Complexity
Volume 2020 (2020), Article ID 9178098, 10 pages
https://doi.org/10.1155/2020/9178098

Research Article

A Novel Conformable Fractional Nonlinear Grey Bernoulli Model and Its Application

Wanli Xie¹ and Guixian Yu²

¹Institute of EduInfo Science and Engineering, Nanjing Normal University, Nanjing, Jiangsu 210097, China
²School of Software & Microelectronics, Peking University, Beijing 100741, China

Correspondence should be addressed to Guixian Yu; yuguixian@pku.edu.cn

Received 29 May 2020; Revised 10 August 2020; Accepted 3 September 2020; Published 19 September 2020

Academic Editor: Hassan Zargarzadeh

Copyright © 2020 Wanli Xie and Guixian Yu. 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.

The fractional nonlinear grey Bernoulli model, abbreviated as FANGBM(1,1), is a successful extension of NGBM(1,1). Although FANGBM(1,1) has numerous excellent characteristics, it has a more complex form of fractional accumulation (FA) operator than raw NGBM(1,1). In this study, we propose a novel fractional nonlinear grey Bernoulli model, named CFNGBM(1,1), which uses conformable fractional accumulation (CFA), which has a simpler form than FANGBM. Using two practical cases, the effectiveness of the proposed CFNGBM(1,1) in practical applications was illustrated. Results show that the CFNGBM(1,1) exhibited higher accuracy than other grey models, thus facilitating its promotion in engineering practices.

1. Introduction

Deng first proposed the grey system theory, which significantly advanced the development of systems science [1]. As a crucial branch of the grey system theory, grey forecasting models have become a research hotspot for scholars. The grey forecasting model is based on data that have some known and unknown information in the real world. Nowadays, this model is widely used in industrial, economic, energy, agricultural, and other fields [2–9]. GM(1,1) is the core model in the grey forecasting models. Moreover, it has a good modeling effect on sequences with exponential law and can reveal future development trends of the system [2–9]. The GM(1,1) modeling process accumulates the original sequence to obtain a first-order accumulative generating operator (1-AGO) sequence, which can weaken the randomness of the original data and fully expose the information contained in the original sequence [9, 10]. Recently, many researchers have conducted a series of fruitful work to further improve the modeling accuracy of the grey models. Considering the varying spans of differential equations, Xie and Liu [10] constructed the DGM(1,1) model and analyzed its unbiased characteristics based on the perspective of discrete to discrete modeling. Cui et al. [11] reported that the grey model based on GM(1,1) can well fit the data series with pure exponential law characteristics; however, in reality, a large amount of data exists with nonhomogeneous exponential characteristics in addition to homogeneous exponential data. To further expand the applicability of the grey model, Cui et al. [11] proposed NGM(1,1, k), which can simulate sequences with nonhomogeneous exponential law. Zeng et al. [12] proposed a novel self-adapting intelligent grey model and used it to predict natural gas demand. Furthermore, the grey Verhulst model can better fit sequences with saturated growth or unimodal characteristics [4]. Recently, Wu et al. [13] proposed the fractional accumulation and fractional difference to improve further grey models, and this method was proved to overcome the drawback by the integer-order accumulation. Subsequently, some novel fractional grey models were proposed gradually. For instance, Yang and Xue [14] investigated the concept of fractional derivative and then used the G-L fractional derivative to deduce a new grey prediction model, and this model was applied to China's electricity prediction. Zhu et al. [15] suggested a new grey model to forecast Jiangsu’s electricity consumption in China, and the results showed that the proposed model had higher accuracy than other models.

