To deal with the forecasting with small samples in the supply chain, three grey models with fractional order accumulation are presented. Human judgment of future trends is incorporated into the order number of accumulation. The output of the proposed model will provide decision-makers in the supply chain with more forecasting information for short time periods. The results of practical real examples demonstrate that the model provides remarkable prediction performances compared with the traditional forecasting model.
China Postdoctoral Science Foundation2018M6305621. Introduction
The supply chain forecasting can be made more accurate when human judgements are incorporated into the forecast system [1]. The performance of purely quantitative forecasting method can be flawed when historical data is limited [2]. In certain cases, the number of available samples is so scarce that providing reliable estimates is a challenging problem [3]. In order to analyze and predict the small samples systems accurately, a large number of studies on supply chain forecasting using grey models and improved grey models have been reported [4–7]. The GM(1,1) (grey first-order differential equation model) forecasting model is formulated for solving limited time series data [8]. There is no strict hypothesis for the distribution of parent data. However, the reliability and validity of the GM(1,1) have never been discussed. First, without considering other causes when using limited time series data, the forecasting of the GM(1,1) is unreliable and provides insufficient information to a decision-maker. Second, human judgment cannot be incorporated into the forecasting systems. A grey prediction model with fractional-order accumulation is newly proposed and has better performance and more freedom compared with the traditional grey model [9]. Multivariable nonequidistance grey model with fractional order accumulation is discussed [10]. The grey model predictor design is modified by using optimal fractional-order accumulation calculus [11]. The air quality indicators in the Beijing-Tianjin-Hebei region are predicted by the grey prediction model with fractional-order accumulation [12]. Therefore, in This paper, three grey models with fractional-order accumulation are put forward. The output of the proposed model can obtain a general interval. This interval is not the grey number [13]. It will provide more forecasting information for a short time period. Most importantly, human judgment for future trend is incorporated into the order number of accumulation.
The rest of the paper proceeds as follows. Grey single variable forecasting models with fractional order accumulation are presented in Section 2. The multivariable grey model with fractional order accumulation is proposed in Section 3. Interval forecasting method of grey models for supply chain management is discussed in Section 4. Some conclusions are given in the last section.
2. Grey Single Variable Forecasting Models with Fractional Order Accumulation
For the original data sequence X(0)={x(0)(1),x(0)(2),…,x(0)(n)}, a new sequence X(p/q)={x(p/q)(1),x(p/q)(2),…,x(p/q)(n)} can be generated by the fractional order accumulated generating operator (FAGO) as x(p/q)(k)=∑j=1kk-j+r-1k-jx(0)(j),k=1,2,…,r [14]. GMp/q(1,1) model and grey exponential smoothing are given in the following way.
2.1. GMp/q(1,1) ModelDefinition 1.
(1)xp/qk-xp/qk-1+azp/qk=bis referred to as the original form of the GMp/q(1,1) model, where z(p/q)(k)=(xp/qk+xp/qk-1)/2,k=2,3,…,n. It is the traditional GM(1,1) model when p/q=1. The ordinary least squares estimate sequence of the GMp/q(1,1) model satisfies(2)a^b^=BTB-1BTYwhere(3)Y=xp/q2-xp/q1xp/q3-xp/q2⋮xp/qn-xp/qn-1,B=-zp/q21-zp/q31⋮⋮-zp/qn1.
The solution of the whitenization equation dx(p/q)(t)/dt+ax(p/q)(t)=b for GMp/q(1,1) is given by (4)x^p/qt=x01-b^a^e-a^t+b^a^,
The p/q(0<p/q≤1) order inverse accumulated generating operator (IAGO) of X(p/q) is (5)X^0=αp/qX^p/q=α1X^1-p/qk=α1x^1-p/q1,α1x^1-p/q2,…,α1x^1-p/qn=x^01,x^1-p/q2-x^01,…,x^1-p/qn-x^1-p/qn-1.
The procedures of GMp/q(1,1) can be summarized as follows.
Step 1.
For the sequence x(0)={1763,2376,2763,2983,3216} from [4], 0.7-order accumulation sequence x(0.7)={1763,3610,5475,7275,9094}.B and Y in (2) are(6)B=-26871-45431-63751-81841,Y=1847186518001819.
Step 2.
Substituting B and Y into (2), we have the parameters a,b. Then GM0.7(1,1) can be represented as (7)dx0.7tdt+0.008x0.7t=1877,t=1,2,⋯.
