\nThank you for raising this insightful observation. To address your concern, we theoretically show that our IF decoder is stable under amplitude and phase estimation/prediction errors.\n\n**Theorem 3** (Stability to Estimation Error)**.** *Let the implicit forecaster predict $\\hat{z}\\_k^{(n)} = \\hat{A}\\_k^{(n)} e^{j \\hat{\\phi}\\_k^{(n)}}$ for $k = 1, \\dots, P$, and let the ground truth coefficients be $z\\_k^{(n)} = A\\_k^{(n)} e^{j \\phi\\_k^{(n)}}$. Define:*\n$$\n\\delta\\_A^{(n)} := \\\\| \\hat{\\mathbf{A}}^{(n)} - \\mathbf{A}^{(n)} \\\\|\\_2, \n\\quad\n\\delta\\_{\\phi}^{(n)} := \\\\| \\hat{\\boldsymbol{\\phi}}^{(n)} - \\boldsymbol{\\phi}^{(n)} \\\\|\\_2,\n\\quad\nA\\_{\\max}^{(n)} := \\max\\_k |A\\_k^{(n)}|.\n$$\n*Then, the total error satisfies:*\n$$\n\\left\\\\| \\mathcal{F}^{-1}(\\hat{\\mathbf{z}}^{(n)}) - \\mathcal{F}^{-1}(\\mathbf{z}^{(n)}) \\right\\\\|\\_2 \n\\leq \\delta\\_A^{(n)} + A\\_{\\max}^{(n)} \\cdot \\delta\\_{\\phi}^{(n)}.\n$$\nThis theorem shows how small errors in amplitude and phase estimates translate into total reconstruction error. The bound is linear in the errors, and decouples the contributions of amplitude and phase. This proves that IF degrades gracefully with bounded amplitude/phase prediction errors, with stability improving as $P$ increases.\n\n*Proof.* We decompose the per-frequency error:\n$$\n\\hat{z}\\_k^{(n)} - z\\_k^{(n)} = \\underbrace{(\\hat{A}\\_k^{(n)} - A\\_k^{(n)}) e^{j \\hat{\\phi}\\_k^{(n)}}}\\_{\\text{amplitude error}} + \\underbrace{A\\_k^{(n)} \\left(e^{j \\hat{\\phi}\\_k^{(n)}} - e^{j \\phi\\_k^{(n)}}\\right)}\\_{\\text{phase error}}.\n$$\nApplying the triangle inequality for complex magnitudes:\n$$\n\\left| \\hat{z}\\_k^{(n)} - z\\_k^{(n)} \\right| \\leq \\left| \\hat{A}\\_k^{(n)} - A\\_k^{(n)} \\right| \\underbrace{\\left| e^{j \\hat{\\phi}\\_k^{(n)}} \\right|}\\_{=1} + \\left| A\\_k^{(n)} \\right| \\left| e^{j \\hat{\\phi}\\_k^{(n)}} - e^{j \\phi\\_k^{(n)}} \\right|.\n$$\nUsing the sharp inequality $ |e^{j\\alpha} - e^{j\\beta}| \\leq |\\alpha - \\beta| $ (which holds because the chordal distance on the unit circle is bounded by angular distance):\n$$\n\\left| \\hat{z}\\_k^{(n)} - z\\_k^{(n)} \\right| \\leq \\left| \\hat{A}\\_k^{(n)} - A\\_k^{(n)} \\right| + \\left| A\\_k^{(n)} \\right| \\left| \\hat{\\phi}\\_k^{(n)} - \\phi\\_k^{(n)} \\right|.\n$$\nDefine $\\mathbf{e}^{(n)} = \\hat{\\mathbf{z}}^{(n)} - \\mathbf{z}^{(n)}$. The vector norm satisfies:\n$$\n\\\\| \\mathbf{e}^{(n)} \\\\|\\_2 \\leq \\left\\\\| (\\hat{\\mathbf{A}}^{(n)} - \\mathbf{A}^{(n)}) \\odot e^{j\\hat{\\boldsymbol{\\phi}}^{(n)}} \\right\\\\|\\_2 + \\left\\\\| \\mathbf{A}^{(n)} \\odot (e^{j\\hat{\\boldsymbol{\\phi}}^{(n)}} - e^{j\\boldsymbol{\\phi}^{(n)}}) \\right\\\\|\\_2.\n$$\nFor the first term:\n$$\n\\left\\\\| (\\hat{\\mathbf{A}}^{(n)} - \\mathbf{A}^{(n)}) \\odot e^{j\\hat{\\boldsymbol{\\phi}}^{(n)}} \\right\\\\|\\_2 = \\| \\hat{\\mathbf{A}}^{(n)} - \\mathbf{A}^{(n)} \\|_2 = \\delta_A^{(n)}\n$$\nsince $|e^{j\\hat{\\phi}_k^{(n)}}| = 1$ for all $k$.\n\nFor the second term:\n$$\n\\left\\\\| \\mathbf{A}^{(n)} \\odot (e^{j\\hat{\\boldsymbol{\\phi}}^{(n)}} - e^{j\\boldsymbol{\\phi}^{(n)}}) \\right\\\\|\\_2 \\leq \\max\\_k |A\\_k^{(n)}| \\cdot \\\\| e^{j\\hat{\\boldsymbol{\\phi}}^{(n)}} - e^{j\\boldsymbol{\\phi}^{(n)}} \\\\|\\_2 \\leq A\\_{\\max}^{(n)} \\\\| \\hat{\\boldsymbol{\\phi}}^{(n)} - \\boldsymbol{\\phi}^{(n)} \\\\|_2\n$$\nwhere the final inequality uses $ |e^{j\\alpha} - e^{j\\beta}| \\leq |\\alpha - \\beta| $ component-wise. Thus:\n$$\n\\left\\\\| \\mathbf{A}^{(n)} \\odot (e^{j\\hat{\\boldsymbol{\\phi}}^{(n)}} - e^{j\\boldsymbol{\\phi}^{(n)}}) \\right\\\\|\\_2 \\leq A\\_{\\max}^{(n)} \\delta\\_{\\phi}^{(n)}.\n$$\nCombining both terms:\n$$\n\\\\| \\mathbf{e}^{(n)} \\\\|\\_2 \\leq \\delta\\_A^{(n)} + A\\_{\\max}^{(n)} \\delta\\_{\\phi}^{(n)}.\n$$\nSince the inverse DFT is unitary ($\\ell\\_2$-norm preserving):\n$$\n\\left\\\\| \\mathcal{F}^{-1}(\\hat{\\mathbf{z}}^{(n)}) - \\mathcal{F}^{-1}(\\mathbf{z}^{(n)}) \\right\\\\|\\_2 = \\\\| \\mathbf{e}^{(n)} \\\\|\\_2 \\leq \\delta\\_A^{(n)} + A\\_{\\max}^{(n)} \\delta\\_{\\phi}^{(n)}.\n$$\nThis completes the proof.\n\n> **W4. Channel-independent forecasting overlooks ... losing multivariate dynamics.**\n\nThis is an excellent suggestion! In our method, we improve the forecasting performance by enhancing the prediction's globality. However, when forecasting, it is also worthwhile to consider **the correlation between the variates of the output series**. This is undoubtedly a valuable research direction to pursue further.