MoFo: Empowering Long-term Time Series Forecasting with Periodic Pattern Modeling
Authors: Jiaming Ma
Binwu Wang
Qihe Huang
Guanjun Wang
Pengkun Wang
Zhengyang Zhou
Yang Wang
Venue: NeurIPS 2025 poster ()
Keywords: Time Series; Deep Learning; Position Information Encoding
Primary area: deep_learning

Abstract: The stable periodic patterns present in the time series data serve as the foundation for long-term forecasting. However, existing models suffer from limitations such as continuous and chaotic input partitioning, as well as weak inductive biases, which restrict their ability to capture such recurring structures. In this paper, we propose MoFo, which interprets periodicity as both the correlation of period-aligned time steps and the trend of period-offset time steps. We first design period-structured patches—2D tensors generated through discrete sampling—where each row contains only period-aligned time steps, enabling direct modeling of periodic correlations. Period-offset time steps within a period are aligned in columns. To capture trends across these offset time steps, we introduce a period-aware modulator. This modulator introduces an adaptive strong inductive bias through a regulated relaxation function, encouraging the model to generate attention coefficients that align with periodic trends. This function is end-to-end trainable, enabling the model to adaptively capture the distinct periodic patterns across diverse datasets. Extensive empirical results on widely used benchmark datasets demonstrate that MoFo achieves competitive performance while maintaining high memory efficiency and fast training speed.

MoFo: Empowering Long-term Time Series Forecasting with Periodic Pattern Modeling
Jiaming Ma', Binwu Wangl2", Qihe Huang', Guanjun Wang' Pengkun Wang Zhengyang Zhou 12 Yang Wang! University of Science and Technology of China (USTC), Hefei, Anhui, China Suzhou Institute for Advanced Research, USTC, Suzhou, Jiangsu, China {Ji (JimmingM,, hqh, always]@mail ustc. edu. cn (wbw2024, pengkun, zzy0929, angyan) wustt.utuc...oo.ooooooooueeceoee edu. cn
Abstract
The stable periodic patterns present the time series data serve as the foundation for long-term forecasting. However, existing models suffer from limitations such as continuous and chaotic input partitioning, as well as weak inductive biases, which restrict their ability to capture such recurring structures. In this paper, we propose MoFo, which interprets periodicity as both the correlation of period- aligned time steps and the trend of period-offset time steps. We first design period- structured patches-21 tensors generated through discrete sampling-wii where each row contains only period-aligned time steps, enabling direct modeling of periodic correlations. Period-offset time steps within a period are aligned columns. To capture trends across these offset time steps, introduce weriod-aware modulator. This modulator introduces an adaptive strong inductive bias through a regulated relaxation function, encouraging the model to generate attention coefficients that align with periodic trends. This function is end-to-end trainable, enabling the model adaptively capture the distinct periodic patterns across diverse datasets. Extensive empirical results on widely used benchmark datasets demonstrate that MoFo achieves competitive performance while maintaining high memory efficiency and fast training speed. Our code is available at official repository.
Introduction
Long-term time series forecasting (LTSF) has found widespread applications across various domains [10, 12, 13, 36, 62, 78, 79], with its core challenge lying in understanding and modeling the inherent periodic patterns present within data [20]. To address this, various cutting-edge models have been proposed, among which Transformer [43] has emerged the de facto backbone for capturing long- range dependencies in LTSF tasks [6, 24, 62, 72]. Despite the promising results achieved, we identify two underexplored potentials that remain be fully harnessed.
Continuous but Chuotic Time Steps of Input. Existing models often input consecutive time steps, and popular patch-based methods are no exception. Specifically, the patching method partitions the input sequence in a continuous manner using either convolutional down sampling [22] or sliding window strategies [5, 32, 75]. which we refer dataset as an example, as shown in the red box in Figure I(a), a conseeutive patch may contain both time steps that phase-aligned across each period (referred to aerine-aligned) and those that misaligned within each period (period-offie -offiet..f.......... We further visualize the pairwise correlations between
the continuous patch. Taking the Electricity
Binwu Wang and Yang Wang are corresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Period-offset time step
High Correlation
Successive Patching
4.424
Low Carrelation @ Even
Structured Ratching
0.193
Tranze
BATN!
ETTM
O
Period aligned time steps (a) Electricity Dataset
Trawffmmr
(b) A utococrelatio Function
B.20
Prisiamer
Posttion 24
ETTAL
Jnn Kores Sarto mogooom
Less Correlation Higher Attention
Position
0.191
Bevetto Weather
Modlo (Dan))
B.19
EYTM
Remoto Decay
0.388
Correct
3
Higher Cornelation But Less Attertion
mV224-3 mod 24, 1-24 24)
Error x
0.31
DEX
24-3-23
(c) Simuous Position Encoding in LTSF
(d) Period Position Calculation
(e) MSE of Transformer based models in LTSF
Figure 1: Case visualization and analysis real-world datasets. (a) Continuous patching strategy. (b) Correlation coefficients within patches. The orange region represents the 95% confidence interval for the null hypothesis from Bartlett's Test [2]. Correlation coefficients that fall within this interval are not statistically significant and cannot be rejected as noise [4]. (c) Attention weights exhibit remote attenuation due to sinuous positional encoding. (d) Absolute and relative position distance in the period perspective. (e) Performance of Transformer based models for LTSF with 720 horizon.
the first time step in the patch and subsequent ones in Figure 1(b). It is evident that period-aligned time steps, such as 8 AM every day, exhibit strong correlations. In contrast, period-offset time steps rapidly decaying correlations and may even introduce noise. Therefore, the internal correlations within a continuous patch are chaotic, which hampers the model's ability to learn periodic patterns.
Weak Inductive Bias for Periodicity. Due to the permutationn invariance of the self-attention mech- anism. sinusoidal positional encodings [16. 32] or timestamp position embeddings [62, 77, 79] are commonly used to inject temporal position information. However, as shown in Figure 1(c). sinusoidal encodings may cause attention weights to decay gradually with increasing position distance [50], which conflicts with the repetitive nature of periodic signals. In contrast, timestamp position em- beddings encode absolute time intervals, leading to large distances between strongly correlated and alige-d time steps. For example, in daily-periood electricity dataset, 8 A.M on Monday and 7 AM on Tuesday may exhibit high correlation due to their similar daily phases. However, as shown in Figure I(d), their absolute time difference is encoded as 23, whereas more reasonable alternative their relative cyclical distance - is only 1. Large temporal distances may mislead the model into assigning diminished attention weights. Such weak inductive bias toward periodicity in Transformer further hampers the ability to capture periodic patterns from consccutive and chaotic input.
To address these limitations, we propose MoFo, a novel TransFormer architecture with Modulator that explicitly models periodic patterns by capturing both the correlations among period-aliy aligned time steps and the trends across period-offset ones. Our approach introduces two key components: Period-structured Patching and the Period-Aware Modulator. The former discretely samples time steps to arrange the input into a 2D tensor where each row represents patch comprising only period-aligned time steps. Period-offset time steps within the period are realigned across columns (i.e., across patches). By modeling the features within each patch, the model can directly learn the underlying periodic correlations. To capture cross-patch periodic trends, we introduce the Period-Awan Modulator, which generates attention modulation term through regulated regulated relaxation function. This function assigns attention coefficients based on the periodic relative distances between time steps, encouraging the resulting attention scores to align with the underlying periodic patterns. As a result, the attention mechanism infused with strong inductive biases that favor periodic dependencies. Crucially, this function is end-to-end trainable, enabling it to dynamically adapt the modulation behavior to the specific periodic characteristics of the input data.
Our contributions are fourfold: ① We propose novel perspective for periodic pattern modeling, termed MoFo, which integrates two innovative strategies for explicit periodicity-awan time series modeling. @ We design Period-Structured Patching strategy that separately manages period-

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aligned and period-offset time steps, enabling the model to directly capture both periodic correlations and trends. ① We introduce a Period-Awart Modulator, which encourages the Transformer to generate attention coefficients that align with underlying periodic trends through a learnable regulated relaxation function. Extensive experiments demonstrate that our model achieves competitive forecasting performance while significantly improving computational efficiency offering up to 14x memory savings and 10x faster training speed compared to state-of-the-an methods.
2
Related Work
In recent years, deep learning methods have achieved remarkable success across broad spectrum of time series tasks, such as forecasting [17, 26, 31, 54, 64, 67, 71J, imputation [52], classification [25, 58], and anomaly detection [35, 66, 70J. Among these, Long-Term Series Forecasting (LTSF) has attracted especially intense interest owing to its foundational role in both academic research and practical applications [30, 51, 55]. Existing deep learning approaches for LTSF can be broadly grouped into four categories according to how they handle the temporal dimension: RNN-based, TCN-based, MLP-based (discussed in Appendix B), and Transformer-bast based models. The Transformer architecture has emerged as the prevailing choice for LTSF, primarily because of its strong ability to capture long-range temporal dependencies. Recent studies have largely concentrated on enhancing both its computational efficiency and modeling power [28, 29], with significant advances driven by two core elements: the self-attentic mechanism and patch-based modeling strategies.
Variants of Self-Attention Mechanism for Time Series Forecasting, One of the most interesting aspects of Transformer in LTSF is its self-attention mechanism. LogTrans [16] introducing convolutional LogSparse attention that ensured long-distance interactions while reducing the number of interactions, an early and influential attempt apply Transformer. Informer [77] proposes
ProbSparse attention while combining with a distillation mechanism to select the most representative query vectors to compute the attention scores. Autoformer [62] utilizes decomposition architecture to discover dependencies for building sequence-level connections based on their aggregation of similar subsequences. FEDformer [79] leverages the attention mechanism in the frequency domain, providing the capture of the underlying oscillatory modes and their intensities. Despite the efficacy of expensse,,
these methodologies in reducing computational expense, they often encounter information bottlenecks in long sequence input [33].
Patch-based Time Series Forecasting Method. Patch-based approaches are widely adopted for efficient time series representation, where the input sequence is divided into consecutive segments (patches). This strategy not only enhances the computational efficiency of the Transformer backbone but also promotes more effective modeling of localized temporal dynamics within each patch [11, 60]. PatchTST [32] explicitly reduces the length of input sequence and preserves the local semantic information by dividing the time series into smaller subsequences. Pyraformer [22] employs pyra- midal attention mechanism, wherein the input sequence is downsampled into patches by multilayer convolution, allowing attention to be applied at coarse scales. Crossformer [75] proposes two-stage attention with a dynamic routing mechanism that performs the patch operation from the sequence dimensionality and the channel dimensionality. Pathformer [5] selects multiple patch volumes at the same time and uses a multi-scale router to determine the interactions between different patches to weaken the impact of the selection of patch volume. All existing patch strategies are sequential along the temporal dimensionality, which does not facilitate effective capture and learning of periodic patterns [11, 60).
3 MoFo
Given X = [x1, [2,,,x,-.... x2...., xr] T
E RT with the look-back window T where X, E IR represents the time series data time step t, the objective of LTSF is to forecast the next L values Y
[XT+1, XT xT+ E R.L. Effective long-term forecasting hinges on the ability to model the intrinsic periodic patterns underlying the data. In this work, shown in Figure 2, we propose MoFo which integrates two key components: Period-structured Patch and the Period-Aware Modulator, which jointly enhance the model's capacity to capture and exploit periodic patterns.
3

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Forecasting Y
Periodic Aware Modulator
Prtodic-Structurees Patch
Linear
Pertod- allggned
Period-offset
Liiear
Flatten
Concat
Mathuul
Transformer With Modulator
Sample
Padding
M
Modulator
Values
Linear
MatMull
Coreer
Pooddin Perood 1st.
Time
Patching
370
4071
Time Series Data
Liber
Ridii Reced MenteOMee
Figure 2: The details of MoFo which contains core contributions for accurate modeling of periodic patterns: Period-structured Patch and Period-Aware Modulator.
3.1 Period-structured Patch in MoFo
Patch-bases input representation method has become a popular approach to improve the efficiency of long-timeeseet modeling of Transformer. However, as discussed in the introduction section, the pairwise correlations between time steps within a continuously partitioned patch can be weak or even negative, which may undermine the model's 's ability to capture periodic patterns. To address this challenge, we propose Period-structured Patching strategy. This approach discretely samples period-aligned time steps from the input sequence form patches. Specifically, given a period length P. which is typically much smaller than the input sequence length T (i.e., P < I) in LTSF settings, we compute Np = [T/P] as the number of period-length segments required to cover the input time series X. Since the length T of the input time series may not be an integer multiple of the period length P. it becomes necessary to apply padding so that all time segments have uniform length without discarding the trailing portion of the input.
Input Padding. Our proposed padding strategy fills incomplete periodic segments with data from adjacent periods. Specifically, we start from the current time step and move backward to delineate periods of length P, and if necessary, we prepend the input time series with the first P (T mod P) time steps of the first complete period, as shown in Figure 2. This ensures seamless continuation of the sequence while retaining its underlying periodic structure. As a result, the input series is extended to X Xpod E RT" with T"=P [T/P], and the padded series can be formally expressed as:
Concat (X mod P):P; X), if mod P 0, if mod 0.
X pad
X
Sampling Patch and Unflatten. We sample time steps at periodic intervals (i.e., period-alignen time steps) from X Xpad and group them into the same patch. For example, for i-th patch, it can be denoted as X = =[2i, Di+ I4 p1| E [T/P|] where 24+ means the data point at the time step (i + P) in Xpod. Then unflatten X to generate the patch-structure input X
X Unflatten
E PPTpP/P
(2)
where P is the number of patches (also equal to the period length). As illustrated Figure 2, the structured input X,, possesses two key properties: O Each row (within patch) contains [T/P] time steps, with all steps period-aligned within their respective periods. This alignment establishes structured temporal dependencies across periods. enabling the explicit modeling of long-range periodic dependencies. @ Fach column (across patches) contains P time steps from a complete period. By capturing correlations among these patches, the model can effectively learn the underlying periodic trends across period-offset time steps.
Periodic Dependency Modeling. Due to this desirable property, even simple neural architec- ture-such as a single-layer MLP-can suffice for capturing the underlying periodic dependency. The design is formally described as follows:
Z X,n W.. + b E R Pxd
where W in E IR xd and bjn d are learnable parameters, And Z is the corresponding output.

