# An Improved Grey Model with Time Power and Its Application

Authors: Jianming Jiang; Caixia Liu; Yuanguo Yao; Yumu Lu; Wanli Xie; Chong Liu
Journal: Complexity
 (2022)

---

## Abstract

The grey system model with time power, which is often called the GM(1,1,tα), appeals considerable interest of research due to its effectiveness in time series forecasting. Aimed to improve further the GM(1,1, tα) model, this paper introduces a new whitening equation with variable coefficient into the original whitening equation which extends applicable scope; as a result, an improved grey model with time power, namely, OGM(1,1, tα), is proposed. Firstly, the time response function of the novel model and the restored values of original series are deduced through grey modelling techniques. Secondly, the variable coefficient in the whitening equation and the time power are determined by particle swarm optimization algorithm. Two empirical examples are then used to verify the validity of the novel model. Finally, the novel model is applied to predict the oil consumption of China from 2004 to 2018. Results show the novel model outperforms other commonly-used competitive models, which can well serve a benchmark model for scholars and decision-makers.


---

## Body

## 1. Introduction

With the deep research of theoretical methods and the extension of application fields, the grey prediction model is gradually applied to predict the fluctuation characteristics from original tendency prediction [1]. Grey forecasting model, as an important branch of grey system theory [2], namely, GM(1,1), has been widely used in various fields, including energy [3], economic, and engineering. Among them, for example, Ding [4] investigated the natural gas consumption of China based on a novel approach of combining a new initial condition and rolling mechanism for enhancing the prediction accuracy. Wang et al. [5] explored the nonlinear relationship between carbon dioxide emissions and economic growth through constructing a PSO algorithm-based grey Verhulst model. Wu et al. [6] forecasted carbon dioxide emissions in the BRICS countries using a novel multivariate grey model. Wang et al. [1] proposed NLS-based nonlinear grey Bernoulli model to predict employee demand of high-tech enterprises in China. Kumar et al. [7] utilized time series models, including grey-Markov, grey model with rolling mechanism, and singular spectrum analysis, to forecast energy consumption in India. Duamn et al. [8] predicted e-waste for reverse logistics operations based on improved univariate grey models. Ma et al. [9] developed a novel time-delayed polynomial grey model to predict the natural gas consumption in China.Under this context, grey forecasting model has been generally received more attention in recent years, numerous grey extensive models have been deduced. For instance, Xie and Liu [10] put forward a discrete grey model which was abbreviated as DGM(1,1); in addition, he also depicted the connection between DGM(1,1) and GM(1,1), which extends applicable scope. Luo and Wei [11] proposed a grey model with polynomial term and its optimization methods. Afterward, Wei et al. [12] estimated the optimal solution for novel grey polynomial prediction model. Wang et al. [13, 14] proposed a series of methods of handling with seasonally fluctuating sequence with small-size characteristics. Wu and Zhang [15] proposed an improved time series interval forecasting method inspired by the literature [16]. Qian et al. [17] proposed a grey model with time term (short for GM(1,1, tα)).Considering the nonlinear characteristics hidden in original series, Chen et al. [18] firstly introduced the Bernoulli model into the traditional grey action quantity, so as to capture the nonlinear trend in raw sequences. Later, Wang et al. [19] optimized this model through changing the background-value coefficient. Wu et al. [20] simultaneously combined a new initial condition and dynamic background value in order to improve further NGBM(1,1) model and applied it to predict China’s GDP. Nguyen et al. [21] proposed nonlinear grey Bernoulli model based on Fourier transformation. Ma et al. [22] propounded a novel nonlinear multivariate grey Bernoulli model to predict the tourist income of China.As many studies reveal, it is a common sense that the fractional accumulation is an effective method to improve further the grey model. Wu et al. [23] originally proposed a grey model with fractional grey model. Later, the fractional grey model has appealed many interesting researches. For example, Wu et al. [24] proposed a fractional multivariate grey model to forecast electricity consumption of Shandong, China. Ma et al. [25] proposed a novel fractional time-delayed grey model with grey wolf optimizer to predict the natural gas and coal consumption in Chongqing, China. Wu et al. [26] investigated the novel fractional grey model, namely, FAGMO(1,1, k), to predict China’s nuclear energy consumption. Due to ease of computation of conformable fractional accumulation and difference, Ma et al. [27] introduced it into grey model; as a result, a novel conformable fractional grey model was proposed. Xie et al. [28] proposed a fractional grey model in opposite direction.Based on the previous literature, this paper proposed a novel grey model which combines a new whitening equation with variable coefficient and the existing GM(1,1,tα). The novelties of this paper are drawn as follows: (i) an improved grey model with time term, namely, OGM(1,1, tα), is proposed; (ii) the time response function of the model and the restored values are deduced in detail; (iii) the model parameters of the novel model are determined by a well-known algorithm, namely, particle swarm optimization (PSO) [29]; (iv) two examples and a real application are used to verify the effectiveness of the novel model.The rest of this paper is organized as follows: Section2 depicts the modelling produces of the existing GM(1,1, tα) and the novel model. Section 3 optimizes the parameters of the novel model based on PSO. Section 4 carries out two empirical examples to prove the validity of the novel model. Section 5 applies the novel model to predict oil consumption of China and the main conclusions are listed in the final section.


