Chapter 2 Literature Review
2.1 Research Papers Related to Improving Grey Models
Grey system theory was initiated in 1982 by Deng. It can effectively deal with incomplete and uncertain information (Deng, 1989; Liu & Forrest, 2007). In grey system theory, if the system information is fully known, the system is called a white system, while the system information is unknown, it is called a black system. A system with partial information known and partial information unknown is grey system (Liu & Lin, 2006, 2010). It included five major parts that are grey prediction, grey relation, grey decision, grey programming, and grey control (Li et al., 2010; Li et al., 2008b).
Grey models are the essential part of the grey system theory, GM(m,n) denotes a grey model, where m is the order of the difference equation and n is the number of variables. In recent years, grey model has been successfully used in finance, physical control, engineering and economics. The advantages of grey model include:
(1) It can be used in circumstances with relatively little data; as low as four observations were reported to estimate the outcome of an unknown system; (2) It can use a first-order differential equation to characterize a system. Therefore, only a few discrete data are sufficient to characterize an unknown system. However, there
12
is a problem that the predicted accuracy of grey model is unsatisfied. The coefficients of the prediction model are not the optimal coefficients. Many researchers have performed a lot of researches for this to improve the predicted accuracy. In this study, research papers related to improving GM(1,1), grey Verhulst model, GM(2,1), and GM(1,n) are introduced as follows:
Table 2-1
Research papers related to improving GM(1,1)
Researchers (Year) The main content of paper
Tien and Chen (1997) The indirect measurement of fatigue limits of structural steel by the deterministic grey dynamic model DGDM(1,1,1).
Hsu and Wen (1998) They combined residual modification and residual Markov-chain sign estimation to improve the accuracy of the original models.
Wen, Huang, and Wen (2000)
They analyzed a predicted error in using GM(1,1) based on the parameter α. Using the criterion of α to decide the optimal value of α has both theoretical and practical possibilities.
Hsu and Chen (2003) Using a technique that combines residual modification with artificial neural network sign estimation is proposed.
Tsaur (2005) The fuzzy system derived from collected data is considered by the fuzzy grey controlled variable to derive a fuzzy GM(1,1).
Li, Yamaguchi, and Nagai (2007)
They proposed a new dynamic analysis model which combines the first-order one-variable grey differential equation model from grey system theory and Markov chain model from stochastic process theory.
They called the improved model as T-MCGM(1,1).
Lin and Lee (2007) They proposed a novel prediction model termed MFGMn(1,1).
Chen (2008) Chen proposed Nonlinear Grey Bernoulli Model, which is a nonlinear differential equation with power n.
Li, Yamaguchi, Nagai, and Masuda (2008a)
They proposed a new prediction analysis model which combines GM(1,1) model from grey system theory and time series ARIMA model from statistics theory. The generated model is called as M3P-ARGM(1,1) model.
They proposed a new model named EFGMm(1,1) by eliminating the error term resulted from the traditional calculation of background value with an integration equation to substitute for such error term.
Tien (2009b) Tien proposed a grey prediction model called first-entry GM(1,1), abbreviated as FGM(1,1)
Chen (2010) Chen proposed a new grey model DPGM(1,1) based on the analysis of the relationship between GM(1,1) and DGM(1,1).
Li et al. (2010) To improve the prediction accuracy of GM(1,1), They proposed a prediction model P-3spGM(1,1).
Pi, Liu, and Qin (2010) The original GM(1,1) model is improved by using three methodologies of the 3-points average technology and the residual modification.
Zeng, Liu, and Xie (2010) Zeng proposed a prediction model of interval grey number based on DGM(1,1).
(table continues)
13
Table 2-1
Research papers related to improving GM(1,1) (continued)
Researchers (Year) The main content of paper
Chen, Guo, and Lo (2011) They proposed a hybrid grey model termed as EGM(1,1), which adopting exponential series to identify the residual error series resulted from grey model.
Kong, Liu, and Wei (2011)
They proposed a kind of data processing method to make the class ratio of the transformed sequence approximate the minimum or maximum class ratio of the original sequence for being better suited to construct R-ODGM(1,1) model and M-ODGM(1,1) model.
Chen, Sun, and Liu (2012)
The parameter is optimized based on genetic algorithm method in the unbiased GM(1,1) power model to minimize the average relative proportional error of accuracy.
Huang (2012) They applied GM(1,1) with adaptive levels of α (hereafter GM(1,1)-α model) to provide a concise prediction model.
Huo and Zhan (2012)
They proposed an improved GM(1,1), which used Fourier series to correct the residual of original value and predictive value, and reconstructed GM(1,1) white background value based on genetic algorithm.
