# Research on a Novel Kernel Based Grey Prediction Model and Its Applications

Authors: Xin Ma
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

The discrete grey prediction models have attracted considerable interest of research due to its effectiveness to improve the modelling accuracy of the traditional grey prediction models. The autoregressive GM(1,1) model, abbreviated as ARGM(1,1), is a novel discrete grey model which is easy to use and accurate in prediction of approximate nonhomogeneous exponential time series. However, the ARGM(1,1) is essentially a linear model; thus, its applicability is still limited. In this paper a novel kernel based ARGM(1,1) model is proposed, abbreviated as KARGM(1,1). The KARGM(1,1) has a nonlinear function which can be expressed by a kernel function using the kernel method, and its modelling procedures are presented in details. Two case studies of predicting the monthly gas well production are carried out with the real world production data. The results of KARGM(1,1) model are compared to the existing discrete univariate grey prediction models, including ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NGBMOP, and it is shown that the KARGM(1,1) outperforms the other four models.


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## Body

## 1. Introduction

The idea of the “Grey Box” modelling is trying to combine the advantages of the “White Box” and the “Black Box.” Deng [1] has pioneered the Grey System Theory based on this idea. The grey prediction models play an important role in the Grey System Theory, and because of their effectiveness in time series prediction the grey prediction models have been widely adopted [2–5].Over three decades of development, many new grey prediction models have been put forward, such as the FGM(1,1) [6], DGMD(1,1,1) [7], NGM(1,1,k) [8], and SAGM(1,1) [9]. Along with these new models, some novel methodologies have also been proposed, and the discrete modelling technique is one of the most efficient methods to build the grey prediction models. The discrete modelling technique has been developed from the research of the DGM(1,1) model [10], which is based on the basic GM(1,1) model. This novel technique has also been used to build the NDGM(1,1,k) model based on the NGM(1,1,k) [11]. And in our previous works, this technique has been extended to build the discrete GM(1,n) models [12, 13]. In these works, the discrete modelling technique has been proved efficient to improve the accuracy of the grey prediction models. Some novel grey prediction models for the nonlinear sequences are developed in recent years. For the univariate regression problems, the nonlinear grey Bernoulli model (NGBM) has been proposed by Chen et al. [14], which is more flexible than the existing grey prediction models and efficient to predict various time series. The NGBM model has attracted considerable research, and some improved grey prediction models based on it have been proposed, such as the Nash NGBM [15], the NGBM with optimal parameter [16], and the optimized NGBM [17] model. As for the multivariate regression problems, the nonlinear GMC(1,n) model [18] has been proved to be more efficient to predict the nonlinear series than the existing models.In recent researches, a novel grey prediction model directly built on the original series has been proposed, which is called the DDGM(1,1) model [19] and is also called the autoregressive GM(1,1) model (ARGM(1,1)) [20] as it is essentially in the autoregressive formulation. It is unnecessary to use the 1-AGO when building the ARGM(1,1) model; thus it is very easy to use. The ARGM(1,1) model has been proved to coincidence with the nonhomogeneous exponential law [19], and it has also been presented to be more efficient than the DGM(1,1) model in some applications [19, 20]. However, the ARGM(1,1) model is essentially a linear model; thus its applicability is limited.In order to improve the applicability of the ARGM(1,1) model, we use the kernel method to build a novel kernel based ARGM(1,1) model, abbreviated as KARGM(1,1). The kernel method has been developed from the Vapnik’s Support Vector Machines (SVM) [21], and it has been proved to be very efficient to convert the classical linear models into nonlinear models in the previous researches [22–24]. The researches of Vapnik’s SVM [21] are the initial works of the kernel method. But the formulation of the Vapnik’s SVM is not easy to use as it involves in a quadric problem with inequivalent constraints. Suykens and Vandewalle [25] have proposed a simplified formulation of kernel method involving a quadric problem with equivalent constraints, which can be converted into a linear system. And the formulation by Suykens and Vandewalle has been proved to be available to extend to linear models into nonlinear models as efficient as the formulation of Vapnik’s SVM [26, 27]. As for the time series regression, the recurrent LS-SVM [28] is a typical model for the nonlinear univariate time series, which can be more easily used to predict the chaotic time series than the recurrent neural networks [29]. In this work, the kernel method by Suykens and Vandewalle will be used to build the KARGM(1,1) model.The rest of this paper is organized as follows. Section2 presents a brief overview of the existing ARGM(1,1) model; Section 3 presents the modelling procedures of the KARGM(1,1); two case studies of predicting the gas well production are presented in Section 4, and conclusions are drawn in Section 5.


