Variational Mode Decomposition and Linear Embeddings are What You Need For Time-Series Forecasting

## Abstract

Abstract Time-series forecasting often faces challenges due to data volatility, which can lead to inaccurate predictions. Variational Mode Decomposition (VMD) has emerged as a promising technique to mitigate volatility by decomposing data into distinct modes, thereby enhancing forecast accuracy. In this study, we integrate VMD with linear models to develop a robust forecasting framework. Our approach is evaluated on 13 diverse datasets, including ETTm2, WindTurbine, M4, and 10 air quality datasets from various Southeast Asian cities. The effectiveness of the VMD strategy is assessed by comparing Root Mean Squared Error (RMSE) values from models utilizing VMD against those without it. Additionally, we benchmark linear-based models against well-known neural network architectures such as LSTM, BLSTM, and RNN. The results demonstrate a significant reduction in RMSE across nearly all models following VMD application. Notably, the Linear + VMD model achieved the lowest average RMSE in univariate forecasting at 0.619. In multivariate forecasting, the DLinear + VMD model consistently outperformed others, attaining the lowest RMSE across all datasets with an average of 0.019. These findings underscore the effectiveness of combining VMD with linear models for superior time-series forecasting. Code is available at this repository: https://github.com/Espalemit/VMD-With-LTSF-Linear.git

## 1 Introduction

Time-series information forecast plays a significant part in estimating future occasions, particularly with the integration of machine learning methods. Machine learning models can be prepared to realize long expectation precision and low error rates by leveraging information from datasets.

Achieving optimal results with machine learning models requires a several preprocessing steps such as data cleaning, data decomposition, hyperparameter tuning, and etc. Additionally, selecting the appropriate time-series forecasting method from a the available strategies is crucial.

One widely used algorithm in time series forecasting is the Transformer, that is well known for its performance in Natural Language Processing (NLP), speech recognition, and computer vision assignments. However, Zeng et al. () has raised questions about the adequacy of Transformers in time-series forecast, inciting comparative execution assessments against easier linear-based calculations such as Linear, NLinear, and DLinear. Comparative tests have appeared that direct models frequently outperformed Transformer-based models like FEDFormer, AutoFormer, Informer, LogTrans, and PyraFormer in terms of Mean Squared Error (MSE) and Mean Absolute Error (MAE) (Zeng et al., ).

Furthermore, data decomposition strategies essentially affect forecast results. Variational Mode Decomposition (VMD) is one such strategy competent of lessening data instability (Dragomiretskiy & Zosso, ). The performance of VMD in time-series information forecast was highlighted in a 2021 study about comparing Recurrent Neural Network (RNN)-based models under different scenarios with PM2.5 air quality data over a few cities in China: 1) without decomposition method, 2) using Empirical Mode Decomposition (EMD), and 3) with VMD utilization. Based on the results, it was shown that VMD consistently achieved the least Root Mean Squared Error (RMSE), especially with Bidirectional LSTM model (Zhang et al., ).

Given VMD’s capability to diminish data volatility and good performance of linear-based models, there exists potential for more compelling time-series data forecasting indeed with less stable data. These models will be compared against other deep learning models such as RNN, Long Short Term Memory (LSTM), and Bidirectional LSTM (BiLSTM), which are less difficult however broadly utilized in time-series forecasting researches.

motivated by the reasons outlined, this aims to investigate ”Variational Mode Deterioration with Straight Embeddings for Successful Time-Series Information Expectation.” The goal of employing VMD for data decomposition and evaluating simplistic linear models (Linear, NLinear, and DLinear) alongside RNN-based models is to develop a predictive framework capable of achieving minimal error rates even under volatile data conditions.

## 2 Related Works

### 2.1 Time-Series Forecasting Models

Linear models. Linear model was originally used as a regression tool in statistics that predict outcomes based on linear relationship between independent variables and dependent variables. Its application on machine learning and pattern recognition were discussed by Bishop () in a broader context. Their simplicity and interpretability made them valuable for many tasks, though they can struggle with capturing complex, non-linear relationships. In a research conducted by Zeng et al. (), linear models were improved into NLinear and DLinear that are capable to compete with many transformer based models in time-series forecasting. Another linear-based model for long term time series forecasting is RLinear which was introduced back in 2023 (Li et al., ). RLinear utilized Reversible Instance Normalization (RevIN) and Channel Independent (CI) strategy to improve overall forecasting performance. RevIN operates by normalizing each instance of the data independently which can lead to improved convergence rate and reduced overfitting (Kim et al., ). Meanwhile CI is a strategy used in multivariate time-series forecasting that normalizes the data separately for each feature (Han et al., ).

