An Improved Nonlinear Grey Bernoulli Model Combined with Fourier Series

Abstract

Grey forecasting is a dynamic forecasting model and has been widely used in various fields. In recent years, many scholars have proposed new procedures or new models to improve the precision accuracy of grey forecasting for the fluctuating data sets. However, the prediction accuracy of the grey forecasting models existing may not be always satisfactory in different scenario. For example, the data are highly fluctuating are with lots of noise. In order to deal with this issue, a Fourier Nonlinear Grey Bernoulli Model (1, 1) (abbreviated as F-NGBM (1, 1)) is proposed to enhance the forecasting performance. The proposed model was established by using Fourier series to modify the residual errors of Nonlinear Grey Bernoulli Model (1, 1) (abbreviated as (NGBM (1, 1)). To verify the effectiveness of the proposed model, fluctuation data of the numerical example in Wang et al.'s paper (Wang et al. 2011) and practical application are used. Both of these simulation results demonstrate that the proposed model could forecast more precisely than several different kinds of grey forecasting models. For future direction, this proposed model can be applied to forecast the performance with the high fluctuation data in the different industries.

Academic Editor:Meng Du

Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road, Kaohsiung 807, Taiwan

Received 28 April 2015; Revised 3 August 2015; Accepted 12 August 2015; 11 October 2015

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.

1. Introduction

Grey forecasting is the main part of grey system theory and an effective method for modeling and forecasting small sample time series. In the early 1980s, Professor Deng [1, 2] proposed the grey model (GM) [figure omitted; refer to PDF] based on control theory. This model utilizes an operator obtained by the first-order accumulation to operate the nonnegative original sequence. It demonstrates the approximate exponential growth laws and achieves short-term forecasting accuracy. With its advantages in dealing with uncertain information and few data required [3-5], the GM [figure omitted; refer to PDF] has been widely and successfully applied to various fields such as tourism [6, 7], transportation [8-10], financial and economic [11-13], integrated circuit industry [14-17], and energy industry [18-20].

In the recent years, many scholars have proposed new procedures or new models to improve the precision accuracy of grey model. For instant, Lin et al. [21] and Wang et al. [22] used different methods to improve the background values. Hsu [17] and Wang and Hsu [23] used different methods to modify the internal parameter estimation, like development coefficient and grey input coefficient. Some scholars had established GM [figure omitted; refer to PDF] model with residuals modification [15, 24]. In addition, many hybrid models based on GM [figure omitted; refer to PDF] were proposed. These included the grey econometric model [25], the grey Markov model [26, 27], and the grey fuzzy model [21]. Despite its improvement in prediction accuracy, the prediction accuracy of the GM [figure omitted; refer to PDF] model is always monotonic. As a result, GM [figure omitted; refer to PDF] model may not be always satisfactory.

The recently developed, Nonlinear Grey Bernoulli Model (NGBM [figure omitted; refer to PDF] ) was named by Chen [28, 29] and firstly appeared in the book [30]. The NGBM [figure omitted; refer to PDF] has greater flexibility than GM [figure omitted; refer to PDF] and Grey-Verhulst model by adjusting power index. Therefore, forecasting of the fluctuation sequence can be performed, as long as the power exponent and structural parameters in the model are known. Because of the flexibility of NGBM [figure omitted; refer to PDF] model, it had a great variety of application to simulate and forecast in different fields. Chen [28] proposed the NGBM [figure omitted; refer to PDF] to forecast the annual unemployment rates of ten selected countries to help governments to develop future policies regarding labor and economic policies. At the same time, Chen et al. [29] also used NGBM [figure omitted; refer to PDF] to forecast foreign exchange rates of twelve Taiwan major trading partners in 2005. Both of the two above studies indicated that the NGBM [figure omitted; refer to PDF] can improve the accuracy of the simulation and forecasting predictions of the original GM [figure omitted; refer to PDF] .

