# Improved CEEMDAN, GA, and SVR Model for Oil Price Forecasting

Authors: Yichun Lu; Junyin Luo; Yiwen Cui; Zhengbin He; Fengchun Xia
Journal: Journal of Environmental and Public Health
 (2022)

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## Abstract

Accurate prediction of crude oil prices (COPs) is a challenge for academia and industry. Therefore, the present research developed a new CEEMDAN-GA-SVR hybrid model to predict COPs, incorporating complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), a genetic algorithm (GA), and support vector regression machine (SVR). First, our team utilized CEEMDAN to realize the decomposition of a raw series of COPs into a group of comparatively simpler subseries. Second, SVR was utilized to predict values for every decomposed subseries separately. Owing to the intricate parametric settings of SVR, GA was employed to achieve the parametric optimisation of SVR during forecast. Then, our team assembled the forecasted values of the entire subseries as the forecasted values of the CEEMDAN-GA-SVR model. After a series of experiments and comparison of the results, we discovered that the CEEMDAN-GA-SVR model remarkably outperformed single and ensemble benchmark models, as displayed by a case study finished based on a time series of weekly Brent COPs.


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## Body

## 1. Introduction

As a vital strategic resource, crude oil has a pivotal role in the economic activities of countries worldwide [1]. However, as COPs are often nonlinear and affected by many unobservable factors, it is difficult to forecast them accurately; thus, exploring new paths and methods to accurately forecast COPs is vital for optimising production and managerial strategies, anticipating future oil price fluctuations, and avoiding market risks [2]. Although experts have not established an agreement on the methods and models used to forecast oil prices, this crucial and difficult problem has been extensively researched. [3–6]. However, COPs are influenced by numerous complex factors, both observed and unobserved [7–9]. Therefore, COP forecast is still a hot spot in the academic literature and in industry. At present, the common and highly accurate methods for predicting COPs can be preliminarily classified into five types: (1) statistic methods, (2) artificial intelligence (AI), (3) decomposition and ensemble, (4) hybrid model methodology, and (5) parameter optimisation.Subsequently scholars have proposed statistical modeling methods to predict COPs, mostly using linear time series models to ameliorate the accuracy of COP prediction. Some have adopted an autoregressive integrated moving average (ARIMA) modeling method to forecast COP [10–12]. The autoregressive model is a classical method that is widely used in economics, energy, and other fields and achieves good prediction results [13]. For example, Mohammadi and Su forecasted crude oil prices using generalised autoregressive conditional heteroskedasticity models [14]. Later, more complex statistical methods including hidden Markov models, dynamic model averaging, and the autoregressive conditional heteroskedasticity approach were employed to forecast the distribution and trends of COPs in the short-term [15–17]. Some progress and breakthroughs have been made in forecasting oil prices using these statistical methods. However, owing to the inherent nonlinear and nonstationary features of COPs, statistical methods are not powerful for crude oil price prediction. The effect of traditional statistical methods on crude oil prediction is very limited.With the advancement of machine learning technology, the application of machine learning algorithms for oil price prediction has become a mainstream trend in current research, and such approaches are becoming increasingly popular. The support vector regression machine (SVR), which can capture nonlinearity, is a popular predictive modeling method for the prediction of COPs [5, 17, 18]. For example, Yu et al. deemed user-defined variables as indeterminate (or random) factors to establish an LSSVR (least squares support vector machine for regression) ensemble training method for oil price forecasting [19]. SVR has the benefit of being able to effectively solve nonlinear and overfitted problems [20]. The artificial neural network (ANN) is another popular model in deep learning [21, 22]. However, although neural network methods have strong self-learning and self-adaptive abilities, they can easily fall into local minima [23]. AI techniques including SVR and ANN display strong capacity for nonlinear modeling. Nevertheless, they are affected by problems such as overfitting and poor stability [24]. Therefore, it is necessary to properly combine the AI techniques mentioned above to uplift the accurateness of COP forecast by exploiting the strengths and avoiding the weaknesses of the various methods.It is hard to realize good forecast results by virtue of certain raw time series owing to their complex characteristics. To address this problem, some researchers have introduced a framework referred to as “decomposition and ensemble” into time series forecasting. This framework is capable of decomposing the time series into simplified parts, uses a predictor to independently predict a single component, and finally integrates all the predictions to form the final prediction results [4, 25–31]. Some researchers have applied this idea to COP forecast. They found that the original complex sequence of COPs could be divided into multiple subseries, after which each single predictor is utilized for prediction and then combined with all single prediction results to form a final prediction. Abdollahi built an ensemble prediction model integrating wavelet decomposition and LSSVM for forecasting oil prices [5]. Wu et al. put forward a new modeling method on the foundation of ensemble empiric mode decomposition (EEMD) and long short-term memory (LSTM) for international crude oil markets [15]. Li and Wang proposed a novel hybrid neuronetwork forecast modeling method based on the combination of EEMD and stochastic recurrent wavelet neuronetwork (SRWNN) for COP prediction [32].For the sake of ameliorating the accurateness of model prediction and deal with the drawbacks of single-model methods, hybrid models are increasingly applied for oil price prediction and have led to progress in field of crude oil price prediction [25, 33–35]. For instance, SVR is often used as a basic model in the framework of a hybrid modeling method to uplift the accurateness of COP prediction. Li et al. developed hybrid modeling methods for COP prediction monthly via variation mode decomposition and SVM optimised by a GA [36]. In addition, the neuronetwork approach has been proved to be relatively suitable for the prediction of residual series containing noise factors. Safari and Davallou combined ARIMA and a nonlinearity autoregression neuronetwork to increase the accuracy of forecasting crude oil prices. They discovered that the neural network modeling method was appropriate for forecasting residual sequences containing substantial amounts of complex information and white noise [37]. Researchers have also shown that the forecast ability of generalised regression neuronetwork (GRNN) models is better than that of ANN models [38, 39]. GRNN models have also been incorporated in hybrid models to uplift the accurateness of forecast [40, 41]. Owing to the different advantages of SVR and GRNN models, combining the two can lead to more accurate predictions.In addition, nonparametric prediction models have been improved from the perspective of parameter optimisation, mainly using GA [42–44]. Li et al. proposed hybrid models containing SVM optimised by GA; the prediction results demonstrated that the optimised models were more robust and accurate [36]. Xiao et al. proposed a hybrid migration learning model (HTLM) for COP prediction and introduced a GA to identify the best match of 2 vital variables in the HTLM [45]. These results show that it is necessary to optimise model parameter by GA in order to achieve better predictions of COPs.Previous work has demonstrated the effectiveness of hybrid and parameter-optimised models (“hybrid and combination”), decomposition and ensemble, and AI approaches. In these frameworks, the selection of a suitable decomposition approach and predictor is essential to improve prediction ability [4]. Given the prediction abilities of CEEMDAN, SVR, GRNN, and GA for parameter optimisation, we have developed a CEEMDAN-GA-SVR-GRNN hybrid modeling method for time series prediction of COPs. First, CEEMDAN is used to realize the decomposition of the complex raw time series of COPs into a group of comparatively simplified subseries. Second, SVR is utilized to predict the target values of every subseries separately. Owing to the intricate parametric settings of SVR, GA is introduced to search for the optimum parametric results for SVR. Afterwards, our team assemble the predicted values of all subseries as the predicted values of the CEEMDAN-GA-SVR model.The primary contributions of our research are stated below:(1)
Our team put forward a new hybrid model incorporating CEEMDAN, GA, and SVR for COP prediction, which fully utilizes the AI arithmetic strengths of GA and SVR. To our knowledge, this is the first time that a CEEMDAN-GA-SVR hybrid modeling method has been utilized for COP prediction.(2)
GA is utilized to optimise the parametric settings for SVR, which aims to further improve forecasting performance.(3)
Experiments demonstrate that our proposed CEEMDAN-GA-SVR hybrid model performs significantly better than single and ensemble benchmark models for COP prediction.The primary innovations of our paper involve these 3 aspects:(1)
Owing to the strong decompositional ability by CEEMDAN, the potent optimisation capability of GA, and the robust forecast ability of SVR, a new ensemble model combining the 4 modeling methods is proposed for COP prediction.(2)
CEEMDAN-GA-SVR are first combined and GA is utilized to optimise the variables of SVR simultaneously.(3)
A CEEMDAN-GA-SVR hybrid modeling method is first proposed for predicting COPs, and the strength of the CEEMDAN-GA-SVR hybrid model is proved by experimental results.The remaining sections of the article are arranged as follows. Section2 briefly introduces CEEMDAN, GA, and SVR and introduces the concept and algorithm of the CEEMDAN-GA-SVR hybrid model. Section 3 reports experimental results involving forecasting of weekly Brent crude oil prices. Section 4 provides some discussion and insights on the foundation of experiment outcomes. Some discoveries are presented in Section 2.


