A novel hybrid optimization ensemble learning approach for energy futures price forecasting

Abstract

Effective energy futures price prediction is an important work in the energy market. However, the existing research on the application of “decomposition-prediction” framework still has shortcomings in noise processing and signal reconstruction. In view of this, this paper first uses PSO to optimize VMD to improve the effectiveness of single decomposition, and further uses SGMD to capture the remaining key information after extracting low-frequency modal components by using PSO-VMD technology. Further, combined with LSTM to predict each component, a new PSO-VMD-SGMD-LSTM hybrid model is innovatively constructed. The empirical research results based on the real energy market transaction price show that compared with the benchmark model, the hybrid model proposed in this paper has obvious forecasting advantages in different forecasting scenarios.

Keywords

Energy futures price forecast secondary decomposition technique long short term memory

1 Introduction

As a key component of the social futures market, the price fluctuation of energy futures has an impact on the changes of the world economic structure, and is related to the social and economic security and healthy development of all countries [6 , 48]. Natural gas and crude oil are key components of the energy market. In the development process of energy market, forecasting the price trend of energy futures is a task for investors and policy makers that cannot to be neglected [23 , 39].

However, the energy market is a typical complex dynamic system, and its price trend is affected by variety uncertain factors, including the social economy, the financial market and market relations of energy commodities, etc [3, 33]. The price data of the energy market is characterized by nonlinearity, high volatility and irregularity. Therefore, it is necessary to establish a model with sufficient learning ability to predict the price of energy futures effectively [1, 22].

The research on market price prediction of energy has become a key issue in the field of energy economy. According to the nature of input variables in forecasting methods, these forecasting methods can be divided into univariate research that only considers price data and multivariate research that covers other factors besides historical price data [46]. This paper belongs to category of univariate research. At present, the methods involved in forecasting energy prices in this field are mainly classified as follows:

The first category is the traditional econometric method. Common models include autoregressive integrated moving average (ARIMA), random walk (RW), and vector autoregressive (VAR), etc [19 , 54]. Nicolau and Palomba used the recursive bivariate VAR models to forecast the price of energy futures. The results show that there are differences in the dynamic relationship between spot prices and futures prices in different energy markets [31]. Liu and Shi proposed a new TZD-GARCH model to predict the price of crude oil futures contracts traded in the NYMEX, and verified the effectiveness of the proposed model in describing the volatility of price series [45].

The second category is the artificial intelligence method. Common models include support vector regression (SVR), random forest and LSTM, etc [9 , 34]. Among them, compared with the traditional machine learning algorithm, deep learning technology can continuously improve its performance with the increase of data scale, and has a stronger ability to extract data features. In recent years, it has been widely used in financial stock market and energy economy [8].

In the application research of financial stock market, Rostamian and O’Hara [2] took the financial transaction data formed by different currencies as the object, built CNN-LSTM model to study its forecasting ability within the framework of directional change, and verified the superiority of the proposed model in forecasting performance. Li et al. [57] combined ARIMA, Fourier transform and LSTM to construct a new framework 13f-LSTM to predict the movement trend of stock prices. Arin and Ozbayoglu [10] combined hybrid deep learning with option pricing model to achieve better pricing than Black-Scholes (BS). Zaheer et al. [43] put forward a hybrid deep learning forecasting model to predict the closing price and the highest price of the stock the next day, and compared it with CNN, RNN, LSTM, CNN-RNN and CNN-LSTM as benchmark models to verify the effectiveness of the model. Lin et al. [51] pay attention to the characteristics of stock data and data preprocessing, and put forward a method based on RNN to predict the opening price, closing price and the difference between them.

In the application research of energy economy, Gundu and Simon [47] put forward a neural network model of LSTM based on improved particle swarm optimization, and used it to predict the closing price of Indian energy exchange. Busar and Lim [13] used AdaBoost-LSTM and AdaBoost-GRU models to predict the fluctuation of crude oil prices, which verified the superiority of the proposed models. Li and Wang [24] constructed a new Saint GRU model to predict energy prices, and used Composite Multiscale Cross-Sample Entropy (CMSCE) algorithm to explore the synchronization between the predicted value and the real value, so as to verify the applicability of the proposed model.

It is worth mentioning that compared with other common network algorithms in deep learning, such as RNN and ANN, LSTM can handle long-distance dependencies in the sequence, reduce the problem of gradient explosion or disappearance, and has stronger robustness. Therefore, in the follow-up study, the LSTM model will be included as a component of the hybrid model.

The third category is the hybrid model. Among them, the “decomposition-prediction” integrated learning framework based on the combination of decomposition technology and prediction model has been as been widely used in recent years [7 , 56]. On the basis of wavelet packet decomposition (WPD), LSTM and stochastic time effective weight (SW) function, Wang and Wang [26] constructed a WPD-SW-LSTM hybrid model to predict the futures price of crude oil, and improved a new error measurement method to evaluate the prediction results of different models, which verified the effectiveness of decomposition technology in improving the prediction accuracy of crude oil prices. Wang and Wang [5] combined EMD with SW-GRU to forecast the futures and spot prices of crude oil and gasoline, and used various error criteria and double-scale complexity invariant distance to evaluate the prediction results of the model. The results showed that the introduction of signal processing method could effectively improve the prediction accuracy of the model. Lin et al. [53] used VMD, the autoregressive model (AR), Elman neural network and improved LSTM model to decompose and reconstruct the energy price, which further improved the performance of model prediction, which improved the forecasting performance of the model and verified the effectiveness and reliability of VMD technology in energy price forecasting and analysis.

Among the three different types of model studies reviewed above, the econometric method of the first type usually assumes that the research data object conforms to the assumption of approximate linearity and stability. However, in the real transaction of energy market, the futures price series contains a lot of noise. Econometric methods can’t effectively identify all kinds of complex factors that affect the price series, and it is difficult to grasp the nonlinear and high volatility characteristics of energy futures price series.

Compared with econometric methods, the second kind of artificial intelligence method can process complex data in parallel and adjust itself [14, 20], and can better describe any degree of mapping relationship. However, this method is easily influenced by parameters and model settings, and there is a probability of over-fitting.

Considering that all models have their limitations in practical application. It is impossible for a single model to perform best in all forecasting situations. Therefore, the third kind of hybrid model, which combines the advantages of various models, has become the mainstream of research. Among them, the Tel@ complex system methodology based on the “decomposition-prediction” framework is a typical representative, and its core advantage is to introduce decomposition technology to preprocess complex price series, obtain corresponding modal components in different time scales, reduce the irregularity of data, and further predict the results of each component in combination with the prediction model [17]. In recent years, it has been widely used in the forecasting research of energy economy [41, 50].

