2.1. Time-Domain and Frequency-Domain Modeling

Traditional time-series models typically operate directly in the time domain, focusing on the original data to capture instantaneous changes, which is essential for identifying abrupt shifts, short-term trends, or periodic patterns. For example, Informer [40] reduces computational complexity through the ProbSparse self-attention mechanism and improves long-sequence forecasting efficiency by incorporating a distillation method [41–43] to compress sequence length. Pyraformer [44] employs a tree structure with pyramid attention to extract features at different resolutions in the time domain. The MTPNet [45] proposes a multiscale transformer pyramid network that combines dimension-invariant embedding with multiscale modeling to capture temporal dependencies at varying granularities. Crossformer [46] introduces cross-dimensional correlations and a two-stage attention mechanism (TSA) for capturing dependencies across both time and dimensions. However, the use of CM embeddings in many time-series models often limits the ability to capture temporal dependencies, as they focus on channel interactions rather than temporal evolution. The CI embedding strategy addresses this by enabling independent processing of each channel while maintaining temporal correlations. PatchTST [47] enhances this by combining CI embedding with patching methods, treating each channel of a multivariate series as an independent univariate time series. This improves memory efficiency and allows the model to handle longer sequences, enhancing long-term forecast accuracy. Despite these advancements, time-domain models still struggle with capturing long-term dependencies and global features, especially in long sequences.

The Autoformer [48] introduces an autocorrelation mechanism based on the Wiener–Khinchin theorem and applies Fourier transforms to uncover periodic dependencies. This frequency-domain approach, applied in the FEDformer [49], uses frequency-enhanced blocks (FEBs) for attention mechanisms, which improves global feature extraction and captures periodic trends. The FiLM [50] uses Fourier analysis combined with low-rank approximation and Legendre projection to address issues such as noise and overfitting, offering deeper insight into the global characteristics of time-series data. However, frequency-domain methods often struggle with short-term fluctuations and local changes.

To leverage the strengths of both domains, a hybrid approach combines frequency-domain modeling for global features with time-domain modeling to capture local seasonal cycles. This integration of global and local information offers a more comprehensive view of time-series dynamics, improving model robustness and adaptability across varying scales and frequencies. The combined approach provides a more effective modeling strategy for time-series forecasting.

2.2. Fourier Component Selection

In the Fourier transform, f represents the frequency, and j denotes the imaginary unit. It decomposes a signal into spectral components, offering insight into the signal’s frequency content. Low-frequency components correspond to slow variations or long-term trends in the time series, while high-frequency components capture rapid fluctuations and short-term changes. In the FEDformer, the time series is transformed into the frequency domain, but retaining all frequency components can lead to overfitting and excessive noise, negatively impacting predictive performance. To mitigate this, a method of randomly selecting a subset of Fourier components is used, which retains both high- and low-frequency components. This improves the model’s effectiveness, though random selection may omit important frequency components, which can degrade prediction accuracy.

Similarly, the FiLM method selects a subset of frequency components, choosing the lowest M to reduce noise and training time. However, incorporating some high-frequency components can improve performance, highlighting that omitting certain frequency components risks losing important features, ultimately affecting predictions.

To address these issues, we propose a strategy that splits the time series into odd- and even-indexed segments, modeled separately in the frequency domain. This approach alleviates overfitting and reduces high-frequency noise, which commonly arises when keeping all Fourier components. In addition, it eliminates the uncertainty introduced by random selection of components [51, 52] and avoids the loss of information linked to focusing only on low-frequency components. Odd–even decomposition, a form of downsampling, preserves the overall structure and key features of the time series, enhancing the model’s ability to focus on the essential patterns while minimizing attention to noise. Since noise tends to be uniformly distributed across the odd and even components, separating these components helps mitigate the noise’s effect on the observed trends and cycles. This separation also improves the accuracy of frequency-domain feature extraction. However, downsampling in odd–even decomposition can result in the loss of high-frequency details, which could affect the model’s ability to capture fast-changing features. To address this, we introduce an odd–even fusion term, combining the benefits of odd–even decomposition while compensating for any lost information, thereby enhancing the model’s robustness.

The inverse Fourier transform, calculated as described above, reassembles the frequency spectrum components in the original time-domain signal.

