Rethinking Fourier Transform from A Basis Functions Perspective for Long-term Time Series Forecasting
Authors: Runze Yang
Longbing Cao
JIE YANG
li jianxun
Venue: NeurIPS 2024 poster (2024)
Keywords: Time series forecasting
Deep learning
Fourier transform
Frequency domain
Primary area: deep_learning_architectures
TL;DR: We propose a Fourier basis mapping model, leveraging the basis functions to provide more implicit frequency features while preserving the temporal characteristic.

Abstract: The interaction between Fourier transform and deep learning opens new avenues for long-term time series forecasting (LTSF). We propose a new perspective to reconsider the Fourier transform from a basis functions perspective. Specifically, the real and imaginary parts of the frequency components can be viewed as the coefficients of cosine and sine basis functions at tiered frequency levels, respectively. We argue existing Fourier-based methods do not involve basis functions thus fail to interpret frequency coefficients precisely and consider the time-frequency relationship sufficiently, leading to inconsistent starting cycles and inconsistent series length issues. Accordingly, a novel Fourier basis mapping (FBM) method addresses these issues by mixing time and frequency domain features through Fourier basis expansion. Differing from existing approaches, FBM (i) embeds the discrete Fourier transform with basis functions, and then (ii) can enable plug-and-play in various types of neural networks for better performance. FBM extracts explicit frequency features while preserving temporal characteristics, enabling the mapping network to capture the time-frequency relationships. By incorporating our unique time-frequency features, the FBM variants can enhance any type of networks like linear, multilayer-perceptron-based, transformer-based, and Fourier-based networks, achieving state-of-the-art LTSF results on diverse real-world datasets with just one or three fully connected layers. The code is available at: https://github.com/runze1223/Fourier-Basis-Mapping.

Rethinking Fourier Transform from A Basis Functions Perspective for Long-term Time Series Forecasting
Runze Yang 1.2 Longhing Cao²*, Jianxun Li', Jie Yang!
Department of Automation, Shanghai Jiao Tong University School of Computing, Macquarie University frunze.y y, jieyang, 1ijx}esjtu. edu. (runze. yang) (adgg.@hhrmmgm. .mq. edu. au, (longbing cao) edu. au
Abstract
The interaction between Fourier transform and deep learning opens new avenues for long-term time series forecasting (LTSF). We propose a new perspective to reconsider the Fourier transform from basis functions perspective. Specifically, the real and imaginary parts of the frequency components can be viewed as the coefficients of cosine and sine basis functions at tiered frequency levels, respec- tively. We argue existing Fourier-based methods do not involve basis functions thus fail interpret frequency coefficients precisely and consider the time-frequency relationship sufficiently, leading to inconsistent starting cycles and inconsistent series length issues. Accordingly, novel Fourier basis mapping (FBM) method addresses these issues by mixing time and frequency domain features through Fourier basis expansion. Differing from existing approaches, FBM (i) embeds the discrete Fourier transform with basis functions, and then (ii) can enable plug-and- play in various types of neural networks for better performance. FBM extracts explicit frequency features while preserving temporal characteristics, enabling the mapping network to capture the time-frequency relationships. By incorpo- rating our unique time-frequene features, the FBM variants can enhance any type of networks like linear, multilayer-perceptron-- Fourier-based networks, achieving state-of-the-art LTSF results diverse real- world datasets with just one or three fully connected layers. The code available at: https://githubb.com comm/uuzeco////e/////////a 1223/Fourier- F Basis-Mapping
transformer-
and
Introduction
Long-term time series forecasting (LTSF) is essential for applications with long-range sequences, which presents significant challenges including long-range temporal dependencies, and frequency- oriented global dynamics. Recently, deep neural networks (DNNs) have thrived to tackle LTSF challenges for the presence of hierarchical effects, varied outliers, and nonlinear dynamics. They use various DNN architectures including recurrent neural networks (RNNs) I 2
3
4 5 6 7
g
convolution neural networks (CNNs) 99] T0
I3 14, multi-layer perceptron (MLP) based networks 19
12
networks [15
16
18), Transformer-basee
20 21
22 23 24
25|
26
27
28
29
30
32 33 However, a recent study NLinear 34) shows that an embarrassingly simple normalized linear model can outperform most of the DNN-based methods. This raises the question: Would complex Transformer-ln
architecture necessarily lead to better LTSF performance? According to CrossGNN [35), DNN- based methods like Transformers suffer from the influence of noise signals and produce high attention scores for unexpected noises.
Cooresponding author
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Mapping Network
bs
Existing Methods:
Information transform paths
Existing Methods: Fourier-based Mapping
Roil
A
B
(
A
(BL)
Taing part
lonn
Frequency Space
Time Series
Inconsistenl Series
Length
Lodi-taac Window
Inconsistent Starling Cycles
Froquencc
Input Time Series
Ti
T diviible
Forccast Time Series
Mapping Network
Fou basi Supensior
HUP
Our Method: Fourier Basis Mapping (FBM)
par
FBM's:
information transform path
keag
C-C-0
Frequency and Time Space
Figure 1: Comparison of Existing Fourier-basse based Methods with Our Approach Fourier basis mapping. Existing methods primarily operate in the frequency domain, with the mapping focusing on frequency features. Our FBM simultaneously operates in both the time and frequency domains, with the mapping focusing on time-frequency features.
Thus, Fourier-based time series modeling emerges as new paradigm to remove noise signals by considering diverse effects hierarchically at different frequency levels. we rethink the Fourier transform from a basis functions perspective, the real and imaginary parts can be interpreted as the coefficients cosine and sine basis functions at tiered frequencies, respectively. However, existing Fourier-based methods do not involve basis functions, thus failing interpret frequency coefficients precisely and do not consider the free-freyynncy relationships sufficiently, as shown in Figure [ They face two main issues: inconsistent starting cycles and inconsistent series length issues. For instance, FEDformer (22), FreTS 18) FiLM [3). FITS (37], FGNet (28). and FL-Net (38) use the real and imaginary parts of frequency components as the input of the mapping network and conduct the mapping in the frequency space. From our new perspective, we find that the amplitude and
arctangent of the real and imaginary components carry more explicit meanings than the components themselves. Furthermore, it is worth noting that the meanings of frequency components are bounded by the series length, and failing to establish the connections will lead to ambiguity in understanding the precise frequency in these models, e.g., Hz sine and cosine wave with series lengths 96 and 336 has distinct meaning. More importantly, Figure shows that Fourier basis function is dimendependent when the input length is not divisible by a certain frequency level, making it even more challenging for the model to correctly interpret those frequency components without the basis functions. Consequently, constructing the mapping in the frequency space is not enough and cannot capture time-frequency relationships, which has been ignored by existing Fourier-based methods. Although some methods like TimesNet (39) introduce time-frequent frequency features, these features are non-interpretable and complex, as the summation over the frequency dimension does not recover the original time series. We provide more detailed discussion of their limitations in Section
Accordingly, we propose the Fourier Basis Mapping (FBM) to address the aforementioned issues, involving learnable and non-leaeeable features. In the first stage, we embed the discrete Fourier transform with basis functions, referred as Fourier basis expansion, to extract frequunyy features. In the second stage, we validate its effectiveness using three types of mapping networks through three FBM variants (FBM-L, FBM-NL, FBM-NP) against four categories of LTSF baseline methods: (1) Linear method: NLinear; (2) Transformer-baset methods: iTrrnsformee 33 and PatchTST 20): (3) MI.P-based methods: N-BEATS [66), FreTS, and TimeMixer [5]: and (4) Fourier-based methods:
N-BEATS, FITS, FreTS, and CrossGNN, The experimental results reveal

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that Fourier basis expansion can enhance any types of DNNs, With time-frequency features, simple mapping networks (e.g. L-vanilla linear networks and NL-three-layer MLPs) can achieve the SOTA of L.TSF on eight real-world datasets and of STSF on PEMS datasets.
2 Related work
Recent studies have investigated the effect of the Fourier transform addressing LTSF challenges. Consequently, we categorize the relevant studies into two groups: (1) Fourier-based architectures, and (2) DNN-based architectures. the first category, we discuss the applications of the Fourier transform, while the second addresses innovative model architectures.
2.1 Fourier-based architectures
Fourier transform has been integrated into a wide range of network architectures, including CNNs [39]. recurrent networks [36], graph neural networks (GNNs) [35] 40), Transformer [22). and
MLP networks [8] 16 17.. However, by rethinking the Fourier transform from a basis functions perspective, we identify the inconsistent starting cycles and series length issues. In path (i) of Figure
1 methods like FEDformer
22 FreTT [8] FiL.M 36), FITS (37, FGNet (28, and FL.-Net [3X
use real and imaginary parts inputs to their mapping networks but the networks cannot easily interpret those coefficients because the crucial information stored in the amplitude, phase and
length of each cycle. In path (ii) of Figure methods like CrossGNN [35, and TimesNet [39 use the
discrete Fourier transform (DFT) to select the top-k amplitudes for noise filtering. However, a higher amplitude does not necessarily indicate a useful frequency and a lower amplitude is not necessarily useless. In addition, CrossGNN only stores the values of the first cycles without considering the effects of phase shifts and dependenn basis, while TimesNet introduces the multi- transform. which compromises the integrity of the temporal information. In path (iii) of Figure T methods like N-BEATS [16] and N-Hits [17] similarly be viewed forecasting through the inverse discrete Fourier transform (IDFT) by computing the output frequency spectrum of the time series, but their networks fail to capture time-dependent effects. In contrast, our method distinguishes itself from existing approaches by leveraging the Fourier basis expansion to provide a mixture of time and frequency domain features, thus avoiding the aforementioned issues.
Fourier
2.2 DNN-based architectures
Firstly, we introduce key techniques for innovative model architectures. [41], initial normalization is introduced to enhance the robustness. Autoformer [24] advocates the trend and seasonal decomposi- tion with the moving average kernel, consistent with other methods like DLinear [3 and TimeMixer [15]. However, the effectiveness of the decomposition relies the choice of kernel size, which is improved by our methods. Methods like VH-NBEATS (42 and BasisFormer [2 involve the pre-trained basis functions to account for hourly, daily, and weekly effects. In contrast, method uses the Fourier basis to capture those effects automatically. Transformer-based methods are one of the most popular LTSF architectures. Some transformer variants improve efficiency by reducing computational complexity and memory cost, such as LogTrans 43 Pyraformer [25] Informer (23]
and Autoformer 24) On the other hand, PStchTTT (20) introduces patching to treat period of local time steps as semantic vector, effectively reducing complexity and achieving state-of-the-ee results over Transformer-bases methods. In a recent study, iTransformer (33) suggests that employing attention layers to capture relationships between variables may yield better results. In this paper, we validate the effectiveness of Fourier basis expansion on linear, MLP, and Transformer based networks, proving that it can enhance any types of networks. The Fourier basis expansion can also be viewed as an enhanced version of trend and seasonal decomposition, as the basis functions, shown in Figure automatically decompose various effects by frequency levels.
3 Rethinking Fourier transform w.r.t. basis functions
In this section, we offer a new perspective of the Fourier Transform for LTSF, Firstly, we discuss the mathematical reasoning behind the DFT and IDFT from the perspective of basis functions. From this new perceptive, we find that the real and imaginary parts of the frequency components can be viewed as the coefficients of cosine and sine basis functions at tiered frequency levels. Consequently,
3