\n\n---\n\nWe have refined our paper by including the above theoretical results and wish our responses have addressed your comments. Once again, thank you for carefully reading our paper and kindly pointing out our oversights! We'd be happy to respond to further questions."}, {"review_id": "lGPek0Iulp", "reviewer": "Reviewer_rnsD", "summary": "The paper introduces a model-agnostic framework for time series forecasting that identifies prediction failure cases through uncertainty estimation and corrects them from local and global perspectives. The method aims to enhance predictive performance while maintaining extensibility, with experimental validation across multiple scenarios.", "questions": "- The paper does not use the widely adopted traffic dataset in this\nfield. Is this omission due to the dataset's large number of variables? Could the authors further analyze how the number of variables impacts the computational overhead of the frequency- domain method (e.g., the relationship between variable count and computation time/resource consumption)?\n- Some models (e.g., TimesNet) exhibit poor performance on certain datasets. Could the authors investigate the root causes of these performance differences? Are they driven by dataset characteristics, model architectural design, or other factors?\n- Further justification is needed to clarify how the proposed method differs from and improves upon prior scale-based decomposition approaches (e.g., TimesNet, TimeMixer). Could the authors supplement experiments on TimeMixer to validate the method's competitiveness?", "limitations": "yes.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths and Weaknesses: Strengths:\n- As a flexible decoding model, the method can adapt to various time\nseries forecasting models, exhibiting model-agnostic\n   characteristics;\n- Experimental results demonstrate effectiveness in forecasting tasks.\nWeaknesses:\n- The frequency-domain approach introduces additional computational overhead, which may be more pronounced in multi-variable datasets, but the paper lacks analysis on how variable counts affect computational costs;\n- The proposed new perspective essentially aligns with the conventional decomposition-based approach (e.g., decomposing time series into periodic components), a methodology already prevalent in works like TimesNet and TimeMixer[1]. The frequency-based MLP method (FIST[2]) is also not novel. The paper provides insufficient comparative analysis or case studies to justify its novelty;\n- No comparison with the strong baseline TimeMixer, which also employs decomposition-based methods.\n[1] Timemixer: Decomposable multiscale mixing for time series forecasting\n[2] FITS: Modeling Time Series with Parameters", "was_revised": true, "final_justification": "According to the authors’ rebuttal and additional experiments, I increase my score. Thanks for the additional explanation. Please revise your paper in the final version.", "rebuttal": "Dear Reviewer rnsD. Thank you for the time and effort you dedicated to reviewing our paper! Below, we provide detailed clarifications on each of your points.\n\n---\n\n> **W1. The frequency-domain approach ... affect computational costs;**\n\nThank you for raising this concern regarding computational complexity. Below, we show that our frequency-domain approach (IF) does not incur additional computational overhead compared to conventional point-wise decoders in multivariate settings.\n\nLet $ \\hat{Y} \\in \\mathbb{R}^{L \\times N} $ denote the predicted future time series over horizon $L$ with $N$ variates. We use the following standard asymptotic notation:\n\n- $\\mathcal{O}(g)$: upper bound up to constant factor, i.e., $f(n) \\in \\mathcal{O}(g(n))$ means there exist constants $c, n_0$ such that $f(n) \\leq c \\cdot g(n)$ for all $n \\geq n_0$.\n\nThen:\n1. **Time Complexity for Point-wise Decoders:**\n\n    Conventional point-wise decoders predict values individually for each variate at each time point, with different time points corresponding to different output neurons, and sharing the same neurons across variates. Thus, their computational complexity is directly proportional to both the forecasting length $L$ and the number of variates $N$, which is simply bounded by $\\mathcal{O}(N \\times L)$, as they must produce $N \\times L$ independent predictions.