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Multiple cyclicity and Wavelet Decomposition with Channel Correlation for Long-term Time Series Forecasting

Multiple cyclicity and Wavelet Decomposition with Channel Correlation for Long-term Time Series Forecasting

arXiv:2606.17996v1 [cs.LG] 16 Jun 2026

Bin Wang1 , Heming Yang1 and Jinfang Sheng1∗ Abstract— Cyclicity and trend are important components of time series data and many studies based on cyclicity and trend have achieved good results in long-term time series forecasting. However, we believe that current work neglects the influence of real-world inter-channel correlations in time series data which leads to suboptimal predictions. Furthermore, these models rely on complex designs to capture diverse information so that resulting in low computational efficiency. To address this challenge, we propose McWC, a long-term time series forecasting model that separately models the cyclicity, trend, and inter-channel correlations. Specifically, McWC first decouples cyclical information from data using a multi-layer cyclicity construction module. Then, it extracts inter-channel correlations using multi-layer perceptron. Next, it models and fuses the multi-layer high-frequency and low-frequency information from data using a multi-level wavelet decomposition module. Finally, it aggregates the results of different components to obtain the output. Simultaneously, we decouple intra-channel autocorrelations by calculating a loss function in the frequency domain. Experiments on six real-world datasets demonstrate that McWC achieves state-of-the-art performance, exhibiting excellent computational efficiency and historical information extraction capabilities.

I. INTRODUCTION Long-term time series forecasting(LTSF) has been an important part in some key areas, such as financial forecasting and analysis [1], IoT device information prediction [2], urban traffic forecast [3], [4], and energy and resource allocation forecasting [5]. However, because long-term time series are real-world sequential data with complex and non-stationary characteristics, it poses significant challenges for researchers to capture meaningful dependencies from extensive historical data. With the development of deep learning, various architectures have been proposed for time series modeling. CNNbased models [6] capture local temporal and channel dependencies via fixed kernels, but perform poorly in modeling global dependencies. Transformer-based models [7] use self-attention to extract intrinsic relations and achieve accurate long-term forecasting, yet they tend to focus on high frequency information while ignoring useful low-frequency components, thus limiting representation diversity. Linearbased models decompose time series into basic components and model them with simple linear operations. Methods such as trend-cycle decomposition and Fourier-based downsampling have achieved competitive results even with low computation and parameters. Therefore, we argue that properly 1 Bin Wang, Heming Yang and Jinfang Sheng are all with School of Computer Science and Engineering, Central South University, HuNan, China. { wb csut, 244712142, jfsheng}@csu.edu.cn *Corresponding author: Jinfang Sheng

modeling dependencies in time series is the key to high performance long-term forecasting. We propose that time series data is composed of cyclical patterns, trend, and inter-channel corrections components. Influenced by the current channel-independent approach, many models ignore inter-channel corrections and still achieve good results by only modeling cyclical patterns and trend. However, we argue that since inter-channel corrections objectively exist, modeling them can improve the model’s forecasting performance to some extent. Meanwhile, most existing models either directly capture temporal relationships in the time series data or construct simple cyclical patterns which prevents temporal relationships from being well represented. Therefore, we believe that constructing multiple cyclical patterns based on the inherent periods of the time series data can effectively model the cyclical patterns in the data. In summary, we propose McWC, which uses the Multicycle Construction Block (McB) to extract cyclical patterns, then captures inter-channel correlations through the ChannelCorrection Extraction Block (CEB) and finally extracts trend through our designed Multi-level Wavelet Decomposition Block (MWB). Through our design, the main components of the time series data are separately extracted and modeled by three interrelated yet independently handled parts, allowing each component to be fully utilized by the model, thereby achieving better forecasting performance. Our tests show that McWC achieves excellent performance on a wide range of long-term forecasting tasks. Our contributions in this paper can be summarized as follows: 1) We proposed McB based on the idea of explicitly constructing cyclical patterns using prior periodic information to capture periodic patterns of different frequencies within the time series data. And we proposed MWB module which uses multi-level wavelet decomposition to extract trend and models frequency-domain information at different scales, eliminating noise interference while capturing trend changes effectively. 2) We proposed CEB, which models dynamic interactions between different scales, handling inter-channel corrections at various scales and enhancing the model’s ability to process multi-scale information. 3) Our designed McWC uses three interrelated but separately modeled components for time series forecasting. McWC demonstrates outstanding performance with remarkable efficiency across multiple long-term time series forecasting tasks and datasets.