Neural networks. Neural network is a more robust algorithm in machine learning that was inspired by the human nervous system, consist of layers of interconnected nodes (neurons) and are designed to model complex, non-linear relationships. These models are trained using backpropagation to adjust weights and improve prediction results (Aggarwal, ). They can range from simple feedforward networks to deep architectures with many layers such as Recurrent Neural Network (RNN). However due to its limitations on long term forecasting, Long Short Term Memory (LSTM) was introduced as an improvement from RNN. LSTM still used RNN architecture but it was designed to handle long-term dependencies in data (Sherstinsky, ).

Transformer As a more recent advancement, an algorithm with self-attention mechanism was introduced, self-attention allows the model to weigh the importance of different parts of an input sequence relative to each other. This architecture are called Transformer and it’s capable of processing all elements simultaneously, rather than sequentially, enhancing both efficiency and performance in capturing complex dependencies (Vaswani et al., ). Transformer is still developed to this day with application on other fields such as AutoFormer and Reformer for time-series forecasting. AutoFormer operates by decomposing the data into trend and seasonal then applied together with Auto-Correlation (Wu et al., ). Meanwhile, Reformer utilizes local attention on a fixed window together with reversible layers to reduce the complexity of computation (Kitaev et al., ). Transformer can be further advanced by unifying the statistics of each input then converting the output with restored statistics, and also by recovering intrinsic non-stationary information into temporal dependencies, those advancement were done in Stationary Transformer (Liu et al., ). Another advancement that transforms the limitations of 1D temporal variation into the 2D space to allow temporal evolution of one feature to be affected by the other (Wu et al., ). In 2023, PatchTST were introduced, a Transformer that utilizes Channel Independance (CI) and segmentation of time series inputs into several subseries-level patches. Each patch represents a portion of the original data (Nie et al., ).

Large Language Models. Large Language Models are models that were designed to understand, work, and generate with various human language. However, it was also recently developed to work with images for Computer Vision (CV) by utilizing its robustness in pattern recognition and reasoning (mirchandani2023). However, well-known LLM such as GPT and BERT can be used in forecasting time series data by fine-tuning it based on temporal data and also unifying their framework so that it can be used in diverse sequential task (classification, anomaly detection, forecasting, etc.) (Zhou et al., ). Time-LLM were developed in this year, a time series LLM model were developed to work with time-series data by reprogramming time series inputs into text prototype representations and guiding LLM’s reasoning by augmenting the input context with declarative prompts (Jin et al., ). It was proven competitive against many recently advanced time series forecasting method such as GPT4TS, DLinear, PatchTST, etc.

### 2.2 Data Mode Decomposition

Data mode decomposition is a technique used to decompose complex time series data into many data modes or Intrinsic Mode Function (IMF) with less complexity. Empirical Mode Decomposition (EMD) is one of data decomposition method that decomposes a signal into a set of intrinsic mode functions (IMFs) and a residual trend, capturing components of varying frequency ranges adaptively. EMD operates by iteratively sifting the data to identify and extract these IMFs, making it particularly effective for non-stationary and nonlinear time series data (Huang et al., ). However, EMD can be sensitive to noise and may suffer from mode mixing, where different modes are incorrectly combined.

To address those limitations, Variational Mode Decomposition (VMD) was introduced as a more robust data decomposition method. VMD decomposes a signal into modes by solving a variational problem that seeks to minimize the bandwidth of each mode while ensuring they are as distinct as possible from each other. It utilized Lagrange function and Alternate Direction Method of Multipliers (ADMM) (Dragomiretskiy & Zosso, ). Lagrange function uses $\lambda$ penalty term to enforce constrain for decomposed components. Lagrange is a quite complex function that can be simplified by implementing ADMM to split the function into several vital parts of function. ADMM works by iterating those functions until it reaches convergence (Boyd et al., ). This method is less prone to noise and mode mixing issues compared to EMD, as it uses a principled optimization approach to separate modes in the frequency domain. Dynamic Mode Decomposition (DMD) is an another data decomposition method that is capable of extracting important dynamic characteristics of the data. DMD decomposes the data into several data modes or IMFs by utilizing matrix analysis and singular value decomposition (Proctor et al., ).

