CFPT: Empowering Time Series Forecasting through Cross-Frequency Interaction and Periodic-Aware Timestamp Modeling
Authors: Feifei Kou
Jiahao Wang
Lei Shi
Yuhan Yao
Yawen Li
Suguo Zhu
Zhongbao Zhang
Junping Du
Venue: ICML 2025 poster (2025)
Keywords: Time series forecasting
Frequency analysis
Timestamp modeling
Primary area: applications->time_series

Abstract: Long-term time series forecasting has been widely studied, yet two aspects remain insufficiently explored: the interaction learning between different frequency components  and the exploitation of periodic characteristics inherent in timestamps. To address the above issues, we propose **CFPT**, a novel method that empowering time series forecasting through **C**ross-**F**requency Interaction (CFI) and **P**eriodic-Aware **T**imestamp Modeling (PTM). To learn cross-frequency interactions, we design the CFI branch to process signals in frequency domain and captures their interactions through a feature fusion mechanism. Furthermore, to enhance prediction performance by leveraging timestamp periodicity, we develop the PTM branch which transforms timestamp sequences into 2D periodic tensors and utilizes 2D convolution to capture both intra-period dependencies and inter-period correlations of time series based on timestamp patterns. Extensive experiments on multiple real-world benchmarks demonstrate that CFPT achieves state-of-the-art performance in long-term forecasting tasks. The code is publicly available at this repository: https://github.com/BUPT-SN/CFPT.

CFPТ: Empowering Time Series Forecasting through Cross-Frequency Interaction and Periodic-Aware Timestamp Modeling
Feifei Kou 123 Jiahao Wang 12
Lei Shi
Yuhan Yao 12 Yawen Li Junping Du
Suguo Zhu
Zhongbao Zhang" *
Abstract
(Low-Fren
Input Time Series
Output T I'me Series
Long-term time series forecasting has been widely studied, yet two aspects remain insufficiently explored: the interaction learning between dif- ferent frequency components and the exploita- tion of periodic characteristics inherent in times- tamps. To address the above issues, propose CFPT. novel method that empowering time series forecasting through Cross- Froos-Frreuency In- teraction (CFI) and Periodic-Aware Timestamp Modeling (PTM). To learn cross-frequency in- teractions, we design the CFI branch to pro- signals in frequency domain and captures their interactions through feature fusion mecha- nism. Furthermore, to enhance prediction per- formance by leveraging timestamp periodicity, we develop the PTM branch which transforms timestamp sequences into 2D periodic tensors and utilizes 2D convolution to capture both intra- period dependencies and inter-period correlations of time series based timestamp patterns. Ex- tensive experiments on multiple real-world bench- marks demonstrate that CFPT achieves of the-art performance long-term forecasting tasks. The code is publicly available at this repository: http://iiuubooo/BUPPP-PSCCC
consist of
Feature Extration
HighhFFeey
Basic Pipeline Time Series Forecasting M Methods
No-Freq
Unified-Freq
Interaction
Low- Freq
High- Free
Weightedd-
Free
Freq
Feature Extraction Methods
Feature Extraction M ethood
(b) D Differences of Frequency Feature Ulilization between een ur ethod and Others
Figure I. Comparison of frequeney component utilization time series forecasting.
1. Introduction
Temporal information data serves as one of the most funda- mental and ubiquitous data types in real-world applications and intelligent systems (Liang et al., 2023; Hu al., 2024; Liang al., 2024a; Ilbert et al., 2024; Liang et al., 2024b). Within the broader domain of temporal data analysis, long- term time series forecasting has emerged as a particularly critical task with applications spanning numerous sectors, including energy consumption (Pinto et al., 2021), trans- portation (He al., 2022), financial markets (He et al., 2023) and so on. Consequently, this task has garnered significant attention from researchers, leading the development of nu- merous advanced methodologies (Nie et al., 2023; Yi et al., 2023; Zeng et al., 2023; Wang et al., 2024b: Lin et al., 2024; Luo & Wang, 2024; Xu et al., 2024a). Despite these ad- vances, current research in long-term time series forecasting leaves two important aspects insufficiently explored.
*Corresponding Author School of Computer Science (National Pilot Software Engineering School). Beijing University of Posts and Telecommunications, Beijing 100876, China Key Lahoratory of Trustworthy Distributed Computing and Service, Beijing Uni- versity of Posts and Telecommunications, Beijing 100876, China Guangxi Key Laboratory of Trusted Software. Guilin Univer- sity of Electronic Technology, Guilin 541004, China 'State Key Laboratory of Media Convergence and Communication, Com- munication University of China, Beijing 100024, China 'Scool of Economics and Management, Beijing University of Posts and Telecomnunications Beijing 1 100876, China "College of Computer Science and Technology, Hangzhou Dianzi University, Hangzhou 310018, China. Correspondence to: Zhongbao Zhang <zhong- baozb @ bupt.edu.cn>>
First, as shown in Figure I(a), time series data inherently contains signals of different frequencies with distinct charac- teristics, where low-frequency components carry fundamen- tal patterns and trends, high-frequenee components reflect short-term dynamics (Ye et al., 2024). The time series pre- diction task is mainly to predict the future time points by taking the historical time series as input and extracting their
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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CFPT: Empowering Time Series Forecasting through Cross-Frequency Interaction and Periodic-Aware Timestamp Modeling
values at 21:00) show strong consistency across weekdays while displaying notable differences between weekdays and weekends. Recent methods have demonstrated encouraging improvements by treating timestamps as enhancement com- ponents for time series forecasting ( (Wang et al., 2024a; Zeng et al., 2024). Some approaches attempt to model timestamps through feature embeddings (Zhou et al., 2021; Wu et al., 2021; 2023) or attention mechanisms (Liu et al., 2024a; Wang et al., 2024b), yet these methods have shown limited effectiveness in practice (Wang et al., 2024a). Despite these attempts in timestamp modeling, the periodic characteristics inherent in timestamps remain insufficiently explored. To fully harness the potential of timestamps for performance enhancement, exploring their periodic patterns becomes a promising direction.
Wechesday
Tuusday
Fridby
Inter-pariod commlations
Saturday
GOBAM
R00
1600PM-
intra-period derendnccss
Figure 2. Illustration of intra-period dependencies and inter-period correlations in Electricity Consuming Load (ECL) dataset.
In this paper, we propose CFPT, novel framework that enhances long-term time series forecasting through two spe- cialized branches. Firstly, We propose the Cross-Frequent Interaction (CFT) branch, it enables separate modeling of different frequency components while capturing their inter- actions through a carefully designed feature fusion mech- anism. Thereby, It retains long-term evolutionary patterns and short-term dynamic features and providing comprehen- sive representation across time scales. Secondly, design the Periodic-aware Timestamp Modeling (PTM) branch. It transforms 1D timestamp sequences into 2D tensors based on fixed period lengths. Through 2D convolution operations, it captures both intra-period dependencies and inter-period correlations, enhancing the temporal context of the predic- tion and improving the prediction results.
features. Based on the way frequency information is uti- lized during feature extraction, the existing methods can be classified into four categories as shown on the left side of Figure 1(b). The predominant approaches, classified No- Freq Methods (Zhou et al., 2021; Liu et al., 2024a; Wang et al., 2024b), operate purely in the time domain and strug- gle to effectively process signals across the full frequency spectrum (Yi et al., 2024a). The other three categories at- tempt to address this limitation through frequency-domain processing: (I) Unified-Freq Methods (Yi al., 2024b) that process frequency components but treat them uniformly without distinguishing their importance; (2) Only Low-Free Methods (Zhou et al., 2022a; Xu et al., 2024b) that exclu- sively utilize low-frequency components while filtering out high-frequency ones; and (3) Weighted-Freet Methods (Zhou et al., 2022b; Zhang et al., 2024; Yi et al., 2024a) that com- bine high and low frequencies through simple weighted sum- mation. Recent empirical studies (Ye et al., 2024; Zhang 2024) reveal that the importance of different frequency components varies across scenarios, with each component potentially beneficial or detrimental to forecasting perfor- mance depending on the specific context, indicating that simply discarding certain frequency components or pro- cessing them independently may be suboptimal. However, although recent studies have introduced frequency-domain processing, they primarily focus on processing different fre- quency components independently, leaving the modeling of cross-frequency interactions unexplored.
Our main contributions can be summarized as follows:
We propose CFPT, a novel dual-branch framework for long-term time series forecasting. The framework leverages both frequency dynamics and timestamp pat- terns through Cross-Frequeet Interaction (CFI) and Periodic-aware Timestamp Modeling (PTM) branches to enhance prediction accuracy.
To effectively model frequency information in time series data, we propose a CFI branch that captures interactions between different frequency components through a feature fusion mechanism for enhanced pre- diction performance.
Second, timestamps naturally reflect the periodic character- istics of time series data (Wang et al., 2024a). As shown in Figure 2. analysis of the data across four consecutive days (three weekdays and one weekend) in 24-hour peri- ods reveals two distinct timestamp patterns. One is the intra-period dependencies, where each day exhibits simi- lar periodic variations with time series values rising and falling at specific timestamps (as illustrated at the marked points 7:00 and 21:00). The other is the inter perr-pe correla- tions, where load patterns corresponding timestamps (e.g.,
To enhance forecasting, we develop a PTM branch that transforms ID timestamps into 2D tensors, capturing intra- and inter-period relations via 2D convolutions. Experiments on multiple real-world benchmarks show method's superiority.