## 2. Materials and Methods

### 2.1. CEEMDAN

Empirical mode decomposition (EMD) is a classical method for decomposing time series. This method decomposes the signal according to the time scale characteristic of the data itself, without setting any basis function in advance. However, a main drawback of EMD was the mode blending issue [46]. To solve such concern, EEMD was developed to average the outcomes of some EMD parameters with the addition of Gaussian white noise based on a raw time series [47]. However, EEMD led to a new concern in signal decomposition, i.e., that residual noise may affect the accuracy of the signal sequence generated from the raw time series by EEMD. For the sake of optimise the ability of EEMD, Torres et al. developed a novel decompositional method referred to as CEEMDAN [48]. Therein, adaptive white noise is supplemented into the raw time series during all decomposition stages; this can enhance the effectiveness of reconstruction of the original signals and enable the method to outperform spectral separation of intrinsic mode functions (IMFs). Compared with EEMD, CEEMDAN requires a smaller quantity of sifting iterative process and has a reduced reconstruction error, leading to a decrease in computational cost. Owing to the validity, the CEEMDAN approach is extensively utilized in energy prediction [4, 25, 26, 49]. Therefore, in our study, we considered utilizing for the purpose of decomposing raw COP series.


### 2.2. Genetic Algorithm

The GA was put forward by Goldberg and Holland, on the foundation of the evolutionary theory [50], and has become an important optimisation algorithm that has been used in many studies [51–53]. In this work, we use GA to find the optimal punishment parameter C, the insensitive loss function, and the radial basis function (RBF) kernel parameters γ in the SVR modeling method and establish a GA-SVR modeling method to forecast time series of crude oil prices. The procedure is stated below.(1)
Select an encoding method and specify the values of genetic parameters such as population size, selection, crossover, mutational method, crossover possibility, and mutation possibility. As GA uses individual fitness values to evaluate the pros and cons of an individual and determine the size of their genetic opportunities, our team set the evolution algebra to 200 generations, the population size to 20, and the fitness function to MSE (mean squared error). This is the MSE generated by the validation subset from the cross-validation (CV) mechanism. It can validly identify the pros and cons of chromosomes in regressive forecast problems and can prevent or reduce the phenomenon of overfitting after CV. In this work, we adopted a 5-fold CV process with the following fitness function formula:(1)MSE=1n∑i=1nyp−yp∗2,whereyp is the observed value, yp∗ is the predicted value, and n is the learning set sample size for the fuzzy information particle. The smaller the fitness value, the superior the individual effect and the greater the probability of being selected.(2)
According to the feature subset encoding of each chromosome, complete the encoding operation and generate the initial populationP randomly. Generally, the choice of encoding strategy relies on the problem feature. The usual encoding strategies include binary encoding and real number coding; binary coding is utilized in most cases.(3)
Compute the fitness values of all individuals within the group as per the fit function. Perform genetic operations using selection, crossover, and mutation operators to produce the next generation of populations.(4)
Estimate if the fitness value satisfies the determined standard; if not, return to the last step or return to step 2, execute the optimisation arithmetic, reach the termination condition, and finally use the individual with the smallest fitness during the evolutionary procedure as the optimum individual.SVR models have excellent performance in categorization or regression, whereas their optimal generalisation performance depends greatly on the setting of parameters. For a given dataset, the most important task is to identify the optimum parameters. Practically, the issue of selecting parameters has not been well resolved. At present, parameters are primarily chosen through assays or a low-efficient grid search approach for CV.As a steady search arithmetic that can be used for optimisation of complex systems, GA has unique advantages compared with other intelligent algorithms for optimisation. GA can easily find the global optimal solution. Because of the utilization of natural selection with survival of the fittest and simplest gene operations, GA is not restricted by conditions like the search space during calculation, and no other auxiliary information is required.


### 2.3. Support Vector Regression Machine

The essence of SVR is the support vector machine, which is a neural network model developed by Vapnik in the 1990s to analyze relatively smaller specimens and smaller probabilistic events [54]. It has been utilized in regressive forecast and applied in many research areas [53]. SVR relies on the principle of structure risk minimisation for regressive estimation; this structural risk is speculated using the insensitive loss function. In addition, SVR uses a risk function that is a combination of penalty terms derived from the principle of empiric error and structure risk minimisation. The establishment principle of the nonlinear ε-SVR used in this work is as follows.Consider a set of dataG=xi,yiin, in which xi denotes the input feature vector, yi denotes the target value, and n denotes the sample size of the time series data. The fundamental purpose of nonlinearity SVR aims at mapping data x to a high-dimensional feature space (HDFS) via a nonlinearity mapping Φ and complete linearity regressive analysis in such space:(2)fx=ωTΦx+b,(3)Φ:Rn⟶F，ω∈F.In equations (2) and (3), b is the liminal value and Φ is the HDFS, which is the nonlinear image of the input space x. We need to estimate ω and b to solve the optimisation problem; the result can be given by the following equation’s minimum value:(4)12w2+C∑i=1iξi+ξi∗,(5)s.tyi−w⋅φx−b≤ε+ξi,w⋅φx+b−yi≤ε+ξi∗,ξ,ξi∗≥0,i=1,…,n.In formula (5), C is the punishment parameter, ξ,ξi∗ is the slack variable, and ε is the insensitive loss function. The utilization of ε uplifts the speculation stability. When conducting empiric research, we need to select the parameters C and. Dual theory is generally used to solve the problems above and then transform the problems into a convex quadratic programming issue. The Lagrange transform of equation (5) can be obtained as follows:(6)L=12ω2+C∑i=1iξi+ξi∗−∑i=1iλiε+ξi−yi+ω,xi+b−∑i=1iλi∗ε+ξi∗+yi−ω,xi−b−∑i=1iηiξi+ηi∗ξi∗.In formula (6), and the partial derivative of the Lagrangian function with respect to the variable ω,b, ξi, ξi∗ is 0. Inputting the Lagrangian operator and optimisation restriction formula, the decision function of formula (6) becomes the following formula:(7)fx=∑i=1lλi−λi∗kx,xi+b.In formula (7), kx,xi is the kernel function of SVR. SVR can use the kernel function to map the low-dimensional nonlinearity raw data to the HDFS when dealing with nonlinear problems, followed by linear processing in the HDFS. Common kernel functions involve linearity kernel functions, multinomial kernel functions, and the Gaussian RBF kernel. Previous research experience indicates that RBF has the best effect when the sample data lacks prior knowledge [38]. Herein, our team utilized RBF as the kernel function in the following form:(8)kx,xi=exp−γx−xi2,γ>0.The core parameter in formula (8) is γ. The selection of the γ value has an important influence on the kernel function. If it is set too large, it will cause overfitting; if it is set too small, it will weaken the generalisation ability of the model.