In the research based on the “decomposition-prediction” framework, decomposition technology is the key to preprocessing complex data. However, there are still the following problems to be solved urgently in the existing research:

Firstly, compared with common decomposition algorithms such as wavelet transform (WT) [4], and ensemble empirical mode decomposition (EEMD), etc [35, 42], VMD [29] can more accurately extract and separate different modal components through Wiener filtering and Hilbert transform, which can more effectively avoid modal mixing. However, in the process of signal reconstruction and restoration, VMD technology still has shortcomings in its ability to recover high-frequency signals.

Secondly, due to various complex factors, there are a lot of intermittent signals and noises in the price signal sequence of energy prices. The secondary decomposition technique combining the advantages of different decomposition techniques can better adapt to the non-stationary price data [27, 30]. However, in the process of combining different decomposition technologies, how to avoid excessive decomposition of signals [49], ensure the effectiveness and rationality of decomposition technologies, and build a decomposition technology more suitable for processing energy futures price data still needs further exploration.

Therefore, in view of the deficiency of existing research, this paper innovatively constructs a new PSO-VMD-SGMD-LSTM hybrid model to forecast the futures prices of natural gas and WTI crude oil. Specifically, this paper first optimizes VMD with PSO, and constructs a new PSO-VMD-SGMD secondary decomposition technology combined with SGMD [18] to overcome the shortcomings of the existing decomposition technology in signal pre-processing and signal reconstruction. Further, LSTM [40] model is used to predict the modal components obtained by decomposition. Compared with the existing research, the PSO-VMD-SGMD decomposition technology proposed in this paper can effectively overcome the shortcomings of the existing decomposition technology in signal reconstruction.

As a supplement to previous research on energy futures price forecasting, the innovations of this paper are as follows:

(1) Aiming at the shortcomings of the existing research on single decomposition technology, an optimized secondary decomposition technology is proposed to preprocess the energy futures price series.

The PSO-VMD-SGMD secondary decomposition technique firstly uses PSO-VMD to extract the low frequency components of the price series, and further uses SGMD to process the remaining modes. SGMD can effectively avoid and overcome the difficulties of mode aliasing and parameter setting sensitivity during data decomposition [16 , 55]. The results based on actual data show that compared with other decomposition techniques (WT, EMD, CEEMDAN, VMD and VMD-SGMD), PSO-VMD-SGMD secondary decomposition technique avoids excessive signal decomposition, and has higher reconstruction accuracy and robustness.

(2) Considering the superiority of deep learning model in dealing with price series, a new PSO-VMD-SGMD-LSTM energy futures price forecasting model is constructed by combining PSO-VMD-SGMD technology and LSTM model.

LSTM is applied to predict the modal components obtained by applying the PSO-VMD-SGMD sec-ondary decomposition technique. Compared with other forecasting models (ELM and BP), LSTM model has better learning and forecasting ability for nonlinear time series by selecting historical information and current information [12, 36]. The PSO-VMD-SGMD secondary decomposition technique reduces the difficulty of forecasting futures price series, and the hybrid model further combined with LSTM model has better prediction ability.

(3) The robustness of the PSO-VMD-SGMD-LSTM hybrid model is tested.

The proposed PSO-VMD-SGMD-LSTM hybrid model is applied to the multi-step ahead forecasting of NYMEX natural gas and WTI futures prices. The empirical results based on real transaction prices indicate that the results of PSO-VMD-SGMD-LSTM hybrid model based on e_MAE, e_MAPE and e_RMSE is superior to other benchmark models, no matter in the scenario of single-step or multi-step ahead forecasting.

Other parts of this paper are arranged as follows: Section 2 expounds the PSO, VMD, SGMD and LSTM models used in the hybrid model of PSO-VMD-SGMD-LSTM, and the concrete construction steps of the hybrid model of PSO-VMD-SGMD-LSTM. Section 3 discusses the prediction performance of different models based on real market transaction data for empirical research, and verifies the effectiveness of the hybrid model constructed. Finally, Section 4 is the conclusion of this paper.

2 Methodology

The PSO-VMD-SGMD-LSTM hybrid model consists of quadratic decomposition technique PSO-VMD-SGMD and LSTM. Firstly, the original futures price series is decomposed by PSO-VMD-SGMD, which reduces the complexity of the sequence and improves the interpretability, and further, the reconstructed sequence is predicted by combining the LSTM model, which significantly improves the performance of the prediction model.

2.1 PSO

Particles in PSO consist of three basic features, which are distributed as position, speed and fitness. Table 1 shows the specific parameters involved. Set the single particle as p_i (i = 1, 2, ⋯ , M), where the position is x_i (t) = [c_i (t) γ_i (t)] ^T and the velocity is v_i (t) = [Δc_i (t) Δγ_i (t)] ^T. The main formulas involved and their explanations are as follows:

Table 1
Symbol abbreviation of PSO

Parameter symbol Parameter meaning

M The size of PSO particle swarm

t _max The largest evolutionary algebra

w _max The largest inertia weight

w _min The smallest inertia weight

c The acceleration factors

ɛ The smallest inertia weight

C The numerical range of regularization

parameter

γ The kernel parameter

e_i (t) The fitness function

p _i The individual optimal value

Parameter symbol	Parameter meaning
M	The size of PSO particle swarm
t _max	The largest evolutionary algebra
w _max	The largest inertia weight
w _min	The smallest inertia weight
c	The acceleration factors
ɛ	The smallest inertia weight
C	The numerical range of regularization
	parameter
γ	The kernel parameter
e_i (t)	The fitness function
p _i	The individual optimal value

$\begin{matrix} v_{i} (t + 1) = w (t) v_{i} (t) + c_{1} r_{1} (p_{i}^{best} (t) - x_{i} (t)) \\ + c_{2} r_{2} (g^{best} (t) - x_{i} (t)) \end{matrix}$ (1) $x_{i} (t + 1) = x_{i} (t) + v_{i} (t + 1)$ (2) $w (t) = w {(wmi n_{\max} t / t_{\max})}_{\max}$ (3)

Where $p_{i}^{best} (t) = {[C_{i}^{best} (t) γ_{i}^{best} (t)]}^{T}$ , the population optimal value is $g_{i}^{best} (t) = {[C_{i}^{gbest} (t) γ_{i}^{gbest} (t)]}^{T}$ , r₁ and r₂ are randomly assigned between [0, 1]. The maximum iteration number t_max and fitness value are tested. The condition for terminating the optimization of parameters is that t_max satisfies the numerical condition or the fitness value is lower than the termination error. At last, the optimal parameters (C^best, γ^best) are obtained.