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CFPT: Empowering Time Series Forecasting through Cross-Prequency Interaction and Periodic-Aware Timestamp Modeling
Cross-Feeuenccy Interaction (CFI)
Pr 2), w
Lirear
Linear
Freq Division
ReLU
Linear
L NayerNorm
High Freq
Linear
The Overall Processing Pipeline of Our Method
Historical Series
Norm
DFT
CFI
IDFT
InverseNorm
Predicted Time Series
1-
Future
THP
PTM
Timestamp
2300:00
Aere Timestamp Modeling (PTM)
intra- peria-perid depen den cles
eg: Hour Day
LayerNorm
Layerroam
())Periodic Slicing
Reshape from 2D to 1D
Linssr
Rel
Linerr
R
00 01 02 (1304 05 -.... 18 19
2D
00 01 03:04 05
18
29
Conv
inter- perir- correlations
(2)Reshape from 1D to 2D
29
00 01 03 05 -- 18 21
F Feature Space Alignment
Period-Aware Pattern Recognition
Figure Overview the proposed CFPT architecture. The middle part shows the overall processing pipeline, while upper and lower parts illustrate the two main branches: Cross-Frequency Interaction (CFI) and Periodic-Aware Timestamp Modeling (PTM), respectively. Detailed meaning the symbols: e, extraction real/imaginary parts; C. concatenation;
S splitting of processed features;
reconstruction of complex numbers.
2. Related Work
2.2. Timestamp Modeling in Time Series Forecasting
2.1. Frequency Modeling in Time Series Forecasting
Timestamps have been increasingly utilized as enhance- ment components for time series forecasting (Wang et al., 2024a: Zeng et al., 2024). Early approaches like In- former (Zhou et al., 2021), Autoformer (Wu al., 2021), FEDformer (Zhou et al., 2022b) and TimesNet (Wu et al., 2023) utilize timestamps by adding timestamp embeddings with data embeddings, while recent methods such as i'Trass- former (Liu et al., 2024a) and TimeXer (Wang et al., 2024b) embed timestamp features as tokens for attention mecha- nisms. Additionally, with the advancement of large lan- guage models in time series forecasting (Zhou al., 2023; Jin et al., 2023; Cao et al., 2024; Pan et al., 2024), some works (Liu et al., 2024b) have attempted to model times- tamps through prompts. However, empirical studies (Wang et al., 2024a; Tan et al., 2024) reveal that such timestamp modeling approaches demonstrate limited effectiveness in practice. Furthermore, despite these attempts in timestamp modeling, the periodic characteristics inherent in times- tamps remain insufficiently explored. To fully harness the potential of timestamps for performance enhancement, we propose to explicitly model both intra-period dependencies and inter-period correlations of time series based on times- tamp patterns.
Time series forecasting with frequency analysis has attracted increasing attention in deep learning research. Recent works have explored various frequency-based approaches to en- hance model performance. As illustrated in Figure ((b), These methods generally fall into four categories based on their frequency processing strategies: (1) No-Freq Meth- ods (Zhou et al., 2021; Liu et al., 2024a; Wang et al., 2024b) that operate purely in the time domain, (2) Unified-Faad -reicc Methods (Yi et al., 2024a) that process all frequencies uni- formly, (3) Only Low-Freq Methods (Zhou et al., 2022a; Xu al., 2024b) that exclusively focus on low-frequency components, and (4) Weighted-Freq Methods (Zhou al., 2022b; Zhang et al., 2024; Yi et al., 2024a) that combine frequencies through weighted summation. While these ap- proaches have shown promising results, they either ignore frequency-domain information entirely or process different frequency components independently. Unlike these meth- ods, we propose to explicitly model tross-frequency interac- tions, recognizing that the importance of different frequency components varies across scenarios and their interactions can significantly impact forecasting accuracy.
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CFPT: Empowering Time Series Forecasting through Cross-Prequency Interaction and Periodic-Aware Timestamp Modeling
3. Preliminaries
3.3. Instance Normalization (Norm & InverseNorm)
3.1. Problem Statement
Time series data often exhibits non-stationarity, where sta- tistical properties like mean and variance shift over time. This can degrade model performance when making predic- tions on future data with different distributions (Kim et al., 2021; Fan et al., 2023; Han et al., 2024; Fan et al., 2025). To address this issue, we apply instance normalization to both input time series X and predicted values X. For input normalization:
Given multivariate time series data X E RX with vari- ables over T time steps and corresponding timestamp infor- mation T E RAxxT where X RRx1 and T., E RMxx denote the observations and timestamps at time step i, re- spectively. The forecasting task aims to predict future r' steps X(+1:++' based on historical observations Xt-rit> accompanied by their corresponding future timestamps T+11++7' to enhance prediction performance. The fore- casting process can be formulated as:
XI-TA - Va
Norm X I-rd
(6)
X++1++T = F(X,-ra; T++1++7)
where and a are the mean and standard deviation of the input window respectively, and is small constant for numerical stability. After obtaining predictions, we apply inverse normalization:
where F()) denotes the proposed forecaster.
3.2. Discrete Fourier Transform (DFT IDFT)
InverseNorm .X+11OTT X
fenorm 1+1+47 X Vo (7)
Time series data can be viewed as discrete samples of con- tinuous signals, which can be transformed into frequency domain for spectral analysis. Given multivariate time series input X RNT, apply discrete Fourier trans- form (DFT) independently to each channel (x()) E RIXT For real-valued signals, we use the single-sided spectrum representation F[k] E ( (1+);
where Xnorm x+1+++'' is the predicted normalized data. This normalization strategy helps maintain consistent statistical properties across different time periods, enabling more ro- bust forecasting performance.
3.4. Timestamp Hierarchical Processing (THP)
T-1 x|n] e 12m Km
To better serve periodic-aware timestamp modeling, we first process the raw timestamp information T441447' E RMxx' through a Timestamp Hierarchical Processing (THP) (Alexandrov et al., 2020) module that extracts in- herent timestamp hierarchies. Specifically, THP a feature engineering module that decomposes timestamps into hier- archical timestamp features at different granularities. Using the notation "X of Y" (written as XOFY) where X repre- sents a finer time unit and Y represents its parent unit, we extract timestamp features including MinuteOfHour in [0, 59] for minute-level fluctuations, HourOIDay in [0, 23] for diurnal patterns, DayOfWeek in [0, 6] for weekly cycles, MonthOrYear in [0. 11] for monthly patterns, and ScasonO- fYear in [O, 3] for quarterly patterns, among others. Each timestamp component is normalized to [-0.5, 0.5) through carefully designed transformations that preserve their cyclic characteristics. The processed hierarchical timestamp fea- tures are denoted as T E R xr',"', where m indicates the number of selected timestamp features.
DFT: F[A]
Σ n=
(2)
where & E [0, 11 is the frequency index and |0,T 1] is the time step index in the original signal. Once in frequency domain, the components F[k] can be charac- terized by their magnitude and phase, calculated from their real part (e(F(k]) E R (1+(+))) and imaginary part Im(F(A)) E (x++1)
Magnitude: A[k] / Re (F (+
(3)
Phase: [kk] atan)(Im(F(())). (k))
(4)
where atan2 is the two-argument arctangent function that determines the angle in all four quadrants. The real and imaginary parts of frequency components represent symmet- ric and asymmetric patterns respectively. After modifying these components to adjust signal characteristics, we can transform the signal back to time domain using IDFT:
T/2
IDFT: x[m]
Σ
Flk]
12x 4p
4. Methodology
4.1. Structure Overview
where E [0, - represents the reconstructed time in- dices. These transformations enable precise control over signal properties while maintaining the fundamental peri- odic patterns. In our implementation, the transformations are performed using Fast Fourier Transform (FFT) with a computational complexity of O T).
The middle part of Figure 3 shows the overall architec- ture of our model, which consists of two branches: Cross- Frequency Interaction (CFI) and Periodic-Aware Timestamp Modeling (PTM). Specifically, as shown in the upper part of Figure 3, the CFI branch first applies DFT to decompose
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CFPT: Empowering Time Series Forecasting through Cross-Frequency Interaction and Periodic-Aware Timestamp Modeling
time series into high and low frequency components. The frequency components are then processed by extracting and concatenating their real and imaginary parts, followed by separate linear projections. A coupling layer with L itera- tions enables interactions between different frequency bands. The interacted features are transformed back to complex do- main and combined through inverse Fourier transform to generate frequency-awen aware predictions.
Initial Feature Processing. The real and imaginary parts of Fr. and Fи are extracted as Rz, Tt. -R and RH, IH E IR respectively, and then concatenated to ob- tain 9L. E RNN22 before passing through separate lin- ear layers, producing feature representations hs. E NRD and h E RNND, where is the hidden dimension. This separation allows independent processing of symmetric and asymmetric patterns while preserving their mathematical relationships.
Meanwhile, as illustrated in the lower part of Figure 3, the PTM branch first processes raw timestamps through THP to obtain m representative timestamp features from MM po- tential hierarchical features that capture patterns at different granularities. These features are then passed through a sim- ple linear feature extraction module to align with the N dimensional time series space. The aligned features are sliced and reshaped according to period length P, enabling 2D convolution to capture both intra-period dependencies (within each period) and inter- enter-period correlations (across different periods) of time series based on timestamp patterns. Two dimension preeeerrnggg linear mappings further refine the timestamp representations.