Li, Masuda, and Nagai
(2012) They proposed a prediction model MC-T-ESGM(1,1).
Li and Wei (2012) They established the optimization of GM(1,1) direct model.
Pao, Fu, and Tseng (2012) They proposed a numerical iterative method to optimize the parameter of NGBM.
Truong and Ahn (2012) They proposed a prediction method based on the modified grey model with first order - one variable - MGM(1,1).
Wang et al. (2012)
To improve the prediction accuracy, an adaptive parameter learning mechanism is applied to SFGM(1,1) model to develop a new model named APL-SFGM(1,1).
Cui et al. (2013) They proposed a novel grey prediction model termed NGM model and its optimized model.
Li, Masuda, and Nagai (2013)
They proposed an improved hybrid optimization model based on GM(1,1) to develop the prediction model in power systems. The improved model is defined as T-MC-RGM(1,1).
Wang (2013) Wang proposed a new method for optimizing Nash nonlinear grey Bernoulli model (Nash NGBM(1,1)).
Guo et al. (2014) A novel grey NGM(1,1,k) self-memory coupling prediction model is put forward in order to promote the predictive performance.
Li, Masuda, and Nagai
(2014c) They proposed an improved grey model to acquire high-control system performance.
Li, Masuda, and Nagai (2014b)
They proposed an improved grey model to predict Japan's domestic and overseas automobile production.
Li, Masuda, and Nagai (2014a)
They proposed a novel grey prediction model to enhance the performance of prediction for the amount of fixed-line and cellular phone subscribers in Japan.
Liu et al. (2014) They developed a optimization model for the GM(1,1) model problem which includes optimization of initial and background values.
Qu, He, and Jia (2014) They proposed an adaptive multi-variable optimized GM(1,1) based on cuckoo search algorithm.
(table continues)
14
Table 2-1
Research papers related to improving GM(1,1) (continued)
Researchers (Year) The main content of paper Xiao, Guo, and Mao
(2014) The extension form GGM(1,1) based on the fractional order accumulated generating is put forward and its theoretical significance is analyzed.
Zhang et al. (2014)
They proposed an improved Nash nonlinear grey Bernoulli model termed PSO–NNGBM(1,1) by using a particle swarm optimization algorithm.
Sheu et al. (2014c) Using the combination of GM(1,1) and Taylor approximation method to Predict the academic achievement of student.
Sheu et al. (2014b) Using Taylor approximation method to improve the predicted accuracy of GM(1,1), GVM, and GM(2,1).
Nguyen et al. (2014b) Using Taylor approximation method in grey system theory to predict the number of teachers and students for admission.
Nguyen et al. (2014a) Using Taylor approximation method in grey system theory to predict the number of foreign students studying in Taiwan.
According to Table 2-1, grey system theory was firstly proposed by Deng in 1982.
Since then, it has become a very popular technique with its applications on the partially unknown parameters, variables etc. The GM(1,1) model is one of the most frequently used grey prediction model. This model is a time series prediction model, encompassing a group of differential equations adapted for parameter variance, rather than a first order differential equation. Its difference equations have structures that vary with time rather than being general difference equations. However, many researchers have pointed that using the GM(1,1) model may face the predicted accuracy of the GM(1,1) model was unsatisfied. First of all, a real system will grow at different speeds during the whole period, but it is difficult for the original GM(1,1) model to reflect real growth trends among the different periods, since it is just suitable for one exponential growth rule.
Secondly, it has been proven that this model is not suitable for long-term prediction, since the absolute value of model coefficient is too large it may lead to a larger prediction error. In order to solve these problems, many researchers have performed a lot of researches for this to improve the predicted accuracy of the GM(1,1) prediction model.
(Hsu and Chen, 2003). The GM(1,1) model has been applied to many real life systems such as social, economic and technical systems (Jian, Wakamatsu, & Feng, 1991; Liang, Liu, & Li, 2014; Wu, Hsiao, & Tsai, 2008; Yang, 1993; Zhu, 2014).
15
Table 2-2
Research papers related to improving Grey Verhulst Model
Researchers (Year) The main content of paper
Guo, Song, and Ye (2005) They used the grey Verhulst model on time series error corrected for the port throughput forecasting.
Hsu (2008)
This paper tended to set up a saturated analysis model of the population of Taiwan by utilizing grey Verhulst model and GM(1,1) model.
Wang and Song (2008)
They imported grey system Verhulst model theory, established the manpower management forecast model for transforming maintenance system.
