Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 740272, 7 pages

http://dx.doi.org/10.1155/2015/740272

## An Improved Nonlinear Grey Bernoulli Model Combined with Fourier Series

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

Academic Editor: Meng Du

Copyright © 2015 Wang Chia-Nan and Phan Van-Thanh. 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

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.

#### 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) 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 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 model with residuals modification [15, 24]. In addition, many hybrid models based on GM 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 model is always monotonic. As a result, GM model may not be always satisfactory.

The recently developed, Nonlinear Grey Bernoulli Model (NGBM ) was named by Chen [28, 29] and firstly appeared in the book [30]. The NGBM has greater flexibility than GM 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 model, it had a great variety of application to simulate and forecast in different fields. Chen [28] proposed the NGBM 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 to forecast foreign exchange rates of twelve Taiwan major trading partners in 2005. Both of the two above studies indicated that the NGBM can improve the accuracy of the simulation and forecasting predictions of the original GM .

Some scholars had tried to improve the NGBM from different aspects recently, such as Zhou et al. [31] who selected the parameter value of 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 and applied it to forecast the economic trends in the integrated circuit industries in Taiwan. Chen et al. [32] proposed a Nash NGBM 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 model to forecast the qualified discharge rate of the industrial waste water in 31 administrative areas in China by improved background interpolation value and exponential value . Wang [34] proposed the optimized Nash NGBM 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 models have been successfully adopted in various fields and they have provided us with promising results, the NGBM 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 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 to get the predicted value and then using Fourier series to modify the residual errors of NGBM . The Fourier series transform the residuals error of NGBM 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 and the F-NGBM . Section 3 demonstrates that F-NGBM has better performances in several numerical examples by comparison with optimized NGBM , original NGBM , optimized GM , and the original GM . 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) 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 can be summarized as follows.

*Step 1. *Let raw matrix stand for the nonnegative original historical time series data where corresponds to the system output at time and is the total number of modeling data.

*Step 2. *Construct by one time accumulated generating operation (1-AGO), which iswhere , .

*Step 3. *The grey differential equation of NGBM is defined as And its whitenization differential equation is as follows:where , ; is called the production coefficient of the background value with a close interval , which is traditionally set to 0.5.

The parameters , and the power of are called the developing coefficient, the named grey input, and an adjustable parameter, respectively, for the power of “” belonging to any real number excluding .

*Step 4. *From (4), the value of parameters and can be estimated by using ordinary least-square method (OLS). That is, where

*Step 5. *The solution of (4) can be obtained after the parameters and have been estimated. That is,

*Step 6. *Applying inverse accumulated generating operation (I-AGO) to , the predicted data of can be estimated as

##### 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 for improving the prediction accuracy. The overall procedure to obtain the modified model is as follows.

Let be the original series of entries and is the predicted series (obtained from NGBM ). Based on the predicted series , a residual series named is defined as where

According to the definition of the Fourier series, the residual sequence of NGBM can be approximately expressed aswhere is called the minimum deployment frequency of Fourier series [6, 35] and only be taken integer number.

Therefore, the residual series is rewritten as where

The parameters are obtained by using the ordinary least squares (OLS) method whose results are in the following equation:

Once the parameters are calculated, the modified residual series is then achieved based on the following expression:

From the predicted series and , the Fourier modified series of series is determined bywhere

##### 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:where and are actual and forecasting values in time period , respectively, and 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 [30, 33], original NGBM [28], optimized GM [6], and original GM [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 and optimized GM models were established by minimizing an objective function of (18) with the constraints being and to get the global optimization of parameters and (just for NGBM ). 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 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 . More details are given below.

##### 3.1. Fluctuating Raw Data Sequence Example

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