Cross-Frequene Interaction. Given the feature represen- tations Jε and hig a coupling network with L layers is designed to model their interactions. Each coupling layer contains four MLPs with identical architecture (p, 17, V, and w) that transform the features through:
k+1 = h" @ exp(p(h))) + (b h+ h @ exp(v(h + (1
(14)
(15)
where @ denotes element-wise multiplication, and k indi- cates the layer index. The multiplicative terms modulate the magnitude of frequency components while preserving their phase relationships, and the additive terms enable ad- justments to both symmetric and asymmetric patterns. This coupling mechanism allows comprehensive feature inter- action between different frequency bands, facilitating the modeling of how long-term trends influence short-term vari- ations and vice versa. The resulting representations ht, hæ E R NxD are then projected through linear layers to (8) obtain gt., In E RNxx2 where w' -
Finally, outputs from both branches are combined through weighted addition. Based the detailed description above of our two complementary branches and their integration, the complete pipeline of CPFT can be mathematically for- mulated as follows:
x--rt = Norm (x)m())))))
7++++++ = THP (T
(9) Frequency Recombination. Finally, the processed fea- (10) tures ÔL and Ôи are split to obtain the real and imagi- (11) nary parts for each frequency component. Specifically, we split ğL E Nx2'' to obtain low-frequency components Rr, E Nxw' and split IN E R.- X2xx to obtain high- (13) frequency components RH, I E NN''' where the first w' features represent the real parts and the remaining w/ fea- tures represent the imaginary parts. This splitting operation exactly reverses the concatenation performed in the initial feature processing stage, ensuring mathematical consistency when reconstructing complex numbers for the IDFT process. These parts are then used to reconstruct complex numbers Fr E Nxw' and F'r gNNx'' respectively. These com- ponents are summed and transformed back through IDFT to obtain the prediction XCFI NNT...
XCFI- = CFI(X
XPTM- PTM +1+47))))
Xnorm = 7 XCFI + (1 71- XPTM
(12)
X44144T' = InverseNorm & (
where E 1) serves as a constant weight coefficient. This dual-branch design leverages both frequency dynamics and timestamp patterns to enhance prediction accuracy.
4.2. Cross-Frequency Interaction (CFI)
To effectively capture and leverage both long-term trends and short-term fluctuations in time series forecasting, we propose the Cross Cross-Frequency Interaction (CFI) branch that explicitly models the interactions between high-frequency and low-frequency components.
4.3. Periodic-Aware Timestamp Modeling (PTM)
While the CFI branch effectively captures frequency- interactions, timestamp patterns and periodicities in times- tamps also provide crucial information for forecasting. To further enhance the prediction accuracy, we propose the Prriddic-Aware Timestamp Modeling (PTM) branch that hi- crarchically processes timestamp information and leverages 2D convolution to capture both intra-period dependencies
Frequency Division. First, we transform the normalized input 1- E N into frequency domain through DFT, yielding F E Nxw, where w 2/2 - 1. Using the median frequency as threshold, we separate F into owwfrrquency components FL E N x and figh-frey components FF E Nxw through complementary binary masks.
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CFPT: Empowering Time Series Forecasting through Cross-Prequency Interaction and Periodic-Aware Timestamp Modeling
and inter-period correlations.
where X44144 is computed following Equation (13) and X++1:++' represents the ground truth values.
Feature Space Alignment. After obtaining the processed hierarchical timestamp features T E I B'x'', we need to align them with the dimensionality of the time se- ries variables to enable effective timestamp pattern model- ing. This alignment is achieved through a lightweight fea- ture extraction module consisting of two sequential Linear- LayerNorm-Ree U blocks, which maps the m-dimensional timestamp features to the N-dimensional time series space. The aligned features A E R retain the original se- quence length while matching the variable dimension for subsequent aeriodic-aware modeling.
5. Experiments
In this section, we conduct extensive experiments on real- world time series datasets to evaluate the effectiveness of our proposed CFPT. We first introduce the experimental setup, followed by comprehensive performance comparisons with etate-offle of-the-art methods. Furthermore, we perform detailed ablation studies on both CFI and PTM branches, analyze hy- perparameter sensitivity and computational efficiency, and visualize prediction results to demonstrate the effectiveness of our framework.
Period-Aware Pattern Recognition. To effectively cap- ture periodic patterns in the aligned timestamp features ZA. we first reorganize them into 2D structure accord- ing to the given period length P. Specifically, re- shape A E Nx into 2 E (N where each slice along the last dimension represents one complete pe- riod. This transformation enables us to apply dimension- preserving 2D convolution operations with kernel size k x k that can simultaneously model both antra-period dependen- cies and inter-period correlations. The 2D convolution struc- ture facilitates the modeling of complex periodic patterss in timestamp features by capturing both local periodic struc- tures and their long-range relationships. After the convolu- tion operations, we employ two dimenssinn-pesservinn linear mappings along the variable and timestamp dimensions respectively to further refine these learned periodic repre- sentations. Finally, we obtain XPTM- V enhance the prediction performance through subsequent integration.
5.1. Experimental Setup
Dataset. We conduct long-term forecasting experiments on seven popular real-world benchmarks including the ETT series (Zhou et al., 2021), ECL. (Wu et al., 2021), Traffic (Wu et al., 2021), and Weather (Wu et al., 2021). These datasets span multiple domains including electricity trans- former data, power consumption patterns, highway traffic monitoring, and metcorological measurements. To maintain consistency and enable direct performance comparison, we adopt identical data preprocessing steps (e.g.. TetesNet (Wu et al., 2023), iTrrnsformer (Liu et al., 2024a)) to ensure fair comparison. The detailed statistics of these datasets are summarized in Appendix A.
Baselines. We compare our proposed CFPT with represen- tative and state-of-the-art models, including: (1) TimeXer (Wang et al., 2024b), a parallel multivariate forecasting model leveraging exogenous variables; (2) FilterNet (Yi et al., 2024a), signal processing based model utilizing frequency filters; (3) i'Transformer (Liu et al., 2024a), which treats series variate tokens capture multivariate correla- tions; (4) PatchTST (Nie et al., 2023), which extracts local patterns through subseries patches with channel indepen- dence; (5) FEDformer (Zhou et al., 2022b), implementing sparse attention in frequency domain; (6) TimesNet (Wu et al., 2023), capturing multi-scale temporal dependencies; (7) DLinear (Zeng et al., 2023) and (8) RLinear (Li et al., 2023), utilizing linear mappings for forecasting.
4.4. Dual-Branch Integration
The CFI and PTM branches are designed to capture com- plementary aspects of time series patterns. While the CFI branch focuses on modeling trequency-domain interactions between different frequency components, the PTM branch specializes in extracting periodic patterns from timestamp dependencics. As shown in Equation (12), these comple- mentary features are integrated through weighted addition with a constant coefficient y, allowing the model to effec- tively combine frequency-domen -omaan knowledge with times- tamp pattern awareness. This dual-branch design creates synergistic effect where the strengths of both frequency- based decomposition and periodic-aware timestamp model- ing are leveraged to enhance the overall prediction accuracy.
Implementation Details. Our experiments are performed with PyTorc 2.0.0 (Paszke et al., 2019) on a single NVIDIA RTX 3090 GPU. For forecasting task, we choose an input sequence length of T = 96 and evaluate on varying fore- casting horizons 7' E (96, 192,336, 720). For timestamp features, we consider MinuteOlHour, HourOIDay, Day- OfWeek, MonthOfYear, and SeasonOfYear in the Times- tamp Hierarchical Processing (THP). And for the period length P in the periodic timestamp slicing of PTM branch, we empirically set it to 24 to maintain a consistent gran-
4.5. Optimization Objective
For our forecasting task, we employ the squared loss (L2) to measure the discrepancy between the prediction and the ground truth. The overall training objective is:
f ==1 ||| || - X441++111 2
(16)
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CFPT: Empowering Time Series Forecasting through Cross-Prequency Interaction and Periodic-Aware Timestamp Modeling
ularity across all datasets. We adopt the mean absolute error (MAE) and the mean square error (MSE) as evaluation metrics. Additional implementation details are provided in Appendix B.
5.3. Ablation Study
Using Frequency Division. In this section, we experiment with the effect of frequency decomposition by comparing CFPT with its variant without separating high- and low- frequency components (w/o CFI-D). As shown in Table 2, the average results demonstrate that removing frequency de- composition consistently leads to performance degradation, indicating the importance of modeling frequency compo- nents separately.
5.2. Main Results
Table shows the strong performance of CFPT in long- term time series forecasting. Compared with forecasters of different structures, CFPT achieves robust performance across various datasets. Although not achieving the best results on the Traffic dataset, CFPT demonstrates superior generalization ability on other benchmarks, validating its effectiveness in handling diverse time series patterns.
Using Cross-Frequenis Interaction. In this section, we study the effectiveness of the cross-frequency interaction by comparing CFPT with its variant without couple layers (w/o CFI-C). As shown in Table 2, removing the coupling layers leads to consistent performance degradation across all datasets, with particularly notable drops on ETTh2 (from 0.364 to 0.394 in MSE). These results demonstrate the im- portance of modeling interactions between different fre- quency components.