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3.2 Period-Aware Modulator for Periodic Trend Modeling
MoFo further incorporates an enhanced Transformer to model the periodie trends across period-offset time steps within periods. To further strengthen the model's representational capability, we first modify the standard Transformer architecture (details Appendix C). Moreover, we introduce Period-ANe Aware Modulator, which integrates strong inductive biases to address the permutation invariance of self-attention, thereby enhancing the model's 's ability to capture period trends.
3.2.1 Period-Aware Modulator
When computing the attention scores, we explicitly introduce a strong inductive bias: the attention generated by the Transformer is encouraged to align with periodic trends. We achieve this by incorporating a modulation term into the attention computation. First, we design periodic positional encodings that effectively capture the relative distances within period - a key distinction from existing approaches such as sinusoidal or timestamp- based positional encodings. Specifically, the relative distance between the i-th period and the j-th period is computed as follows,
Yiy min 5) mod P, (j i) mod P) [0, [P/2]],
(4)
this strategy ensures that time steps that are periodically close remain proximal. For example, in traffic dataset with an hourly sampling frequency and a daily period (i.e., 24 time steps), the relative period distance between 8:00 AM Monday and 7:00 AM on Tuesday is while their absolute distance is 23. This allows the model to better perceive and learn periodic trends. and we can get relative periodic distance matrix, which is denoted r (Tig)1-=1 PXP.
We proceed by introducing modulation term within the attention computation, designed to encourage the model to prioritize temporal dependencies that align with the underlying cyclical trend of the data. Specifically, we initially define a modulation matrix M E {0, 1)PxP" where each entry M.j is an indicator that reveals whether time steps i and j are close in terms of cyclical distance. A natural way to construct this matrix is by using the Heaviside Step function H : IR + {0, 1}: Mij- = H (3 - and its logarithmic value can be defined as:
Mij = (o. Yh I 3, - log Mij 0. if > 3,
if ih $ B -00, if Mij B.
0,
(5)
where Yr is the periodic distance between i-th period and j-th period calculated above. And 3 is a distance penalty threshold controlling the penalization distance. Hence, the attention coefficient between the i-th patch and the j-th patch is satisfying,
Q.K.J Vdh
+ log M,,) = exp
Q.K, Van
if Yij S B, )
exp
(6)
0,
if 7is > B.
The attention score is considered valid only if the cyclical distance between time step j and i is no more than B. Otherwise, the attention coefficient penalized due to the distance and set to zero Finally, our attention mechanism can be written as,
QKT
Attention (Q, V) Softmax
+ log M)V,
ddh
where Q ZW ER Pxdr, K
Q
ZW K E BRRxx V zW, E Pxd,
(8)
where W W k. and WA e dddx are learnable parameters.
3.2.2 Regulated Relaxation Function for Smooth Approximation
There are still two points with the potential for improvement: 1 DDue to the non-differentiabiliyy of the discrete truncation operation in the step function, 33 cannot be optimized through backpropagation, which limits the flexibility and adaptability of the method. @ Although the modulated attention mech- anism implicitly incorporates temporal positional information by preserving attention coefficients only between time steps with small periodic distances, it imposes uniform inductive biases on nearby steps. As result, the model still faces challenges related to permutation invariance [15].
5

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To address these limitations, we proposed regulated relaxation function to approximate the Heaviside Step function to generate the modulator. In contrast to the sharp Heaviside Step function, our function S exhibits a smooth attenuation trend, shown in Figure 2, which is defined as follows:
Theorem 1. Regulated Relaxation Function
Define a continuous differentiable function S (;a,8): R U (0) - [0, 1] as follows,
exp (-7)
S 07) = + exp (a (7 3)) + 1 exp (a.3)
E 1].
where the regulated parameter a > control the gradient of attenuation and 3 the distance penalty threshold. This function has following properties:
(1) S (y;,, B) is the smooth approximation of H. (3 Ky) for arbitrary Y 2 satisfies,
S (0; a, 3) = (+o0; a, B) 0, Va, >
(10)
(2) The cumulative error upper bound of this smooth approximation satisfies,
2log a
(B - -) (n;
o+
+20) (11)
1 1 expa
where the proof of Theorem is provided in Appendix A.I. Regulated relaxation function takes relative periodic distance matrix I {ry H R²* generate the attention modulation term. Finally, the formula of our attention with Regulated Relaxation Function (RRF) is as follows,
QK
RRF (Q, V) Sootmax
log a(T;,, e R Pxdi,
(12)
Vdh
where Q ZW ER Pxdr K
ZW
Pxdx, V = ZW Pxdi,
(13)
where a and are learnable parameters, and Q,
K, and V are the linear projections Z.
Advantages for Periodic Modeling. O The key distance penalty threshold 33 is defined learnable parameter, which can be adaptively learned from the data. This enables more accurate and dynamic modeling of periodic trend. @ The modulation term varies smoothly with the periodic relative distance, implicitly encoding discriminative positional information of time steps. This effectively addresses the permutation-invariant limitation inherent in the standard attention mechanism. & The function exhibits a smoother trend and can be flexibly controlled through the learnable parameter This adaptability allows Transformer architecture to customize personalized modulation behaviors for time series with diverse periodic characteristics, thereby refining the attention process to better capture periodic dependencies.
3.3 Forecasting and Optimization
The Final time series prediction values is obtained by summing the flattened and linearly transformed outputs of the Transformer backbone Z E Pxd as follows:
Y Flatten Z)W out bout IR
(14)
where W out E ( and Dout E TR. are learnable parameters. When the time series involves multiple variables (i.e., C 1), adopt
learning and compute the relative loss weights based on the maximum loss across channels, dynamically adjusting the loss weights during training promote equal learning with stable convergence. This process is formulated as follows:
L(Y Y.)
L -"=
== max {ILCYY Yallic (1,2,....0)
(15)
C c: Y.Il"
where Y Y, the ground-truth values and prediction values of the channel
Complexity Analysis. MoFo's computational complexity has quadratic dependence on period length but is independent of the input sequence length, making its efficiency highly favorable.

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For example, in datasets with hourly sampling (e.g. 24 time steps per period), the computational cost is minimal relative to the total sequence length in long-term forecasting tasks, such as 720 time steps.
4 Experiment
4.1 Experimental Setup
Datasets. We conduct our experiments on widely used real-world time series datasets with periodic pattern from 4 different domains, including ETTh1, ETTh2, ETTmI, ETTm2, Weather, Solar Energy, Electricity, and Traffic. A summary of all datasets is provided in Table 1.
Table 1: Statistics of used datasets.
Dataset
ETTh1
ETTh2
ETTml
ETTm2
Weather
Solar Energy
Electricity
Traffic
# Channels
7
7
7
21
137
321
862
# Samples
14,400
14,400
57,600
57,600
52.696
52,560
26,304
17,544
Frequency
1 hour
1 bour
15 mins
15 mins
10 mins
10 mins
hour
hour
Split ratio
6:2:2
6:2:2
6:2:2
6:2:2
7:1:2
6:2:2
7:1:2
7:1:2
Settings. Our experiments are conducted on an NVIDIA A100 GPU with 40GB memory, us- ing PyTorch under Python 3.11.5. We implement our method within the TFB platform [34] to ensure a fair comparison. Following the evaluation protocol in TFB [34], we report the best per- formance achieved over look-back window lengths T E {96, 336, 512) and forecasting horizons L E {96, 192, 336, 720}. Model performance is evaluated using two standard metries: mean squared error (MSE) and mean absolute error (MAE). To maintain fairness in evaluation, we disable the "Drop Last" batch-sampling trick[18]. We use the Adam optimizer [14] with the L1 loss function from the FreDF strategy [53].
Table 2: Performance comparisons for LTSF. The best and second best are marked in corresponding colors. All experimental results are selected from the best performance under the look-back window
length T {96, 336, 512].
MoFo
DUET (2025)
PDF
Trranformmrr (2024)
Pathformme (2024)
CycleNet (2024)
TimeMixer (2024)
PatchTST (2023)
Crossformer (2023)
DLinear
Method
(Ours)
Metric
|SS
MAE
|
MSE
MAE MAE|MSE MSE
MAE|
MSE
MAE
MSE
MAE
MSE MAE | MSE MAE| MSE
MAE| MSE
MAE
MSE
MAE
96
10:300
0. 1.389
0.352
0.384
0
0.391
00338
0.405
10.372
0.392
00332
0.394
0.372
0.401
0.377
0.397
00441
0.435
0.379
0.4m3
192
0.397
0.413
0.398
0.409
0.392
0.414
0.424
0.440
0.408
0.418
0.434
0.417
0.413
0.430
0.409
0.425
0.409
0.438
0.408
0.419
0.407
0.424
0.426
0.435
0.449
0.460
0.434
0.400
0.429
0.438
0.433
720
0:447
0.454
0.429
0.455
0.456
0.462
0.495
0.487
0.450
0.463
0.488
0.464
0.446
0.484
0.457
0.477
0.501
0.514
0.471
0.493
10.275
0.334
0.270
0.336
0.276
0.341
002297
0.348
0.277
0.336
0.277
0.341
002811
0.351
00.274
0.337
00.728
0.603
0.300
0.364
E7712
192
0.327
0.373
0.332
0.374
0.339
0.387
0.372
0.403
0.345
0.380
0.341
0.385
0.349
0.387
0.348
0.384
0.723
0.607
0.387
0,423
3566
0.400
0.353
0.397
0.374
0.406
0.388
0.417
0.37%
01408
0.370
0.411
0.366
0.413
0.377
0.416
0.740
0.628
(1449)
0..87
720
0..379
0.425
0:382
0.425
0.398
0.433
0.424
0.444
0.437
0.455
0.424
0.451
0:401
0.436
0.446
0.441
1.386
0.882
0.704
0.597
96
10.286
0.335
0.279
0.333
0.286
0.340
0.300
0.353
00.290
0.335
[0:297
0.311
[0.293
0.345
0.2889
0.343
00.314
0.367
0.300
0.345
192
0.320
0. 16.63
0.320
0.35%
0.521
0.364
0.341
0.390
0.357
0.363
0.332
0.365
0.335
0.372
0.329
0.368
0.374
0.410
0.336
0.366
336
0.347
0.382
0.348
0.377
0.354
0.383
0.374
0.396
0.374
0.384
0.366
0.386
0.365
0.386
0.362
0.390
0.413
0.432
0.367
0,386
00.388
0:405
0.408
0.408
0.415
0.429
0.430
0.428
0.416
0.417
0.411
0.422
0.412
0.413
0.753
0.419
677m2
96
10.155
0.240
[0.161
0.248
0..13
0.251
10.175
0.266
10.164
0.250
0.157
0.247
[0.165
0.256
0.165
0.255
00.296
0.391
0.164
0.255
192
0.211
0.283
02214
0.267
0.219
0.290
0.242
0.312
0.219
0.288
0.214
0.286
0:225
0.298
0.221
0.293
0.369
0.416
0.224
0.304
0.158
0.314
0.267
0.320
0.282
0.337
0.267
0.319
0.269
0.322
0.277
0.588
0,337
720
0.342
0.368
0
0.348
0.374
0.349
0.382
0.375
0.394
0.361
0.377
0.363
0.382
0.350
0.387
0.362
0.381
0.750
0.612
0.371
0.401
Weather
00.41
0.186
10.16
0.191
0.147
0.196
0.115
0.207
0.144
0.195
0.166
0.222
0..14
0.198
0..550
0.200
0.143
0.210
0.170
0,230
192
0.186
0.230
0.188
1.:11
0.193
0.240
(.200)
0.248
0.191
0.235
00211
0.259
0.192
0.243
0.191
0.239
0.195
0.261
0.216
0,273
336
0.233
0.234
0.268
0.245
0.280
0.282
0.287
0.243
0.274
0.262
0.291
0.247
0.284
0.242
0.279
0.254
0.319
0.258
0.307
720
0.312
0.331
0.305
0.319
0.323
0.334
0.320
0.356
0.318
0.326
0.329
0.338
0.318
0.330
0.312
0.330
0.335
0.365
0.323
0,362
96
0.169
0.214
0..69
0.195
0.181
0.247
(.199)
0.244
20.288
0.235
[0.201
0.252
101779
0.232
0.177
0.234
0.183
0,200
0.199
0.265
192
0.177
0.251
0.187
0.207
0.199
0.257
0.195
0.257
0.196
0.220
0.221
0.261
0.201
0.259
0.2004
0.302
0.208
0.226
0.220
0.282
3986
0.186
0.258
0.199
0.213
0.208
0.269
0.203
0.266
0.195
0.220
0.233
0.269
0.190
0.256
0.212
0.293
0.212
0.239
0.234
0.295
0.193
00202
0.212
0.223
0.208
0.257
0.236
0.271
0.205
0.215
0.215
Electriity
96
10.122
0.215
(0.188
0.219
00.188
0.222
00.134
0.230
10.135
0.222
0.126
0.221
[0.153
0.256
0.143
0.247
0.247
00.134
0.231
0.111
0.237
192
0.140
0.234
0.145
0.235
0.147
0.212
0.154
0.250
0.157
0.253
0.144
0.239
0.168
0.269
0.158
0.260
0.146
0.243
0.154
0.251
0.157
0.252
0.163
0.169
0.265
0.170
01267
0.161
0.253
0.189
0.264
0.169
0.268
00.191
0.284
0.19%
0.281
0.199
0.289
0..194
0.288
0.211
0.302
0.119
0.286
00.228
0.320
0.214
0.307
0.237
0.314
0.2204
0.301
10.362
0.247
0.360
0.238
0.368
0.252
0.363
0.265
00388
0.250
0.389
0.276
0.369
0.257
0.370
0.262
0.526
0.288
0.395
0.275
Tralicc
0.379
0.261
0.384
0.273
0.257
0.406
0.280
0.400
0.272
0.280
336
0.390
0.258
0.395
0.259
0.393
0.268
0.396
0,277
0.424
0.265
0.425
0.291
0.407
0.272
0.396
0.275
0.505
0.276
0.417
0.286
720
0.424
0.281
0:135
0.278
0.138
0.297
0.44
0.308
0.452
0.283
0.150
0.303
0:161
0,316
0.135
0.295
0.552
0,301
0.154
0,308
7