Step 3.
The 0.7-order accumulated generating operation values x^(0.7) can be obtained by employing (4): (8)x^0.7=1763,3618,5458,7283,9094.The 2-order accumulated generating operation values x^(1) is (9)x^1=x^0.70.3=1763,4147,6888,9890,13101.
The predicted values x^1(0) are x^1(0)={1763,2384,2740,3002,3211}, which are listed in Table 1. Mean absolute percentage error (MAPE=100%(1/n)∑k=1n|(x0k-x^0k)/x(0)(k)|) compares the actual values with the forecasted values to evaluate the precision. The use of intelligent methods is a new trend in the grey models [15, 16]. In this paper, particle swarm optimization is adopted to find the optimal order which produces the minimum MAPE. The experiments are conducted in the MATLAB R2015b.
The predictive values of different models.
Time
resilience performance
GM(1,1)
GM0.7(1,1)
1
1763
1763
1763
2
2376
2441
2384
3
2763
2685
2740
4
2983
2954
3002
5
3216
3250
3211
MAPE
2.0
0.4
The results of Table 1 mean that GM0.7(1,1) have better performance than the traditional GM(1,1). To investigate the feasibility of the GMp/q(1,1) model in the supply chain, the following cases are given.
The Supply Chain Performance Resilience Forecasting Example [4]. The periodic resilience performance indicators of case firm are listed in Table 2. By GMp/q(1,1), respectively, the forecasting values and MAPE of different models are given in Table 3.
The periodic resilience performance indicators of a firm [4].
Time
RPI 1
RPI 2
RPI 3
RPI 4
RPI 5
1
2386
1345
674
6784
1763
2
2399
1876
567
5647
2376
3
3453
2345
453
4563
2763
4
3645
2784
363
3743
2983
5
4064
3764
278
3345
3216
The error analysis of different models.
RPI 1
RPI 2
RPI 3
RPI 4
RPI 5
MAPE of GM(1,1)
6
3
1
3
2
MAPE of GMp/q(1,1)
4.1
2.3
0.4
1.2
0.4
In Table 3, the error analysis shows that GMp/q(1,1) ensure the best fit of the data to achieve strong prediction capability. GMp/q(1,1) can predict the supply chain resilience of an Indian electronics manufacturer.
The Sales Volume of Printers Forecasting Example [20]. Shih et al used the sales volume of printers from 2002 to 2007 as training data. The volume of 2008 is the testing data. Actual values are listed in Table 4 and the errors of eight models are shown in Table 5. The results show that the forecasting accuracy of the GM0.4(1,1) is better than traditional GM(1,1).
Actual data of the sales volume of registered printers in Taiwan by year.
Year
2002
2003
2004
2005
2006
2007
2008
Sales volume
1047957
1138440
1091705
1083829
981984
838822
820161
The forecasting values and MAPE of different models.
Forecasting technique
Forecasting value for 2008
Actual value
MAPE
Moving average
910403
11.0
Exponential smoothing
854168
4.1
Quadratic trend
646699
21.2
Cubic trend
665629
820161
18.8
Non-linear trend
876998
6.9
ANP
830797
1.3
GM0.4(1,1)
817142
0.4
GM(1,1)
836018
1.9
The LCD TV Demand Forecasting Example [19]. Tsaur collected the LCD TV demand data from 2001 to 2006 as shown in Table 6. It was obvious that the collected data are limited time series data, and therefore the possible methods such as GM(1,1) and single exponential smoothing model were used for forecasting. The forecasting results are shown in Table 6. In Table 6, we find that the GMp/q(1,1) has better forecasting performance of MAPE than the single exponential smoothing model.
The predictive values of different models (units:ten thousand).
Year
LCD TV demand
single exponential model
AR(2) model
GM0.9(1,1)
2001
81
59
81
2002
150
151
95
197
2003
393
149
183
423
2004
970
567
505
898
2005
2018
1086
1092
1903
2006
4300
2682
2551
4026
MAPE
37.6
41.1
10.9
2.2. Grey Exponential Smoothing Model
FAGO is widely used in grey models for its ability to smooth the randomness of original data [9]. Through FAGO, the disorderly data may be converted into regular form. Actually, by using FAGO, the disorderly sequence may be converted into an approximately increased sequence. For example, X(0)={16,3,7,2,4,5}, the 0.81-AGO sequence is X(0.81)={16,15.96,21.16,20.86,23.28,26.54}. The lines of these sequences are illustrated in Figures 1 and 2, respectively. Comparing the two lines, it is clear that the trend of sequence X(0.81) in Figure 2 is more obvious than the sequence X(0) in Figure 1.
