Nonlinear grey Bernoulli model based on Fourier transformation and its application in forecasting the electricity consumption in Vietnam

Abstract

In recent decades, the Nonlinear Grey Bernoulli Model “NGBM (1, 1)” has been applied in various fields and achieved positive results. However, its prediction results may be inaccurate in different scenarios. In order to expand the field of application and to improve the predictive quality of the NGBM (1, 1) model, this paper proposes an effective model (named Fourier-NGBM (1, 1)). This model includes two main stages; first, we get the error values based on the actual data and predicted value of NGBM (1, 1). Then, we use a Fourier series to filter out and to select the low-frequency error values. To test the superior ability of the proposed model, two numerical data sets were used. One is the historical data of annual water consumption in Wuhan from 2005 to 2012 in He et al. ’s paper, and the other is example data from Wang et al. ’s paper. The forecasted results prove that the performance of the Fourier-NGBM (1, 1) model is better than three other forecasting models, namely GM (1, 1), NGBM (1, 1) and the improved Grey Regression model. Furthermore, this study also applied the proposed model to forecast the electricity consumption in Vietnam up to the year 2020. The empirical results can offer valuable insights and provide basic information for model building to develop future policies regarding electrical industry management. In subsequent research, more methodologies can be used to reduce the residual error of the NGBM (1, 1) model, such as Markov chain or different kinds of Fourier functions. Additionally, the proposed model can be applied in different industries with fluctuating data and uncertain information.

Keywords

Fourier series nonlinear grey Bernoulli model prediction accuracy residual error electricity consumption Vietnam

1 Introduction

The grey prediction model is the central part of grey system theory [1]. It is an efficient method for dealing with uncertain information and building time series forecasting models with limited data. In the early 1980s, the grey model GM (1, 1) was introduced by Deng based on control theory [2, 3]. It was soon more commonly used than other forecasting models in time series because this model is easy to simulate and needs only small sample sizes. It was constructed based on the first-order accumulation method and first-order differential equation. Fundamentally, it follows the exponential growth laws. Based on the parameter value of a, we can make a decision in short-term, mid-term as well as long-term forecasting. Previous studies have evidenced the advantages of the grey forecasting model in dealing with the problem of uncertain information and limited data [3 –5]. Therefore, many scientists have used this model in different areas such as tourism demand forecasting [6, 7], cargo throughput or motor vehicle forecasting [8 –10], gold or stock index price forecasting [11 –13], the output value forecasting of the high technology industry [14 –17], energy demand forecasting [18 –20], etc.

Recently, many scholars have used different ways to expand the application areas and to improve the prediction quality of grey models. Specifically, Lin et al. [21] and Wang et al. [22] provided new ways to calculate the background values of the grey forecasting model. Hsu [17] and Wang et al. [23] proposed a systematic approach to optimize the development coefficient or the grey input coefficient or both of these parameters. Some researchers focused on the residual error modification of the GM (1, 1) model, such as Hsu [15] and Wang et al. [24]. Additionally, many scientists proposed hybrid models based on the GM (1, 1) model. These include the grey econometric model [25], grey Markov model [26, 27], and the grey fuzzy model [21], etc. All of the aforementioned research concentrated on the modification of background values or residual errors.

On the other hand, some scholars used different equations to modify the GM (1, 1) model to become new models. For example, Chen [28] proposed the nonlinear grey Bernoulli model NGBM (1, 1) to forecast the unemployment rate. Gou et al. [9] proposed a grey Verhulst model to forecast cargo throughput. Chen et al. [29] used the NGBM (1, 1) model to predict the annual unemployment rates of ten countries. Zhou et al. [30] also applied the proposed model to forecast the foreign exchange rates of Taiwan’s major trading partners. The above real case studies evidenced that the NGBM (1, 1) significantly improved the efficient prediction of the original GM (1, 1) model.

With the flexible changing of the exponential parameter n in the NGBM (1, 1) forecasting model, this feature has made the model more suitable where there are linear or nonlinear characteristics of real systems. Relying on this feature, Zhou et al. [30] applied the novel NGBM (1, 1) model to forecast the power load. Hsu [16] proposed the novel NGBM (1, 1) based genetic algorithm to forecast the output values of high tech industries in Taiwan. Chen et al. [31] proposed a Nash NGBM (1, 1) model based on the Nash equilibrium concept to forecast the monthly stock indices in Taiwan. Wang et al. [32] proposed an optimized NGBM (1, 1) model to forecast the qualified discharge rate in 31 provinces in China. Pao et al. [33] proposed a model named the NGBM-OP model to forecast three important indicators of the clean energy economy in China, which are the amount of CO₂ emissions, energy consumption, and economic growth. The empirical results proved that the prediction ability of the proposed model was remarkably improved in comparison with GM (1, 1), ARIMA and the NGBM (1, 1) model.

In fact, because the initial condition is a part of the predictive function, this is a main factor that affects the performance of grey prediction, which is the reason why most researchers have concentrated on the modification of the background values. In this paper, the author provides a different approach to reduce the model error by modifying the error sequence obtained from the NGBM (1, 1) with a Fourier series. Two numerical example and the results of a real case show that the Fourier-NGBM (1, 1) model can significantly improve the prediction accuracy.