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the effect of the IDFT for frequency based mapping can be seen as analogous to the design of the N-BEATS seasonal block. As such, we identify the inconsistent starting cycles and inconsistent series length issues in existing studies.
Let X and Y be the input and output time series, respectively, and T and represent the look-back window and forecast horizon, assuming both of them are even numbers. H and H* refer to the frequency spectrum of the input and the output, respectively. Then, DFT and IDFT of the input time series X can be expressed as follows:
T-1
(-i2Ther)
H((k = DFT(X)
Σ
X(n]exx exp
k= 0,1,,..,
(I)
n=0
T-1
2 T
X(n] = IDFT(H) = T Hik
Σ
exp
n 0, o........,TT
(2)
From the perspective of basis functions, both DFT and IDFT can be expressed by T + orthogonal cosine basis functions and -/2 orthogonal sine basis functions. This is because the DFT of real-valued signal is Hermitian symmetric. Their proofs and further explanations can be found in Appendices B.1 and B.2 Subsequently, we can rewrite the IDFT w.r.t. basis functions, and the connection between X and H can be expressed as follows:
2
1
(2 T
2xkk )) n= -.0,.....T
X
Σ
ak
b& sin
T k=
T
HR [b]
0, T 2
H1 [k] = 0, k=
(3)
ak 12 HR[K],
k =
1
T
bk
2 [r(k),
k == 1
T
HR[k] and Hy[k] represent the real and imaginary parts of H[k] respectively, where k refers the frequency level, and H[} HR[K; +HHr[k]] Eq. ( provides an essential insight that the real and imaginary parts of the frequency spectrum can be interpreted as the coefficients of cosine and sine basis functions, respectively. Hence, computing the frequency spectrum of the output time series equivalent to computing the coefficients between cosine and sine basis functions, which can be considered analogous to the N-BEATS--casoe found in Appendix B.
design. Further explanation and proof's can be
Consequently, we identify two issues in existing Fourier-based studies: inconsistent starting cycles and inconsistent series length issues. The inconsistent starting cycles issue arises since the real and imaginary parts do not carry explicit meanings without the basis functions. This is because that adding sine wave and cosine wave with the same frequency results in shifted cosine with the same frequency, which can be expressed as follows:
(tt) A coss(utt B sin(wt) R cos(wt- o),
(4)
R = V A B2, & = aacctan (rrtan(B.40 A).
Therefore, the key information is embedded in the amplitude and arctangent of the real and imaginary values rather than the values themselves. The inconsistent series length issue arises since the meaning of frequency in hertz (Hz) is bound by the series length.
Figure 2 illustraate these two issues as Cases and II through manually generated time series with pure sine and cosine basis functions, assuming using time series X to forecast time series Y. In Case X and Y have the same frequency and series length but different starting cycles. If the real and imaginary parts of the frequency spectrum of the input are used to compute that of the output through independent channels, we cannot find such a mapping as there no mathematical solution. However, their relationships can be easily identified through the amplitude and arctangent of the real and imaginary values, which are embedded in basis functions. As shown in Figure 2 a mathematical solution exists which achievable through Eq. 44.
In Case II, X and Y have almost the same frequency and starting cycles but different series lengths. Although the landscapes of the frequency spectrum look similar, their components (in Hz) have

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Case I: Inconsistent Starting Cycles Issue
Case II: Inconsistent Series Length Issue
lik..... Boxs)
[Rby Paxs
Y bi
Our Perspective
Math Solation
With Explict Frequency
squieeleen 8
8ile
Previous Perspective
Solution
Without Rack Ambiguos Fraquecy
Irapery
Figure 2: Two Issues of Existing Fourier-based LTSF Models: Inconsistent Starting Cycles Issue and Inconsistent Series Length Issue. Case I, X and Y have a starting cycle gap of 104 over 336. In Case II, X and Y have different series lengths of 336 and 192, respectively.
distinct meanings. For instance, an 8Hz cosine function with a series length of 192 is equivalent to 14Hz cosine function with series length of 336. Unfortunately, such frequency connection is not provided by existing models. Hence, the precise interpretation of frequency components is challenging for the model due varying series lengths. The presence of time-dependent basis functions makes it even harder for the model to accurately interpret the frequency components. Instead, these issues can be solved easily by the incorporation of basis functions, which will be discussed in Section
FBM: Fourier basis mapping
To address two aforementioned issues, we introduce the Fourier basis mapping (FBM) Figure shows the architecture of FBM. The primary strength of FBM lies in constructing time- freeffreyenccy features, which capture explicit frequency information while preserving temporal characteristics. Subsequently, the downstream mapping considers the time-frequency space rather than solely the time or frequency space for forecasting.
First, we apply data normalization to enhance the robustness of FBM, the same as the implementation suggested in [4]] Then, we generate the frequency spectrum H using DFT. Next, we multiply the real part of H (denoted as HR) with the orthogonal cosine basis C and the imaginary part of H (denoted as Hy) with the orthogonal sine basis obtain the mixture of frequency and temporal
features. Let N = [0, ,,...,T T 1), then C and S can be expressed as follows:
2xN 7 [1, cos(
(T 1)TN
cos(
.cos(TN)],
(5)
1 10. 10,2
2mN ).... .2.22 ssin
(T 1)TN
S -
), sin(TTN)
T
5

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X gele
Fourier Basis Mapping
E i txt.
Nemoliiztton
Orbiggena Conine Recis
De-Normaliztoo
DFT
Decoder
NL
Hi e +0
Ortlogonal Saa Busis
H T +0
G C
R B
NP
Multayy @
Figure 3: Architecture of Fourier Basis Mapping (FBM): with three feature-output mapping methods, denoted as L. vanilla linear network, NL nonlinear MLP, and NP - nonlinear PStchTTST.
Further, we combine the cosine and sine basis functions to obtain the shifted starting cycles of the time series, forming G. The scalar 2 is derived in Eq. (4) for Hermitian symmetric. Thus, by summing over the 2 + 1 frequency domain, the model can recover the original time series without compromising the integrity of the temporal information. The process is analogous to decomposing the original time series into T + components with tiered frequency levels.
decoder to map the features G to the output time series. Our time- feeeefffruency features automatically decompose hierarchical effects by frequency levels, accounting for their interactive effects while filtering out associated noise through the mapping network. The Fourier basis expansion provides more effective and interpretable initial features, facilitating a simpler downstream mapping process. To this end, further propose three mapping methods: linear (L), nonlinear three-layer perception (NL), and nonlinear PatchTST (NP). Consequently, three FBM variants are generated: FBM-L with linear network. FBM-NL with MLP network, and FBM-NP with transformer-bice based network with patching. The first one includes a single vanilla linear layer. the second one consists of three fully connected layers with activation functions, and the last one follows the same structure as PStchTST, but we conduct the patching after the Fourier basis expansion. It is worth noting that FBM-NP employs fewer patches than PacthTST reduce complexity, making more efficient while delivering superior performance. Details of our three mapping networks, including hyperparameters and layer specifications, are provided Figure ( Table 5 in Appendix C
Finally,
5 Experiment results
5.1 Baselines and experiment settings
We compare FBM with eight baseline methods: NLinear, PatchTST, N-BEATS, CrossGNN, FED- former, FreTS, FiLM, and BasisFormer. These methods are chosen because they represent four categories of LTSF modeling methods: (1) Linear method: NLiiear; (2) Transformer- based methods: iTransformer and PatchTST; (3) MLP-bascd methods: N-BEATS, FreTS, and TimeMixer; and (4) Fourier-based methods: N-BEATS, FITS, FreTS, and CrossGNN. Furthermore, FBM is evaluated w.r.t. its three variants: FBM-L, FBM-NL, and FBM-NP. The mean squared error (MSE) and mean absolute error (MAE) are used as evaluation metrics. The look-back window is set to 336, and the forecasting horizons are set to 96, 192, 336, 720. In Appendix A.3] window of 336 is always associated with better results than that of 96 for most baseline methods. Eight datasets are used for the evaluation: ETThI, ETTh2, ETTm ETTm2, Electricity, Traffic, Weather, and Exchange. In addition, we also evaluate the STSF of four PEMS datasets: PEMS03, PEMS04, PEMS07, PEMS08. More details about these baselines and datasets be found in Appendix A. We split the ETT dataset into 12/4/4 months and the other datasets into training, validation, and test sets by the ratio of 6.5/1.5/2. Here, we use a smaller training set and a larger validation set compared to baseline methods to prevent early stopping and ensure a fairer comparison. With time- frequeeyy features, the model can achieve saate-ofthe-an performance for both LTSF and STSF most of the time using either one linear layer or three non- non-linear layers.
also show that look-back
5.2 Effects of FBM mechanisms for LTSF
We evaluate the forecasting performance of three FBM variants against eight diversified baselines incorporated by different architectures: Linear, MLP, Transformer, and Fourier-based architectures. Table T shows the quantitative results on eight datasets. While preserving simple structure, FBM variants achieve the best performance on most datasets across most prediction horizons.