## 2. Methodologies

### 2.1. The Traditional GM(1,1,tα) Model

The modelling processes of the GM(1,1,tα) model proposed by Qian et al. [17] can be described as follows.Assume a nonnegative series to be(1)X0=x01,x02,…,x0n,n≥4,and then the first-order accumulative generating operation 1−AGO series is written as(2)X1=x11,x12,…,x1n,where x1k=∑i=1kx0i.The equation(3)dx1tdt+ax1t=btα+c,α>0,is called the whitening equation of GM(1,1, tα). a and btα+c refer the development coefficient and grey action quantity.The basic form of GM(1,1,tα) is given by(4)x0k+az1k=bkα+c,where z1k is called the background value and z1k=0.5×x1k+x1k−1.Using the least square method, the model parametersa, b, and c can be estimated, there is(5)a,b,cT=BTB−1BTY,where(6)B=−z122α1−z133α1⋮⋮⋮−z1nnα1,Y=x02x03⋮x0n.


### 2.2. The Improved GM(1,1,tα) Model

Based on the above description of the GM(1,1,tα) model, a new whitening equation with a variable coefficient is introduced in (3) so as to improve further the forecasting ability of the GM(1,1, tα) model; as a result, the improved GM(1,1, tα) model, namely, OGM(1,1, tα), is proposed. The modelling procedure of OGM(1,1, tα) is summarized as follows.Definition 1.
Letu>0; the whitening equation of a new grey forecasting model is given by(7)dx1tdtu+ax1t=btα+c,α>0.
Obviously, (7) turns to be (1) as u=1, and so the novel model should be regarded as a general form of GM(1,1, tα).
Here, (7) is easily rewritten as(8)dx1tdt+autu−1x1t=btαutu−1+cutu−1.
Integrating both sides of (8) over the interval k−1,k, one can write(9)∫k−1kdx1t+∫k−1kautu−1x1tdt=∫k−1kbtαutu−1dt+∫k−1kcutu−1dt.
By using two-point trapezoidal formula, (9) becomes(10)x0k+az11k=buu+αku+α−k−1u+α+cku−k−1u,where z11k is so-called the background value and z11k=0.5×uku−1x1k+uk−1u−1x1k−1.
Let(11)G=−z112uu+α2u+α−1u+α2u−1u−z113uu+α3u+α−2u+α3u−2u⋮⋮⋮−z11nuu+αnu+α−n−1u+αnu−n−1u,H=x02x03⋮x0n.
Similar to subsection2.1, the model parameters a, b and c are calculated through the least square method, there is,(12)a,b,cT=GTG−1GTH.Theorem 1.
GivenX0, X1 and a,b,cT, the solution to of (7) can be given by(13)x1t=e−atueax01+b∫1tfu,τdτ+caeatu−ea,where fu,t=e−atuutu+α−1.
Proof. First of all, multiplying both sides in (7) by eatu, there is,(14)eatudx1tdt+autu−1x1t=eatubtαutu−1+cutu−1.That is,(15)deatux1tdt=eatubtαutu−1+cutu−1.Afterward, integrating both sides of (15) over the interval 1,t, then(16)∫1tdeatux1τ=∫1teaτubταuτu−1+cuτu−1dτ.By simplifying (16), which yields(17)eatux1t−eax01=b∫1tfu,τdτ+caeatu−ea.This completes the proof.In particular, the integral item∫1tfu,τdτ in (13) could be approximately estimated using the following equation:(18)∫1tfu,τdτ≈∑i=21000△ifu,εi,where △i is taken as 0.001 for any i.And then sett=k, the time response function of the novel model is obtained and the restored values of X0,k=2,3,… can be acquired using the first-order inverse accumulative generating operation 1−IAGO, there is,(19)x^0k=x^1k−x^1k−1.