## 2. Overview of the Autoregressive GM(1,1) Model

With a given time series{x(1),x(2),…,x(n)}, the autoregressive GM(1,1) (ARGM(1,1)) model is represented as the following linear difference equation [20]:(1)xk=αxk-1+β,k=2,3,…,n.The parametersα and β can be obtained using the least squares method as follows:(2)α,βT=BTB-1BTY,where(3)Y=x2x3⋮xn,B=x11x21⋮⋮xn-11.The solution of the ARGM(1,1) model can be obtained using the recursive method, which is given as the following discrete function:(4)xk=αkx1+1-αk1-αβ.The discrete function (4) is used to compute the predicted values.


## 3. The Proposed KARGM(1,1) Model

In this section, the modelling procedures of the novel kernel based autoregressive GM(1,1) model, abbreviated as KARGM(1,1), will be presented.


### 3.1. Representation of the KARGM(1,1) Model

With a given time series{x(1),x(2),…,x(n)}, the KARGM(1,1) model is represented as the following difference equation:(5)xk=αxk-1+wTφk+β,where φ is a nonlinear mapping, which is defined as(6)φ:R⟶F,and F is a higher dimensional feature space; w is a vector in F.The linear combinationwTφ(k) is a nonlinear function of k. For example, if we consider a nonlinear function(7)gk=0.3sin⁡k∗π+k,and define the nonlinear mapping as(8)φk=sin⁡k∗π,kT,and set w=[0.3,1]T, then we have(9)gk=wTφk.


### 3.2. Parameters Estimation of the KARGM(1,1) Model

Being different from the ARGM(1,1) model, we cannot simply use the least squares method to estimate the parameters of the KARGM(1,1) model, because it is not always computationally feasible to find the formulation of the nonlinear mapping φ [30]. Firstly, we consider the regularized problem as follows:(10)minJα,w,e=12α2+12w2+γ2∑k=2nek2s.t.xk=αxk-1+wTφk+β+ek,where γ is the regularized parameter to balance the flatness of the fitting curve and the error bound, and · is the 2-norm. The optimization problem (10) can be solved using the KKT conditions, which have been presented in Appendix A.We firstly define the Lagrangian function as(11)Lα,w,e;λ=Jα,w,e-∑k=2nλkαxk-1+wTφk+β+ek-xk,where λk is the Lagrangian multiplier. Then the KKT conditions can be given as(12)∂L∂α=0⟹α=∑k=2nλkxk-1∂L∂w=0⟹w=∑k=2nλkφk∂L∂β=0⟹∑k=2nλk=0∂L∂ek=0⟹ek=λkγ-1∂L∂λk=0⟹xk=αxk-1+wTφk+β+ek.By eliminating theα, w and ek, the KKT conditions can be converted to the following linear system:(13)01→n-1T1→n-1Ω+Q+γ-1In-1βλ→=0Y,where(14)1→n-1=1,1,…,1n-1T,λ→=λ2,λ3,…,λnT,Y=x2,x3,…,xnT,Qij=xixj,Ωij=φi+1·φj+1,and In-1 and n-1 dimensional identity matrix with all the diagonal elements are 1 and others are zero. The inner product Ωij can be expressed using a kernel function which satisfies the Mercer’s condition, that is,(15)Ωij=φi+1·φj+1=Ki+1,j+1.The Gaussian kernel is often employed, which is defined as(16)Ki,j=exp⁡-i-j22σ2,where σ is the kernel parameter.The linear system (13) can be solved within the inner product (15) expressed by the Gaussian kernel (16), and then the parameter β and the Lagrangian multipliers λk can be obtained. The parameter α can then be computed using the first equation in the KKT conditions (12).


### 3.3. The Solution of the KARGM(1,1) Model

The KARGM(1,1) model (5) can be easily solved using the recursive method as follows:(17)xk=αxk-1+wTφk+β=ααxk-2+wTφk-1+β+wTφk+β=α2xk-2+∑τ=k-1kαk-τwTφτ+β=⋯=αk-1x1+∑τ=2kak-τwTφτ+β.Noticing the second equation in the KKT conditions (12), which can be rewritten as w=∑j=2n‍λjφ(j), we have(18)wTφτ=∑j=2nλjφj·φτ.Using the inner product (15), we can rewrite the nonlinear function (18) as follows:(19)wTφτ=∑j=2nλjKj,τ.For convenience, we note the following discrete function:(20)ϕτ=wTφτ+β=∑j=2nλjKj,τ+β.Then the solution (17) can be rewritten as(21)xk=αk-1x1+∑τ=2kak-τϕτ.The discrete function (21) can be used to compute the predicted series.It should be noticed that we do not need to know the expression of the nonlinear mappingφ, because all the computational procedures of the KARGM(1,1) model only involve in the inner product φ(i)·φ(j), and it can be expressed by a proper kernel function.Actually, the KARGM(1,1) represents a dynamical system which contains a “White” part and a “Black” part. The “White” part is the linear recursion which is knowna priori, and the “Black” part is the linear combination wTφ(k), which is expressed by the kernel function and finally determined by the raw data. So we can say that the KARGM(1,1) model is a real “Grey” model.