Some scholars had tried to improve the NGBM [figure omitted; refer to PDF] from different aspects recently, such as Zhou et al. [31] who selected the parameter value of [figure omitted; refer to PDF] by using a particle swarm optimization algorithm and used the model to forecast the power load of the Hubei electric power network. Hsu [16] used the genetic algorithm to optimize the parameters of the NGBM [figure omitted; refer to PDF] and applied it to forecast the economic trends in the integrated circuit industries in Taiwan. Chen et al. [32] proposed a Nash NGBM [figure omitted; refer to PDF] based on the Nash equilibrium concept. This strengthens the adaptability of the model and eventually improves the accuracy of the model. Later, Wang et al. [33] proposed optimized NGBM [figure omitted; refer to PDF] model to forecast the qualified discharge rate of the industrial waste water in 31 administrative areas in China by improved background interpolation value [figure omitted; refer to PDF] and exponential value [figure omitted; refer to PDF] . Wang [34] proposed the optimized Nash NGBM [figure omitted; refer to PDF] by optimizing the initial conditions to forecast the main economic indices of high technology enterprises in China. Performance evaluation of this results showed that the optimized model can fit the data well and provide guidance for policy making decisions for the development of high technology enterprise and so on.

Although those improved NGBM [figure omitted; refer to PDF] models have been successfully adopted in various fields and they have provided us with promising results, the NGBM [figure omitted; refer to PDF] is not always satisfactory in some special scenarios. For example, the data are highly fluctuating or are with lots of noise. In order to deal with these issues, this paper based on the advantages of Nonlinear Grey Bernoulli Model [figure omitted; refer to PDF] and Fourier series to build an effectiveness model aims to increase the predictive accuracy. The proposed model is a two-stage procedure; the first stage is using the NGBM [figure omitted; refer to PDF] to get the predicted value and then using Fourier series to modify the residual errors of NGBM [figure omitted; refer to PDF] . The Fourier series transform the residuals error of NGBM [figure omitted; refer to PDF] into frequency spectra, and then the researchers select the low-frequency term. This way can filter out high-frequency terms, which are supposed to be noisy, and then have better performance. To verify the effectiveness of the proposed model, both fluctuation data of the numerical example in Wang et al.'s paper [33] and practical application are used. All these simulation results indicated that the proposed model could offer a more precise forecast than several different kinds of grey forecasting models. Through simulation results, this study offers an effective model in order to deal with the high fluctuation sequence.

The remainder of this paper is organized as follows. Section 2 briefly introduces the original NGBM [figure omitted; refer to PDF] and the F-NGBM [figure omitted; refer to PDF] . Section 3 demonstrates that F-NGBM [figure omitted; refer to PDF] has better performances in several numerical examples by comparison with optimized NGBM [figure omitted; refer to PDF] , original NGBM [figure omitted; refer to PDF] , optimized GM [figure omitted; refer to PDF] , and the original GM [figure omitted; refer to PDF] . Finally, the conclusions are made in Section 4.

2. Methodology

2.1. A Brief Introduction to the Nonlinear Grey Bernoulli Model

The Nonlinear Grey Bernoulli Model (NGBM) [figure omitted; refer to PDF] is a first-order single-variable grey Bernoulli model with an interpolated coefficient in the background value [28, 29]. According to Zhou et al. [31], the procedures involved in using the NGBM [figure omitted; refer to PDF] can be summarized as follows.

Step 1.

Let raw matrix [figure omitted; refer to PDF] stand for the nonnegative original historical time series data [figure omitted; refer to PDF] where [figure omitted; refer to PDF] corresponds to the system output at time [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the total number of modeling data.

Step 2.

Construct [figure omitted; refer to PDF] by one time accumulated generating operation (1-AGO), which is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Step 3.

The grey differential equation of NGBM [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] And its whitenization differential equation is as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] is called the production coefficient of the background value with a close interval [figure omitted; refer to PDF] , which is traditionally set to 0.5.

The parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and the power of [figure omitted; refer to PDF] are called the developing coefficient, the named grey input, and an adjustable parameter, respectively, for the power of " [figure omitted; refer to PDF] " belonging to any real number excluding [figure omitted; refer to PDF] .