## 2.1. The Traditional GM(1,1,tα) Model

The modelling processes of the GM(1,1,tα) model proposed by Qian et al. [17] can be described as follows.Assume a nonnegative series to be(1)X0=x01,x02,…,x0n,n≥4,and then the first-order accumulative generating operation 1−AGO series is written as(2)X1=x11,x12,…,x1n,where x1k=∑i=1kx0i.The equation(3)dx1tdt+ax1t=btα+c,α>0,is called the whitening equation of GM(1,1, tα). a and btα+c refer the development coefficient and grey action quantity.The basic form of GM(1,1,tα) is given by(4)x0k+az1k=bkα+c,where z1k is called the background value and z1k=0.5×x1k+x1k−1.Using the least square method, the model parametersa, b, and c can be estimated, there is(5)a,b,cT=BTB−1BTY,where(6)B=−z122α1−z133α1⋮⋮⋮−z1nnα1,Y=x02x03⋮x0n.


## 2.2. The Improved GM(1,1,tα) Model

Based on the above description of the GM(1,1,tα) model, a new whitening equation with a variable coefficient is introduced in (3) so as to improve further the forecasting ability of the GM(1,1, tα) model; as a result, the improved GM(1,1, tα) model, namely, OGM(1,1, tα), is proposed. The modelling procedure of OGM(1,1, tα) is summarized as follows.Definition 1.
Letu>0; the whitening equation of a new grey forecasting model is given by(7)dx1tdtu+ax1t=btα+c,α>0.
Obviously, (7) turns to be (1) as u=1, and so the novel model should be regarded as a general form of GM(1,1, tα).
Here, (7) is easily rewritten as(8)dx1tdt+autu−1x1t=btαutu−1+cutu−1.
Integrating both sides of (8) over the interval k−1,k, one can write(9)∫k−1kdx1t+∫k−1kautu−1x1tdt=∫k−1kbtαutu−1dt+∫k−1kcutu−1dt.
By using two-point trapezoidal formula, (9) becomes(10)x0k+az11k=buu+αku+α−k−1u+α+cku−k−1u,where z11k is so-called the background value and z11k=0.5×uku−1x1k+uk−1u−1x1k−1.
Let(11)G=−z112uu+α2u+α−1u+α2u−1u−z113uu+α3u+α−2u+α3u−2u⋮⋮⋮−z11nuu+αnu+α−n−1u+αnu−n−1u,H=x02x03⋮x0n.
Similar to subsection2.1, the model parameters a, b and c are calculated through the least square method, there is,(12)a,b,cT=GTG−1GTH.Theorem 1.
GivenX0, X1 and a,b,cT, the solution to of (7) can be given by(13)x1t=e−atueax01+b∫1tfu,τdτ+caeatu−ea,where fu,t=e−atuutu+α−1.
Proof. First of all, multiplying both sides in (7) by eatu, there is,(14)eatudx1tdt+autu−1x1t=eatubtαutu−1+cutu−1.That is,(15)deatux1tdt=eatubtαutu−1+cutu−1.Afterward, integrating both sides of (15) over the interval 1,t, then(16)∫1tdeatux1τ=∫1teaτubταuτu−1+cuτu−1dτ.By simplifying (16), which yields(17)eatux1t−eax01=b∫1tfu,τdτ+caeatu−ea.This completes the proof.In particular, the integral item∫1tfu,τdτ in (13) could be approximately estimated using the following equation:(18)∫1tfu,τdτ≈∑i=21000△ifu,εi,where △i is taken as 0.001 for any i.And then sett=k, the time response function of the novel model is obtained and the restored values of X0,k=2,3,… can be acquired using the first-order inverse accumulative generating operation 1−IAGO, there is,(19)x^0k=x^1k−x^1k−1.