### 2.4. CEEMDAN-GA-SVR: Developed Method for COP Prediction

Based on the idea of “decomposition and ensemble,” our team propose a hybrid modeling method combining CEEMDAN, GA, and SVR, termed CEEMDAN-GA-SVR, to forecast COPs. This hybrid modeling method includes 3 phases, as presented by Figure1.Stage 1. Decomposition. CEEMDAN is utilized to realize the decomposition of a raw series of COPs into (1)N IMFs, denoted IMFi (i=1,2,…,N), and (2) one residue R.Stage 2. Individual prediction. Each IMF or residue is divided into a learning dataset and testing dataset in an equal manner. Then, every SVR forecast model on the foundation of GA optimisation is trained on every learning dataset in an independent manner, and the forecast modeling method is utilized on every testing dataset.Stage 3. Ensemble. Addition aggregation is used to assemble the forecasted value of every decomposed part as the eventual forecasted outcome, which are referred to as the predicted results of the CEEMDAN-GA-SVR model.Figure 1
COPs and the relevant decomposed parts by CEEMDAN.The modeling method’s flowchart is shown in Figure2.Figure 2
Flowchart of the developed CEEMDAN-GA-SVR combined model algorithm.The developed CEEMDAN-GA-SVR hybrid modeling method uses not only the “decomposition and ensemble” approach that is widely employed in energy economics but also a “hybrid and combination” approach [55–59]. First, CEEMDAN is employed to separate the volatility and complexity series of COPs into a group of comparatively simplified subseries, comprising multiple IMFs and one residue. Second, SVR utilizing GA optimisation is utilized for every decomposed subseries for forecasting. We chose SVR as the predictor as it had been demonstrated to be suitable for COP prediction in previous researches [60–62]. As CEEMDAN and SVR have many parameters, it is hard to set the optimal values of these variables beforehand. Hence, GA is employed to find the optimal variables for SVR; this can obviously enhance the prediction effectiveness of every separate subseries. Eventually, the values forecasted by SVR modeling methods for every decomposed subseries are utilized to produce the CEEMDAN-GA-SVR forecasted COPs via addition aggregation. The “decomposition and ensemble” and “hybrid and combination” aspects of CEEMDAN-GA-SVR are expected to contribute to improving the ability of COP prediction.Certain previous researches also used SVR modeling methods to predict crude oil price series. Our study is different from previous work mainly regarding the decompositional process, hybrid model, and parameter-optimised method, in that (1) previous studies decomposed raw COP series based on the EEMD method and (2) previous studies constructed the SVR model using constant parameter settings. In contrast to the previous literature, our study employs CEEMDAN to realize the decomposition of raw COP series and applies GA to rapidly search for optimal variables for SVR in a simultaneous manner.


## 2.1. CEEMDAN

Empirical mode decomposition (EMD) is a classical method for decomposing time series. This method decomposes the signal according to the time scale characteristic of the data itself, without setting any basis function in advance. However, a main drawback of EMD was the mode blending issue [46]. To solve such concern, EEMD was developed to average the outcomes of some EMD parameters with the addition of Gaussian white noise based on a raw time series [47]. However, EEMD led to a new concern in signal decomposition, i.e., that residual noise may affect the accuracy of the signal sequence generated from the raw time series by EEMD. For the sake of optimise the ability of EEMD, Torres et al. developed a novel decompositional method referred to as CEEMDAN [48]. Therein, adaptive white noise is supplemented into the raw time series during all decomposition stages; this can enhance the effectiveness of reconstruction of the original signals and enable the method to outperform spectral separation of intrinsic mode functions (IMFs). Compared with EEMD, CEEMDAN requires a smaller quantity of sifting iterative process and has a reduced reconstruction error, leading to a decrease in computational cost. Owing to the validity, the CEEMDAN approach is extensively utilized in energy prediction [4, 25, 26, 49]. Therefore, in our study, we considered utilizing for the purpose of decomposing raw COP series.


## 2.2. Genetic Algorithm

The GA was put forward by Goldberg and Holland, on the foundation of the evolutionary theory [50], and has become an important optimisation algorithm that has been used in many studies [51–53]. In this work, we use GA to find the optimal punishment parameter C, the insensitive loss function, and the radial basis function (RBF) kernel parameters γ in the SVR modeling method and establish a GA-SVR modeling method to forecast time series of crude oil prices. The procedure is stated below.(1)
Select an encoding method and specify the values of genetic parameters such as population size, selection, crossover, mutational method, crossover possibility, and mutation possibility. As GA uses individual fitness values to evaluate the pros and cons of an individual and determine the size of their genetic opportunities, our team set the evolution algebra to 200 generations, the population size to 20, and the fitness function to MSE (mean squared error). This is the MSE generated by the validation subset from the cross-validation (CV) mechanism. It can validly identify the pros and cons of chromosomes in regressive forecast problems and can prevent or reduce the phenomenon of overfitting after CV. In this work, we adopted a 5-fold CV process with the following fitness function formula:(1)MSE=1n∑i=1nyp−yp∗2,whereyp is the observed value, yp∗ is the predicted value, and n is the learning set sample size for the fuzzy information particle. The smaller the fitness value, the superior the individual effect and the greater the probability of being selected.(2)
According to the feature subset encoding of each chromosome, complete the encoding operation and generate the initial populationP randomly. Generally, the choice of encoding strategy relies on the problem feature. The usual encoding strategies include binary encoding and real number coding; binary coding is utilized in most cases.(3)
Compute the fitness values of all individuals within the group as per the fit function. Perform genetic operations using selection, crossover, and mutation operators to produce the next generation of populations.(4)
Estimate if the fitness value satisfies the determined standard; if not, return to the last step or return to step 2, execute the optimisation arithmetic, reach the termination condition, and finally use the individual with the smallest fitness during the evolutionary procedure as the optimum individual.SVR models have excellent performance in categorization or regression, whereas their optimal generalisation performance depends greatly on the setting of parameters. For a given dataset, the most important task is to identify the optimum parameters. Practically, the issue of selecting parameters has not been well resolved. At present, parameters are primarily chosen through assays or a low-efficient grid search approach for CV.As a steady search arithmetic that can be used for optimisation of complex systems, GA has unique advantages compared with other intelligent algorithms for optimisation. GA can easily find the global optimal solution. Because of the utilization of natural selection with survival of the fittest and simplest gene operations, GA is not restricted by conditions like the search space during calculation, and no other auxiliary information is required.