2.2 VMD

The decomposition process of VMD is as follows:

(1) A variational model is constructed. The original signal is decomposed into K components and the following constrained variational function is constructed. Where {w_k} = { w₁, ⋯ , w_k } is the central frequency spectrum of each mode; u_k is the k-th IMF component obtained by decomposition; ∂_t is the partial derivative of t; δ (t) is the impact function; * is the computational symbol for convolution. $\begin{matrix} \min_{{u_{k}}, {w_{k}}} {\sum_{k = 1}^{K} | [\partial_{t} (δ (t) + \frac{j}{π t}) * u_{k} (t)] e^{- j w_{k} t} ∥_{2}^{2}}, \\ s . t = \sum_{k = 1}^{K} u_{k} = f (t) \end{matrix}$ (4)

(2) The expression of augmented Lagrange function is as follows. $\begin{matrix} L ({u_{k}}, {w_{k}}, λ) = α \sum_{k = 1}^{K} ∥ [\partial_{t} (δ (t) + \frac{j}{π t}) * u_{k} (t)] \\ e^{- j w_{k} t} ∥_{2}^{2} + ∥ C (t) - \sum_{k = 1}^{K} u_{k} (t) ∥_{2}^{2} \\ + 〈 λ (t), C (t) - \sum_{k = 1}^{K} u_{k} (t) 〉 \end{matrix}$ (5)

Where α is the quadratic penalty factor and λ (t) is Lagrange multiplier.

(3) Initialize ${u_{k}^{1}}$ , ${w_{k}^{1}}$ and {λ¹}, let n = 1. In the Fourier transform domain, ${{\hat{u}}_{k}^{n + 1}}$ , ${{\hat{w}}_{k}^{n + 1}}$ and ${{\hat{λ}}^{n + 1}}$ are updated by alternating direction multiplier algorithm, where τ Is the degree of noise tolerance. ${\hat{u}}_{k}^{n + 1} (w) = \frac{\overset{\land}{C} (w) - \sum_{i \neq k} {\hat{u}}_{i} (w) + \frac{\hat{λ} (w)}{2}}{1 + 2 α {(w - w_{k})}^{2}}$ (6) ${\hat{w}}_{k}^{n + 1} = \frac{\int_{0}^{\infty} w {| {\hat{u}}_{k} (w) |}^{2} dw}{\int_{0}^{\infty} {| {\hat{u}}_{k} (w) |}^{2} dw}$ (7) ${\hat{λ}}^{n + 1} = {\hat{λ}}^{n} (w) + τ [\overset{\land}{C} (w) - \sum_{k = 1}^{K} | {\hat{u}}_{k}^{n + 1} (w) |]$ (8)

(4) The update will be stopped after the stop conditions are met. The stop conditions are as follows. $\sum_{k = 1}^{K} \frac{{∥ {\hat{u}}_{k}^{n + 1} - {\hat{u}}_{k}^{n} ∥}_{2}^{2}}{{∥ {\hat{u}}_{k}^{n} ∥}_{2}^{2}} < ɛ (ɛ > 0)$ (9)

2.3 SGMD

The main realization process of SGMD is as follows.

For any original signal x = (x₁, x₂, ⋯ , x_n), n is the data length. According to Takens theorem, the signal is reconstructed to obtain the following trajectory matrix X. $X = [\begin{matrix} x_{1} \\ ⋮ \\ x_{m} \end{matrix} \begin{matrix} x_{1 + τ} \\ ⋮ \\ x_{m + τ} \end{matrix} \begin{matrix} \dots \\ ⋱ \\ \dots \end{matrix} \begin{matrix} x_{1 + (d - 1) τ} \\ ⋮ \\ x_{m + (d - 1) τ} \end{matrix}]$ (10)

Where d is the embedding dimension, τ is the delay time, and m = n - (d - 1) τ. Let A = X^TX, and construct the following Hamilton matrix M. $M = [\begin{matrix} A^{T} \\ 0 \end{matrix} \begin{matrix} 0 \\ - A \end{matrix}]$ (11)

Let N = M², where M and N are Hamilton matrices, and construct the following symplectic orthogonal matrix Q. $Q^{T} NQ = [\begin{matrix} B \\ 0 \end{matrix} \begin{matrix} R \\ A^{T} \end{matrix}]$ (12)

Where B is the upper triangular matrix. The eigenvalues of the matrix B are λ₁, λ₂, ⋯ , λ_d. $σ_{i} = \sqrt{λ_{i}} (i = 1, 2, \dots, d)$ is the eigenvalue of matrix A, Q_i is the eigenvector corresponding to matrix A. Furthermore, $S_{i} = Q_{i}^{T} X^{T}$ , Z_i = Q_iS_i, where S is the conversion coefficient matrix, Z is the reconstructed trajectory matrix. $Z = Z_{1} + Z_{2} + \dots + Z_{d}$ (13)

The reconstructed matrix Z_i (1 ≤ i ≤ d) is transformed into a set of symplectic geometric components Y_i with length n by diagonal averaging. In which Z_m×d = (z_ij) _m×d, 1 ≤ i ≤ m, 1 ≤ j ≤ d. Let d^* = min(m, d), m^* = max(m, d), n = m + (d - 1) τ. When m < d, $y_{ij}^{*} = z_{ij}$ ; Otherwise, $y_{ij}^{*} = z_{ji}$ . The form of diagonal average transformation matrix is as follows. $y_{k} = {\begin{matrix} \frac{1}{k} \sum_{p = 1}^{k} y_{p, k - p + 1}^{*}, 1 \leq k \leq d^{*} \\ \frac{1}{d^{*}} \sum_{p = 1}^{d^{*}} y_{p, k - p + 1}^{*}, d^{*} \leq k \leq m^{*} \\ \frac{1}{n - k + 1} \sum_{p = k - m^{*} + 1}^{n - m^{*} + 1} y_{p, k - p + 1}^{*}, m^{*} \leq k \leq n \end{matrix}$ (14)

Further, the initial SGC components of group d are determined, namely [Y₁, Y₂, ⋯ , Y_d], where Y_i = (y₁, y₂, ⋯ , y_n). The initial SGC components are not completely independent and may have the same characteristics. That is, the first component SGC₁ is obtained by adding the first component Y₁ and its related components, and the remaining matrix is represented as G₁. The first column of G₁ and its related components are represented as SGC₂. Finally, the residual signal is assigned as res, where n is the number of iterations. $S = \sum SGC (n) + res$ (15)

2.4 LSTM

The modal obtained by PSO-VMD-SGMD decomposition technology has obvious trend components, showing nonlinear and non-stationary characteristics. As a special RNN, LSTM can capture the changing rules of sequences within parameters. Table 2 shows the related symbol abbreviations. The specific formula is as follows:

Table 2
Symbol abbreviation of LSTM

Parameter symbol	Parameter meaning
f _t	The forgetting gate at time t
σ (·)	The sigmoid activation function
X _t	The input of the prediction model
h _t-1	The output vector at t - 1 (also the
	state h vector)
W _f	The weight matrix
b _f	The deviation of the forgetting gate
[·]	The vector connection operator
i _t	The input gate
o _t	The output gate
c _t	The current state

f_{t} = σ (W_{f} \cdot [h_{t - 1}, X_{t}] + b_{f})

(16)

i_{t} = σ (W_{i} \cdot [h_{t - 1}, X_{t}] + b_{i})

(17)

\tilde{c_{t}} = \tanh (W_{c} \cdot [h_{t - 1}, X_{t}] + b_{c})

(18)

c_{t} = f_{t} * c_{t - 1} + i_{t} * \tilde{c_{t}}

(19)

o_{t} = σ (W_{0} \cdot [h_{t - 1}, X_{t}] + b_{0})

(20)

h_{t} = o_{t} * \tanh (c_{t})

(21)

Where W_i and b_i represent the weight matrix and bias of the input gate, respectively. The forgetting gate f_t determines the information components saved from the previous state c_t-1. Where tanh (·) represents the tanh activation function. h_t is the final output form of LSTM unit.

