Natural Gas Futures Price Prediction Based on Variational Mode Decomposition–Gated Recurrent Unit/Autoencoder/Multilayer Perceptron–Random Forest Hybrid Model

Abstract

Forecasting natural gas futures prices can help to promote sustainable global energy development, as the efficient use of natural gas as a clean energy source has become key to the growing global demand for sustainable development. This study proposes a new hybrid model for the prediction of natural gas futures prices. Firstly, the original price series is decomposed, and the subsequences, along with influencing factors, are used as input variables. Secondly, the input variables are grouped based on their correlations with the output variable, and different models are employed to forecast each group. A gated recurrent unit (GRU) captures the long-term dependence, an autoencoder (AE) downscales and extracts the features, and a multilayer perceptron (MLP) maps the complex relationships. Subsequently, random forest (RF) integrates the results of the different models to obtain the final prediction. The experimental results show that the model has a mean absolute error (MAE) of 0.32427, a mean absolute percentage error (MAPE) of 10.17428%, a mean squared error (MSE) of 0.46626, a root mean squared error (RMSE) of 0.68283, an R-squared (R²) of 93.10734%, and an accuracy rate (AR) of 89.82572%. The results demonstrate that the proposed decomposition–selection–prediction–integration framework reduces prediction errors, enhances the stability through multiple experiments, improves the prediction efficiency and accuracy, and provides new insights for forecasting.

1. Introduction

By accurately forecasting natural gas futures prices, it is possible not only to provide decision-making support for energy market participants [1,2] but also to enhance the market competitiveness of natural gas as a clean energy source, thus promoting the sustainable optimization of the energy structure and the development of a low-carbon economy [3]. However, the prediction of natural gas futures prices presents various challenges. To effectively tackle challenges such as non-stationarity and noise in data, scholars have introduced the concept of “decomposition–integration” [4]. This approach decomposes complex price time series into multiple simpler and more manageable subsequences [5]. Each subsequence is then forecast independently [6], and their predictions are integrated to obtain the final forecast value [7]. Specifically, techniques like the completely adaptive ensemble empirical modal decomposition of noise [8], the improved ensemble empirical modal decomposition (IEEMD) of noise [9], and variational modal decomposition (VMD) [10], among others, can decompose natural gas futures price time series into modal components with different frequency characteristics. The advantages of the decomposition–integration approach include the following: decomposing complex problems to enhance the prediction performance and reduce error accumulation [11]; handling complex data issues such as nonlinearity, randomness, and high noise levels [12]; improving the computational efficiency by processing in frequency bands to enhance the interpretability [13]; and enhancing models’ accuracy and robustness [14].

When constructing forecasting models [15], various factors influencing natural gas futures prices are thoroughly considered [16,17], including but not limited to historical price data [18], global economic indicators [19], policy changes [20], unforeseen events, and energy consumption trends. Related indicators such as volatility [21] and market sentiment indicators [22,23] are also incorporated. These factors are appropriately processed to serve as input features for the model [24], thereby enhancing its predictive ability and robustness [25]. The advantages of incorporating influencing factors include gaining a deeper understanding of the data relationships [26]; supporting multivariate analysis, applicable to multi-input single-output model forms [27]; providing rich information, where how each influencing factor addresses different forecasting needs [28,29]; and integrating multiple methodologies [30,31].

Hybrid models combine advanced algorithms and tools such as long short-term memory (LSTM) [32], autoencoders (AEs), autoregressive integrated moving average (ARIMA) [33,34], gated recurrent units (GRUs) [35], convolutional neural networks (CNNs) [36], generalized autoregressive conditional heteroskedasticity (GARCH) [37], multilayer perceptron (MLP), random forest (RF), and others for final prediction purposes [38,39]. The advantages of hybrid models include enhancing models’ robustness and stability against noise [40], offering flexible combinations with multiple choices [41], conducting multidimensional analyses to overcome the limitations of single models, and being widely applicable to address complex issues [42].

To address frequent unexpected events in the energy market, such as global pandemics, geopolitical conflicts [43], and natural disasters, models incorporate dynamic adjustment mechanisms and anomaly detection algorithms to capture the real-time impacts of these events on prices and make corresponding adjustments, thereby enhancing the model’s predictive capabilities under extreme conditions [44]. To comprehensively reflect the dynamic changes in natural gas futures prices, scholars typically categorize the input features based on their varying degrees of influence on the prices into input feature sets. These include, but are not limited to, historical prices [45], trading volumes, open interest [46], macroeconomic indicators [47], seasonal factors, and market sentiment indices extracted through natural language processing [48]. During model training and testing [49], factors such as the prediction periods, time steps [50], and sequence length segmentation ratios also influence the prediction outcomes. Adjusting these periods or steps can aid in the study of the impact of different prediction windows on model performance to find optimal prediction strategies [51].

To objectively assess model performance, new models are typically compared with traditional models in terms of predictive performance [52]. A comparative analysis using evaluation metrics highlights the significant advantages of the new model in terms of prediction accuracy, stability, and generalization capabilities [53]. Comparing different combinations of models and experimental results helps to validate the accuracy of the proposed model, identifies deficiencies in the control group models [54], optimizes innovative settings [55], and promotes interdisciplinary integration.

Table 1. Reference methodologies.