### 3.4. Summary of Computational Steps

The computational steps of the KARGM(1,1) model are summarized as follows.Step 1.
Select an appropriate kernel functionK(·,·) and the regularized parameter γ in (10).Step 2.
Compute the parametersβ and λ2,λ3,…,λn by solving the linear system (13), and then compute the parameter α using the first equation in the KKT conditions (12).Step 3.
Compute the values of the predicted series using the discrete function (21).


## 3.1. Representation of the KARGM(1,1) Model

With a given time series{x(1),x(2),…,x(n)}, the KARGM(1,1) model is represented as the following difference equation:(5)xk=αxk-1+wTφk+β,where φ is a nonlinear mapping, which is defined as(6)φ:R⟶F,and F is a higher dimensional feature space; w is a vector in F.The linear combinationwTφ(k) is a nonlinear function of k. For example, if we consider a nonlinear function(7)gk=0.3sin⁡k∗π+k,and define the nonlinear mapping as(8)φk=sin⁡k∗π,kT,and set w=[0.3,1]T, then we have(9)gk=wTφk.


## 3.2. Parameters Estimation of the KARGM(1,1) Model

Being different from the ARGM(1,1) model, we cannot simply use the least squares method to estimate the parameters of the KARGM(1,1) model, because it is not always computationally feasible to find the formulation of the nonlinear mapping φ [30]. Firstly, we consider the regularized problem as follows:(10)minJα,w,e=12α2+12w2+γ2∑k=2nek2s.t.xk=αxk-1+wTφk+β+ek,where γ is the regularized parameter to balance the flatness of the fitting curve and the error bound, and · is the 2-norm. The optimization problem (10) can be solved using the KKT conditions, which have been presented in Appendix A.We firstly define the Lagrangian function as(11)Lα,w,e;λ=Jα,w,e-∑k=2nλkαxk-1+wTφk+β+ek-xk,where λk is the Lagrangian multiplier. Then the KKT conditions can be given as(12)∂L∂α=0⟹α=∑k=2nλkxk-1∂L∂w=0⟹w=∑k=2nλkφk∂L∂β=0⟹∑k=2nλk=0∂L∂ek=0⟹ek=λkγ-1∂L∂λk=0⟹xk=αxk-1+wTφk+β+ek.By eliminating theα, w and ek, the KKT conditions can be converted to the following linear system:(13)01→n-1T1→n-1Ω+Q+γ-1In-1βλ→=0Y,where(14)1→n-1=1,1,…,1n-1T,λ→=λ2,λ3,…,λnT,Y=x2,x3,…,xnT,Qij=xixj,Ωij=φi+1·φj+1,and In-1 and n-1 dimensional identity matrix with all the diagonal elements are 1 and others are zero. The inner product Ωij can be expressed using a kernel function which satisfies the Mercer’s condition, that is,(15)Ωij=φi+1·φj+1=Ki+1,j+1.The Gaussian kernel is often employed, which is defined as(16)Ki,j=exp⁡-i-j22σ2,where σ is the kernel parameter.The linear system (13) can be solved within the inner product (15) expressed by the Gaussian kernel (16), and then the parameter β and the Lagrangian multipliers λk can be obtained. The parameter α can then be computed using the first equation in the KKT conditions (12).


## 3.3. The Solution of the KARGM(1,1) Model

The KARGM(1,1) model (5) can be easily solved using the recursive method as follows:(17)xk=αxk-1+wTφk+β=ααxk-2+wTφk-1+β+wTφk+β=α2xk-2+∑τ=k-1kαk-τwTφτ+β=⋯=αk-1x1+∑τ=2kak-τwTφτ+β.Noticing the second equation in the KKT conditions (12), which can be rewritten as w=∑j=2n‍λjφ(j), we have(18)wTφτ=∑j=2nλjφj·φτ.Using the inner product (15), we can rewrite the nonlinear function (18) as follows:(19)wTφτ=∑j=2nλjKj,τ.For convenience, we note the following discrete function:(20)ϕτ=wTφτ+β=∑j=2nλjKj,τ+β.Then the solution (17) can be rewritten as(21)xk=αk-1x1+∑τ=2kak-τϕτ.The discrete function (21) can be used to compute the predicted series.It should be noticed that we do not need to know the expression of the nonlinear mappingφ, because all the computational procedures of the KARGM(1,1) model only involve in the inner product φ(i)·φ(j), and it can be expressed by a proper kernel function.Actually, the KARGM(1,1) represents a dynamical system which contains a “White” part and a “Black” part. The “White” part is the linear recursion which is knowna priori, and the “Black” part is the linear combination wTφ(k), which is expressed by the kernel function and finally determined by the raw data. So we can say that the KARGM(1,1) model is a real “Grey” model.