Based on the above analysis, these models are mostly noted to be linear models or only suitable for a specific type of data. To characterize the complex laws of the system in the real world, the model structure must be flexibly adjusted according to the actual background of the modeling and should describe the nonlinear characteristics of data. Thus, to achieve these objectives, the nonlinear grey Bernoulli model (NGBM) was proposed [16]. The NGBM is a first-order univariate grey system model that can be used to describe the nonlinear development of data. The main advantage of the NGBM(1,1) is that the power index in the grey interaction can better reflect the nonlinear characteristics of the original data [16]. Furthermore, particle swarm optimization and genetic algorithm were used to optimize the power index in the NGBM(1,1) and achieve good forecast results [17, 18]. Wang et al. [19] established a nonlinear optimization model to simultaneously optimize the background value and power index, further enhancing the predictive ability of the model; this optimization model was used to facilitate early warning of the attainment rate of industrial wastewater in 31 administrative regions in China. Liu and Xie [20] proposed a novel nonlinear grey model, namely, WBGM(1,1), which was based on the Weibull cumulative distribution function, in which the nonlinear parameter was optimized by the genetic algorithm, and thereby, many valuable results were found. Although the grey Bernoulli model has numerous applications in reality and solves many difficult problems, it heavily relies on the selection of parameters. Although many existing methods have been proposed for optimizing the parameters, the order number of these models is an integer. The order of integer may prevent the model from achieving the best effect because the appropriate order must be selected based on actual data. Wu et al. [21] reported a new modeling method of fractional nonlinear grey Bernoulli model, called FANGBM(1,1). This model was successfully applied to China’s short-term energy consumption forecasting. However, the fractional accumulation (FA) of FANGBM(1,1) has more complicated mathematical expressions. Moreover, its implementation in programming is challenging when the mathematical foundation of engineers is relatively weak. To better extend the NGBM to engineering practices, herein, we present a novel fractional nonlinear grey Bernoulli model, named CFNGBM(1,1), in which the construction of FA is easily extended to engineering. To verify the effectiveness of the novel model, the model is verified by the case of carbon dioxide emissions in China.

The emission of carbon dioxide directly affects ecosystem as well as the sustainable economic development. The reliable projection of CO₂ emissions can assist decision-makers in shaping the associated energy policy. For this, many scholars have carried out a lot of effective works [22–24]. Due to the rapid development of China’s economy, carbon dioxide emissions are inevitably affected by economic policies, with dynamic changes. In accordance with the previous literature, the grey Bernoulli model with conformable fractional accumulation is developed for China's carbon dioxide forecasting, in which the nonlinear parameter of the novel model is determined by a bioinspired algorithm, namely, the whale optimization algorithm (denoted as WOA for short).

On these theoretical bases, we summarize the contributions of our work as follows:

(1)A case is presented to demonstrate that FA and CFA are close in value(2)The novel fractional nonlinear grey Bernoulli model, CFNGBM(1,1), is discussed(3)The parameter estimation and optimization methods of the order and power index of the CFNGBM(1,1) are provided(4)Two empirical cases are used to verify the accuracy of the proposed model

The remaining of this paper is organized as follows. First, Section 2 provides an introduction to the previous fractional nonlinear grey Bernoulli model. Next, the proposed grey model is presented in Section 3. Section 4 presents the optimization method of our model. Section 5 demonstrates the verification method of our model. Finally, Section 6 summarizes the conclusions of this study.

2. Description of the FANGBM(1,1)

In this section, we provide a brief review of the FANGBM(1,1). Wu et al. [21] proposed the fractional nonlinear grey Bernoulli model and successfully applied it to an engineering practice. The FANGBM(1,1) is defined as follows:

with

Assume

to be the nonnegative series; then, the r-order accumulative generating operator (r-AGO) series [16] is

where . The equation

is called the whitening equation of FANGBM(1,1). Here, to obtain the structural parameters of FANGBM(1,1), the discrete form of equation (5) should be deduced. For this, a crucial result (Ref. [8]) is introduced to discretize equation (5). Let be the mapping function, and if is a grey mapping function, it should satisfy the following:

(i)(ii)(iii) is the basic mapping of whitening differential equations,

where , is the grey item set and is the grey item mapping, is the white item set, and is the white item mapping.