\n\n2. **Time Complexity for our IF Decoder:**\n\n    In contrast, our IF decoder predicts frequency-domain features, with a fixed frequency pool size $P \\geq L$ and $N$ variates. As discussed in Paragraph **Computational efficiency**, IF uses the symmetric properties of frequency-domain signals, resulting in $\\frac{3}{2}P + 3$ output neurons per variate, independent of the forecasting length $L$, **and shared across variates**. Therefore, the forward pass complexity of IF is simply bounded by $\\mathcal{O}(N \\times P)$. However, after predicting frequency-domain features, an inverse DFT (IDFT) is applied to reconstruct the time-domain signal. The IDFT per variate can be implemented efficiently with complexity $\\mathcal{O}(P\\log P)$ using FFT-based algorithms. Hence, for all $N$ variates, the complexity of IDFT is $\\mathcal{O}(N \\times P\\log P)$. Combining both terms, the total time complexity of IF = decoder + IDFT complexity = $\\mathcal{O}(N \\times P + N \\times P\\log P)$ = $\\mathcal{O}(N \\times P\\log P)$.\n\nComparing the time complexity of point-wise decoders and our IF decoder, we observe that both decoders scale linearly with the number of variates. Thus, our approach does not introduce additional computational overhead related to variate counts.\n\n> **W2. The proposed new ... justify its novelty;**\n\nWe appreciate your comments concerning the perspective of our method. However, we would like to clarify that the core novelty of our method does **not** lie in adopting a decomposition-based forecasting approach. As we stated in Lines 39-46 of Section 1 **Introduction**, our method is *motivated* by decomposition forecasting. Instead, our key contribution is to **improve the decoding stage of time series forecasting** and inspire further work on this point, which is often overlooked in previous methods. Specifically, the proposed IF decoder is designed to enhance the **globality and pattern clarity** of the predicted time series.\n\nCompared with prior decomposition-based approaches such as TimesNet and TimeMixer, our method offers several advantages:\n- **Global Prediction**: Since frequency features are computed for the entire time series, our IF decoder can **have better global views of the prediction**, which allows it to express long-range temporal patterns in a more compact and coherent form. In contrast, TimeMixer and TimesNet still make predictions in a point-wise manner.\n- **Precise Pattern Control**: We think previous decomposition-based forecasting methods can hardly guarantee that their predictions align accurately with trend-seasonal components. However, in our approach, since each output neuron in IF is specialized in predicting one particular wave at a certain frequency, it inherently **ensures that the predicted components have well-defined, distinct periodic patterns** (i.e., low frequencies correspond to trends, and high frequencies represent seasonalities), offering component distinctness.\n- **Efficiency**: Given that our method predicts underlying dynamics as frequency components, each component can be **compactly** represented by its amplitude and phase, thereby enabling more efficient forecasting. In contrast, previous decomposition-based approaches still rely on point-wise predictions for each component in the time domain, which leads to higher computational overhead.\n\nThanks for the references to prior frequency-based methods such as FITS. However, the core innovation with FITS is in learning the **mappings** of input-output time series in the complex plane using complex-valued linear layers, while our approach is focused on predicting encoded time series representations into **frequency features** of various dynamics to construct the future series (such dynamics can be longer than the future series), which is a critical distinction that contributes to the novelty of our approach. Furthermore, our method is compatible with a broad range of backbones and can be plugged into existing forecasting models.\n\n> **W3. No comparison with ... employs decomposition-based methods.**\n\nFor each model type, we select the **most competitive method** as the baseline. MLP-based models like TimeMixer have fallen behind SOFTS [1], which is why we did not choose TimeMixer as our baseline. Moreover, there is a compatibility issue between IF and TimeMixer. Given that **TimeMixer separately learns and predicts decomposed long-short term patterns with different predictors** (i.e., it has a list of encoder representations and a list of predicted patterns), we can hardly find a fused encoder representation (i.e., a unified `enc_out`) that is suitable to be fed into our IF decoder. We regret not being able to provide the corresponding experimental results.\n\n[1] SOFTS: Efficient Multivariate Time Series Forecasting with Series-Core Fusion. In The Thirty-eighth Annual Conference on Neural Information Processing Systems. 2024.\n\n> **Q1P1. The paper does ... number of variables?