II. R ELATED W ORK

III. M C WC A. OverView

The primary process of LTSF involves analyzing and extracting features from past time series data to predict future values. Therefore, how to properly decompose the data into different components and extract inter-channel corrections to achieve accurate future predictions has always been a core concern for researchers. In terms of decomposition, FEDformer [8] employs a multi-kernel moving average to enhance its decomposition capability. As a Linear-based model, DLinear [9] employs a moving average method to decompose time series data into seasonal and trend components and uses two separate single-layer linear networks to model these components for prediction, achieving better results than Autoformer [10] which uses a Transformer architecture. Guided by cyclicity and trend decomposition, MICN [11] decomposes data into Seasonal and Trend-Cyclical terms and combines correlations between local features to achieve high-quality forecasting. TimesNet [12] employs Fourier transform to detect temporal periodicities, converts 1D sequences into 2D tensors and leverages Inception CNN to capture intra-period and inter-period variations. It then achieves high-precision prediction through temporal unfolding and adaptive aggregation. PatchTST [13] decomposes continuous time series data into individual patches and uses a Transformer model for forecasting with good results. TimeMixer [14], which adopts a multi-level decomposition pattern obtains time series at different scales through downsampling and decomposes them into seasonal and trend series at each scale to model the different components. TimeKAN [15] utilizes the FreTS [16] time-frequency conversion concept, decomposes frequencydomain information at different scales and uses the modeling capability of KAN networks to extract trend. WPMixer [17] uses wavelet transforms to model the high-frequency and low-frequency components of the time series data separately. For inter-channel dependency extraction: despite strong performance of channel-independent strategies, researchers explore encoders capturing both intra and inter-channel interactions. Crossformer [18] uses two-stage attention to hierarchically process temporal and variable dimensions, enhancing model comprehension. iTransformer [19] inverts time series and uses self-attention to capture inter-channel dependencies. CARD [20] introduces channel-aligned attention to capture temporal and multi-channel dynamic dependencies, plus a token mixing module for multi-resolution tokens and multi-scale knowledge utilization. SDE [21] leverages state space models for LTSF, capturing temporal dynamics and inter-channel dependencies simultaneously. Different from the previous methods, our proposed McWC combines the strengths of cyclicity-trend decomposition and inter-channel corrections extraction and leverages the powerful capability of multi-scale analysis in extracting trend to enhance the model’s forecasting accuracy.

The overall architecture of McWC is shown in Figure 1. It consists of three modules: the McB module for constructing cyclical patterns, the CEB module for extracting interchannel correlations, and the backbone forecasting module MWB. Through these modules, McWC decomposes, learns, and integrates long-term time series data, thereby achieving high-quality forecasting performance. B. Normalization & Multi-cycle Construction Block First, we use RevIN method to normalize time series data. Then we identify the top-k period lengths p = {p1 , p2 . . . pk } from data using prior knowledge, where pi denotes the period of the i-th highest weight cyclical pattern. For each period pi , we construct a globally shared, learnable matrix Mi ∈ Rpi ×C initialized to zero. By iteratively copying Mi , we obtain a periodic component Cyclisti ∈ RL×C with the same length as the input sequence and subtract the periodic component from the normalized data. All periodic components and the prediction backbone are jointly optimized via backpropagation to update the learnable matrices, enabling accurate modeling of cyclical patterns. This procedure is formalized in formula(1): X X = RevIN (X) − Cyclisti (1) This allows subsequent models to ignore periodic patterns and focus on building trend and inter-channel corrections and improve prediction quality. C. Channel-Correction Extraction Block In CEB, we first split the input into patches based on the time series locality principle. For each patch, an MLP aggregates information to efficiently process both global and local details. We then transpose data dimensions and apply another MLP to blend cross-channel information, reverse the patch-splitting operation to restore the original dimensions, and add the result to the original input as supplementary information for subsequent steps. The entire block process is represented by Formula (2) and Formula (3): XP atch = P atch (X) + M LP (P atch (X))