By the basic mapping of the whitening differential equation, equation (5) becomes

where denotes the background value and is obtained as

Therefore, after using the least-squares method to estimate the parameters , the solution of equation (5) with becomes

Using the r-order inverse AGO (r-IAGO), the simulated values of are solved as follows:

3. Methodology

3.1. Conformable Fractional Accumulation and Difference

In this subsection, we briefly review conformable fractional accumulation (CFA) and conformable fractional difference (CFD). Ma et al. [25] presented the following definitions of CFA and CFD:

Thus, the CFA operator has a simpler form than the CFD operator. In particular, when , it has the following form:

To compare the difference between the FA and CFA operators, the sequence [1, 2, 1.5, 1.3] provided in the literature [26] is used. FA and CFA are observed to be close in value with different orders (Table 1). Moreover, the mean and standard deviations are quite similar, which shows that CFA can be a good substitute for FA.

Table 1: Comparison of FA and CFA with different orders.

Raw data
FA
CFA

Value
0 = 0.2
r = 0.6
r = 0.1
r = 1.4
r = 1.8
r = 0.2
r = 0.6
r = 0.1
r = 1.4
r = 1.8

1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0

2
2.2
2.6
3.0
3.4
3.8
2.1
2.5
3.0
3.3
3.7

1.5
2.0
3.1
4.5
5.9
7.6
2.7
3.4
4.5
6.4
7.6

1.3
3.6
5.2
7.5
10.3
13.9
3.7
5.2
7.5
10.8
13.9

Average
2.2
2.9
4
5.1
6.5
2.3
3.0
4
5.3
6.5

Standard
1.0
1.7
2.7
3.9
5.5
1.1
1.7
2.7
4.2
5.5

3.2. The Proposed CFNGBM(1,1) Model

In this subsection, we provide the definition of the CFNGBM(1,1). Let the original sequence be ; thus, its r-order CFA is , where

The CFNGBM(1,1) is

By substituting the data into the CFNGBM(1,1), we achieve

Within discrete form (13), we can easily get the parameter of the CFNGBM(1,1) using the least-squares method with given samples (14) as

where

We use instead of . Thus, we achieve the error sequence , where , , and are defined in equation (15). Set

To minimize s, and should satisfy

Theorem 1. We set

When , the parameters of the model also can be obtained using the following method, , where

proof. From the expression of matrix B, we have

Then,

Thus,

The parameters of the model can be obtained using the least-squares method:

The whitening equation of the CFNGBM(1,1) is

The time response function of the CFNGBM(1, 1) with is expressed as

The predicted value of the model can be obtained as follows:

Therefore, we can observe that CFNGBM(1,1) has a simpler structure than FANGBM(1, 1).

4. Parameter Optimization of the CFNGBM(1,1)

In the above descriptions, order r and power index are assumed to be known; however, they are often changeable in a different system that requires flexible adjusting over a given modeling background. To optimize the parameters of the model, we used the whale optimization algorithm (WOA), proposed by Mirjalili and Lewis [27] to automatically determine the order and power index. According to the hunting behavior of humpback whales, this intelligent algorithm lets the current best candidate solution (agent) be the target prey or near the optimum. Once the best agent is determined, the other agents attempt to change their positions toward the best agent.

The optimization process of the WOA algorithm is summarized as follows:

Step 1. Set algorithm parameters.

Step 2. Initialize the whales’ population .

Step 3. Calculate the fitness of each search agent .

Step 4. Update the parameters of the algorithm.

Step 5. Generate a random number in . Select different strategies of location update according to .

Step 6. Return to Step 3 until optimal values r and are determined.