This paper is an extended version of our conference paper published in the ACIIDS 2019 proceedings [34]. We have significantly expanded the paper, adding the following elements: Firstly, we use highly fluctuating data and lots of noise to test the performance of the presented model and we propose to apply it to a real case related to forecasting electricity consumption in Vietnam. The remainder of this paper is organized as follows. Section 2 analyzes the related work. Section 3 briefly introduces the NGBM (1, 1) model and proposes a systematic approach for improving the prediction accuracy of the NGBM (1, 1) by a Fourier series. Section 4 proves the effectiveness of the proposed model with two numerical examples; one is water consumption forecasting in He and Tao [35] and the other is fluctuation data in Wang et al. Section 5 presents the application of the proposed model in a real case. Finally, the paper concludes with some comments and suggestions.

2 Related work

As mentioned in section 1, the NGBM (1, 1) model expresses a superior forecasting ability when compared with the GM (1, 1) and grey Verhulst models. This is because it does not require a specific number in the exponential value of n (n ≠ 1). So, it can be flexible in determining the shape of the model’s curve with fluctuating sequence data. This characteristic has attracted the interest of many scientists and prompted further research for improving the prediction accuracy of this model. To deal with the solution of the optimal background interpolation value p, Zhou et al. used a particle swarm optimization algorithm [30]. To show the superior forecasting ability of the proposed model, the long-term power load during 1996 to 2007 is used for the modelling. According to the MAPE in-sample and out-of-sample indexes, the outcome of the simulation clearly indicated that the proposed model produces better forecasted results than the other four mentioned models which are the GM (1, 1), the optimal p value of GM (1, 1), the NGBM (1, 1) and the proposed model. These results once again confirmed that the prediction accuracy of NGBM (1, 1) can be improved by using a PSO algorithm.

Hsu [16] applied a genetic algorithm to get the optimal values in the NGBM (1, 1) model in order to enhance the performance prediction with a small amount of data. Then, the author applied it to forecast the economic trends in the integrated circuit industries in Taiwan during the period 1990 to 2007. The empirical results indicated that the error values of the proposed model in this study were lower than the traditional GM (1, 1) model and grey Verhulst model in terms of the MAPE index, which for in-sample was 12.48% and out-of-sample 4.64%. Chen et al. [31], based on the Nash equilibrium concept, proposed a Nash NGBM (1, 1) model which is an optimization of the exponential value n and the background interpolation value p in the NGBM (1, 1). For testing the proposed model’s adaptability, the monthly stock indices in Taiwan were used. The forecasting results show that the proposed model actually improved the forecasting precision.

Wang et al. [32] proposed a novel NBGM (1, 1) model by optimizing the two parameters p (background interpolation value) and n (exponential value). To solve this problem, the authors used the LINGO software. In order to verify the efficiency of the novel NGBM (1, 1) model in an environment with fluctuating data, the authors used a numerical example from Wen’s work and the output values of the electronics industry in Taiwan. The simulated results demonstrated that the forecast performance of the novel NBGM (1, 1) model is significantly improved compared to the original NGBM (1, 1) model. In addition, the authors also applied the improved model to forecast the annual qualified discharge rate in 31 provinces in China during the period 2001 to 2011. Empirical results show that the proposed model is more effective and reliable in prediction, with an average RPE index (relative percentage error) of less than 4% for 27 provinces and more than 4% for 4 provinces.

In summary, all these previous studies focused on the modification of the background value and parameters by using different mathematical algorithms. Based on the idea of Wang and Phan [24], this study presents a systematic approach and tries to enhance the performance of the NGBM (1, 1) model by using the Fourier series.

3 Methodology

3.1 Nonlinear grey bernoulli model “NGBM (1, 1)”

According to Chen et al. [29, 31], the NGBM (1, 1) is a first-order one-variable grey Bernoulli differential equation. For predictions related to nonlinear time series with a small sample, the forecasted results of NGBM (1, 1) are better than the original grey forecasting models such as GM (1, 1) and grey Verhulst. The algorithm of NGBM (1, 1) is expressed as follows:

Step 1. Assume that we have the original time series data X⁽⁰⁾ $X^{(0)} = {x^{(0)} (t_{i})}, i = 1, 2, . . ., m$ (1) where x⁽⁰⁾ (t_i) is the system output at time t_i, x⁽⁰⁾ (t_i) ≥0

m is the total number of modelling data.

Step 2. To construct X⁽¹⁾, we use the first-order accumulated generating operation (1-AGO) in the following equation: $X^{(1)} = {x^{(1)} (t_{i})}, i = 1, 2, . . ., m$ (2) where $x^{1} (t_{k}) = \sum_{i = 1}^{k} x^{(0)} (t_{i}), k = 1, 2, . . ., m$ (3)

Step 3. X⁽¹⁾ is modelled by the Bernoulli differential equation: $\frac{{dX}^{(1)}}{dt} + {aX}^{(1)} = b {[X^{(1)}]}^{n}$ (4) where a is called the developing coefficient,

b is the grey input,

n is any real number and n ≠ 1.