## 3 Model

### 3.1 VMD-Linear Framework

Figure: Figure 1: VMD-Linear Framework
Refer to caption: extracted/5826460/images/ProposedFramework.png

The structure of VMD-Linear shown in Figure 1 was inspired by VMD-BLSTM framework introduced by Zhang et al. (). The goal is to reduce data bandwidth in all IMF or data modes with the sum of all IMF equals to that of the original data. LTSF-Linear models will be assigned on each IMF for time-series forecasting. As the final step, the individual forecasts will be aggregated to produce the overall forecast result.

### 3.2 Variational Mode Decomposition

Variational Mode Decomposition (VMD) is a method commonly used to decompose complex signals into several k parts of signals (modes) that have been modulated in terms of amplitude and frequency, also known as Intrinsic Mode Function (IMF).

$$ $IMF\;or\;u_{k}(t)=A_{k}(t)\cos(\phi_{k}(t))$ (1) $$

The symbol $\phi_{k}(t)$ is a non-decreasing function (can be considered a wave) with the condition that $\phi_{k}(t)\geq 0$, and $A_{k}(t)$ is the envelope of the wave $\phi_{k}(t)$. All IMF formed must satisfy Equation 1. VMD applies Hilbert transform to obtain the analytic frequency spectrum ($Au_{k}$) of each mode $u_{k}$. In the transformation process, each mode is multiplied by the center frequency ($e^{-jw_{k}t}$) to shift its frequency spectrum towards the baseband (Dragomiretskiy & Zosso, ).

$$ $Au_{k}(t)=[\delta(t)+\frac{j}{\pi t}]*u_{k}(t)e^{-jw_{k}t}$ (2) $$

The Dirac distribution is symbolized as $\delta$ and j is the imaginary number satisfying $j^{2}=-1$. The symbol $w_{k}$ in the center frequency ($e^{-jw_{k}t}$) denotes the central pulse/vibration of each mode. The signal’s bandwidth will be estimated using $H^{1}$ Gaussian Smoothness of the demodulated signal. In this case, the estimation is performed by squaring the $L^{2}$ normalization of the gradient.

$$ $min_{u_{k},w_{k}}\{\Sigma_{k}\parallel\partial_{t}[(\delta(t)+\frac{j}{\pi t}) \ast u_{k}(t)]e^{-jw_{k}t}\parallel_{2}^{2}\}$ (3) $$

Equation 3 only applies when $\Sigma_{k}\;u_{k}=original\;signal$.The main objective of VMD is to minimize the bandwidth around the center frequency in each mode $m_{k}$ while maintaining the condition that the sum of all modes equals the original signal. It is recommended to use Lagrange multiplier ($\lambda$) and quadratic penalty. The quadratic penalty is used to address additional Gaussian noise. In noise-free conditions, infinite weights are required to enforce strict data accuracy. The Lagrange multiplier is a way to impose strict constraints. By combining the strict constraint imposition of the Lagrange multiplier and the quadratic penalty ($\alpha$) with finite weights, the enhanced Lagrangian function L is formulated in Equation 4 to transform the constrained problem.

$$ $\begin{split}&L({u_{k}},{w_{k}},\lambda)=\alpha\Sigma_{k}\parallel\partial_{t} [(\delta(t)+\frac{j}{\pi t})\ast u_{k}(t)]e^{-jw_{k}t}\parallel_{2}^{2}+\\ &\parallel f(t)-\Sigma_{k}u_{k}(t)\parallel_{2}^{2}+[\lambda(t),f(t)-\Sigma_{k }u_{k}(t)]\\ \end{split}$ (4) $$

The minimization problem in Equation 4 can be addressed using the Lagrange function L by employing the Alternate Direction Method of Multipliers (ADMM) iteratively on $u_{k}$, $w_{k}$, and $\lambda$. Iteration will stop if the condition is met: $\Sigma_{k}(\parallel m_{k}^{n+1}-m_{k}^{n}\parallel_{2}^{2}/\parallel m_{k}^{n
}\parallel_{2}^{2})<\varepsilon$ where $\varepsilon$ is the threshold value.