From the frequency modeling perspective, models like TimeXer (Wang et al., 2024b), iTransformer (Liu et al., 2024a), and RLinear (Li et al., 2023) that operate solely in the time domain show limited capability in capturing com- plex frequency patterns. While FilterNet (Yi et al., 2024a) and FEDformer (Zhou et al., 2022b) attempt to process different frequency components through weighted summa- tion, they fail to model the intricate interactions between frequeney bands. Taking the ETT series as an example, where both high-frequency fluctuations and low-frequency trends coexist, our cross- frosuefre interaction mechanism effectively captures the dynamic relationships between dif- ferent frequency components. This advantage is particularly evident on the ECL dataset, which exhibits rich periodic pat- terns at multiple frequency scales due to daily and weekly electricity consumption cycles, leading to more accurate predictions.
Using CFI Branch. In this section, we investigate the neces- sity of frequency-based modeling by completely removing the CFI branch (w/o CFI), As shown in Table 2, the model performance significantly degrades without frequency anal- ysis, with MSE increasing from 0.374 to 0.391 on ETTmI, 0.364 to 0.390 ETTh2, and 0.240 to 0.251 on Weather. The substantial performance drop validates the effectiveness of our frequency-based design in CFPT.
Using Future Timestamps. In this section, we compare CFPT with its variant using historical timestamps (CFPT- HT), where the final linear layer in PTM branch maps from historical to future sequence length instead of directly mod- cling future timestamps. As shown in Figure 4, the perfor- mance gap between these two models widens as the pre- diction length increases from 96 to 720 steps, with CFPT consistently outperforming CFPT-HT (0.499 vs 0.544 in MSE at 720 steps). This suggests that using future times- tamps becomes increasingly crucial for longer horizons, likely due to the information loss when mapping historical patterns to extended forecasting windows.
From the timestamp modeling perspective, methods such as DLinear (Zeng et al., 2023) and FilterNet (Yi et al., 2024a) that do not utilize timestamp information show limitations in capturing timestamp patterns. While TimesNet (Wu et al., 2023) incorporates timestamps through simple addition with positional encodings, and TimeXer (Wang et al., 2024b) and iTransformer (Liu et al., 2024a) treat timestamps as attention tokens, these approaches cannot fully exploit the inherent pe- riodic characteristics of time series. This is particularly cru- cial for the ECL dataset, which contains complex temporal dependencies due to its regular daily patterns overlaid with weekly variations and seasonal trends. Our periodic-aware timestamp modeling explicitly captures both intra- period dependencies and inter-period correlations, enabling better understanding of these hierarchical timestamp patterns. The experimental results on ECL validate the effectiveness of our timestamp modeling approach.
Using 2D Timestamp Modeling, In this section, we com- pare CFPT with its variant using ID convolution without periodic slicing (CFPT-IDT). As shown in Table 3, imple- menting 2D timestamp modeling consistently yields better performance across all datasets, with MSE improving from 0.376 to 0.374 ETTm1, 0.369 to 0.364 on ETTh2, and 0.242 to 0.240 on Weather. These results demonstrate the advantage of our period- 2D modeling strategy.
The superior performance across different benchmarks demonstrates that our dual-branch architecture, which com- bines frequency interaction and periodic-aware timestamp modeling, provides a more comprehensive solution for long- term time series forecasting.
Using PTM Branch. In this section, we study the effec- tiveness of timestamp modeling by completely removing the PTM branch. As shown in Table 3. removing times- tamp modeling leads to significant performance degradation
7

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CFPT: Empowering Time Series Forecasting through Cross-Frequency Interaction and Periodic-Aware Timestamp Modeling
Table Forecasting results with look-back window r = 96 and prediction lengths r' E {96, 192, 336, 720). The MSE and MAE metrics averaged across all prediction horizons, with lower values indicating better performance. The best results are shown in bold. Our full results are in Appendix D.
Methods
CFPT
TimeXer
FilterNet
|irrasfoormrr
PatchTST
FEDformer
TimesNet
DLincar
RLinear
Metries
MSE
MAE|
MSE
MAE|
MSE
MAE|
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
ETTml
0.374
0.393 0.382 0.397 0.384 0.398 0.407 0.410
0.387 0,400 0,448 0.452 0.400 0,406
0,403 0.407 0.414 0.407
ETTm2
0.269
0.315 0.274 0.322 0.276 0.322
0.288
0.332
0.281 0.326
0.305 0.349
0.291 0.333
0.350 0,401 0.286 0.327
ETThl
0.433
0.429 0.437 0.437
0.440 0.432 0,454
0.448
0.469
0.454
0.440 0.460 0.458 0.450
0.456 0.452 0.446 0.434
ETTh2
0.364
0.393 0.367 0.396 0.378 0.404 0,383
0.407
0.387 0.407
0.437 0.449 0.414 0.427
0.559 0.515 0.374 0.398
ECL
0.164 0.259 0.171 0.270 0.201 0.285 0.178
0.270
0,205
0.290
0.214 0.327 0.192 0.295
0.212
0.300 0.219 0.298
Traffic
0.470
0.289
0.466
0.287
0.521
0.340
0.428
0.282
0,481
0.304
0,610
0.376
0.620
0.336
0.625
0.383 0.626 0.378
Weather
0.240
0.267
0.241
0.271
0.248
0.274
0.258
0.278
0.259
0.281
0.309
0.360
0.259 0.287
0.265
0.317 0.272 0.291
1 Count
6
6
0
0
0
0
1
0
0
0
0
Table Ablation study frequency modeling. Results are aged from all forecasting horizons.
Table 3. Ablation study on timestamp modeling. Results aver- aged from all forecasting horizons.
Methoos Metrics
CFFT
w/o CFL.D
wlo CFI-C
wio CFI
Methods Metris
CFPT
CFPT-IDT
wie PTM
MSE
MAE
MSE
MAR
MSE
MAE
MAE
MSE
MAF
ETTml
0,374
0,393
0.379
0.394
0,391
0.398
0.391
0.398
ETTm]
0,374
0,393
0,376
0.394
0.385
0.395
BFTh2
0,364
0,393
0,369
0.397
0.394
0.410
0,390
0,407
ETTh2
0,364
0,393
0,369
0.398
0,370
0.396
Weather
0.240
0.267
0,342
0.269
0.250
0.275
0,251
0.276
Weather
0.240
0,267
0,242
0.269
0.255
0.276
0.54
CFPT
CFPT-HT
efficiency, Detailed analyses on hyperparameter sensitivity and computational efficiency are referred to Appendix C.
MSE
0.50
0.492
5.5. Visualization of Prediction Performance
0.481
We present a prediction showcase on the ECL. dataset, as shown in Figure 5. We select TimeXer (Wang al., 2024b), FilterNet (Yi et al., 2024a), and iTransformer (Liu et al., 2024a) as the representative comparison methods. Com- paring with these
0.46
0.465
0.644
0.44
models, we observe that CFPT delivers more accurate predictions of future series variations, demonstrating superior forecasting performance.
96
192
336
720
Prediction Length
Figure 4. MSE comparison between CFPT (using future times- tamps) and CFPT-HT (using history timestamps) on Traffic dataset.
6. Conclusion
In this paper, propose CFPT, novel framework that empowers long-term time series forecasting through cross- frequency interaction and periodic- aware timestamp mod- eling. The Cross-Frequency Interaction (CFI) branch and Periodic-Aware Timestamp Modeling (PTM) branch work collaboratively, where CFI explicitly models interactions be- tween different frequency components through dedicated coupling mechanism, while PTM enhances prediction by capturing both intra-period dependencies and inter-period correlations based on timestamp patterns via 2D convolution on period- period-based tensors. Our comprehensive experiments across multiple benchmarks demonstrate the superiority of CFPT, achieving robust performance especially on datasets
across all datasets, with MSE increasing from 0.374 to 0.385 on 1, 0.364 to 0.370 on ETTh2, and 0.240 to 0.255
Weather. These results validate the importance of explicit timestamp modeling in our framework.
5.4. Hyperparameter and Efficiency Analysis
We analyze CFPT's hyperparameter sensitivity by examin- ing critical parameters including weight coefficient 7, kernel size k, number of coupling layers L, and model dimension D. as well as evaluate its computational efficiency in terms of training time and resource consumption. CFPT achieves robust performance while maintaining high computational
8

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CFPT: Empowering Time Series Forecasting through Cross-Prequency Interaction and Periodic-Aware Timestamp Modeling
Türkmen, A. C.,, and Wang, Y. GluonTS:: Probabilistic and Neural Time Series Modeling in Python. Journal of Machine Learning Research, 21(116):1-6, 2020.
ghanchaa
Cao, D., Jia, F., Arik, S. O., Pfister, T., Zheng. Y., Ye, W., and Liu, Y. TEMPO: Prompt-based generative pre-trained transformer for time series forecasting. In The Twelfth International Conference on Learning Representations, 2024.
(a) CFPT
(b) TimeXer
Fan, W. Wang, P., Wang. D., Wang. D., Zhou, Y., and Fu, Y. Dish-ts: general paradigm for alleviating distribution shift in time series forecasting. In Proceedings of the AAAI conference on artificial intelligence, volume 37, pp. 7522-7529, 2023.