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Table 1: Performance of FBM-L, FBM-NL, and FBM-NP in Multivariate Long-term Time Series Forecasting Compared to Eight Baseline Methods on Eight Datasets.
Meciel
FBM-L
FBM.SP
PacNTST
Trexhlan
K-BEATS
COSANNS
FITS
FreT5
XISC
56
0.355
0.290
0.365
0.395
0.22
0.335
0.911
0.435
0.314
0,590
11393
0.417
0.355
0.406
13387
0,410
0.576
0.400
0,365
1L222
0.454
DA2S
KTTH
11-23
01411
0.435
0415
11415
D-21
01425
0.417
C/222
14435
0.440
1422)
0,422
14425
13434
BA19
11427
11454
0.453
2/400
01413
0.430
0,225
0431
0433
0.43
0.431
0.445
0.451
0435
0.500
0448
0.447
0439
Sa12
0.35
730
04414
01435
04400
04439
D.445
0.453
0.442
0.433
14902
0.10
M4477
0.462
16450
(3473
BA47
14455
16431
14456
11356
2.55
0.271
0,391
0.277
0343
0220
0.340
0.253
0.342
0.216
0,358
0,303
0.362
0.275
0.359
0.303
0.963
0223
0.344
0.76
0.22
0.327
0.56
356
01321
0.376
0,392
0,394
0.354
06401
0.36
0.355
0,312
0,585
(1401
0.424
0.362
0.404
13350
0.407
0.351
0.435
0.308
0.372
0.499
0.09
11399
01412
0442
03596
01424
0.315
0.435
0.310
C.221
0.446
01395
0.433
14425
13465
11373
DAST
3.501
0.355
0.550
0.367
0,305
03337
0.364
0.324
0.362
01355
0.347
0.374
0.322
0.500
01345
0.583
0.355
0.385
(1363
0.508
0,235
0.399
0.233
0.355
0.359
386
0.371
0.394
0.399
5.355
0371
0.399
0,377
0.390
0.262
0,5%%
0.383
0.401
0.365
0.392
0.400
0.406
0.375
0.39
0.373
0.396
0.399
0.406
0:556
ITTTW?
192
00219
0.230
0.255
1.2%
02224
0..35
0.225
0.254
0.255
0,50
(1233)
0.311
11235
0.500
1X225
0.200
022
0.314
10222
0.250
0,522
0.271
0.522
01232
0.356
0.56
01354
0.363
5.66
0.396
0.336
0:511
0.365
0.506
0L374
0.02
0.300
0.376
0.354
0.222
0L302
0.366
0.431
54666
0.847
0.337
0132
0:200
01222
B.145
0.335
0:133
0.297
0.337
0.353
01131
0.250
0.340
01:07
0.366
01:45
0.247
0.315
0.545
0.155
0.248
0.140
0243
0149
0.342
5.557
0.250
0.151
0.244
0.135
0.340
0.133
0.245
0L138
0.352
01151
0.258
0.155
0.239
0.158
5.238
0.172
0.361
0.711
0.175
0.369
0.178
0.174
0.269
0.755
720
06212
0.297
0.22
0.205
0205
11215
0.214
0.299
0.210
0.297
ILU95
0.208
0.212
0.26
1L217
0.204
0.14
0L273
0L213
00301
0.229
0.315
0:571
0.711
0 134
0:96
0.755
0.753
D-025
0.788
0.181
0.77%
0.763
0.311
0.761
D:274
0.795
D:175
0.741
0:571
0:434
5.11
0.299
0.402
0.417
0.50
0.419
0.20
0.776
0:653
0.380
0.433
0.583
11:007
0.765
0:031
0.797
D.555
0.307
0:055
0.302
0.318
01243
0.433
720
0.477
0.309
0.45
0277
0242
0.231
0.422
0.337
0.454
5.2%
0.443
0.506
0.499
0.319
0.496
0.326
04496
0.313
0475
0,310
0.335
0.390
01:54
01%
0.701
B1.66
0:104
5:166
Weater
11203
0.347
0.194
0312
11245
0:231
0.362
0.200
0.216
IL204
0.11
14227
0.44
14227
12.39
11215
11200
11241
0.200
3270
T96
(1757
0.785
0.5
01745
0:745
0.766
0.753
0.785
0.755
0.751
D7756
10300
D.79)
017055
0:345
0.553
5.700
0.355
0.542
Eachungs
192
16325
0,339
0.210
0.336
1119
01312
3.315
0.331
0.31
0.525
(L241
0.555
11235
0.354
16312
0.406
111.5
11335
11222
14350
11346
2447
0.347
9.79
11537
0.533
0/943
0.31
0.583
0.732
1,400
0070
1412
0.507
0031
1.855
Avrrage
11223
0.329
0.235
0.34
1021
01281
0:005
0.314
0335
035
14:41
0.335
0.343
14300
0.311
11331
14:47
0.431
5.006
FBM-L vs linear network (e.g. NLinear): We compare FBM-L with the NLinear model, revealing improvement across all datasets and prediction horizons, reducing the average MSE and MAE from 0.333 and 0.349 to 0.323 and 0.339, respectively. This demonstrates the effectiveness of decomposing the original time series by Fourier basis expansion. The frequency domain features allow the model to distinguish noises from hierarchical effects. We find that the improvement is more notable on datasets ETTh1 and ETTh2 due to their richer frequency components and higher prevalence of noise signals. Furthermore, our experiments reveal that even a simple vanilla linear network FBM-L has the potential to outperform state-of-the-an DNN-based architectures. More explanation is provided in Section 5.3
FBM-NL vs MLP-based network and FBM-NP VS Transformer- network: FBM-NL performs better than TimeMixer and FBM-NP performs better than PatchTST at most of the time, where the former two are both MLP-based networks and the latter two are both Transformer-batt based networks with patching. The experimental results further demonstrate the effectiveness of Fourier basis expansion, suggesting that extracting time-frequee feneffruuency features efficiently is more advantageous than relying solely on deeper architectures. It is worth noting that TimeMixer also decomposes the original time series into trend and seasonal effects, but its performance largely depends on the choice of the moving average kernel size. In contrast, our Fourier basis expansion automatically decomposes various effects hierarchically within distinct frequency levels. Although FBM-NP uses fewer patches than PSt it still achieves better performance. Our FBM variants also consistently outperform other Transformer-bast based and MLP-based architectures, including iTransformer and N-BEATS.
FBM variants vs. Fourier-based network: We also observe that FBM variants make a significant improvement over all Fourier-based methods, including N-BEATS, FreTS, CrossGNN, and FITS. This discrepancy may largely be attributed to the inconsistent starting cycles and series length issues discussed in Section 3 Notably, FITS and CrossGNN also emphasize the importance of the amplitude of real and imaginary values, similar to FBM. However, they overlook the fact that Fourier basis functions are time-dependent when the input length is not divisible by a certain frequency level. Consequently, mapping in the frequency space cannot capture the time-frequency relationship effectively. For instance, CrossGNN retains only the values from the first cycle but fails to account for variations in the subsequent cycles as well as shifts in the phase of the cycles. In contrast, FBM
7

---

Table 2: Short Term Time Series Forecasting on Dataset PEMS with Forecast Horizon L 12.
Method
FRM-L
FBM-NI.
FBM-NP
NLineir
PaschTSTT
TimeMiser
iTranffrmer
RILM
TimesNet
Erur
MSC
MAB
MSE
MAE
MSC
MAE
MSE
MAE
MSE
MAD
MSE
MAB
MSC
MAB
MAPE
MAE
MSE
MAE
PEMS04
D3888
0.195
0.071
0.172
0.171
0,170
0.089
0.196
0.055
0.179
0.021
0.173
0.033
0.176
0,118
0.257
0.091
0.196
PEMSO8
0:011
0,180
0,000
0.159
0.061
0.19
0.081
0.190
0.062
0.165
0.060
0.160
0.02
0.165
0.108
0.236
0.034
0.234
PFMSO3
0.077
0.183
0.060
0.161
0.061
0162
0.078
0.184
0.065
0.174
0.022
0.164
0.011
0.163
0.108
0,223
0.089
0.199
PEMS07
13073
10130
0353
0.14%
0.054
0,150
01073
0,180
0U057
0.164
0.053
0.151
0.533
0.148
0.101
0.221
0,079
0,182
variants demonstrate better representation of fime-freuueney domain features, as evidenced by their superior performance. In the following sections, we empirically discuss the reasons for their effectiveness through visualization.
Although our model primarily focuses on addressing LTSF challenges, we also evaluate the perfor- mance of STSF on PEMS dataset in Table under the same settings as previous baseline methods. Our FBM variants consistently achieve the best performance in most cases.
5.3 Effect of weight on time-frequency features
In Figure we visualize the effect of the weights of the one-layer vanilla linear network to show how FBM-L considers the time-frequency relationship. We decompose the weights W into L time- specific parts, denoted W [Wo. o. Wi,.... WL-1]. where W, represents the impact of the time-frequency features the i-th time step of Y. We flatten the weights into array similar to G in Figure where the horizontal axis represents time, and the vertical axis represents frequency. Specifically, we visualize the heat maps of Wo for the ETThl, Electricity, and Traffic datasets, comparing them with the Fourier basis. Since the datasets Electricity and Traffic are more stable than ETTh1, the weights of the linear layer for the former datasets are closer the Fourier basis than those for the latter datasets. This implies that FBM-L considers more time umeefrequency relationships in ETTh1 than Electricity and Traffic, leading to more sigmificant improvement. The further visualization of the weights Wo. W10. W20. W Wa. Wan. and W50 for Electricity is provided in Figure 4 Appendix C It is not surprising to see that the heatmap shifts over time. conclusion, our approach allows the model to eliminate noises across a mixture of time and frequency domains and consider time-frequency relationships. In contrast, existing studies only eliminate noises in the frequency domain.
108
(8)
Tme
ETThl
Electricity
Traffic
Basis
Figure 4: Effect of Weight Wo Datasets ETTh1, Electricity, and Traffic.
5.4
Linear networks vs non-linear networks
We compare the performance of three FBM-variants on eight datasets, as shown in Table I FBP-NL and FBM-NP perform similarly across the datasets, as both are nonlinear. In contrast, FBM-L and FBM-NL reveal great performance differences on distinct datasets, which may be attributed their varied data characteristics. To further explain such differences, Figure shows the visualization of the frequency spectrum for a random input on the last seven dimensions across eight datascts. shows that trending effects usually appear on high-frequency levels, scasonal effects usually fall on intermediate frequency levels, and noise signals typically manifest on low-frequency levels. This explains why the Fourier transform is beneficial for LTSF, as it hierarchically separates various effects, with specific effects (e.g., hourly, daily, weekly) aligning with their corresponding frequency levels. For instance, there is consistently high energy the multiples of 14 on all hourly-leve data ETTh1, ETTh2, Electricity, and Traffic in Figure 5] This is attributed to the day effect falling into these frequency levels, given the repeating cycle of 24 with look-back window of 336.
By examining the frequency spectrums of all hourly-level datasets, we also find that ETTh1 and ETTh2 display much richer frequency spectrum compared to other datasets. Figures 8 and 9 in
8