**\n\nBoth the Traffic dataset and the PEMS datasets are widely used time series forecasting datasets for transportation. However, the PEMS datasets have four different subsets (e.g., PEMS03, PEMS04, PEMS07, and PEMS08), which **cover a broader range of scenarios**, including varying road types, traffic densities, and sensor distributions. Moreover, the PEMS datasets exhibit **super long-term patterns mixed with rapid, short-term changes**, making them more suitable for evaluating a decoder’s ability to express ultra-long-range fluctuations and represent compounded temporal dynamics. Therefore, we chose the PEMS datasets to better demonstrate our model's strengths in handling more challenging forecasting scenarios.\n\n> **Q1P2. Could the authors ... computation time/resource consumption)?**\n\nWe have analysed the relationship between the variate count and computation time/resource consumption. Please kindly refer to the response: **W1.**.\n\n> **Q2. Some models (e.g., ... or other factors?**\n\nThank you for your valuable question. We would like to clarify that, for experimental fairness, we did not perform any hyperparameter tuning for the baseline models or the models augmented with IF throughout the entire experiments. Instead, to ensure consistency and accurately assess our approach, all models used the **same hyperparameters** before and after applying IF, which are provided by their original code repositories. This constraint could affect the performance of IF, making it sub-optimal. For example, TimesNet’s relatively poor performance on certain datasets (e.g., ETTh1 and ETTh2). We believe this is largely due to its hyperparameter configurations, where the model dimension and the feedforward dimension are set to 32, which is very small and will limit IF’s learning capacity.\n\n> **Q3P1. Further justification is ... (e.g., TimesNet, TimeMixer).**\n\nPlease kindly refer to the response: **W2.**.\n\n> **Q3P2. Could the authors ... the method's competitiveness?**\n\nPlease kindly refer to the response: **W3.**.\n\n---\n\nWe have clarified and discussed the above points further in the paper revisions and hope we have addressed your comments. If you have any further questions and comments, we'd be delighted to address them."}, {"review_id": "5HPdi2cI9Z", "reviewer": "Reviewer_ZFna", "summary": "This paper addresses a limitation in current time series forecasting (TSF) models: while existing methods mainly focus on effectively encoding historical data, they often overlook the decoding (forecasting) phase, which translates learned information into future predictions. To tackle this, the authors propose the Implicit Forecaster (IF) — a novel decoding module that improves the forecasting phase by predicting the future series as a combination of implicit frequency-based components. Instead of forecasting each future time point independently, IF models the time series globally by predicting the amplitude and phase of multiple constituent waves, leveraging Fourier analysis for signal reconstruction. Extensive experiments show that integrating IF into mainstream TSF models consistently improves their forecasting accuracy across diverse real-world datasets, setting new state-of-the-art results. Overall, the work highlights the importance of refining the forecasting stage in TSF pipelines and demonstrates the effectiveness of a frequency-domain approach to better capture global trends and periodic dynamics.", "questions": "1. I am quite curious about the scalability of this method. Could it be applied in some time series foundation models (e.g., \\[1\\]\\[2\\])? \n\n[1] Timer: Generative Pre-trained Transformers Are Large Time Series Models\n\n[2] Unified Training of Universal Time Series Forecasting Transformers", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The Implicit Forecaster is well motivated, combining frequency-domain analysis with implicit decomposition, and the design choices (e.g., amplitude and phase heads, skip-connections) are thoughtfully justified.\n\n2. The authors conduct extensive experiments on 13 diverse, real-world datasets, demonstrating consistent performance gains over strong baselines, which shows the method’s generality and practical relevance.\n\n3. The paper is well written, well structured, and easy to follow. The figures and tables effectively illustrate key points and results.\n\nWeaknesses: \n\n1. The choice of a fixed-size frequency pool might not be optimal across all domains. Some discussion on adaptive or learnable frequency selection could make the approach more flexible.