Wang, Dang, and Liu (2009) They proposed unbiased grey Verhulst model. Recursive solutions are given under two initial conditions of the unbiased model.
Liu and Bi (2010) The Verhulst model with remedy and its application in forecasting quantity of student taking entrance examination to college.
Zhang (2012)
Zhang analyzed the cause of grey Verhulst model’s inaccuracy, the new formula of grey derivative is strutted and the unbiased grey Verhulst model is given in this paper.
Wang et al. (2013)
They proposed the time-delayed Verhulst model and then establish a grey time-delayed Verhulst model using the method of grey differential equations.
Zeng et al. (2013) They proposed the Verhulst model of interval grey number based on Information decomposing and model combination.
Zhou (2013) Zhou proposed a new time series prediction model for the time series growth in S-type or growth being saturated.
Kordnoori, Mostafaei, and Kordnoori (2014)
They suggested a new Grey Verhulst model and Fourier residual Grey Verhulst model to improve the predicted accuracy.
Sheu et al. (2014b) Using Taylor approximation method to improve the predicted accuracy of GM(1,1), GVM, and GM(2,1).
Verhulst model was first proposed by Germany biologist Verhulst to describe some increasing process like “S” curve which has saturation. It has been extensively used in numerous applications to explain the phenomenon of population increasing, living creature breeding and its individual growth. Grey Verhulst model, which is a first-order one-variable grey differential equation and also a time series model. It is a special grey prediction model which is developed to deal with the simulation for small sample data sequence with the characteristic of approximate single peak. This model is capable of simulating the time sequence data with the characteristic of saturated S curved. In recent years, it has been widely applied in some research fields asthroughput forecasting, the population prediction of Taiwan, andpredicting the road traffic accident (Guo et al., 2005;
Hsu, 2008;Mohammadi et al., 2011).
16
Table 2-3
Research papers related to improving GM(2,1)
Researchers (Year) The main content of paper Li, Yamaguchi, and Nagai
(2005)
They proposed a new method of GM according to Laplace transform in frequency domain. This proposal method is solved to nth order differential equation in respect of the solution of the
Based on the grey differential equation of DGM(2,1), the solution was inconsistent with the connotation expression, and the differential restored value was also inconsistent with the inverse-accumulating restored value. So the optimized DGM(2,1) was presented, which had the white exponential superposition.
Huang and Wei (2010)
For S-shaped sequence, they proposed GM(2,1) and time series combined model while non-monotonic oscillations and saturated the residual series model were set, sequence meeting the conditions of residual modeling.
Li et al. (2010)
They proposed a new calculation of initial value and derivative to enhance the predicted power according to cubic spline function.
They called the improved prediction as T-3spGM(2,1).
Li et al. (2011) They proposed the improved grey dynamic model GM(2,1), a second order single variable grey model, to enhance the forecasted accuracy. They called the proposed model as 3spGM(2,1) model.
Xu et al. (2011)
They presented an approach to the least squares solution to grey Verhulst model, and verified its feasibility by numerical examples.
They also presented the least squares solutions of grey models GM(1,1) and GM(2,1).
Yong and Wei (2011)
They optimized the grey derivative and background value of non-equigap DGM(2,1) model by calculating the definite integral of the whitened differential equation, and then, established a new non-equigap DGM(2,1) model.
Deng et al. (2012)
They improved the classical non-equidistant GM(1,1). To be more specific, n-AGO transformation took the place of the original data, which was called as mGM(n,1).
Shao and Su (2012) Based on the solution structure of white differential equation of DGM(2,1) model, they deduced the new 2-order grey derivative expression.
Nguyen et al. (2014a) Using Taylor approximation method in grey system theory to predict the number of foreign students studying in Taiwan.
Sheu et al. (2014a) Using GM(2, 1) and T-GM(2, 1) to predict the number of students for admission.
According to Table 2-3, the GM(2,1) model is a kind of grey model which is constructed by grey derivative and second-order grey derivative. GM(2,1) model can make up some defect of GM(1,1) model, but it still has its own defects. In order to solve
17
these problems, many researchers have performed a lot of researches for this to improve the predicted accuracy of the GM(2,1) prediction model.
Table 2-4
Research papers related to improving GM(1,n)
Researchers (Year) The main content of paper
Wu and Chen (2005) They proposed an integrated prediction method using GMC(1,n) combined with the improved GRA.
Han and Xu (2007) The parameter q of MGM(1,n) has been optimized, and a multi-variable grey model (MGM(1,n,q)) based on genetic algorithm.