The line of X0.
The line of X0.81.
An irregular and increased sequence can be predicted by using double exponential smoothing (GDES). Then we give the following definition.
Definition 2 (see [21]).
For the original time series X(0)={x(0)(1),x(0)(2),…,x(0)(n)}, r-AGO is given in Definition 1. GDES follows the equations (10)S′k=αxrk+1-αS′k-1S′′k=αS′k+1-αS′′k-1bk=α1-αS′k-S′′kak=2S′k-S′′kwhere 0≤α≤1,S′(k) and S′′(k) are the single and double exponential smoothing values for time k, respectively. The forecasting form is x(r)(k)=ak+kbk.
If r=0, GDES is the traditional double exponential smoothing. The process of calculating GDES can be summarized as follows.
Step 1.
Set the order number r and obtain the r-FAGO sequence of X(0) according to Definition 1.
Step 2.
Calculate the parameters (ak and bk) by using Definition 2.
Step 3.
Compute the predictive value using x^(r)(k+m)=ak+mbk, where m is the out-of-sample size.
Step 4.
Transform the prediction value back to the original sequence by means of IAGO [21].
In this paper, the predictive values are calculated by using different α′s. The α that produces a small mean square error for the fitted values and shows an expected future growth is chosen. We assumed that S′(0) and S′′(0) are equal to the initial historical values.
In this case, the data from [17] is the spare demand of a firm. The accuracy of demand forecast significantly affects the firm‘s sustainability and profitability. To compare the GDES with traditional double exponential smoothing, the smoothing constant of GDES and traditional double exponential smoothing is set to α=0.69. The predictive values of two models are shown in Table 7.
The predictive values of different models.
Year
Actual demand [17]
GDES
traditional double exponential smoothing
2006
16
16
16
2007
3
3
-2
2008
7
9
4
2009
2
2
-1
2010
4
4
3
2011
5
6
5
MAPE
8.5
62.4
As shown in Table 7, GDES can enhance the accuracy of small data forecasting.
3. The Multivariable Grey Model with Fractional Order Accumulation
The existing multivariable grey models (GMC(1,N)) all used first-order accumulated generating operation sequence [22–26]. In this section, the fractional order accumulated generating operation sequence is introduced into the GMC(1,N) model.
Definition 3.
It is assumed that X1(0)={x1(0)(1),x1(0)(2),…,x1(0)(r)} is the sequence of system characteristic and Xi(0)={xi(0)(1),xi(0)(2),…,xi(0)(r)}(i=2,3,…,N) are the sequences of relevant factors. Then (11)dx1p/qtdt+b1x1p/qt=b2x2p/qt+b3x3p/qt+⋯+bnxnp/qt+uis the GMCp/q(1,N) model, where xi(p/q)(k) is the p/q order accumulation of xi(0)(k), xi(p/q)(k)=∑j=1kk-j+r-1k-jxi(0)(j),k=1,2,…,r [14]. It is the traditional GMC(1,N) model when p/q=1. Using ordinary least squares, the parameters of GMCp/q(1,N) model are estimated: (12)b^1,b^2,…,b^n,u^T=BTB-1BTYwhere(13)Y=x1p/q2-x1p/q1x1p/q3-x1p/q2⋮x1p/qn-x1p/qn-1,B=-0.5x1p/q2+x1p/q1x2p/q2⋯xnp/q21-0.5x1p/q3+x1p/q2x2p/q3⋯xnp/q31⋮⋮⋮⋮⋮-0.5x1p/qr-1+x1p/qrx2p/qr⋯xnp/qr1.
Set x^1(0)(1)=x1(0)(1); when t≥2, the convolution integral of the right hand of (12) can also be discretised as (14)x1p/qt=x101e-b1t-1+∑τ=2te-b1t-τ+0.5fτ+fτ-12,where f(t)=b2x2(p/q)(t)+b2x2(p/q)(t)+⋯+bnxn(p/q)(t)+u. Then the p/q(0≤p/q≤1) order inverse accumulated generating operator of the predicted value is the same as (5).