---

Appendix C further show that the frequency spectrums of Electricity and Traffic are more stable than those of ETTh1 and ETTh2. This suggests that there a larger proportion of noise signals in datasets ETTh and ETTh2,, but less in datasets Electricity and Traffic. The model is more likely to overfit when the data is mixed with significant amount of noise. This is why FBM-L achieves better performance on datasets ETTh1 and ETTh2, while FBM-NL performs better on datasets Electricity and Traffic. On the other hand, the frequency spectrums for ETTml I and ETTm2 shift to the left, contrasting with those for ETThl and ETTh2. This is because the granularity of ETThl and ETTh2 four times larger than that of ETTml I and ETTm2. This leads to the actual forecast horizon of the time being reduced, resulting in less noise in the data. This explains why FBM-NL performs slightly better than FBM-L on ETTml when the granularity changes. This also applies to datasets with other levels of granularity, such the Exchange and Weather data. This observation validates original hypothesis that providing the frequency spectrum is insufficient, as the real-world frequency can be ambiguous for the model when changes in granularity series length alter the meanings of the spectrum. In conclusion, when a dataset consists a large proportion noise signals with a limited amount of data, simpler network is preferred, and vice versa.
63
1.0
ETTh1
BTTb2
Electricity
Traftic
02
04
18
-5
88
100
68-
ETTml
BTTm2
Weather
Exchange
Figure 5: Frequency Spectrum of A Random Input of Last Seven Dimensions on Eight Datasets.
Conclusion
We make the first attempt to theoretically and empirically discuss several issues associated with the existing research on the deep Fourier transform for LTSF from the perspective of basis functions. Our insights and findings reveal that the real and imaginary parts of the frequency spectrum can be interpreted as the coefficients of cosine and sine basis functions, disclosing two issues commonly appearing in existing studies: inconsistent starting cycles and inconsistent series length issues. We further propose Fourier basis mapping model Fourier basis mapping (FBM) to address these issues by extracting more efficient time-freyuency features that carry explicit frequency information while retaining temporal patterns. Extensive experiments empirically verify the effectiveness of the FBM approach, to enhance various types of mapping networks and achieve state-of-the-an performance with simple one or three layers. The visualization analysis also reveals that deep neural mapping can replace the effect of inverse discrete Fourier transform and allow for more flexible consideration of frequencyccc relationships. The ablation study further shows that extracting time-frequency features is more valuable than simply increasing the depth of the network. The primary focus of this study to refine Fourier-bact based methods in a way that is more reasonable and interpretable. The time- freeufryy features hold great potential for future time series analysis tasks by enhancing various types of mapping networks (e.g., linear, MLP, Transformer networks). For example, Fourier basis expansion can enhance the performance of Transformer-basse networks by only adjusting the initial projection layer after Fourier basis expansion. In future, we will also consider applying Fourier basis expansion CNN- and RNN-based networks, well as to tasks such as anomaly detection.

---

Acknowledgments and Disclosure of Funding
This research is partly supported by NSFC (No. 62376153), Australian Research Council (ARC) Linkage Grant LP230201022, the ARC Discovery Grant DP240102050, and the ARC Council Linkage Infrastructure, Equipment and Facilities (LIEF) Grant LE240100131.
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A Supplement materials for experiment results
A.1 Datasets
We conduct our experiments on !welve real-world datasets: ETT (ETThl, ETTh2, ETTml, ETTm2), Electricity 3) Traffic Weather Exchange rate (Exchange and PEMS (PEMS04, PEMSO8M, PEMS03, PEMS07). The granularity of ETTh1, ETTh2, Electricity, and Traffic is an hourly time scale; the granularity of ETTml and ETTm2 is at fffteen-miute time scale; the granularity of Weather is at a ten minmmntte time scale; the granularity of Exchange the granularity of PEMS is at a five-minute time scale. The ETTh datasets involve seven oil and load features of electricity transformers, spanning from July 2016 July 2018. Traffic comprises hourly road occupancy rates measured by 862 sensors in the San Francisco Bay area from 2015 to 2016. Electricity records hourly electricity consumption (in kWh) of 321 clients from 2012 to 2014. Weather includes 21 weather indicators for 2020 in Germany, such as air temperature, humidity, and so on. Exchange tracks the daily exchange rates of eight countries from 1990 to 2016. PEMS contains the public traffic network data in California with four public subsets.
daily time scale; and
A.2 Related work
NLinear 34 is a simple one-layer linear network with data normalization 41.. PatchTST (20
introduces a Transformer-based model for time series forecasting with two crucial improvements:
patching and channel independence. FEDformer [22] adopts a structure similar to Transformer but converts time-domain features into srqquency-domaine features through the Fourier transform.
CrossGNN (35] creates multi-scale identifier to address noise in time series by applying the Fourier transform and utilizes graph neural network (GNN) to study cross-variable interactions. FreTS [18] projects the time series into high dimensions and characterizes the frequency space by utilizing interaction channels for the real and imaginary parts with a nuatilayer-perceptron based network. TimesNet [39] transforms miedimensional time domain information into 2D tensor based multiple time scales through Fourier transform, and then utilizes the convolution neural network to
analyze the transformed information. N-BEATS (16) and N-HiTS 117 forecast by computing the coefficients of sine and cosine basis functions using multilayer perceptron network. FiLM [36] combines the power of the Fourier transform with the recursive memory unit (LMU) 44 where
LMU is introduced to provide robust representation for long time series through the Legendre projection, variant version of recurrent neural network. BasisFormer [2] computes basis functions through adaptive self-supervised contrastive learning and employs a Transformer-bant
architecture
study the connection between input time series and its basis functions. Autoformer 24 a model based on Transformer that employs decomposition architecture along with an auto-correlation mechanism to capture dependencies across time. Informer (33 introduces a probability sparse self- attention mechanism to study the temporal relationship. LogTrans 43 develops log sparse attention mechanism to improve the efficiency of Transformer-basse networks. Pyraformer [25) is Transformer-based model that learns a multi-resolltion representation of time series data using pyramidal attention module to capture cross-time dependencies for forecasting. iTransformer [3 applies attention study the inverted dimensions of the input sequence, capturing the dependencies between time series. TimeMixer [15) introduces a multi-scale decomposition using a bottom-up and top-down MLP network. FITS frequency domain.
37 proposes complex frequency linear interpolation to study the
A.3 Computational efficiency
In Table 3 and we evaluate the complexity of our method in comparison to other sophisticated approaches and discuss the influence of the look-back window the final prediction results. In Table we provide a comparison of the training time for one epoch cycle on the ETTh! dataset, as well
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13

---

Table 3: Hidden Layer Complexity and Average Training Time per Epoch the ETTh1 Dataset.
FBMAL
FBM-NP
NLinear
PochTST
N-BEATS
CrosSNNN
Thunstommer
TimeMierr
FIMM
TimesNet
Time
34.21
12.38
148.50
Complexny
(,5)
(+4)
003
((7)
022
D 0,725
(:2]
0/7
00)
0.7"
Table 4: The Effect of Look-back back Window on ETThI.
Model
FBM-LL
NLinear
PachTST
BEATS
CrosstiN
iTrnnsfrmmer
Timeblier
Frror
MAR
MAR
MSE
MAR
MSE
MAE
90.6
0.381
0.394
0.389
0.401
0.387
0.401
0.395
0.400
0.397
0.410
0.386
0.405
0.404
0.417
0.389
0.395
0.371
0.399
536/96
0.366
0.300
0.391
0.416
0.374
0.309
0.387
0.410
0.376
0.400
0.399
0.416
0:404
0.423
0.373
0.307
0.385
0.408
96/192
D.434
0.421
0.450
0.437
0.441
0.431
0.447
0.433
0.437
0.429
0.446
0.438
0.469
0.455
0.440
0.425
0.434
0.429
336/192
0,403
0.411
0.432
0.421
0.426
0.422
0.428
0.434
0.419
0.427
0:436
0.642
0.461
0.460
0.405
0.416
0:429
0.432
96/336
0-471
0.438
0.495
0.462
0.487
0.455
0.482
0.447
0:482
0.449
0.494
0.463
0.521
0.456
0.477
0.441
0.496
0.451
336/366
0.418
0.420
0,435
0.435
0.431
0.448
0.647
0.439
0.42
0.436
0.451
0.488
0.439
0.422
0.456
0.450
96/30
0.486
0.488
0:485
as an increase in complexity the layer. Our FBM model achieves the best prediction results at relatively fast training speed. In contrast, attention mechanisms typically suffer from higher memory costs and slower calculation speeds. All the experiments are conducted on a GeForce GTX 3090 GPU and an Intel(R) Core(TM) i7- 17-6850K CPU. In the past literature, substantial amount of work selects a look-back window of 96. In Table 4 we conduct a comparison of prediction results with back windows of 96 and 336 the ETThl dataset. Remarkably, the look-back window of 336 consistently yields better performance for almost every baseline method. While certain methods, such as Transformer, and FreTS, exhibit lower sensitivity to the look-back window, the majority experiences significant increase in both MSE and MAE when a look-back window of 96 used rather than that of 336.
B Supplemental materials for rethinking Fourier transform w.r.t. basis functions
B.1 Proof of real-valued discrete Fourier transform via basis functions
The discrete Fourier transform (DFT) and inverse discrete Fourier transform (IDFT) can be expressed by _ cosine basis funetions and T 1 sine basis functions. We provide the proof that the real-valued discrete Fourier transform can be represented by sine and cosine functions as follows:
T-1 x [n] exp
2 T
H(k) = Σ
-
n=0
F F-1
T
( 2m
nxpexp exp ( 2
-X[0] + (-
+
X exp
(-i
+ + -n]
2
T
T
#=1
T
20knn 2mkkn i isin( Xcns(cos(
2m ))
x + (-1)
2
T
n=1
(6)
T-1
27Tkin
2m
Σ
X n](cos((
isin(
T
T
T
2mkn
X(0] + cos(TT))
+
Σ
[1
X(T -n)) cos
2
T
2T )
Σ [x]n]
XIT - nl) sin
k == O....,
T
14