2.5 The PSO-VMD-SGMD-LSTM hybrid model

Figure 1 is the frame diagram of the PSO-VMD-SGMD-LSTM hybrid model, which includes the decomposition part of data, the prediction part and the comparison part of results. The decomposition part is completed by PSO-VMD-SGMD technology, and the prediction part is completed by LSTM model. The specific steps are described as follows:

Fig. 1

Flow chart of PSO-VMD-SGMD-LSTM hybrid model.

Step 1: Decomposition of PSO-VMD. The VMD technology optimized by PSO algorithm is used to decompose the original sequence to obtain the main low-frequency part and residual components. The number of PSO-VMD levels is set to 3. The main low-frequency parts extracted from the original data are represented as IMF1 and IMF2, respectively. The remaining high-frequency parts are given by the difference between the original sequence and IMF1 and IMF2.

Step 2: Apply SGMD to decompose the obtained residual components to obtain a series of independent symmetric geometric components (SGC) and the residual term. PSO-VMD-SGMD secondary decomposition technology integrates the advantages of VMD and SGMD technology to avoid noise interference in the original price signal and obtain more satisfactory decomposition results. Figure 2 shows the modal components obtained by applying the PSO-VMD-SGMD secondary decomposition technology to natural gas price data. Accordingly, Fig. 3 shows the results of WTI crude oil prices. The order of all modal components is from low frequency to high frequency.

Fig. 2

Components obtained by applying PSO-VMD-SGMD technology (natural gas price).

Fig. 3

Components obtained by applying PSO-VMD-SGMD technology (WTI crude oil price).

Step 3: The LSTM model is used to predict a series of modes obtained by the decomposition technology. The predicted values of all modes are added to obtain the final prediction result. The LSTM model can predict the modal components with different characteristics, ensuring the accuracy of the results and avoids the complexity of the model.

Step 4: Obtain the final prediction result. Based on the above steps, the numerical results of forecasting based on PSO-VMD-SGMD-LSTM model are obtained, and further compared with a series of benchmark models to verify the effectiveness of the proposed model.

3 Results and discussion

3.1 Analysis of sample data

Natural gas and crude oil are important energy components in the world energy market. This paper selects the weekly futures prices of natural gas traded on the NYMEX and WTI crude oil for case study, and the data are from (https://www.eia.gov/index.php). Specifically, the futures price of natural gas is set as Case I, and the time range of sample data is from January 14th, 1994 to June 10th, 2022, with a total of 1483 samples. The futures price data of WTI crude oil is set as Case II, and the time range of sample data is from April 8, 1983 to June 10, 2022, with a total of 2045 samples. Besides, the case data includes training set and test set. The ratio of training set to sample set is 0.8, and that of testing set is 0.2. That is, in Case I, the first 1198 samples are training sets and the last 285 samples are testing sets. Accordingly, the first 1640 samples of Case II are training sets, and the last 405 samples are testing sets. Figure 4 shows the price data of Case I and Case II during the sample period. Table 3 shows the specific sample information of the case. Table 4 is descriptive statistics of case data. It can be seen from the results of the relevant charts and tables that the futures prices of natural gas and crude oil show obvious nonlinear, non-stationary and non-cyclical characteristics during the sample period.

Fig. 4

Trend chart of sample data (Case I is shown in the upper picture and Case II is shown in the lower picture).

Table 3

The size and date range of the sample

		Sample size	Date range
Case I	Sample set	1483	1994.01.14-2022.06.10
	Training set	1198	1994.01.14-2016.12.30
	Test set	285	2017.01.06-2022.06.10
Case II	Sample set	2045	1983.04.08-2022.06.10
	Training set	1640	1983.04.08-2014.09.12
	Test set	405	2014.09.19-2022.06.10

Table 4

Descriptive statistics of sample data

	Mean	Maximum	Minimum	Standard	Skewness	Kurtosis
				deviation
Case I	4.0155	14.4620	1.3870	2.1967	1.6283	6.1773
Case II	44.2768	142.4600	3.9200	28.5141	0.9073	2.7494

3.2 Evaluation indicators of the model

This paper adopts three mainstream evaluation indicators, namely e_MAE (mean absolute error), e_MAPE (mean absolute percentage error) and e_RMSE(root mean square error). Using different indicators to evaluate the prediction performance of each model can reflect the capacity of models more comprehensively and objectively, and avoid the model selection deviation caused by the results of using only a single evaluation indicator. Relevant evaluation indicators are defined as follows:

$\begin{matrix} e_{MAE} = \frac{1}{n} \sum_{i = 1}^{n} | y_{i} - {\hat{y}}_{i} |, \\ e_{MAPE} = \frac{1}{n} \sum_{i = 1}^{n} | y_{i} - {\hat{y}}_{i} | / y_{i}, e_{RMSE} \\ = \sqrt{1 / n \sum_{i = 1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}} \end{matrix}$ (22)

Specifically, n is the number of observation points, y_i and ${\hat{y}}_{i}$ represent the real value and the predicted value at time i. The numerical value of the index and the prediction accuracy are in the opposite direction.

3.3 Benchmark model

A series of corresponding benchmark models are constructed based on the framework of “decomposition-prediction”. Specifically, in order to test the difference of prediction performance between “decomposition-prediction” framework and single prediction model, and to verify the superiority of LSTM prediction model adopted in this paper, ELM, BP and LSTM models are included in the single benchmark model. In order to verify the superiority of the secondary decomposition technology proposed in this paper, a series of benchmark models (WT-LSTM, EMD-LSTM, EEMD-LSTM, CEEMDAN-LSTM, VMD-LSTM, VMD-EMD-LSTM, VMD-EEMD-LSTM, VMD-CEEMDAN-LSTM and VMD-SGMD-LSTM) are further constructed, which are combined with single decomposition technology or secondary decomposition technology. Table 5 shows the characteristics of different models. All models in different case studies are applied to the scenarios of 1-step, 3-step and 5-step ahead forecasting through Matlab 2019b.