5. Validation of the CFNGBM(1,1)

We present two cases to verify the efficacy of the proposed model compared with five competing models, including the traditional FANGM [28], FAGM [13], NGMO [29], NGM [11], and FANGBM [21]. Moreover, to evaluate the forecasting accuracy of these grey models, we applied certain statistical indicators to assess the quality of the model, which are defined as follows:

Case 1. Forecasting of CO₂ emissions in China
With China’s rapid economic advancement, the use of energy has considerably increased, leading to a significant increase in CO₂ emissions. Increased CO₂ emission warms the climate and causes natural disasters, such as melting glaciers. Effective CO₂ emissions forecasting can enable early warning and help decision-makers to devise rational plans. Here, data from 2000 to 2014 were used to fit the model and data from 2015 to 2018 were used to validate the model (source: BP Statistical Review of World Energy 2019). To optimize the parameters of the CFNGBM model with all data, we used the WOA. From Table 2, we observe that the CFNGBM’s order r and power index are 0.951 and 0.520, respectively; its MAPE is 2.446, which is the smallest of the four models (the results of FAGM and FANGM are obtained from the work of Wu et al. [22].)
In this case, to verify the advantages of WOA in CFNGBM, the classic optimization algorithms including PSO and GA are selected as benchmarks. Table 3 shows that the MAPE of WOA, PSO, and GA is 2.446%, 2.043%, and 2.1405%, respectively, and computation times are taken as 6.211 s, 62.618 s, and 16.386 s, respectively, which indicates that although the accuracy of WOA is not the best (close to the best), the time taken by WOA is significantly lower than that of other competitors.
The obtained parameters are used to fabricate a model and make predictions. Table 4 shows the results of the simulation and prediction of CO₂ emissions in China using six models.
To verify the accuracy of each model, equations (28)–(32) were used. We can observe from Table 5 that during the simulation phase, the MAE, MSE, MAPE, RMSPE, and R of the CFNGBM are 152.5, 41945, 2.6866, 4.749, and 0.9961, respectively. Although the indicators of CFNGBM are not the best, they are close to the best. In the forecast stage, the MAE, MSE, MAPE, and RMSPE of the CFNGBM are 300.2, 96338, 3.256, 3.374, and 0.737, respectively, thus achieving the best results among all models.

Table 2: The minimum MAPE and optimal order, power index of FAGM(1,1), FANGM(1,1,k,c), FNGBM(1,1), and CFNGBM(1,1), and CO₂ emissions in China.

CFNGBM
FNGBM
FAGM
FANGM

MAPE (%)

MAPE (%)

MAPE (%)

MAPE (%)

0.951
0.520
2.446
0.23
0.59
2.87
0.268
4.9512
0.8681
3.7164

Table 3: The minimum MAPE and time consumption of WOA, PSO, and GA in Case 1.

WOA
PSO
GA

MAPE(%)
Time consumption(s)
MAPE(%)
Time consumption(s)
MAPE(%)
Time consumption(s)

2.446
6.211
2.0431
62.618
2.1405
16.386

Table 4: Results of CO₂ emissions in China using six models.

Year
Raw data
FANGM
FAGM
NGMO
NGM
CFNGBM
FNGBM

2000
3362.7000
3362.700
3362.700
3362.700
3362.700
3362.700
3362.700

2001
3525.0000
3292.203
3737.913
3266.673
2734.660
2924.815
3630.167

2002
3845.4000
4046.790
4313.449
4050.256
3565.453
3781.706
4196.168

2003
4534.4000
4758.969
4882.322
4763.443
3565.453
4570.822
4807.091

2004
5337.0000
5411.520
5421.771
5412.557
5009.980
5291.335
5411.459

2005
6099.5000
6005.557
5929.415
6003.355
5636.505
5944.246
5988.919

2006
6677.9000
6545.085
6406.808
6541.077
6206.782
6531.700
6529.440

2007
7240.3000
7034.704
6856.271
7030.490
6725.863
7056.520
7027.942

2008
7378.5000
7479.004
7280.102
7475.935
7198.342
7521.915
7482.242

2009
7708.8000
7882.348
7680.392
7881.361
7628.404
7931.286
7892.007

2010
8135.2000
8248.796
8058.993
8250.365
8019.857
8288.104
8258.135

2011
8805.8000
8582.073
8417.533
8586.217
8376.1678
8595.829
8582.339

2012
8991.5000
8885.570
8757.449
8891.896
8700.489
8857.853
8866.861

2013
9237.7000
9162.358
9080.014
9170.114
8995.695
9077.469
9114.256

2014
9223.7000
9415.204
9386.364
9423.337
9264.398
9257.840
9327.248

2015
9174.6000
9646.596
9677.513
9653.810
9508.978
9401.992
9508.613

2016
9119.0000
9858.765
9954.379
9863.578
9731.601
9512.802
9661.103

2017
9229.8000
10053.71
10217.79
10054.50
9934.2376
9592.993
9787.389

2018
9428.7000
10233.21
10468.51
10228.27
10118.68
9645.137
9890.026

Table 5: Error metrics of CO₂ emissions in China.