---

Baselines. We compare MoFo with 17 advanced baselines in long-term time series forecasting
comprising DUET [37]. PDF [7], iTransformer [24]. Pathformer [5], CycleNet [20], TimeMixer [59].
PatchTST [32], Crossformer [75], DLinear [73], NLinear [73]. FITS [68], FiL. [78], MICN [57],
FEDformer [79], Triformer [6], Non-stationary Transformer [23], and Informer [77].
4.2 Forecasting Performance of MoFo
Experimental results are summarized in Table 2. Due to space constraints, we compare against a larger set of baselines in Appendix D.3. CycleNet explicitly models periodic patterns, yet achieves lower forecasting performance compared to DLinear, which merely employs simple fully connected layers. TimeMixer, on the other hand, utilizes multi-scale modeling techniques comprehensively capture complex temporal dynamics, Compared to methods such as PatchTST, Pathformer, and other approaches that rely on continuous patching strategies, our model achieves superior forecasting performance. This improvement is attributed to the proposed deriodicity-based Discrete Patching, which enables the model to better capture temporal dependencies across time steps. PDF introduces a multi-scale decomposition framework that models temporal dependencies from both long-term and short-term perspectives. DUET employs bidirectional clustering over both temporal and channel dimensions to adaptively capture spatio-temporal dependencies, achieving the best performance among existing baselines. However, it does not explicitly model periodic patterns and suffers from high computational complexity. Benefiting from the Period-Aware Modulator, our model focuses explicitly on periodic pattern learning, leading to the overall best forecasting accuracy. These results demonstrate the effectiveness of our design choices in capturing Long-range periodic dependencies.
4.3 Efficiency Analysis of MoFo
We compare the computational efficiency with Transformer based baselines on the Traffic dataset. As shown in Table 3 and Table 11, MoFo achieves the best forecasting accuracy among all Transformer- based models. while demonstrating the lowest computational complexity and highest efficiency. Similarly, PatchTST, which also adopts patching strategy, employs an independent channel learning approach that significantly increases model complexity. Its parameter count grows by more than 10x, FLOPs increase by 25x, and training speed slows down by 17x. Pathformer enhances prediction accuracy through dynamic path adaptation and a patching strategy, albeit at the cost of increased learning overhead. Compared to DUET, one of the por-prrfrnnnng baseline models, MoFo reduces the number of parameters by more than 3x, the computational cost by more than 10x, and significantly lowers both memory consumption and training time. This is because our model's 's complexity grows quadratically with the period length, which is significantly shorter than the input sequence length. Moreover, show that only single Transformer layer is sufficient for effective learning.
Table 3: Efficiency comparison of MoFo and SOTA baselines with I, = 720 in Traffic dataset. All
results of each model are under the optimal hyperparameters for fair comparison. Parameters: All learnable parameters requiring gradient descent. MACs: multiply-accumulate operations. FLOPs: floating point operations. M: Million (10%). B: Billion (10"). T: Trillion (10*2). MB: Megabyte. s:
Second. indicates the relative percentage increasing regarding MoFo.
Models
MSE
# Parameters
MACss
FLOPS
Memory Usage
Epoch Time
Traffic LL T20]
Crossformer
3.23
54.20%
85.03
B
13,556
171.77%
368 Syoaa 1y10
PaccTTT
27.8 1058
2.02
Tpsourms
2.08
44,782
297.0955
876 Sy 535
Pathformer
0.452
9.61 M 47300-095
110.46
B Bearans ams
117.76
Besa
36,602 MB
voss 8055
1 Sei200
iTansformer
0.445 (000
5.37 M. 128.789
297.96 B 1200
446.30
19,608 MB
52
PDP
0.40
2.48 M M-20055
637.05
R R0r0
662.47
H
38,014 MB
002
76
DUET
2.435
11.2 0997
137.33
B 2005
975.96
B arm
75,616 1416.305
516
Syoor 87%
MoFo
0.424
2.40
74.96 B
76.74 D
4,988 MD
47
4.4 Hyperparameters Sensitivity Experiments
We investigate the sensitivity of MoFo to its two core hyperparameters the number of Transformer layers and model dimensionality- horizon L E (96, 192, 336, 720}. we use the best-performing hyperparameter configuration and vary only the target hyperparameter, as shown in Figure 4.4. We report both MSE and MAE for evaluation. The number of Transformer layers is varied from to 6, and the model dimensionality is tested in
the ETTh2 and Electricity datasets. For each forecasting
8

---

The number of Transtormer Layers
The number of dimensionality
ET122
E1122
Electricity
Electrictty
C
12
1 96
L =
192
=
336
=
L
720
96
Figure 3: Hyperparameter sensitivity experiments.
Ours
Cpatch
Sinuous Pos
Leaamalle Pos
Modulator
0.450
0,425
0
MSB
0.20
0.19
Trafficc
ETT22
hm
BTTm2
Weather
Solar-Energy
Ellctricity
Figure 4: Ablation study on all datasets with forecasting horizons L 712.
the range 64 (ETTh2) and 64 304 (Electricity), with the number of attention heads fixed at H 4. Results show that MoFo achieves strong performance even with a single Transformer layer, while additional layers yield marginal gains. This suggests that the model efficiently captures inter- patch dependencies without requiring deep architectures. Furthermore, increasing the dimensionality generally improves performance; however, the gains plateau beyond certain threshold, indicating diminishing returns from further capacity expansion.
4.5 Ablation Study
We design ablation experiments validate the soundness of the each component of MoFo: '+ Cpatch' which used continuous patch technology; '+Sinuous Pos' and 't Learnable Pos' which use Sinuous and Learnable position instead of our periodic relative position, respectively; "- Modulator' which removes the period-awart modulator; '+ Mean Loss' which only use L1 loss function. As shown in Figure 3, the ablation study reveals that 'Moodulaoor' variant achieves the worst performance. This because the Period-Aware Modulator plays crucial role in guiding the model to focus on extracting periodic patterns. '+epatch' variant also suffers from higher prediction errors, which can be attributed to the fact that our proposed discrete patching strategy enables direct modeling of dependencies among periodically aligned time steps. In summary, all variants perform worse than MoFo model to varying degrees, demonstrating the effectiveness and necessity of each component in capturing long-range periodic dependencies.
4.6 Scalability for Look-back Window of MoFo
We evaluate the scalability of MoFo under varying look-back window lengths. Specifically, we construct an ultra-long look-back setting on the ETTm2 dataset, where the input sequence length reaches up to 10K, one hundred times of the forecasting horizon: L 96, T 96 (5k), k = (1,2.... 20}. We report the maximum memory consumption during training, per-epoch training time, and FLOPs. As shown in Figure 5, as the look-back window increases, both DUET and PTt exhibit a sharp rise in memory usage and training time, indicating that their computational complexity scales closely with the input length. In contrast, even when the input sequence length grows by 100x, MoFo sees less than a 10% increase peak memory usage and less than 2% increase in training time. Its computational cost depends primarily on the period length rather than

---

Maximum Memory Usage (MB)
Training Speed (s/epoch)
Flonting Point Operations (FLOPs)
5000
Malto PucATST
30
1.5
PackiST
25
1.0
2000
1500
0.5
1000
900
0.0
4000
6000
8000
10000
10000
4000
5000
Figure 5: Scalability for look-back window of models ETTm2 dataset.
full sequence length. These results highlight the strong scalability MoFo and demonstrate its great potential for modeling ultra-long sequences with minimal overhead.
Attention Visualization
We extract and visualize the attention coefficient matrices from models trained four datasets with periodic lengths (P) of 24, 96, 144, and 24, respectively, all spanning a time window of one day. As shown in Figure 6, the coefficients in each attention head exhibit distinct periodic patterns. Notably, when the cyclical distance between two time steps is the largest (i.e., equal to P/2), the attention scores reach their minimum values. These attention coefficients that align with the underlying periodic trends significantly enhance the model's performance in long-term time series forecasting.
ETThl
ETTm
Weather
Traffic
0:90
435
0.0
Figure 6: Visualization of attention scores.
Conclusion
In this paper, we propose MoFo, a novel framework for time series forecasting that leverages the inherent periodic structure of temporal data to explicitly model both periodic correlations and tem- poral trends. Through our proposed Period-strrctturd patches, the model able to directly capture correlations among time steps that share the same phase across periods. We further introduce period-aware modulator, which enhances the attention mechanism with an adaptive inductive bias guided by underlying periodic trends. Experimental results demonstrate that MoFo achieves competi- tive forecasting performance compared to state-of-the-ar methods, while significantly improving computational efficiency and scalability, especially for long input sequences.
Acknowledgment
This paper is partially supported by the National Natural Science Foundation China (No. 12227911.. The AI-driven experiments, simulations and model training were performed on the robotic Al Al-Scientist platform of Chinese Academy Sciences.
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[72] Wenzhen Yue, Yong Liu, Xianghua Ying, Bowei Xing, Ruohao Guo, and Ji Shi. Freeformer: Frequency enhanced transformer for multivariate time series forecasting. arXiv preprint wrXiv:2501.1398) 2025.
[73] Ailing Zeng, Muxi Chen, Lei Zhang, and Qiang Xu. Are transformers effective for time series forecasting? In Proceedings of the AAAI conference on artificial intelligence, volume 37, pages 11121-11128, 2023.
[74] Biao Zhang and Rico Sennrich. Root mean square layer normalization. Advances in Neural Information Processing Systems, 32, 2019.
[75] Yunhao Zhang and Junchi Yan. Crossformer: Transformer utilizing tross-dimension dependency for multivariate time series forecasting. In The eleventh international conference on learning representations, 2023.
[76] Zheng Zhao, Weihai Chen, Xingming Wu, Peter CY Chen, and Jingmeng Liu. Lstm network: deep learning approach for short-term traffic forecast. IET Intelligent Transport Systems, 2017.
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15

---

A Mathematics Justification
A.1 The Proof of Theorem
Proof of (1); For arbitrary a, B > O, there are S (0; a, = 1,8 (+00; = O.
In fact.
1
exp (-0)
1
S a0; 1 1+ exp (a (0)
+ 1 1 exp (a3)
+ exp (a.) + exp (xp(-a3)
(16)
exp(a.))
1
+
1.
1 exp (a:3) 1+ exp (a:3
and
exp (-00
1
0
8 (+00; a,
1 1 exp (a (+00 - B)) +
+
0. (17)
1+ exp (aB)
+ 00
1+ exp (a
Proof of (2): The upper error bound of the smooth approximation satisfies,
1 (3 (1; a, 3) S
log
1
+0 (a + +00).
(18)
a
e
In fact, the Heaviside step function H(3 7) can be viewed as the differentiation of the maximum value function max (0, 33 7 as follows,
d
d max - 7
- H(3 - 7) d(3 >) ma -3 = dr
(19)
Our sigmoidal attenuation function can be seen follows,
dl log (1 + exp(a - -))) d 1 + cxp (a3)
exp (-7)
S (); = dr a
(20)
Hence, the cumulative error Jt H7 (B - 7) - S (1: B) dr satisfies,
[B -2-) 7) - (1 dy
H (B -7) ()
13 ((8--) 7) a
l HB 7) -s(7,, dr
9
+00
-H ())) a
9 d
dr log (1 + exp (a(B 7))) + d exp (-7) dp
d 1
max 3- +
Jo
dy
dy + exp (a.)
to0 d dy
d
d exp (-7)
max }}
log (1 exp exp (a(3 7))) d7 + exp (a:3) dr
dr a
(21)
7) log (1 (a 7))) + exp (-7) max
1
a
1 exp (a:3)
[max
- -
1
log (1 + exp (a(3 7))) 1 exp (a.3)
(-7)
1000
log2
exp (-8)
1
0 +
+ 3 a
log (1 + cxp((()))) - 1 exp (a:3)
a
1 exp (a.
log 2
exp (-3)
0 0 - 0 0 +
+ exp (a.3)
2 logg
2 2exp (- 3)
1
+ 3 a
log (1 + exp (a.3).
a
+ exp (a:3)
16

---

Since,
a3 = 3 B 0,
B log (1 + exp (a:3)) < 3 log (exp (a.3)) =
(22)
and
2 log 2
eexp (-3) -1
2 log
1
2log 2
a 1 exp (a:3) - a
(23)
1 1 exp (a:3)
a
1 exexxpp (a)
then the the cumulative error has upper bound satisfies
J (3 S a,
2log2 dr 3) a
+ e
+0
+ +0o).
(24)
A.2 Autocorrelation Function and Bartlett's Test with Null Hypothesis
Autocorrelation Function (ACF). The Autocorrelation Function [3] measures the linear correlation between a time series and a lagged version of itself. For the complete observable time series values X E IR with total observable time steps Nr, the autocorrelation function Ag of X at lag the ratio of the estimator of the covariance between the time series and the series lagged by k to the estimator of the variance the time series as follows:
is
N,- [X, (X 11) [-1,11 T=
A
(25)
Σ
X - 11)
where μ Ni EES X E is the expected value of X.
Bartlett's Test with Null Hypothesis. The autocorrelation coefficients Au can be viewed as random variables. However, even in the time series consisting of pure random noise, there may exist non-zero autocorrelation coefficients at some lags [39]. Hence, we require method to ascertain whether an observed autocorrelation A represents truly non-zero population or is merely due this inherent randomness. This is typically achieved by performing confidence intervals derived from Bartlett's test [2] on null hypothesis [38]. The null hypothesis denoted as Ho is a fundamental concept in statistical inference [1]. Its primary purpose is serve as a base assumption for hypothesis testing. In that the ground-true autocorrelation coefficient A₂ at a specific lag k is zero. The Bartlett's 's test constructs confidence intervals to test this null hypothesis for individual lags. These intervals are based on an estimate of the standard deviation of Aş under the assumption that Ho is true. Specifically, the 1 confidence interval for the autocorrelation A& under the null hypothesis Hp : A, is centered by =Za/2 SE(AA). Using Bartlett's 's formula for the variance of Ag. the standard error SE(AA)) is approximated by:
context of ACF, the relevant null hypothesis is Ho : AA
with boundaries given
SE(At)
(1
+
A7),
(26)
N
j=1
where A are the autocorrelations for lags j = 1.2,..., --1 and Za/2 is the 1 a/2 quantile of the standard normal distribution (e.g., Zo 025 R= 1.96 for a 95% confidence interval, corresponding to
0.05) [61]. Thus, the approximate a confidence interval boundaries for testing Ho : Ar
are:
1 N +
Za/2 2
(1
Σ A3), +2
(1
2
A3)
(27)
N
Any autocorrelation value A₁ that falls within this confidence interval is considered consistent with the null hypothesis (Ho AA = 0). In such cases, we do not have sufficient statistical evidence to
II is important to note that the alpha a here is different from the learnable parameter alpha mentioned in the RRF in the MoFo. The alpha a here is specific parameter notation used in statistics [41]. We maintain consistency here reduce potential confusion with specialized terminology.
17