2.3. Trend-Seasonality Decomposition

Trend-seasonality decomposition is a commonly used method in time-series analysis, forecasting, and modeling. It involves separating a sequence into its trend, seasonal, and residual components, providing a deeper understanding of the sequence’s behavior and fluctuations. By breaking down time-series data, the structure of the sequence becomes more apparent, supporting the development of more precise predictive models. Seasonal components capture recurring patterns within fixed intervals, while the goal of seasonal decomposition is to extract these periodic variations. Trend components represent the long-term changes in the data, and trend decomposition focuses on isolating these long-term patterns to better capture the sequence’s overall evolution.

However, these three methods have a significant drawback: they require continuous adjustments to the pooling window size. Since the choice of pooling window size directly affects the ability to capture seasonal changes, in practical applications, it can be challenging to determine an appropriate window size in advance. Consequently, it cannot be guaranteed that a highly accurate seasonal periodicity of the time series will be found. To address this problem, we presented a SCASeriesDecomp strategy, using a Fourier transform to discover the primary cycles in the time series as pooling windows. By employing this method, we can accurately capture seasonal patterns at different scales and variations, reducing dependence on the manual adjustment of hyperparameters.

2.4. Encoder–Decoder

In this approach, we eliminate the traditional encoder stack and rely on a single-layer decoder to jointly perform both encoding and decoding functions. This technique enables both the extraction of time-series characteristics and the forecasting of future data points. We introduce a one-step generative decoder architecture, which removes the need for an encoder and utilizes a single layer for generating predictions. Unlike the generative decoder in the Informer, this decoder can sample the full input sequence as L_token, incorporating both the sequence length for predictions and the input data during training. By handling information modeling in a unified manner, it can compute all forecasted values in one forward pass, significantly enhancing the model’s efficiency in both space and time. Experimental results show that this one-layer generative decoder can still deliver outstanding forecasting performance.

3.1. Preliminary

Time-series forecasting leverages historical data to predict upcoming observations. This task can be formulated as a mapping function f, such that Y = f(X). In this context, the input sequence X = [X₁, X₂, …, X_t, …, X_T] represents historical data points, where X_t is the observed value at time t and T is the total sequence length. The target sequence is expressed as Y = [Y_1+h, Y_2+h, …, Y_t+h, …, Y_T+h], with Y_t+h representing the forecasted value at time t + h, and h indicating the prediction horizon.

3.2. Overall Architecture

This paper introduces a dual-branch model framework based on seasonal cycle awareness. To verify the effectiveness of this framework, we applied it to two mainstream embedding methods: CI and CM, and conducted model predictions. Subsequently, we compared the prediction results with existing classic models based on CI and CM, providing a comprehensive evaluation of our framework’s advantages under different embedding strategies. This comprehensive evaluation confirms that the proposed framework is capable of retaining essential features within time-series data, while also showcasing enhanced performance in modeling interactions across different channels. It presents a well-balanced approach for addressing multivariate time-series forecasting tasks.

The proposed model utilizes a decomposition technique to separate the time series into seasonal and trend components, employing a dual-branch feature extraction framework to handle each component independently. Unlike the Autoformer and FEDformer models, where the trend-seasonality decomposition struggles to identify a precise seasonal cycle, we introduce a seasonal cycle-adaptive decomposition model that gradually breaks the time series into its seasonal and trend parts. Consequently, the previous approach of modeling only the seasonal component in Autoformer and FEDformer is abandoned. In response, we develop a parallel architecture that separately predicts periodic and trend-related patterns, leveraging their unique temporal characteristics. The final forecast is obtained by combining predictions from both components.

For capturing complex seasonal patterns, we incorporate a SFB for feature extraction. Instead of relying on conventional self- and cross-attention mechanisms, this block uses a frequency-based attention method called parity correction attention. Together with multiscale dilated convolutions, this enables efficient global information modeling in the frequency domain while enhancing local feature extraction in the time domain, thereby improving the model’s ability to learn latent patterns in the time series. To process the trend component, we use a trend-cyclical forecasting block (TFB), which extracts trend-related features via a simple fully connected layer. This block identifies internal correlations, uncovering hidden relationships in the data to better understand the trend behavior. As shown in Figure 2, the “odd” and “even” components represent the odd and even parts from the time-series decomposition, while “mean” refers to the fused odd–even component. “FA” indicates the frequency-domain attention mechanism, and “S” and “T” represent the seasonal and trend components, respectively.