(2)


T  X = X + α · XP atch + β · M LP XPT atch

(3)

Here, α is a trainable model parameter, while β is a manually defined scaling weight fixed during training. This design lets inter-channel dependencies in sequential data either be learned automatically from historical data or explicitly regulated via prior knowledge. Consequently, the model prediction remains robust when inter-channel correlations are weak or absent, thus enabling effective modeling of interchannel dependencies.

Fig. 1.

The architecture of McWC.

D. Backbone(MWB)

F. Train Loss

In the MWB, we first use multi-level wavelet decomposition to split the input into a low-frequency component and multiple high-frequency components. We then assign an independent MLP to each decomposed component; these MLPs operate within their respective frequency domains to independently map historical data to forecasted time series. Finally, the model reconstructs the complete time series via inverse wavelet decomposition of all high and low frequency forecast results. This process is represented by Formula (4) and Formula (5):

To solve the problem that MSE loss struggles to handle intra-channel autocorrelation, we introduce FreDF [22] as the loss of our model to better address autocorrelation in sequence data.

[XA , XD1 . . . XDm ] = W aveletDecomp (X)

(4)

 X = WaveletReconstruction MLP(XA ), MLP(XD1 ), 
. . . , MLP(XDm ) (5) Here, XA is the low-frequency signal obtained from the multilevel wavelet decomposition of the input data, and XD1 . . . XDm are the high-frequency signals obtained at each level of the decomposition. In this way, the multilevel wavelet decomposition module achieves the extraction of trend variations from the sequential data. E. Component Fusion Before producing the final output, we need to fuse the components and apply the inverse normalization of RevIN to align the distributional characteristics of the result with the original sequence as formula(6) : X = iRevIN (X +

X

Cyclist′i )

(6)

In this context, Cyclist′i denotes the cyclical pattern for the k-th cycle and Y is the final forecast generated by the model.

Loss = α × FFT(pred) − FFT(real) 1 + (1 − α) × MSE

(7)

Here, α is the weight for the frequency-domain loss, F F T () denotes the Fast Fourier Transform operation, pred represents the prediction and real represents the ground truth value. This enables the model to better handle autocorrelation issues within channels by modifying the model’s loss function by transforming time-domain data into the frequencydomain. IV. E XPERIMENTS A. Experiment setting Experimental Datasets: To validate the forecasting accuracy of our mode, we selected six commonly used real-world datasets in LTSF (ETTh1, ETTh2, ETTm1, ETTm2, Weather, Electricity) for experimentation. Baseline Models: Based on recency, innovation and forecasting performance, we selected seven well-regarded time series models in the field as our baselines. These include both Linear-based and Transformer-based models: (1) WPMixer (2) SDE (3) TimeMixer (4) iTransformer (5) FreTS (6) PatchTST and (7) DLinear. Evaluation Metrics: This experiment uses Mean Squared Error (MSE) and Mean Absolute Error (MAE) as the evaluation metrics for the models. Experimental Setup: All experiments were implemented with PyTorch on a single NVIDIA 3090 24GB GPU and baseline model metrics were derived from local testing via their original codebase scripts.

TABLE I P ERFORMANCE COMPARISON OF DIFFERENT TIME SERIES FORECASTING MODELS ON BENCHMARK DATASETS . T HE BEST RESULTS ARE HIGHLIGHTED IN BOLD RED AND THE SECOND - BEST RESULTS ARE SHOWN IN BOLD BLACK .