Categorization	Reference Number	Method
Decomposition	[8]	EEMD
	[9]	IEEMD
	[10]	VMD
Model	[32]	LSTM
	[33,34]	ARIMA
	[35]	GRU
	[36]	CNN
	[37]	GARCH

compares the different methods used in the studies cited in this paper for the forecasting of futures prices; these are divided into decomposition and modeling categories, with the decomposition category mainly used to split the price series and enrich the data features; the modeling category is mainly applicable when learning the patterns of price changes and is used for forecasting.

The VMD-GRU/AE/MLP-RF hybrid model proposed in this study is significantly innovative in time series forecasting. The research framework is shown in Figure 1. The model adopts the “decomposition–selection–forecasting–integration” framework, which not only includes the traditional decomposition and forecasting steps but also introduces a grouping strategy based on the correlation coefficients and model integration to systematically deal with the input variables. Through VMD decomposition, the time series is split into multiple subseries with different frequencies to reduce the complexity. The correlation coefficient grouping strategy quickly filters important variables in the preprocessing stage, reduces the input dimensions, and is more efficient in computation compared to the attention mechanism and SHAP values, making it suitable for large-scale data. In terms of model combination, the pairing of the GRU, AE, and MLP gives full play to the advantages of each model: the GRU efficiently captures the long-term dependence of the time series and has a concise structure; the AE reduces the dimensionality and extracts the key features to remove noise; and the MLP performs nonlinear combination to enhance the expressive ability. This combination ensures a balance between computational efficiency and performance, satisfying the requirement for low computational complexity and small data volumes, being suitable for long-series prediction. Finally, the prediction results are integrated via random forest to further improve the model’s robustness and accuracy. The experiments verify the effectiveness of the model and prove that it has significant advantages in terms of accuracy and robustness [56].

This article is structured as follows. Section 2 introduces the theoretical framework and the proposed models used in the study. Section 3 covers the data preprocessing, introduces the evaluation metrics, and discusses the grouping of the input variables. Section 4 presents the experimental correlations and compares the results. Section 5 concludes the study.

2. Methodology

2.1. Variational Mode Decomposition

Variational mode decomposition is a signal processing method. It decomposes complex signals into multiple intrinsic mode functions (IMFs), each with specific frequency characteristics and stability. The formula is given by (1), where $A_k(t)$ represents the instantaneous amplitude and ${\varphi }_k(t)$ represents the instantaneous phase.

$$s_k(t)=A_k(t)\cos ({\varphi }_k(t)).$$

(1)

2.2. Pearson Correlation Coefficient

The Pearson correlation coefficient is a widely used statistical measure in statistics, assessing the linear correlation between two variables X and Y. It is represented by (2). The coefficient ranges within [−1, 1], where a value closer to 1 indicates a strong positive correlation between the variables, a value closer to 0 indicates a weak correlation, and a value closer to −1 indicates a strong negative correlation.

$${\rho }_{X,Y}={\displaystyle \frac{\mathrm{cov}(X,Y)}{{\sigma }_X{\sigma }_Y}}.$$

(2)

2.3. Gated Recurrent Unit

The gated recurrent unit (GRU) is a popular type of recurrent neural network architecture. It retains much of LSTM’s effectiveness while reducing the model complexity and computational burden. The GRU consists of two gate units, the update gate and the reset gate, defined by (3) and (4), respectively. Here, W and b denote the weights and biases of the gates, $\sigma$ represents the sigmoid activation function, $h_{t-1}$ is the previous hidden state, $x_t$ is the current input, and $\left[·,·\right]$ denotes vector concatenation.

$$z_t=\sigma (W_z·\left[h_{t-1},x_t\right]+b_z).$$

(3)

$$r_t=\sigma (W_r\left[h_{t-1},x_t\right]+b_r).$$

(4)

2.4. Autoencoder

An autoencoder (AE) is a type of neural network model; it is used primarily for unsupervised or semi-supervised learning tasks such as data dimensionality reduction, compression, and feature extraction. The AE consists of two main components: the encoder and the decoder. The encoder maps the input data to a lower-dimensional encoded space, as shown in (5), while the decoder attempts to reconstruct the original data from the encoded space, as shown in (6). Here, f and g represent nonlinear transformation functions in the encoder and decoder, respectively.

$$z=f(x).$$

(5)

$$x^{\prime }=g(z).$$

(6)

2.5. Multilayer Perceptron

A multilayer perceptron (MLP) is a type of feedforward artificial neural network model that consists of at least three layers of nodes: an input layer, one or more hidden layers, and an output layer. In an MLP, input signals pass through the input layer, are weighted and transformed nonlinearly by neurons in the hidden layers, and finally reach the output layer to produce prediction results. The equations for the MLP are shown in (7) and (8), where u denotes the weighted sum and y represents the result.

$$u=\sum _{i=1}^n{w}_i{x}_i+b.$$

(7)

$$y=f(u).$$

(8)

2.6. Random Forest

Random forest (RF) is a type of machine learning model based on the ensemble learning principle. It enhances the prediction accuracy and stability by constructing multiple decision trees and combining their predictions. The formula is shown in (9), where $w_i$ represents individuals and $h_i$ represents the weights corresponding to predictions.

$$H(x)={\displaystyle \frac{1}{T}}\sum _{i=1}^T{w}_i{h}_i(x).$$

(9)

2.7. VMD-GRU/AE/MLP-RF Hybrid Model

This study constructs a hybrid model named VMD-GRU/AE/MLP-RF. Figure 2 illustrates the framework of the VMD-GRU/AE/MLP-RF hybrid model, with the following procedural steps.