Step 4.

From (4), the value of parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] can be estimated by using ordinary least-square method (OLS). That is, [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Step 5.

The solution of (4) can be obtained after the parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have been estimated. That is, [figure omitted; refer to PDF]

Step 6.

Applying inverse accumulated generating operation (I-AGO) to [figure omitted; refer to PDF] , the predicted data of [figure omitted; refer to PDF] can be estimated as [figure omitted; refer to PDF]

2.2. The Residual of NGBM (1, 1) Modification by Fourier Series

Because Fourier series can transform the residuals error into frequency spectra and then select the low-frequency terms, moreover, Fourier technique can filter out high-frequency terms, which are supported to be noise, and then have better performance. Therefore, this study uses the Fourier series [6] to modify the residual of the NGBM [figure omitted; refer to PDF] for improving the prediction accuracy. The overall procedure to obtain the modified model is as follows.

Let [figure omitted; refer to PDF] be the original series of [figure omitted; refer to PDF] entries and [figure omitted; refer to PDF] is the predicted series (obtained from NGBM [figure omitted; refer to PDF] ). Based on the predicted series [figure omitted; refer to PDF] , a residual series named [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

According to the definition of the Fourier series, the residual sequence of NGBM [figure omitted; refer to PDF] can be approximately expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is called the minimum deployment frequency of Fourier series [6, 35] and [figure omitted; refer to PDF] only be taken integer number.

Therefore, the residual series is rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

The parameters [figure omitted; refer to PDF] are obtained by using the ordinary least squares (OLS) method whose results are in the following equation: [figure omitted; refer to PDF]

Once the parameters are calculated, the modified residual series is then achieved based on the following expression: [figure omitted; refer to PDF]

From the predicted series [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the Fourier modified series [figure omitted; refer to PDF] of series [figure omitted; refer to PDF] is determined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

2.3. Evaluative Precision of Forecasting Models

In order to evaluate the forecast capability of the model, Means Absolute Percentage Error (MAPE) index is used in this study to evaluate the performance and reliability of forecasting technique [36]. It is expressed as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are actual and forecasting values in time period [figure omitted; refer to PDF] , respectively, and [figure omitted; refer to PDF] is the total number of predictions.

Wang and Phan [35] interpret the MAPE results as a method to judge the accuracy of forecasts, where more than 10% is an inaccurate forecast, 5%-10% is a reasonable forecast, 1%-5% is a good forecast, and less than 1% is an excellent forecast.

3. Validation of the F-NGBM (1 , 1 )

In this section, two examples are given to compare the proposed model with several different kinds of grey forecasting models, which are the optimized NGBM [figure omitted; refer to PDF] [30, 33], original NGBM [figure omitted; refer to PDF] [28], optimized GM [figure omitted; refer to PDF] [6], and original GM [figure omitted; refer to PDF] [6, 8], to show the effectiveness of proposed model in the high fluctuation data sets. The first example in this study is proposed in Wang' paper [33] and the second example is the real case study of the gold price (GP) in the afternoon from the London Fix.

The procedures of the optimized NGBM [figure omitted; refer to PDF] and optimized GM [figure omitted; refer to PDF] models were established by minimizing an objective function of (18) with the constraints being [figure omitted; refer to PDF] and [figure omitted; refer to PDF] to get the global optimization of parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] (just for NGBM [figure omitted; refer to PDF] ). More detailed procedures of these models were comprehensively illustrated in Chen et al. [29] and Wang et al. [33]. In terms of the proposed prediction model, the procedure has two stages. The first stage is to build the NGBM [figure omitted; refer to PDF] to roughly predict the next data from a set of the most recent data. The second stage is to use Fourier series to refine the residual error by the NGBM [figure omitted; refer to PDF] . More details are given below.