$$ $\widehat{u}_{k}^{n+1}=\frac{\hat{f}(w)-\Sigma_{i\neq k}\;\widehat{u}_{i}(w)+ \frac{\widehat{\lambda}(w)}{2}}{1+2\alpha(w-w_{k})^{2}}$ (5) $$

$$ $\widehat{w}_{k}^{n+1}=\frac{\int_{0}^{\infty}w|\widehat{u}_{k}(w)|^{2}dw}{\int _{0}^{\infty}|\widehat{u}_{k}(w)|^{2}dw}$ (6) $$

$$ $\hat{\lambda}^{n+1}(w)=\hat{\lambda}^{n}(w)+\tau(\hat{f}(w)-\Sigma_{k}\widehat {u}_{k}^{n+1})$ (7) $$

For multivariate forecasting, the main idea still identical but most variables is represented in vectors. Thus, increasing the complexity and computational demands (Rehman & Aftab, ).

### 3.3 Long-term Time Series Forecasting Linear

Long-term Time Series Forecasting Linear or LTSF-Linear is a sequence of simple one-layer linear models consisting of Linear, NLinear, and DLinear (Zeng et al., ). LTSF-Linear models already support multivariate forecasting by alternating the number of channels in the parameter. LTSF-Linear models is more efficient in terms of time and resource due to having no more than two linear layers, allowing training with smaller memory and fewer parameters possible. Linear model is the simplest model with a simple one layer linear model, unlike NLinear and DLinear.

Figure: Figure 2: NLinear Framework
Refer to caption: extracted/5826460/images/NLinear.png

NLinear As shown in Figure 2, NLinear model is used to anticipate distribution shifts in the data by reducing the last input passed to the Linear layer and then adding back the reduced input before making the final prediction.

Figure: Figure 3: DLinear Framework
Refer to caption: extracted/5826460/images/DLinear.png

DLinear DLinear enhances the simple linear model by utilizing moving average to split the data into Trend and Seasonal with each going through their seperate linear layer. This model should be used if there is a clear trend in the data. DLinear framework is shown in Figure 3.

## 4 Experiments

In this section, we present experiments using LTSF-Linear models and few neural network models such as Recurrent Neural Network (RNN), Long Short Term Memory (LSTM), and Bidirectional LSTM (BLSTM) on 13 real-world datasets. In this research, we compare each model performance using RMSE values.

### 4.1 Dataset

We used 13 different real-word dataset obtained from various sources in the internet. The datasets are described as below:

- 1.
ETTm2 Dataset, this dataset were compiled by Beijing Guowang Fuda Science and Technology Department (Zhou et al., ). It consist electricity distribution dynamics and were captured in 15-minutes intervals with a total of 69,680 data points.
- 2.
Wind Turbine, The following data contains information about a certain wind turbine in Turkey that were captured using the SCADA system (Erisen2019). Wind Turbine related data such as wind speed, wind direction, and power generated were captured in 10-minute intervals with a total of 50,500 data points.
- 3.
M4 Dataset, The M4 dataset were originally used in M4 Competition as a benchmark to evaluate various models in real word scenarios. It consist many features and can be used for multivariate forecasting (Makridakis et al., ). This dataset were recorded in 7-days intervals.
- 4.
SEA Air Quality Dataset We also used various real-world Air Quality Index (AQI) datasets from many cities recorded from 27 January 2022 to 22 Mei 2023 (AQI2023). Those cities were Central Singapore, Banjarbaru, Batam, Jakarta, South Jakarta, Jambi, Malang, Medan, Samarinda, and Semarang.

ETTm2, Wind Turbine, and M4 dataset were used because of its relatively high variance in their features (Arthur et al., ). Those 3 datasets will be used in multivariate forecasting together with Central Singapore AQI dataset. As for univariate forecasting, all dataset will be used including mentioned multivariate datasets.

We divide the dataset into training data and test data with a certain ratio. Dataset fewer than 500 rows will be split with 90:10 ratio, and dataset with more than 10000 rows will be trimmed to the first 10,000 rows then split with 80:20 ratio. We conducted two different testing scenario based on number of features, namely univariate and multivariate. On univariate testing scenario, we only used one feature on each dataset in this testing scenario. Meanwhile, multivariate testing scenario used 4 datasets that contain multiple feature such as ETTm2, Wind Turbine SCADA, M4 Competition, and Central Singapore Air Quality dataset.