---

Then, we can deduce that the frequency spectrum is Hermitian symmetric, where H H - k).
B.2 Proof of real-valued inverse discrete Fourier transform via basis functions
The real-valued inverse discrete Fourier transform can be expressed by sine and cosine basis functions as follows:
T
2mkn
25kkn
2 2 ))
Xin]
H
exp
H
(cos((
+ i isnn
-2
2 1))
H kj(cos(
sin
7
T
F-1
2xkn T
2m })
(HR[0) + HR
(TS( +
Σ
HR
[k] + (H))))))sss)
+ isin(
T
T
kml
T-1
2
2
HR [k]
HH [k))s
isin(
)));
T
T
(7)
2
(2tkn
1
HR [k] cos
H1
[k] sin
+
(Hn
T
T
HR!
cos cos(TT))):
2xcnn
2mkkn
0g CO8
bk sin
n = O,...,
T
T
T
k=
HR [+),
k = 0, I
[H[[ 0,
k k= 0, T k
dş
2
[k]
k: = 1,....
T
bk
H (k),
1.
B.3 Equivalence between the IDFT and NBEAATS-Sasson--Blockk design for frequency side mapping.
We provide further explanation of why IDFT and SeasSnnlannl-BBa are equivalent for the frequency space mapping. The general structure of Fourier-base based method can summarized as follows:
H DFT(X),
H V = Encoder )
(8)
Y IDFT(H").
The encoder can refer to any deep neural architecture, such as a CNN-based network, RNN-based network, GNN-based network, Transformer-basee network, MLP-based network. Therefore, the model aims to make forecasting by mapping H to H However, if we rewrite the IDFT in obtain the following equation:
Equation 8 by Equation (4) and assume M
10............... ... L 1],
2mTM L
a H sin (TTMM)
(2TMM)
0g HoHHRKk [k]] cos
L
L
k==0
1, 0,
L 2
0
2,
1,..., 1.
This is equivalent to the N-BEAAT-SCasonall Seasonal-Blod design, their structure can be expressed as follows:
0 MLP(X),
2
2TiM L
(10)
008
0, sin
0,4 4+1"
1=0
15

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Table 5: The hyperparameter of FBM-NL, FBM-NP and PatchTST
ETTh1
ETTh2
ETTml
ETTm2
Electricity
Traffic
Weather
Exchange
PFMS
FBM-NL
1440/1400
14401440
1440/1440
1440/1440
1440/1440
1440/1400
14401440
1440/1440
720/660
FBM-NP
P/K
14/16
14/16
14/128
14/128
14/128
28/128
14/128
14/128
14/128
PachTST
P/K
42/16
42/16
42/128
42/128
42/128
42/128
42/128
42/128
42/128
In the N-BEATS model, they use a multilayer perceptron to compute the coefficients of sine and cosine waves, denoted as 0. Although the dimension of R4+2 is twice as large the dimension of HY E C if consider the real part and imaginary part of HY separately. then the dimension is exactly the same. As result, the objective of the deep Fourier-based method and the N-BEATS model in Equation (9) and Equation (10) are equívalent:
compute the coefficients of sine and cosine basis functions with tiered frequeney, as the scalar of basis function doesn't 't provide any difference in deep learning. The only difference is that the former one maps the frequency domain to the frequency domain, and the latter one maps the time domain to the frequency domain.
C Supplemental results to support experimental analysis
Figure is provided as supplementary material for Section 4 Figure FBM-L represents linear network, FBM-NL is three-layer MLP network with dropout and ReLU as the activation function, and FBM-NP has the same structure as PatchTST but differs in the number of patches and initial projection layer. All the rest hyperparameters for FBM-NP are the same as those for PatchTST. The key hyperparameters are provided T You can always tune them to achieve better results than us. Since validation set is used to determine when to early stop the training, adjusting hyperparameters based on the test set can undermine generalization. Additionally, tuning random seeds to achieve better results may compromise reproducibility across different devices. Therefore, hyperparameters in most cases and maintain the same random seed throughout the entire experiment.
fixed
Figure 7 is provided as supplementary material for Section 5.3 In Figure 7 we visualize the weights Wo. o. W 10. W20. 20. W Wso. so. W40. 40. and Wso su for the Electricity dataset and observe that the heatmaps of weights shift over time.
Figures [ and Pare provided as supplementary material for Section 5.4 In these figures, we provide the frequency spectrums of three random inputs for each dataset. We observe that the frequency spectrums of the Electricity and Traffic datasets quite stable, followed by the ETTm1, ETTm2, and Weather datasets. In contrast, the frequency spectrums of ETThl and ETTh2 are not stable, as reflected by the changing landscape over time with less similarity certain levels.
16

---

G nox15 -1xx
G goxt +xxr
Linear(11 T
+ 2), Hi)
Dividing into Patches
Linear( T 1) K)
T
Dropout(p 0.15)
2
NP
ReLU
Stack Multi-Heat Heat Attention
Linear H4,, Ha)
Linear (K x P,L)
8 e RDL
Dropout(p 0.15)
G Rox(] Jjjr
ReWU
Linear T x11)
L
Linear Linear(H,,L)
Y R
P c R0xl
Figure 6: The Layers of FBM-L,, FBM-NI, and FBM-NP in Detail. FBM-L represents a vanilla linear network, FBM-NL denotes a three-layer non-linear network, while FBM-NP shares the same structure as PatchTST but differs in the number of patches and applies the patching after the Fourier basis expansion.
-386
100
X0
X1
-135
-4.3%
-11
Wo
W10
W
0006
1005
0385
000
100
C30
X0
30-
155
-23
125
<20
200
100
Wa0
Wan
W00
Figure 7: The Visualization of Weights Wo. Wi. Wo0. W30. Wao and W00 on the Electricity Dataset.
17

---

075
05
875
3.0
225
11
200
09
225
220
e.g. 1: ETTh1
c.g. 2: ETThl
c.g. 3: ETThl
05
0.0
4.9
00
03
05
00
.98
00
0.25
01
200
00
00
225
025
15
135
1: ETTh2
e.g. 2: ETTh2
e.g. 3: ETTh2
05
85
05-
05
05
00
100
e.g. 1: Traffic
e.g. 2: Traffic
e.g. 3: Traffic
025
00
000
05
C5
20
35
00
00
08
0
e.g. 1: Electricity
e.g. 2: Electricity
e.g. 3: Electricity
Figure 8: The Frequency Spectrum of Three Extra Random Inputs of the Last Seven Dimensions the ETThl, ETTh2, Traffic and Weather Datasets.
18

---

025
0.5
000
00
3.0
030
00
000
02
125
0.5
0.00
49
115
025
200
0.0
00
025
125
825
135
c.g. 1: ETTml
c.g. 2: ETTml
c.g. 3: ETTml
875
000
00
0300
035
038
0.0
01
325
000
00
0.00
225
498
075
200
2.00
02
025
105
e.g. 1: ETTm2
e.g. 2: ETTm2
e.g. 3: ETTm2
0051
005
018
0000
0000
800
000
025
835
0.00
83
899
3.00
12:
05
800
00
330
05
111
e.g. 1: Weather
e.g. 2: Weather
e.g. 3: Weather
375
01
81
00
08
01
300
30
100
68
300
0025
000
0000
600g
015
300
01
00
175
025
155
e.g. 1: Exchange
2: Exchange
e.g. 3: Exchange
Figure 9: The Frequency Spectrum of Three Extra Random Inputs of the Last Seven Dimensions the ETTml, ETTm2, Electricity and Exchange Datasets.
19