8
150
16
180
125
195
Fan, w., Zheng. S., Wang, P., Xie, R., Yi, K., Zhang, Q., Bian, J., and Fu, Y. In-flow: Instance normalization flow for non-stationary time series forecasting. In Proceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining V. 1, pp. 295-306, 2025.
(c) FilterNet
(d) iTTrasformer
Figure 5. Visualization of prediction on the ECL. dataset with look- back and horizon length as 96.
Han, L., Ye, H.-J., and Zhan, D.-C. SIN: Selective and inter- pretable normalization for long-term time series forecast- ing. Forty-first International Conference on Machine Learning, 2024.
with complex periodic patterns like ECL. For future work, incorporating adaptive period detection mechanisms could further enhance our PTM branch to better accommodate datasets with irregular periodicity patterns. We hope our work can inspire future research to explore more sophis- ticated frequency interaction mechanisms and timestamp modeling strategies for enhancing long-term time series forecasting capabilities.
He, H., Zhang, Q., Bai, S., Yi, K., and Niu, Z. Catn: Cross attentive tree-aware network for multivariate time series forecasting. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 4030-4033 2022.
He, Q.-Q.. Siu, S. W. I., and Si, Y.W. Instance-based deep transfer learning with attention for stock movement prediction. Applied Intelligence, 53(6) 2023.
Acknowledgements
This work was supported by the National Key Research and Development Program of China (2023YFF0725103),
Hu, X., Fan, W., Chen, H., Wang, P., and Fu, Y. Recon- structing missing variables for multivariate time series forecasting via conditional generative flows. In Proceed- ings of the 33rd International Joint Conference on Artifi- cial Intelligence (IJCAI, Jeju, Korea, 2024.
the National Natural Science Foundation of China (No.62002027, 62472042, 62072488, 62272058, U23A20319), the 8th Young Elite Scientists Spon- sorship Program by CAST (2022QNRC001), Guangxi Key Laboratory of Trusted Software (No. NX..33023,4) and the Beijing Natural Science Foundation (Grant N233033.
Ilbert, R., Odonnat, A., Feofanov, V., Virmaux, A.. Paolo, G., Palpanas, T., and Redko, I. SAMformer: Unlocking the potential of transformers in time series forecasting with surpness-awarr minimization and channel-wise atten- tion. In Forty-firs International Conference on Machine Learning, 2024.
Impact statement
This paper presents work whose goal is advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Jin, M., Wang, S., Ma, L., Chu, Z., Zhang, J. Y., Shi, X., Chen, P.-Y,, Liang, Y., Li, Y.-F,, Pan, S., et al. Time-llm: Time series forecasting by reprogramming large language models. arXiv preprint arXiv:2310.01728. 2023.
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CFPT: Empowering Time Series Forecasting through Cross-Prequency Interaction and Periodic-Aware Timestamp Modeling
A. Dataset Descriptions
evaluate CFPT on seven widely-used time series benchmarks: ETT series (Zhou et al., 2021) (ETThl, ETTh2, ETTml, ETTm2). ECL (Wu al., 2021), Traffic (Wu et al., 2021), and Weather (Wu et al., 2021). The ETT series contains transformer temperature records with different sampling rates (hourly and 15-minute). ECL consists of electricity consumption data from multiple clients, Traffic records road occupancy rates from highway sensors, and Weather includes various meteorological measurements. These datasets cover diverse scenarios with different sampling frequencies and variable dimensions, providing a comprehensive testbed for long-term forecasting. The detailed statistics of these datasets are summarized in Table 4.
Table Statistical Properties of Experimental Datasets.
Dataset
ETTm1&2
ETTh1&2
ECL
Traffic
Weather
Timesteps
69,680
17,420
26,304
17,544
52,696
Channels
7
7
321
862
21
Dataset Size (34465, 11521, 11521)
(8545, 2881, 2881
(18317, 2633, ) (12185, 1757, 3509)
(36792, 5271, 10540)
Granularity
15 mins
hour
hour
hour
10 mins
Periodicity
Daily
Daily
Daily & Weekly
Daily & Weekly
Daily & Seasonly
B. Implementation Details
All experiments are implemented in PyTorch 2.0.0 (Paszke et al., 2019) and conducted on single NVIDIA RTX 3090 GPU, We utilize ADAM optimizer (Kingma, 2014) with learning rate searched from (0.01, 0.005, 0.0001, 0.0005)0 and L.2 loss for model optimization. The training process fixed to 10 epochs with batch size selected from {4, 8, 16, 128]. For model architecture, we conduct grid search over several key hyperparameters:
(a) Weight coefficient y from 0.2, 0.9). (b) Kernel size from (2.3, 3, 4, 5). (c) Number of coupling layers L from {1,3, 6, 12), and (d) Model dimension D from {128, 256, 512). For timestamp features, consider MinuteOfHour, HourOfDay, DayOfWeek, MonthOrYear, and SeasonOFYear in the Timestamp Hierarchical Processing (THP). Additionally, fixed period length P = 24 is applied in PTM branch. All experiments are repeated with seed 2025. We reproduced all baseline models following the official implementation of TimesNet (Wu et al., 2023).
C. Model Analysis
C.1. Hyperparameter Sensitivity Analysis
To evaluate the parameter sensitivity of CFPT, we conduct ablation studies by varying key hyperparameters: (a) Weight
coefficient y from (0.1, 0.2, ..., 0.9), (b) Kernel size from {2, 4, 5), (c) Number of coupling layers L from {1,3, 6, 12), and (d) Model dimension D from {128, 256, 512). The results on ETTm1, ETTh2 and Weather datasets with lookback length of 96 and horizon length of 720 are illustrated in Figure 6. The model shows optimal performance when 7 is in [0.1, 0.7], with relatively stable or degraded performance beyond 0.7. kernel size
generally yields better results. The impact of coupling layers L varies across datasets, which may be attributed to their different frequency characteristics. The model achieves optimal performance with hidden dimension = 512.
12

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CFPT: Empowering Time Series Forecasting through Cross-Prequency Interaction and Periodic-Aware Timestamp Modeling
0.440
0.440
0.420
0.420
0.400
0.380
ET1m1
0.380
ET1ml
0.360
Weather
0.360
Weather
0.340
0.340
0.1
0.5
2
(a)
(b)
0.440
0.420
0.400
0.380
ETTm3
Weather
0.360
256
512
(c)
(d)
Figure 6. Sensitivity analysis of hyperparameters ETTml, ETTh2 and Weather datasets with lookback length of 96 and horizon length of 720.
C.2. Efficiency Analysis
As shown in Table 5, we evaluate CFPT's efficiency based on model parameters, GPU memory usage, and training time per iteration on ETTml, ETTh2 and Weather datasets with lookback window size fixed 96 while varying prediction horizons from 96 to 720 timesteps. The results demonstrate that our model maintains stable training speed around 0.8s per iteration and exhibits low resource requirements with moderate parameter sizes and GPU memory usage across all settings, even for the Weather dataset with higher dimensionality and longer prediction horizons. These results indicate that CFPT achieves strong performance with modest computational costs, making it suitable for practical deployment.
Table 5. Model efficiency analysis with fixed lookback window size of 96 and varying prediction horizons, evaluated on different datasets terms of model parameters, GPU memory usage, and training time.
Dataset
Horizon
Parameter (MB)
GPU Memory
Running Time (s iter)
(MiB)
96
48,971
75.675
0,813
192
4.744
15.982
0.804
ETTml
336
1,879
12.780
0.805
720
7.631
20.341
0.831
96
1,243
10,925
0.781
192
17.344
33.619
0.789
ETTh2
336
25,460
190.327
0.819
21.262
52.163
0.789
96
12,507
248.786
0.793
192
3,474
148.505
0.812
Weather
336
3,924
167.071
0.800
720
3.246
169.071
0.804
13

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CFPT: Empowering Time Series Forecasting through Cross-Frequency Interaction and Periodic-Aware Timestamp Modeling
D. Performance of Long-term Multivariate Forecasting
To comprehensively evaluate CFPT, we conduct long-term multivariate forecasting experiments on various real-world benchmarks. With lookback window size fixed as 96, we test the model's performance across different prediction horizons ranging from (96, 192, 336, 7203. The detailed results are presented in Table 6.
Table 6. Full results of the long- term multivariate forecasting task.
Methods
CFPT
TimeXer
FitterNet
[iransformmr
PStchTST
FEDformer
TimesNet
DLinear
R
Metrics
MSE
MAE|
MSE
MAE|
MSE
MAE/
MSE
MAE
MSE
MAE
MSE
MAF] MSE MAE MSE MAE]
MSE MAE
96
0.316 0.356 0.318 0.356 0.318 0.358 0.334 0.368 0.329 0.367 0.379 0.419 0.3338 0.375 0.345 0.372 0.355 0.376
192
0.354 0.380 0.362 0.383 0.364 0.383 0.377 0.391 0.367 0.385 0.426 0.44] 0.374 0.387 0,380 0.389 0.391 0.392
ETTml
336
0.383 0.400 0.395 0.407 0.396 0.406 0.426 0.420 0.399 0.410 0.445 0.459
0.410
0.411 0.413 0.413 0.424 0.415
720
0.444 0,434 0,452
0.441 0.456 0.444 0.491 0,459 0.454 0,439 0,543 0,490 0.478 0.450 0,474 0,453 0.487 0,450
Avg.