## 3.4. Summary of Computational Steps

The computational steps of the KARGM(1,1) model are summarized as follows.Step 1.
Select an appropriate kernel functionK(·,·) and the regularized parameter γ in (10).Step 2.
Compute the parametersβ and λ2,λ3,…,λn by solving the linear system (13), and then compute the parameter α using the first equation in the KKT conditions (12).Step 3.
Compute the values of the predicted series using the discrete function (21).


## 4. Applications

Two case studies of predicting the gas well production are carried out in this section to validate the effectiveness of the KARGM(1,1) model. The monthly production data are collected from two gas wells in Sichuan, China. In order to compare the performance of the KARGM(1,1) model and other existing discrete grey models, the ARGM(1,1), NDGM(1,1,k) [11], DGM(1,1) [10], and the nonlinear grey Bernoulli model with optimal parameter i (NGBMOP) [16] are also applied in the case studies. The mean absolute percentage error is used as an overall measurement of accuracy of the prediction models, which is defined as(22)MAPE=1n∑k=1nx^k-xkxk×100%,where x(k) is the real value and x^(k) is the predicted value, respectively.


### 4.1. Case Study 1

The raw data used in case study 1 are collected from the gas well B51 in Sichuan, China. Twenty points of monthly gas production (104m3) are listed in Table 1. The first 15 points are used to build the models, and the last 5 points are used for testing. The Gaussian kernel (16) is used to build the KARGM(1,1) model.Table 1
Monthly gas production data of the B51 in Sichuan, China.
Month Gas production Month Gas production Month Gas production Month Gas production 1 90.2837 6 63.9314 11 34.2398 16 53.2518 2 72.1744 7 30.8003 12 44.0861 17 50.3282 3 54.7787 8 60.5955 13 49.6398 18 37.8982 4 54.1369 9 58.2767 14 51.7017 19 36.4632 5 64.6249 10 7.5851 15 44.5744 20 48.0127The kernel parameterσ and the regularized parameter γ are tuned using the cross validation, which is widely used in the other kernel methods, such as the Partially Linear LS-SVM [31]. The 5-fold cross validation is employed in this case. The MAPE of validation is used to assess the chosen kernel parameter and the regularized parameter. A brief summary of the 5-fold cross validation has been described in Appendix B. The results of the MAPE for each pair of σ and γ are plotted in Figure 1. It can be seen that the minimum MAPE is found at the point σ=0.9 and γ=14; thus this pair will be used to build the KARGM(1,1) in this case.Figure 1
The MAPE of cross validation of KARGM(1,1) with different σ and γ using the 5-fold cross validation in Case Study 1.The predicted values by the KARGM(1,1), ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NGBMOP are listed in Table 2, along with the absolute percentage error of each point and the MAPE of each model. The results are also plotted in Figure 2.Table 2