Step 4. In order to estimate the parameters a and b, Equation (4) is approximated as: $\frac{Δ X^{(1)} (t_{k})}{{dt}_{k}} + {aX}^{(1)} (t_{k}) = b {[X^{(1)} (t_{k})]}^{n}$ (5) where $Δ X^{(1)} (t_{k}) = x^{(1)} (t_{k}) - x^{(1)} (t_{k - 1}) = x^{(0)} (t_{k})$ (6) $Δ t_{k} = t_{k} - t_{k - 1}$ (7)

Let Δt_k = 1, then $z^{(1)} (t_{k}) = {px}^{(1)} (t_{k}) + (1 - p) x^{(1)} (t_{k - 1}), k = 2, 3, . ., n$ (8)

To replace X⁽¹⁾ (t_k) in Equation (5), we obtain $x^{(0)} (t_{k}) + {az}^{(1)} (t_{k}) = b {[z^{(1)} (t_{k})]}^{n}, k = 2, 3, . . ., m$ (9) where z⁽¹⁾ (t_k) is the background value,

p is the production coefficient (p ∈ [0, 1]), and normally the p value is set to 0.5.

Step 5. We use the ordinary least squares method (OLS) to calculate the parameters a and b in Equation (9) as follows: $[\begin{matrix} a \\ b \end{matrix}] = (B^{T} B)^{- 1} B^{T} Y$ (10) where $B = [\begin{matrix} - z^{(1)} (t_{2}) (z^{(1)} (t_{2}))^{n} \\ \begin{matrix} z^{(1)} (t_{3}) \\ . . . . . . \\ - z^{(1)} (t_{m}) \end{matrix} \begin{matrix} (z^{(1)} (t_{3}))^{n} \\ . . . . . . \\ (z^{(1)} (t_{m}))^{n} \end{matrix} \end{matrix}]$ (11) and $Y = {[x^{(0)} (t_{2}), x^{(0)} (t_{3}), . . ., x^{(0)} (t_{m}))]}^{T}$ (12)

Step 6. Replacing the parameters a and b into Equation (4), we can get the ${\hat{x}}^{(0)} (t_{k})$ by the equation below: $\begin{matrix} {\hat{x}}^{(1)} (t_{k}) = {[(x^{(0)} (t_{1})^{(1 - n)} - \frac{b}{a}) e^{- a (1 - n) (t_{k} - t_{1})} + \frac{b}{a}]}^{\frac{1}{1 - n}}, \\ n \neq 1, k = 1, 2, 3, . . . \end{matrix}$ (13)

Step 7. To get the predicted value of ${\hat{x}}^{(0)} (t_{k})$ , we adopt the inverse accumulated generating operation (IAGO) as the function below: ${\begin{matrix} {\hat{x}}^{(0)} (t_{1}) = x^{(0)} (t_{1}) \\ {\hat{x}}^{(0)} (t_{k}) = {\hat{x}}^{(1)} (t_{k}) - {\hat{x}}^{(1)} (t_{k - 1}) \end{matrix}$ (14)

3.2 Residual error modification of NGBM (1, 1) by Fourier series

The Fourier series [6] was used to filter out and to select the low-frequency error values of the NGBM (1, 1) to improve the performance in the prediction. The overall procedure is as follows:

Step 1. Call ɛ a residual error series. ɛ is defined by: $ɛ (k) = x (k) - \hat{x} (k), k = 2, 3, . . . m$ (15)

Step 2. Use the Fourier series for modifying the error values of NGBM (1, 1) as in the following equation:

$\begin{matrix} \hat{ɛ} (k) = \frac{1}{2} a_{(0)} + \sum_{i = 1}^{Z} [a_{i} cos (\frac{2 π i}{m - 1} (k)) \\ + b_{i} sin (\frac{2 π i}{m - 1} (k))], k = 1, 2, 3, ., m \end{matrix}$ (16) where $Z = (\frac{m - 1}{2}) - 1$ is called the minimum deployment frequency and only takes integer numbers [36]. Therefore, the residual error is estimated as follows: $ɛ = P \times C$ (17) where $P = [\begin{matrix} \frac{1}{2} & cos (\frac{2 π \times 1}{m - 1} \times 2) sin (\frac{2 π \times 1}{m - 1} \times 2) & . . . & cos (\frac{2 π \times Z}{m - 1} \times 2) sin (\frac{2 π \times Z}{m - 1} \times 2) \\ \frac{1}{2} & cos (\frac{2 π \times 1}{m - 1} \times 3) sin (\frac{2 π \times 1}{m - 1} \times 3) & . . . & cos (\frac{2 π \times Z}{m - 1} \times 3) sin (\frac{2 π \times Z}{m - 1} \times 3) \\ . . . & . . . & . . . & . . . \\ \frac{1}{2} & cos (\frac{2 π \times 1}{m - 1} \times m) sin (\frac{2 π \times 1}{m - 1} \times m) & . . . . & cos (\frac{2 π \times Z}{m - 1} \times m) sin (\frac{2 π \times Z}{m - 1} \times m) \end{matrix}]$ (18) and $C = [a_{0}, a_{1}, b_{1}, a_{2}, b_{2}, . . ., a_{Z}, b_{Z}]$ (19)

