Complexity

Volume 2018, Article ID 3869619, 12 pages

https://doi.org/10.1155/2018/3869619

## Forecasting Crude Oil Consumption in China Using a Grey Prediction Model with an Optimal Fractional-Order Accumulating Operator

^{1}College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China^{2}Chongqing Haifu Medical Technology Co. Ltd., Chongqing 400035, China

Correspondence should be addressed to Huiming Duan; moc.361@naudgnimiuh

Received 12 April 2018; Accepted 5 July 2018; Published 1 August 2018

Academic Editor: Mahardhika Pratama

Copyright © 2018 Huiming Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Crude oil, which is an important part of energy consumption, can drive or hinder economic development based on its production and consumption. Reasonable predictions of crude oil consumption in China are meaningful. In this paper, we study the grey-extended SIGM model, which is directly estimated with differential equations. This model has high simulation and prediction accuracies and is one of the important models in grey theory. However, to achieve the desired modeling effect, the raw data must conform to a class ratio check. Unfortunately, the characteristics of the Chinese crude oil consumption data are not suitable for SIGM modeling. Therefore, in this paper, we use a least squares estimation to study the parametric operation properties of the SIGM model, and the gamma function is used to extend the integer order accumulation sequence to the fractional-order accumulation generation sequence. The first-order SIGM model is extended to the fractional-order FSIGM model. According to the particle swarm optimization (PSO) mechanism and the properties of the gamma function of the fractional-order cumulative generation operator, the optimal fractional-order particle swarm optimization algorithm of the FSIGM model is obtained. Finally, the data concerning China’s crude oil consumption from 2002 to 2014 are used as experimental data. The results are better than those of the classical grey GM, DGM, and NDGM models as well as those of the grey-extended SIGM model. At the same time, according to the FSIGM model, this paper predicts China’s crude oil consumption for 2015–2020.

#### 1. Introduction

Energy is an important material basis for global economic growth and human social development. As an important component of energy consumption, the production and consumption of crude oil can drive or hinder economic development. At present, China is facing rapid economic growth, changes in consumer spending structures, and an economic development with an increasing dependence on crude oil resources [1, 2]. Crude oil supply and demand imbalances are becoming increasingly prominent. Low utilization of crude oil, irrational consumption structures, serious pollution, and other issues can restrict the development of China’s economy. With China’s industrialization, its urbanization, energy, and environmental constraints will increase. The settling of the contrast between the energy and economic development is related to the sustainable development of China’s economy and society.

Crude oil demand forecasting is an important part of the development of crude oil development strategies and the scientific, reasonable, and accurate analysis of China’s crude oil demand, which is needed not only to protect China’s energy security and effectively prevent the bottlenecking of crude oil supplies but also for the realization of China’s economic health. Sustainable and rapid development will have important impacts on these processes. China’s rapidly growing energy consumption and its structural changes continue to challenge China’s energy supply security. Therefore, effective methods of addressing the demand for crude oil are expected to become the basis for the policy formulation of China’s energy supply security and will directly affect the stability of social production and national energy security in addition to helping the Chinese government establish an independent demand forecasting mechanism for crude oil and the energy sector to achieve an effective market transformation.

There are many ways to forecast crude oil demands, including the autoregressive moving average (ARMA) model [3], autoregressive conditional heteroscedasticity (ARCH) model [4], generalized ARCH (GARCH) model [5], and other time series methods as well as via artificial neural networks [6], fuzzy theory predictions [7, 8], and grey system methods [9, 10]. Liu et al. [11] used a time series approach to forecast the US West Texas Lightweight (WTI) crude oil prices based on crude oil demands. Liang et al. [12] predicted China’s crude oil price using wavelet decomposition. Zhang [13] used the quadratic moving average method to predict the annual consumption of the next five years of oil consumption. Guo et al. [14] used soft computing and hard computing to forecast China’s crude oil demand. Azadeh et al. [15] analyzed the oil consumption of Canada, United States, Japan, and Australia from 1990 to 2005 using fuzzy-regression data envelopment. Azadeh et al. [16] predicted the crude oil prices using a fuzzy-regression algorithm. Park and Yoo [17] studied the dynamics of oil consumption and economic growth in Malaysia.