## 2.3. Support Vector Regression Machine

The essence of SVR is the support vector machine, which is a neural network model developed by Vapnik in the 1990s to analyze relatively smaller specimens and smaller probabilistic events [54]. It has been utilized in regressive forecast and applied in many research areas [53]. SVR relies on the principle of structure risk minimisation for regressive estimation; this structural risk is speculated using the insensitive loss function. In addition, SVR uses a risk function that is a combination of penalty terms derived from the principle of empiric error and structure risk minimisation. The establishment principle of the nonlinear ε-SVR used in this work is as follows.Consider a set of dataG=xi,yiin, in which xi denotes the input feature vector, yi denotes the target value, and n denotes the sample size of the time series data. The fundamental purpose of nonlinearity SVR aims at mapping data x to a high-dimensional feature space (HDFS) via a nonlinearity mapping Φ and complete linearity regressive analysis in such space:(2)fx=ωTΦx+b,(3)Φ:Rn⟶F，ω∈F.In equations (2) and (3), b is the liminal value and Φ is the HDFS, which is the nonlinear image of the input space x. We need to estimate ω and b to solve the optimisation problem; the result can be given by the following equation’s minimum value:(4)12w2+C∑i=1iξi+ξi∗,(5)s.tyi−w⋅φx−b≤ε+ξi,w⋅φx+b−yi≤ε+ξi∗,ξ,ξi∗≥0,i=1,…,n.In formula (5), C is the punishment parameter, ξ,ξi∗ is the slack variable, and ε is the insensitive loss function. The utilization of ε uplifts the speculation stability. When conducting empiric research, we need to select the parameters C and. Dual theory is generally used to solve the problems above and then transform the problems into a convex quadratic programming issue. The Lagrange transform of equation (5) can be obtained as follows:(6)L=12ω2+C∑i=1iξi+ξi∗−∑i=1iλiε+ξi−yi+ω,xi+b−∑i=1iλi∗ε+ξi∗+yi−ω,xi−b−∑i=1iηiξi+ηi∗ξi∗.In formula (6), and the partial derivative of the Lagrangian function with respect to the variable ω,b, ξi, ξi∗ is 0. Inputting the Lagrangian operator and optimisation restriction formula, the decision function of formula (6) becomes the following formula:(7)fx=∑i=1lλi−λi∗kx,xi+b.In formula (7), kx,xi is the kernel function of SVR. SVR can use the kernel function to map the low-dimensional nonlinearity raw data to the HDFS when dealing with nonlinear problems, followed by linear processing in the HDFS. Common kernel functions involve linearity kernel functions, multinomial kernel functions, and the Gaussian RBF kernel. Previous research experience indicates that RBF has the best effect when the sample data lacks prior knowledge [38]. Herein, our team utilized RBF as the kernel function in the following form:(8)kx,xi=exp−γx−xi2,γ>0.The core parameter in formula (8) is γ. The selection of the γ value has an important influence on the kernel function. If it is set too large, it will cause overfitting; if it is set too small, it will weaken the generalisation ability of the model.


## 2.4. CEEMDAN-GA-SVR: Developed Method for COP Prediction

Based on the idea of “decomposition and ensemble,” our team propose a hybrid modeling method combining CEEMDAN, GA, and SVR, termed CEEMDAN-GA-SVR, to forecast COPs. This hybrid modeling method includes 3 phases, as presented by Figure1.Stage 1. Decomposition. CEEMDAN is utilized to realize the decomposition of a raw series of COPs into (1)N IMFs, denoted IMFi (i=1,2,…,N), and (2) one residue R.Stage 2. Individual prediction. Each IMF or residue is divided into a learning dataset and testing dataset in an equal manner. Then, every SVR forecast model on the foundation of GA optimisation is trained on every learning dataset in an independent manner, and the forecast modeling method is utilized on every testing dataset.Stage 3. Ensemble. Addition aggregation is used to assemble the forecasted value of every decomposed part as the eventual forecasted outcome, which are referred to as the predicted results of the CEEMDAN-GA-SVR model.Figure 1
COPs and the relevant decomposed parts by CEEMDAN.The modeling method’s flowchart is shown in Figure2.Figure 2
Flowchart of the developed CEEMDAN-GA-SVR combined model algorithm.The developed CEEMDAN-GA-SVR hybrid modeling method uses not only the “decomposition and ensemble” approach that is widely employed in energy economics but also a “hybrid and combination” approach [55–59]. First, CEEMDAN is employed to separate the volatility and complexity series of COPs into a group of comparatively simplified subseries, comprising multiple IMFs and one residue. Second, SVR utilizing GA optimisation is utilized for every decomposed subseries for forecasting. We chose SVR as the predictor as it had been demonstrated to be suitable for COP prediction in previous researches [60–62]. As CEEMDAN and SVR have many parameters, it is hard to set the optimal values of these variables beforehand. Hence, GA is employed to find the optimal variables for SVR; this can obviously enhance the prediction effectiveness of every separate subseries. Eventually, the values forecasted by SVR modeling methods for every decomposed subseries are utilized to produce the CEEMDAN-GA-SVR forecasted COPs via addition aggregation. The “decomposition and ensemble” and “hybrid and combination” aspects of CEEMDAN-GA-SVR are expected to contribute to improving the ability of COP prediction.Certain previous researches also used SVR modeling methods to predict crude oil price series. Our study is different from previous work mainly regarding the decompositional process, hybrid model, and parameter-optimised method, in that (1) previous studies decomposed raw COP series based on the EEMD method and (2) previous studies constructed the SVR model using constant parameter settings. In contrast to the previous literature, our study employs CEEMDAN to realize the decomposition of raw COP series and applies GA to rapidly search for optimal variables for SVR in a simultaneous manner.


## 3. A Case Study in the Brent Oil Market

### 3.1. Experimental Dataset Source and Evaluation Criteria

Brent crude oil is manufactured in the Brent area of the North Atlantic and the North Sea. Its crude oil futures account for more than 2/3 of the crude oil futures trading volume across the globe, and it is the benchmark for futures prices of crude oil in the international market [63]. The data utilized herein were acquired from Bloomberg and consisted of the weekly settlement prices of North Sea Brent (Brent) crude oil futures between June 2, 2017, and May 21, 2021. There were an overall 204 specimens in the dataset. For the sake of confirming the validity of our approach, the weekly COP time series for Brent was separated into learning and test datasets. Following some previous reports in the literature, the first 80% of the total observations in each time series were utilized as the learning dataset, while the remaining 20% were utilized as the test dataset [4, 29, 30]. Of the total samples, 163 observations were used as the training set, and 41 observations were utilized as a testing dataset to check the model’s effectiveness [64].Five indicators were used to evaluate the experimental results: MSE, root MSE (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and the Diebold-Mariano (DM) test [52]:(9)MSE=1N∑t=1NObservedt−Predictedt2,RMSE=1N∑t=1NObservedt−Predictedt2,MAE=1N∑t=1NObservedt−Predictedt,MAPE=1N∑t−1NObservedt−PredictedtObservedt∗100,in which N is the size of the assessed specimens, and Observedt and Predictedt represent the real and forecasted results at time t, separately. The DM analysis was utilized to calculate the statistic differences in the prediction accurateness of model pairs.


### 3.2. Description of COP Time Series

The weekly COP time series for Brent has obvious nonlinear characteristics, and its change trend shows strong volatility. For example, the price of Brent crude oil was approximately 60.2 USD/barrel in Jan 2020, before plummeting to 35.88 USD/barrel in April 2020; this change in just a few months is an example of the dramatic fluctuations and nonstationary features of COP time series [63].As shown in Table1, the average of the weekly Brent COP time series was 56.7140, indicating that the weekly Brent crude oil prices fluctuate at around 56 USD/barrel. The highest value of the time series was 71.7300 USD/barrel, while the lowest was 35.8800 USD/barrel. There was thus a large difference between the maximum and minimum prices; moreover, the standard deviation was 8.0580. These results indicate that the weekly Brent COP time series fluctuates violently.Table 1
Descriptive statistics for weekly brent COPs.
ObservationsMeanStandard deviationMinimumMaximumOil price20456.71408.058035.880071.7300Figure1 shows the original COPs and the decomposed parts. Clearly, the raw COPs show remarkable fluctuations. Amongst the decomposed parts, IMF1 to IMF4 show obvious high-frequency features in narrow ranges, whereas IMF5 to IMF6 and the residue show obvious low-frequency features in wide ranges. After obtaining the decomposed parts, the initial complex COP series prediction can now be divided into predictions of several simpler components.