### 4.2 Experimental Protocols

First, dataset were cleaned then normalized by using Standard Scaler. After normalization, variational mode decomposition were implemented on the dataset and will create K data modes or IMFs. Each data modes will be going through separate forecasting model, then the forecasting results of each modes is summed together to attain the final forecasting result. We used LASSO regularization in loss calculation to avoid vanishing gradients that could happen while training the models (Mairal & Yu, ). RMSE were used as a metric evaluation method for emphasizing each model’s impact on reducing error rates (Hodson, ).

**Table 1: Multivariate RMSE Results**
| Dataset | Linear | DLinear | NLinear | LSTM | BiLSTM | RNN |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| No VMD | VMD | No VMD | VMD | No VMD | VMD | No VMD | VMD | No VMD | VMD | No VMD | VMD |  |
| M4 Weekly | 1,041 | 0.102 | 1,063 | 0.040 | 1,116 | 0.064 | 1,070 | 0.092 | 1,126 | 0.089 | 1,092 | 0.080 |
| Central Singapore | 1,203 | 0.098 | 1,221 | 0.024 | 1,331 | 0.051 | 1,165 | 0.124 | 1,218 | 0.203 | 1,204 | 0.077 |
| Wind Turbine | 1,397 | 0.007 | 1,404 | 0.005 | 1,414 | 0.011 | 1,379 | 0.025 | 1,394 | 0.038 | 1,381 | 0.330 |
| ETTm2 | 1,380 | 0.010 | 1,406 | 0.007 | 1,398 | 0.014 | 1,355 | 0.252 | 1,379 | 0.031 | 1,321 | 0.243 |
| Average | 1,255 | 0,054 | 1,273 | 0,019 | 1,314 | 0,035 | 1,242 | 0,123 | 1,279 | 0,090 | 1,249 | 0,182 |

**Table 2: Univariate RMSE Results**
| Dataset | Linear | DLinear | NLinear | LSTM | BiLSTM | RNN |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| No VMD | VMD | No VMD | VMD | No VMD | VMD | No VMD | VMD | No VMD | VMD | No VMD | VMD |  |
| Banjarbaru PM2.5 | 1,011 | 0.620 | 1,12 | 0.703 | 1,188 | 0.908 | 1,216 | 0.445 | 1,127 | 0.726 | 1,083 | 0.494 |
| Batam PM2.5 | 1,027 | 0.605 | 1,006 | 0.687 | 1,428 | 0.858 | 1,336 | 0.398 | 1,306 | 0.866 | 1,208 | 0.894 |
| Jakarta PM2.5 | 1,042 | 0.563 | 1,045 | 0.779 | 1,133 | 0.713 | 1,396 | 1.274 | 1,297 | 0.827 | 1,183 | 0.694 |
| South Jakarta PM2.5 | 1,051 | 0.662 | 1,002 | 0.841 | 1,209 | 0.922 | 1,143 | 0.814 | 1,261 | 1.122 | 1,295 | 1.141 |
| Jambi PM2.5 | 0,962 | 0.529 | 0,951 | 0.521 | 1,130 | 0.712 | 1,135 | 1.108 | 1,154 | 1.011 | 1,028 | 1.101 |
| Malang PM2.5 | 1,002 | 0.581 | 1,007 | 0.855 | 1,166 | 0.953 | 1,221 | 1.222 | 1,172 | 1.260 | 0,999 | 1.161 |
| Medan PM2.5 | 1,000 | 0.545 | 0,994 | 0.714 | 1,079 | 0.962 | 1,164 | 0.572 | 1,080 | 0.415 | 1,75 | 0.440 |
| Samarinda PM2.5 | 0,830 | 0.510 | 0,832 | 0.652 | 0,988 | 0.634 | 0,897 | 0.844 | 0,935 | 1.435 | 0,853 | 0.796 |
| Semarang PM2.5 | 1,020 | 0.824 | 1,016 | 0.780 | 1,091 | 1.122 | 1,231 | 0.901 | 1,228 | 0.959 | 1,236 | 0.631 |
| M4 Weekly V2 | 0,591 | 0.631 | 0,592 | 0.753 | 0,636 | 0.613 | 0,682 | 0.307 | 0,635 | 0.151 | 0,602 | 0.698 |
| Central Singapore<br>PM10 | 1,056 | 0.574 | 1,046 | 0.605 | 1,152 | 0.906 | 1,285 | 0.642 | 1,322 | 0.938 | 1,376 | 0.922 |
| WindTurbine<br>ActivePower | 1,020 | 0.786 | 1,025 | 1.022 | 1,398 | 1.295 | 1,421 | 1.940 | 1,424 | 0.865 | 1,392 | 0.919 |
| ETTm2 HUFL | 1,014 | 0.788 | 1,004 | 0.908 | 1,373 | 0.794 | 1,417 | 0.744 | 1,387 | 1.129 | 1,352 | 1.004 |
| Average | 0,967 | 0.619 | 0,969 | 0.742 | 1,133 | 0.883 | 1,177 | 0.872 | 1,161 | 0.881 | 1,167 | 0.824 |