For example, the customer perception indicators of engine product are listed in Table 8 from [18]. To obtain the sales volume forecasting value, the engine sales volume is X1(0). Energy efficiency, product safety, cost performance, and service promptness are X2(0),X3(0),X4(0), and X5(0), respectively. This case has limited data. Thus we can build GMCp/q(1,5). GMC0.15(1,5) has the smaller MAPE than the GMCp/q(1,5) with the other accumulated order numbers. Thus the results of GMC0.15(1,5) are given in Table 9.
The customer perception indicators of engine product [18].
Year
sales volume
energy efficiency
product safety
cost performance
service promptness
2005
850.15
8.2112
1
1
1
2006
972.18
8.5027
1.0326
1.0786
1.0405
2007
1078.47
8.2711
1.0352
1.1029
1.007
2008
1069.28
8.0806
1.0521
1.0624
0.9818
2009
1259.26
8.1915
1.0846
1.0277
1.0028
2010
1454.51
9.0849
1.0911
1.0705
1.1229
2011
1413.92
8.7934
1.0404
0.9908
1.0796
2012
1672.11
8.9042
1.0612
0.9769
1.0992
2013
1948.23
9.3961
1.1289
1.0243
1.1704
2014
2063.35
9.938
1.2214
0.9711
1.3701
The predictive values of different models.
Year
sales volume
Quadratic polynomial model [18]
GMC0.15(1,5)
2005
850.15
850.15
850.15
2006
972.18
861.7971
965.79
2007
1078.47
929.5540
1055.96
2008
1069.28
1058.6918
1144.18
2009
1259.26
1103.4097
1264.41
2010
1454.51
1285.5118
1399.63
2011
1413.92
1525.6792
1555.71
2012
1672.11
1567.0815
1743.53
2013
1948.23
1786.3352
1934.37
2014
2063.35
2095.0247
2100.82
MAPE
8.24
3.0
By comparing the MAPE in Table 9, the example of demand forecasting for engine products demonstrated the applicability and validity of the GMCp/q(1,5).
Actually, X1(0) is taken as the supply chain demand. Xi(0)={xi(0)(1),xi(0)(2),…,xi(0)(r)}(i=2,3,…,N) are the sequences of relevant factors. GMCp/q(1,N) are suitable to predict the supply chain demand, because GMCp/q(1,N) can depict the model more precisely with new degrees of freedom and performances.
4. Interval Forecasting of Grey Models for Supply Chain Management
Fractional derivatives accumulate the whole history of the system in weighted form and it is referred to as the memory effect. As we know, big samples forecasting models depend on statistical laws. In this section, the small samples forecasting models which depend on the memory effect are a new path. For example, x(1)(k) in grey forecasting models denotes the weight of x(0)(i)(i=1,2,…,k) as 1. The larger p/q of x(p/q)(k) is, the larger the weight of old data is. The smaller p/q of x(p/q)(k) is, the smaller the weight of old data is. Reducing p/q can reduce the weights of old data, which can put more emphasis on the newer data.
Although the grey models (including GMp/q(1,1), GDESp/q, and GMCp/q(1,N)) have better forecasting performance, the input data is too limited and the forecasting values are point estimations which supply too limited information for a decision-maker. In this section, in order to obtain a better fitted model with smaller estimated errors and a smaller support forecasting interval, a novel fractional grey modelling mechanism is applied for solving limited time series data using the following steps.
Actually, for many forecasting cases in the supply chain, the accumulated order number often satisfied 0<p/q<1. If the future trend is similar to the newer data, the setting value of p/q is smaller. If the future trend is similar to the order data, the setting value of p/q is bigger. In the supply chain. for many forecasting cases with limited information, if it is difficult to judge the future trend, the forecasting value must be in a forecasting interval. The upper value and the lower value of forecasting interval can be obtained by different fractional grey models.