### 3.3. Experimental Settings

The proposed CEEMDAN-GA-SVR model was evaluated and analyzed in two ways in our study. First, without any decomposition and ensemble, we compared the GA-SVR single model with other single models, which involved 1 important statistic model (ARIMA), three classical AI approaches (GRNN, back propagation neuronetwork [BPNN], and particle swarm optimisation SVR [PSO-SVR]), and the original SVR. Second, as previously finished studies have revealed that ensemble models using the “decomposition and ensemble” framework show better forecast abilities in contrast to single models for COP prediction, we compared forecast abilities between the developed CEEMDAN-GA-SVR and the rest of ensemble forecast modeling methods. Hence, the entire single models were employed to the forecast phase in ensemble models. Based on the identical COP series, our team tested if the developed CEEMDAN-GA-SVR model could significantly ameliorate the forecast ability. To prove the capacity of the proposed CEEMDAN-GA-SVR in prediction and CEEMDAN in decomposition, our study also compared CEEMDAN-GA-SVR with CEEMDAN-PSO-SVR, CEEMDAN-SVR, CEEMDAN-GRNN, CEEMDAN-BPNN, and CEEMDAN-ARIMA. The parameters of the GA, PSO, BPNN, and ARIMA methods and the parameter ranges of CEEMDAN used in the assays are presented by Table2. The parametric values for CEEMDAN, GA, PSO, BPNN, and ARIMA were taken from the literature [26, 60]. All experiments were performed in the MATLAB R2018b environment.Table 2
Parameter settings.
MethodDescriptionParametersCEEMDANComplete EEMD with adaptive noiseNoise standard deviation: 0.2Number of realizations: 100GAGenetic algorithmNumber of evolutionary algebras: 200Size of population: 20Fitness function: MSEPSOParticle swarm optimisationNumber of iteration generations: 200Size of particle: 20Fitness function: MSEBPNNBack propagation neural networkSize of the hidden layer: 10Maximal training epochs: 1000Learning rate: 0.001ARIMAAutoregressive integrated moving averageAkaike information criterion to decide parameters (p-d-q)


### 3.4. Results and Analyses

#### 3.4.1. Single Models

Without any decompositional process, the single models were applied straightly to the raw series of COPs. Our team compared the initial SVR and GA-SVR approaches with 1 important statistic modeling method, ARIMA, and three classical AI modeling methods, PSO-SVR, GRNN, and BPNN. The experiment outcomes are presented by Table3, in which the optimum forecast outcomes were displayed in bold.Table 3
Outcomes of single models.
Test datasetModelMSERMSEMAEMAPE (%)GA-SVR2.84361.68631.32282.4389PSO-SVR4.93322.22111.63652.9448SVR4.69382.16651.95373.6541GRNN7.47422.73393.00924.5098BPNN8.91352.98552.55454.8120ARIMA12.90693.59263.00925.5907As shown in Table3, amongst the entire single models, GA-SVR obtained the lowest MSE, RMSE, MAE, and MAPE values, whereas the ARIMA modeling method achieved the greatest MSE, RMSE, MAE, and MAPE values. Among the AI models, PSO-SVR achieved the lowest MSE, RMSE, MAE, and MAPE values. Among the SVR-related modeling methods, GA-SVR achieved lower MSE, RMSE, MAE, and MAPE values in contrast to the PSO-SVR and SVR models, revealing that the former outperformed the latter in terms of COP prediction. To put it in another way, use of the GA optimisation approach to search the optimal parametric results for SVR can uplift the forecast ability.Regarding the directional statistics, as presented by Table3, the GA-SVR modeling method achieved the highest values, indicating that it performed best in direction prediction among all the single prediction models. In addition, the DM analysis was utilized to evaluate if the forecast ability of GA-SVR remarkably outperformed other single modeling methods. Table 4 displays the statistical results of the DM analysis and p values (in brackets).Table 4
Results of DM test for single models.
Tested modelBenchmark modelPSO-SVRSVRGRNNBPNNARIMAGA-SVR−1.996 (0.0459)∗∗−3.376 (0.0007)∗∗∗−3.929 (0.0001)∗∗∗−4.258 (0.0000)∗∗∗−6.691 (0.0000)∗∗∗PSO-SVR−0.562 (0.5741)−0.9291 (0.3528)−1.505 (0.1324)−2.652 (0.0080)∗∗∗SVR−1.178 (0.2388)−1.827 (0.0677)∗∗−4.684 (0.0000)∗∗∗GRNN−1.01 (0.3125)−5.417 (0.0000)∗∗∗BPNN−2.356 (0.0185)∗∗∗∗∗statistical significance at 1%, ∗∗statistical significance at 5%.The DM test outcomes presented by Table4 reveal that the GA-SVR modeling method remarkably outperformed the statistic modeling method ARIMA and the AI modeling methods PSO-SVR, SVR, GRNN, and BPNN, as the relevant DM statistics were far lower than 0 and every p value was <0.05. The GA-SVR model also outperformed the PSO-SVR and SVR modeling methods as per the DM statistics, and the relevant p values were again <0.05, demonstrating that GA-SVR is significantly superior to PSO-SVR and the SVR in the majority of cases.


#### 3.4.2. Ensemble Models

Given the effectiveness of the “decomposition and ensemble” approach, our team introduced the decompositional approach CEEMDAN into the developed ensemble model herein. Thus, using the same decomposition approach (i.e., CEEMDAN), we compared the CEEMDAN-GA-SVR predictor with CEEMDAN-PSO-SVR, CEEMDAN-SVR, CEEMDAN-GRNN, CEEMDAN-BPNN, and CEEMDAN-ARIMA. The experiment simulation outcomes for the ensemble models are presented by Table5.Table 5
Outcomes for ensemble models.
Test datasetModelMSERMSEMAEMAPE (%)CEEMDAN-GA-SVR0.17090.41340.31580.6311CEEMDAN-PSO-SVR0.77650.88120.47150.8773CEEMDAN-SVR2.57471.60461.03551.8520CEEMDAN-GRNN2.39511.54761.24312.3158CEEMDAN-BPNN4.98012.23161.83983.4331CEEMDAN-ARIMA7.78602.79032.27694.2513CEEMDAN-GA-SVR achieved the best prediction results with the lowest MSE, RMSE, MAE, and MAPE values in every case, demonstrating that the developed CEEMDAN-GA-SVR modeling method outperformed every other ensemble model. The superior forecast performance of the developed CEEMDAN-GA-SVR model could be attributed to 2 primary reasons: the valid decompositional process of CEEMDAN, the better forecast capability of SVR with GA optimisation. Overall, the proposed CEEMDAN-GA-SVR achieved better prediction performance in contrast to the other prediction modeling methods.To enhance the persuasiveness of the outcomes, we introduced the DM test to study the forecast outcomes of the ensemble models; the statistic results and the relevant p values are presented by Table6.Table 6
Outcomes of DM test for ensemble models.
CEEMDANTested modelGA-SVRPSO-SVRSVRGRNNBPNNARIMACEEMDANGA-SVR−1.972 (0.0486)∗∗−2.363 (0.0181)∗∗−4.776 (0.0000)∗∗∗−4.077 (0.0000)∗∗∗−4.27 (0.0000)∗∗∗PSO-SVR−1.742 (0.0816)∗−3.111 (0.0019)∗∗∗−3.547 (0.0004)∗∗∗−3.673 (0.0002)∗∗∗SVR0.2545 (0.7991)−2.324 (0.0201)∗∗−4.431 (0.0000)∗∗∗GRNN−2.808 (0.0050)∗∗∗−3.764 (0.0002)∗∗∗BPNN−6.081 (0.0000)∗∗∗∗∗∗statistical significance at 1%, ∗∗statistical significance at 5%.As shown in Table6, when we compared the prediction outcomes of the developed CEEMDAN-GA-SVR modeling method with other models, the DM statistics were much smaller than 0 and the relevant p values were near 0 (p<0.05), indicating that CEEMDAN-GA-SVR significantly outperforms other models for crude oil price forecasting. Furthermore, CEEMDAN-GA-SVR was significantly superior to CEEMDAN-PSO-SVR, CEEMDAN-SVR, CEEMDAN-GRNN, CEEMDAN-BPNN, and CEEMDAN-ARIMA. In addition, the AI models (CEEMDAN-GA-SVR, CEEMDAN-PSO-SVR, CEEMDAN-SVR, CEEMDAN-GRNN, and CEEMDAN-BPNN) showed similar prediction performance to each other but all significantly outperformed the statistic modeling method ARIMA, indicating that AI modeling methods are superior to statistic models for COP prediction. In addition, CEEMDAN-GA-SVR was remarkably better than the initial SVR. For instance, the DM test statistic value between CEEMDAN-GA-SVR and SVR was −2.648, and the corresponding p value was approximately 0, proving that the former was significantly better than the latter. The DM analysis outcomes have evidenced that combining CEEMDAN decomposition, SVR prediction, GA optimisation, and the GRNN residual correction model can remarkably reinforce the forecast ability of crude oil price forecasting [65, 66].