---

conclude that the true autocorrelation at lag k is different from zero; the observed A₁ is likely due to random variation inherent in noise process. Conversely, if Ar exceeds these boundaries, we reject the null hypothesis and conclude that the autocomelation at lag & is statistically significant.
We present in the Fig the average ACF values and from some selective channel for all datasets used in this study, calculated across each channel with maximum lag of 512, and report the null hypothesis region with 95% confidence interval. We observe that the same positions within each period of the time series, the ACF values exhibit significant peaks. Therefore, it is highly rational to apply patching based on periodic positions for the time steps. The orange region represents the 95% confidence interval for the null hypothesis from Bartlett's 's Test [2]. Correlation coefficients that fall within this interval are not statistically significant and cannot be rejected as noise [4]
B Related Work
Time series modeling serves as a core task in numerous domains, such transportation and atmo- spheric science [21, 29, 44, 45, 46, 47, 48, 49, 69]. Early approaches primarily relied on recurrent neural networks (RNNs) and temporal convolutional networks (TCNs). recent years, models based on multilayer perceptrons (MLPs) have garnered significant attention due to their lightweight and efficient performance.
RNN-based Models. RNNs, among the earliest deep learning architectures for sequential data, have been widely adopted for long-term time series forecasting, with notable variants such LSTM [76] and GRU [8]. To mitigate the problem of too many recurrent steps, SegRNN [19] introduces a segmented recurrence mechanism combined with parallel multi-step prediction, substantially cutting
down the number of iterations.
TCN-Based Model. TCNs employ convolutional operations to effectively model local contextual patterns in time series, offering good trade-off between computational efficiency and forecasting accuracy. Recent advances have extended TCNs to better capture long-range temporal dependencies. For instance, ModemnTCN [27] adopts large convolutional kernels to greatly expand the receptive field, allowing the model to capture broader temporal structures. Likewise, Pyraformer [22] integrates TCN layers with Transformer framework; it uses stacked TCN layers for downsampling to obtain coarse-grained time series representations, which then processed by the Transformer enhance both scalability and performance.
MLP-Based Model. MLP--baed models, when thoughtfully designed, have shown strong perfor- mance in time series forecasting. DLinear [73] illustrates this by using a moving average kernel to decompose the input series into trend and scasonal parts, each modeled separately by dedicated linear layers. PatchMLP [42] adopts a patching strategy that incorporates channel mixing to improve cross-variable information exchange. Extending this idea, HDMixer [11] employs adaptive patch lengths to capture both intra-patch short-term dynamics and inter- inter-patch long-term dependencies while modeling intricate variable interactions. Meanwhile, FITS [68] operates MLPs in the frequency domain, leveraging spectral analysis to emphasize dominant signal components and better capture global temporal relationships.
C
The Transformer Layer in MoFo
We the pre-norm Transformer Layer [43, 56) of multi-head attention with Regulated Relaxation Function (Eq. 12).
Vanilla Transformer Layer. Transformer [43] consists of the self-attention function MultiHcad(.)) with feedforward networks F-F(-),, and two distinct normalization layers Norm; Assuming the input hidden representation is Z Pxd with period length P and model dimensionality d, the output hidden representation Z e iR nx of one Transformer layer as follows,
Ž MNdro (Z)) Z,
(28)
Z FNo(N (Z)) Z.
Here we depict the pre-norm structure [56]. The multi-head attention mechanism is used in Trans- former to improve the representation performance. Let Z -Nomm (Z), the multi-head attention
18

---

function is weighted combination of outputs from different head as follows,
(alti-Head(2)) ( Concat Concat(hcadee (head1. heady, ...,, headd Wo E R Pxd h Attention Zw q. 2w K. Zw;, E R Pxds
(29)
where H is the number of heads. W W W Rdxdd and Wo E Rdxdd are learnable projections parameters with head dimensionality d, = d/H. And self-attention function vanilla Transformer is defined follows,
QK dd
Attention (, K, V) = Softmax
VER
(30)
where Softmax is an exponential activation with 4 normalization [9] in the last dimensionality. And the attention scores between the i-th token and all other tokens after softmax operation follows,
Q.K
Q.K
exp
Softmax
Van
ERP.
(31)
Q.K
1ー= exp
Vah
Modification in MoFo. However, we leverage Regulated Relaxation Function to instead the attention function in MoFo introduced Section 3.2.2 as follows,
QKT
Attention (Q. K, V) =RRF (Q, K, V) Softmax
+ log a( E R Pxds
v vdh.
(32)
where ZW ER Pxda K zw E RRPx V ZW E Pxd,
The feedforward networks in MoFo are gated linear units [40] with Swish activation.Let 2 Norm, the FFN(-) is defined as follows,
FFN (SwiGLU(2W, +bi) @ (ZW2+ b2)) W3 by E Pxd
(33)
where weight matrices WT, W2 E Rax4d W3 E Radxd and bias parameters WT, W2 e Rdd W3 E Id are learnable. The normalization layer is root mean square normalization [74] as follows,
Z 2 gi
2 g2
Norm, (Z) =
E Pxd
Norm2(22 (2)
E ER
(34)
V Z.,
Vi Z
where B1, 82 E Rd are learnable scale parameters of normalization.
D
Experiments
D.1 Dataset Analysis
We further visualize the temporal correlations across multiple datasets. As shown in Figure 7, most datasets exhibit clear periodic patterns, with relatively high correlations between time steps separated by fixed intervals. This suggests that commonly adopted sequential input strategies- such as those used in patch-based methods tend group temporally adjacent but semantically unrelated time steps, thereby hindering the model's 's ability to capture intrinsic periodic structures.
D.2 Settings
Our experiment is based on the TFB platform [34] for a fair comparison. Following the set- tings in TFB [34], we report the best performance within the optional historical sequence length T' E {96, 336, 512} of the multiple forecasting length L E {96, 192, 336, 720} with two common generic metrics including the mean square error (MSE) and mean absolute error (MAE) judge the performance of our model. Our experiments are executed on an NVIDIA A100 with 40GB memory. Our code environment is based on the PyTorch framework using Python 1.1. The 'Drop Last' trick is closed to ensure a fair comparison [18]. We adopt Adam [14] optimizer. The training process guided by the L, loss function of the FreDF strategy [53]. The penalization distance parameter (3 > 0 is restricted in (0, P). We utilize only one layer of Transformer with attention head H = 4 for each setting in all datasets. LTSF datasets often have multiple channels (or variates), and we adopt the channel independence approach [32] simultancous independent learning of all channels.
19

---

Ascoomtti Fooie W31
Alcardil Fooee W01
Adoccn alc Noy 903
P 24
Average
Channel
Channel
#5
Ascombtiee frode 1em
ETThZ
P P=24 24
Avarage
Channol
Chanael #2
Channel #1
ETTml
P P 96
Average
Chanñel #3
Channel #9
Channel #5
ETTm2
p p= 96
Average
Channel 01
Channel #2
Channel M1
1433)
Kallo Fastiee (AC))
Adlooesiiio Powile 9001
Woather
P P= 144
Average
Channel
Channel #17
Channel 820
Solar
=
144
Average
Channell
Channel #100
Channel #136
201
201
Eloctricity
P p=2 24
Average
Channel
Charnel 4100
Channel
Adk uoneobaiii Perate 1901
Traffic
P=24
Average
Channell 82000
Channel #500
Channel #800
Figure The autocorrelation visualization of all datasets.
20

---

Table 4: Performance comparisons for LTSF. The best results are marked in corresponding colors. All experimental results are selected from the best performance under the historical sequence length
T E {9,, 336, 512).
MaFo (Ours)
FITS (2023)
Niinaar (2023)
TimesNet (2022))
FRDiormer (2022)
Triformer (2022)
MICN (2022)
FILM (2022)
Stationary (2022)
Informer (2021)
Method
Metric
MSE
MAE]
MSE
MAE| MSE
MAE|
MSE
MAB|
MSE
MAE MSE
MAE MSE
MAE] MSE
MAE|
MSE
MAE|
MSE
MAE
96
0.360
0.389
01.376
0. 0.396
0.385
03,403
0.389
0,412
0.379
0.419
0 09399
0.425
0.37%
0,412
0.370
0.394
(1.91
0.524
0.571
0.39
0.397
0.413
0.400
0.418
0:122
0.126
0.140
0.413
0.420
0.449
336
0.407
0.424
0.419
0:435
0.431
0.429
0.523
0.487
0.445
0.466
0.492
0.479
0.428
0.447
0.434
0.435
0.632
0.551
0.588
0,492
720
0.447
0.454
0.435
0.458
0.439
0.452
0.521
0.495
0.474
0.488
0.549
0.529
0.474
0.499
0.463
0.474
0.828
0.658
0.623
0.519
0.273
0.334
(0:277
0:345
[0:276
0.338
0.334
0.370
00.337
0.380
0.936
0.660
0.313
0.372
0.2282
0.346
(1.47
0,936
117T2
192
0.327
0.373
0.331
0.379
0.345
0.382
0.404
0.413
0.415
0.428
1.290
0.768
0.419
0.439
0.358
0.401
0.379
0.418
0.448
1,290
336
0.361
0,405
01.350
03.3%
0.366
01408
0.389
01443
0.389
0.457
1:325
0.781
0.477
0.475
0.372 0.425
01358
0.413 0.464
1.325
0.425
0.382
0:425
00:10
0.111
0.134
0.418
0.418
0.188
1.500
0.723
0.600
0.425
0.412
96
0.286
0.335
00.303
0.345
[0:301
0.343
0.340
0.378
(0.463
0.463
0.349
0.388
0.303
0.349|
0.301
0.343
0.415
0.410
0.422
0.349
B77m2 mm7mm
192
0.320
0.363
0.337
0.365
0.355
0.379
0.392
0.404
0.575
0.516
0.387
0.410
0.336
0.369
0.339
0.365
0.494
0.451
0.480
0.387
0.347
0.382
00,368
0.384
(:372
0.385
014233
0,426
00.118
0.544
0.426
0.446
0.370
0.391
0.174
0.385
(1577
0.490
0.531
0,426
720
0.388
0.411
0.420
0.413
00440
0.418
0.:45
0.453
[0.612
0.551
0.482
0.476
0.410
0.421
0.423
0.414
0.636
0.535
0.578
0.482
0.155
0.240
00.165
0.254
[0.163
0.252
0.189
0.265
0.211
0.309
0.276
0.344
0.173
0.254
0.210
0,276
192
0.211
0.283
0.219
0.291
0.211
0.290
0.254
0.310
0.297
0.360
0:473
0.453
0.232
0.313
0.220
0.291
0.338
0.373
0.365
0.473
336
0.255
0.114
0.222
0.326
(0.273
0.326
0.313
0.345
0.366
0.400
0.692
0.549
0.303
0.367
0.277
0.329
0.432
0.416
0.414
0.669
720
0.342
0.368
0.359
0.381
0.366
0.382
0.413
0.402
0.459
0.450
1.936
0.856
0.467
0.477
0.363
0.386
0.584
0.476
0.468
1.936
Weaher
96
0.141
0.186
[1.172
0. 0.225
(0.180
0,226
0.168
0.214
0.229
0.298
0.170
0.236
0.172
0.232
0.17%
0.229
00.188
0.242
0.256
0.170
192
0.186
0.230
0.215
0.261
0.218
0.261
0.219
0.262
0.265
0.334
0.216
0.277
0.214
0.271
0.218
0.263
0.240
0.290
0,300
0.216
0.233
0.272
0.2661
0.295
0.266
0.296
0.278
0.302
0.330
0:272
0.324
0.259
0.266
0.295
0.322
0,272
720
0.312
0.331
0.326
0.341
0.334
0.345
0.353
0.351
0.423
0.418
0.350
0.379
0.309
0.343
0.332
0.341
0.396
0.378
0.388
0.350
0.169
0.214
00.208
0.255
002022
0.245
0.19%
0.270
00.485
0.570
0.225
0.279
0,190
0.25
0.214
0.259
0.366
0.390
0.368
0.225
Solar
192
0.177
0.231
0.229
0.267
0..233
0.258
0.206
0.276
0.415
0.477
0.250
0,295
0.226
0,284
0,236
0.257
0.400
0.386
0,388
0,250
396
0.186
0.238
(1.241
0.273
012258
0,265
0.208
0.284
1
1,008
0.839
0.261
0.297
0.259
0.90%
0.241
0.265
01.41
0.394
0:420
0.261
720
0,193
0.248
0.211
0.277
0.246
0.268
0.232
0.291
0.655
0.627
0.259
0.292
0.341
0.365
0.241
0.268
0.379
0.377
0.105
0.259
Eleerritty
96
0.122
0.215
[0.139
0.237
[0.140
0.236
0.169
0.271
[0.191 0.305
0.2011
0.298
0.158
0.266
0.154
0.246
00.17
0.274
0.321
0.201
192
0.140
0.234
0.115
0.250
0.155
0.248
0.180
0,280
012203
0.316
0.209
0.307
0,175
0.287
0,168
0.261
0.180
0.283
0.362
0,209
396
0.157
0.252
(1.170
0.268
0.177
0.264
0.204
0.304
0.2211
0.333
0.225
0.323
0.184
0.296
0.189
0.284
0.204
0.225
720
0.191
0.284
00212
0.304
002210
0.297
0.205
0.304
00,259
0.364
0.264
0.353
0.200
0.310
0,249
0.340
0.221
0.319
0.525
0.264
96
0.362
0.247
0,,47
0.290
(0.395
0.272
0.595
0.312
0.593
0.365
0.589
0.323
0.517
0.313
0.412
0.284
0.603
0.330
0.392
0.589
Traffcc
192
0.379
0.254
01418
0.2944
(14007
0.277
0:613
0.322
0.611
0:381
0.597
0.325
0.526
0.302
0,415
0.285
0611
0.338
1,280
0,597
336
0.390
0.258
0.433
0.308
0.417
0.282
0.626
0.332
0.627
0.389
0.617
0.332
0.545
0.307
0.430
0.299
0.628
0.342
0.477
0.617
720
0.424
0.281
01,486
0.347
0.453
0,302
0.635
0.340
01.646
0.394
0.650
0.350
0,569
0,328
0,525
0.371
01646
0.350
1,294
0.650
D.3 Performance Comparison with More Baselines
Considering readability, we only compared our approach with some representative SOTA baselines in Section 4.2. Here, we include more additional LTSF baselines to provide a more comprehensive eval- uation of the performance of MoFo. Specifically, we add the following baselines: ML.P-based models including FITS [68] NLincar [73] and FiLM [78]; TCN-based Models including TimesNet [63] and MICN [57]; Transformer- based models including FEDformer [79]. Triformer [6], Non-stationary Transformer (Stationary) [23] and Informer [77]. As shown in Table 4, NLinear remains a powerful baseline as linear model. FITS and FiLM extract time series representation in the frequency domain and in combining Legendre memory models, respectively. MICN utilizes convolutional networks to capture local and global contexts, while FEDformer enhances the Transformer in the frequency domain. Triformer introduces efficient triangular attention with convolutional down-sampling on coarse-time series, and Non-stationary Transformer (Stationary) focuses on addressing the non- stationarity of time series. Finally, Informer effectively tackles the computational challenges of long sequence prediction through its ProbSparse attention and distillation techniques. However, MoFo demonstrates superiority across almost all metrics comparing to all the baselines by reasonable beriod-based Discrete Patching strategy. The Modulator in MoFo not only dynamically models the periodicity of time series data but also addresses permutation invariance, enhancing the representation capabilities of the Transformer in MoFo.
D.4 Performance None Periodicity Datasets
To further evaluate the generalization ability of MoFo beyond strictly periodic signals, we investigate its performance on datasets that lack explicit periodicity of a publicly available Influenza-Like Illness (II.I) dataset released by the U.S. Centers for Disease Control and Prevention (CDC), which contains weekly reports of the proportion of II--related visits from 2002 to 2021. The III dataset exhibits weak or no clear scasonality, making it an appropriate dataset for testing models under
21