Table 5
Characteristics of different models

Model number Model name Single decomposition technique Quadratic decomposition technique Optimized quadratic decomposition technique

Model 1 ELM

Model 2 BP

Model 3 LSTM

Model 4 WT-LSTM √

Model 5 EMD-LSTM √

Model 6 EEMD-LSTM √

Model 7 CEEMDAN-LSTM √

Model 8 VMD-LSTM √

Model 9 VMD-EMD-LSTM √

Model 10 VMD-EEMD-LSTM √

Model 11 VMD-CEEMDAN-LSTM √

Model 12 VMD-SGMD-LSTM √

Model 13 PSO-VMD-SGMD-LSTM √ √

Model number	Model name	Single decomposition technique	Quadratic decomposition technique	Optimized quadratic decomposition technique
Model 1	ELM
Model 2	BP
Model 3	LSTM
Model 4	WT-LSTM	√
Model 5	EMD-LSTM	√
Model 6	EEMD-LSTM	√
Model 7	CEEMDAN-LSTM	√
Model 8	VMD-LSTM	√
Model 9	VMD-EMD-LSTM		√
Model 10	VMD-EEMD-LSTM		√
Model 11	VMD-CEEMDAN-LSTM		√
Model 12	VMD-SGMD-LSTM		√
Model 13	PSO-VMD-SGMD-LSTM		√	√

3.4 Analysis of results

Tables 6-8 record the specific values of evaluation indicators of different models in the prediction study of Case I. Specifically, Table 6 shows the results of a single prediction model. Table 7 shows the performance of a series of prediction models combining single decomposition technology. Table 8 shows the performance of the prediction model combined with secondary decomposition technology. Table 9 shows the statistical test results of the benchmark model relative to the proposed model. Figures 5-7 show the results comparison of e_MAE, e_MAPE and e_RMSE of different models respectively. Accordingly, Tables 10-13, Figs. 8-10 are the prediction results of Case II.

Fig. 5

The results comparison of e_MAE (Case I).

Fig. 6

The results comparison of e_MAPE (Case I).

Fig. 7

The results comparison of e_RMSE (Case I).

Fig. 8

The results comparison of e_MAE (Case II).

Fig. 9

The results comparison of e_MAPE (Case II).

Fig. 10

The results comparison of e_RMSE (Case II).

Table 6

Prediction results of models without decomposition technique (Case I)

Model number	One-step ahead			Three-step ahead			Five-step ahead
	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE
Model 1	0.1519	4.5420	0.2381	0.2265	7.0802	0.3425	0.3077	9.0000	0.4715
Model 2	0.2565	8.3918	0.3856	0.3221	10.4048	0.4567	0.3846	12.1751	0.6068
Model 3	0.1466	4.4481	0.2173	0.2231	6.7589	0.3177	0.2954	8.9410	0.4434

Note: The bold values indicate that the corresponding model achieves the best prediction performance.

Table 7

Prediction results of the model combined with single decomposition technique (Case I)

Model number	One-step ahead			Three-step ahead			Five-step ahead
	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE
Model 4	0.0476	1.5004	0.0640	0.0737	2.3353	0.1161	0.1068	3.2968	0.1647
Model 5	0.1609	5.1141	0.2428	0.1422	4.5470	0.2135	0.1774	5.8182	0.2475
Model 6	0.1392	4.3747	0.2041	0.1324	4.1308	0.2086	0.2022	6.7520	0.2827
Model 7	0.0824	2.6785	0.1077	0.0920	2.8957	0.1280	0.1104	3.5081	0.1615
Model 8	0.0428	1.3535	0.0654	0.0539	1.7376	0.0736	0.0939	2.8252	0.1429

Note: The bold values indicate that the corresponding model achieves the best prediction performance.

Table 8

Prediction results of the model combined with quadratic decomposition technique (Case I)

Model number	One-step ahead			Three-step ahead			Five-step ahead
	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE
Model 9	0.0419	1.3261	0.0582	0.0521	1.7166	0.0729	0.0820	2.6625	0.1229
Model 10	0.0391	1.2165	0.0528	0.0519	1.6611	0.0786	0.0781	2.4367	0.1130
Model 11	0.0384	1.1688	0.0530	0.0512	1.6539	0.0730	0.0662	2.2779	0.1012
Model 12	0.0361	1.0814	0.0507	0.0511	1.5834	0.0705	0.0644	2.0104	0.0965
Model 13	0.0198	0.6434	0.0257	0.0297	0.9714	0.0423	0.0393	1.2102	0.0572

Note: The bold values indicate that the corresponding model achieves the best prediction performance.

Table 9

The p-value results of Wilcoxon signed-rank test. (Case I)

Model	One-step	Three-step	Five-step
number	ahead	ahead	ahead
Model 1	0	0	0.0034
Model 2	0	0	0
Model 3	0	0	0
Model 4	0	0	0
Model 5	0	0	0
Model 6	0	0	0
Model 7	0	0	0
Model 8	0	0.0085	0
Model 9	0	0	0
Model 10	0.0001	0	0
Model 11	0	0.0004	0.0236
Model 12	0	0.0089	0.0094

Table 10

Prediction results of models without decomposition technique (Case II)

Model number	One-step ahead			Three-step ahead			Five-step ahead
	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE
Model 1	2.0679	5.0704	2.9899	3.2629	7.3822	4.7494	4.0688	9.5352	6.0013
Model 2	4.3726	9.3605	6.1390	5.5363	12.0172	8.0850	6.8116	14.8462	9.7702
Model 3	1.9020	4.5985	2.9290	2.9977	6.6979	4.3234	4.0616	8.3654	5.7730

Note: The bold values indicate that the corresponding model achieves the best prediction performance.

Table 11

Prediction results of the model combined with single decomposition technique (Case II)

Model number	One-step ahead			Three-step ahead			Five-step ahead
	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE
Model 4	0.8829	2.1542	2.2984	1.4713	3.2423	2.1113	1.8153	4.5046	2.7159
Model 5	1.4113	3.1725	2.2121	2.1533	4.3393	3.2904	3.0415	5.8967	4.6727
Model 6	1.9405	3.3253	2.6220	2.4893	4.7354	3.8710	2.9984	5.7502	4.5815
Model 7	1.4296	3.3242	1.9971	1.4120	2.9325	1.9772	1.4797	3.4257	2.0552
Model 8	0.8629	2.1341	1.8972	0.9994	2.4622	1.8900	1.4233	3.8095	2.0571

Note: The bold values indicate that the corresponding model achieves the best prediction performance.