3.1. Fluctuating Raw Data Sequence Example

First example, F-NGBM [figure omitted; refer to PDF] is used to predict an example proposed in the Wang et al.'s paper [33]. In this example, the raw data sequence jumps randomly [figure omitted; refer to PDF] in this case. Wang et al. used as an example to demonstrate the improvement in the accuracy of the optimized NGBM [figure omitted; refer to PDF] . In this section, we also adopt this example to compare the forecasting performance of the F-NGBM [figure omitted; refer to PDF] with the optimized NGBM [figure omitted; refer to PDF] and the original NGBM [figure omitted; refer to PDF] in Wang et al. [33]. Forecasting results are shown in Table 1 and Figure 1.

Table 1: Forecasted results from the grey models.

Original value	GM (1, 1)	Optimized GM (1, 1) [figure omitted; refer to PDF]	Original NGBM (1, 1) [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Optimized NGBM (1, 1) [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	F-NGBM (1, 1) [figure omitted; refer to PDF] , [figure omitted; refer to PDF]
	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	MAPE (%)	[figure omitted; refer to PDF]	MAPE (%)	[figure omitted; refer to PDF]	MAPE (%)	[figure omitted; refer to PDF]	MAPE (%)	[figure omitted; refer to PDF]	MAPE (%)
[figure omitted; refer to PDF]	5	5	0	5	0	5	0	5	0.1	5	0
[figure omitted; refer to PDF]	6	5.084	15.26	5.488	8.527	6.499	8.31	6.00	0.00	6	0
[figure omitted; refer to PDF]	4	5.634	40.86	5.866	46.66	4.921	23.03	4.828	20.70	4	0
[figure omitted; refer to PDF]	7	6.920	10.80	6.271	10.42	6.986	0.2	6.946	0.77	7	0
MAPE (%)			16.73		16.42		7.89		7.16		0.00

Figure 1: Original fluctuation sequence curves versus forecasts.

[figure omitted; refer to PDF]

Table 1 reveals that the optimized NGBM [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] has a higher accuracy than the original NGBM [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] as well as the optimized GM [figure omitted; refer to PDF] and original GM [figure omitted; refer to PDF] . By adopting the F-NGBM [figure omitted; refer to PDF] with parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the MAPE of the F-NGBM [figure omitted; refer to PDF] decreased from 7.16% to 0%. This result clearly indicated that the F-NGBM [figure omitted; refer to PDF] is the best fitting model among five forecasting models on the same sequence. Much clearer visualization is showed in Figure 1.

3.2. The Gold Price Forecasting

To give more evidence about the forecasting ability of the proposed model, this paper uses the real case in the life. This study uses the historical data of gold price (GP) from the London PM Fix, from day 03/11/2014 to 28/4/2015, to verify the effectiveness of F-NGBM [figure omitted; refer to PDF] . The data of gold price is obtained from the daily statistical data published on the website of Kitco [37]. There are totally 122 observations available as illustrated in Figure 2. From Figure 2, the trend of gold price (GP) is a wild fluctuation and is highly nonlinear over the period. To demonstrate the superiority of proposed model for both of interpolation and extrapolation data, this study sets the samples from 03.11.2004 to 24.04.2015 (120 data points) for in-sample estimation. And the remainders of the sample are reserved for out-of-sample forecasting purposes (or validation data set).

Figure 2: Gold price time series from the day 03/11/2014 to 28/4/2015.

[figure omitted; refer to PDF]

In order to find out the parameters in five forecasting models, which are original GM [figure omitted; refer to PDF] , optimized GM [figure omitted; refer to PDF] , original NGBM [figure omitted; refer to PDF] , and optimized NGBM [figure omitted; refer to PDF] as well as the proposed model, Microsoft Excel is used. Beside a basic function in Excel, Excel software also offers two useful functions named MMULT (array 1, array 2) to return the matrix product of two relevant arrays and Minverse (array) to return the inverse matrix. For the sake of convenience, the detailed calculation and modeling process are omitted here. All the parameters estimation results of these models are listed in Table 2. Only the MAPE of these models for both in-sample and out-of-sample forecasting are shown in Tables 3 and 4, respectively.