We use the OLS method to calculate the parameters a₀, a₁, b₁, a₂, b₂ . . . . a_z, b_z by the equation below: $C = {(P^{T} P)}^{- 1} P^{T} {[ɛ]}^{T}$ (20)

Replacing the parameters just estimated in Equation (21) into Equation (17), we can get the modified residual series $\hat{ɛ}$ as follows:

$\begin{matrix} \hat{ɛ} (k) = \frac{1}{2} a_{(0)} + \sum_{i = 1}^{Z} [a_{i} cos (\frac{2 π i}{m - 1} (k)) \\ + b_{i} sin (\frac{2 π i}{m - 1} (k))] \end{matrix}$ (21)

Step 3. From the predicted series $\hat{x}$ in Equation (15) and $\hat{ɛ}$ in Equation (22), the predicted series value of the Fourier NGBM (1, 1) “ $\hat{v}$ ” is determined by: $\hat{v} = {{\hat{v}}_{1}, {\hat{v}}_{2}, {\hat{v}}_{3}, . . . ., {\hat{v}}_{k}, . . ., {\hat{v}}_{m}}$ (22) where $\hat{v} = {\begin{matrix} {\hat{v}}_{1} = v_{1} \\ {\hat{v}}_{k} = {\hat{x}}_{k} + {\hat{ɛ}}_{k} (k = 2, 3, . . . m) \end{matrix}$ (23)

3.3 Forecast model evaluation

In this paper, the Mean Absolute Percentage Error (MAPE) index is used to estimate the accuracy and reliability of the prediction models [36]. It is expressed as follows: $MAPE = \frac{1}{m} \sum_{k = 2}^{m} | \frac{x^{(0)} (k) - {\hat{x}}^{(0)} (k)}{x^{(0)} (k)} | \times 100 {%$ (24) where x⁽⁰⁾ (k) indicates the actual value while ${\hat{x}}^{(0)} (k)$ are the forecasting values in time period k, and m is the total number of predictions. According to Wang et al. [37, 38], the MAPE is divided into four levels of forecasting accuracy: level 1 is an excellent forecast (less than 1%), level 2 is a good forecast (from 1% to 5%), level 3 is a reasonable forecast (from 5% to 10%), and more than 10% is an inaccurate forecast.

4 Model validation

In this section, two examples with highly fluctuating data or lots of noise are given to illustrate the effectiveness of the proposed model. The first example is the historical data of annual water consumption in He et al.’s paper [35] and the second example is the example data in Wang et al.’s paper [32]. The details of the two simulations are presented below.

4.1 Example data from He et al. [35]

To show the effectiveness of the Fourier-NGBM (1, 1), this study first takes the annual water consumption demand in Wuhan as a numerical example of data and then compares the performance among the Fourier-NGBM (1, 1), a coupled model of the grey system and multivariate linear regression, and the original NGBM (1, 1) [35].

According to the historical data of the annual water consumption demand from 2005 to 2012 in Wuhan (see Table 1), this paper uses Microsoft Excel to simulate the mathematical algorithm of grey forecasting models. This software was provided by Microsoft Corporation. Excel software not only provides some basic functions but also provides two useful functions, Mmult and Minverse, to calculate the parameters in the grey forecasting model. In particular, the function of Mmult (array 1, array 2) can be used for multiplying two relevant arrays matrices. The function of Minverse (array) is to calculate the inverse matrix. Moreover, Excel software includes the efficiency tool named Excel-solves for finding the optimal values of parameters in forecasting models. For the sake of convenience, this paper just shows the detailed results of NGBM (1, 1) and Fourier-NGBM (1, 1) in Table 1. The other predicted results are omitted here, and only the MAPE indexes of these models are listed in Table 2.

Table 1
Forecasting results from the NGBM (1, 1) and Fourier-NGBM (1, 1) models

Year Actual value Original NGBM (1, 1) Fourier-NGBM (1, 1)

n = -0.11, p = 0.5 n = -0.11, p = 0.5

Forecasted value % error Forecasted value % error

2005 367,204 367,204 0 367,204 0

2006 369,355 369,355 0 369,355 0

2007 363,883 364,331.6 0.1 363,883 0

2008 360,467 366,777.6 1.8 360,467 0

2009 377,944 372,695.6 1.4 377,944 0

2010 379,346 380,754.6 0.4 379,346 0

2011 397,345 390,359.2 1.8 397,345 0

2012 395,713 401,199.8 1.4 395,713 0

MAPE 0.8 0

Year	Actual value	Original NGBM (1, 1)	Fourier-NGBM (1, 1)
2005	367,204	367,204	0	367,204
2006	369,355	369,355	0	369,355
2007	363,883	364,331.6	0.1	363,883
2008	360,467	366,777.6	1.8	360,467
2009	377,944	372,695.6	1.4	377,944
2010	379,346	380,754.6	0.4	379,346
2011	397,345	390,359.2	1.8	397,345
2012	395,713	401,199.8	1.4	395,713
MAPE			0.8