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non con-periodic temporal conditions. We forecast future prediction using four prediction horizons: L E {24, 36, 48, 60) with optional look-back window T E (36, 1043.
To emphasize the modeling of non-periodic time series, we follow the preprocessing strategy of TimesNet [63] by identifying the main pseudo-period as the reciprocal of the dominant frequency derived via Fast Fourier Transform (FFT) for each input sequence. As shown in Table 5, although MoFo was originally designed to leverage explicit periodic structures, its superior results on ILI demonstrate strong adaptability and robustness in capturing complex temporal dependencies even in the absence of clear periodic patterns. This highlights that MoFo not only excels in periodic forecasting but also generalizes effectively to irregular, non-stationary time series domains.
Table 5: Performance comparison on non-periodicity dataset ILI.
Methods
MoFo
FITS
Temes.et
Metries
MSE
MAE
MSE
MAE
MSE
MAE
24
2,113
0.927
2.176
0.928
2.255
0.936
36
1.952
0.924
2.166
0.993
2.132
0.940
1.714
0.824
2.011
0.928
2.182
0,944
60
1,800
0.906
2.010
0.967
2.169
0.940
D.5 Ablation Study on Padding Strategy
Our padding strategy, detailed in Section 3.1 formalized in Eq. 1, is designed to preserve temporal continuity when the sequence length T is not an integer multiple of the detected period length P. Specifically, we start from the current time step and move backward to delineate periods of length P. Any prefix that does not form complete period is left-paddes with the leftmost elements of the nearest full period on its right. This scheme ensures that all time steps participate in subsequent computations without discarding boundary information. To evaluate the effect of this design, introduce a variant termed *+ Zero Padding', where incomplete periods are instead filled with zeros. We conduct a comparative analysis the ETTml and ETTm2 datasets under identical settings. The results are summarized in Table 6, where the best best- pest-performin metrics are highlighted in bold.
Table 6: Ablation Stuyy on Padding Strategy
Methods
MoFo
+ zeros padding
Metric
MSE
MAE
MSE
MAE
96
0.286
0.335
0.292
0.345
ETTm2 ETTml
192
0.320
0.363
0.328
0.372
336
0.347
0.382
0.348
0.386
720
0.388
0.411
0.401
0.435
96
0.155
0.240
0.1.66
0.247
192
0.211
0.283
0.215
0.293
336
0.258
0.314
0.261
0.318
720
0.342
0.368
0.345
0.372
Empirically, our proposed period-aware padding consistently outperforms zero padding across both benchmarks. We attribute this improvement to its ability re-use the immediately preceding historical patterns, thereby maintaining local temporal coherence and facilitating smoother periodic transitions. In contrast, zero padding introduces abrupt discontinuities that disrupt the temporal rhythm, leading to inferior generalization. These findings confirm that our padding mechanism not only preserves data integrity but also serves as an implicit temporal regularizer that enhances long-tern termtermmnn forecasting stability.
D.6 Look-back Window Sensitivity Experiments
In time series forecasting, the look-back window length-i.e., the number of historical steps fed into the model-is a critical hyperparameter that directly affects performance. Different architectures exhibit varying sensitivities to the length of historical context: models emphasizing long-term dependencies may benefit from longer input sequences, while those designed for short-term dynamics
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might suffer from redundant or noisy inputs when the window is excessively long. Therefore, treating the look-back length as tunable hyperparameter rather than fixed setting is essential for a fair and comprehensive evaluation.
We perform experiments under multiple look-back window configurations to systematically assess this sensitivity. In the original setup, look-back lengths are predefined for each dataset, and models are trained and tested under all candidate configurations. The best-performing results are then reported, reflecting each model's optimal temporal receptive field. This protocol ensures a fair comparison among models with heterogeneous design principles and differing dependency ranges.
In line with this methodology. we examine the performance of our model and baselines under varying look-back window lengths. For the forecasting horizon of 720, adopt {96, 336, 512} as candidate look-back windows, corresponding to commonly used temporal spans in long-term forecasting. The experimental results, summarized in Table 7, reveal that our model maintains consistently strong performance across different window lengths, demonstrating both its robustness and its capacity to adaptively leverage available historical information. These findings justify our choice of treating the look-back window length as tunable hyperparameter in the main experiments and highlight the stability of MoFo under diverse temporal contexts.
Table 7: Look-back window sensitivity experiments of all optional look-back window length T E {96, 336, 512) on the forecasting length setting 720.
Method
MoFo
FITS
DLinear
TimesNet
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
T 96
0.447
0.454
0.547
0,518
0,515
0,511
0,521
0.495
T 336
0.459
0.469
0.475
0.487
0.471
0.493
0,542
0.519
T
512
0.443
0.463
0.435
0.458
0.464
0.488
0.560
0.531
T
96
0.416
0.442
0.439
0.452
0.650
0.571
0.434
0.448
T 336
0.393
0.428
0.397
0.431
0.704
0.597
0.472
0.480
T 512
0.379
0.425
0,382
0,425
0.786
0.623
0,480
0.468
D.7 Period Sensitivity Experiments
The accurate identification of periodic structures plays a crucial role in MoFo's ability to capture long-term dependencies and recurrent temporal dynamics. However, in real-world time series, period selection is often uncertain due to noise, seasonal drift, or data heterogeneity. To comprchensively examine MoFo's s robustness to such variations, we conduct three complementary sensitivity analyses: adjusting the given period length to test the impact of under- and over-estimation of period length, introducing multiple coexisting periodicities to evaluate the model's adaptability to pixed-period settings, and @ comparing robustness under complex dynamic period settings. The other experimental settings are kept consistent with the main experiments in Section 4.
Sensitivity to period length P. To assess the importance of accurate period calibration in MoFo, we conduct series of sensitivity experiments on the given period length P. We first design two vari- ants-"+ Half" and "+ Double"-in which the detected period is halved or doubled, respectively. As summarized in Table 8, results on three representative datasets with distinct periodicities demonstrate that a well-calibrated period is crucial for forecasting accuracy. When P is substantially under- or over-estimated, the model's ability to capture intrinsic temporal regularities degrades significantly, while the original configuration preserves the correct periodic structure and yields optimal perfor- mance. To further verify the robustness of this observation, we introduce four additional fine-grained perturbations: "+5%" and "+15%", where P is increased by 5% and 15%, and "-5%" and "-15%", where P is decreased by 5% and 15% As shown in Table 8, even minor deviations from the detected period lead to measurable performance drops across datasets with different dominant periods. These results collectively highlight that the accuracy of period estimation plays a pivotal role in MoFo's ability to effectively model periodic dependencies in time series data.
Sensitivity to multiple coexisting periods. In addition to single-period sensitivity, we further investigate MoFo's behavior under multiple periodic structures. We adopt the Traffic dataset, which exhibits possible two major periods-dai daily (P 24) and weekly (P 168) patterns. In practical multi-period scenarios, we typically use the shorter period as the baseline configuration. Our
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Table 8: Sensitivity experiments of period length P.
Method
MoFo
+5%
+15%
-5%
-15%
Half
Double
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.286
0.335
0.292
0.335
0.297
0.338
0.291
0.337
0.297
0.341
0.302
0.346
0.295
0.359
Weather F11ml
192
0.320
0,363
0.329
0.368
0.336
0.370
0.333
0.369
0.344
0.386
0.335
0.379
0,339
0.388
336
0.347
0.382
0.357
0.388
0.372
0.391
0.354
0.385
0.376
0.402
0.374
0.404
0.372
0.408
720
0
0.411
0.406
0.420
0.403
0.416
0.402
0.415
0.411
0.420
0.414
0.425
0.419
0.429
96
0.141
0.186
0.143
0.192
0.155
0.206
0.157
0.221
0.146
0.188
0.152
0.199
0.157
0.206
192
0.186
0.230
0.196
0.236
0.202
0.252
0.199
0.240
0.201
0.242
0.193
0.239
0.194
0.234
336
0.233
0.272
0.238
0.277
0.245
0.286
0.235
0,280
0.243
0.286
0.241
0.279
0.244
0.284
720
0.312
0,331
0.335
0.358
0.323
0.358
0.332
0.352
0.335
0.355
0.348
0.363
0.342
0.355
96
0.362
0.247
0.388
0.266
0.395
0.265
0.383
0.255
0.382
0.265
0.376
0.258
0,370
0.252
Trfficc
192
0.379
0.254
0.397
0.275
0.408
0.287
0.398
0.284
0.409
0.288
0.380
0.262
0.388
0.275
336
0.399
0.258
0.395
0.260
0.405
0.275
0.395
0.266
0.417
0.294
0.406
0.286
0.395
0.283
720
0.424
0.281
0.424
0.281
0.443
0.298
0.437
0.283
0.443
0.303
0.434
0.297
0.438
0.302
experiments show that this choice enables MoFo to capture fine-grained temporal variations while retaining stable performance across longer periods, as shown in Table 9. To explore whether integrating multiple MoFo models could further improve prediction, we introduce three variants: (1) MoFo-1, trained with 24-hour (daily) period (default setting of MoFo); (2) MoFo-7, trained with a 7-y (weekly) period; and (3) Mix-MoFo, which ensembles both models and combines their outputs through learnable fusion layer. The results indicate that using the shorter period configuration provides the most precise periodic alignment, while the ensemble model further stabilizes performance under complex multi-period signals. These findings confirm MoFo's 's robustness and adaptability when modeling time series with heterogeneous or nested periodic patterns.
Table 9: Sensitivity experiments of multiple coexisting periods.
Method
MoFo-1
Mix-MoFo
MoFo-7
Metric
MSE
MAE
MSE
MAF
MSE
MAF
96
0.362
0.247
0.365
0.254
0.381
0.257
Traffic
192
0.379
0.254
0.380
0.255
0.394
0.262
336
0.390
0.258
0.395
0.263
0.409
0.272
720
0.424
0.281
0.430
0.285
0.451
0,289
@ Sensitivity to dynamic periods. To further evaluate robustness of MoFo under complex temporal settings, we conduct experiments on datasets with mixed and time-varying periodicities. To sys- tematically investigate this issue, we construct synthetic dataset named Mixed-ETT by combining two standard datasets, ETTh1 (period = 24) and ETTml (period = 96). Specifically, each dataset evenly divided into four temporal segments, and these segments are alternately concatenated along the time axis form the new sequence as Mixed-ETT
{ETTh1(: 4), ET.mI[ 3l. ETThi[
31, E | 1]}. This design produces a dataset with alternating periodic structures of 24, 96, 24, 96, mimicking the temporal heterogeneity commonly observed in real applications. The other experimental settings are kept consistent with the main experiments in Section 4. We compare MoFo against two representative baselines, DUET and DLinear. As shown in Table 10, experimental results demonstrate that our straightforward MoFo implementation achieved strong performance. When dealing with multi-period time series, we set the smallest period length as our baseline configuration. The "Mix-MoFo" variant added assumptions regarding additional periods, which increased the risk of overfitting and consequently led decline in performance. These results confirm that MoFo can effectively handle dynamic and mixed periodic behaviors without explicit retraining or manual period adjustment.
D.8 Efficiency Analysis
We compare the computational efficiency of Transformer-basee baseline models on the Solar dataset with L 96. As shown in Table 11, MoFo achieves the least MSE among all branef models, while demonstrating the least parameters number with fastest training speed. All the benefits of MoFo arise from its effective Period-based Discrete Patching strategy, which reduces the complexity of the Transformer to quadratic in relation to the period length, while utilizing only
24