### 4.3 Forecasting Results

Baseline we choose other well-known neural network models such as Recurrent Neural Network (RNN), Long Short Term Memory (LSTM), and Bidirectional LSTM (BiLSTM). RNN were developed in 1986 by utilizing Backpropagation Through Time (BPTT) to recognize patterns in sequences of data. RNN were further upgraded to LSTM as an algorithm that can recognize patterns in long sequences of data while maintaining useful information (Sherstinsky, ). To enable LSTM to use future contexts of the data, Bidirectional LSTM were developed (Graves et al., ). We will also be utilizing VMD together with mentioned neural network models.

Results Multivariate forecasting results were listed in Table 1. RMSE values reduction in multivariate predictions were shown after applying VMD with all models having average RMSE values below 0.2. On average, the RMSE decreased by 1.067 to 1.279 when using VMD. The lowest RMSE values were obtained by the DLinear + VMD model in all prediction cases, with an average RMSE of 0.019. The second smallest average RMSE values were held by the NLinear + VMD model with a value of 0.035.

Univariate results were shown in Table 2. In most datasets, the use of VMD decomposition methods has proven to significantly enhance the performance of prediction models. However, there are still some cases where the model did not experience a decrease in RMSE, such as in the prediction of WindTurbine ActivePower data with LSTM, which saw an increase in RMSE from 1.421 to 1.940. The best univariate prediction model based on average RMSE across all prediction cases was the Linear model with an average RMSE of 0.619, followed by the DLinear model in second place with an average RMSE of 0.742.

## 5 Conclusion And Future Work

This study demonstrates the effectiveness of Variational Mode Decomposition (VMD) in enhancing the performance of time-series forecasting models by decomposing the original signal into multiple data modes. We integrated VMD with the LTSF-Linear family, which includes Linear, DLinear, and NLinear models, known for their simplicity and competitive edge against Transformer-based models. To rigorously evaluate our approach, we benchmarked it against established neural network models, including RNN, LSTM, and BLSTM, using RMSE as the evaluation metric. The results unequivocally show that VMD significantly reduces RMSE across both univariate and multivariate forecasting tasks. Notably, the Linear + VMD model achieved the best performance on all univariate datasets, while the DLinear + VMD model excelled across all multivariate datasets. These findings highlight the superior performance and robustness of VMD-enhanced linear models in time-series forecasting, providing a promising direction for future research.

Further research should explore optimal forecasting models and try various preprocessing methods. LTSF-Linear has a limited capacity in handling long term time series forecasting but can be further improved by applying various preprocessing methods.

## Peer Reviews

confidence: 5
contribution: 1
initial_rating: 1
presentation: 1
questions: See weaknesses.
soundness: 1
strengths: * The paper explores enhancing time series forecasting using Variational Mode Decomposition (VMD).
* The proposed approach is tested across 13 diverse time series datasets.
summary: This study addresses the challenge of data volatility in time-series forecasting by employing Variational Mode Decomposition (VMD) to enhance prediction accuracy. VMD decomposes data into distinct modes, which, when integrated with linear models, creates a robust forecasting framework. Tested on 13 datasets (ETTm2, WindTurbine, M4, and air quality data from Southeast Asia), the approach shows that models using VMD outperform those without it, as evidenced by lower Root Mean Squared Error (RMSE) values. The Linear + VMD model achieved the best RMSE in univariate forecasting (0.619), while the DLinear + VMD model excelled in multivariate forecasting with an average RMSE of 0.019. This highlights the advantage of combining VMD with linear models for effective time-series forecasting.
weaknesses: * The focus is on point forecasting, whereas probabilistic forecasting would be more relevant and challenging for many practical applications.
* The statement "Data decomposition techniques also significantly influence forecasting outcomes," supported by only two citations, lacks sufficient evidence for this claim.
* Phrases such as "to mitigate data volatility" and "under unstable data conditions" are used without clarifying how the proposed method directly addresses these issues.
* Key references are missing in the related work, such as [https://dl.acm.org/doi/full/10.1145/3533382](https://dl.acm.org/doi/full/10.1145/3533382).
* Section 3, the main one, is challenging to read, with multiple variables missing math mode formatting and variables not introduced.
* In Tables 1 and 2, the substantial increase in accuracy from "No VMD" to "VMD" is unclear, suggesting that factors beyond VMD may be influencing the performance.
* Simple yet effective forecasting baselines like ARIMA and exponential smoothing were not considered in the comparisons.