Step 1. Data Collection. Collect raw data including daily natural gas futures: open, close, high, low prices, trading volume, amplitude, production, import volume, export volume, and storage volume. Output variable is the median price between the open and close prices.

Step 2. Decomposition. Use the VMD model to decompose price data sequences, determining the optimal number of modes. This produces 5 intrinsic mode functions (IMFs) and 1 residual (Res), with the 5 IMFs serving as new input variables.

Step 3. Feature Selection. Employ Pearson correlation coefficients to filter variables. Sort input variables by their correlation coefficients with the price in descending order.

Step 4. Grouping. Divide variables into groups based on correlation coefficient intervals. Strong correlation group: open, close, high, low prices, IMF5. Moderate correlation group: import volume, IMF4, IMF3. Weak correlation group: IMF2, IMF1, amplitude. Negative correlation group: trading volume, production, export volume, storage volume

Step 5. Prediction. Predict results for different correlation groups using different models. Use GRU, AE, MLP to predict strong, moderate, and weak correlation groups based on evaluation metrics. Generate 3 sets of prediction sub-results.

Step 6. Integration. Integrate sub-results using RF to obtain the final prediction result. Train decision trees with bootstrap sampling from the three sets of sub-results. Weight integration of sub-results is performed with weights of a, b, c as defined in (10), aimed at reducing randomness and incorporating the data characteristics of the sub-results.

$$Y=a{y}_1+b{y}_2+c{y}_3$$

(10)

3. Data and Preliminary Analysis

3.1. Data Source

The data for this study are sourced from the official website of the U.S. Energy Information Administration (EIA), including variables such as the opening, closing, high, and low prices; trading volume; amplitude; production; import volume; export volume; and storage volume. For simplicity, the output variable, the original price sequence, is taken as the median between the open and close prices. The dataset spans from 4 April 1990 to 28 May 2024. The opening price, closing price, high price, low price, trading volume, and amplitude are daily frequency data, which are recorded as a daily time unit. Meanwhile, production, imports, exports, and storage are monthly frequency data, which are recorded in each month as a time unit. In order to unify the time frequencies, we split the monthly frequency data into daily frequency data: this is achieved by copying the monthly frequency data of a particular month to each day of the month. In this way, the monthly frequency data are converted into daily frequency data, resulting in a total of 8706 data points.

3.2. Data Preprocessing

3.2.1. Handling Missing Values

The collected dataset includes 11 variables over 8706 time points with daily frequency data. Missing values are handled using linear interpolation, defined by (11).

$$y=y_0+\left(x-x_0\right){\displaystyle \frac{y_1-y_0}{x_1-x_0}}.$$

(11)

3.2.2. Normalization

Min–max normalization is applied in this study. Due to the vast differences in the ranges of price-related variables (very small) and quantity-related variables (very large), normalization helps to simplify calculations and improve the computational efficiency. Normalized data are mapped to the range [0, 1], as shown in (12).

$$x^{\prime }={\displaystyle \frac{\left(x-x_{min}\right)}{\left(x_{max}-x_{min}\right)}}.$$

(12)

3.2.3. Data Splitting

The dataset is split into training and testing sets in an 85%/15% ratio. The training set consists of 7400 data points, covering the period from 4 April 1990 to 18 June 2019. The testing set comprises 1306 data points, spanning from 19 June 2019 to 28 May 2024.

3.3. Evaluation Metrics

This study evaluates the predictive performance of the model using several metrics: the mean absolute error (MAE), mean absolute percentage error (MAPE), mean squared error (MSE), root mean squared error (RMSE), coefficient of determination ($R^2$), and accuracy rate (AR). Among these, smaller values of the MAE, MAPE, MSE, and RMSE indicate the stronger predictive capabilities of the model. A value of $R^2$ and AR closer to 1 signifies the better fit of the model to the data. The formulas for these evaluation metrics are given in (13)–(18).

$$MAE={\displaystyle \frac{1}{n}}\sum _{t=1}^n\left|{\widehat{y}}_t-y_t\right|.$$

(13)

$$MAPE={\displaystyle \frac{100\%}{n}}\sum _{t=1}^n\left|{\widehat{y}}_t-y_t\right|.$$

(14)

$$MSE={\displaystyle \frac{1}{n}}\sum _{t=1}^n{\left({\widehat{y}}_t-y_t\right)}^2.$$

(15)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{t=1}^n{\left({\widehat{y}}_t-y_t\right)}^2}.$$

(16)

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{t=1}^n}{\left(y_t-{\widehat{y}}_t\right)}^2}{{\displaystyle \sum _{t=1}^n}{\left(y_t-{\overline{y}}_t\right)}^2}}.$$

(17)

$$AR=1-{\displaystyle \frac{100\%}{n}}\sum _{t=1}^n\left|{\displaystyle \frac{{\widehat{y}}_t-y_t}{y_t}}\right|.$$

(18)

3.4. Decomposition

This study employs the VMD model to decompose the original price sequence into five sub-series and a residue based on the stability of the central frequencies, as illustrated in Figure 3.

Table 2. Center frequency.