## 3. Optimization of Parameters by PSO

To assess the prediction accuracy of the novel model, two statistical indices are employed in this paper, which are the mean absolute percentage error (MAPE) and the root mean square error (RMSE), and defined as follows.(20)MAPE=1n−1x^0k−x0kx0k×100%,RMSE=1n−1∑i=2nx^0k−x0k2.In the above modelling procedures, the variable coefficientu and time-item coefficient α are assumed to be know. In this regard, this subsection is to determine these parameters, based on a well-known algorithm, namely Particle Swarm Optimization (PSO) algorithm [29]. this subsection constructs a simply optimization problem for searching the best parameters through minimizing the sum of simulated errors between simulated and actual values, which can be mathematically formulated as follows,(21)minu,αMAPE=1n−1x^0k−x0kx0k×100%.The detailed computational steps of the novel model based on PSO are summarized as follows.Step 1.
Initialize population. Setm = 50 particle and 2-dimensional space, each particle in this search space has initial position xi=xi,1,xi,2 and velocity vi=vi,1,vi,2,i=1,2,…,m.Step 2.
Calculate fitness values using (21).Step 3.
Update velocity and position of each particle using the following equations(22)vi=wvi+c1×rand×pbest−xi+c2×rand×gbest−xi,xi=vi+xi,where w refers inertia factor, c1 and c2 are acceleration factors (generally c1=c2=2), rand are random numbers that change in the range of 0,1.Step 4.
Update the best positionpbest and the best position of the whole particles gbest.Step 5.
Stopping criteria, ift≥Tmax or the current optimum position meets a pre-set minimum threshold, print the best values, otherwise return to Step 2.
The flowchart of the novel model by PSO are also given in Figure1 so as to provide a clear visualization.Figure 1
The flowchart of the novel model by PSO.