Fitting
FANGM
FAGM
NGMO
NGM
CFNGBM
FNGBM

MAE
153.5
220.3
155.8
316.9881
152.5
150.6

MSE
26872.8
65406.3
0.8231
138420.3
41945
29616

MAPE
2.5842
3.7676
2.6513
5.5964
2.6866
2.54

RMSPE
3.0
4.8804
3.2305
7.6696
4.749
3.35

R
0.99
0.9948
0.9968
0.9953
0.9961
0.997

Prediction
FANGM
FAGM
NGMO
NGM
CFNGBM
FNGBM

MAE
710.0
841.5226
712.014
585.3498
300.2
473.7

MSE
524020.8
752026.2
525870.6
364849.21
96383
232290

MAPE
7.6790
9.09
7.7009
6.3281
3.256
5.12

RMSPE
7.8225
9.356
7.8375
6.5233
3.374
5.22

R
0.8144
0.823
0.8117
0.8117
0.737
0.784

Case 2. Forecasting of CO₂ emissions in India
India is the second-most populous country in the world. The utility of energy increases with environmental pressure. In particular, the increase in CO₂ emissions causes many adverse changes. Thus, a reasonable prediction of CO₂ emissions is significant for the country to formulate sustainable development strategies. In this case, we used all data to search the parameter order and power exponent. The results are shown in Table 6. CFNGBM’s order r and power index are 1.1519 and 0.3081, respectively; its MAPE is 1.074. (The results of FAGM and FANGM are obtained from the work of Wu et al. [22]).
Table 7 shows that the MAPE of WOA, PSO, and GA is 1.074%, 0.8559%, and 0.848%, respectively, and corresponding computation times are 6.0027s, 58.882s, and 0.8488s, respectively. Similar to Case1, WOA still has a short running time in comparison with PSO and GA.
The obtained optimized parameters are used to establish a model and make predictions. Table 8 shows the results of the simulation and prediction of CO₂ emissions in India using six models.
We can observe from Table 9 that during the simulation phase, the MAE, MSE, MAPE, RMSPE, and R of the CFNGBM model are 152.5, 41945, 2.6866, 4.749, and 0.9961, respectively. Although the indicators of CFNGBM are not the best, they are close to the best. In the forecast stage, the MAE, MSE, MAPE, and RMSPE of the CFNGBM are 300.2, 96338, 3.256, 3.374, and 0.737, respectively, achieving the best results among all models.

Table 6: The minimum MAPE and optimal order, power index of FAGM(1,1), FANGM(1,1,k,c), FNGBM(1,1), and CFNGBM(1,1), and CO₂ emissions in India.

CFNGBM
FANGBM
FAGM
FANGM

MAPE (%)

MAPE (%)

MAPE (%)

MAPE (%)

1.1519
0.3081
1.074
0.129
0.03
1.148
0.1649
1.7519
1.2269
1.3275

Table 7: The minimum MAPE and time consumption of WOA, PSO, and GA in Case 2.

WOA
PSO
GA

MAPE(%)
Time consumption(s)
MAPE(%)
Time consumption(s)
MAPE(%)
Time consumption(s)

1.074
6.0027
0.8559
58.882
0.8488
17.598

Table 8: Results of CO₂ emissions in India using six models.