0.374 0.393
0.382 0.397 0.384 0.398 0.407
0.410
0.387 0.400 0.448 0.452
0.400
0.406 0.403 0.407 0.414 0.407
96
00.167 0.249
0.1 0.256 0.4 0.257 0,0 0.264
00.175 0.259 2.3 0.287
0.187
0.267 0.193 0.292 0.265
192
0.232 0.292 0.237 0.299 0.240 0.300 0.250 0.309
0.241 0.302 0.269 0.328
0.249 0.309 0.284 0.362 0.246 0.304
ETTm2
336
0.290 0.331 0.296
0.338 0.297 0.339 0.311 0.348
0.305
0.343 0.325 0.366
0.321
0.351 0.369 0.427 0.307 0.342
720
0.385 0.389 0.392
0.394 0.392 0.393 0.412 0.407
0.402 4.0 0,421 0.415 0.408
0.403 0.554 0.522 0.407 0.398
Avg.
0.269
0.315
0.274
0.322 0.276 0.322
0.288
0.332
0.281 0.326 0.305 0.349
0.291 0.333 0.350 0.401 00.286 0.327
96 00.322 0.391
00,382
0.403
0.375 0.394
0.386
0,405
0.414
0.419 0.376
0.419
0.402 0,386 0.400 0.395
192
0.425 0.421
0.429
0.435 0.436 0.422 0.441
0,436
0.460 0.445 0,420 0.448 0.436
0.429 0.437 0.432 0.437 0.424
E
336
0.467 0.442 0.468
0.448 0.476 0.443 0.487 0.458
0.501 0.466 0,459 0.465
0.491 0.469 0.481 0.459 0.479 0.446
720
0.466 0.461 0.469
0.461 0.474 0.469 0.503
0.491
0.500
0.488 0.506 0.507 0.521 0.500 0.519 0.516 0.481 0.470
Avg.
0.433
0.429 0.437
0.437 0.440 0.432 0.454
0.448
0.469
0.454 0.440 0.460
0.458
0.450 0.456 0.452 0.446 0.434
96
0.285 0.336 0.286
0.338 0.292 0.343 0.297
0.349
0.302
0.348 0.358
0.397
0.374 0.333 0.387 0.2288 0.338
192 0.363 0.388 0,363 0.389 0.369 0.395 0,380
0.400
0,388 0.400 0.429 0.439 0.402 0.414 0.477 0.476 0.374 0.390
ETTh2
336
0.413 0.426 0.414 0.423 0.420 0.432 0.428
0.432
0.426 0.433 0.496 0.487 0.452 0.452 0.594 0.541 0.415 0.426
720
0.396 0.422 0,408 0.432 0.430 0.446 0.427
0,445
0,431 0.446 0.463 0.474 0.462 0.468 0.831 0.657 0.420 0.440
Avg.
0.364 0.393 0.367 0.396 0.378 0.404 0.383
0.407
0.387 0.407 0.437 0.449
0.414 0.427 0.559 0.515 0.314 0.374 0.398
96
00113 0.231
0.114
0.242
0.6 0.264
0.148
0.240
0.181
0.270 193 0.308
0.166
0.272 0.197 0.282/0.201 0.281
192
0.153 0.246 0.1 0.256 185 0.270 12162 0.253
0.188 0.274 0.201 0.315
0.184 0.289 0.196 0.285 0.201 0.283
ECL
336
0.168 0.265
0.176 0.275 0.202 0.286 0.178 0.269
0.204
0.293 0,214 0.329
0.198 0,300 0.209 0.301 0.215 0.298
720
0.199 0.293
0,211 0.306 0.242 0.319 0.225
0.317
0,246 0.324 0.246 0.355
0.220 0.320 0,245 0.333 0.257 0.331
Avg.
0.164 0.259
0.171 0.270 0.201 0.285
0.178
0.270
0.205 0.290 0.214 0.327
0.295 0.212 0.300 0.219 0.298 0.192
96
00.444 0.274
00.428
0.271
0.336
0.395
0.268
0.462
0.295 0.587 0.366
0.593
0.321 0.650 0.396
0.699 0.669 0.389
192 0.460 0.280 0.448 0.282 0.508 0.333
0.417
0.276
0.466 0.296 0.604 0.373
0.617
0.336 0.598 0.370 0.601 0.366
Traffic
336 0.477 0.289 0.473 0.289 0.518 0.335
0.433
0.283
0.482 0.304 0.621 0.383
0.629 0.336 0.60S 0.373 0.609 0.369
720 0.499 0.313 0.516 0.307 0.553 0.354
0.467
0.302
0.514
0.322 0.626 0.382 0.640
0.350 0.645 0.394 0.647 0.387
Avg.
00.470 0.289 0.466 0.287 0,521 0.340
0.428
0.282
0.481 0.304 0.610 0.376
0.620 0,336 0.625 0.383 0.626 0.378
0.154 0.200
0.1
0.205
064 0.210
0.174
0.214
0.177
0.218
0.217 0.296
0.172
0.220 0.196 0.255 1. 0.232
192
0.203 0.242
0.204 0.247 0.214 0.252 0.221 0.254
0.225 0.259 0.276 0.336 0.219 0.261 0.237 0.296 0.240 0.271
Weather
336
0.261 0.286
0.261 0.290 0.268 0.293 0.278 0.296
0.278
0.297 0.339
0.380 0,280
0.306 0.283 0.335 0.292 0.307
720
0.340 0.339
0.340
0.341
0.344 0.342
0.358
0.347
0.354
0.348
0.403 0.428
0.365
0.359 0.345 0.381
0.364 0.353
Avg.
0.240
0.267
0.241
0.271
0.248
0.274
0.258
0.278
0.259
0.281
0.309
0.360
0.259
0.287 0.265
0.317
0.272 0.291
1st Count
29
30
3
3
4
4
0
0
0
14


Reviews:
[{"review_id": "DBgSPUYElf", "reviewer": "Reviewer_RXr7", "summary": "This article investigates time series tasks and proposes a method called CFPT. Its main idea is to improve prediction results by capturing the complex relationships between different frequency components. The paper is written very clearly and the proposed modules have good motivation.\n\n## Update after rebuttal\n\n> I've read the rebuttal and other reviewers' comments, my final rating is weak accept. The reason why I give this score is that although the authors have already responded to most of the questions, I feel that the description of the CFI module is still not detailed enough and needs further elaboration.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Many thanks for your review and precious comments and advises. Specific responses are presented below:\n\n**Response to Question 1:**\n\nThank you for suggesting a more comprehensive introduction to Section 5. We will add the following sentence at the end of the opening paragraph: \"Furthermore, we perform detailed ablation studies on both CFI and PTM branches, analyze hyperparameter sensitivity and computational efficiency, and visualize prediction results to demonstrate the effectiveness of our framework.\"\n\n**Response to Question 2:**\nThank you for pointing out the need for more details on the splitting operation in the CFI branch. In Section 4.2 under \"Frequency Recombination,\" we will replace the current text:\n\"Specifically, we get low-frequency components$\\hat{R}_L, \\hat{I}_L \\in \\mathbb{R}^{N \\times w'}$ and high-frequency components$\\hat{R}_H, \\hat{I}_H \\in \\mathbb{R}^{N \\times w'}$.\"\nWith this more detailed description:\n\"Specifically, we split $\\hat{g}_L \\in \\mathbb{R}^{N \\times 2w'}$  to obtain low-frequency components $\\hat{R}_L, \\hat{I}_L \\in \\mathbb{R}^{N \\times w'}$ and split $\\hat{g}_H \\in \\mathbb{R}^{N \\times 2w'}$ to obtain high-frequency components $\\hat{R}_H, \\hat{I}_H \\in \\mathbb{R}^{N \\times w'}$, where the first $w'$ features represent the real parts and the remaining $w'$ features represent the imaginary parts. This splitting operation exactly reverses the concatenation performed in the initial feature processing stage, ensuring mathematical consistency when reconstructing complex numbers for the IDFT process.\"\n\n**Response to Question 3:**\n\nThank you for your valuable comments. In the revised version, we will add an explanatory statement of the \"predicted normalized data\"($\\hat X_{t + 1:t + \\tau }^{norm}$).\n\n**Response to Other Comments Or Suggestions:**\n\nThank you for your valuable comments. In the revised version, we will ensure consistency by using 'L' throughout the paper, including in Figure 3, to avoid any confusion for readers.", "claims_and_evidence": "Yes, I believe that the paper is supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The article conducted extensive experiments on the seven commonly used datasets across diverse domains with comparing state of art baseline methods. The evaluation metrics used MSE and MAE. Overall, the method and evaluation criteria make sense for the time series forecasting problems.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "Yes, the experiment is reasonable. Ablation research, hyperparameter sensitivity analysis, efficiency evaluation, and performance visualization can validate the performance of the model.", "relation_to_broader_scientific_literature": "The paper advances the time series forecasting field, as evidenced by their related work discussion. In frequency-domain time series modeling, the proposed cross-frequency interaction mechanism represents an advancement over existing methods. In timestamp modeling research, the periodic-aware timestamp modeling provides a novel perspective compared to the existing methods that show limited effectiveness in practice.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "**Strengths**:\nThis work is significant for advancing time series forecasting by recognizing the crucial need to model both frequency component interactions and timestamp periodicities, the aspects overlooked in current methods. And the experimental results are reliable, while the evaluation demonstrates the method's strong performance through comprehensive evaluations in diverse forecasting scenarios under fair and rigorous evaluation protocols.\n\n**Weaknesses**:\n1. The introduction to Section 5 (Experiments) at its beginning could be more comprehensive. It is encouraged to briefly outline all experimental components, including the Ablation Study, Hyperparameter Analysis, and Visualization sections.\n2. In addition to the current explanation, it is encouraged to provide further details on the splitting operation in the CFI branch. \n3. In Section 3.3 (Instance Normalization), the symbol for the predicted normalized data in Equation 7 should be clarified. \n4. In the Timestamp Hierarchical Processing (THP) section, the paper mentions that \"each timestamp component is normalized to [-0.5, 0.5] through carefully designed transformations that preserve their cyclic characteristics\", but doesn't specify the actual transformations used or explain how they maintain the cyclical nature of the temporal features.", "other_comments_or_suggestions": "The notation for coupling layers should be consistent throughout the paper. Figure 3 uses 'N' while the text uses 'L'.", "questions_for_authors": "Based on the Weaknesses, I have the following questions:\n(1)\tCould you expand the introduction to Section 5 (Experiments) to include a brief overview of all experimental components?\n(2)\tHow is the splitting operation in the CFI branch performed? Clarifying these details would enhance the reproducibility and understanding of the methodology.\n(3)\tWhat does the symbol in Equation 7 represent in the context of normalized data? Clarifying its meaning would enhance the understanding of the instance normalization process.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VT7CS91EFy", "reviewer": "Reviewer_LJCA", "summary": "This paper targets the problem of long-term time series forecasting with attention to frequency components and timestamp patterns. The proposed method is based on two observations: the importance of different frequency components varies across scenarios and their interactions may impact forecasting accuracy, and timestamps naturally reflect periodic characteristics that remain insufficiently explored in existing approaches. Then a dual-mechanism forecasting framework named CFPT is proposed to capture both cross-frequency interactions through a dedicated coupling mechanism and periodic timestamp patterns using 2D convolution on period-based representations. Experiments are conducted across seven real-world benchmark datasets with varying prediction horizons, and the results demonstrate the superiority of the proposed framework compared to state-of-the-art methods.