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Table 10: Sensitivity experiments of period varying synthetic dataset.
Method
MoFo
DUET
DLinear
Metric
MSE
MAB
MSE
MAE
MSR
MAE
Mixxd-EIT
96
0.178
0.223
0.186
0.229
0.194
0.232
0.185
0.232
0.184
0.241
0.190
0.247
336
0.180
0.237
0.187
0.247
0.196
0.256
720
0.191
0.253
0.218
0.278
0.219
0.282
a single layer of the Transformer, which is sufficient. Similarity, the competitive baseline DUET with dual clusting strategy on both temporal dimension and channel dimension exhibits over 14x increase in parameters number, 4x higher FLOPs, and slower training speed. Compared to PStchTST, classic Transformer-baur based model that employs successive patching strategy on the time series, MoFo reduces the number of MACs by more than 80x, accelerates the training speed by over 10x, and significantly decreases both the parameter requirements and memory usage. This primarily because PatchT'ST stacks multiple layers of the Transformer, which is necessary for its architecture, yet lacks a reasonable positional encoding for time series data. Although iTransformer, FEDformer, and Informer have fewer computations than MoFo, they require longer training times well as larger parameters, and their performance lags behind MoFo by up to 1.8x since their complexity architectures.
Table 11:
Efficiency comparison of MoFo and SOTA baselines with L = 720 in Traffic dataset. All results of each model are under the optimal hyperparameters for fair comparison. Parameters: All learnable parameters requiring gradient descent. MACs: multiply-accumun floating point operations. M: Million (109.. B: Billion (10%). Tr Trillion (10*2). MB: Megabyte, s:
operations. FLOPs:
Second. indicates the relative percentage increasing regarding MoFo and + indicates the relative
percentage decreasing.
Models
MSR
Parameters
MACCs
FT.OPs
Memmor Usage
Epoch Time
Informer
2.26 M 765s
7.13 B purresg
7.18
852 MB ps.ans
58 S-inn nats
Stationary
M M UI
passa 4075
1,710 00.5445
24 8-sa 5905
Triformer
0.225
1.62 o000
33.56
38.45
11.714 MB
yan 7066
147
Solar 96]
FEDiormer
0.485
3.63 M aus
1.86 B P1LA000
1.52
858 MB
204
aats
Crossformer
0.183
3.82 M
Mesol 11955
159.3
B
166.1 B
10,698 MB
101 S-a01
PatchTST
0.170
2.62 M
607.4
B
644.9 1548880655
29,726 MB
you ca56
215 X 2
Pathformer
0.218 yas
5.72 M das AR0g
24,84 B
27.99
11.00. 440%
12,754 MB
rann 6064
614
Transformmrr
0.190 10000...00 c795
0.51 M s7.
2.32
2.66
(77 17.9445
788 MB (7..919
18 8Ho 000g
PDF
0.181 rr.06
5.82 1ooooso omm
204.1
B 45
208.0
5,616
25
DUET
0.169
5.64 M
ome
19.23
B 100
60.88
4,422 MB
35
MoFo
0.169
0.36
7.33
11.69
3,458 MB
18 s
E
Discussion and Future Work
MoFo performs discrete patching based on periodic position, grouping the most correlevant time steps within the same patch for prioritized interaction. However, our periodic positional embedding provided by the Regulated Relaxation Function is currently only applicable to vanilla self-attention mechanisms, which exhibit quadratic complexity (although our specific implementation' is complexity is quadratic with respect to the period length P rather than the sequence length T with P < T). Exploring how to adapt this method attention mechanisms with linear complexity a direction worthy of future investigation. Simultaneously, applying the core ideas of MoFo within time series LLM foundational models to empower their development in time series learning another promising avenue for subsequent research.
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Reviews:
[{"review_id": "qoZmIYIjBy", "reviewer": "Reviewer_EWUL", "summary": "The paper introduces MoFo, a Transformer-based long-term time-series forecasting model comprising\n1. Period-Structured Patching (PSP): reshapes an input length-$T$ series into $P$ rows of phase-aligned \"patches\", reducing the number of tokens processed;\n2. Period-Aware Modulator (PAM): multiplies attention logits by a learnable sigmoid mask of phase distances, theoretically shown to approximate a hard phase gate.\nThe authors claim significant memory and runtime savings, as well as competitive accuracy, on eight periodic benchmarks, and provide a theoretical bound on the mask approximation error.", "questions": "How does accuracy change if the supplied $P$ is perturbed by ±5–15 % or if multiple periods (e.g., 24 h & 7 d) coexist?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\nS1) PSP and PAM are easy to implement and interpret; the paper is mostly well written.\nS2) On datasets with a single, stable period, MoFo yields substantial token reduction and lower memory/FLOPs.\nS3) The paper provides a concise proof that the sigmoid mask converges to a binary phase filter.\n\nWeaknesses\nW1) The proposed method relies on the assumption of a period length $P$. However, real-world time series often exhibit complex, time-varying periodic patterns. For example, in Figure 1(a), the \"Successive Patching\" clearly shows multiple peaks representing mixed periodicities. Why is it reasonable to treat these as belonging to the same period? When dealing with multiple samples from different time periods, how can we ensure accurate and meaningful period selection? When data faces mixed periodicities and time-varying periods (e.g., summer vs. winter electricity consumption patterns), can MoFo effectively handle such complex periodic variations?", "was_revised": true, "final_justification": "The authors have addressed most of my other concerns. I suggest that the authors provide detailed explanations from the rebuttal in the revised paper for clarity. I have raised my score to 4.", "rebuttal": "We sincerely thank you for your precious time and insightful comments. We will now address your concerns point by point below.\n\n---\n## **W1.**\n\n**W1.1. Confusion about period**\n\n\nIn Figure 1(a), the “Successive Patching” does not represent the coexistence of mixed periods; rather, it includes multiple peaks, which are dynamic components within a single period and together constitute a complete period. \n\n**We cannot simply assume that a single peak corresponds to one complete period.** For instance, in traffic data, there are distinct morning and evening peak periods; however, the periodicity of traffic is typically considered to be one full day rather than 12 hours. To accurately determine the period length, we employ the autocorrelation function (ACF). As shown in Figure 1(b), the autocorrelation coefficients of the sequence in Figure 1(a) indicate that a truly complete period often contains multiple peaks.\n\n**W1.2. Accurate period selection**\n\nAs stated in Appendix Section A.2 of the manuscript, we utilize three steps to accurately determine periodicity.\n\nFirstly, we identify potential period lengths of repeating patterns within the sequence using the autocorrelation function (ACF). The ACF is a widely used tool in the field of time series analysis for examining periodic patterns [1-3].\n\nSecondly, we verify the existence of these periods through statistical significance testing, such as utilizing a 95% confidence interval, to eliminate any noise interference.\n\nFinally, we also conduct hyperparameter experiments to assess the sensitivity of model performance with respect to different period lengths.\n\n**W1.3. Time-varying periods** \n\nThe electricity dataset (ETT) used in our experiments aligns perfectly with your requirements. The ETT dataset records electricity consumption from two counties in China over a continuous period from July 2016 to July 2018, thus encompassing both winter and summer patterns. However, it is worth noting that the periodic lengths for the two seasons do not differ significantly. For your convenience, we have included the relevant performance results below.\n\n| Dataset |Method  | MoFo         | DUET         | DLinear      |\n|:--------:|:--------:|:--------------:|:--------------:|:--------------:|\n|          | Prediction Window    |       MSE, MAE      |     MSE, MAE        |     MSE, MAE        |\n| ETTh1  | 96     | 0.360, 0.389 | **0.352**, **0.384** | 0.379, 0.403 |\n|        | 192    | **0.397**, 0.413 | 0.398, **0.409** | 0.408, 0.419 |\n|        | 336    | **0.407**, **0.424** | 0.414, 0.426 | 0.440, 0.440 |\n|        | 720    | 0.447, **0.454** | **0.429**, 0.455 | 0.471, 0.493 |\n| ETTm1  | 96     | 0.286, 0.335 | **0.279**, **0.333** | 0.300, 0.345 |\n|        | 192    | **0.320**, 0.363 | **0.320**, **0.358** | 0.336, 0.366 |\n|        | 336    | **0.347**, 0.382 | 0.348, **0.377** | 0.367, 0.386 |\n|        | 720    | **0.388**, 0.411 | 0.405, **0.408** | 0.419, 0.416 |\n\nFurthermore, to address your concerns more comprehensively, we have created a synthetic dataset (Mixed-ETT) by systematically combining ETTh1 (with a period of 24) and ETTm1 (with a period of 96). We divided each dataset into four equal parts along the time axis. Then, we alternately sampled each segment to construct the new dataset Mixed-ETT, represented as {ETTh1[:$\\frac{1}{4}$], ETTm1[$\\frac{1}{4}$:$\\frac{1}{2}$], ETTh1[$\\frac{1}{2}$:$\\frac{3}{4}$], ETTm1[$\\frac{3}{4}$:1]}. Consequently, the Mixed-ETT dataset displays a mixed periodic length of (24, 96, 24, 96). We compare the performance of state-of-the-art models, DUET and DLinear, as shown below.\n\n|  Dataset    | Method| MoFo         | Duet         | DLinear     |\n|:------------:|:--------:|:--------------:|:-------------:|:------------:|\n|          | Prediction Window    |       MSE, MAE      |     MSE, MAE        |     MSE, MAE        |\n| Mixed-ETT | 96     | **0.178**, **0.223** | 0.186, 0.229 | 0.194, 0.232 |\n|            | 192    | 0.185, **0.232** | **0.184**, 0.241 | 0.190, 0.247 |\n|            | 336    | **0.180,** **0.237** | 0.187, 0.247 | 0.196, 0.256 |\n|            | 720    | **0.191**, **0.253** | 0.218, 0.278 | 0.219, 0.282 |\n\nEven under time-varying periods, MoFo retains the competitive performance, underscoring the soundness of its architectural design and its robust capability to capture dynamic temporal patterns.\n\n---\n\n## **Q1.**\n\n**Q1.1. Performance under perturbed period setting**\n\nTo resolve any remaining concerns, we introduced four variants: \n\n- **\"+5 %\"** and **\"+15 %\"**: the period setting is increased by 5 % and 15 %, respectively. \n- **\"–5 %\"** and **\"–15 %\"**: the period setting is decreased by 5 % and 15 %, respectively. \n\nWe evaluated these perturbations on three datasets with distinct period lengths; the quantitative results are reported below.\n\n| Dataset | Method   | MoFo | + 5% | + 15% | - 5% | - 15% |\n|:---------:|:---------:|:---------:|:---------:|:---------:|:---------:|:-----------------:|\n|          | Prediction Window   |       MSE, MAE      |     MSE, MAE        |     MSE, MAE        |     MSE, MAE        |     MSE, MAE        |\n| ETTm1   | 96     | **0.286**, **0.335**     | 0.292, 0.335     | 0.297, 0.338     | 0.291, 0.337     | 0.297, 0.341     |\n|         | 192    | **0.320**, **0.363**     | 0.329, 0.368     | 0.336, 0.370     | 0.333, 0.369     | 0.344, 0.386     |\n|         | 336    | **0.347**, **0.382**     | 0.357, 0.388     | 0.372, 0.391     | 0.354, 0.385     | 0.376, 0.402     |\n|         | 720    | **0.388**, **0.411**     | 0.406, 0.420     | 0.403, 0.416     | 0.402, 0.415     | 0.411, 0.420     |\n| Weather | 96     | **0.141**, **0.186**     | 0.143, 0.192     | 0.155, 0.206     | 0.157, 0.221     | 0.146, 0.188     |\n|         | 192    | **0.186**, **0.230**     | 0.196, 0.236     | 0.202, 0.252     | 0.199, 0.240     | 0.201, 0.242     |\n|         | 336    | **0.233**, **0.272**     | 0.238, 0.277     | 0.245, 0.286     | 0.235, 0.280     | 0.243, 0.286     |\n|         | 720    | **0.312**, **0.331**     | 0.335, 0.358     | 0.323, 0.358     | 0.332, 0.352     | 0.335, 0.355     |\n| Traffic | 96     | **0.362**, **0.247**     | 0.388, 0.266     | 0.395, 0.265     | 0.383, 0.255     | 0.382, 0.265     |\n|         | 192    | **0.379**, **0.254**     | 0.397, 0.275     | 0.408, 0.287     | 0.398, 0.284     | 0.409, 0.288     |\n|         | 336    | **0.390**, **0.258**     | 0.395, 0.260     | 0.405, 0.275     | 0.395, 0.266     | 0.417, 0.294     |\n|         | 720    | **0.424**, **0.281**     | 0.424, 0.281     | 0.443, 0.298     | 0.437, 0.283     | 0.443, 0.303     |\n\n---\n\n**Q1.2. Performance under multiple periods setting**\n\nWe utilized the traffic dataset, which includes the multiple time intervals you mentioned: 1 day and 7 days. In practice, when dealing with multi-period data, we typically set the shorter period as our baseline configuration. Our experimental results were very impressive, as shown in Table 2 in the manuscript.\n\nTo address your concern regarding the performance when integrating multiple MoFo models to handle different periods, we introduced three models. One original model (MoFo-1) excels in sequences with a period length of 24 hours (1 day), another (MoFo-7) is designed for a weekly period of 7 days, and the final ensemble model (Mix-MoFo) learns and predicts in parallel using the two MoFo models, combining their outputs to produce the final prediction. Below are the results from our experiments on the Traffic dataset.\n\nThe experimental results indicate that using the shorter period as the configuration allows MoFo to capture more precise periodic patterns, leading to enhanced performance.\n\n| |  | MoFo-1|Mix-MoFo| MoFo-7|\n|--------|------|----------------|------|------|\n|          | Prediction Window   |       MSE, MAE      |     MSE, MAE        |     MSE, MAE        |\n| Traffic | 96    | **0.362**, **0.247** | 0.365, 0.254 | 0.381, 0.257 |\n|        | 192   | **0.379**, **0.254** | 0.380, 0.255 | 0.394, 0.262 |\n|        | 336   | **0.390**, **0.258** | 0.395, 0.263 | 0.409, 0.272 |\n|        | 720   | **0.424**, **0.281** | 0.430, 0.285 | 0.451, 0.289 |\n\nExperimental results demonstrate that our straightforward MoFo implementation achieved strong performance. When dealing with multi-period time series, we set the smallest period length as our baseline configuration. The “Mix-MoFo” variant added assumptions regarding additional periods, which increased the risk of overfitting and consequently led to a decline in performance.\n\nRef: \n\n[1] Wu H, Xu J, Wang J, et al. Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting[J]. Advances in neural information processing systems, 2021, 34: 22419-22430.\n\n[2] Lin S, Lin W, Hu X, et al. Cyclenet: Enhancing time series forecasting through modeling periodic patterns[J]. Advances in Neural Information Processing Systems, 2024, 37: 106315-106345.\n\n[3] Lin S, Lin W, Wu W, et al. SparseTSF: Modeling Long-term Time Series Forecasting with* 1k* Parameters[C]//International Conference on Machine Learning. PMLR, 2024: 30211-30226."