For the original data sequence X(0)={x(0)(1),x(0)(2),…,x(0)(n)}, the forecasting values of GM0.1(1,1) model (or GDES0.1 and GMC0.1(1,N)) are denoted as (15)X^0.10=x^0.101,x^0.102,…,x^0.10n,and the forecasting values of GM0.9(1,1) model (or GDES0.9 and GMC0.9(1,N)) are denoted as (16)X^0.90=x^0.901,x^0.902,…,x^0.90n.For the same point k, the smaller of x^0.9(0)(k) and x^0.1(0)(k) is set to the lower value of the forecasting interval. The bigger of x^0.9(0)(k) and x^0.1(0)(k) is set to the upper value of the forecasting interval. Thus we can obtain a grey forecasting interval. For example, (17)X0=1763,2376,2763,2983,3216,the forecasting values of GM0.1(1,1) model (or GDES0.1 and GMC0.1(1,N)) are (18)X^0.10=1763,2353,2756,3023,3197,3307,and the forecasting values of GM0.9(1,1) model (or GDES0.9 and GMC0.9(1,N)) are (19)X^0.90=1763,2418,2709,2973,3233,3497.Therefore, the grey forecasting interval values are (20)1763,1763,2353,2418,2709,2756,2973,3023,3197,3233,3307,3497.The above method is called the grey interval model (GIM). To investigate the feasibility of GIM in supply chain forecasting, seven cases are listed as follows.
Case 1.
The data are from [27]. The data come from the sales volume of semiconductor components in a firm. To compare the proposed model with the other model, the data before November are the training set. The fitting values of different models are given in Table 10.
The predictive results (December) of different models are given in Table 11.
We can clearly see from the results given in Table 10 that GMp/q(1,1) can obtain higher fitting accuracy. From the results given in Table 11, we can see that GIM can obtain shorter intervals including the actual value. But the simple linear regression derives a wide forecasting interval which provides too little information for the decision-maker.
The fitting values of different models.
Month
sales volume
GM0.1(1,1)
GM0.9(1,1)
1
103148
103148
103148
2
119066
119269
124066
3
137468
139992
143842
4
155892
162736
164238
5
186370
187063
186099
6
216230
212887
209924
7
249776
240225
236108
8
266829
269135
265023
9
294475
299698
297046
10
334595
332004
332578
11
365988
366158
372051
MAPE
1.4
2.4
The predictive results of different models.
model
sales volume
predictive values
interval value
simple linear regression (Xue 2015)
414032
436402
[425763, 494570]
GIM
414032
[402269, 415940]
Case 2.
The data come from [28]. The data refer to the numbers of end-of-life vehicles in the West Anatolia. Forecasting the return flow of an end-of-life product is important for all decision levels of the reverse supply chain. In this paper, the predictive results of GIM are given in Table 12. The results of interval value can provide guidance to the managers and practitioners of recovery and recycling systems.
The predictive results of number of end-of-life vehicles.
Year
number of end-of-life vehicles
interval value
2008
1977
[1977, 1977]
2009
2455
[2407, 2567]
2010
2818
[2460, 2527]
2011
2266
[2422, 2432]
2012
2241
[2319, 2369]
2013
2265
[2199, 2320]
Case 3.
The data come from [29]. In recent years, China’s high production of thin film transistor liquid crystal display (TFT-LCD) panels has led to intense competition in this industry, causing its supply to be greater than the actual demand. Under such circumstances, reducing inventory levels and inventory turnover is a critical issue faced by panel manufacturers. The production quantity is maintained at an appropriate balance point considering the total cost. To achieve this balance in production marketing coordination, an accurate short-term demand forecast is essential. Because the demand for TFT-LCD panels is affected tremendously by the global business cycle, whereas the business cycle has substantially changed in recent years, the use of numerous long-term historical observations does not satisfy the needs of the short-term forecasts. Therefore, this study applied data produced by a leading Taiwanese TFT-LCD panel manufacturer to verify the forecasting performance of GIM [29]. In this paper, the predictive results of GIM are given in Table 13. The actual value is in the forecasting interval. It demonstrates that GIM can provide more information with predictive values.
The predictive results of demand of TFT-LCD panels.
Months
Actual value
GM0.1(1,1)
GM0.9(1,1)
interval value
4
1.718
1.718
1.718
[1.718, 1.718]
5
1.728
1.714
1.715
[1.714, 1.715]
6
1.714
1.735
1.740
[1.735, 1.740]
7
1.753
1.746
1.741
[1.741, 1.746]
MAPE
0.6
0.7
8
1.742
1.748
1.729
[1.729, 1.748]
Case 4.
The data come from [30]. They are the return quantity for a third party e-waste firm in Turkey. The predictive results are listed in Table 14.