\n\n## update after rebuttal\n\nAll my concerns have been addressed. I find this work both novel and solid, so I’d be glad to keep my score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Many thanks for your review and precious comments and advises. Specific responses are presented below:\n\n**Response to Question 1:**\n\nThank you for your valuable comments. In the revised version, we will update Figure 3 by carefully adjusting the layout to make the expression of Figure 3 more aesthetically pleasing and clear.\n\n**Response to Question 2:**\n\nThank you for identifying the mathematical notations that need clarification. In Section 3.2, we will add clear explanations for parameters k and n in the DFT formulation: \"where k ∈ [0, T/2] is the frequency index in the transformed domain (representing frequencies from zero to Nyquist frequency) and n ∈ [0, T-1] is the time step index in the original signal.\" Additionally, for Equation (5), we will update the upper limit of the summation from T-1 to T/2 to correctly reflect our implementation. For Section 4.5, we will refine the optimization objective by removing the summation parameter S to present a cleaner formulation of the loss function, focusing on the core squared loss between prediction and ground truth.\n\n**Response to Other Comments Or Suggestions:**\n\nThank you for your valuable comments. In the revised version, we will correct the errors in the quotation marks and thoroughly review the entire text to ensure that such issues do not recur.", "claims_and_evidence": "The claims in the submission are well-supported by clear evidence. The comprehensive benchmark results across multiple datasets demonstrate strong model performance. The thorough ablation studies effectively validate the contribution of each model module.", "methods_and_evaluation_criteria": "(1)The proposed methods are well-suited for long-term time series forecasting. The model's design thoughtfully handles frequency components and periodic patterns, effectively capturing both temporal trends and fluctuations through its specialized branches.\n(2)The evaluation criteria are appropriate and comprehensive. The testing utilizes diverse real-world benchmarks from environmental monitoring, power systems, transportation, and meteorological domains. The evaluation employs standard metrics and multiple prediction horizons, providing solid validation through baseline comparisons.", "theoretical_claims": "The theoretical claims about frequency decomposition and reconstruction are correct. These claims are verified through Equations 2-5, showing that signal transformation between time and frequency domains is reversible and preserves information (via DFT and IDFT), frequency components can be described by magnitude and phase, and complex signals can be decomposed and reconstructed using real and imaginary parts. These results confirm the model's frequency domain processing validity.", "experimental_designs_or_analyses": "The experimental design is methodologically sound. The authors evaluated CFPT on seven diverse datasets (ETT series, ECL, Traffic, Weather) against eight state-of-the-art baselines using standard metrics (MSE and MAE). Multi-horizon testing (96, 192, 336, 720 steps) effectively assessed long-term forecasting capabilities. Ablation studies systematically isolated each component's contribution through carefully designed variants (w/o CFI-D, w/o CFI-C, w/o CFI, CFPT-HT, CFPT-1DT, w/o PTM), confirming the necessity of frequency division, cross-frequency coupling, future timestamps, and 2D periodic modeling. Hyperparameter sensitivity analysis and computational efficiency evaluations further supported the model's robustness and practicality.", "relation_to_broader_scientific_literature": "(1) The paper advances the broader scientific literature on time series forecasting through its novel treatment of frequency components. Previous approaches can be categorized into four types based on their frequency handling: No-Freq Methods like [R1] that operate purely in the time domain, Unified-Freq Methods like [R2] that process all frequencies uniformly, Only Low-Freq Methods like [R3] that exclusively focus on low-frequency components, and Weighted-Freq Methods like [R4] that combine frequencies through weighted summation. This paper introduces a novel cross-frequency interaction mechanism that explicitly models the relationships between different frequency bands, building on recent empirical findings that demonstrate how the importance of frequency components varies across different forecasting scenarios.\n(2) The paper contributes to timestamp modeling in time series forecasting. Early approaches like [R1] and [R5] incorporated timestamps through basic embeddings, while recent methods have explored more complex approaches: some like [R6] and [R7] treat timestamps as attention tokens, and others like [R8] have attempted to model timestamps through prompts. However, empirical studies reveal that these approaches show limited effectiveness. This paper uncovers the potential of explicitly modeling the periodic characteristics inherent in timestamps through a 2D convolution architecture that captures both intra-period dependencies and inter-period correlations.\n\nReferences:\n[R1] Zhou et al., \"Informer: Beyond efficient transformer for long sequence time-series forecasting\", AAAI 2021\n[R2] Yi et al., \"Frequency-domain mlps are more effective learners in time series forecasting\", NeurIPS 2024\n[R3] Zhou et al., \"Film: Frequency improved legendre memory model for long-term time series forecasting\", NeurIPS 2022\n[R4] Zhou et al., \"FEDformer: Frequency enhanced decomposed transformer for long-term series forecasting\", ICML 2022\n[R5] Wu et al., \"Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting\", NeurIPS 2021\n[R6] Liu et al., \"iTransformer: Inverted transformers are effective for time series forecasting\", ICLR 2024\n[R7] Wang et al., \"TimeXer: Empowering transformers for time series forecasting with exogenous variables\", arXiv 2024\n[R8] Liu et al., \"Autotimes: Autoregressive time series forecasters via large language models\", arXiv 2024", "essential_references_not_discussed": "No essential references appear to be missing from this paper.", "other_strengths_and_weaknesses": "Strengths:\nS1. The paper presents an innovative dual-branch architecture that addresses two insufficiently explored aspects in time series forecasting: the interaction learning between different frequency components and the exploitation of periodic characteristics inherent in timestamps.\nS2. The authors provide clear motivation by systematically identifying limitations in existing approaches, classifying frequency-based methods into four categories and demonstrating how current timestamp modeling techniques show limited effectiveness in practice.\nS3. The experimental evaluation demonstrates exceptional rigor with comprehensive testing across seven diverse real-world benchmarks and thorough ablation studies that validate each component's contribution to the overall performance improvements.\nS4. CFPT shows excellent potential for real-world applications in domains like energy consumption and transportation, maintaining competitive computational efficiency while delivering superior performance on datasets with complex periodic patterns.\n\nWeaknesses:\nW1. The layout of Figure 3 should be improved to enhance readability. For example, the text \"Reshape from 2D to 1D\" should be repositioned directly above its corresponding arrow for better visual flow.\nW2. Some mathematical notations should be clarified. Parameters k and n in Section 3.2 and summation parameter S in Section 4.5 would benefit from clearer explanations, which would improve clarity and enhance understanding of the methodology.", "other_comments_or_suggestions": "In Section 3.4, there is incorrect usage of quotation marks where two right quotation marks appear instead of a proper pair of left and right quotation marks.", "questions_for_authors": "Q1. Could the authors improve the layout of Figure 3 to enhance readability?\nQ2. Could the authors address the mathematical notation issues that have been outlined in the weaknesses section?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FwA6d8i45i", "reviewer": "Reviewer_PJn9", "summary": "This paper addresses the challenge of long-term time series forecasting by introducing CFPT, a method that integrates frequency component analysis with timestamp pattern recognition. The key innovation lies in modeling cross-frequency interactions while simultaneously capturing periodic characteristics in timestamps. Experiments are conducted on multiple tasks, and the results prove the effectiveness of the proposed framework.\n\n# After rebuttal:\n\nThe authors solve my concerns and I vote for acceptance.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Many thanks for your review and precious comments and advises. Specific responses are presented below:\n\n\n**Response to Question 1:**\n\nThank you for your valuable comment. We will clarify that we implement the DFT using FFT algorithms in our work. Specifically, we will add the following statement at the end of Section 3.2: \"In our implementation, these transformations are performed using Fast Fourier Transform (FFT) algorithms with a computational complexity of O(T log T).\" This clarification will help readers understand our actual implementation approach.\n\n**Response to Question 2:**\n\nThank you for your valuable comments. We are so sorry that we have mistaken the arrows. In the revised version, we will update Figure 2 to make all arrows follow chronological order.