The grey model is simple and adaptable, can handle mutations of parameters, and does not require many data points for predictive updates. The forecasting model GM (1,1) [18] has been widely used in many fields, such as those of transportation, medicine, industry, agriculture, and military [19–21], since its introduction. Researchers have expanded a variety of new models, such as DGM (1,1), NDGM (1,1), and GM (1, N) [22–28], from the classic GM (1,1) model. Concurrently, the grey prediction model has been studied in detail, including its background value, modeling mechanism, combinatorial model, and model optimization [29–33]. Grey forecasting models have been successfully applied for crude oil demand forecasting: Huang et al. [34] have used the grey prediction model to predict global crude oil consumption. Xu [35] used the grey model to forecast China’s crude oil consumption. Mu Hailin et al. also used the grey model to predict China’s crude oil consumption.

The SIGM model [10] is an extended version of the classical GM (1,1) model. The SIGM model can optimize the model parameters, which are directly estimated from the differential equation, making its simulations and predictions more accurate. However, the parameters in the literature [10] are too cumbersome to estimate, so this paper uses the least squares estimation method to simplify the parameter estimations of the SIGM model and to obtain the corresponding formula. At the same time, the modeling data of the SIGM model is a first-order cumulative generation sequence. To achieve the desired modeling results, the raw data must conform to the class ratio test, but the data characteristics of China’s crude oil consumption do not meet the class ratio test. Therefore, this paper will promote the use of the SIGM model, which uses the gamma function to extend the integer order cumulative generation operator into the fractional-order cumulative generation operator, to extend the first-order cumulative generation sequence to the fractional-order cumulative sequence and to establish the FSIGM model of the fractional-order operator. At the same time, by using the mechanism of the particle swarm optimization (PSO) and the properties of the gamma function of the fractional-order generation operator, the optimal fractional particle swarm optimization algorithm of the FSIGM model is obtained, and the optimal fractional order is obtained using different data. Finally, the data describing the consumption of the crude oil in China from 2002 to 2014 are analyzed. The results show that the newly proposed FSIGM model has an improved accuracy and prediction accuracy over those of the original SIGM; however, its simulation accuracy is much higher than the classic GM, DGM, and NDGM models. The accuracy of the prediction is not much different from that of the GM and FSIGM models, but the simulation accuracy is obviously better than the DGM and NDGM models.

The sections of this paper are organized as follows: In Section 2, the basic concepts and properties of the GM (1,1) and SIGM models are introduced. In Section 3, the fractional-order SIGM model is proposed and its important properties are analyzed. Based on the mechanisms of the particle swarm optimization, the particle swarm optimization algorithm is obtained. In Section 4, the crude oil consumption in China from 2002 to 2014 is used for empirical analysis. The simulation results and prediction results of the FSIGM model are compared with the classical grey model GM, DGM, and NDGM models and the grey-extended SIGMD model. In Section 5, conclusions are drawn.

#### 2. Preliminaries

This section mainly introduces the definition and basic properties of the GM (1,1) model and the definition of the SIGM model. The least squares estimation is used to estimate the parameters of the SIGM model, which is simpler than the method used in the literature [10].

##### 2.1. GM (1,1) Model

Assume that the sequence:
is an original data sequence, and the sequence:
is the accumulated generation sequence of *X* (0), where

is the mean sequence of . where

*Definition 1. *Assume that the sequence , , and is shown as (1), (2), and (3), then
is a first-order equation with a variable grey system prediction model, which is referred to as GM (1,1) model [18]. Its parameter estimation:
where
The intrinsic reduction value of the GM (1,1) model is

##### 2.2. SIGM Model

*Definition 2 (see [10]). *For , , and given by (1), (2), and (3), and is a constant, then the following equation:
is the expanded form of GM (1,1) model.

By definition, we can get the following.

*Property 1. *The parameter vector of SIGM model is , using least squares estimation.
where are

*Definition 3. *The equation:
is the whitening equation of FSIGM model .

Thus, we can get the following theorem.

Theorem 1. *Assume that , and are given by Definition 1 and Property 1, and
*(1)*The time response function of the whitening (12) is
*(2)*The time response function of the whitening (28) is
*(3)*Restore value is
*

*Proof 1. *From (11):
When , there is . Thus, we can get (12), then from Definition 2, we can get (13) and (14).