\n\n2. The authors should add more visualization results to demonstrate the effectiveness of the model design. For example, I suggest the authors to provide the visualization results for both Frequency prediction and Amplitude prediction.", "was_revised": true, "final_justification": "I think authors have addressed my concerns. So I raise my score to 4 to support this paper.", "rebuttal": "Dear Reviewer ZFna. We appreciate your positive comments and valuable suggestions regarding our paper. We respond to each of your points and questions as follows.\n\n---\n\n> **W1. The choice of ... approach more flexible.**\n\nThank you for your insightful and thoughtful comment. We agree that the use of a fixed-size frequency pool may not be optimal across all the data domains. In fact, we are adding discussions on the possibility of learnable frequency selection in the appendix of the revised paper, where we explore directions such as:\n- **Gated Frequency Pool**: One natural idea is that we can scan through the entire training dataset and find the most significant frequencies under a fixed input length based on the average amplitude (a similar method has been used in Koopa [1]). Then, combined with the amplitude of each frequency of the current input, it is possible to use a simple neural network and a gating mechanism to filter out less contributive frequencies in the output, thereby deactivating the corresponding weights in AHead and PHead by masking to achieve an adaptive frequency pool. Note that the computational complexity of the gating mechanism must be lower bounded by the cumulative cost of the pruned frequencies; otherwise, the adaptive frequency selection will fail to bring any real efficiency improvement.\n\n[1] Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors. In The Thirty-seventh Annual Conference on Neural Information Processing Systems. 2023.\n\n> **W2. The authors should ... and Amplitude prediction.**\n\nWe have added more visualization results in the appendix of our revised paper to demonstrate both phase and amplitude predictions, including the results showing the effect of errors in phase prediction on the final time-domain output. However, we regret that we are unable to share the results at this time as NeurIPS guidelines explicitly prohibit any figures during the rebuttal period. Nevertheless, we assure the reviewers that these visualizations and the corresponding analyses will be included in the final version of our paper.\n\n> **Q1. I am quite ... models (e.g., [1][2])?**\n\nWe have reviewed the time series foundation models mentioned in the references. From the perspective of model architecture, our method can indeed be applied to such models. Since these models typically learn a unified time series encoder representation, which is well-suited to be used as inputs to our IF decoder for prediction. We are currently running experiments to verify this, but due to time limitations, we have not obtained the final results. Once the experiments are completed, the performance of IF on time series foundation models will be reported in the final version of our paper as \"further generality analysis\".\n\n---\n\nWe hope our clarifications addressed your concerns, and we are looking forward to answering any further questions."}]

Author rebuttal:

Meta-review: This work proposes a decoding module named Implicit Forecaster. It decomposes time series into frequency components (amplitude, phase, frequency) for implicit prediction and reconstructs future series via inverse DFT.  Instead of forecasting each future time point independently, the forecaster models the time series globally by predicting the amplitude and phase of multiple constituent waves, leveraging Fourier analysis for signal reconstruction. 

The paper received 4 reviews and the reviewers appreciated the overall quality of the work and the experimental results. Despite this there were several major points of contention that the reviewers pointed out:

* A lack of theoretical analysis.
* The choice of a fixed-size frequency pool might not be optimal across all domains. Also this might get inefficient for larger domains
* The work closely resembles conventional decomposition-based approaches and the differences between them and the proposed method was not very clear.

The rebuttal provided was detailed and the authors did a great job especially in providing the critical missing theoretical links. The new experiments on the Traffic data set and the comparison with TimeMixer was also well appreciated. Overall, all the reviewers agreed on accepting the paper and I concur. I request the authors to include the new results and the theoretical analysis from the rebuttal in the final version of the paper.
Decision: Accept (poster)