The proposed model comprises two blocks, that is, the SFB and the TFB. The SFB includes the SCASeriesDecomp, FPCA, multiscale dilated convolutional (MSDC), and three main blocks. The detailed descriptions of the SCASeriesDecomp, FPCA, MSDC, and TFB are provided in Sections 3.3.1, 3.3.2, 3.3.3, and 3.4, respectively.

3.3. SFB

3.3.1. SCASeriesDecomp

A critical aspect of trend-seasonality decomposition is the accurate estimation of the sequence’s seasonal period, which directly influences the separation of trend and seasonal components. To achieve this, selecting a suitable pooling window becomes essential. This module uses the Fourier transform to identify the primary period, which is subsequently applied as the kernel size for one-dimensional pooling, aiding in the separation of the time series into its trend and seasonal components. This approach improves the model’s ability to detect and react to periodic patterns in the data.

The Fourier transform serves as a powerful tool for transitioning between time and frequency domains, uncovering the latent cyclic patterns present in time-series data. By examining the spectral output, one can pinpoint the key frequencies that dominate the series. Figure 3 illustrates this process, where frequency-domain analysis reveals the structural composition of the input. The amplitude spectrum is averaged to gauge the significance of the frequencies, with the most prominent one being selected to estimate the sequence’s period. This identified period is then employed as the window size for average pooling, enabling the model to effectively extract and isolate the trend and seasonal features with high accuracy. The specific steps of this process are as follows:

$$ ()

In this context, X refers to the time-series input of the SCASeriesDecomp module. The dominant frequency F_top1 is derived from X via the Fourier transform, N denotes the input sequence length, and Period corresponds to the accurately identified seasonal cycle. This period is then utilized as the window size for one-dimensional pooling to extract the trend component (T), while the seasonal component (S) is derived by subtracting T from the original sequence. The fundamental idea of this module is to determine the periodic pattern of the sequence using the Fourier transform and use the resulting period as the kernel size for convolution, aligning with the data’s inherent cyclical structure. A moving average is applied to smooth the sequence and separate it into trend and seasonal elements. The trend is extracted via smoothing, and the difference between the original sequence and the trend represents the seasonal component. This approach improves the model’s ability to accurately disentangle trend and periodic fluctuations in the data.

3.3.2. FPCA

For capturing global information, FPCA is utilized for global modeling, as shown in Figure 4. Initially, the input $$ undergoes transformation through fully connected layers, resulting in Q in a higher-dimensional space. This linear transformation enriches the feature representation, thereby enhancing the model’s expressive capability. The resulting high-dimensional sequence Q is subsequently split into two lower-resolution subsequences, Q_odd and Q_even, which preserve most of the informative content from the original sequence despite the downsampling. To address the potential information loss owing to odd–even decomposition, an odd–even fusion term (Q_mean) can be introduced to correct the information loss. According to sampling theory [58], compared to odd–even fusion terms, odd and even terms, respectively, lose some information. Odd–even fusion terms have the following two main characteristics: First, because each data point in the odd–even fusion terms is the average of adjacent odd and even terms, they contain comprehensive information from both odd and even terms. Therefore, odd–even fusion terms contain more complete information, which can lead to more effective features during subsequent processing. Consequently, this allows for the correction of features obtained from odd and even terms, thereby improving their accuracy. Second, odd–even fusion terms represent the average of adjacent even and odd term data. Morphologically, the fluctuation of odd–even fusion terms is relatively small, demonstrating a smoother and more continuous trend. This aids in reducing data noise and instability, thereby enhancing the accuracy and stability of feature extraction.

The schematic in Figure 5 shows the odd–even decomposition process. By comparing the odd–even components with the original sequence, it is clear that decomposing the time series into odd and even components reduces the sampling interval. Consequently, this reduction allows for the attenuation of fine-scale noise within the time series, facilitating a more accurate modeling of its overall features. Moreover, the odd–even fusion terms aim to address potential local information loss resulting from the decomposition process. Specifically, these fusion terms integrate information from both odd and even components, effectively retaining the primary characteristics of the original sequence. Consequently, they serve as a corrective measure for the results obtained from the odd and even components, enabling a more comprehensive representation of the seasonal variations inherent in the original sequence. By employing this approach of odd–even decomposition and feature correction, the algorithm is endowed with enhanced generalization capabilities. The detailed implementation process can be expressed as

$$ ()