## 4. Validation of the Novel Model

This section provides two examples to examine the prediction performance of the novel model compared with other commonly-used grey system models, which are the traditional GM(1,1) model, the discrete grey model (DGM(1,1)), the non-homogeneous grey model (NHGM(1,1)) and the grey model with time-item (GM(1,1,tα)).Case 1.
(Forecasting the output values of the high technology industry of China) This subsection takes the output values of the high technology industry of China [30] as an example to verify the novel model. The raw data set is divided into two groups, the data from 2005 to 2012 are utilized to establish the novel model and competitive models, and the left 2 samples are used to examine the prediction performance.
First of all, the time-item coefficientα of GM(1,1, tα) and the variable coefficient u and the time-item coefficient α are all determined by PSO, the minimum MAPE values and the corresponding parameters can be seen in Table 1, the tracks of seeking for the optimum values of these parameters are graphed in Figure 2.
It is observed from Table1 that the RMSE and MAPE values of the novel model are 0.15 and 2.03% for simulation period, and those of the novel model are 0.22 and 1.32% for prediction period, respectively. These values are smaller than those of other competitors either in simulation or prediction period, this indicates that the novel model has a relatively higher accuracy in this case.Table 1
The simulated results and indices by five grey models.
YearRaw dataGM(1,1)DGM(1,1)NHGM(1,1)GM(1,1,tα)OGM(1,1,tα)α=1.37α=2u=1.6920053.3920064.164.064.074.514.164.3720074.974.724.735.114.734.9420085.575.495.515.855.485.4220095.966.396.416.786.406.2520107.457.447.467.947.487.4220118.758.668.699.398.728.77201210.2310.0710.1111.2010.1110.19RMSE0.210.200.600.200.15MAPE(%)2.702.537.662.272.03201311.611.7211.7713.4611.6511.62201412.7413.6413.6916.2813.3413.05RMSE0.640.682.830.430.22MAPE(%)4.064.4621.892.581.32The bold values represent the RMSE and MAPE (%) of the model proposed in this paper for model evaluation in the example.Figure 2
Track of seeking for the optimal parameters of the GM(1,1,tα) model (a) and the novel model (b).
(a)(b)Case 2.
(Forecasting the annual per capita electricity consumption in China) The data, gathered from the literature [12], are taken as the second example to prove the validity of the novel model. Similar to Case 1, the data are empirically grouped into the training and testing sets, the first 13 data are used to build the prediction models and the left 3 observations are used to examine the prediction accuracies of these models. The optimal parameters of the novel model and the GM(1,1, tα) model by PSO are listed in Table 2, and the tracks of seeking the best values are plotted in Figure 3.
It is easily seen that the novel model outperforms other competing models due to its lowest RMSE and MAPE values in simulation and prediction period, and so the novel model is suitable for prediction problem of the annual per capita electricity consumption in China at hand.Table 2
The simulated results and indices by five grey models.
YearRaw dataGM(1,1)DGM(1,1)NHGM(1,1)GM(1,1,tα)OGM(1,1,tα)α=1.21α=0.05u=0.1320001066.920011157.61265.961267.791051.451133.781156.1420021286.01399.871401.931236.871308.181286.1920031477.11547.951550.261430.351499.581476.8820041695.21711.701714.281632.231704.561691.5420051913.01892.771895.661842.881921.281918.3220062180.62092.992096.232062.682148.582153.1620072482.22314.392318.022292.042385.662394.3520082607.62559.212563.282531.362631.942641.1120092781.72829.922834.492781.072886.982893.0120103134.83129.283134.393041.643150.463149.8320113497.03460.303466.023313.533422.113411.4320123684.23826.343832.753597.233701.753677.73RMSE87.8088.24104.3049.9249.68MAPE(%)3.713.714.181.581.1520133993.04231.094238.273893.263989.203948.6920144132.94678.674686.704202.154284.354224.2620154321.05173.585182.574524.464587.104504.43RMSE600.42608.04136.80176.78121.04MAPE(%)12.9713.162.963.312.52Figure 3
Track of seeking for the optimal parameters of the GM(1,1,tα) model (a) and the novel model (b).
(a)(b)