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26


Reviews:
[{"review_id": "id8hblrbuc", "reviewer": "Reviewer_Z6Ax", "summary": "The paper introduces a novel approach that rethinks the application of the Fourier Transform for time series prediction, proposing a unique Fourier Basis Mapping (FBM) method. It combines both learnable and non-learnable components and demonstrates potential improvements through comprehensive experiments. This paper provides cool solutions to the inconsistent starting cycle and series length issues practically and theoretically. However, significant concerns remain. The paper does not clearly link the identified problems with the proposed solutions, resulting in fragmented logic. Figure 1 and the Introduction section lack clarity and focus. The methods section's reliance on non-learnable structures without clear integration with learnable components raises questions about their effectiveness. The unclear title and purpose of Section 5.3, along with insufficient explanation of denoising capabilities, further weaken the paper. Additionally, the absence of comparative studies and lack of ablation experiments undermine the validation of the proposed method's effectiveness. Addressing these issues could potentially render the paper more acceptable (7 scores).", "questions": "At first, I want to give this paper an Accept score. However,  I have the following three main concerns after reading this paper:\n1. Eq. (3) concludes the main technical contribution towards the long-term time series prediction problem. I agree that we should represent a time series with both the Time and Frequency domains to avoid the inconsistent starting cycle and length issue. However, when I look at the overall framework of the proposed method, I find that the function of this Time-Frequency joint representation is somewhat like extracting features. In this case, I want to know the relationship between the non-parameter feature extractor and the learnable decoder. As your ablation study in Sec 5.3 claims, the decoder has a noise-tuning effect for this module. I doubt this and expect the authors to provide a more comprehensive analysis, or the real effectiveness of the proposed module should be viewed as not fully explored.\n\n2. Experiments are not fair. The comparative experiment should be built with the same architecture. FBM has three variants, and they should be compared with the baseline models separately. The most important two comparisons are FBM-L v.s. NLinear; FBM-NP v.s. PatchTST.  I hope a more fair comparisons can be conducted.\n\n3. The writing should be improved, see Weekness for details.\n\nWith the following issues all addressed, this paper can score 7.", "limitations": "Experiments are not fair, and the proposed methods are not well elaborated.", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "1. The paper presents an innovative approach to time series prediction by utilizing the Fourier Transform (FT), which provides a novel perspective on leveraging frequency domain information for temporal data.\n\n2. The proposed Fourier Basis Mapping (FBM) method effectively integrates both learnable and non-learnable components, showcasing a unique combination that could potentially enhance the feature extraction process and improve prediction accuracy.\n\n3. The extensive experiments conducted demonstrate the potential of the proposed method to improve performance on various time series benchmarks, as evidenced by the quantitative results provided in the study.", "weaknesses": "1 The Introduction section inadequately discusses the issues with existing methods. The descriptions are obscure and hard to follow. It is recommended to replace these with more understandable explanations. Two separate paragraphs to elaborate on the identified problems and the corresponding proposed solutions are expected. The current version does not clearly articulate how the method is specifically designed to address each problem, leading to a fragmented logic. Readers may find it difficult to directly correlate the proposed method with the problem-solving benefits claimed.\n\n2 Figure 1 lacks focus and merely replicates descriptions from the text without emphasizing the unique aspects of the proposed method. The distinctive features of the method are not prominently highlighted.\n\n3 The methods section primarily consists of non-learnable structures, which raises significant concerns. The paper does not explain how these non-learnable components integrate with learnable structures to enhance feature extraction, leading to questions about their effectiveness. Therefore,  the title \"Effect of Weights for Linear Mapping\" in Section 5.3 becomes unclear and feels disjointed, making it difficult to grasp the purpose of this section at a glance. The paper fails to clearly explain why this visualization was conducted and how it supports the stated conclusions. Additionally, the explanation of the model's denoising capabilities is sparse and lacks clarity, seeming somewhat self-serving.\n\n4 A major issue in the experiments is the absence of comparative studies with the same decoder, making it hard to confirm the effectiveness of the proposed frequency domain method. Furthermore, for the key contributions such as denoising and problem-solving, it is suggested to design ablation experiments to validate these effects.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank you for your constructive suggestions, which will be fully reflected in the final version.  Please also refer to the one-page PDF attached and our reply within the global response to all reviewers for the major changes.\n\n**Weakness 1**\n\nWe take your kind suggestion and revise the Introduction. As this is important and beneficial for all reviewers to better understand the paper, we have put our answer in **Introduction and Figure 1** within the global response. \n\n**Weakness 2**\n\nWe update Figure 1 in the PDF file attached to the response to all reviewers, illustrating, comparing, and summarizing existing Fourier-based methods with FBM and emphasis our main contributions. \n\n**Weakness 3**\n\nYou may misread/misunderstand part of our work. Our main contribution is on extracting better initial features enable more effective downstream mapping. With time-frequency features, very simple mapping networks (L-vanilla linear network and NL-three-layer MLP) can achieve the SOTA in any circumstance. The internal mechanisms of FBM variants are provided in Figure 6 in Appendix. Assuming the look-back window $T=336$ and forecast horizon $L=96$, FBM-L decomposes the original time series into $\\frac{T}{2}+1$ pieces at different frequency granularities. Thus, our FBM-L mapping uses a “nn.Linear (336 $\\times$ 169, 96)”, and the FBM-NL includes two additional nn.Linear layers with activation functions, see Figure 6 in Appendix for details.\n\nThe Fourier transform is to decompose a time series hierarchically to remove noise signals. For example, for a two-week hourly time series, $T=336$, with day effects and noises only, after Fourier transform, day effects fall in the frequency levels that are multiples of $14$, and noises fall in all the other frequency levels, making mapping much easier. However, the existing mapping in the frequency space completely ignores the time dependent frequency information and faces two inconsistency issues. This is because a Fourier basis function is time-dependent when the input length is not divisible by the frequency level (see Figure 1 in the PDF). The mapping in the frequency space cannot capture the time-frequency relationships. Thus, Section 5.3 visualizes how FBM-L considers the time-frequency relationships. Since the datasets Electricity and Traffic are more stable than ETTh1 and ETTh2, the weights of the linear layer for the former datasets are closer to the Fourier basis than those for the latter datasets. This implies that FBM-L considers more time-frequency relationships in ETTh1 and ETTh2 than in Electricity and Traffic, leading to more significant improvement. We add a noise attack example to further verify the denoising effect: random Gaussian noises are added to ETTh1 with a 0.1 probability at each input time point during training, resulting in the performance below:\n\nETTh1|MSE|MAE\n-|-|-\nFBM-L with noise L=192|0. 403|0.411\nFBM-L without noise L=192|0.403|0.411 \nFBM-L with noise L=336|0. 418|0.420\nFBM-L without noise L=336|0.418|0.420\n\n\n**Weakness 4**\n\nWe add experiments for a fairer comparison of FBM-L, FBM-NL, and FBM-NP in the PDF:\n\n1.\tFBM-L vs NLinear in Table 1\n2.\tFBM-NP vs PatchTST in Table 1 \n3.\tFBM-NL vs TimeMixer [1] in Table 2 \n\n(1) both with linear networks, (2) both with PatchTST networks, and (3) both with MLP networks. \n\nFBM-L outperforms NLinear on all datasets and forecast horizons. The average MSE and MAE of NLinear are 0.3135 and 0.3395, respectively, which drops to 0.3034 and 0.3297 for FBM-L. For PatchTST, the average MSE and MAE are 0.3079 and 0.3360, respectively, which drops to 0.3058 and 0.3340 for FBM-NL. FBM-NP performs better than PatchTST in most cases. The improvement on PatchTST is less significant because PatchTST primarily considers the time domain. It is worth mentioning that TimeMixer's structure and hyperparameters are well-tuned for each dataset across different forecast horizons, but FBM-NL uses the same structure and hyperparameters all the time for better reproducibility. However, FBM-NL still performs better than TimeMixer in most cases (6 out of 8). These three comparisons demonstrate the effectiveness of the time-frequency features.\n\nRegarding the ablation test in the paper, three layers MLP are sufficient to capture deep time-frequency relationships because FBM-NL always achieves better or the same performance as FBM-NP when non-linear mapping is preferred for a dataset. With our time-frequency features, FBM achieves SOTA performance in almost all circumstances by simply using either a linear layer or a three-layer MLP without tuning.\n\n**Question 1**\n\nFBM considers time-frequency space mapping rather than frequency space mapping. For internal mechanism and empirical thinking, you can refer to our response to Weakness 3. For the effectiveness of time-frequency features, you can refer to our response to Weakness 4. \n\n**Question 2**\n\nPlease refer to our response to Weakness 4.  \n\n**Question 3**\n\nPlease refer to our response to Weakness 1 and the global responses for major changes.\n\n**Limitation**\n\nSee our response to Weakness 4 for a fairer comparison and to Weakness 3 for elaborating the proposed methods.\n\nRef:\n\n[1] TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting\n\nWe have substantially revised the paper addressing all of your comments, please let us know any other kind suggestions.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "iQC4qWb3cW", "reviewer": "Reviewer_akVH", "summary": "This work rethinks the discrete Fourier transform from a basis functions perspective, identifying two key issues in existing Fourier-based methods: inconsistent starting cycles and series lengths. To address these, the paper proposes the Fourier basis mapping model (FBM), which leverages Fourier basis expansion to obtain a mixture of time and frequency features. Experiments show FBM outperforms diverse baselines on long-term forecasting tasks, with the linear FBM-L performing better for noisy data and nonlinear FBM-NL/NP better for less noisy data, highlighting the importance of combining time and frequency information.", "questions": "Q1: How does your proposed FBM model perform in short-term forecasting tasks, especially in comparison to its performance in long-term time series prediction? Are there significant differences? Do you have plans to explore the applicability of FBM in short-term forecasting tasks in future research, such as in evaluations on the M4 dataset and PEMS dataset?\n\nQ2: You mentioned that FBM has better interpretability compared to complex Transformer models. Could you further analyze the internal mechanism of FBM and elaborate on how the spatiotemporal mixed features affect the model's predictive performance? This is crucial for a better understanding of the advantages of FBM.\n\nQ3: It would be helpful to provide more analysis on model parameter sensitivity, efficiency analysis of model computational complexity (including runtime and memory overhead), detailed settings of hyperparameters, and visual showcases.\n\nQ4: Could more comparative experiments with state-of-the-art models such as N-Hits[1], iTransformer[1], TimeMixer[2], and TimesNet[3] be provided? This is crucial for a comprehensive evaluation of this work.\n\n[1]N-Hits: Neural Hierarchical Interpolation for Time Series Forecasting\n\n[2]iTransformer: Inverted Transformers Are Effective for Time Series Forecasting\n\n[3]TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting\n\n[4]TimesNet: TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis", "limitations": "The authors have not addressed limitation of their work. I would suggest the authors to include such section in the final version of their work.", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "S1: The Fourier basis mapping (FBM) model effectively combines time and frequency domain features, outperforming a diverse set of baselines including linear, Transformer, and other Fourier-enhanced methods on long-term time series forecasting tasks.\n\nS2: The different FBM variants (linear and nonlinear) can adaptively select the appropriate model based on the noise characteristics of the data, demonstrating the importance of the time-frequency mixed features.", "weaknesses": "W1: The method is primarily validated on long-term time series forecasting tasks, and its applicability to short-term forecasting tasks requires further exploration.\n\nW2: While the FBM model exhibits better interpretability compared to more complex models like Transformers, further analysis of its internal mechanisms is needed to better understand the role of the time-frequency mixed features.\n\nW3: There is a lack of visual showcase demonstrations, making it difficult to provide a qualitative evaluation of the experiments. Additionally, there is a lack of efficiency analysis regarding the computational complexity.