	IMF1	IMF2	IMF3	IMF4	IMF5	IMF6	IMF7	IMF8	IMF9	IMF10
K = 1	3.7512	0	0	0	0	0	0	0	0	0
K = 2	$9.0\times {10}^{-5}$	3.7511	0	0	0	0	0	0	0	0
K = 3	$2.0\times {10}^{-6}$	$1.0\times {10}^{-4}$	3.7510	0	0	0	0	0	0	0
K = 4	$6.0\times {10}^{-7}$	$5.0\times {10}^{-6}$	$2.0\times {10}^{-4}$	3.7510	0	0	0	0	0	0
K = 5	$3.0\times {10}^{-7}$	$2.0\times {10}^{-6}$	$3.0\times {10}^{-5}$	$5.0\times {10}^{-4}$	3.7507	0	0	0	0	0
K = 6	$2.0\times {10}^{-7}$	$7.0\times {10}^{-7}$	$3.0\times {10}^{-6}$	$3.0\times {10}^{-5}$	$5.0\times {10}^{-4}$	3.7507	0	0	0	0
K = 7	$2.0\times {10}^{-7}$	$5.0\times {10}^{-7}$	$2.0\times {10}^{-6}$	$8.0\times {10}^{-6}$	$4.0\times {10}^{-5}$	$5.0\times {10}^{-4}$	3.7506	0	0	0
K = 8	$4.0\times {10}^{-8}$	$1.0\times {10}^{-7}$	$5.0\times {10}^{-7}$	$2.0\times {10}^{-6}$	$8.0\times {10}^{-6}$	$4.0\times {10}^{-5}$	$5.0\times {10}^{-4}$	3.7506	0	0
K = 9	$4.0\times {10}^{-8}$	$1.0\times {10}^{-7}$	$2.0\times {10}^{-7}$	$6.0\times {10}^{-7}$	$2.0\times {10}^{-6}$	$9.0\times {10}^{-6}$	$4.0\times {10}^{-5}$	$5.0\times {10}^{-4}$	3.7506	0
K = 10	$5.0\times {10}^{-8}$	$1.0\times {10}^{-7}$	$2.0\times {10}^{-7}$	$4.0\times {10}^{-7}$	$9.0\times {10}^{-7}$	$2.0\times {10}^{-6}$	$9.0\times {10}^{-6}$	$4.0\times {10}^{-5}$	$5.0\times {10}^{-4}$	3.7506

shows the center frequency distribution of VMD decomposition under different modal numbers. From the table data, it can be seen that, when the number of IMFs is increased to 5, the center frequency shows a more stable distribution, and the change in the center frequency is not significant when the number of IMFs is further increased, which suggests that the decomposed modal components can effectively separate out the different frequency components in the signal without modal overlapping or over-decomposition when K = 5. Too many modal components will increase the computational complexity, while too few modal components may not be able to fully capture the details of the signal. Therefore, we choose to decompose the original price series into 5 IMFs and 1 residual.

In Figure 3, the original price sequence is represented by the black line, the sub-series by blue lines, and the residue by the red line. This decomposition separates the characteristics of the original price sequence, enabling multiscale analysis. The frequency decreases from IMF1 to IMF5: high-frequency sub-series such as IMF1 and IMF2 reflect short-term rapid fluctuations in the price sequence; medium-frequency sub-series like IMF3 contain transitional information from high to low frequencies; and low-frequency sub-series like IMF4 and IMF5 capture long-term trends and major features of the original sequence.

3.5. Variable Selection Based on Correlation

Figure 4 illustrates the correlation coefficients between variables obtained using the Pearson correlation method. The coefficients range within [−0.2, 1], where a deeper red indicates a stronger positive correlation and a deeper blue indicates a stronger negative correlation between the variables. Lighter colors indicate weaker correlations.

From Figure 4, it is observed that the variables with strong correlations include the open price, close price, high price, low price, IMF5, and the price itself, all exhibiting correlation coefficients above 0.9 due to their reflection of price data changes. Additionally, the import volume shows a correlation coefficient of 0.66 with price-related variables. Strong correlations are also evident among the production, storage volume, and trading volume. Most variables exhibit weak correlations, with a few variables displaying negative correlations, such as the production and import volume, reflecting an inverse relationship where increased production enhances the supply and may weaken the import volume.

We translate the last row or column data from the correlation matrix into groups to generate

Table 3. Correlation table.

Correlation Group	Variable	Correlation Coefficient
Strong correlation group	Low price	0.99955
	High price	0.99946
	Open price	0.99944
	Close price	0.99943
	IMF5	0.94435
Moderate correlation group	Import volume	0.66065
	IMF4	0.43602
	IMF3	0.17683
Weak correlation group	IMF2	0.07291
	IMF1	0.03973
	Amplitude	0.00880
Negative correlation group	Storage volume	−0.00547
	Trading volume	−0.01164
	Export volume	−0.02157
	Production	−0.07833

, displaying the correlation levels between different variables and prices. The variables are grouped based on their correlation strength with prices into the strong correlation, moderate correlation, weak correlation, and negative correlation groups. The strong correlation group’s correlation coefficients range within (0.7, 1]; it contains variables that are highly correlated with the target variable and can be considered almost linear. The medium correlation group’s correlation coefficients range within (0.1, 0.7]; it contains variables that have some correlation but not enough to be considered strongly correlated. This interval was chosen to capture variables that may have a potential effect on the target variable but are not significant. The weak correlation group’s range of correlation coefficients is (0, 0.1], which indicates that the variable is correlated with the target variable but at a low level. For the negative correlation group, the correlation coefficients range within [−1, 0].The correlation coefficient of 0.17683 for IMF3 indicates a weak correlation but is still above the threshold for weak correlations and therefore it cannot be ignored completely. Its categorization into the medium correlation group is intended to retain this part of the potential impact in the model and avoid losing important information due to over-screening.