Table 2

Summary of evaluation indexes of model accuracy

Models	MAPE (%)	Forecasting accuracy (%)	Performance
GM (1, 1)	1.163	98.837	Good
Coupled model of grey system and	1.156	98.84	Good
multivariate linear regression model in [35]
Original NGBM (1, 1)	0.8	99.2	Excellent
Fourier-NGBM (1,1)	0	100	Excellent

According to the average of MAPE in Table 1, this results reveals that the residual error of Fourier- NGBM (1, 1) with n = -0.11 and p = 0.5 is lower than the original one. The average MAPE index decreased from 0.8% to 0%. This result was also compared with the coupled model of the grey system and multivariate linear regression model in [35], as well as GM (1, 1). The overall MAPE indexes as well as the performance forecasting of these models is shown in Table 2.

The results from Table 2 show that the MAPE of the Fourier-NGBM (1, 1) model is nearly 0%. This feature confirms that the performance of the Fourier-NGBM (1, 1) model is excellent. From comparison with the three remaining models, the Fourier-NGBM (1, 1) shows the highest accuracy in forecasting the total annual water consumption demand in Wuhan.

4.2 Example data from Wang et al. [32]

In this example, the raw data sequence jumps randomly X⁽⁰⁾ = (5, 6, 4, 7). In this case, Wang et al. [32] used the example to demonstrate the accuracy of the optimized NGBM (1, 1). In this section, we also used the same example to compare the forecasting performance of the Fourier-NGBM (1, 1) model with the optimized NGBM (1, 1) and the original NGBM (1, 1) in Wang’s paper [32]. The results of the forecasting models are shown in Table 3 and Fig. 1.

Table 3
Forecasted results from the different grey models

No Actual value Original NGBM (1, 1) Optimized NGBM (1, 1) Fourier-NGBM (1,1)

n = -10, p = 0.5 n = -19.58, p = 0.569 n = -10, p = 0.5

Forecasted value % error Forecasted value % error Forecasted value % error

k = 1 5 5 0 5 0.1 5 0

k = 2 6 6.499 8.31 6.00 0.00 6 0

k = 3 4 4.921 23.03 4.828 20.70 4 0

k = 4 7 6.986 0.2 6.946 0.77 7 0

MAPE (%) 7.89 7.16 0.00

No	Actual value	Original NGBM (1, 1)	Optimized NGBM (1, 1)	Fourier-NGBM (1,1)
k = 1	5	5	0	5	0.1	5	0
k = 2	6	6.499	8.31	6.00	0.00	6	0
k = 3	4	4.921	23.03	4.828	20.70	4	0
k = 4	7	6.986	0.2	6.946	0.77	7	0
MAPE (%)			7.89		7.16		0.00

Fig.1

Original fluctuation sequence curves versus forecasts.

Fig.2

Curves of actual and forecasted values using Holt’s-TCES, GM (1, 1), original NGBM (1, 1) and Fourier-NGBM (1, 1).

The results from Table 3 show that the optimized NGBM (1, 1) with n = -19.58 and p = 0.569 has a higher accuracy than the original NGBM (1, 1) with n = -10 and p = 0.5. However, by using the Fourier-NGBM (1, 1) with parameters n = -10 and p = 0.5, the MAPE of the Fourier-NGBM (1, 1) decreased from 7.16% to 0%. This result clearly indicated that the Fourier-NGBM (1, 1) is the best fitting model among the three forecasting models on the same sequence. A clearer visualization is shown in Fig. 1.

5 Application of proposed model in forecasting the electricity consumption in Vietnam

5.1 Data collection

The data of electricity consumption from 1990–2015 in Vietnam were obtained from the US Energy Information Administration [39] and are shown in Table 4.

Table 4
Electricity consumption in Vietnam

Year Electricity consumption Year Electricity consumption

(GWH) (GWH)

1990 8,678 2003 38,461

1991 9,152 2004 43,414

1992 9,654 2005 49,008

1993 10,665 2006 53,845

1994 12,284 2007 59,159

1995 14,636 2008 64,998

1996 16,946 2009 71,415

1997 19,151 2010 78,466

1998 21,665 2011 94,658

1999 23,739 2012 105,474

2000 26,745 2013 115,069

2001 30,187 2014 128,435

2002 34,073 2015 141,800

Year	Electricity consumption	Year	Electricity consumption
1990	8,678	2003	38,461
1991	9,152	2004	43,414
1992	9,654	2005	49,008
1993	10,665	2006	53,845
1994	12,284	2007	59,159
1995	14,636	2008	64,998
1996	16,946	2009	71,415
1997	19,151	2010	78,466
1998	21,665	2011	94,658
1999	23,739	2012	105,474
2000	26,745	2013	115,069
2001	30,187	2014	128,435
2002	34,073	2015	141,800

5.2 Forecasting models: Testing and results

In this case, four forecasting models, which are Holt’s trend corrected exponential smoothing method, traditional GM (1, 1), the nonlinear grey Bernoulli model NGBM (1, 1) and the proposed model, named Fourier-NGBM (1, 1), were used to find the best model in forecasting the electricity consumption in Vietnam as well as to test the proposed forecasting solution. First, we ran the data with the four forecasting models to find the best model among them based on the MAPE index.