## 5. Application of Forecasting Oil Consumption of China

China is currently in a state of developing structural adjustment and steady growth. The manufacturing and service sectors constitute the pillars of the economy. Besides, Economic growth act as the most important factor leading to carbon emission increase, this is because Chinese rapid economic growth in previous decades were heavily dependent on the resource-driven industries, which consumes enormous fossil energy and generates plenty of pollutants. Therefore, accurately forecasting oil consumption becomes crucial in energy systems, especially for fossil-fuel market.The data of oil consumption of China is collected from the official website National Bureau of Statistics of China (https://data.stats.gov.cn/english), as shown in Table 3. Similar to Case 1 and Case 2 (elaborated in Section 4), the parameters of the novel model and the GM(1,1, tα) model are determined by PSO, the minimum MAPE values and the corresponding parameters are given in Table 4, and the tracks of seeking for best values are graphed in Figure 4.Table 3
Annual oil consumption of China form 2004 to 2018 (Mote).
YearRaw dataYearRaw data2004322.62012487.62005328.12013508.92006352.12014529.52007369.62015561.82008376.82016574.02009393.62017597.52010448.92018628.02011465.6Table 4
The simulated results and indices by five grey models.
YearRaw dataGM(1,1)DGM(1,1)NHGM(1,1)GM(1,1,tα)OGM(1,1,tα)α=1.25α=1.25u=0.202004322.62005328.1329.39329.53321.90328.47328.402006352.1347.82347.96341.92345.25345.352007369.6367.28367.42362.62365.67367.402008376.8387.82387.97384.04388.16390.722009393.6409.52409.66406.20411.81414.382010448.9432.43432.58429.12436.04438.112011465.6456.62456.77452.83460.47461.792012487.6482.17482.31477.36484.83485.392013508.9509.14509.29502.74508.91508.922014529.5537.62537.77528.99532.59532.38RMSE9.219.2110.488.559.05MAPE(%)1.751.752.231.591.582015561.8567.70567.84556.15555.76555.792016574.0599.45599.60584.24578.36579.162017597.5632.99633.13613.30600.35602.492018628.0668.40668.54643.36621.69625.82RMSE29.8730.0212.4425.734.78MAPE(%)4.464.491.960.830.78The bold values represent the RMSE and MAPE (%) of the model proposed in this paper for model evaluation in the example.Figure 4
Track of seeking for the optimal parameters of the GM(1,1,tα) model (a) and the novel model (b).
(a)(b)The simulated results can be obtained once the unknown parameters are searched, the prediction results are tabulated in Table4.From Table4, in simulation period, the RMSE values of the novel model and other models are 9.21, 9.21, 10.48, 8.55 and 9.05, and the MAPE values of these model are 1.75%, 1.75%, 2.23%, 1.59% and 1.58%, respectively; in prediction period, the RMSE values of the novel model and other models are 29.87, 30.02, 12.44, 25.73 and 4.78, and the MAPE values of these model are 4.46%, 4.49%, 1.96%, 0.83% and 0.78%, respectively. It is not doubt that the novel model is superior to other competitors, followed by GM(1,1, tα), NHGM(1,1), GM(1,1) and DGM(1,1) in this case. The same findings can be found from Figure 5 that fitted values of the novel model are much closer than those of other models. In summary, the proposed model is appreciated model for forecasting oil consumption of China so as to provide a valuable reference for energy sectors and related enterprises, which asstanting decision-makers understand future information and formulate corresponding strategies in advance.Figure 5
Curves of raw data and simulated values in oil consumption of China from 2004 to 2018.


## 6. Conclusion and Future Work

Aimed to improve further the existing grey model with time item, abbreviated as GM(1,1,tα), a new whitening equation with variable coefficient is introduced for acquiring extensive whitening equation, as a consequence, an improved grey model with time item, namely OGM(1,1, tα), is proposed. The model has many advantages. First, OGM(1,1, tα) can be directly extended to fractional order. In addition, the background value coefficient can be adjusted dynamically. Accordingly, it has a good scalability and can be directly extended to other new types of models. Second, OGM(1,1, tα) has a simple structure and relatively few parameters, and the time consumption is usually tiny. For example, OGM(1,1, tα) is implemented in MATLAB environment on a computer with windows 10 system, Intel i5 CPU, and 8 GB of RAM. The time costs of the first case and the second case are only 0.029 934 s and 0.047 817 s, respectively. Thus, the model does not have the problem of occupying large computing resources.The main contributions of this paper can be summarized as follows.(1)
An improved GM(1,1,tα) is put forward, which is based on a novel whitening equation with variable coefficient.(2)
PSO algorithm is utilized to determine the unknown parameters, which are variable coefficient of the whitening equation and time-item parameter.(3)
GM(1,1,tα) has a good scalability and can be directly extended to other new types of models.(4)
The novel model is applied to predict the oil consumption of China due to its flexible forecasting ability.The scalability of this model is very good. For example, the grey model based on the proposed method can be directly extended to fractional order. In addition, the background value coefficient can be adjusted dynamically.Although the superiorities of the novel model have been discussed, there exist some issue, which should be noticed. For example, the inherent error caused by leap from the whitening equation to the basic form should be considered in the coming research. Besides, the fractional accumulation would be received more attention in next work.



---

*Source: 6910865-2022-01-29.xml*