4. Results

4.1. Model Selection Experiment

Due to the different characteristics of the variable data in the different correlation groups, the optimal predictive model may vary. Therefore, this section applies GRU, MLP, RNN, CNN, AE, LSTM, and MLP models to predict the three correlation groups. By comparing the prediction errors, the most suitable models for each correlation group are selected.

Table 4. Strong correlation model errors and rankings.

Model	MAE	MAPE	MSE	RMSE	${\mathit{R}}^2$	AR
GRU	0.1127	3.07%	0.0287	0.1695	99.62%	96.93%
BP	0.1144	3.26%	0.0301	0.1734	99.57%	96.74%
RNN	0.1543	4.16%	0.0545	0.2334	99.23%	95.84%
CNN	0.1550	4.25%	0.0537	0.2318	99.25%	95.75%
LSTM	0.1790	4.82%	0.0711	0.2666	99.00%	95.18%
AE	0.1770	4.85%	0.0679	0.2605	99.05%	95.15%
MLP	0.2230	5.77%	0.1132	0.3364	99.01%	94.23%
Rank	MAE	MAPE	MSE	RMSE	${\mathbf{R}}^{\mathbf{2}}$	AR
GRU	1	1	1	1	1	1
BP	2	2	2	2	2	2
RNN	3	3	4	3	3	3
AE	5	6	5	5	5	6
LSTM	6	5	6	6	7	5
MLP	7	7	7	7	6	7

,

Table 5. Medium correlation model errors and rankings.

Model	MAE	MAPE	MSE	RMSE	${\mathit{R}}^2$	AR
AE	1.1581	37.76%	2.1102	1.4527	68.75%	62.24%
MLP	1.1941	42.00%	2.2114	1.4871	71.69%	58.00%
LSTM	1.2051	40.55%	2.2368	1.4956	68.14%	59.45%
BP	1.2395	43.26%	2.2820	1.5106	68.44%	56.74%
GRU	1.2476	44.15%	2.3408	1.5300	70.07%	55.85%
RNN	1.2661	46.03%	2.3880	1.5453	71.80%	53.97%
CNN	1.2746	46.64%	2.3462	1.5317	72.30%	53.36%
Rank	MAE	MAPE	MSE	RMSE	${\mathbf{R}}^{\mathbf{2}}$	AR
AE	1	1	1	1	7	1
MLP	2	3	2	2	6	3
LSTM	3	2	3	3	2	2
BP	4	4	4	4	5	4
GRU	5	5	5	5	1	5
RNN	6	6	6	7	4	6
CNN	7	7	7	6	3	7

and

Table 6. Weak correlation model errors and rankings.

Model	MAE	MAPE	MSE	RMSE	${\mathit{R}}^2$	AR
MLP	1.2579	44.79%	2.3546	1.5345	64.96%	55.21%
LSTM	1.2964	41.35%	2.8063	1.6752	44.92%	58.65%
CNN	1.3023	48.14%	2.3895	1.5458	70.49%	51.86%
BP	1.3270	42.80%	2.8266	1.6812	55.17%	57.20%
RNN	1.4733	50.64%	3.1759	1.7821	39.56%	49.36%
GRU	1.5476	53.72%	3.4118	1.8471	37.68%	46.28%
AE	2.0208	48.27%	7.5699	2.7514	51.00%	51.73%
Rank	MAE	MAPE	MSE	RMSE	${\mathbf{R}}^{\mathbf{2}}$	AR
MLP	1	3	1	1	2	3
LSTM	2	1	3	3	5	1
CNN	3	4	2	2	1	4
BP	4	2	4	4	3	2
RNN	5	6	5	5	6	6
GRU	6	7	6	6	7	7
AE	7	5	7	7	4	5

present the evaluation metrics for prediction with various models for each correlation group. The upper part shows numerical values, and the lower part ranks the models based on the prediction errors under each evaluation metric.

Table 4 displays the prediction errors and rankings for the strong correlation group using different models. Considering all factors, the GRU model is selected for the prediction of data in the strong correlation group.

Table 5 evaluates various models for the prediction of variables in the medium correlation group. The AE model is chosen based on comprehensive consideration for the prediction of data in the medium correlation group.

Table 6 evaluates various models for the prediction of variables in the weak correlation group. The MLP model is selected based on comprehensive consideration for the prediction of data in the weak correlation group.

After an internal comparison within Table 4, Table 5 and Table 6 to select models suitable for the different correlation groups, vertical comparisons across these tables reveal that the prediction effectiveness is the highest for the strong correlation group, achieving an accuracy rate of over 90%, while the other two groups have accuracies of only 50% to 60%.

4.2. Robustness Test

The robustness of the samples is tested by dividing them into in-sample training and out-of-sample prediction groups. In-sample training involves using the model to train on 85% of the training set data and predict the results. Out-of-sample prediction involves using the model to predict on the remaining 15% test set data. To eliminate randomness, each experiment is conducted 10 times, and the average values are shown in

Table 7. Robustness test.

Experimental Group	MAE	MAPE	MSE	RMSE	${\mathit{R}}^2$	AR
In-sample	0.2587	7.29%	0.4119	0.6414	95.78%	92.71%
Out-of-sample	0.2961	9.62%	0.4012	0.6328	94.18%	90.38%

.