5.2.1 Holt’s trend corrected exponential smoothing method (Holt’s-TCES)

Holt’s trend corrected exponential smoothing method was extended with simple exponential smoothing to allow the forecasting of data with a trend by Holt in 1957 [40]. This method involves a forecast equation and two smoothing equations (one for the level and one for the trend). The specific algorithm of Holt’s trend corrected exponential smoothing method was illustrated in [40]. We ran Holt’s-TCES with the smoothing parameter for the level alpha = 0.4 and the smoothing parameter for the trend beta = 0.3. The forecasted results and residual errors are tabulated in Table 5.

Table 5
Forecasted value and residual error by the GM (1, 1) and Holts’-TCES models

Years Actual value Traditional GM (1, 1) Holt’s-TCES

Forecasted value % error Forecasted value % error

1990 8,678 8,678 0.00 8,678 0.00

1991 9,152 10,120.57 0.11 9,152 0.00

1992 9,654 11,315.89 0.17 9,626 0.00

1993 10,665 12,652.39 0.19 10,114.56 0.05

1994 12,284 14,146.73 0.15 10,878.15 0.11

1995 14,636 15,817.58 0.08 12,152.6 0.17

1996 16,946 17,685.76 0.04 14,156.08 0.16

1997 19,151 19,774.59 0.03 16,616.96 0.13

1998 21,665 22,110.12 0.02 19,279.57 0.11

1999 23,739 24,721.50 0.04 22,168.99 0.07

2000 26,745 27,641.31 0.03 24,920.64 0.07

2001 30,187 30,905.97 0.02 27,992.96 0.07

2002 34,073 34,556.21 0.01 31,476.43 0.08

2003 38,461 38,637.57 0.00 35,432.5 0.08

2004 43,414 43,200.98 0.00 39,924.77 0.08

2005 49,008 48,303.36 0.01 45,020.03 0.08

2006 53,845 54,008.37 0.00 50,793.35 0.06

2007 59,159 60,387.18 0.02 56,558.34 0.04

2008 64,998 67,519.39 0.04 62,455.01 0.04

2009 71,415 75,493.97 0.06 68,633.77 0.04

2010 78,466 84,410.41 0.08 75,241.58 0.04

2011 94,658 94,379.95 0.00 82,413.59 0.13

2012 105,474 105,526.97 0.00 94,662.93 0.10

2013 115,069 117,990.54 0.03 107,636.3 0.06

2014 128,435 131,926.17 0.03 120,150.2 0.06

2015 141,800 147,507.69 0.04 133,999.1 0.06

MAPE (%) 4.67% 7.33 %

ρ = 100 - MAPE(%) 95.27 % 92.06 %

Years	Actual value	Traditional GM (1, 1)	Holt’s-TCES
1990	8,678	8,678	0.00	8,678	0.00
1991	9,152	10,120.57	0.11	9,152	0.00
1992	9,654	11,315.89	0.17	9,626	0.00
1993	10,665	12,652.39	0.19	10,114.56	0.05
1994	12,284	14,146.73	0.15	10,878.15	0.11
1995	14,636	15,817.58	0.08	12,152.6	0.17
1996	16,946	17,685.76	0.04	14,156.08	0.16
1997	19,151	19,774.59	0.03	16,616.96	0.13
1998	21,665	22,110.12	0.02	19,279.57	0.11
1999	23,739	24,721.50	0.04	22,168.99	0.07
2000	26,745	27,641.31	0.03	24,920.64	0.07
2001	30,187	30,905.97	0.02	27,992.96	0.07
2002	34,073	34,556.21	0.01	31,476.43	0.08
2003	38,461	38,637.57	0.00	35,432.5	0.08
2004	43,414	43,200.98	0.00	39,924.77	0.08
2005	49,008	48,303.36	0.01	45,020.03	0.08
2006	53,845	54,008.37	0.00	50,793.35	0.06
2007	59,159	60,387.18	0.02	56,558.34	0.04
2008	64,998	67,519.39	0.04	62,455.01	0.04
2009	71,415	75,493.97	0.06	68,633.77	0.04
2010	78,466	84,410.41	0.08	75,241.58	0.04
2011	94,658	94,379.95	0.00	82,413.59	0.13
2012	105,474	105,526.97	0.00	94,662.93	0.10
2013	115,069	117,990.54	0.03	107,636.3	0.06
2014	128,435	131,926.17	0.03	120,150.2	0.06
2015	141,800	147,507.69	0.04	133,999.1	0.06
MAPE (%)	4.67%		7.33 %
ρ = 100 - MAPE(%)	95.27 %		92.06 %