Predicted values of monthly gas production of B51 by the ARGM(1,1), NDGM(1,1,k), DGM(1,1), NGBMOP, and KARGM(1,1).
Month Gas production ARGM(1,1) Error NDGM(1,1,k) Error DGM(1,1) Error NGBMOP Error KAGM(1,1) Error 1 90.2837 90.2837 0 90.2837 0 90.2837 0 90.2837 0.00 90.2837 0.00 2 72.1744 59.1583 18.03 73.2911 1.55 63.5896 11.89 72.8510 0.94 70.0022 3.01 3 54.7787 51.0645 6.78 64.7782 18.25 61.0010 11.36 65.2404 19.10 55.9178 2.08 4 54.1369 48.9598 9.56 57.9739 7.09 58.5177 8.09 61.4602 13.53 53.5025 1.17 5 64.6249 48.4125 25.09 52.6425 18.54 56.1356 13.14 58.5062 9.47 65.0060 0.59 6 63.9314 48.2702 24.50 48.5808 24.01 53.8504 15.77 55.9700 12.45 61.2468 4.20 7 30.8003 48.2332 56.60 45.6134 48.09 51.6582 67.72 53.6995 74.35 34.3068 11.38 8 60.5955 48.2236 20.42 43.5893 28.07 49.5553 18.22 51.6200 14.81 59.2332 2.25 9 58.2767 48.2211 17.25 42.3785 27.28 47.5380 18.43 49.6887 14.74 55.7955 4.26 10 7.5851 48.2204 535.73 41.8686 451.98 45.6028 501.22 47.8783 531.21 12.5928 66.02 11 34.2398 48.2202 40.83 41.9629 22.56 43.7464 27.76 46.1701 34.84 32.4912 5.11 12 44.0861 48.2202 9.38 42.5780 3.42 41.9656 4.81 44.5509 1.05 45.1528 2.42 13 49.6398 48.2202 2.86 43.6422 12.08 40.2572 18.90 43.0106 13.35 49.4121 0.46 14 51.7017 48.2202 6.73 45.0933 12.78 38.6184 25.31 41.5412 19.65 51.3527 0.68 15 44.5744 48.2202 8.18 46.8780 5.17 37.0463 16.89 40.1364 9.96 45.2044 1.41  MAPE   52.13  45.39  50.63  51.30  7.00  16 53.2518 48.2202 9.45 48.9502 8.08 35.5382 33.26 38.7911 27.16 47.2950 11.19 17 50.3282 48.2202 4.19 51.2703 1.87 34.0915 32.26 37.5008 25.49 50.3266 0.00 18 37.8982 48.2202 27.24 53.8041 41.97 32.7037 13.71 36.2619 4.32 50.5842 33.47 19 36.4632 48.2202 32.24 56.5221 55.01 31.3724 13.96 35.0710 3.82 50.5825 38.72 20 48.0127 48.2202 0.43 59.3988 23.71 30.0953 37.32 33.9254 29.34 50.5832 5.35  MAPE   14.71  26.13  26.10  18.02  17.75Figure 2
Prediction results of the monthly gas production of B51 by the prediction models. The raw data are plotted using the mark “+” and the predicted values are plotted using the solid line and the mark “o”. (a) ARGM(1,1); (b) NDGM(1,1,k); (c) DGM(1,1); (d) NGBMOP; (e) KARGM(1,1).
(a) (b) (c) (d) (e)It can be seen in Table2 that the minimum MAPE for fitting of ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NBGMOP is more than three times larger than that of KARGM(1,1). The MAPE for prediction of KARGM(1,1) is much smaller than the NDGM(1,1,k), DGM(1,1), and NBGMOP, and it is slightly larger than that of the ARGM(1,1). It is shown in Figure 2 that the predicted values of KARGM(1,1) are quite close to the real values, and these values are presented to follow the overall trend of the monthly production of B51; but the distances between the predicted values of the other four models are very large, and they have failed in catching the overall trend of the gas production, which indicates that the modelling accuracy of these four models is not acceptable. In summary, the KARGM(1,1) performs best in the presented models.