}, {"review_id": "x9jnzPkO70", "reviewer": "Reviewer_7GM5", "summary": "This paper proposes MoFo, a Transformer-based forecasting method that models time series in a two-dimensional structure by leveraging their periodic patterns. The authors conduct correlation-based experiments in an attempt to validate the effectiveness of the proposed approach.", "questions": "Is MoFo still effective on time series data without clear periodicity? Has the method been explored on such datasets, for example, the Human Connectome Project (HCP), which involves non-periodic time series? I would like to see more discussion on the applicability and performance of the proposed method in non-periodic scenarios.", "limitations": "The authors mentioned limitations regarding computational complexity.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\nS1: The method for handling periodicity in the paper is reasonable and demonstrates a certain degree of novelty.\n\nS2: The figures and tables throughout the paper are clear, and the presentation is well-structured.\n\nWeaknesses:\n\nW1: The paper lacks comparisons with important baseline methods, especially since the proposed method also relies on 2D modeling. Therefore, it should be compared with previous classic SOTA methods that also use 2D modeling, such as TimesNet[1] (CNN-based), WITRAN[2] (RNN-based), and TPGN[3] (PGN-based, where PGN is a new paradigm based on RNN). Moreover, in the related work section, these key methods are not addressed, and other emerging approaches, such as TimeKAN[4] (KAN-based, where KAN is a new paradigm based on Linear), are also omitted.\n\n[1] Wu, H., Hu, T., Liu, Y., Zhou, H., Wang, J., & Long, M. (2023). Timesnet: Temporal 2d-variation modeling for general time series analysis. In The Eleventh International Conference on Learning Representations.\n\n[2] Jia, Y., Lin, Y., Hao, X., Lin, Y., Guo, S., & Wan, H. (2023). WITRAN: Water-wave Information Transmission and Recurrent Acceleration Network for Long-range Time Series Forecasting. In Thirty-seventh Annual Conference on Neural Information Processing Systems.\n\n[3] Jia, Y., Lin, Y., Yu, J., Wang, S., Liu, T., & Wan, H. (2024). PGN: The RNN's New Successor is Effective for Long-Range Time Series Forecasting. In The Thirty-eighth Annual Conference on Neural Information Processing Systems.\n\n[4] Huang, S., Zhao, Z., Li, C., & Bai, L. (2025). TimeKAN: KAN-based Frequency Decomposition Learning Architecture for Long-term Time Series Forecasting. In The Thirteenth International Conference on Learning Representations.\n\nW2: The experimental setup contains clear flaws. Although the authors claim to follow configurations from previous studies, the actual design lacks fairness and rigor.\n\n(1) Improper handling of historical look-back length\n\n(1.1) Different look-back lengths may alter the data splitting process, thereby affecting model training. This represents a lack of control over experimental variables.\n\n(1.2) Different models exhibit varying sensitivity to look-back lengths, so treating it uniformly as a hyperparameter is clearly inappropriate. A more fair and rigorous approach would be to predefine several common look-back lengths and treat them as separate experimental groups, enabling a more systematic comparison of how each method performs under different historical horizons.\n\n(2) Limited hyperparameter search space\nThe TFB paper cited by the authors only expands to 8 parameter configurations. Taking Transformers as an example, even commonly used parameters like d_model, e_layer, and d_layer are barely covered—let alone other hyperparameters. Such a narrow search space makes it difficult to draw convincing conclusions.\n\n(3) Why an extensive hyperparameter search is necessary\n\n(3.1) Even with identical random seeds and hyperparameter values, experimental results may still vary across hardware platforms.\n\n(3.2) Different models have different levels of sensitivity to hyperparameters. When the experimental platform changes, directly reusing published settings cannot guarantee that baseline models operate at their optimal capacity, nor that results are consistent with those reported originally.\n\nTherefore, to ensure a fair comparison, it is essential to perform a fresh hyperparameter search for all models within a unified and sufficiently broad search space by selecting optimal configurations on the validation set and then evaluating and reporting results on the test set. While models with more hyperparameters may lead to a larger search space, this process is critical for ensuring fairness and rigor. Only after eliminating platform differences and hyperparameter biases, and ensuring that all models perform at their best, can a trustworthy and meaningful comparison be made.\n\nFurthermore, the authors do not compare their approach with the baselines mentioned in W1. Consequently, the current experiments are insufficient to convincingly establish the effectiveness of the proposed method.\n\nW3: The efficiency experiments likewise omit comparisons with the baselines listed in W1, as well as with FITS and DLinear. In addition, the paper does not specify how the hyperparameters were chosen for these tests. It must be emphasized that efficiency studies are intended to measure the resource consumption inherent to each model architecture; hence, all experimental variables should be controlled to ensure a fair comparison. In particular, shared hyperparameters should be kept identical so as to eliminate confounding effects, such as one method being run with very small settings while another is evaluated with much larger ones.", "was_revised": true, "final_justification": "The authors have addressed my concern. Therefore, I am raising my score to 4.", "rebuttal": "Thank you very much for your time and effort; they are crucial for improving the quality of our manuscript.\n\n----\n\n## **W1. More baselines**\n\nWe have already compared TimesNet in Table 4. Next, we added comparative experiments with PGN, TimeKAN, and WITRAN. The performance results are shown in the table below.\n\n| Datasetr | Method | MoFo | TimeKAN  | PGN  | TimesNet  | WITRAN  |\n|--------|--------|-----------------|--------------------|----------------|---------------------|-------------------|\n|          | Prediction Window       |       MSE,  MAE |       MSE, MAE |          MSE, MAE  |          MSE, MAE  |          MSE, MAE  |\n| ETTh1  | 96     | **0.360**, **0.389**| 0.369, 0.396       | 0.402, 0.407   | 0.389, 0.412        | 0.660, 0.627      |\n|        | 192    | **0.397**, **0.413**| 0.402, 0.417       | 0.446, 0.433   | 0.440, 0.443        | 0.844, 0.723      |\n|        | 336    | **0.407**, **0.424**| 0.419, 0.430       | 0.494, 0.456   | 0.523, 0.487        | 0.967, 0.766      |\n|        | 720    | 0.447, **0.454**| **0.442**, 0.463       | 0.490, 0.478   | 0.521, 0.495        | 0.965, 0.768      |\n| ETTh2  | 96     | 0.273, **0.334**| **0.279**, 0.343       | 0.323, 0.358   | 0.334, 0.370        | 0.601, 1.285      |\n|        | 192    | 0.327, **0.373**| **0.326**, 0.380       | 0.426, 0.420   | 0.404, 0.413        | 4.193, 1.611      |\n|        | 336    | **0.361**, **0.405**| 0.368, 0.411       | 0.452, 0.457   | 0.389, 0.435        | 5.036, 1.819      |\n|        | 720    | **0.379**, **0.425**| 0.419, 0.443       | 0.438, 0.449   | 0.434, 0.448        | 4.556, 1.817      |\n\n> **Fairness statement**. Although the TFB-benchmark setting we adopted may differ from your expectations, we would like to clarify that all models are run under the same experimental environment and unified configuration to ensure basic fairness in evaluation. We chose to conduct experiments on the latest TFB mainly because it has been extensively validated and is academically recognized—the benchmark was nominated for the Best Paper Award at VLDB 2024 and has been cited over 62 times on Google Scholar.\n\n----\n\n## **W2. Historical look-back length**\n\nPlease allow us to clarify TFB’s approach regarding historical window selection. According to the TFB GitHub guidelines, multiple look-back window lengths are preset in the experiments, covering typical settings commonly used in time series forecasting tasks. For each model, tests are conducted independently on all candidate historical window lengths, and the best-performing result is ultimately reported.\n\nWe believe this approach does not introduce fairness issues; on the contrary, it is fairer to models with varying sensitivities. Forcing all models to use exactly the same look-back length might disadvantage those that are particularly sensitive to this parameter, causing them to be unfairly regarded as “underperforming”. **In contrast, as long as the device conditions and the amount of historical data are consistent, allowing each model to flexibly choose its optimal historical window according to its structural characteristics both ensures a fairer evaluation of predictive performance and better reflects the flexible practices found in real-world applications.**\n\nDespite having a different perspective, in response to your suggestion, we conducted experiments using several commonly adopted predefined look-back window lengths. Specifically, for forecasting with a future window length of 720, we used {96, 336, 512} as the look-back windows. Due to time constraints, partial experimental results are shown below.\n\n|| Metric | MoFo  | TimeKAN  | FITS  | PGN   | TimesNet  | WITRAN  | DLinear |\n|-------|--------|-----------------------|-------------------------|---------------------|---------------------|-------------------------|--------------------|---------------------|\n|          |   Look-back Window    |       MSE,  MAE |       MSE, MAE |          MSE, MAE  |          MSE, MAE  |          MSE, MAE  |          MSE, MAE  |          MSE, MAE  |\n| ETTh1 | 96     | **0.447**, **0.454**         | 0.463, 0.470           | 0.547, 0.518         | 0.490, 0.478         | 0.521, 0.495         | 0.965, 0.768       | 0.515, 0.511         |\n|       | 336    | **0.459**, **0.469**         | 0.509, 0.490           | 0.475, 0.487         | 0.522, 0.515         | 0.542, 0.519         | 0.976, 0.771       | 0.471, 0.493         |\n|       | 512    | 0.443, 0.463         | 0.442, 0.463           | **0.435**, **0.458**         | 0.692, 0.598         | 0.560, 0.531         | 1.034, 0.802       | 0.464, 0.488         |\n| ETTh2 | 96     | **0.416**, **0.442**         | 0.438, 0.451           | 0.439, 0.452         | 0.438, 0.450         | 0.434, 0.448         | 4.556, 1.817       | 0.650, 0.571         |\n|       | 336    | **0.393**, **0.428**         | 0.420, 0.449           | 0.397, 0.431         | 0.466, 0.474         | 0.472, 0.480         | 4.272, 1.745       | 0.704, 0.597         |\n|       | 512    | **0.379**, **0.425**         | 0.408, 0.443           | 0.382, 0.425         | 0.440, 0.465         | 0.480, 0.468         | 4.668, 1.787       | 0.786, 0.623         |\n\n----\n\n## **W3. Hyperparameter search**\n\n(1) According to the TFB guidelines, the authors follow the hyperparameters specified in the original papers. And they further performed additional hyperparameter searches on 16 configuration sets to achieve a more comprehensive evaluation, which covers the key hyperparameters sufficiently.\n\n(2) TFB has fully open-sourced the necessary environment configuration, including detailed version information, along with a fixed running protocol, enabling users to reproduce the same performance.\n\n(3) Please refer to W1.\n\n----\n\n## **W4. Complexity**\n\n**Our efficiency comparison, as stated in the caption of Table 3, is conducted under the condition that each model achieves its optimal predictive performance—a standard and widely accepted practice in academic research. Since the ultimate goal of time series forecasting is to enhance prediction accuracy, efficiency comparisons that consider performance are both meaningful and practical.** For instance, if all models are configured with extreme parameter settings, like compressing the feature dimension to 1, computational efficiency might seem significantly improved, but prediction performance would suffer greatly. Therefore, a fair and informative efficiency evaluation must be conducted at comparable performance levels to ensure both validity and practical significance.\n\nIn response to your suggestion, we used the ETTh1 dataset as an example and optimized this configuration for the best performance, while keeping certain hyperparameters such as batch size and learning rate fixed. We report the efficiency for fixed batch sizes (specifically 4 and 128), learning rate 1e-4 and look-back window 512, as well as for each model under its best-performing setting. Please refer to the table below for details (K: Kilo-$10^3$, M: Million-$10^6$). We observe that our model achieves significant performance improvements with only a slight increase in complexity.\n\n| ETTh1       | Model   | MSE    | MAE    | Memory   | # Para   | # MACs      | # FLOPs      |\n|-------------|---------|--------|--------|----------|----------|-------------|--------------|\n| bs=4        | MoFo    | **0.372**  | **0.399**  | 542 MB   | 33.0 K   | 2.5 M   | 2.9 M    |\n| | FITS    | 0.381  | 0.406  | 540 MB   | 24,0 K | 0.7 M     | 0.7 M      |\n|  | DLinear | 0.387  | 0.415  | 532 MB   | 98,5 K   | 2.8 M   | 2.8 M    |\n| bs=128 | MoFo    | **0.380**  | **0.390**  | 636 MB   | 22.4 K   | 42.3 M  | 50.6 M   |\n| | FITS    | 0.492  | 0.478  | 584 MB   | 24,0 K   | 21.4 M  | 21.4 M   |\n|  |   DLinear | 0.389  | 0.418  | 558 MB   | 98.5 K   | 88.5 M  | 88.1 M   |\n\n----\n\n## **Q1.**\n\nThe Human Connectome Project (HCP) is **a large-scale dataset with over 1TB of data**, requiring **mandatory institutional approval for access**. Due to time constraints, we were unable to use and study it during the rebuttal period. To address your concerns, we opted to utilize the Influenza-Like Illness (ILI)  dataset for evaluation instead. The ILI dataset, published by the US Centers for Disease Control and Prevention (CDC) from 2002 to 2021, comprises health-related data that record the weekly proportion of ILI patients relative to the total number of visits, without any obvious seasonality. We predict future values for four windows: $L\\in$ {24,36,48,60}.\n\nTo emphasize our modeling of non-periodic time series, we follow the approach of TimesNet by computing the main pseudo-period as the reciprocal of the dominant frequency obtained via Fast Fourier Transform (FFT) on each input time series. Although MoFo was designed with periodic patterns in mind, the results below demonstrate that it still achieves remarkably high accuracy on non-periodic data, highlighting its strong capability in capturing complex temporal dynamics even in the absence of clear periodicity.