The predictive results in Table 14 indicate that GM0.1(1,1) and GM0.9(1,1) can obtain small fitting errors. By GIM, the obtained forecasting interval indeed includes the actual value. Thus, we can conclude that GIM is effective for limited sample forecasting problems.
The predictive results of return quantity for third party.
Time
Actual value
GM0.1(1,1)
GM0.9(1,1)
interval value
1
536.4
536.4
536.4
[536.4, 536.4]
2
538.61
538.2
535.8
[535.8, 538.2]
3
569.35
569.6
554
[554, 569.6]
4
613
612.8
565.3
[565.3, 612.8]
MAPE
0.03
2.7
5
633.32
664.6
572.6
[572.6, 664.6]
Case 5.
Shih et al. used the sales volume of printers from 2002 to 2007 as the training data. The volume of 2008 is the testing data [20]. Actual values are listed in Table 4 and the results of GIM are shown in Table 15. The results show that the actual value of 2008 is in the forecasting interval. Thus, we can conclude that GIM is effective for the limited sample forecasting problem.
The forecasting interval value of the proposed model.
Forecasting technique
Actual value
interval value
GIM
820161
[813065, 963371]
Case 6.
Actual values are listed in Table 7 and the results of GIM are shown in Table 16. As shown in Table 16, the results show that the actual demand values are all in the forecasting interval. It also indicates that highly volatile demand can obtain volatile intervals. These results can provide more information for the supply chain manager. Thus, we can conclude that GIM is effective for limited sample forecasting problems.
The forecasting interval value of the proposed model.
Year
Actual demand [17]
GDES0.1
GDES0.9
interval value
2006
16
16
16
2007
3
-1
4
[-1,4]
2008
7
5
9
[5,9]
2009
2
0
2
[0,2]
2010
4
3
4
[3,4]
2011
5
5
6
[5,6]
Case 7 (see [19]).
Tsaur collected the LCD TV demand data from 2001 to 2006 as shown in Table 6. The forecasting results are shown in Table 17. In Table 17, we find that the interval of GIM is shorter than that of the fuzzy autoregressive model.
The predictive results of different models (units:ten thousand).
Year
LCD TV demand
the proposed model
fuzzy autoregressive model [19]
2001
81
2002
150
[197,210]
[82,276]
2003
393
[423,476]
[318,532]
2004
970
[898,1024]
[687,969]
2005
2018
[1903,2156]
[1748,2163]
2006
4300
[4026,4493]
[4041,4779]
5. Conclusion
In theory, GMp/q(1,1), GDESp/q, and GMCp/q(1,N) models are discussed in this paper, respectively. Their application scopes are listed in Table 18. The results show that these models outperform the traditional grey models in the prediction precision.
The application scopes summary.
Situation
Method
GMp/q(1,1)
exponential trend
GDESp/q
volatile trend
GMCp/q(1,N)
multi-variable limited data
Using traditional grey models, the forecasting values of limited data are point estimations which supply too limited information for a decision-maker. It is easy to arouse suspicion for this kind of point estimations. Thus, a novel interval forecasting method is put forward. From the empirical results, we found that the forecasting capability of the GIM is quite encouraging, but those of the time series models are not. In addition, the interval of GIM is smaller than that of the simple linear regression and fuzzy autoregressive model. In the real world, the environment is uncertain and we can only use a limited amount of data to provide future forecasts for a short time period. In this situation, GIM is more satisfactory than the time series model.
In practice, GIM is summarized in Table 19. This paper contributes to the literature with an interval model for small samples forecasting in supply chain. It significantly improves small sample forecasting due to the interval result carrying more information. This paper demonstrates that grey modelling can be successfully applied to the forecasting problem in supply chain. Moreover, the proposed forecasting system can be used as a strategic tool for forecasting under uncertain conditions with a small amount of recent data.
The proposed model summary.
Situation Judgement
Method
the future trend is similar to the newer data
p/q is smaller
the future trend is similar to the order data
p/q is bigger
difficult to judge the future trend
interval forecasting
For future research, in order to investigate the feasibility of the novel model in supply chain, it may be used for other real world cases for forecasting and the performances of the methods can be compared.
Data Availability
All the data are from the references in this paper.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
The relevant researches carried out in this paper are supported by the National Natural Science Foundation of China (No. 71871084, 71401051, and 71801085). We also acknowledge the Project funded by China Postdoctoral Science Foundation (2018M630562).
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