\n\nW4: More comparisons with state-of-the-art models are missing, which is crucial for evaluating the quality of the work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank you for your constructive suggestions, which will be fully reflected in the final version. Please refer to the new experiment results in the one-page PDF attached to the global response.\n\n**Weakness W1 and Question Q1**\n\nTable 3 in the PDF shows new short time-series forecasting (STSF) results on the PEMS dataset, but time constrains completing experiments on the M4 dataset (LTSF methods including Pathformer, iTransformer, CrossGNN, FiLM, PatchTST, NLinear and FreTS do not test on M4). FBM variants continuously achieve the SOTA performance for STSF on the PEMS datasets. \n\n\n**Weakness W2 and Question Q2**\n\nYes, the main contribution of our work is on extracting more interpretable time-frequency features enabling more effective downstream mapping. Consequently, we can use very simple mapping networks (L-vanilla linear network and NL-three-layer MLP) to achieve the SOTA.\n\nThe internal mechanisms of FBM variants are provided in Figure 6 in Appendix. Assume the look-back window $T=336$ and forecast horizon $L=96$, FBM-L decomposes a time series into $\\frac{T}{2}+1$ pieces at different frequency granularities by Eq. (3). Thus, our FBM-L mapping uses a “nn.Linear (336 $\\times$ 169, 96)”, and FBM-NL includes two additional nn.Linear layers with activation functions.\n\n1.\tPerformance: Our ablation study evaluates the comparisons between FBM-L vs NLinear and FBM-NP vs PatchTST in Table 1 and FBM-NL vs TimeMixer in Table 2 in the attached PDF, showing the effectiveness of the time-frequency features. The first compares two linear networks, the second compares two PatchTST networks, and the last compares two MLP networks. FBM-L outperforms NLinear on all datasets and forecast horizons. The average MSE and MAE of NLinear are 0.3135 and 0.3395, respectively, which drops to 0.3034 and 0.3297 for FBM-L. For PatchTST, the average MSE and MAE are 0.3079 and 0.3360, respectively, which drops to 0.3058 and 0.3340 for FBM-NL. FBM-NP performs better than PatchTST in most cases. The improvement on PatchTST is less significant because PatchTST primarily considers the time domain. It is worth mentioning that TimeMixer's structure and hyperparameters are well-tuned for each dataset across different forecast horizons, but FBM-NL uses the same structure and hyperparameters all the time for better reproducibility. However, FBM-NL still performs better than TimeMixer in most cases (6 out of 8). With our time-frequency features, our model achieves SOTA performance on all datasets for both LTSF and STSF simply with one or three layers even without tuning. \n\n2.\tInterpretability: We empirically explain why the time-frequency features are better. The previous frequency mapping suffers from issues of inconsistent starting cycles and series lengths. This is because crucial information is stored in the amplitude and phase of the cycle, validated in Eq. (3) and (4). Additionally, a Fourier basis function is time-dependent when the input length is not divisible by the frequency level (see Figure 1 in the PDF). The mapping in frequency space cannot capture time-frequency relationships. Thus, Section 5.3 visualizes how FBM-L considers time-frequency relationships. Since the datasets Electricity and Traffic are more stable than ETTh1 and ETTh2, the weights for the former datasets are closer to the Fourier basis than those for the latter datasets. This implies that FBM-L considers more time-frequency relationships in ETTh1 and ETTh2 than in Electricity and Traffic, leading to more significant improvement.\n\n**Weakness W3 and Question Q3** \n\nWe revise the paper for better visual showcase. FBM-L and FBM-NL are listed as two independent columns in Table 1, with a new baseline TimeMixer added. \n\nThe efficiency analysis is conducted on ETTh1 with the same setting, here is the training speed for one epoch:\n\n|Model|FBM-L|FBM-NL|FBM-NP|NLinear|PatchTST|N-BEATS|CrossGNN|FEDformer|FreTS|FiLM|BasisFormer|TimeMixer|Pathformer| iTransformer|TimesNet|\n|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|Time|4.86|10.7|34.21|1.36|33.30|5.21|23.50|161.29|28.69|148.50|6.75|10.91|241.23|11.64|12.18|\n\nFBM-L and FBM-NL rank second and fifth in terms of training speed, respectively, because they just need to train one and three nn.Linear layers, respectively. The memory is $\\mathbf{O}(L^2/2)$, but one can easily combine some frequency levels to reduce it. For FBM-NL, we use the same hyperparameters for each dataset across different horizons for better reproducibility, see Figure 6 in Appendix. FBM-NP uses the same optimal hyperparameters as PatchTST.\n\n\n**Weakness W4 and Question W4**\n\nTable 2 in the PDF shows the experiment results of new baselines iTransformer, TimeMixer, TimesNet, and Pathformer [1]. Due to time constraints, we cannot finish experiments on N-Hits. However, we can refer to experiment results on the same dataset reported in their paper for comparison. For example, they report 0.401 MSE and 0.413 MAE on ETTm2 (L=720), our FBM-L achieves 0.364 MSE and 0.381 MAE. Table 2 shows our FBM variants continuously achieve the SOTA performance against these new baselines. \n\n**Limitation**\n\nFor the limitation part, there may be some misunderstanding. Our method does consider the monotonic trend effect, as the Fourier basis has time dependent and independent frequency basis, which helps separate the trend and seasonal effects like DLinear. However, how to use the previous trend to predict future trends is still a major challenge, which hasn’t been solved for every method, as real-world data usually don’t have a patternable trend effect.\n\nRef:\n\n[1] Pathformer: Multi-scale Transformers with Adaptive Pathways for Time Series Forecasting\n\nWe have thoroughly addressed all comments and produced a new version of the paper, thank you for any other suggestions.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "AwO6xHsrJg", "reviewer": "Reviewer_8ZRG", "summary": "The authors propose a Fourier basis mapping model, named FBM, for long-term time series forecasting. FBM embeds the discrete Fourier transform with basis functions, and then introduces a mapping network to replace the inverse discrete Fourier transform. This approach allows FBM to capture implicit frequency features while preserving temporal characteristics. Experiments demonstrate the effectiveness of the proposed framework.", "questions": "See Weaknesses.", "limitations": "No limitations or potential negative societal impacts are discussed in the paper.", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "(1)The paper is easy to follow.\n\n(2)The experimental results on eight real-world datasets prove that FBM achieves competitive performance.", "weaknesses": "(1)The technical contributions are incremental and the overall impact is not significant enough, as some key components are similar to existing works. For example, the Fourier basis mapping is similar to FEDformer. In addition, the three mapping methods used in the decoder look like a simple combination of linear network, nonlinear MLP, and PatchTST.\n\n(2)The introduction of the model architecture is confusing, as many implementation details of the modules in Figure 1 and Figure 3 are not clearly explained or even introduced (e.g., the Fourier basis expansion operation, the normalization operation, and the de-normalization operation).\n\n(3)Several groups of comparative experiments are carried out, but some SOTA works [1, 2, 3] are not compared or mentioned. The authors should compare their model with SOTA works to demonstrate its effectiveness.\n\n(4)There are many typos and writing mistakes in the manuscript. For example, on page 2, \"Huange et. al.\" should be \"CrossGNN\". On page 6, \"Table 2\" should be \"Table 1\". On page 12, \"Crossgnn\" should be \"CrossGNN\", \"N-beats\" should be \"N-BEATS\", and \"Nhits\" should be \"N-HITS\". The manuscript requires a thorough proofreading.\n\n(5)Some figures provided are too small to allow for a thorough investigation of the details. The authors should use vector figures instead. In addition, in Figure 5, the meaning of the x- and y-axes need to be explained and the meaning of the number \"5\" in the right part should also be clarified.\n\n[1] Liu Y, Hu T, Zhang H, et al. iTransformer: Inverted Transformers Are Effective for Time Series Forecasting. ICLR 2024.\n[2] Wang S, Wu H, Shi X, et al. Timemixer: Decomposable multiscale mixing for time series forecasting. ICLR 2024.\n[3] Chen P, Zhang Y, Cheng Y, et al. Pathformer: Multi-scale transformers with Adaptive Pathways for Time Series Forecasting. ICLR 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your constructive suggestions, which will be fully reflected in the final version, including clarifying misreading/misunderstanding. Extra experiment results and the updated Figure 1 are attached in the one-page PDF within the global response for all reviewers. \n\n**Weakness (1)**\n\nThere could be misreading/misunderstanding of the motivation and main contributions of our work. We redraw Figure 1 and revise Introduction as explained in **Introduction and Figure 1** of the global response to clarify the workflow and compare our FBM with others. Our contribution is to extract more interpretable features to enable much more effective downstream mapping. \n\nThus, we disagree with the statement that “some key components are similar to existing works.” Instead, our FBM opens a new perspective for time series forecasting using the FBM with the time-frequency features, making the following main differences:\n\n1.\tThe Existing Fourier-based mapping is in frequency space, while FBM is conducted in time-frequency space. \n2.\tExisting methods face inconsistent starting cycles and inconsistent series length issues, addressed by FBM.\n3.\tExisting methods ignore that Fourier basis could be time dependent, addressed by FBM. \n\nIn fact, existing Fourier-based mapping methods (e.g. FEDformer) do not involve Fourier basis mapping like ours. They use real and imaginary parts as inputs to their mapping networks and conduct in frequency space. In contrast, FBM conducts mapping in both time and frequency space.\n\nWe provide a new perspective that real and imaginary parts can be interpreted as the coefficients of cosine and sine basis functions at different frequencies, as shown in Eq. (3). However, existing Fourier-based mappings do not involve such basis functions, thus failing to interpret these coefficients correctly. Eq. (4) shows that adding a cosine wave and a sine wave at the same frequency results in a shifted cosine wave of the same frequency. This means that the crucial information is embedded in the amplitude and phase of each cycle rather than in the real and imaginary parts. This leads to issues of inconsistent starting cycles and inconsistent series lengths in existing methods, addressed by our FBM.\n\nIn addition, a Fourier basis function is time-dependent when the input length is not divisible by a frequency level (see new Figure 1 in PDF ). Consequently, the mapping in the frequency space fails to capture this time-dependent frequency information. Our FBM addresses these issues. It decomposes an input time series into $\\frac{T}{2}+1$ pieces at hierarchical frequency granularity, where T is the input sequence length. We show that time-frequency features enable simplified downstream mapping, and very simple mapping networks (L-vanilla linear network and NL-three-layer MLP) can beat complicated Transformers, etc. FBM achieves SOTA performance using either a vanilla linear layer or a three-layer MLP on eight real-world datasets for LTSF and PEMS datasets for STSF.\n\nWe sincerely appreciate if you could read our response to Weakness W2 for Reviewer akVH as well, as it is important to understand the unique contribution of time-frequency features.\n\n\n**Weakness (2)**\n\nPlease refer to new Figure 1 and the updated Introduction in the global response. \n\nAs discussed in line 167, we use the normalization and denormalization same as the following reference:\n\nRef: Reversible Instance Normalization for Accurate Time-Series Forecasting against Distribution Shift.\n\nSpecifically, normalization subtracts the average value of the input time series and denormalization adds that back to the forecast time series at the end, as widely used in LTSF and demonstrated effective by the reference. \n\nThe Fourier basis expansion operation follows Eq. (3) and aims to solve the issues discussed in Section 3 of the paper. \n\nThe internal mechanisms of the three mapping methods are provided in Figure 6 in Appendix.\nAssuming the look-back window $T=336$ and forecast horizon $L=96$, FBM-L decomposes the time series into $\\frac{T}{2}+1$ pieces at different frequency granularities by Eq (3). Thus, our FBM-L mapping uses a “nn.Linear (336 $\\times$ 169, 96)”, and the FBM-NL includes two additional nn.Linear layers with activation functions. FBM-NL uses the same structures and hyperparameters for each dataset across different horizons for better reproducibility. \n\n**Weakness (3)**\n\nFour SOTA baseline methods Pathformer, iTransformer, TimeMixer and TimesNet [1] are compared in Table 2 of the PDF for the global response. Due to time constraints, we only finish LTSF experiments on datasets ETTh1 and ETTh2 but also add extra experiments for STSF on the PEMS dataset. FBM-L and FBM-NL consistently achieve the SOTA performance for both LTSF and STSF without tuning, as shown in the PDF.\n\n**Weakness (4)**\n\nMany thanks for pointing out the typos, we’ve revised the paper and thoroughly polished the paper for the final version.\n\n**Weakness (5)**\n\nFigure 1 is updated in the PDF to emphasize the distinctive features of FBM against existing methods, pointing out the issues of previous methods and highlighting our main contribution. All other figures are adjusted for better visualization. In Figure 5, the x-axis represents the frequency level (Hz), and the y-axis represents the amplitude of real and imaginary values. ``5’’ refers to the last value of 175, removed from all figures. \n\n**Limitations**:\n\nThis paper on time series forecasting does not involve negative social impact.\n\nRef:\n\n[1] TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis\n\nWe have thoroughly addressed all comments and produced a new version of the paper and thank you for any further kind suggestions.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
**Please refer to the one-page PDF file attached here for more information.** 