Next, we apply Fourier transforms to Q_odd, Q_even, and Q_mean individually, transforming the signals from the time domain to the frequency domain, where features are extracted through a weighted operation. The output of the Fourier transform, represented as $$, $$, and $$ for each component, can be obtained by multiplying each frequency-domain component by randomly initialized parameter kernel weights, that is, $$, $$, and $$. followed by the addition of learnable biases $$, $$, and $$, respectively. This weighting operation modifies the contribution of different frequency components, allowing the model to better capture the signal’s distinctive features.

$$ ()

where terms denote odd, even, and mean, respectively; d_i = 1, 2.., d represents the input channel; and d_o = 1, 2.., d indicates the output channel. After feature extraction, the results corresponding to F_odd, F_even, and F_mean are Y_odd, Y_even, and Y_mean. The obtained odd feature sequence, even feature sequence, and odd–even fusion feature sequence are denoted as Y_odd, Y_even, and Y_mean, respectively. Next, we discuss how to correct Y_odd and Y_even based on the Y_mean. Let Y_odd+res = Y_odd + Y_res and Y_even+res = Y_even + Y_res represent the corrected Y_odd and Y_even sequences, respectively. Now, we discuss how to compute Y_res. Given the relatively high precision of the Y_mean, after modification, we aim for the following equation to hold true:

$$ ()

In the following, we substitute Y_odd+res = Y_odd + Y_res and Y_even+res = Y_even + Y_res into equation (16) above, resulting in the following equation:

$$ ()

By rearranging the terms, we can obtain the corrected sequence Y_res, with the specific expression shown as follows:

$$ ()

Compared to the odd–even fusion feature sequence, the Y_res contains components in the original sequence that are not entirely captured by the odd and even components, including subtle, complex, or irregular patterns. Therefore, utilizing Y_res to adjust the odd and even components can enhance the capture of seasonal and correlated information in the data, better representing the global information in the original seasonal patterns. This method of feature correction helps enhance the expressive capacity of features, enabling the model to adapt more effectively to the complexity of the data. The complete procedure for correcting the features extracted from odd and even terms using Y_mean is described as follows:

$$ ()

Subsequently, the learned frequency-domain representations (Y_odd+res and Y_even+res) are mapped back into the temporal space via inverse Fourier operations. The generated segments ($$ and $$) are then integrated into a unified sequence according to their original odd–even arrangement.

$$ ()

3.4. TFB

4.1. Baselines

This paper selects PatchTST and Crossformer, based on CI(CI-Based) models, as well as FEDformer, Autoformer, DLinear, MICN, and Informer, based on CM (CM-Based) models, as baseline models for performance comparison with the proposed SCA-Net, which incorporates both CI and CM strategies. For CI-based models, we choose the current state-of-the-art (SOTA) model, PatchTST, as the primary benchmark. For CM-based models, since our model utilizes Fourier transforms, FEDformer is chosen as the main comparison target. To ensure fairness, we adopt the four forecasting horizons (96, 192, 336, and 720) used by PatchTST and FEDformer in their optimal performance settings, with the input length uniformly set to 96. In addition, the input and output sequence lengths are kept consistent across all baseline models to ensure the comparability of the experimental results.

4.2. Evaluation Metrics

4.3. Multivariate Results

For multivariate forecasting, we conducted experiments on eight datasets across four prediction lengths, as shown in Table 1. Our model produced two sets of experimental results based on CI and CM, namely, SCA-Net (CI) and SCA-Net (hereafter, results for SCA-Net refer to the CM experiments). The CI model SCA-Net (CI) achieved the best results in the majority of cases compared to PatchTST. In contrast, the CM model SCA-Net outperformed the baseline model FEDformer across all datasets and four prediction lengths, with an overall MSE reduction of 12.1% and an overall MAE reduction of 7.9%. Specifically, for a prediction length of 96, the overall MSE decreased by 19.5%, and MAE decreased by 12.9%; for a prediction length of 192, the overall MSE and MAE decreased by 11.7% and 8.6%, respectively; for a prediction length of 336, the overall MSE and MAE decreased by 8.5% and 6.8%, respectively; and for a prediction length of 720, the overall MSE and MAE decreased by 10.1% and 6.1%, respectively. Based on an analysis of these experimental results, it is evident that the proposed SCA-Net exhibited a considerable advantage in terms of prediction performance. On the ETT dataset, the overall MSE and MAE decreased by 7.4% and 4.1%, respectively; on the Electricity dataset, the overall MSE and MAE decreased by 6.2% and 4.3%, respectively; on the Exchange dataset, the overall MSE and MAE decreased by 12.7% and 6.2%, respectively; on the Traffic dataset, the overall MSE and MAE decreased by 11% and 15%, respectively; and on the ILI dataset, the overall MSE and MAE decreased by 15.2% and 12.5%, respectively. In summary, the SCA-Net exhibited considerable performance improvements across various datasets and scenarios. Even when dealing with time-series data, such as the Exchange and ILI datasets, that lacked obvious seasonal cyclical features, the model demonstrated enhanced accuracy and stability in sequence prediction through the exploration of both local and global information.