Table 7 compares the results of in-sample training and out-of-sample prediction. They are close, but in-sample training involves optimizing the model by adjusting the parameters and structure, while out-of-sample prediction tests the model’s performance on entirely new data, thus being slightly inferior to in-sample training. Figure 5 illustrates a comparison of 10 predictions each for in-sample training and out-of-sample prediction. It shows that the in-sample training results closely align with the actual values, especially when the data undergo sudden changes, whereas the out-of-sample prediction results show greater deviations from the actual values.

Table 8. Variance and confidence intervals.

	In-Sample		Out-of-Sample
Pred	Variance	Confidence Interval	Variance	Confidence Interval
Pred_1	0.4122	[−0.0027, 0.0266]	0.4484	[−0.1141, −0.0412]
Pred_2	0.4405	[0.0083, 0.0386]	0.3642	[−0.1136, −0.0479]
Pred_3	0.4537	[0.0096, 0.0404]	0.4321	[−0.1223, −0.0508]
Pred_4	0.4046	[−0.0047, 0.0243]	0.4438	[−0.0934, −0.0209]
Pred_5	0.4091	[−0.0044, 0.0248]	0.3675	[−0.1055, −0.0395]
Pred_6	0.3753	[−0.0095, 0.0184]	0.415	[−0.1122, −0.0421]
Pred_7	0.4063	[−0.0058, 0.0233]	0.3504	[−0.1044, −0.04]
Pred_8	0.4488	[0.0076, 0.0382]	0.3655	[−0.1135, −0.0477]
Pred_9	0.3855	[−0.003, 0.0253]	0.4494	[−0.134, −0.061]
Pred_10	0.3809	[−0.0083, 0.0198]	0.4004	[−0.1326, −0.0638]

demonstrates the in-sample and out-of-sample variances and confidence intervals for the 10 experiments. The in-sample variance ranges from 0.3753 to 0.4537 with small fluctuations, with Pred6 being the most stable and Pred3 the most volatile. The confidence intervals are narrower, such as for Pred6, which has a small and precise prediction error, and Pred9, which is also narrower but close to zero, indicating a concentrated distribution of errors. The out-of-sample variance ranges from 0.3504 to 0.4494, with slightly larger fluctuations than in the in-sample case. The confidence intervals are generally wide and mostly negative, such as [−0.1141, −0.0412] for Pred1, indicating a wide margin of error and the possible underestimation of the actual value. Overall, the degree of volatility is similar inside and outside the sample, but the stability outside the sample is slightly lower.

4.3. Correlation Comparison Experiment

The models selected in Section 4.1 are used to predict the strong, medium, and weak correlation groups separately. Specifically, the GRU is used to predict the strong correlation group’s test set data, the AE for the medium correlation group’s test set data, and the MLP for the weak correlation group’s test set data. The prediction results are shown in Figure 6 and

Table 9. Comparison of experimental errors in correlation.

Correlation Group	MAE	MAPE	MSE	RMSE	${\mathit{R}}^2$	AR
Strong	0.1127	3.07%	0.0287	0.1695	99.62%	96.93%
Medium	1.1581	37.76%	2.1102	1.4527	68.75%	62.24%
Weak	1.2579	44.79%	2.3546	1.5345	64.96%	55.21%

.

Figure 6 depicts the prediction results for the different correlation groups. The black line represents the actual natural gas futures prices, the blue line represents the predictions for the strong correlation group using the GRU model, the red line represents the predictions for the medium correlation group using the AE model, and the yellow line represents the predictions for the weak correlation group using the MLP model. It can be observed that the blue line closely overlaps with the black line, with slight variations during changes. The red line reflects the long-term trend of the black line’s variations, while the yellow line reflects the short-term fluctuations in the black line’s variations. Thus, the predictions for the strong correlation group closely match the actual data, the predictions for the medium correlation group reflect the long-term characteristics of the actual data, and the predictions for the weak correlation group reflect short-term data fluctuations.

Table 9 presents the error results for the three correlation groups. The first four evaluation metrics (MAE, MAPE, MSE, RMSE) should be minimized, while the last two ($R^2$, AR) should approach 1 for better performance. Comparing the MAE, MAPE, MSE, and RMSE among the three groups reveals that the prediction errors for the medium and weak correlation groups are approximately 10 times higher than those for the strong correlation group. The MAE, MAPE, MSE, and RMSE for the medium and weak correlation groups are quite similar, indicating significantly reduced prediction accuracy. Comparing the $R^2$ and AR values, the prediction error for the strong correlation group approaches 1, indicating very high accuracy and a negligible difference from the actual data. In contrast, the prediction errors for the medium and weak correlation groups are approximately half those for the strong correlation group, indicating similar results with significantly reduced prediction accuracy. Based on the comparison of the prediction error values, the prediction accuracy for the strong correlation group is ten times higher than that for the other two groups, and the accuracy for the medium and weak correlation groups is reduced by half compared to the strong correlation group.

Figure 7 shows the convergence curve of the GRU model with 20 iterations and a loss function of MSE. The strong correlation group dataset is used, of which 85% is used as the training set and then 20% is separated from it as the validation set. The loss values for both the training set (blue line) and the validation set (red line) decrease with the number of iterations, decreasing rapidly at the beginning and slowing down at the end, indicating that the model gradually converges. The loss of the validation set is lower than that of the training set, showing that the model is not overfitted, has a strong generalization ability, learns data features effectively, and performs well on new data.