### 4.2. Case Study 2

The raw data used in case study 2 are collected from the gas well B41 in Sichuan, China. Twenty points of monthly gas production (104m3) are listed in Table 3. The first 15 points are used to build the models, and the last 5 points are used for testing. The Gaussian kernel (16) is also used to build the KARGM(1,1) model.Table 3
Monthly gas production data of the B41 in Sichuan, China.
Month Gas production Month Gas production Month Gas production Month Gas production 1 12.8 6 9.3 11 12.2 16 7.5 2 9.2 7 10.3 12 11 17 6.8 3 8.8 8 9.4 13 9 18 7.5 4 9.2 9 9.3 14 8.2 19 7.7 5 8.5 10 11.2 15 6.4 20 8.5In this case, the kernel parameterσ and the regularized parameter γ are tuned using 5-fold cross validation, and the results of the MAPE for each pair of σ and γ are plotted in Figure 4. The minimum MAPE is found at the point σ=2.0 and γ=11; thus this pair will be used to build the KARGM(1,1) in this case.The predicted values by the KARGM(1,1), ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NBGMOP are listed in Table 4, along with the absolute percentage error of each point and the MAPE of each model. The results are also plotted in Figure 3.Table 4
Predicted values of monthly gas production of B41 by the ARGM(1,1), NDGM(1,1,k), DGM(1,1), NGBMOP, and KARGM(1,1).
Month Gas production ARGM(1,1) Error NDGM(1,1,k) Error DGM(1,1) Error NGBMOP Error KAGM(1,1) Error 1 12.8 12.8 0 12.8 0 12.8 0 12.8 0 12.8 0.00 2 9.2 10.83 17.75 7.95 13.60 9.60 4.35 9.09 1.19 9.22 0.21 3 8.8 9.89 12.33 8.56 2.70 9.57 8.79 9.41 6.91 8.88 0.96 4 9.2 9.43 2.48 9.11 0.99 9.55 3.77 9.50 3.23 8.80 4.33 5 8.5 9.21 8.33 9.58 12.72 9.52 12.00 9.53 12.15 9.00 5.89 6 9.3 9.10 2.13 9.97 7.20 9.49 2.08 9.54 2.61 9.41 1.22 7 10.3 9.05 12.13 10.26 0.38 9.47 8.08 9.54 7.40 9.60 6.81 8 9.4 9.03 3.98 10.44 11.11 9.44 0.44 9.52 1.31 9.54 1.52 9 9.3 9.01 3.07 10.50 12.96 9.41 1.23 9.50 2.17 9.93 6.76 10 11.2 9.01 19.57 10.43 6.91 9.39 16.17 9.47 15.41 10.95 2.21 11 12.2 9.01 26.18 10.19 16.48 9.36 23.26 9.44 22.59 11.64 4.60 12 11 9.00 18.14 9.77 11.14 9.34 15.12 9.41 14.46 11.04 0.35 13 9 9.00 0.04 9.16 1.74 9.31 3.45 9.37 4.14 9.41 4.59 14 8.2 9.00 9.80 8.31 1.34 9.28 13.22 9.33 13.83 7.76 5.42 15 6.4 9.00 40.68 7.20 12.55 9.26 44.67 9.29 45.21 6.78 5.94  MAPE   11.77  7.45  10.44  10.17  3.39  16 7.5 9.00 20.05 5.80 22.63 9.23 23.10 9.25 23.36 6.65 11.34 17 6.8 9.00 32.40 4.07 40.17 9.21 35.40 9.21 35.42 7.14 5.07 18 7.5 9.00 20.05 1.96 73.90 9.18 22.42 9.16 22.20 7.86 4.78 19 7.7 9.00 16.93 −0.58 107.55 9.16 18.91 9.12 18.44 8.44 9.61 20 8.5 9.00 5.92 −3.60 142.40 9.13 7.42 9.07 6.76 8.77 3.14  MAPE   19.07  77.33  21.45  21.24  6.79Figure 3
Prediction results of the monthly gas production of B41 by the prediction models. The raw data are plotted using the mark “+” and the predicted values are plotted using the solid line and the mark “o”. (a) ARGM(1,1); (b) NDGM(1,1,k); (c) DGM(1,1); (d) NGBMOP; (e) KARGM(1,1).
(a) (b) (c) (d) (e)Figure 4
The MAPE of cross validation of KARGM(1,1) with different σ and γ using the 5-fold cross validation in Case Study 2.It can be seen in Table4 that the minimum MAPE for fitting of ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NBGMOP is more than two times larger than that of KARGM(1,1). The minimum MAPE for prediction of ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NBGMOP is more than three times larger than that of the KARGM(1,1). In Figure 3 it is also shown that the predicted values of KARGM(1,1) are very close to the real values and the KARGM(1,1) can accurately catch the overall trend of the monthly production of B41; but the distances between the predicted values of the other four models are still very large. In this case study it is clearly that the KARGM(1,1) has be best performance.