Table 12

Prediction results of the model combined with quadratic decomposition technique (Case II)

Model number	One-step ahead			Three-step ahead			Five-step ahead
	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE	e _MAE	e _MAPE	e _RMSE
Model 9	0.8542	2.0126	1.6536	0.9625	2.3651	1.7360	1.3969	3.7058	2.0387
Model 10	0.8534	1.9566	1.6047	0.9526	2.2685	1.6871	1.3875	3.6726	2.0324
Model 11	0.8449	1.9135	1.3456	0.9328	2.1775	1.4923	1.3833	3.5826	1.9846
Model 12	0.8392	1.8131	1.2030	0.9306	2.0852	1.3648	1.3664	3.2266	1.9736
Model 13	0.5938	1.3075	0.7620	0.8038	1.7434	1.0443	0.8016	1.8798	1.1243

Note: The bold values indicate that the corresponding model achieves the best prediction performance.

Table 13

The p-value results of Wilcoxon signed-rank test. (Case II)

Model	One-step	Three-step	Five-step
number	ahead	ahead	ahead
Model 1	0	0	0
Model 2	0	0	0
Model 3	0	0	0
Model 4	0.0019	0	0
Model 5	0	0	0
Model 6	0	0	0
Model 7	0	0	0
Model 8	0.0019	0	0
Model 9	0	0	0
Model 10	0	0.0071	0
Model 11	0.0054	0.0395	0.0166
Model 12	0.0131	0.0645	0.0235

The PSO-VMD-SGMD-LSTM hybrid model has the best forecasting accuracy in the forecasting scenarios of 1-step, 3-step and 5-step ahead forecasting, which indicates the effectiveness of the model in the energy futures price forecasting. The specific analysis is as follows:

3.4.1 Analysis of the prediction results of Case I

(1) Analysis of single prediction model.

Compared with ELM and BP models, the LSTM model has obtained the optimal numerical results of e_MAE, e_MAPE and e_RMSE in the scenarios of 1-step, 3-step and 5-step ahead forecasting. Taking the value of e_MAE in the scenario of 1-step ahead forecasting as an example, compared with the prediction results of ELM and BP, the results of LSTM are improved by 3.4% and 42.84%, respectively.

The reason for this result is that the futures price data has the characteristics of nonlinearity and high volatility. The LSTM model is more suitable for learning the nonlinear patterns in futures price data and capturing their changing rules.

(2) Analysis of the application of decomposition technology.

The numerical results of e_MAE, e_MAPE and e_RMSE of the hybrid model combined with decomposition technology (WT-LSTM, EMD-LSTM, EEMD-LSTM, CEEMDAN-LSTM, VMD-LSTM, VMD-EMD-LSTM, VMD-EEMD-LSTM, VMD-CEEMDAN-LSTM, VMD-SGMD-LSTM and PSO-VMD-SGMD-LSTM) are mostly obviously better than those of the single prediction model (ELM, BP and LSTM) in the scenarios of 1-step, 3-step and 5-step ahead forecasting. Taking the value of e_MAE in the scenario of 1-step ahead forecasting as an example, compared with the predicted results of LSTM, the results of WT-LSTM, EMD-LSTM, EEMD-LSTM, CEEMDAN-LSTM, VMD-LSTM, VMD-EMD-LSTM, VMD-EEMD-LSTM, VMD-CEEMDAN-LSTM, VMD-SGMD-LSTM and PSO-VMD-SGMD-LSTM improved by 67.53%, -9.7%, 5.04%, 43.79%, 70.8%, 71.41%, 73.32%, 73.8%, 75.37% and 86.49% respectively. That is, except EMD-LSTM model, other models combined with decomposition technology have effectively improved the prediction accuracy.

The reason for this result is that the complexity and random fluctuation of the original sequence are reduced by the application of decomposition technology, and the mutual interference of various modes is suppressed to a certain extent, so that the unique characteristics of different modes can be better identified and the prediction accuracy of the whole model can be effectively improved.

(3) Application analysis of single decomposition technology.

The VMD technology has the best performance among a series of single decomposition technologies. The results of e_MAE, e_MAPE and e_RMSE of VMD-LSTM model are better than those of hybrid models combined with other single decomposition techniques (WT-LSTM, EMD-LSTM, EEMD-LSTM and CEEMDAN-LSTM) in the scenarios of 1-step, 3-step and 5-step ahead forecasting. Taking the value of e_MAE in the scenario of 1-step ahead forecasting as an example, compared with WT-LSTM, EMD-LSTM, EEMD-LSTM and CEEMDAN-LSTM, the results of VMD-LSTM are improved by 10.08%, 73.39%, 9.64% and 48.05%, respectively.

The reason for this result is that, compared with other single decomposition algorithms, VMD iteratively updates the central frequencies of modes and signal signals for the time series of energy futures prices, and demodulates different modes to corresponding fundamental frequency bands step by step. Therefore, different modes and their corresponding central frequencies can be extracted more effectively.

(4) Application analysis of secondary decomposition technology.

The results of e_MAE, e_MAPE and e_RMSE of PSO-VMD-SGMD-LSTM model using PSO-VMD-SGMD secondary decomposition technology are better than those of VMD-EMD-LSTM, VMD-EEMD-LSTM, VMD-CEEMDAN-LSTM and VMD-SGMD-LSTM models without PSO in the scenarios of 1-step, 3-step and 5-step ahead forecasting. Taking the value of e_MAE in the scenario of 1-step ahead forecasting as an example, Compared with VMD-EMD-LSTM, VMD-EEMD-LSTM, VMD-CEEMDAN-LSTM and VMD-SGMD-LSTM, the results of PSO-VMD-SGMD-LSTM are improved by 52.74%, 49.36%, 48.43% and 45.15%, respectively.

The reason for this result is that although the model using VMD technology has a certain fitting accuracy, the energy futures price contains a lot of disorderly noise, and it is difficult to completely extract the price characteristics by single decomposition technology. Through further decomposition of price series and reasonable signal reconstruction by SGMD, valuable information can be further mined and the fluctuation features in the price series can be extracted more efficiently. In addition, the performance of PSO-VMD-SGMD-LSTM model compared with VMD-SGMD-LSTM model further verifies the superiority of PSO.

(5) The statistical test analysis of the hybrid model.

Referring to Wilcoxon signed-rank test used by Li et al. [32], Table 9 shows the results of p values tested by different benchmark models in 1-step, 3-step and 5-step ahead forecasting scenarios respectively. The hypothesis of this test is that there is no significant difference between the prediction accuracy of the two models. According to the results in Table 9, compared with a series of benchmark models, the model proposed in this paper has significant differences in most forecasting scenarios.

In addition, in the scenarios of 1-step, 3-step and 5-step ahead forecasting, the results of e_MAE, e_MAPE and e_RMSE of the hybrid model proposed in this paper based on the PSO-VMD-SGMD secondary decomposition technique are optimal, which shows that the proposed PSO-VMD-SGMD-LSTM hybrid model is an effective method for energy futures price prediction.