Models Metric

McWC

WPMixer

SDE

Ours

2025

2025

TimeMixer iTransformer 2024

FreTS

PatchTST

Dlinear

2024

2023

2023

2024

MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE

ETT h1

96 0.368 0.388 0.374 0.387 0.387 0.402 0.381 0.398 0.394 0.409 0.395 0.407 0.376 0.397 0.396 0.410 192 0.419 0.417 0.428 0.414 0.443 0.432 0.441 0.430 0.448 0.441 0.490 0.477 0.426 0.432 0.445 0.440 336 0.454 0.434 0.462 0.437 0.492 0.457 0.500 0.459 0.492 0.465 0.510 0.480 0.469 0.457 0.487 0.465 720 0.459 0.456 0.482 0.466 0.504 0.484 0.552 0.507 0.521 0.504 0.568 0.538 0.518 0.504 0.512 0.510 avg 0.425 0.423 0.436 0.426 0.456 0.443 0.468 0.448 0.463 0.454 0.490 0.475 0.447 0.447 0.460 0.456

ETT h2

96 0.277 0.327 0.277 0.330 0.296 0.344 0.286 0.339 0.300 0.349 0.332 0.387 0.308 0.359 0.341 0.395 192 0.348 0.375 0.351 0.377 0.381 0.395 0.391 0.404 0.381 0.399 0.451 0.457 0.380 0.406 0.481 0.479 336 0.394 0.409 0.363 0.394 0.429 0.433 0.421 0.432 0.423 0.432 0.466 0.473 0.412 0.429 0.592 0.542 720 0.410 0.432 0.405 0.427 0.435 0.444 0.468 0.468 0.426 0.445 0.485 0.471 0.435 0.456 0.840 0.661 avg 0.357 0.385 0.349 0.382 0.385 0.404 0.391 0.410 0.382 0.406 0.433 0.447 0.383 0.412 0.563 0.519

96 0.304 0.344 0.334 0.368 0.322 0.363 0.327 0.364 0.341 0.376 0.337 0.374 0.323 0.364 0.345 0.373 192 0.355 0.373 0.358 0.375 0.361 0.385 0.367 0.386 0.380 0.394 0.382 0.398 0.371 0.391 0.381 0.391 ETT m1 336 0.385 0.395 0.384 0.397 0.401 0.414 0.393 0.403 0.419 0.418 0.420 0.423 0.398 0.408 0.415 0.415 720 0.443 0.431 0.456 0.435 0.452 0.443 0.451 0.442 0.486 0.455 0.490 0.471 0.457 0.444 0.472 0.450 avg 0.371 0.385 0.383 0.393 0.384 0.401 0.384 0.398 0.406 0.410 0.407 0.416 0.387 0.401 0.403 0.407 96 0.161 0.240 0.170 0.251 0.177 0.263 0.174 0.257 0.183 0.266 0.186 0.275 0.184 0.267 0.193 0.292 192 0.224 0.283 0.235 0.295 0.248 0.311 0.236 0.299 0.252 0.312 0.259 0.323 0.246 0.304 0.284 0.361 ETT m2 336 0.284 0.322 0.300 0.336 0.313 0.353 0.301 0.339 0.314 0.351 0.349 0.386 0.311 0.348 0.384 0.429 720 0.382 0.381 0.391 0.392 0.418 0.415 0.400 0.400 0.411 0.406 0.559 0.511 0.418 0.414 0.556 0.523 avg 0.262 0.306 0.274 0.318 0.289 0.335 0.277 0.323 0.290 0.333 0.338 0.373 0.289 0.333 0.354 0.401 96 0.157 0.202 0.163 0.205 0.165 0.213 0.161 0.208 0.175 0.215 0.171 0.227 0.175 0.217 0.196 0.256 192 0.202 0.242 0.207 0.245 0.214 0.255 0.207 0.251 0.225 0.257 0.218 0.280 0.220 0.255 0.238 0.299 Weather 336 0.260 0.286 0.267 0.291 0.273 0.297 0.264 0.293 0.279 0.298 0.265 0.317 0.279 0.297 0.281 0.330 720 0.341 0.339 0.338 0.337 0.353 0.352 0.345 0.345 0.361 0.350 0.326 0.351 0.356 0.348 0.345 0.381 avg 0.240 0.267 0.243 0.269 0.251 0.279 0.244 0.274 0.260 0.280 0.245 0.293 0.257 0.279 0.265 0.316 96 0.135 0.227 0.166 0.260 0.147 0.245 0.156 0.247 0.148 0.240 0.171 0.260 0.180 0.272 0.210 0.301 192 0.152 0.242 0.175 0.261 0.161 0.257 0.170 0.260 0.164 0.256 0.177 0.268 0.187 0.279 0.210 0.304 Electricity 336 0.170 0.260 0.193 0.282 0.176 0.274 0.187 0.278 0.177 0.270 0.190 0.284 0.204 0.295 0.223 0.319 720 0.207 0.293 0.233 0.314 0.207 0.304 0.227 0.312 0.228 0.313 0.228 0.316 0.245 0.328 0.257 0.349 avg 0.166 0.255 0.191 0.279 0.172 0.270 0.185 0.274 0.179 0.269 0.191 0.282 0.204 0.293 0.225 0.318 Total AVG