\n\n| | MoFo | TimeKAN | FITS | PGN  | TimesNet  | WITRAN |\n|----------|-----------------|--------------------|-----------------|---------------|---------------------|-------------------|\n|    $L$      |       MSE,  MAE |       MSE, MAE |          MSE, MAE  |          MSE, MAE  |          MSE, MAE  |          MSE, MAE  |          MSE, MAE  |\n| 24       | **2.113**, **0.927**    | 2.176, 0.928       | 2.182, 1.002    | 2.255, 0.936  | 2.131, 0.958        | 3.486, 1.218      | 2.208, 1.031       |\n| 36       | **1.952**, **0.924**    | 2.166, 0.993       | 2.330, 1.051    | 2.132, 0.940  | 2.612, 0.974        | 5.129, 1.596      | 2.032, 0.981       |\n| 48       | **1.714**, **0.824**    | 2.011, 0.928       | 2.761, 1.184    | 2.182, 0.944  | 1.916, 0.897        | 3.862, 1.332      | 2.209, 1.063       |\n| 60       | **1.800**, 0.906    | 2.010, 0.967       | 2.929, 1.217    | 2.169, 0.940  | 1.995, **0.905**        | 4.812, 1.543      | 2.292, 1.086       |"}, {"review_id": "H3qyk3iiEb", "reviewer": "Reviewer_XTBU", "summary": "This paper introduces MoFo, a Transformer-based approach for long-term sequence forecasting. The method employs a periodicity-structured patching strategy along with a periodicity-aware modulator, inspired by theoretical mathematical insights. Extensive experimental results demonstrate that MoFo achieves highly competitive performance and superior efficiency.", "questions": "Please kindly see the Weaknesses above.", "limitations": "YES", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strength:\nS1. The paper has a strong and clearly articulated motivation. The authors have effectively identified and explained the gap in existing work on time series forecasting.\n\nS2. The proposed method is novel, incorporating a periodicity-aware modulator and an input representation strategy that emphasize the modeling of temporal periodicity. And the approach is driven by theoretical mathematical insights.\n\nS3. Extensive experiments validate the model’s effectiveness, showcasing significant performance improvements and approximately 15× efficiency gains. \n\nS4.  The paper is well-written and well-structured, and The mathematical formulas are presented in a clear and explicit manner.\n\n### Weakness:\nW1. Some time series forecasting benchmarks need to be supplemented. For example, how does the model compare to FITS and SparseTF models?\n\nW2. Why does the time complexity of MoFo remain stable as the input time series becomes longer?\n\nW3. What does MACs mean in Table 3?  What is the purpose of this indicator?", "was_revised": true, "final_justification": "Since authors have targeted and addressed my questions in rebuttal I keep the score conservatively.", "rebuttal": "We appreciate your guidance—your suggestions have helped us produce a more precise, readable, and broadly accessible manuscript.\n\n## **W1. More Baselines**\n\nThank you for your valuable suggestion. The FITS [5] model is already included in Table 4 of Appendix E.3. To further address your concern, we have conducted additional experiments comparing FITS [5] and SparseTF [6] on the ETTm1 and ETTm2 datasets; the results are reported below. For clarity, the best performance in each experiment is highlighted in bold. Across all evaluations, MoFo consistently achieves the most competitive results, demonstrating superior performance.\n\n|  Dataset| Method| MoFo              | FITS            | SparseTSF            |\n|---------|--------|-------------------|-----------------|----------------------|\n|         | Prediction Window    |       MSE, MAE    |        MSE, MAE |        MSE, MAE      |  \n| ETTm1   | 96     | **0.286**, **0.335**  | 0.303, 0.345    | 0.316, 0.355         |\n|         | 192    | **0.320**, **0.363**  | 0.337, 0.365    | 0.348, 0.376         |\n|         | 336    | **0.347**, **0.382**  | 0.368, 0.384    | 0.373, 0.387         |\n|         | 720    | **0.388**, **0.411**  | 0.420, 0.413    | 0.434, 0.422         |\n| ETTm2   | 96     | **0.155**, **0.240**  | 0.165, 0.254    | 0.166, 0.256         |\n|         | 192    | **0.211**, **0.283**  | 0.219, 0.291    | 0.220, 0.292         |\n|         | 336    | **0.258**, **0.314**  | 0.272, 0.326    | 0.273, 0.327         |\n|         | 720    | **0.342**, **0.368**  | 0.359, 0.381    | 0.361, 0.381         |\n\nFurthermore, as requested by Reviewer 7GM5, we have added comparisons with the TimesNet [1], WITRAN [2], PGN[3], and TimeKAN[4] models. Please refer to our response to W1 for details.\n\n## **W2. Why does the time complexity of MoFo remain stable as the input time series becomes longer?**\n\nIn MoFo, due to the unique patch strategy, the number of patches is equal to the period length and thus remains constant. Each patch contains exactly $\\lceil T / P\\rceil$ elements, where $T$ is the input sequence length and $P$ is the period length. Since the time complexity depends solely on the number of patches (i.e., the period length), MoFo is not sensitive to increases in the input sequence length.\n\n## **W3. What is MACs**\n\nMACs (Multiply–Accumulate operations) quantify the total number of fused multiply-add operations executed during a single forward pass. This metric serves as a hardware-agnostic proxy for computational cost: fewer MACs generally translate to lower latency and reduced energy consumption on both CPUs and accelerators. By reporting MACs alongside accuracy, we provide a clear, hardware-independent view of the efficiency–performance trade-off offered by each model.\n\nRef:\n\n[1] Wu, H., Hu, T., Liu, Y., Zhou, H., Wang, J., & Long, M. (2023). Timesnet: Temporal 2d-variation modeling for general time series analysis. In The Eleventh International Conference on Learning Representations.\n\n[2] Jia, Y., Lin, Y., Hao, X., Lin, Y., Guo, S., & Wan, H. (2023). WITRAN: Water-wave Information Transmission and Recurrent Acceleration Network for Long-range Time Series Forecasting. In Thirty-seventh Annual Conference on Neural Information Processing Systems.\n\n[3] Jia, Y., Lin, Y., Yu, J., Wang, S., Liu, T., & Wan, H. (2024). PGN: The RNN's New Successor is Effective for Long-Range Time Series Forecasting. In The Thirty-eighth Annual Conference on Neural Information Processing Systems.\n\n[4] Huang, S., Zhao, Z., Li, C., & Bai, L. (2025). TimeKAN: KAN-based Frequency Decomposition Learning Architecture for Long-term Time Series Forecasting. In The Thirteenth International Conference on Learning Representations.\n\n[5] Xu, Z., Zeng, A., & Xu, Q. (2023). FITS: Modeling time series with $10 k $ parameters. arXiv preprint arXiv:2307.03756.\n\n[6] Lin, S., Lin, W., Wu, W., Chen, H., & Yang, J. (2024, July). SparseTSF: Modeling Long-term Time Series Forecasting with* 1k* Parameters. In International Conference on Machine Learning (pp. 30211-30226). PMLR."}, {"review_id": "gJNoRIubrN", "reviewer": "Reviewer_Q3nE", "summary": "The paper proposes the MoFo framework for long-term time series forecasting, which models periodicity as the correlation of period-aligned time steps and the trend of period-offset time steps. By designing a periodic structural patch and a periodic-aware modulator, the framework captures periodic patterns effectively. Experiments demonstrate that it outperforms 17 baselines, achieving both a 14x improvement in memory efficiency and a 10x faster training speed.", "questions": "The authors can refer to the weakness listed above.", "limitations": "The authors can refer to the weakness listed above.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n1. Modeling periodicity as the correlation of period-aligned time steps and the trend of period-offset time steps provides a novel modeling perspective for long-term time series forecasting.\n2. MoFo demonstrates competitive predictive performance compared to 17 state-of-the-art baselines on multiple popular benchmark datasets, validating the effectiveness of the proposed method.\n3. The model's computational complexity is related to the period length rather than the input sequence length, achieving up to a 14$\\times$ improvement in memory efficiency and a 10$\\times$ speedup in training.\n\n**Weaknesses**\n\n1. Presentation Issues \n   - Grammatical errors. For example, there is a subject-verb agreement issue on line 182 regarding singular and plural forms. \n   - The names of all designed modules should be in lowercase.\n2. Clarifications Needed \n   - The description of the patch-based structured input is unclear, especially the padding strategy. Using data from the last period for padding may introduce potential information redundancy —could this affect model generalization? \n   - How to determine the cycle length? What does formula H mean in Eq.10?\n3. Baselines \n   - How many layers of Transformer did the author use? Did the author perform a sensitivity analysis on this parameter?\n   - A hyperparameter study on the period length is suggested, for example, using double the default period length, to evaluate its impact on performance.", "was_revised": true, "final_justification": "The paper proposes the MoFo framework for long-term time series forecasting, which models periodicity as the correlation of period-aligned time steps and the trend of period-offset time steps. By designing a periodic structural patch and a periodic-aware modulator, the framework captures periodic patterns effectively. Experiments demonstrate that it outperforms 17 baselines, achieving both a 14x improvement in memory efficiency and a 10x faster training speed. The author's response has largely addressed my concerns. Considering the significance of MoFo for long-term time series forecasting, I have raised my score to 5.", "rebuttal": "Dear Reviewer Q3nE,\n\nThank you for your kind words. Your feedback and guidance have been instrumental in improving our paper, and we sincerely appreciate your support and constructive suggestions. Next, we will respond to your concerns point by point.\n\n## **W1. Presentation Typos**\n\nWe sincerely appreciate your constructive feedback and will thoroughly revise and proof-read the manuscript in the next version to address every point you raised.\n\n## **W2.1. Padding Strategy**\n\nOur padding strategy is already described in detail in **Section 3.1, lines 109–119** and formalized in **Equation (1)** of the manuscript. In fact, Given an input sequence $\\mathbf{X} = [\\mathbf{x}_1, …, \\mathbf{x}_T]^\\top$, we start from the current time step $\\mathbf{x}_T$ and move past to delineate periods of length $P$. Any prefix that does not form a complete period is left-padded with the leftmost elements of the nearest full period on its right; this yields a padded input length of $\\lceil T / P\\rceil * P$ and exactly $\\lceil T / P\\rceil$ periods.  \n\nTo evaluate the impact of our strategy, we introduce a variant \"+ zero padding\" that pads the value with zeros and conpare on ETTm1 and ETTm2 datasets. The results are shown below and the best-performing metric for each experiment is bolded.\n\n\n| Dataset | Method  | MoFo              | + zeros padding           |\n|---------|--------|-------------------|---------------------------|\n|         | Prediction Window    |       MSE, MAE    |    MSE, MAE               |\n| ETTm1   | 96     | **0.286, 0.335**  | 0.292, 0.345              |\n|         | 192    | **0.320, 0.363**  | 0.328, 0.372              |\n|         | 336    | **0.347, 0.382**  | 0.348, 0.386              |\n|         | 720    | **0.388, 0.411**  | 0.401, 0.435              |\n| ETTm2   | 96     | **0.155, 0.240**  | 0.156, 0.247              |\n|         | 192    | **0.211, 0.283**  | 0.215, 0.293              |\n|         | 336    | **0.258, 0.314**  | 0.261, 0.318              |\n|         | 720    | **0.342, 0.368**  | 0.345, 0.372              |\n\n\nOur padding strategy consistently outperforms zero padding, demonstrating that re-using the immediately preceding historical pattern enhances generalization rather than introducing redundancy.\n\n\n## **W2.2.1. Determine the cycle length**\n\nWe determine the period length by calculating the distance between the peaks of the autocorrelation coefficients of the time series data on the training set. The specific method for this calculating can be found in **Appendix A.2**.\n\n\n## **W2.2.2. What is $\\mathcal{H}$ ?**\n\nWe apologize for omitting the precise definition. In our notation, $\\mathcal{H}$ denotes the Heaviside step function satisfying\n$$\n\\mathcal{H}(s) = 0 \\text{ if } s < 0; \\mathcal{H}(s) = 1 \\text{ if } s > 0.\n$$\nWe will add this clarification to the next version.\n\n## **W3.1. Transformer layers in MoFo**\n\n**MoFo attains best performance with only one Transformer layer**. The corresponding ablation is provided in **Section 4.6** and **Figure 5** of the manuscript, where we systematically vary the number of Transformer blocks. Across all settings, stacking additional layers yields negligible to zero gains, confirming that a single layer already saturates the model’s capacity.\n\n## **W3.2. Period length of MoFo**\n\nFollowing your suggestion, we designed two variants—**\"+ Half\"** and **\"+ Double\"**—to investigate MoFo’s sensitivity to the chosen the half and the double period length. As shown the below Table, the results on three diverse datasets with different period reveal that a well-calibrated period is essential: an inappropriate choice disrupts the model’s ability to capture the intrinsic periodicity of the time series, whereas the original setting preserves this structure and retains optimal performance. \n\n\n| Dataset| Method   | MoFo                | + Half              | + Double            |\n|----------|--------|---------------------|---------------------|---------------------|\n|          | Prediction Window    |       MSE, MAE      |     MSE, MAE        |     MSE, MAE        |\n| ETTm1    | 96     | **0.289, 0.336**    | 0.302, 0.346        | 0.295, 0.359        |\n| | 192    | **0.326, 0.367**    | 0.335, 0.379        | 0.339, 0.388        |\n|          | 336    | **0.361, 0.391**    | 0.374, 0.404        | 0.372, 0.408        |\n|          | 720    | **0.405, 0.419**    | 0.414, 0.425        | 0.419, 0.429        |\n| Weather  | 96     | **0.141, 0.186**    | 0.152, 0.199        | 0.157, 0.206        |\n| | 192    | **0.186, 0.230**    | 0.193, 0.239        | 0.194, 0.234        |\n|          | 336    | **0.233, 0.272**    | 0.241, 0.279        | 0.244, 0.284        |\n|          | 720    | **0.312, 0.331**    | 0.348, 0.363        | 0.342, 0.355        |\n| Traffic  | 96     | **0.362, 0.247**    | 0.376, 0.258        | 0.370, 0.252        |\n| | 192    | **0.379, 0.254**    | 0.380, 0.262        | 0.388, 0.275        |\n|          | 336    | **0.390, 0.258**    | 0.406, 0.286        | 0.395, 0.283        |\n|          | 720    | **0.424, 0.281**    | 0.434, 0.297        | 0.438, 0.302        |"}]

Author rebuttal:

Meta-review: This paper proposes a modulated method for period modeling in long-term time series forecasting. All reviewers acknowledged the strong motivation, clear presentation, concise model design, and (after rebuttal) comprehensive evaluation. Following the rebuttal, two reviewers increased their ratings to positive scores, and the overall consensus was in favor of acceptance.

I strongly encourage the authors to revise the paper in line with the reviewers’ comments and to release their code to the community to enhance reproducibility.

Recommendation: Accept as poster.
Decision: Accept (poster)