We thank all reviewers for constructive comments and suggestions, and expect positive feedback on the unique innovation, extensive experiments, and competitive performance. During the rebuttal, we have addressed all comments and produced a new version of the paper. Major works include:

1.	Clarifying the motivation and workflow of our Fourier Basis Mapping (FBM) model by redrawing Figure 1 (shown in the PDF file) and revising Introduction. 
2.	Adding ablation studies for FBM-L vs NLinear and FBM-NP vs PatchTST, with results in Table 1 in the PDF.
3.	Comparing FBM with four extra SOTA methods: iTransformer, TimeMixer, TimesNet, and Pathformer, with results in Table 2.
4.	Testing short time series forecasting (STSF) on the PEMS datasets with results in Table 3.

**Introduction and Figure 1**

First, we revise Introduction to improve readability:

1. para 1 summarizes the significance and challenges of long time-series forecasting (LTSF) and introduces the existing deep LTSF methods.
2. para 2 analyzes the gaps of the Fourier-based method, disclosing two major issues that are ignored: inconsistent starting cycles, and inconsistent series lengths. 
3. para 3 proposes our unique insight into addressing the above issues from the basis function perspective, and summarizes the main ideas and contributions of FBM as well as the evaluations of FBM against existing LTSF methods.

Second, we redraw Figure 1 as shown in the PDF to emphasize the distinctive features of FBM against existing methods. Existing Fourier-based methods involve mapping in frequency space, whereas FBM is conducted in both time and frequency space.

Existing methods like FEDformer, FreTS, and FiLM use real and imaginary parts as inputs to their mapping networks but cannot easily interpret their coefficients because the crucial information is stored in the amplitude and phase of each cycle. This leads to inconsistent starting periods and series lengths issue, which are often ignored in existing research. Other methods e.g. CrossGNN, TimesNet use the top-k amplitude to filter noise. However, a higher amplitude does not necessarily indicate useful frequency, and a lower amplitude is not necessarily useless. More importantly, Figure 1 shows that a Fourier basis function is time-dependent when the input length is not divisible by a certain frequency level. Consequently, the mapping in frequency space fails to capture the time-frequency relationships.

By interpreting real and imaginary parts as the coefficients of cosine and sine waves at different frequencies, FBM decomposes an input time series into $\frac{T}{2}+1$ pieces with hierarchical frequency granularity, and T is the sequence length. With time-frequency features, FBM achieves state-of-the-art (SOTA) performance using either a vanilla linear layer or a three-layer MLP on eight real-world datasets for LTSF and PEMS datasets for STSF.

**Table 1: new results on effectiveness**

Table 1 in the PDF shows the results of FBM-L vs. NLinear and FBM-NP vs. PatchTST. FBM-L outperforms NLinear on all datasets and forecast horizons, demonstrating the effectiveness of time-frequency features, as both are linear networks. The average MSE and MAE of NLinear are 0.3135 and 0.3395, respectively, which drops to 0.3034 and 0.3297 for FBM-L. FBM-NP performs better than PatchTST in most of the time, as both are PatchTST networks. For PatchTST, the average MSE and MAE are 0.3079 and 0.3360, respectively, which drop to 0.3058 and 0.3340 for FBM-NL. The improvement on PatchTST is less significant because PatchTST primarily considers the time domain. 

**Table 2: new results for more baselines**

Requested by Reviewers akVH and 8ZRG, Table 2 includes results by comparing methods: TimesNet, PathFormer, iTransformer, and TimeMixer on the ETTh1 and ETTh2 datasets. It shows that either FBM-L or FBM-NL consistently achieves the SOTA performance in all circumstances. Additionally, by comparing FBM-NL with TimeMixer, we find that FBM-NL performs better than TimeMixer most of the time (6 out of 8), as both use the MLP network. It is worth mentioning that TimeMixer's structure and hyperparameters are well-tuned for each dataset across different forecast horizons. However, FBM-NL uses the same three-layer MLP and the same hyperparameters for each dataset across different horizons for better reproducibility, see Figure 6 in Appendix.

All the experiment results are conducted with an input sequence length of 336, rather than 96, except for PathFormer, as its hyperparameters do not work with 336. This is consistent with the settings in the paper. A length of 336 always produces better results than 96. Regarding the score reported in the following papers, Group A (Seq len: 336) consistently reports lower MSE and MAE than Group B (Seq len: 96).

Group A:

1.	PatchTST: A Time Series is Worth 64 Words: Long-term Forecasting with Transformers
2.	NLinear: Are Transformers Effective for Time Series Forecasting

Group B: 

1.	PathFormer: Multi-scale Transformers with Adaptive Pathways for Time Series Forecasting
2.	iTransformer: Inverted Transformers Are Effective for Time Series Forecasting
3.	TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting
4.	TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis

**Table 3: new results on STSF**

Table 3 shows new STSF results on the PEMS datasets, verifying the effectiveness of our FBM. Although FBM primarily addresses the LTSF challenges, it also achieves the best for STSF. In STSF, we change the last hidden state of FBM-NL from 1440 to 720.

We share the LTSF codes in supplementary, allowing to verify our reported experimental results, and will be released on GitHub later with STSF. 

**Conclusion**

FBM-L and FBM-NL are listed as two independent columns in Table 1 for a fairer comparison. Four SOTA baseline methods and STSF results are added to the paper.

Meta-review: The paper leverages the Fourier Transform to frequency domain information in predictive models for temporal data. Reviewers Z6Ax and akVH appreciated the novelty of the method, its combination of learnable and non-learnable components, the model selection tailored to the data, and the extensive experiments showing the potential of the method. However, both reviewers indicated lapses in the presentation. Additionally, reviewer Z6Ax recommended comparative studies with the same decoder, while Reviewer akVH requested more comparisons to the state of the art. The authors have provided extensive additional experiments and have greatly improved their introduction and figure. The changes were appreciated by the two reviewers, who agreed to accept the paper. Moreover, reviewer akVH has replicated some of the results reported in the paper (comparison of the model with patchTST) and found that the numbers reported are reliable. Reviewer 8ZRG considered the technical contribution to be incremental and the method not clearly explained, while also pointing out some s.o.t.a models for comparison. The authors have provided an extensive response, which, by my assessment, resolves at least the issues with the comparisons against s.o.t.a., and clarifies the contributions. Notably, Reviewer Z6Ax also pointed Reviewer 8ZRG to a part of the response that appears to clarify the contribution and address a concern. Reviewer 8ZRG has chosen to decrease their score, which is extremely unusual at this stage, and I did not find the justification for it convincing. From their statement "this paper is not ready for being published in NeurIPS", one might gather that the decrease in score is meant to prevent this paper from being published. Reviewer 8ZRG's final set of concerns was also addressed by the authors. It seems that at least some of the results were reproduced by the other reviewer (relevant to point 3), while MAE and MAPE were indeed used by the contender method (point 1), then two of the 3 final points are addressed.
Overall, my assessment is that this paper constitutes a valuable contribution to the field and warrants acceptance. However, the authors should be extremely mindful when preparing the camera-ready and incorporate all the reviewers' suggestions to improve the clarity of the presentation and present the new experiments.
Decision: Accept (poster)