The prediction results on the ETT dataset for horizons of 96, 192, 336, and 720 are illustrated in Figure 6. It can be observed that the proposed model consistently achieves superior alignment with the actual values, especially at peak and valley points. Furthermore, the accompanying error plots indicate that the model maintains the lowest error levels overall, highlighting its improved performance.

4.4. Univariate Results

Extensive experiments on univariate forecasting were also conducted. In comparison to the baseline FEDformer, as depicted in Table 2, the SCA-Net achieved the best predictive results across all datasets and prediction lengths, with the overall MSE and MAE decreasing by 15.6% and 10.2%, respectively. Analyzing different prediction lengths, for a prediction length of 96, the SCA-Net exhibited a 20.5% decrease in overall MSE and a 12.4% decrease in overall MAE; for a prediction length of 192, the overall MSE and MAE decreased by 14.1% and 9.6%, respectively; and for a prediction length of 336, the overall MSE and MAE decreased by 14.4% and 9%, respectively. Based on this analysis, it is evident that the proposed model achieved excellent predictive performance in univariate forecasting. On the ETT dataset, the MSE and MAE decreased by 11.1% and 7.7%, respectively; on the Electricity dataset, the MSE and MAE reduced by 11.4% and 7.7%, respectively; on the Exchange dataset, the MSE and MAE dropped by 28.5% and 25.9%, respectively; on the Traffic dataset, the MSE and MAE declined by 20.8% and 12.8%, respectively; and on the ILI dataset, the MSE and MAE decreased by 13.2% and 7%, respectively. Overall, the SCA-Net exhibited considerable performance improvements in univariate forecasting tasks across different prediction lengths and datasets. Consequently, the presented method could be considered to be applicable in various scenarios. Through extensive experimental validation, the SCA-Net demonstrated particularly good performance improvements compared to other forecasting models with specific datasets or in specific scenarios. This indicates that the SCA-Net possesses good versatility and performance advantages when addressing time-series forecasting problems.

4.5. Ablation Experiments

In this work, we performed ablation experiments to assess the contribution of the proposed modules to the model’s performance. The key modules under investigation include the SCASeriesDecomp, FPCA, MSDC, and TFB. Experiments were designed from different perspectives to assess the impact of each module.

4.5.3. Ablation Experiment on MSDC

Next, we evaluate the impact of MSDC to verify the importance of capturing local dependencies within seasonal cycles. To achieve this, we adopt two complementary strategies. Approach 1: We embed the MSDC module into both components of FEDformer, resulting in FEDformer-v2. If this modified model demonstrates enhanced forecasting performance, it validates the value of explicitly modeling seasonal cycles. Approach 2: We ablate the MSDC component from the architecture, retaining only the FPCA module for extracting global characteristics. A noticeable decline in performance in this case would further support the necessity of integrating both global and local modeling mechanisms tailored to seasonal structures.

The experimental outcomes presented in Tables 8 and 9 suggest that MSDC contributes substantially to enhancing predictive accuracy. This confirms the rationality of incorporating seasonal pattern modeling and capturing local temporal dependencies. Moreover, evidence from Table 9 and Figure 7 shows that even in the absence of MSDC, our framework, featuring a lightweight single-layer, one-step generative decoder, outperforms the encoder–decoder–based FEDformer. These results affirm the efficacy of the proposed SSGD design and underscore its strong potential for future development.

Table 8. Ablation experimental results on the MSDC in FEDformer.