---

confidence: 1
contribution: 2
initial_rating: 3
presentation: 2
questions: 1. Why do the authors trim the dataset with more than 10000 rows down to the first 10,000 rows? 
2. Why do authors only use M4 weekly dataset in the evaluations? M4 datasets consists of time series with various periodicities (hourly, daily, weekly, monthly). They should all be used in the evaluations.   
3. What is the value of K (the number of modes in VMD) used in model training? How is K selected? How does the model performance change with different values of K? Ablation studies are needed to address these questions. 
4. Why are the forecasting results of each mode summed to get the final prediction (page 6), instead of using a weighted average with fixed or learned weights?
5. In additional to the aggregate RMSE, what are the common failure modes of VMD (e.g., fail to capture outliers)? This error audit is important to informing whether the proposed method is applicable in other domains.
soundness: 2
strengths: 1. Overall, the paper is clearly written and easy to understand, although additional technical details need to be provided (see below). 
2. Using Variational Mode Decomposition (VMD) with linear embedding for time series forecast is moderately novel and potentially applicable in other domains.
summary: This paper proposes a technique of time series forecast using Variational Mode Decomposition (VMD) and linear post-processing. VMD decomposes the original time series into different intrinsic mode functions. The light-weight linear post-processing provides additional model capacity for different forecasting tasks. The authors evaluate their proposed method on 4 datasets and demonstrate that the proposed technique (combined with the base model) improves the forecast accuracy of each base model.
weaknesses: 1. The authors should use additional metrics to comprehensively evaluate the proposed method, such as MAE, MAPE, etc. 
2. The baseline methods used in the evaluations are too generic (e.g., linear, RNN, LSTM) and not used in SOTA time series forecasts. The authors should compare the proposed method against several Transformer-based models with SOTA forecast accuracy, such as the ones reviewed in Section 2.1. 
3. The dataset size is too small since the authors only use up to 10,000 rows in each dataset (page 6). This may potentially lead to bias in the datasets and the reported evaluation metrics. The authors should use complete datasets to train and evaluate the proposed method, which should be able to handle larger datasets than 10,000 rows.
4. Ablation studies are needed to understand the impact of different model components and hyperparameters in the model architecture. For instance, the number of modes K is an important hyperparameter to analyze via ablation studies. 
5. The authors should discuss the limitations of their work and how these limitations can be addressed in future research.
6. The metrics for No VMD reported in Tables 1 and 2 are incorrect (probably due to typo). The value should be 1.041 instead of 1,041 (and similarly for all other values).
7. The authors should report the size of each of the 13 datasets in Table 2.

---

confidence: 5
contribution: 2
initial_rating: 3
presentation: 2
questions: As provided in weakness.
soundness: 1
strengths: By decomposing data into modes, the VMD-LTSF (Long-Term Series Forecasting) framework aims to improve forecast accuracy.
summary: This paper proposes integrating Variational Mode Decomposition (VMD) with linear models for time-series forecasting to address data volatility challenges. Experimental evaluation is given to show its performance.
weaknesses: 1. Although VMD is integrated, the motivation is missing in this paper. The authors are suggested to give more theoretical discussions and an intuitionistic example to strongly motivate the work. Notably, based solely on the results in Tables 1 and 2, which compare primarily with classic models like LSTM and RNN, it is difficult to discern the authors’ claimed conclusion that -- ***VMD and Linear are what you need for TSF***.

2. This work lacks a comprehensive analysis of existing time series forecasting and decomposition techniques. Additionally, the paper overemphasizes benchmarks against neural network models like LSTM and RNN, overlooking other competitive decomposition-based techniques or hybrid methods, such as N-BEATS (ICLR2020), OneNet (NeurIPS2023), etc.