## 4.1. Case Study 1

The raw data used in case study 1 are collected from the gas well B51 in Sichuan, China. Twenty points of monthly gas production (104m3) are listed in Table 1. The first 15 points are used to build the models, and the last 5 points are used for testing. The Gaussian kernel (16) is used to build the KARGM(1,1) model.Table 1
Monthly gas production data of the B51 in Sichuan, China.
Month Gas production Month Gas production Month Gas production Month Gas production 1 90.2837 6 63.9314 11 34.2398 16 53.2518 2 72.1744 7 30.8003 12 44.0861 17 50.3282 3 54.7787 8 60.5955 13 49.6398 18 37.8982 4 54.1369 9 58.2767 14 51.7017 19 36.4632 5 64.6249 10 7.5851 15 44.5744 20 48.0127The kernel parameterσ and the regularized parameter γ are tuned using the cross validation, which is widely used in the other kernel methods, such as the Partially Linear LS-SVM [31]. The 5-fold cross validation is employed in this case. The MAPE of validation is used to assess the chosen kernel parameter and the regularized parameter. A brief summary of the 5-fold cross validation has been described in Appendix B. The results of the MAPE for each pair of σ and γ are plotted in Figure 1. It can be seen that the minimum MAPE is found at the point σ=0.9 and γ=14; thus this pair will be used to build the KARGM(1,1) in this case.Figure 1
The MAPE of cross validation of KARGM(1,1) with different σ and γ using the 5-fold cross validation in Case Study 1.The predicted values by the KARGM(1,1), ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NGBMOP are listed in Table 2, along with the absolute percentage error of each point and the MAPE of each model. The results are also plotted in Figure 2.Table 2
Predicted values of monthly gas production of B51 by the ARGM(1,1), NDGM(1,1,k), DGM(1,1), NGBMOP, and KARGM(1,1).
Month Gas production ARGM(1,1) Error NDGM(1,1,k) Error DGM(1,1) Error NGBMOP Error KAGM(1,1) Error 1 90.2837 90.2837 0 90.2837 0 90.2837 0 90.2837 0.00 90.2837 0.00 2 72.1744 59.1583 18.03 73.2911 1.55 63.5896 11.89 72.8510 0.94 70.0022 3.01 3 54.7787 51.0645 6.78 64.7782 18.25 61.0010 11.36 65.2404 19.10 55.9178 2.08 4 54.1369 48.9598 9.56 57.9739 7.09 58.5177 8.09 61.4602 13.53 53.5025 1.17 5 64.6249 48.4125 25.09 52.6425 18.54 56.1356 13.14 58.5062 9.47 65.0060 0.59 6 63.9314 48.2702 24.50 48.5808 24.01 53.8504 15.77 55.9700 12.45 61.2468 4.20 7 30.8003 48.2332 56.60 45.6134 48.09 51.6582 67.72 53.6995 74.35 34.3068 11.38 8 60.5955 48.2236 20.42 43.5893 28.07 49.5553 18.22 51.6200 14.81 59.2332 2.25 9 58.2767 48.2211 17.25 42.3785 27.28 47.5380 18.43 49.6887 14.74 55.7955 4.26 10 7.5851 48.2204 535.73 41.8686 451.98 45.6028 501.22 47.8783 531.21 12.5928 66.02 11 34.2398 48.2202 40.83 41.9629 22.56 43.7464 27.76 46.1701 34.84 32.4912 5.11 12 44.0861 48.2202 9.38 42.5780 3.42 41.9656 4.81 44.5509 1.05 45.1528 2.42 13 49.6398 48.2202 2.86 43.6422 12.08 40.2572 18.90 43.0106 13.35 49.4121 0.46 14 51.7017 48.2202 6.73 45.0933 12.78 38.6184 25.31 41.5412 19.65 51.3527 0.68 15 44.5744 48.2202 8.18 46.8780 5.17 37.0463 16.89 40.1364 9.96 45.2044 1.41  MAPE   52.13  45.39  50.63  51.30  7.00  16 53.2518 48.2202 9.45 48.9502 8.08 35.5382 33.26 38.7911 27.16 47.2950 11.19 17 50.3282 48.2202 4.19 51.2703 1.87 34.0915 32.26 37.5008 25.49 50.3266 0.00 18 37.8982 48.2202 27.24 53.8041 41.97 32.7037 13.71 36.2619 4.32 50.5842 33.47 19 36.4632 48.2202 32.24 56.5221 55.01 31.3724 13.96 35.0710 3.82 50.5825 38.72 20 48.0127 48.2202 0.43 59.3988 23.71 30.0953 37.32 33.9254 29.34 50.5832 5.35  MAPE   14.71  26.13  26.10  18.02  17.75Figure 2
Prediction results of the monthly gas production of B51 by the prediction models. The raw data are plotted using the mark “+” and the predicted values are plotted using the solid line and the mark “o”. (a) ARGM(1,1); (b) NDGM(1,1,k); (c) DGM(1,1); (d) NGBMOP; (e) KARGM(1,1).
(a) (b) (c) (d) (e)It can be seen in Table2 that the minimum MAPE for fitting of ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NBGMOP is more than three times larger than that of KARGM(1,1). The MAPE for prediction of KARGM(1,1) is much smaller than the NDGM(1,1,k), DGM(1,1), and NBGMOP, and it is slightly larger than that of the ARGM(1,1). It is shown in Figure 2 that the predicted values of KARGM(1,1) are quite close to the real values, and these values are presented to follow the overall trend of the monthly production of B51; but the distances between the predicted values of the other four models are very large, and they have failed in catching the overall trend of the gas production, which indicates that the modelling accuracy of these four models is not acceptable. In summary, the KARGM(1,1) performs best in the presented models.