## 3.1. Experimental Dataset Source and Evaluation Criteria

Brent crude oil is manufactured in the Brent area of the North Atlantic and the North Sea. Its crude oil futures account for more than 2/3 of the crude oil futures trading volume across the globe, and it is the benchmark for futures prices of crude oil in the international market [63]. The data utilized herein were acquired from Bloomberg and consisted of the weekly settlement prices of North Sea Brent (Brent) crude oil futures between June 2, 2017, and May 21, 2021. There were an overall 204 specimens in the dataset. For the sake of confirming the validity of our approach, the weekly COP time series for Brent was separated into learning and test datasets. Following some previous reports in the literature, the first 80% of the total observations in each time series were utilized as the learning dataset, while the remaining 20% were utilized as the test dataset [4, 29, 30]. Of the total samples, 163 observations were used as the training set, and 41 observations were utilized as a testing dataset to check the model’s effectiveness [64].Five indicators were used to evaluate the experimental results: MSE, root MSE (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and the Diebold-Mariano (DM) test [52]:(9)MSE=1N∑t=1NObservedt−Predictedt2,RMSE=1N∑t=1NObservedt−Predictedt2,MAE=1N∑t=1NObservedt−Predictedt,MAPE=1N∑t−1NObservedt−PredictedtObservedt∗100,in which N is the size of the assessed specimens, and Observedt and Predictedt represent the real and forecasted results at time t, separately. The DM analysis was utilized to calculate the statistic differences in the prediction accurateness of model pairs.


## 3.2. Description of COP Time Series

The weekly COP time series for Brent has obvious nonlinear characteristics, and its change trend shows strong volatility. For example, the price of Brent crude oil was approximately 60.2 USD/barrel in Jan 2020, before plummeting to 35.88 USD/barrel in April 2020; this change in just a few months is an example of the dramatic fluctuations and nonstationary features of COP time series [63].As shown in Table1, the average of the weekly Brent COP time series was 56.7140, indicating that the weekly Brent crude oil prices fluctuate at around 56 USD/barrel. The highest value of the time series was 71.7300 USD/barrel, while the lowest was 35.8800 USD/barrel. There was thus a large difference between the maximum and minimum prices; moreover, the standard deviation was 8.0580. These results indicate that the weekly Brent COP time series fluctuates violently.Table 1
Descriptive statistics for weekly brent COPs.
ObservationsMeanStandard deviationMinimumMaximumOil price20456.71408.058035.880071.7300Figure1 shows the original COPs and the decomposed parts. Clearly, the raw COPs show remarkable fluctuations. Amongst the decomposed parts, IMF1 to IMF4 show obvious high-frequency features in narrow ranges, whereas IMF5 to IMF6 and the residue show obvious low-frequency features in wide ranges. After obtaining the decomposed parts, the initial complex COP series prediction can now be divided into predictions of several simpler components.


## 3.3. Experimental Settings

The proposed CEEMDAN-GA-SVR model was evaluated and analyzed in two ways in our study. First, without any decomposition and ensemble, we compared the GA-SVR single model with other single models, which involved 1 important statistic model (ARIMA), three classical AI approaches (GRNN, back propagation neuronetwork [BPNN], and particle swarm optimisation SVR [PSO-SVR]), and the original SVR. Second, as previously finished studies have revealed that ensemble models using the “decomposition and ensemble” framework show better forecast abilities in contrast to single models for COP prediction, we compared forecast abilities between the developed CEEMDAN-GA-SVR and the rest of ensemble forecast modeling methods. Hence, the entire single models were employed to the forecast phase in ensemble models. Based on the identical COP series, our team tested if the developed CEEMDAN-GA-SVR model could significantly ameliorate the forecast ability. To prove the capacity of the proposed CEEMDAN-GA-SVR in prediction and CEEMDAN in decomposition, our study also compared CEEMDAN-GA-SVR with CEEMDAN-PSO-SVR, CEEMDAN-SVR, CEEMDAN-GRNN, CEEMDAN-BPNN, and CEEMDAN-ARIMA. The parameters of the GA, PSO, BPNN, and ARIMA methods and the parameter ranges of CEEMDAN used in the assays are presented by Table2. The parametric values for CEEMDAN, GA, PSO, BPNN, and ARIMA were taken from the literature [26, 60]. All experiments were performed in the MATLAB R2018b environment.Table 2
Parameter settings.
MethodDescriptionParametersCEEMDANComplete EEMD with adaptive noiseNoise standard deviation: 0.2Number of realizations: 100GAGenetic algorithmNumber of evolutionary algebras: 200Size of population: 20Fitness function: MSEPSOParticle swarm optimisationNumber of iteration generations: 200Size of particle: 20Fitness function: MSEBPNNBack propagation neural networkSize of the hidden layer: 10Maximal training epochs: 1000Learning rate: 0.001ARIMAAutoregressive integrated moving averageAkaike information criterion to decide parameters (p-d-q)