According to the test set evaluation error metrics in Table 9, the errors of the medium correlation group and the weak correlation group in the first four metrics are about 10 times those of the strong correlation group, while the errors of the medium correlation group and the weak correlation group in the last two metrics are only half those of the strong correlation group. This suggests that the strong correlation group has significantly higher prediction accuracy for most indicators, especially for key indicators, where the errors are much lower than those of the other groups. Therefore, the strong correlation group is assigned an 80% weight to ensure that it plays a dominant role in the overall prediction. The medium correlation group and the weak correlation group also contribute to some indicators, but their overall performance is not as strong as that of the strong correlation group, and their error values are not very different, so each group is assigned a 10% weight. The final weighting ratio is 8:1:1, which highlights the advantages of the strong correlation group and retains the contributions of the other groups to ensure that the prediction results are mainly driven by the strong correlation group, while taking into account the overall prediction ability of the model. The formula for the determination of the weights is given in (19).

$$Y=0.8y_1+0.1y_2+0.1y_3$$

(19)

4.4. Ablation Experiment

To demonstrate the effectiveness of this model, four sets of experiments were designed with each group, removing one of the correlation groups to assess its overall impact. The purpose of the “strong + medium” group, “strong + weak” group, and “medium + weak” group was to test the effects of the weak, medium, and strong correlation groups, respectively, on the prediction. The “full model” group included input variables from all correlation groups, aiming to evaluate the optimization level of the model.

Figure 8 displays the prediction results for the first three experimental groups. The black line represents the actual values, the blue line represents the predictions using the strong + medium group variables, the red line represents the predictions using the strong + weak group variables, and the green line represents the predictions using the medium + weak group variables. From Figure 8, it can be observed that the blue line deviates the least from the black line, indicating the highest accuracy. The green line shows the greatest deviation, especially at price inflection points, while the red line falls between the other two. This suggests that the predictions are the least accurate with the medium + weak group variables, followed by the strong + weak group, and they are the most accurate with the strong + medium group variables. This also confirms that the GRU predictions for the strong correlation group contribute the most to the predictions, whereas the predictions using the AE for the medium correlation group and the MLP for the weak correlation group are less effective and similar in performance.

Table 10. Comparison of errors in ablation experiments.

Experimental Group	MAE	MAPE	MSE	RMSE	${\mathit{R}}^2$	AR
Strong + Medium	0.4361	13.91%	0.6654	0.8157	90.25%	86.09%
Strong + Weak	0.4231	14.75%	0.6071	0.7792	91.25%	85.25%
Medium + Weak	1.2015	41.06%	2.1892	1.4796	67.42%	58.94%
Full model	0.3243	10.17%	0.4663	0.6828	93.11%	89.83%

presents the error evaluation metrics for the four experimental groups in the sensitivity analysis. Comparing the MAE, MAPE, MSE, and RMSE, the medium + weak group exhibits the highest errors, ranging from 2 to 10 times higher than the other experiments. The errors for the strong + weak, strong + medium, and full model groups are relatively close, but, overall, the full model outperforms the strong + weak group, which in turn outperforms the strong + medium group. Comparing the $R^2$ and AR across the four groups, the predictions for the medium + weak group are significantly less accurate than those of the other three groups, while the remaining three groups show similar performance, with the full model being superior to the strong + medium group, which is superior to the strong + weak group.

The ablation experiment demonstrates that the full model group achieves the highest prediction accuracy and best overall performance. Within the predictive model structure, the GRU predictions for the strong correlation group are identified as the most valuable, followed by the predictions using the AE for the medium correlation group and the MLP for the weak correlation group.

4.5. Step Length Experiment

This section explores the impact of the time step length on the prediction results. Three experimental groups were set up with sliding time windows of 3, 5, 7, 9, 11, and 13 time steps for the input sequences, while keeping the other settings constant.

In Figure 9, the black line represents the actual data, and the blue solid line, red solid line, light blue solid line, green solid line, purple solid line, and yellow solid line represent the sliding time windows of 3, 5, 7, 9, 11, and 13 steps, respectively. The light blue line follows the black line, indicating that the predicted value for the seven-step window is closest to the actual value. The largest deviations from the black line are the blue and yellow lines, indicating that the 3- and 13-step prediction windows are the least accurate, and that a too long or too short window affects the prediction results. Combined with

Table 11. Comparison of time window errors.

Experimental Group	MAE	MAPE	MSE	RMSE	${\mathit{R}}^2$	AR
3-step	0.4205	13.43%	0.5677	0.7535	91.72%	86.57%
5-step	0.3245	10.18%	0.4677	0.6839	93.12%	89.82%
7-step	0.2941	9.54%	0.4248	0.6518	93.85%	90.46%
9-step	0.3760	10.28%	0.6487	0.8054	90.34%	89.72%
11-step	0.4122	12.31%	0.4898	0.6999	92.77%	87.69%
13-step	0.4514	13.58%	0.5521	0.7431	91.85%	86.42%

(which lists the errors in detail), it can be seen that the prediction error decreases and then increases with the increase in the time step, which indicates that a time window with a certain length will allow the prediction to generate the optimal results, an increase in the optimal step will lead to redundancy in the amount of data, and the model will not be able to find the key points in learning the data. Moreover, a decrease in the optimal step will lead to a reduction in the amount of data and reduce the feature richness of the data, and the model will have limited access to information, leading to a decrease in the prediction performance. From the results in Table 11, the optimal step size range is around 7.

4.6. Experiment with Addition of Negative Variable

This section explores the predictive effects of variables negatively correlated with prices. A control experiment is set up where one group uses the predictions from this model, while another group adds the negatively correlated variables as input variables to this model. Section 4.1 compares the prediction errors of the negatively correlated group using the MLP model for the prediction of negatively correlated variables.