5.2.2 The traditional GM (1, 1) model

As mentioned in the introduction, the traditional GM (1, 1) model is the core of the grey forecasting model. It is constructed from a first-order differential model with one input variable. The overall modelling algorithm of the traditional GM (1, 1) forecasting model was shown in [24]. Based on the mathematical algorithm, this study can calculate the parameters: a = -0.112 and b = 8597.37. The function of GM (1, 1) for the electricity consumption is as follows: ${\hat{x}}^{(1)} (k) = [(8678 + \frac{8597.37}{0.112}) e^{0.112 \times k} - \frac{8597.37}{0.112}]$

All the forecasted values and the residual error of the traditional GM (1, 1) model are recorded in Table 5. As can be seen in Table 5, the MAPE index of the traditional GM (1, 1) model and Holt’s-TCES model in forecasting the electricity consumption is 4.67% and 7.33%, respectively. This indicates that the accuracy of the traditional GM (1,1) is higher than Holt’s-TCES model.

5.2.3 The NGBM (1, 1) and fourier-NGBM (1, 1) models

According to the historical data (in Table 4), this study can calculate the parameters: a = -0.1078, b = 2203.49, n = 0.123 and p = 0.5 based on the algorithms of the original NGBM (1, 1) model in section 3.1. The modelling of NGBM (1, 1) for the electricity consumption in Vietnam was found as follows: ${\hat{x}}^{(1)} (k) = [(8678^{0.877} + \frac{2203.49}{0.1078}) {e^{0.1078 \times 0.877 \times k} - \frac{2203.49}{0.1078}]}^{1.14025}$ , with k = 1, 2, 3 . . . m.

The residual error series attained from NGBM (1, 1) in Equation (16) is then modified with a Fourier series (in Equation (19)), which forecasted the results of the Fourier-NGBM (1, 1) as per the algorithms stated in section 3.2. The detailed modelling and computing procedures have been omitted due to limited space. The forecasted values of traditional NGBM (1, 1) and Fourier-NGBM (1, 1) are summarized in Table 6.

Table 6
Forecasted value and residual error by the NGBM (1, 1) and Fourier-NGBM (1, 1) models

Year Actual data Original NGBM (1, 1) n = 0.123, p = 0.5 Fourier-NGBM (1, 1) n = 0.123, p = 0.5