## 3.4. Results and Analyses

### 3.4.1. Single Models

Without any decompositional process, the single models were applied straightly to the raw series of COPs. Our team compared the initial SVR and GA-SVR approaches with 1 important statistic modeling method, ARIMA, and three classical AI modeling methods, PSO-SVR, GRNN, and BPNN. The experiment outcomes are presented by Table3, in which the optimum forecast outcomes were displayed in bold.Table 3
Outcomes of single models.
Test datasetModelMSERMSEMAEMAPE (%)GA-SVR2.84361.68631.32282.4389PSO-SVR4.93322.22111.63652.9448SVR4.69382.16651.95373.6541GRNN7.47422.73393.00924.5098BPNN8.91352.98552.55454.8120ARIMA12.90693.59263.00925.5907As shown in Table3, amongst the entire single models, GA-SVR obtained the lowest MSE, RMSE, MAE, and MAPE values, whereas the ARIMA modeling method achieved the greatest MSE, RMSE, MAE, and MAPE values. Among the AI models, PSO-SVR achieved the lowest MSE, RMSE, MAE, and MAPE values. Among the SVR-related modeling methods, GA-SVR achieved lower MSE, RMSE, MAE, and MAPE values in contrast to the PSO-SVR and SVR models, revealing that the former outperformed the latter in terms of COP prediction. To put it in another way, use of the GA optimisation approach to search the optimal parametric results for SVR can uplift the forecast ability.Regarding the directional statistics, as presented by Table3, the GA-SVR modeling method achieved the highest values, indicating that it performed best in direction prediction among all the single prediction models. In addition, the DM analysis was utilized to evaluate if the forecast ability of GA-SVR remarkably outperformed other single modeling methods. Table 4 displays the statistical results of the DM analysis and p values (in brackets).Table 4
Results of DM test for single models.
Tested modelBenchmark modelPSO-SVRSVRGRNNBPNNARIMAGA-SVR−1.996 (0.0459)∗∗−3.376 (0.0007)∗∗∗−3.929 (0.0001)∗∗∗−4.258 (0.0000)∗∗∗−6.691 (0.0000)∗∗∗PSO-SVR−0.562 (0.5741)−0.9291 (0.3528)−1.505 (0.1324)−2.652 (0.0080)∗∗∗SVR−1.178 (0.2388)−1.827 (0.0677)∗∗−4.684 (0.0000)∗∗∗GRNN−1.01 (0.3125)−5.417 (0.0000)∗∗∗BPNN−2.356 (0.0185)∗∗∗∗∗statistical significance at 1%, ∗∗statistical significance at 5%.The DM test outcomes presented by Table4 reveal that the GA-SVR modeling method remarkably outperformed the statistic modeling method ARIMA and the AI modeling methods PSO-SVR, SVR, GRNN, and BPNN, as the relevant DM statistics were far lower than 0 and every p value was <0.05. The GA-SVR model also outperformed the PSO-SVR and SVR modeling methods as per the DM statistics, and the relevant p values were again <0.05, demonstrating that GA-SVR is significantly superior to PSO-SVR and the SVR in the majority of cases.


### 3.4.2. Ensemble Models

Given the effectiveness of the “decomposition and ensemble” approach, our team introduced the decompositional approach CEEMDAN into the developed ensemble model herein. Thus, using the same decomposition approach (i.e., CEEMDAN), we compared the CEEMDAN-GA-SVR predictor with CEEMDAN-PSO-SVR, CEEMDAN-SVR, CEEMDAN-GRNN, CEEMDAN-BPNN, and CEEMDAN-ARIMA. The experiment simulation outcomes for the ensemble models are presented by Table5.Table 5
Outcomes for ensemble models.
Test datasetModelMSERMSEMAEMAPE (%)CEEMDAN-GA-SVR0.17090.41340.31580.6311CEEMDAN-PSO-SVR0.77650.88120.47150.8773CEEMDAN-SVR2.57471.60461.03551.8520CEEMDAN-GRNN2.39511.54761.24312.3158CEEMDAN-BPNN4.98012.23161.83983.4331CEEMDAN-ARIMA7.78602.79032.27694.2513CEEMDAN-GA-SVR achieved the best prediction results with the lowest MSE, RMSE, MAE, and MAPE values in every case, demonstrating that the developed CEEMDAN-GA-SVR modeling method outperformed every other ensemble model. The superior forecast performance of the developed CEEMDAN-GA-SVR model could be attributed to 2 primary reasons: the valid decompositional process of CEEMDAN, the better forecast capability of SVR with GA optimisation. Overall, the proposed CEEMDAN-GA-SVR achieved better prediction performance in contrast to the other prediction modeling methods.To enhance the persuasiveness of the outcomes, we introduced the DM test to study the forecast outcomes of the ensemble models; the statistic results and the relevant p values are presented by Table6.Table 6
Outcomes of DM test for ensemble models.
CEEMDANTested modelGA-SVRPSO-SVRSVRGRNNBPNNARIMACEEMDANGA-SVR−1.972 (0.0486)∗∗−2.363 (0.0181)∗∗−4.776 (0.0000)∗∗∗−4.077 (0.0000)∗∗∗−4.27 (0.0000)∗∗∗PSO-SVR−1.742 (0.0816)∗−3.111 (0.0019)∗∗∗−3.547 (0.0004)∗∗∗−3.673 (0.0002)∗∗∗SVR0.2545 (0.7991)−2.324 (0.0201)∗∗−4.431 (0.0000)∗∗∗GRNN−2.808 (0.0050)∗∗∗−3.764 (0.0002)∗∗∗BPNN−6.081 (0.0000)∗∗∗∗∗∗statistical significance at 1%, ∗∗statistical significance at 5%.As shown in Table6, when we compared the prediction outcomes of the developed CEEMDAN-GA-SVR modeling method with other models, the DM statistics were much smaller than 0 and the relevant p values were near 0 (p<0.05), indicating that CEEMDAN-GA-SVR significantly outperforms other models for crude oil price forecasting. Furthermore, CEEMDAN-GA-SVR was significantly superior to CEEMDAN-PSO-SVR, CEEMDAN-SVR, CEEMDAN-GRNN, CEEMDAN-BPNN, and CEEMDAN-ARIMA. In addition, the AI models (CEEMDAN-GA-SVR, CEEMDAN-PSO-SVR, CEEMDAN-SVR, CEEMDAN-GRNN, and CEEMDAN-BPNN) showed similar prediction performance to each other but all significantly outperformed the statistic modeling method ARIMA, indicating that AI modeling methods are superior to statistic models for COP prediction. In addition, CEEMDAN-GA-SVR was remarkably better than the initial SVR. For instance, the DM test statistic value between CEEMDAN-GA-SVR and SVR was −2.648, and the corresponding p value was approximately 0, proving that the former was significantly better than the latter. The DM analysis outcomes have evidenced that combining CEEMDAN decomposition, SVR prediction, GA optimisation, and the GRNN residual correction model can remarkably reinforce the forecast ability of crude oil price forecasting [65, 66].


## 3.4.1. Single Models

Without any decompositional process, the single models were applied straightly to the raw series of COPs. Our team compared the initial SVR and GA-SVR approaches with 1 important statistic modeling method, ARIMA, and three classical AI modeling methods, PSO-SVR, GRNN, and BPNN. The experiment outcomes are presented by Table3, in which the optimum forecast outcomes were displayed in bold.Table 3
Outcomes of single models.
Test datasetModelMSERMSEMAEMAPE (%)GA-SVR2.84361.68631.32282.4389PSO-SVR4.93322.22111.63652.9448SVR4.69382.16651.95373.6541GRNN7.47422.73393.00924.5098BPNN8.91352.98552.55454.8120ARIMA12.90693.59263.00925.5907As shown in Table3, amongst the entire single models, GA-SVR obtained the lowest MSE, RMSE, MAE, and MAPE values, whereas the ARIMA modeling method achieved the greatest MSE, RMSE, MAE, and MAPE values. Among the AI models, PSO-SVR achieved the lowest MSE, RMSE, MAE, and MAPE values. Among the SVR-related modeling methods, GA-SVR achieved lower MSE, RMSE, MAE, and MAPE values in contrast to the PSO-SVR and SVR models, revealing that the former outperformed the latter in terms of COP prediction. To put it in another way, use of the GA optimisation approach to search the optimal parametric results for SVR can uplift the forecast ability.Regarding the directional statistics, as presented by Table3, the GA-SVR modeling method achieved the highest values, indicating that it performed best in direction prediction among all the single prediction models. In addition, the DM analysis was utilized to evaluate if the forecast ability of GA-SVR remarkably outperformed other single modeling methods. Table 4 displays the statistical results of the DM analysis and p values (in brackets).Table 4
Results of DM test for single models.
Tested modelBenchmark modelPSO-SVRSVRGRNNBPNNARIMAGA-SVR−1.996 (0.0459)∗∗−3.376 (0.0007)∗∗∗−3.929 (0.0001)∗∗∗−4.258 (0.0000)∗∗∗−6.691 (0.0000)∗∗∗PSO-SVR−0.562 (0.5741)−0.9291 (0.3528)−1.505 (0.1324)−2.652 (0.0080)∗∗∗SVR−1.178 (0.2388)−1.827 (0.0677)∗∗−4.684 (0.0000)∗∗∗GRNN−1.01 (0.3125)−5.417 (0.0000)∗∗∗BPNN−2.356 (0.0185)∗∗∗∗∗statistical significance at 1%, ∗∗statistical significance at 5%.The DM test outcomes presented by Table4 reveal that the GA-SVR modeling method remarkably outperformed the statistic modeling method ARIMA and the AI modeling methods PSO-SVR, SVR, GRNN, and BPNN, as the relevant DM statistics were far lower than 0 and every p value was <0.05. The GA-SVR model also outperformed the PSO-SVR and SVR modeling methods as per the DM statistics, and the relevant p values were again <0.05, demonstrating that GA-SVR is significantly superior to PSO-SVR and the SVR in the majority of cases.