0.303 0.336 0.312 0.344 0.322 0.355 0.324 0.354 0.330 0.358 0.350 0.381 0.327 0.360 0.378 0.402

1st Times

26

25

4

6

1

0

0

B. Main results All results in this experiment were obtained by running the original code scripts locally(except for FreTS) and the complete results are shown in Table 1. The results clearly show that our model’s performance is significantly better than the baseline models. Specifically, compared to the best results from other models, our model achieves an average reduction in MSE of 2.5% on ETTh1, 3.1% on ETTm1, 4.3% on ETTm2, 1.2% on Weather, and 3.4% on Electricity. Similarly, for these datasets, the MAE is reduced by 0.7%, 2.0%, 2.8%, 0.7% and 5.2% respectively. Although our model does not achieve the best prediction performance on every dataset, McWC’s accuracy remains very close to

0

0

0

1

0

0

0

0

0

the top-performing models in those cases. Furthermore, in terms of the number of top rankings achieved, our proposed McWC obtained 51 best results and 9 second-best results, far surpassing all other models in the experiment. This demonstrates that McWC possesses accurate and versatile forecasting capabilities for the vast majority of natural time series. C. Ablation experiment To demonstrate the effectiveness of the modules we designed and added, we conducted ablation studies on four datasets: ETTh1, ETTm2, Weather, and Electricity. This experiment consists of configurations with different module combinations and the specific modules included in each case

Fig. 2. Experiments of the information extraction performance by extending the sequence length on datasets(a) Weather and (b) Electricity. Comparison of forecasting performance between ours and baseline models by varying look-back windows on datasets(c) Weather and (d) Electricity.

TABLE II A BLATION STUDY OF M C WC: TL MEANS TRAIN LOSS ; !MEANS MODULE INCLUDED IN BACKBONE , # MEANS NOT INCLUDED . B EST

TABLE III C OMPARISON OF COMPUTATIONAL COST.T HE BEST RESULTS ARE HIGHLIGHTED IN BOLD RED .

RESULTS IN BOLD RED .

Modules

ETTh1

ETTm2

Weather

Electricity

Models

McWC 2025

WPMixer 2025

SDE 2025

GFLOPs GFLOPs GFLOPs

TimeMixer 2024

CEB

McB

TL

MSE

MSE

MSE

MSE

Metric

! # ! ! ! # #

! ! # ! # ! #

! ! ! # # # !