Furthermore, by observing the values of predicted results of models in the scenarios of 1-step, 3-step and 5-step ahead forecasting, it can be found that the numerical results of e_MAE, e_MAPE and e_RMSE of all models are gradually increasing, which reflects the problem of error accumulation in the prediction process. The PSO-VMD-SGMD-LSTM model constructed in this paper has lower and more stable error.

3.4.2 Analysis of the prediction results of Case II

In the analysis of prediction results of Case II, comparing the numerical results of e_MAE, e_MAPE and e_RMSE of each model in Case I and Case II, similar conclusions can be drawn. the validity of the PSO-VMD-SGMD-LSTM hybrid model constructed in this paper for energy futures price prediction was further verified.

4 Conclusions

Due to the high volatility and randomness of energy futures price series, accurate data decomposition methods are essential to reduce the complexity of data and improve the prediction accuracy of models. In this paper, the PSO-VMD-SGMD-LSTM hybrid model is constructed by combining SGMD with VMD optimized by PSO. The empirical results based on actual transaction data show that PSO-VMD-SGMD secondary decomposition technique can effectively depict the detailed and valuable signal characteristics in the price series, obtain more complete and valuable decomposition results, and has better prediction accuracy and robustness. In the future research, we can consider introducing or adding other factors into the neural network to improve the prediction accuracy of the model for energy futures price time series.

Funding

This research was funded by the National Natural Science Foundation of China, No.71973028.

Conflicts of interest

The authors declare no conflict of interest.

References

Alhassan

, Basher

S.A.

and Kabir Hassan

, Oil subsidies and the risk exposure of oil-user stocks: Evidence from net oil producers, Resources Policy 61 (2019), 461e72.

Rostamian

and O’Hara

J.G.

, Event prediction within directional change framework using a CNN-LSTM model, Neural Computing & Applications 34(20) (2022), 17193–17205.

, Chen

C.Q.

, Nie

and Jiang

Y.H.

, Visiting effects of crude oil price on economic growth in BRICS countries: fresh evidence from wavelet-based quantile-on quantile tests, Energy 178 (2019), 234e51.

Walczak

and Massar

D.L.

, Noise suppression and spinal compression using the wavelet packet transform, Chemometrics and Intelligent 36(2) (1997), 81–94.

Wang

and Wang

, Energy futures and spots prices forecasting by hybrid SW-GRU with EMD and error evaluation, Energy Economics 90 (2020), 104827.

Wang

and Wang

, Energy futures prices forecasting by novel DPFWR neural network and DS-CID evaluation, Neurocomputing 338 (2019), 1–15.

Zhu

B.Z.

, Shi

X.T.

, Chevallier

, Wang

and Wei

Y.M.

, An adaptive multiscale ensemble learning paradigm for nonstationary and nonlinear energy price time series forecasting, Journal of Forecasting 35(7) (2016), 633–651.

Zhang

, Sjarif

N.N.A.

and Ibrahim

, Deep learning models for price forecasting of financial time series: A review of recent advancements, Wiley Interdisciplinary Reviews-data Mining and Knowledge Discovery (2023), e1519.

Keles

, Scelle

, Paraschiv

and Fichtner

, Extended forecast methods for day-ahead electricity spot prices applying artificial neural networks, Applied Energy 162 (2016), 218–230.

10.

Arin

and Ozbayoglu

A.M.

, Deep learning based hybrid computational intelligence models for options pricing, Computational Economics 59(1) (2022), 39–58.

11.

Saghi

and Rezaee

M.J.

, An ensemble approach based on transformation functions for natural gas price forecasting considering optimal time delays, Peerj Computer Science (2021), e409.

12.

Zhou

F.T.

, Huang

Z.H.

and Zhang

C.H.

, Carbon price forecasting based on CEEMDAN and LSTM, Applied Energy 311 (2022), 118601.

13.

Busari

G.A.

and Lim

D.H.

, Crude oil price prediction: A comparison between AdaBoost-LSTM and AdaBoost-GRU for improving forecasting performance, Computers & Chemical Engineering 155 (2021), 107513.

14.

Yang

, Yang

and Dai

, Time-series prediction modelling based on an efficient self-organization learning neural network, IFAC Papers OnLine 48 (2015), 248e53.

15.

H.J.

, Huang

X.Y.

, Zhou

D.H.

, Cao

A.D.

, Su

M.Y.

, Wang

Y.F.

and Guo

L.L.

, Forecasting carbon price in China: A multimodel comparison, International Journal of Environmental Research and Public Health 19(10) (2022), 6217.

16.

Pan

, Yang

, Zheng

and Cheng

, Symplectic transformation based variational Bayesian learning and its applications to gear fault diagnosis, Measurement 147 (2019), 106827.

17.

, Tang

and Lao

, Oil price forecasting based on EMD and BP_AdaBoost neural network, Open Journal of Statistics 8(4) (2018), 660–669.

18.

Pan

H.Y.

, Yang

, Li

, Zheng

D.J.

and Cheng

J.S.

, Symplectic geometry mode decomposition and its application to rotating machinery compound fault diagnosis, Mechanical Systems And Signal Processing 114 (2019), 189–211.

19.

Zhang

, Zhao

, Liu

, Xu

, Chen

and Huang

, Finding potential lenders in P2P lending: a hybrid random walk approach, Journal of Information Science 432 (2018), 376–391.

20.

Sadgali

, Sael

and Benabbou

, Performance of machine learning techniques in the detection of financial frauds, Procedia Computer Science 148 (2019), 45e54.

21.

Barunik

and Krehlik

, Combining high frequency data with non-linear models for forecasting energy market volatility, Expert Systems With Applications 55 (2016), 222–242.

22.

J.C.

, Zhu

S.W.

and Wu

Q.Q.

, Monthly crude oil spot price fore-casting using variational mode decomposition, Energy Economics 83 (2019), 240–253.

23.

Wang

J.L.

, Lei

C.R.

and Guo

M.Y.

, Daily natural gas price forecasting by a weighted hybrid data-driven model, Journal of Petroleum Science and Engineering 192 (2020), 107240.

24.

J.M.

and Wang

, Forcasting of energy futures market and synchronization based on stochastic gated recurrent unit model, Energy 213 (2020), 118787.

25.

Wang

, Zhou

, Hong

, Li

and Wang

S.Y.

, A multi-granularity heterogeneous combination approach to crude oil price fore-casting, 91 (2020), 104790.

26.

Wang

and Wang

, A new hybrid forecasting model based on SW-LSTM and wavelet packet decomposition: A case study of oil futures prices, Computational Intelligence (2021), 7653091.

27.

Sun

J.Y.

, Zhao

P.P.

and Sun

S.L.

, A new secondary decomposition-reconstruction-ensemble approach for crude oil price forecasting, Resources Policy 77 (2022), 102762.

28.

Zhao

, Exploring the influence of the main factors on the crude oil price volatility: An analysis based on GARCH-MIDAS model with Lasso approach, Resources Policy 79 (2022), 103031.

29.