Methods	Dataset metrics	ETTh1				ETTh2				ETTm1				ETTm2
		96	192	336	720	96	192	336	720	96	192	336	720	96	192	336	720
FEDformer	MSE	0.376	0.420	0.459	0.506	0.346	0.429	0.496	0.463	0.379	0.426	0.445	0.543	0.203	0.269	0.325	0.421
	MAE	0.419	0.448	0.465	0.507	0.388	0.439	0.487	0.474	0.419	0.441	0.459	0.490	0.287	0.328	0.366	0.415
FEDformer-v2	MSE	0.373	0.415	0.453	0.487	0.338	0.426	0.486	0.456	0.355	0.394	0.420	0.471	0.197	0.265	0.323	0.419
	MAE	0.412	0.441	0.461	0.497	0.380	0.433	0.483	0.473	0.402	0.427	0.441	0.465	0.285	0.326	0.362	0.413

• Note: FEDformer-v2 integrates the MSDC directly into FEDformer (lower MSE/MAE indicates better performance, with bold and italic representing the best and second-best results).

Table 9. Ablation experimental results on the MSDC in SCA-Net.

Methods	Dataset metrics	ETTh1				ETTh2				ETTm1				ETTm2
		96	192	336	720	96	192	336	720	96	192	336	720	96	192	336	720
FEDformer	MSE	0.376	0.420	0.459	0.506	0.346	0.429	0.496	0.463	0.379	0.426	0.445	0.543	0.203	0.269	0.325	0.421
	MAE	0.419	0.448	0.465	0.507	0.388	0.439	0.487	0.474	0.419	0.441	0.459	0.490	0.287	0.328	0.366	0.415
SCA-Net	MSE	0.361	0.402	0.435	0.460	0.331	0.410	0.445	0.453	0.313	0.368	0.401	0.455	0.183	0.254	0.316	0.418
	MAE	0.406	0.433	0.452	0.476	0.375	0.426	0.457	0.471	0.365	0.412	0.435	0.460	0.275	0.317	0.357	0.413
SCA-Net-v4	MSE	0.375	0.417	0.454	0.480	0.332	0.411	0.447	0.478	0.323	0.384	0.403	0.463	0.185	0.256	0.318	0.432
	MAE	0.417	0.442	0.459	0.505	0.376	0.428	0.460	0.485	0.380	0.424	0.441	0.462	0.276	0.319	0.359	0.426

• Note: SCA-Net-v4 directly removes the MSDC from SCA-Net (lower MSE/MAE indicates better performance, with bold and italic representing the best and second-best results).

4.5.4. Ablation Experiment on TFB

Next, we validated the effectiveness of the TFB, specifically focusing on the regression of the trend components. In SCA-Net-v5, we refrained from applying regression to the trend components decomposed by the SCASeriesDecomp. Instead, we performed a simple addition of the decomposed trend components with the final seasonal components through residual connections. If the predictive performance of SCA-Net-v5 deteriorated considerably compared to the SCA-Net, it would indicate that modeling the trend components through regression was effective. Table 10 highlights the specific outcomes of the experiment.

Table 10. Ablation experimental results on TFB.

Methods	Dataset metrics	ETTh1				ETTh2				ETTm1				ETTm2
		96	192	336	720	96	192	336	720	96	192	336	720	96	192	336	720
SCA-Net	MSE	0.361	0.402	0.435	0.460	0.331	0.410	0.445	0.453	0.313	0.368	0.401	0.455	0.183	0.254	0.316	0.418
	MAE	0.406	0.433	0.452	0.476	0.375	0.426	0.457	0.471	0.365	0.412	0.435	0.460	0.275	0.317	0.357	0.413
SCA-Net-v5	MSE	0.368	0.407	0.440	0.462	0.361	0.427	0.446	0.522	0.314	0.369	0.409	0.459	0.183	0.255	0.321	0.430
	MAE	0.410	0.435	0.456	0.479	0.402	0.439	0.458	0.525	0.373	0.414	0.437	0.462	0.276	0.319	0.363	0.425

• Note: SCA-Net-v5 directly removes TFB from SCA-Net (lower MSE/MAE indicates better performance, with bold and italic representing the best and second-best results).