3. The mathematical foundation of the VMD model is briefly presented, but lacks an in-depth explanation of how the VMD decomposition can theoretically improve linear predictive models, for example, by connecting the VMD components to specific improvements in fluctuation mitigation or pattern separation. Besides, specific parameters and architecture choices, such as the number of modes in VMD, aren’t fully justified. Also, it’s unclear how sensitive the VMD-Linear model is to hyperparameter tuning (such as Lagrange multiplier $ \lambda$ and penalty parameter $ \alpha$), which could impact model generalizability.

4. The experimental part only includes comparisons with a few classic models, making it resemble an ablation study rather than a comprehensive performance evaluation. Consequently, it is difficult to discern the actual contribution of the proposed model. Besides, the authors emphasize multivariate forecasting improvements but mainly compare against linear models without showing performance relative to advanced multivariate models or ensemble methods.

---

confidence: 5
contribution: 4
initial_rating: 8
presentation: 3
questions: 1. In the Univariate RMSE Results, it is observed that VMD does not consistently outperform non-VMD models. Could this be due to specific characteristics or structures of the datasets?

2. If possible, could you provide more mathematical details explaining why VMD tends to perform better than non-VMD models? Additionally, why do linear models consistently outperform neural networks in the multivariate cases?
soundness: 3
strengths: 1. Innovatively introduce the VMD method for time-series datasets and apply it to two distinct types of models: Linear models and Neural Networks

2. Implement both VMD and Non-VMD approaches across 13 large datasets, presenting results for both multivariate and univariate cases

3. Provide comprehensive background on the datasets and the development of time-series analysis techniques
summary: In this paper, the authors propose using Variational Mode Decomposition (VMD) to mitigate volatility by decomposing data into distinct modes. VMD is applied to three linear models (Linear, DLinear, and NLinear) and three neural networks (LSTM, BiLSTM, and RNN), and tested on 13 diverse datasets, including ETTm2, WindTurbine, M4, and 10 air quality datasets from Southeast Asian cities. Performance is evaluated using Root Mean Squared Error (RMSE). The results show that in multivariate settings, VMD consistently outperforms non-VMD approaches across all datasets and models, with DLinear+VMD achieving the smallest errors. However, in univariate settings, models without VMD generally perform better, with both linear models and neural networks showing the lowest errors for their respective datasets.

The authors also provide an overview of linear models, neural networks, transformers, and large language models, alongside a detailed description of the 13 datasets used. In Section 3.2, they present the mathematical foundation of Variational Mode Decomposition (VMD) and explain how it is solved using the Alternating Direction Method of Multipliers (ADMM) within a penalized Lagrange function framework.
weaknesses: 1. In Table 1 and Table 2, some data use commas (e.g., "1,041") while others use periods (e.g., "0.005"). Please standardize these formats or provide an explanation for the differences.

2. Add more explanations and visualizations for the three neural network models, similar to the detailed descriptions provided for the linear models in Section 3.3.

3. I am not familiar with the ADMM algorithm. Are Equations (5), (6), and (7) derived specifically from the problem, or are they part of a standard algorithm? If they are derived, please include more mathematical details. If they follow a fixed iterative algorithm, kindly provide additional references.

Meta-review: This paper introduces a time series forecasting technique combining Variational Mode Decomposition (VMD) with lightweight linear post-processing. VMD decomposes the original time series into intrinsic mode functions (IMFs), and the linear post-processing step enhances the model's capacity to handle various forecasting tasks.

However, as noted by reviewers, the paper has several shortcomings:

Limited analysis of existing techniques: It lacks a thorough review of existing time series forecasting and decomposition methods, which is essential for positioning the proposed approach within the broader context.

Overemphasis on neural network benchmarks: The evaluation focuses heavily on comparisons with neural network models like LSTM and RNN, while neglecting competitive decomposition-based techniques or hybrid approaches.

Theoretical explanation gap: The paper does not provide a detailed theoretical analysis of how VMD decomposition specifically enhances the performance of linear predictive models, such as by mitigating fluctuations or separating patterns in the data.

Insufficient experimental scope: The experiments primarily compare the proposed method with a few classical models, resembling an ablation study rather than a comprehensive performance evaluation across a diverse set of benchmarks.

Decision: Reject