Dragomiretskiy

and Zosso

, Variational mode decomposition, IEEE Trans. Signal Process 62(3) (2014), 531–544.

30.

Lin

, Jiang

, Xiao

and Zhou

, Crude oil price forecasting based on a novel hybrid long memory GARCH-M and wavelet analysis model, Physica A: Statistical Mechanics and its Applications 543 (2020), 123532.

31.

Nicolau

and Palomba

, Dynamic relationships between spot and futures prices. The case of energy and gold commodities, Resources Policy 45 (2015), 130–143.

32.

M.W.

, Xu

D.Y.

, Geng

and Hong

W.C.

, A hybrid approach for forecasting ship motion using CNN-GRU-AM and GCWOA,84, Applied Soft Computing 114 (1080).

33.

Kok

M.V.

, Varfolomeev

M.A.

and Nurgaliev

D.K.

, Determination of SARA fractions of crude oils by NMR technique, Journal of Petroleum Science and Engineering 179 (2019), 1e6.

34.

Yahsi

, Canakoglu

and Agrali

, Carbon price forecasting models based on big data analytics, Carbon Management 10(2) (2019), 175–187.

35.

Huang

N.E.

, Shen

, Long

S.R.

, Wu

M.C.

, Snin

H.H.

, Zheng

, Yen

N.C.

, Tung

C.C.

and Liu

H.H.

, The empirical mode decomposition and the Hubert spectrum for nonlinear and non-stationary time series analysis, Proceedings Mathematical Physical & Engineering Sciences 454(1971) (1998), 903–995.

36.

Vuong

P.H.

, Dat

T.T.

, Mai

T.K.

, Uyen

P.H.

and Bao

P.T.

, Stock-price forecasting based on XGBoost and LSTM, Computer Systems Science and Engineering 40(1) (2022), 237–246.

37.

Jiang

, Liu

Z.K.

, Wang

J.Z.

and Zhang

L.F.

, Decomposition-selection-ensemble forecasting system for energy futures price forecasting based on multi-objective version of chaos game optimization algorithm, Resources Policy 73 (2021), 102234.

38.

Qin

Q.D.

, Xie

K.Q.

, He

H.D.

, Li

, Chu

X.H.

, Wei

Y.M.

and Wu

, An effective and robust decomposition-ensemble energy price forecasting paradigm with local linear prediction, Energy Economics 83 (2019), 402–414.

39.

Gao

, Hou

C.H.

and Nguyen

B.H.

, Forecasting natural gas prices using highly flexible time-varying parameter models, Economic Modelling 105 (2021), 105652.

40.

Hochreiter

and Schmidhuber

, Long short-term memory, Neural Computing and Applications 9(8) (1997), 1735–1780.

41.

Sun

S.L.

, Jin

, Li

H.T.

and Li

Y.W.

, A new hybrid optimization ensemble learning approach for carbon price forecasting, Applied Mathematical Modelling 97 (2021), 182–205.

42.

Wang

, Zhang

, Wu

and Wang

, Wind speed forecasting based on the hybrid ensemble empirical mode decomposition and GA-BP neural network method, Renewable Energy 94 (2016), 629–636.

43.

Zaheer

, Anjum

, Hussain

, Algarni

A.D.

, Iqbal

, Bourouis

and Ullah

S.S.

, A multi parameter forecasting for stock time series data using LSTM and deep learning model, Mathematics 11(3) (2023), 590.

44.

Bollerslev

, Generalized autoregressive conditional heteroskedasticity, Journal of Economics 31(3) (1986), 307–27.

45.

Liu

and Shi

Y.L.

, Forecasting crude oil future volatilities with a threshold Zero-Drift GARCH model, Mathematics 10(15) (2022), 2757.

46.

Zhang

T.T.

, Tang

Z.P.

, Wu

J.C.

, Du

X.X.

and Chen

K.J.

, Multi-step-ahead crude oil price forecasting based on two-layer decomposition technique and extreme learning machine optimized by the particle swarm optimization algorithm, Energy 229 (2021), 120797.

47.

Gundu

and Simon

S.P.

, PSO-LSTM for short term forecast of heterogeneous time series electricity price signals, Journal of Ambient Intelligence and Humanized Computing 12(2) (2021), 2375–2385.

48.

Liu

W.P.

, Wang

C.Z.

, Li

Y.G.

, Liu

Y.S.

and Huang

K.K.

, Ensemble forecasting for product futures prices using variational mode de-composition and artificial neural networks, Chaos Solitons & Fractals 146 (2021), 110822.

49.

Zhang

, Li

, Wang

, Wu

, Anovel fault diagnosis proce-dure based on improved symplectic geometry mode decomposition and optimized SVM, Measurement (2020), 108644.

50.

Zhang

, Yang

K.L.

, Lu

, Wu

J.Y.

, Yu

and Lin

, Predicting carbon futures prices based on a new hybrid machine learning: Comparative study of carbon prices in different periods, Journal of Environmental Management 346 (2023), 118962.

51.

Lin

Y.F.

, Huang

T.M.

, Chung

W.H.

and Ueng

Y.L.

, Forecasting fluctuations in the financial index using a recurrent neural network based on price features, IEEE Transactions on Emerging Topics in Computational Intelligence 5(5) (2021), 780–791.

52.

Jiang

, Chen

, Li

, Symplectic geometry decomposition-based features for automatic epileptic seizure detection, Computers in Biology and Medicine (2020),116103549.

53.

Lin

, Lu

, Tan

and Yu

Y.Y.

, Forecasting energy prices using a novel hybrid model with variational mode decomposition, Energy 246 (2022), 123366.

54.

Lee

Y.S.

and Tong

L.I.

, Forecasting time series using a methodology based on autoregressive integrated moving average and genetic programming, Knowledge-based Systems 24(1) (2011), 66–72.

55.

Cheng

and Wang

, Enhanced symplectic characteristics mode decomposition method and its application in fault diagnosis of rolling bearing, Measurement 166 (2020), 108108.

56.

Hajiabotorabi

, Samavati

F.F.

, Ghaini

F.M.M.

and Shahmoradi

, Multi-WRNN model for pricing the crude oil futures market, Expert Systems with Applications 182 (2021), 115229.

57.

Z.J.

, Liao

Y.P.

, Hu

, Ni

L.Y.

and Lu

Y.T.

, A financial deep learning framework: Predicting the values of financial time series with ARIMA and LSTM, International Journal of Web Services Research 19(1) (2022).

A novel hybrid optimization ensemble learning approach for energy futures price forecasting

Abstract

Keywords

1 Introduction

2 Methodology

2.1 PSO

Table 2 Symbol abbreviation of LSTM

3.1 Analysis of sample data

3.4.2 Analysis of the prediction results of Case II

4 Conclusions

Funding

Conflicts of interest

References

Table 2
Symbol abbreviation of LSTM