It can be seen from Table 10 that the original SCA-Net, which incorporates trend regression, achieves better forecasting performance than SCA-Net-v5. In addition, by plotting the difference bar chart, as shown in Figure 8, it provides a clearer visualization of the predictive differences between the SCA-Net-v5 and SCA-Net. This allows one to gain a more precise understanding of the effectiveness of regression processing for trend components. In time series, trend components typically encompass long-term and global patterns, with fully connected layers excelling in establishing global correlations across the entire feature set. Consequently, opting for regression methods can help in capturing global correlations among features within trend components, thereby enhancing the effectiveness of their modeling. Moreover, regression methods offer advantages in handling continuous data and predicting future trends, facilitating better fitting and forecasting of long-term trend changes. The integration of trend components processed through regression with seasonal components through residual connections can help compensate for missing trend information during the seasonal-trend decomposition process in time-series data. This integration enables the model to better capture the correlations between trend and seasonal components, thereby improving the overall feature set. This integration allows the model to more effectively capture the relationships between trend and seasonal components, leading to an improved feature set. By strengthening the connection between these components, residual connections can help fill information gaps, ultimately improving the model’s accuracy and robustness when handling complex time-series data.

4.6. Performance Analysis as Strong Plugins

Strong plugins refer to additional components or modules that can be integrated into existing models, aimed at significantly enhancing model performance, particularly by addressing specific limitations or weaknesses of the base model. In this study, we focus on analyzing the impact of SCASeriesDecomp and FPCA as strong plugins on the performance of the PatchTST model. The experimental results, shown in Table 11, demonstrate that integrating these two plugins into the model leads to significant performance improvements. Specifically, the introduction of SCASeriesDecomp and FPCA results in a general reduction in predictive errors, validating their effectiveness in enhancing model accuracy. In the ETTh1 dataset, the inclusion of SCASeriesDecomp and FPCA reduces the overall MSE from 0.449 to 0.437 and 0.435, respectively, and the overall MAE from 0.449 to 0.439 and 0.437. This trend is similarly observed across other datasets, further confirming their efficacy. In summary, SCASeriesDecomp and FPCA serve as valuable plugins that notably enhance the performance of PatchTST on diverse datasets and forecasting periods. This discovery offers a robust foundation for future work aimed at incorporating these plugins into other models to improve predictive performance.

4.7. Parameter Sensitivity Analysis

Here, we conducted a sensitivity analysis of the kernel size in the MSDC feature output layer, based on experiments with the ETT datasets. Owing to variations in the seasonal patterns of time series across different datasets, a key objective of the sensitivity analysis on the MSDC’s kernel is to identify the optimal seasonal cycles corresponding to each dataset. This design facilitates precise modeling of temporal periodicity. By modulating the convolutional kernel’s scale, the model can more effectively extract local temporal patterns, increase responsiveness to cyclical fluctuations, and ultimately improve predictive accuracy.

Different kernel scales in the convolutional layer can capture features across varying ranges. Thus, we performed a sensitivity analysis on these parameters using a radar chart, as depicted in Figure 9. As depicted in the radar chart, the choice of kernel scale substantially influences the model’s overall performance. The chart depicts a polygon for each scale, with each vertex representing a specific kernel size. The shape of the polygon reflects the model’s performance at different scales, whereas the convexity and size of the polygon correspond to the model’s performance under different parameter settings. Through this visualization, one can easily evaluate performance fluctuations under changing conditions, which assists in choosing the best kernel size. According to the experimental results, adjusting seasonal cycles according to dataset-specific characteristics is essential. This enables the extraction of accurate local features, which in turn improves feature fusion.

4.8. Complexity Analysis

Here, we analyzed the spatiotemporal complexity in detail and conducted experiments on a 3060Ti graphics card. By comparing the training duration and forecasting accuracy of the models at a horizon length of 96 on the ETTh1 dataset, their overall performance can be assessed. A comprehensive evaluation strategy was adopted, taking into account parameter quantity, training duration, and prediction accuracy of the baseline methods to holistically measure model effectiveness. According to the statistics in Table 12, a corresponding visualization was generated in Figure 10, illustrating comparative efficiency and capability. As shown, our model achieves a well-balanced compromise between resource consumption and predictive performance when compared to its counterparts. The importance of focusing on the efficiency of the models in training and inference should be noted, while also considering the model’s expressive power to ensure an adequate representation ability while maintaining reasonable computational costs. However, objectively speaking, the proposed model still exhibits a relatively high overall parameter count compared to other models, which is one of the key problems that needs to be addressed in future work. In future studies, the next step should focus on reducing spatiotemporal complexity while improving forecasting accuracy, thereby further enhancing the practicality and overall effectiveness of the proposed model.