Table 2: The parameter evaluation of the fitting models.

Model	Parameter value
GM (1, 1)	[figure omitted; refer to PDF] = 0, [figure omitted; refer to PDF] = 0.5
Optimized GM (1, 1)	[figure omitted; refer to PDF] = 0, [figure omitted; refer to PDF] = 0.001
Original NGBM (1, 1)	[figure omitted; refer to PDF] = 0.0381, [figure omitted; refer to PDF] = 0.5
Optimized NGBM (1, 1)	[figure omitted; refer to PDF] = 0.0385, [figure omitted; refer to PDF] = 0.001
F-NGBM (1, 1)	[figure omitted; refer to PDF] = 0.0381, [figure omitted; refer to PDF] = 0.5

Table 3: In-sample comparisons among five grey forecasting models.

Model	MAPE (%)	Forecasted accuracy (%)	Performance
GM (1, 1)	2.1113	97.8887	Good
Optimized GM (1, 1)	2.1108	97.8892	Good
Original NGBM (1, 1)	1.7948	98.2052	Good
Optimized NGBM (1, 1)	1.7791	98.2209	Good
F-NGBM (1, 1)	0.0008	99.9992	Excellent

Table 4: Out-of-sample comparisons among five grey forecasting models.

Model	MAPE (%)	Forecasted accuracy (%)	Performance
GM (1, 1)	1.4475	98.5525	Good
Optimized GM (1, 1)	1.4450	98.5520	Good
Original NGBM (1, 1)	0.8074	99.1926	Good
Optimized NGBM (1, 1)	0.8073	99.1925	Good
F-NGBM (1, 1)	0.2499	99.7501	Excellent

Tables 3 and 4 show that the MAPE indexes of proposed model for in-sample and out-of-sample forecast are 0.0008% and 0.2499%, respectively. These results indicate that the forecasted performance of proposed model is the best fitting performance among four forecasting models. In addition, Figures 3 and 4 emphasize that the curve of F-NGBM [figure omitted; refer to PDF] extremely closed with actual data than the curve of the remaining fours forecasting models. Similarly, the curves of optimized NGBM [figure omitted; refer to PDF] and the original NGBM [figure omitted; refer to PDF] have similar forecasting results (Figure 3) and the curves of optimized GM [figure omitted; refer to PDF] and the original GM [figure omitted; refer to PDF] are nearly the same (Figure 4).

Figure 3: Curves of actual and simulated values using traditional and optimized GM [figure omitted; refer to PDF] models and F-NGBM [figure omitted; refer to PDF] for gold price forecasting.

[figure omitted; refer to PDF]

Figure 4: Curves of actual and simulated values using traditional and optimized NGBM [figure omitted; refer to PDF] and F-NGBM [figure omitted; refer to PDF] for gold price forecasting.

[figure omitted; refer to PDF]

4. Conclusion

In this paper, the effectiveness model named F-NGBM [figure omitted; refer to PDF] is proposed for improving the prediction accuracy in the high fluctuation data sets. The proposed prediction model approach uses the NGBM [figure omitted; refer to PDF] to roughly predict the next data from a set of the most recent data and then uses the Fourier series to fit the residual error proceeded by the NGBM [figure omitted; refer to PDF] . From Tables 1, 3, and 4, it is evident that F-NGBM [figure omitted; refer to PDF] could offer a better precise forecast than several different kinds of grey forecasting models, such as optimized-NGBM [figure omitted; refer to PDF] , original NGBM [figure omitted; refer to PDF] , optimized GM [figure omitted; refer to PDF] , and the original GM [figure omitted; refer to PDF] .

Future researchers can be using different equations or different methodologies like the Markov Chain, neural network to improve the accuracy of F-NGBM [figure omitted; refer to PDF] . Furthermore, the proposed model can be applied in many other industries with the high fluctuation data to forecast the performances.

Acknowledgments

The authors are grateful to the editors and the anonymous reviewers for their helpful and constructive comments and suggestions in editing this paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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