## 3.4.2. Ensemble Models

Given the effectiveness of the “decomposition and ensemble” approach, our team introduced the decompositional approach CEEMDAN into the developed ensemble model herein. Thus, using the same decomposition approach (i.e., CEEMDAN), we compared the CEEMDAN-GA-SVR predictor with CEEMDAN-PSO-SVR, CEEMDAN-SVR, CEEMDAN-GRNN, CEEMDAN-BPNN, and CEEMDAN-ARIMA. The experiment simulation outcomes for the ensemble models are presented by Table5.Table 5
Outcomes for ensemble models.
Test datasetModelMSERMSEMAEMAPE (%)CEEMDAN-GA-SVR0.17090.41340.31580.6311CEEMDAN-PSO-SVR0.77650.88120.47150.8773CEEMDAN-SVR2.57471.60461.03551.8520CEEMDAN-GRNN2.39511.54761.24312.3158CEEMDAN-BPNN4.98012.23161.83983.4331CEEMDAN-ARIMA7.78602.79032.27694.2513CEEMDAN-GA-SVR achieved the best prediction results with the lowest MSE, RMSE, MAE, and MAPE values in every case, demonstrating that the developed CEEMDAN-GA-SVR modeling method outperformed every other ensemble model. The superior forecast performance of the developed CEEMDAN-GA-SVR model could be attributed to 2 primary reasons: the valid decompositional process of CEEMDAN, the better forecast capability of SVR with GA optimisation. Overall, the proposed CEEMDAN-GA-SVR achieved better prediction performance in contrast to the other prediction modeling methods.To enhance the persuasiveness of the outcomes, we introduced the DM test to study the forecast outcomes of the ensemble models; the statistic results and the relevant p values are presented by Table6.Table 6
Outcomes of DM test for ensemble models.
CEEMDANTested modelGA-SVRPSO-SVRSVRGRNNBPNNARIMACEEMDANGA-SVR−1.972 (0.0486)∗∗−2.363 (0.0181)∗∗−4.776 (0.0000)∗∗∗−4.077 (0.0000)∗∗∗−4.27 (0.0000)∗∗∗PSO-SVR−1.742 (0.0816)∗−3.111 (0.0019)∗∗∗−3.547 (0.0004)∗∗∗−3.673 (0.0002)∗∗∗SVR0.2545 (0.7991)−2.324 (0.0201)∗∗−4.431 (0.0000)∗∗∗GRNN−2.808 (0.0050)∗∗∗−3.764 (0.0002)∗∗∗BPNN−6.081 (0.0000)∗∗∗∗∗∗statistical significance at 1%, ∗∗statistical significance at 5%.As shown in Table6, when we compared the prediction outcomes of the developed CEEMDAN-GA-SVR modeling method with other models, the DM statistics were much smaller than 0 and the relevant p values were near 0 (p<0.05), indicating that CEEMDAN-GA-SVR significantly outperforms other models for crude oil price forecasting. Furthermore, CEEMDAN-GA-SVR was significantly superior to CEEMDAN-PSO-SVR, CEEMDAN-SVR, CEEMDAN-GRNN, CEEMDAN-BPNN, and CEEMDAN-ARIMA. In addition, the AI models (CEEMDAN-GA-SVR, CEEMDAN-PSO-SVR, CEEMDAN-SVR, CEEMDAN-GRNN, and CEEMDAN-BPNN) showed similar prediction performance to each other but all significantly outperformed the statistic modeling method ARIMA, indicating that AI modeling methods are superior to statistic models for COP prediction. In addition, CEEMDAN-GA-SVR was remarkably better than the initial SVR. For instance, the DM test statistic value between CEEMDAN-GA-SVR and SVR was −2.648, and the corresponding p value was approximately 0, proving that the former was significantly better than the latter. The DM analysis outcomes have evidenced that combining CEEMDAN decomposition, SVR prediction, GA optimisation, and the GRNN residual correction model can remarkably reinforce the forecast ability of crude oil price forecasting [65, 66].


## 4. Conclusions

Accurate forecast of COPs is a common problem faced in theoretical research on energy economics and in industry. The present work focuses on the weekly North Sea Brent crude oil futures settlement price from June 2, 2017, to May 21, 2021. To uplift the prediction of COPs, we established a CEEMDAN-GA-SVR hybrid model incorporating CEEMDAN, GA, and SVR. This model enriches the current research on time series forecasting of international COPs and has certain practical and theoretical significance. First, CEEMDAN is used to realize the decomposition of the complex raw time series of COPs into a group of comparatively simpler subseries. Second, SVR is utilized to predict the target values of every decomposed subseries separately. Owing to the intricate parametric settings of SVR, GA is introduced to search for the optimum parametric values for SVR. Subsequently, our team assemble the predicted values of all individual subseries as the predicted values of the CEEMDAN-GA-SVR model. As far as we know, this is the first time that such a CEEMDAN-GA-SVR hybrid model has been introduced in the field of COP prediction.First, the experiment outcomes reveal the following: (1) in contrast to benchmark models, our CEEMDAN-GA-SVR hybrid model shows significantly enhanced forecast ability for COP prediction; (2) CEEMDAN performs better than EEMD for the decomposition of raw COP series; (3) GA can efficiently search for the optimal parameters for SVR, thereby improving the prediction of COPs.Second, the primary benefit of our CEEMDAN-GA-SVR is that it takes full advantage of the benefits of CEEMDAN, GA, and SVR, respectively, and can remarkably ameliorate the ability of COP prediction in contrast to certain latest forecast models. As SVR is suitable for the prediction of complex nonlinear series, the ensemble prediction model has comparatively strong interpretability in contrast to conventional regressive models. However, as we utilize GA to search for the best parametric settings for SVR, the overall execution time of the developed model is longer in contrast to other prediction models based on fixed parameters. In summary, our CEEMDAN-GA-SVR model shows significantly enhanced prediction performance and has promise for applications in crude oil price forecasting.Third, through empirical study on the weekly data of Brent crude oil futures settlement price in the North Sea, we predicted the trend of crude oil prices with relative accuracy. As an important commodity and strategic material, crude oil has important practical significance for the productivity and activities of countries and enterprises. The research results presented here could help government authorities to better forecast global COPs and to form more accurate oil price expectations in order to plan production and business activities more scientifically, which is vital for optimising the production structure of the national government authorities and preventing the risk of oil price fluctuations.



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*Source: 3741370-2022-06-27.xml*