\n\n\n**Response to Other Comments Or Suggestions:**\n\nThank you for your valuable comments. In the revised version, we will rename Section 3.2 to \"Discrete Fourier Transform (DFT & IDFT)\" to align with Section 3.3.", "claims_and_evidence": "The submission provides strong and convincing support for its claims. The model achieves superior performance across seven benchmarks compared to baseline models, particularly on complex periodic datasets, while ablation studies demonstrate clear performance degradation when removing key components.", "methods_and_evaluation_criteria": "The proposed methods make good sense for the time series forecasting problem. The approach addresses two insufficiently explored aspects in long-term forecasting: the interaction between different frequency components and the periodic characteristics inherent in timestamps. Moreover, the paper employs a well-designed evaluation strategy for time series forecasting. The choice of benchmark datasets captures diverse real-world scenarios. The evaluation includes meaningful comparisons with leading baselines and examines the effectiveness of different model components.", "theoretical_claims": "I have verified the correctness of the theoretical claims and their proofs in the paper. The formulations of the Discrete Fourier Transform (DFT) and its inverse (IDFT) in equations (2) and (5) are mathematically sound, correctly establishing the foundation for frequency analysis and reconstruction. Additionally, the instance normalization proofs in equations (6) and (7) properly demonstrate the basis for time series data normalization and inverse normalization.", "experimental_designs_or_analyses": "The experimental design is generally sound and follows standard practices in time series forecasting. The authors use the input length of 96 timesteps and evaluate on multiple prediction horizons across seven widely-used benchmark datasets. The comparison with eight recent baseline models and the use of standard metrics is appropriate. The ablation studies are particularly thorough, systematically isolating the contributions of each component. For example, removing the Cross-Frequency Interaction (CFI) branch results in significant performance degradation, highlighting the importance of modeling frequency interactions for accurate forecasting.", "relation_to_broader_scientific_literature": "The key contributions of this paper connect to several important developments in time series forecasting literature. First, while previous works like iTransformer process time series in time domain, and recent methods like Fedformer and FilterNet explore frequency-domain processing, CFPT innovatively models cross-frequency interactions through a coupling network rather than processing components independently or using simple weighted summation. Second, the paper improves timestamp modeling beyond current methods that rely on embeddings, attention mechanisms, or prompts. Its periodic-aware approach using 2D convolutions effectively captures both intra-period dependencies and inter-period correlations, providing a more comprehensive solution to timestamp modeling.", "essential_references_not_discussed": "No essential references appear to be missing from the paper's discussion. The references are up-to-date, and appropriately contextualize the paper's contributions within the current state of research.", "other_strengths_and_weaknesses": "Strengths:\n\n1.\tThe quality of the work is commendable, as the paper not only identifies key limitations in current forecasting paradigms but provides a principled solution that aligns with the interplay between different frequency components and the inherent periodic patterns in timestamps.\n\n2.\tThe paper is well-organized with clear logic and thorough experimental design.\n\n3.\tThe implementation is practically valuable, maintaining computational efficiency while achieving state-of-the-art performance. The stable training speed and moderate resource requirements make it suitable for real-world deployment.\n\nWeaknesses:\n\n1.\tThe author should provide justification for choosing DFT over the more computationally efficient FFT algorithm, while the FFT could reduce computational complexity from O(n²) to O(n log n).\n\n2.\tThe visualization in Figure 2 could be improved. Arrows should follow chronological order to better represent temporal dependencies in the time series data, while the current arrows point in the opposite direction.", "other_comments_or_suggestions": "For consistency, consider renaming section 3.2 to \"Discrete Fourier Transform (DFT & IDFT)\" to align with section 3.3's naming pattern, as both discuss paired operations.", "questions_for_authors": "1.\tWhy does the paper use DFT instead of FFT for frequency analysis? Would using FFT affect the model's performance or implementation?\n\n2.\tCould you correct the visualization of inter-period correlations in Figure 2? The arrows should follow chronological order rather than pointing in the opposite direction to accurately reflect temporal dependencies.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "NA"}}, {"review_id": "34OTS0cxSq", "reviewer": "Reviewer_7jYd", "summary": "The paper introduces CFPT method including two branches to address two key limitations in existing methods: inadequate modeling of interactions between different frequency components and insufficient exploitation of timestamp periodicity. The CFI branch processes signals in the frequency domain and captures interactions between different frequency components, and the PTM branch transforms timestamp sequences into 2D tensors to identify both intra-period and inter-period patterns. Experiments show that CFPT offers an effective solution for time series forecasting. \n\n## update after rebuttal\nI keep the overall recommendation of \"4 accept\", as the authors have addressed my major questions and concerns.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Many thanks for your review and precious comments and advises. Specific responses are presented below:\n\n**Response to Question 1:**\n\nThank you for your valuable comments. We did notice that the text in Figure 5 appears too small. In the revised version, we will increase the font size and enhance the clarity of text in Figure 5 to ensure better readability for readers.\n\n**Response to Question 2:**\n\nThank you for your valuable suggestion about discussing our approach's limitations. We appreciate the recommendation to include this important aspect in our paper. In the revised version, we will add the following to our conclusion section: \"For future work, incorporating adaptive period detection mechanisms could further enhance our PTM branch to better accommodate datasets with irregular periodicity patterns.\" This addition acknowledges a limitation of our current approach while suggesting a clear direction for improvement.", "claims_and_evidence": "The claims are generally well-supported. The paper provides comprehensive experimental results across multiple datasets showing CFPT outperforms baseline models. Ablation studies effectively demonstrate the contribution of each component. The visualizations and efficiency analysis further strengthen the evidence for the model's effectiveness.", "methods_and_evaluation_criteria": "The methods and evaluation aspects are well-designed for the time series forecasting task. The architecture provides a reasonable approach for handling both frequency and temporal patterns, while the evaluation follows common practices in time series prediction by using appropriate benchmarks across diverse domains with standard performance metrics.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "Yes, I have checked the soundness of the experimental designs, and I believe they are well-designed for the time series forecasting task.", "relation_to_broader_scientific_literature": "The paper makes significant contributions to the broader scientific literature on frequency-based modeling and timestamp-based modeling in time series forecasting.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n1)\tThe CFPT framework is a novelty. It can advance the long-term time series forecasting domain by considering both the complex interrelationships between frequency components and the periodic patterns in timestamps, providing a new perspective for addressing challenging forecasting tasks and inspiring future research on sophisticated frequency interaction mechanisms and timestamp modeling strategies.\n2)\tThe paper demonstrates a well-structured presentation. It presents a logical progression from problem formulation to experimental validation, effectively communicating both theoretical concepts and implementation details.\n3)\tThe experiments are reproducible, with detailed implementation information provided. The authors clearly describe all experimental details including software versions, hardware specifications, and hyperparameter settings.\n4)\tThe method's superior performance across multiple tasks demonstrates its potential for practical application in real-world environments where accurate long-term forecasting is critical.\n\nWeaknesses:\n1.\tIt is encouraged to increase the font size in Figure 5 to improve readability.\n2.\tThe conclusion lacks discussion of limitations, which would have provided a more balanced assessment of the work and potential directions for improvement.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1)\tCould you increase the font size in Figure 5 to improve readability?\n2)\tWhat do you consider to be the limitations of your approach? Understanding potential constraints would help contextualize your results and suggest directions for future work.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: **Summary:** This paper proposes CFPT, a novel architecture for long-term time series forecasting that explicitly models (1) cross-frequency interactions via a frequency-domain coupling mechanism, and (2) timestamp periodicity through a periodic-aware timestamp modeling branch using 2D CNN. The proposed method is evaluated on several real-world datasets across multiple prediction horizons and compared with eight recent state-of-the-art models. Extensive ablations and efficiency analyses demonstrate the utility and robustness of the proposed components.

**Decision:** All reviewers rated the paper positively and praised its novelty and solid empirical evaluation. The proposed cross-frequency interaction mechanism offers a fresh perspective beyond existing frequency-domain methods, and the periodic timestamp modeling adds a unique contribution to timestamp representation in time series forecasting. Minor concerns regarding clarity of figures, notation, and implementation details were effectively addressed in the rebuttal. As such, I recommend acceptance.
Decision: Accept (poster)