0.425 0.425 0.434 0.443 0.447 0.443 0.434

0.262 0.263 0.274 0.265 0.274 0.267 0.278

0.240 0.244 0.245 0.241 0.244 0.242 0.261

0.1660 0.167 0.190 0.1667 0.192 0.1667 0.201

96 192 ETT m2 336 720 avg

0.18 0.26 0.38 0.71 0.38

14.24 12.57 9.20 14.78 12.69

2.47 2.59 2.78 3.27 2.77

2.58 3.06 3.76 5.65 3.760

0.451

0.280

0.260

0.200

96 192 Weather 336 720 avg

0.53 0.77 1.14 2.13 1.14

14.03 7.29 3.73 12.02 9.26

37.62 38.00 38.55 160.10 68.56

5.43 6.14 7.20 10.03 7.20

Backbone (without modules)

are detailed in Table 2. The experimental results clearly show that the McB module, which constructs periodic patterns, enhances performance on most datasets. The TL and CEB modules function by addressing intra-channel autocorrelation and inter-channel correlation in different dataset respectively. Additionally, we compared the Backbone against an MLP by conducting forecasting across all datasets to validate the effectiveness of the Backbone. Their average MSE and MAE were 0.327, 0.355 and 0.333, 0.359 respectively. Compared to the MLP, our Backbone achieved an improvement of 1.8% in average MSE and 1.1% in average MAE across all

GFLOPs

datasets, demonstrating its effectiveness. Therefore, through the combined effect of our designed and introduced modules, McWC achieves outstanding performance on the majority of datasets. This sufficiently proves the importance and necessity of the modules we proposed and incorporated into McWC. D. Model Efficiency To evaluate the efficiency of our proposed McWC, we measured its computational cost in Giga Floating Point Operations (GFLOPs) under different testing conditions across

various datasets. For the experiments, we calculated GFLOPs using the parameters that yielded the optimal results for each model under its respective test conditions. As shown in Table 3, McWC requires nearly an order of magnitude less computational resources than the other three baseline models across all tested scenarios. This demonstrates that our proposed McWC model achieves stable and highly efficient operational performance while maintaining forecasting accuracy. E. Varing look-back window It is evident that the length of the look-back window is directly proportional to the amount of historical information it contains. However, without strong information extraction and integration capabilities, excessive historical data may lead to overfitting and adversely affect forecast performance. As a deep learning model that explicitly models the intrinsic components of sequential data, our proposed McWC possesses powerful capabilities for extracting historical information. To validate this, we conducted experiments on the Weather and Electricity datasets with varying look-back window lengths: look-back length = {48, 96, 192, 336} and forecast length = {96, 192, 336, 720}. As shown in Figure 2(a) and 2(b), the forecasting quality of McWC improves consistently as the look-back window increases. Furthermore, we compared the information extraction capability of McWC against multiple baseline models by setting a fixed prediction length of 192 and testing different look-back window lengths. The results in Figure 2(c) and 2(d) demonstrate that McWC’s prediction quality is positively correlated with the look-back window length and its information extraction capability outperforms the baseline models. V. C ONCLUSION In this paper, we propose McWC, a computationally efficient model for long-term time series forecasting. Based on the idea of component decomposition, McWC adopts the McB, CEB, and MWB modules to model periodicity, trend and channel correlation, improving its ability to capture complex patterns and data mutations. We further introduce FreDF to optimize the loss function, enhancing intra-channel autocorrelation modeling and prediction performance. Extensive experiments on real-world datasets show that McWC effectively exploits historical information and achieves highquality forecasting. ACKNOWLEDGMENT This work was supported by the National Natural Science Foundation of China (Grant No. 82574111). R EFERENCES [1] S. Zhang, X. Ma, Z. Fang, H. Pan, G. Yang, and G. R. Arce, “Financial time series forecasting based on momentum-driven graph signal processing,” Applied Intelligence, vol. 53, no. 18, pp. 20 950– 20 966, Sep. 2023. [2] Z. Zhang, L. Meng, and Y. Gu, “Sageformer: Series-aware framework for long-term multivariate time-series forecasting,” IEEE Internet of Things Journal, vol. 11, no. 10, pp. 18 435–18 448, 2024.

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