Forecasted value % error Forecasted value % error

1990 8,678 8,678.00 0.00 8,678.00 0.00

1991 9,152 8,406.19 8.15 9,152.00 0.00

1992 9,654 9,884.23 2.38 9,654.00 0.00

1993 10,665 11,410.07 6.99 10,665.00 0.00

1994 12,284 13,044.68 6.19 12,284.00 0.00

1995 14,636 14,824.73 1.29 14,636.00 0.00

1996 16,946 16,779.78 0.98 16,946.00 0.00

1997 19,151 18,937.52 1.11 19,151.00 0.00

1998 21,665 21,326.09 1.56 21,665.00 0.00

1999 23,739 23,975.24 1.00 23,739.00 0.00

2000 26,745 26,917.15 0.64 26,745.00 0.00

2001 30,187 30,187.00 0.00 30,187.00 0.00

2002 34,073 33,823.55 0.73 34,073.00 0.00

2003 38,461 37,869.67 1.54 38,461.00 0.00

2004 43,414 42,372.86 2.40 43,414.00 0.00

2005 49,008 47,385.87 3.31 49,008.00 0.00

2006 53,845 52,967.34 1.63 53,845.00 0.00

2007 59,159 59,182.44 0.04 59,159.00 0.00

2008 64,998 66,103.73 1.70 64,998.00 0.00

2009 71,415 73,811.93 3.36 71,415.00 0.00

2010 78,466 82,396.90 5.01 78,466.00 0.00

2011 94,658 91,958.69 2.85 94,658.00 0.00

2012 105,474 102,608.69 2.72 105,474.00 0.00

2013 115,069 114,470.95 0.52 115,069.00 0.00

2014 128,435 127,683.60 0.59 128,435.00 0.00

2015 141,800 142,400.47 0.42 141,800.00 0.00

MAPE 2.20 0.00

Year	Actual data	Original NGBM (1, 1) n = 0.123, p = 0.5	Fourier-NGBM (1, 1) n = 0.123, p = 0.5
1990	8,678	8,678.00	0.00	8,678.00
1991	9,152	8,406.19	8.15	9,152.00
1992	9,654	9,884.23	2.38	9,654.00
1993	10,665	11,410.07	6.99	10,665.00
1994	12,284	13,044.68	6.19	12,284.00
1995	14,636	14,824.73	1.29	14,636.00
1996	16,946	16,779.78	0.98	16,946.00
1997	19,151	18,937.52	1.11	19,151.00
1998	21,665	21,326.09	1.56	21,665.00
1999	23,739	23,975.24	1.00	23,739.00
2000	26,745	26,917.15	0.64	26,745.00
2001	30,187	30,187.00	0.00	30,187.00
2002	34,073	33,823.55	0.73	34,073.00
2003	38,461	37,869.67	1.54	38,461.00
2004	43,414	42,372.86	2.40	43,414.00
2005	49,008	47,385.87	3.31	49,008.00
2006	53,845	52,967.34	1.63	53,845.00
2007	59,159	59,182.44	0.04	59,159.00
2008	64,998	66,103.73	1.70	64,998.00
2009	71,415	73,811.93	3.36	71,415.00
2010	78,466	82,396.90	5.01	78,466.00
2011	94,658	91,958.69	2.85	94,658.00
2012	105,474	102,608.69	2.72	105,474.00
2013	115,069	114,470.95	0.52	115,069.00
2014	128,435	127,683.60	0.59	128,435.00
2015	141,800	142,400.47	0.42	141,800.00
MAPE			2.20

Tables 5 and 6 reveal that the original NGBM (1, 1) with n = 0.123 and p = 0.5 shows a higher accuracy than that of the traditional GM (1, 1) in terms of the MAPE of the NGBM (1, 1) decreased from 4.67% to 2.20 %.

As can be seen in Table 6, the Fourier-NGBM (1, 1) with n = 0.123 and p = 0.5 shows a higher accuracy than the original NGBM (1, 1) as the MAPE of the Fourier-NGBM (1, 1) is 0%. This result clearly indicates that the Fourier-NGBM (1, 1) is the best fitting model for forecasting the electricity consumption in Vietnam.

5.3 Implication

After comparison of the performance of the four forecasting models in forecasting the electricity consumption in Vietnam, the empirical results clearly showed that the Fourier-NGBM (1, 1) is the best model. Therefore, the study strongly suggests that the Fourier-NGBM (1, 1) is used for estimation of electricity consumption demand in the future. The forecasted value of electricity consumption in Vietnam up to 2020 is shown in Table 7. Because of the real values of electricity consumption demand in 2016 and 2017 are unaccessible. Therefore, this paper cannot compare the forecasted values of Fourier-NGBM (1, 1) model. Forecasted value of the electricity consumption up to the year 2020 are shown in Table 7.

Table 7
Forecasted value of electricity consumption by Fourier-NGBM (1, 1)

Year Electricity consumption (GWH)

2016 159,538.68

2017 176,821.35

2018 196,644.05

2019 219,281.45

2020 245,085.43

Year	Electricity consumption (GWH)
2016	159,538.68
2017	176,821.35
2018	196,644.05
2019	219,281.45
2020	245,085.43

In Table 7, the forecasted electricity consumption in 2019 and 2020 will be over 219,281 and 245,085 GWH, respectively. This figure indicates that electricity consumption in Vietnam will grow significantly in the future. This empirical result will provide a basic scenario for decision-makers to make a good decision in the future regarding electrical industry management and building the developing strategy.

6 Conclusion

NGBM (1, 1) is an important extension of the traditional grey prediction model, and can be used to predict the nonlinear small sample time sequences encountered in practical data. In this paper, after the original NGBM (1, 1) model is reviewed, a novel systematic approach for improving the forecasting accuracy of NGBM (1, 1) is introduced. The results from the simulation of a numerical example in He et al. ’s and Wang et al. ’s papers indicate that Fourier-NGBM (1, 1) gives a superior modelling performance. To demonstrate further the advantages of the Fourier-NGBM (1, 1) model, electricity consumption demand in Vietnam is forecasted. The simulation results show that Fourier-NGBM (1, 1) is more suitable for electricity consumption demand forecasting than the other models compared with it, and it can also be used to make highly accurate predictions for other nonlinear forecasting tasks.

Furthermore, this study found the model suitable to forecast the electricity consumption in Vietnam update to 2020. The contribution of the current study provides some insights and basic information for decision-makers in model building for developing future policies regarding electrical industry management.

Further work is recommended to apply different methods such as Markov chain or different kinds of Fourier function to modify the residual error of the NGBM (1, 1) model and to expand the application of the proposed model to forecasting performance in different industries.

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Nonlinear grey Bernoulli model based on Fourier transformation and its application in forecasting the electricity consumption in Vietnam

Abstract

Keywords

1 Introduction

2 Related work

3 Methodology

3.1 Nonlinear grey bernoulli model “NGBM (1, 1)”

4.1 Example data from He et al. [35]

5.1 Data collection

5.2.1 Holt’s trend corrected exponential smoothing method (Holt’s-TCES)

5.2.3 The NGBM (1, 1) and fourier-NGBM (1, 1) models

Table 7 Forecasted value of electricity consumption by Fourier-NGBM (1, 1) Year Electricity consumption (GWH) 2016 159,538.68 2017 176,821.35 2018 196,644.05 2019 219,281.45 2020 245,085.43

References

Table 7
Forecasted value of electricity consumption by Fourier-NGBM (1, 1)

Year Electricity consumption (GWH)

2016 159,538.68

2017 176,821.35

2018 196,644.05

2019 219,281.45

2020 245,085.43