*3. The FSIGM Model*

*In this section, we propose a new FSIGM model based on fractional-order accumulation generation, which uses the gamma function [36] to represent the parameter estimation of the fractional-order cumulative generation sequence and finds the optimal order using the adaptive particle swarm optimization [37] method.*

*3.1. Fractional Extension Operator*

*In Section 2.1, we have assumed that is 1-AGO; the -order cumulative generation sequence is defined below.*

*Definition 4. *Let by (1) be r-AGO, where
Equation (16) can be expressed as

*When , is called as integer order accumulation sequence; when , is called as fractional-order accumulation generation sequence.*

*In order to express the -order cumulative generation sequence with the gamma function, the definition and nature of the gamma function are given below.*

*Definition 5. * and ; is the gamma function of the real number defined as
Through the integral points, we can deduce the properties of the gamma function as follows:

*Property 2. *, when ,

*Through Definition 5 and Property 2, (17) can be expressed as
*

*Particularly, when , expanded coefficient is
*

*The grey reducing generation corresponds to the grey accumulating generation, which can be viewed as a process of grey release; it is the grey cumulative generation sequence to restore. Therefore, the grey accumulating generation operator and the grey reducing generation operator must satisfy the reciprocity.*

*Definition 6 (see [36]). *For given by (1), an -order reducing generation operator (RGO) sequence can be generated by -RGO as follows:
is called as fractional reducing generation operator -RGO ().

*3.2. The FSIGM Model*

*This section mainly introduces the fractional-order SIGM model, which is the FSIGM model, and studies its important properties. First, define the FSIGM model.*

*Definition 7. *Let be the original sequence, from Definition 1, and is the -order accumulation generation sequence of , which is given by Definition 4.
is called as FSIGM model, where is given by (19) and

*Specifically, when , (19) becomes ; it is the original form of the SIGM model.*

*According to the definition of the model FSIGM model, we can get the following properties.*

*Property 3. *The parameter vector of the FSIGM model , using least squares estimation:
where are
then

*Property 4. *The matrix in Property 3 and Property 4 can be represented by the gamma function as follows:

*Definition 8. *is the whitening equation of FSIGM model . The following theorem:

*Theorem 2. , and are given by Definition 7 and Definition 5, then
(1)The time response function of the whitening (28) is
(2)The time response function of the whitening (28) is
(3)Restore value is
where *

*Proof 2. *The FSIGM and SIGM models have the same structures, such that the SIGM model is a special case of FSIGM. The difference between the two models is that the FSIGM model uses the -order cumulative sequence of the original sequence as its modeling sequence, and the SIGM model uses the first-order accumulation sequence of the original sequence as the modeling sequence, so the conclusion is true.

*3.3. Optimization of the FSIGM Model*

*3.3. Optimization of the FSIGM Model*

*Particle swarm optimization (PSO) is a type of global optimization evolution algorithm and was proposed by Kennedy and Eberhart in 1995 [36]. The concept of the PSO algorithm is simple, needing adjustments of a small number of parameters, and is also easy to program. The method has been widely used in function optimization, neural network training, and other fields.*

*From Theorem 2, the restored value can be calculated. Next, the mean absolute percentage error (MAPE) is defined.
where in represents the raw data and represents a simulation value or a predicted value.*

*We want to obtain the optimal order , which minimizes the MAPE between and , by solving the following optimization problem:
*

*The PSO algorithm based on adaptive mutation of population fitness variance [37] is used to optimize the order, such that (33) is used as the fitness of the particle. The order of the minimum mean relative error can then be obtained. The adaptive mutation particle swarm optimization algorithm of the optimal sequence is as follows:*

*Step 1. *Randomly initialize the position and velocity of the particle swarm, taking , which is the mean of the FSIGM model.

*Step 2. *Set in the particle to the current position; thus, is set to the best particle position in the initial population.

*Step 3. *Calculate the average relative error of the fractional operator FSIGM model when = . The specific steps are as follows:
(1)Calculate the -order cumulative generation sequence of the original sequence , produce the mean generation sequence with consecutive neighbors of , and calculate the first-order cumulative generation operator of .(2)Solve the parameter and then calculate the reduction value according to (31) to find the simulation value of .(3)Calculate the average relative error of () according to (32).(4)Determine whether is less than the given convergence value ; if this condition is satisfied, then implement the ninth step; otherwise, implement the fourth step.

*Step 4. *For all particles of the particle group, do the following.
(1)Update the position and speed of the particle:
where (2)If the particle fit is better than the fit of , can be set as the new position.(3)If the particle fit is better than the fitness of , can be set as the new position.

*Step 5. *Calculate the population variance fit and

*Step 6. *Calculate the probability of the variation , where

*Step 7. *Generate a random number ; if , perform a mutation operation according to (36); otherwise, perform the eighth step.

*Step 8. *Determine whether the algorithm convergence criteria are met; if these conditions are satisfied, perform the ninth step; otherwise, return to the third step.

*Step 9. *Output such that is the optimal value. Output . At the same time, determine the predictive value of the FSIGM model and the average relative error MAPE.

*It can be seen that the modeling steps of the FSIGM model are shown in Figure 1.*