Figure 10 presents the prediction results for the experimental and control groups. The black line represents the actual results, the red line represents the predictions from the experimental group using this model, and the blue line represents the predictions from the control group after adding negatively correlated group variables. From Figure 10, it can be seen that both experimental groups have varying degrees of error compared to the actual results. The vertical colored lines in Figure 10 indicate the magnitude of the errors, with the addition of negatively correlated group variables resulting in decreased prediction accuracy, demonstrating that factors that are negatively correlated with the output variable reduce the prediction accuracy.

Table 12. Comparison of errors with negative variables added.

Experimental Group	MAE	MAPE	MSE	RMSE	${\mathit{R}}^2$	AR
Full Model	0.3243	10.17%	0.4663	0.6828	93.11%	89.83%
Add negative correlation	0.4615	14.12%	0.8383	0.9156	87.21%	85.88%

shows the prediction errors for the two experimental groups considered in this section. Comparing these results reveals that adding negatively correlated group variables significantly increases the prediction errors compared to the full model experimental group without these variables, thus validating the hypothesis.

5. Conclusions and Discussion

This paper proposes a VMD-GRU-AE-MLP-RF hybrid model for the forecasting of natural gas futures. The approach involves decomposing the price original series using VMD and utilizing the decomposed sub-series along with influencing factors as the input variables. By conducting a correlation analysis, the input variables are categorized into groups based on their degrees of correlation. Different prediction models—the GRU, AE, and MLP—are then applied to forecast each group according to their respective characteristics. Finally, an RF model integrates the predictions from the three groups of sub-results to generate the final forecast, effectively enhancing the prediction accuracy and reducing forecasting errors.

The innovation of this study lies in enriching the input variable information by incorporating sub-series with different frequencies and influencing factors. Furthermore, the adaptation of different prediction models to different input dimensions enhances the model flexibility. The use of the RF model to integrate diverse prediction outputs is another innovative aspect, contributing to improved forecasting performance. Additionally, we paper conducted numerous experiments to empirically analyze and validate the proposed model, and, according to the evaluation indices, all of them achieve good prediction results. For example, in the case of the RMSE, the best result in the model selection experiment was 0.1695; the best result in the stability test was 0.6328; the best result in the correlation experiment was 0.1695; the best result in the ablation experiment was 0.7792; the best result in the time span experiment was 0.6503; and the best result in the all-variable experiment was 0.6828. These experimental results effectively prove the feasibility and validity of the model.

In contrast to previous research methods, the model proposed in this study no longer relies solely on a single advantage, such as enriching data features only through serial decomposition methods, adding only factors that affect prices to aid in forecasting, or relying only on advanced models to improve the forecasting accuracy. Instead, this model employs a variety of advanced algorithms to optimize every aspect of the analytical framework.

In terms of findings, the model screening test employs eight different models to predict three groups of input variables with different degrees of correlation and selects the model that best matches the input variables through the prediction error. This method is more flexible and adaptable than the use of a single model to predict the correlation groups of the input variables. Robustness experiments were also used to comprehensively examine both in-sample and out-of-sample data, and the results showed that this multilink model prediction structure was able to output more stable results, further proving its superiority. The correlation and ablation experiments deeply analyzed the prediction results of different models for the corresponding correlation groups, and the results showed that the high correlation group had the best prediction effect due to its close correlation with the price; the prediction results for the medium correlation group mainly presented the long-term trends of the data; and the prediction results of the low correlation group reflected the short-term trends of the data. By presenting the different information contained in the prediction results in groups, the present study provides a superior perspective for an understanding of the internal logic of the prediction results. In addition, the step size experiment explored the effect of the length of the sliding time window on the prediction results, and it was found that the longer the sliding time window, the richer the historical information included, but too long a window will instead make it difficult for the model to recognize important information. This type of experiment is less common in previous studies, and it provides a new perspective for the design of the prediction model. Finally, a negative variable experiment was conducted by adding negative correlation group data to the input variables, and the results showed that deleting the negative correlation group data could significantly improve the prediction accuracy. This indicates that eliminating inappropriate input variables can optimize the prediction effect. Most of the previous research has “added” the input variables through subsequence decomposition or adding related factors, while this study further optimized the quality of the input variables through “subtracting” the input variables, which provides a direction for the improvement of prediction models.

The hybrid VMD-GRU-AE-MLP-RF model proposed in this paper shows high accuracy and flexibility in natural gas futures forecasting, but there are some limitations. First, the number of subsequences generated in the price decomposition part affects the input variable grouping. Second, changes in the weights assigned to the final forecasts of each correlation group of the input variables can also affect the final results, but the current model does not yet provide a mechanism to dynamically adjust the weights.

The future research directions include the following: firstly, relevant algorithmic mechanisms can be introduced to adapt to different data sets by assigning weights according to the prediction error; secondly, the data characteristics can be enriched by introducing more influencing factors.

Natural gas futures prediction modeling not only aids in market decision-making but also provides support for sustainable development goals. By accurately predicting the natural gas price trend, the model can provide data support for policymakers to optimize the energy structure and reduce the dependence on traditional fossil fuels. In addition, the optimization directions of the model, such as dynamic weight adjustment and multimodal data fusion, are in line with the requirements of sustainable development in terms of flexibility and adaptability. Future research could further explore the application of the model in energy transition and ecosystem protection and promote the deep integration of natural gas futures forecasting with sustainable development goals.