Li et al. (2008b)
This improved prediction model combined the first-order two variables grey differential equation model (abbreviated as GM(1,2) model) from grey system theory and Taylor approximation method from approximation optimization theory.
Tien (2008)
Tien proposed a new prediction model, the interval grey dynamic model by convolution integral with the first-order derivative of the 1-AGO data and n series related, abbreviated as IGDMC(1,n).
Shao and Wei (2009) They reconstructed a new GM(1,n) model equation by using a kind of optimized background value.
Tien (2009a) Tien proposed a new prediction model called the deterministic grey dynamic model with convolution integral (DGDMC(1,n)).
Li et al. (2011)
They developed a prediction model of steel's tensile strength with the Brinell hardness acting as a leading indicator for a high temperature. They abbreviated the combined model as T-GM(1,2).
Tien (2011) Tien proposed a new model, which is called the first-pair-of-data GMC(1,n), abbreviated as FGMC(1,n).
Tien (2012) The algorithm of GMC(1,n) is applied to explain why the existing GM(1,n) model is incorrect.
Che, Luo, and Liu (2013) They proposed a new non-equidistant multivariable new information MGRM(1,n) model.
Guo, Xiao, and Forrest (2013)
They proposed a novel comprehensive adaptive grey model CAGM(1,n) in order to overcome the disadvantages existing in the traditional GM(1,n).
Hui, Li, and Shi (2013) They proposed a new method to choose optimal forecast variable number and data sample size for multi-variable grey model.
Luo (2013) Luo established a new information optimizing model can be used to build model in non-equal interval & equal interval time sequences.
Luo and Liu (2013b) They established MGRM(1,n) model can be used in non-equal interval & equal interval time series.
Luo and Liu (2013a) They established a non-equidistant multivariable MGRM(1,n).
Luo, Liu, and Yi (2013)
Based on accumulated generating operation of reciprocal number, a non-equidistant multivariable new information optimization MGRM(1,n) model was put forward which was taken the mth component as the initialization.
Shen and Sun (2013) They proposed an optimized grey system multivariate GM(1,n) model based on the optimization of background values of GM(1,n).
(table continues)
18
Table 2-4
Research papers related to improving GM(1,n) (continued)
Researchers (Year) The main content of paper
Wang (2013) In order to create the model of differential equations of system using limited and poor datum in the simulation of system dynamic the inverse GM model is proposed by Wang.
He, Che, and Liu (2014) They established a new non-equidistant multivariable MGM(1,n) model can be used in equidistance & non-equidistance model.
Li et al. (2014) They established MGRM(1,n) model can be used in non-equal interval and equal interval time series.
Luo, Che, and Liu (2014) A non-equidistant multivariable grey model MGRM(1,n) was built through applying reciprocal accumulated generating operation.
Luo and Liu (2014b)
Non-equidistant multivariable MGRM(1,n) model based on accumulated generating operation of reciprocal number was put forward which was taken the first component as the initialization.
Luo and Liu (2014a) They analyzed the building method of background value in grey model MGRM(1,n).
Wang (2014) Wang introduced nonlinear parameters into GMC(1,n) model and additionally apply a convolution integral to produce an improved forecasting model here designated as NGMC(1,n).
Wang and Pei (2014)
To improve the modelling accuracy of GDMC(1,n), n interpolation coefficients (taken as unknown parameters) are introduced into the background values of the n variables.
Yang, Deng, and Jiang (2014b)
They constructed the new-information background values of the multivariable non-equidistant new-information grey model MGRM(1,n) using the modeling method of progressive optimization.
Yang, Deng, and Jiang (2014a)
They proposed a constructing method of the background values of non-equidistant MGRM(1,n) model, and the background value constructing formula was given.
Nguyen et al. (2014b) Using Taylor approximation method in grey system theory to predict the number of teachers and students for admission.
According to Table 2-4, Li and co-workers proposed the T-GM(1,2) model in 2008 (Li et al., 2008b). In 2011, Li and co-workers used the T-GM(1,2) model to develop a prediction model of steel's tensile strength with the Brinell hardness acting as a leading indicator for a high temperature (Li et al., 2011). In 2014, Sheu and co-workers used Taylor approximation method to improve the predicted accuracy of GM(1,1), GVM, and GM(2,1) (Sheu et al., 2014b). They also used the combination of GM(1,1) and Taylor approximation method to predict the academic achievement of student (Sheu et al., 2014c). In 2014, Nguyen and co-workers used Taylor approximation method in grey system theory to predict the number of teachers and students for admission (Nguyen et al., 2014b), and predict the number of foreign students studying in Taiwan (Nguyen et al., 2014a).
19