## 4.2. Case Study 2

The raw data used in case study 2 are collected from the gas well B41 in Sichuan, China. Twenty points of monthly gas production (104m3) are listed in Table 3. The first 15 points are used to build the models, and the last 5 points are used for testing. The Gaussian kernel (16) is also used to build the KARGM(1,1) model.Table 3
Monthly gas production data of the B41 in Sichuan, China.
Month Gas production Month Gas production Month Gas production Month Gas production 1 12.8 6 9.3 11 12.2 16 7.5 2 9.2 7 10.3 12 11 17 6.8 3 8.8 8 9.4 13 9 18 7.5 4 9.2 9 9.3 14 8.2 19 7.7 5 8.5 10 11.2 15 6.4 20 8.5In this case, the kernel parameterσ and the regularized parameter γ are tuned using 5-fold cross validation, and the results of the MAPE for each pair of σ and γ are plotted in Figure 4. The minimum MAPE is found at the point σ=2.0 and γ=11; thus this pair will be used to build the KARGM(1,1) in this case.The predicted values by the KARGM(1,1), ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NBGMOP are listed in Table 4, along with the absolute percentage error of each point and the MAPE of each model. The results are also plotted in Figure 3.Table 4
Predicted values of monthly gas production of B41 by the ARGM(1,1), NDGM(1,1,k), DGM(1,1), NGBMOP, and KARGM(1,1).
Month Gas production ARGM(1,1) Error NDGM(1,1,k) Error DGM(1,1) Error NGBMOP Error KAGM(1,1) Error 1 12.8 12.8 0 12.8 0 12.8 0 12.8 0 12.8 0.00 2 9.2 10.83 17.75 7.95 13.60 9.60 4.35 9.09 1.19 9.22 0.21 3 8.8 9.89 12.33 8.56 2.70 9.57 8.79 9.41 6.91 8.88 0.96 4 9.2 9.43 2.48 9.11 0.99 9.55 3.77 9.50 3.23 8.80 4.33 5 8.5 9.21 8.33 9.58 12.72 9.52 12.00 9.53 12.15 9.00 5.89 6 9.3 9.10 2.13 9.97 7.20 9.49 2.08 9.54 2.61 9.41 1.22 7 10.3 9.05 12.13 10.26 0.38 9.47 8.08 9.54 7.40 9.60 6.81 8 9.4 9.03 3.98 10.44 11.11 9.44 0.44 9.52 1.31 9.54 1.52 9 9.3 9.01 3.07 10.50 12.96 9.41 1.23 9.50 2.17 9.93 6.76 10 11.2 9.01 19.57 10.43 6.91 9.39 16.17 9.47 15.41 10.95 2.21 11 12.2 9.01 26.18 10.19 16.48 9.36 23.26 9.44 22.59 11.64 4.60 12 11 9.00 18.14 9.77 11.14 9.34 15.12 9.41 14.46 11.04 0.35 13 9 9.00 0.04 9.16 1.74 9.31 3.45 9.37 4.14 9.41 4.59 14 8.2 9.00 9.80 8.31 1.34 9.28 13.22 9.33 13.83 7.76 5.42 15 6.4 9.00 40.68 7.20 12.55 9.26 44.67 9.29 45.21 6.78 5.94  MAPE   11.77  7.45  10.44  10.17  3.39  16 7.5 9.00 20.05 5.80 22.63 9.23 23.10 9.25 23.36 6.65 11.34 17 6.8 9.00 32.40 4.07 40.17 9.21 35.40 9.21 35.42 7.14 5.07 18 7.5 9.00 20.05 1.96 73.90 9.18 22.42 9.16 22.20 7.86 4.78 19 7.7 9.00 16.93 −0.58 107.55 9.16 18.91 9.12 18.44 8.44 9.61 20 8.5 9.00 5.92 −3.60 142.40 9.13 7.42 9.07 6.76 8.77 3.14  MAPE   19.07  77.33  21.45  21.24  6.79Figure 3
Prediction results of the monthly gas production of B41 by the prediction models. The raw data are plotted using the mark “+” and the predicted values are plotted using the solid line and the mark “o”. (a) ARGM(1,1); (b) NDGM(1,1,k); (c) DGM(1,1); (d) NGBMOP; (e) KARGM(1,1).
(a) (b) (c) (d) (e)Figure 4
The MAPE of cross validation of KARGM(1,1) with different σ and γ using the 5-fold cross validation in Case Study 2.It can be seen in Table4 that the minimum MAPE for fitting of ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NBGMOP is more than two times larger than that of KARGM(1,1). The minimum MAPE for prediction of ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NBGMOP is more than three times larger than that of the KARGM(1,1). In Figure 3 it is also shown that the predicted values of KARGM(1,1) are very close to the real values and the KARGM(1,1) can accurately catch the overall trend of the monthly production of B41; but the distances between the predicted values of the other four models are still very large. In this case study it is clearly that the KARGM(1,1) has be best performance.


## 5. Conclusions

In this paper, a novel kernel based grey model (KARGM(1,1)) has been proposed, and its effectiveness has been assessed in the two case studies of the gas well production forecasting in comparison to the existing discrete grey models, including the ARGM(1,1), NDGM(1,1,k), DGM(1,1), and NBGMOP. The results of the case studies have shown that the novel KARGM(1,1) model outperformed the other three discrete grey models.Essentially, the existing ARGM(1,1), NDGM(1,1,k), and DGM(1,1) are linear model, and their solutions are all exponential functions. And this is also the reason why they only produce simple curves in the case studies, which cannot be used to express complex time series. The NBGMOP is a nonlinear model, but its applicability is still limited according to the results shown in the case studies. On the other hand, the KARGM(1,1) model contains a general nonlinear function wTφ, which is expressed by a proper kernel function as shown in (19) and determined by the raw data. Thus the KARGM(1,1) model can be more effective to deal with complex time series.What is more, this paper has illustrated a new way of building the discrete grey model using the kernel method and also proved the effectiveness of this new methodology. Thus, the relative researches can be carried out based on this research in the future.



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*Source: 5471748-2016-12-13.xml*