Year
Raw data
FANGM
FAGM
NGMO
NGM
CFNGBM
FANGBM

2000
962.500
962.500
962.500
962.500
962.500
962.500
962.500

2001
970.3000
980.592
933.755
946.740
942.075
967.941
946.603

2002
1021.900
999.9330
979.664
1008.260
1003.321
989.666
990.837

2003
1062.300
1055.790
1045.513
1073.200
1067.978
1049.63
1052.89

2004
1116.600
1055.790
1120.686
1141.770
1136.236
1125.931
1124.093

2005
1204.600
1202.766
1201.597
1214.160
1208.296
1208.185
1201.427

2006
1252.500
1285.234
1286.709
1290.580
1284.370
1293.661
1283.573

2007
1365.500
1371.716
1375.272
1371.260
1364.680
1381.476
1369.885

2008
1466.900
1461.625
1466.901
1456.430
1449.464
1471.344
1460.032

2009
1595.600
1554.655
1561.393
1546.340
1538.971
1563.200
1553.849

2010
1661.000
1650.652
1658.650
1641.270
1633.462
1657.074
1651.268

2011
1735.700
1749.547
1758.634
1741.490
1733.217
1753.031
1752.279

2012
1849.200
1851.329
1758.634
1847.280
1838.527
1851.151
1856.909

2013
1930.000
1956.018
1966.813
1958.980
1949.704
1951.521
1965.214

2014
2083.300
2063.660
2075.077
2076.890
2067.072
2054.228
2077.268

2015
2147.800
2174.317
2186.197
2201.370
2067.072
2159.356
2193.161

2016
2234.200
2288.064
2300.237
2332.790
2321.785
2266.991
2312.993

2017
2316.900
2404.985
2417.272
2471.540
2459.878
2377.216
2436.875

2018
2479.100
2525.170
2537.379
2471.540
2605.662
2490.109
2564.927

Table 9: Error metrics of CO₂ emissions in China.

Fitting
FANGM
FAGM
NGMO
NGM
CFNGBM
FANGBM

MAE
14.7581
18.8077
17.8016
18.5138
15.190
15.611

MSE
349.5036
4826.52
493.6639
13950.48
396.77
410.78

MAPE
1.0421
1.4362
1.3239
4.7964
1.0909
1.150

RMSPE
1.0421
1.9368
3.2305
1.6429
1.4800
1.520

R
0.9987
0.9948
0.9982
0.9982
0.9985
0.9986

Prediction
FANGM
FAGM
NGMO
NGM
CFNGBM
FANGBM

MAE
53.6342
65.7712
712.0148
111.4319
28.918
82.48

MSE
3371.48
4826.5
13950.48
11499.00
1242
7506

MAPE
2.3264
2.8566
4.7964
4.3017
1.2633
3.569

RMSPE
2.512
3.0094
5.0409
4.5704
1.5344
3.731

R
0.9868
0.9867
0.9885
0.9885
0.9866
0.9872

6. Conclusion and Future Research

The FANGBM, a successful generalization of the nonlinear grey Bernoulli model, generalizes the integer-order accumulation to a more general fractional-order accumulation and determines a more suitable accumulation order through intelligent optimization. Because of the optimization of order and power index, FANGBM has higher precision than some previous grey models. Although FANGBM(1,1) has high accuracy and numerous excellent characteristics, its FA operator is relatively complicated to build model. Hence, it is difficult for general engineering technicians to master it. Herein, we propose the CFNGBM(1,1), wherein the fractional-order accumulation is implemented by CFA, which has a simpler structure. Using two practical examples of CO₂ prediction in China and India, we determine that CFNGBM(1,1) has the same accuracy (or higher prediction accuracy) as FANGBM(1,1) and is a good substitute.

Although CFNGBM(1,1) has many excellent characteristics, further work is required in the future. We list them as follows: (1) optimizing the background value of the CFNGBM(1,1) model to achieve more high accuracy and (2) using more efficient intelligent optimization algorithms to further improve the modeling accuracy.

Data Availability

The data used to support the findings of this study are included in the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

Both the authors equally contributed to this work.

Acknowledgments

This work was supported by grants from the Postgraduate Research & Practice Innovation Program of Jiangsu Province (grant no. KYCX20_1144).

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