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

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. Section 2 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]:

x(k) = [alpha]x(k - 1) + [beta], k = 2, 3, ..., n. (1)

The parameters a and [beta] can be obtained using the least squares method as follows:

[[[alpha], [beta]].sup.T] = [([B.sup.T]B).sup.-1] [B.sup.T]Y, (2)

where

[mathematical expression not reproducible]. (3)

The solution of the ARGM(1, 1) model can be obtained using the recursive method, which is given as the following discrete function:

x(k) = [[alpha].sub.k]x(1) + [1 - [[alpha].sup.k]]/[1 - [alpha]] [beta]. (4)

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:

x(k) = [alpha]x(k - 1) + [w.sup.T][phi](k) + [beta], (5)

where [phi] is a nonlinear mapping, which is defined as

[phi] : R [right arrow] F, (6)

and F is a higher dimensional feature space; w is a vector in F.

The linear combination [w.sup.T][phi](k) is a nonlinear function of k. For example, if we consider a nonlinear function

g(k) = 0.3 sin (k * [pi]) + [square root of (k)], (7)

and define the nonlinear mapping as

[phi](k) = [[sin (k * [pi]), [square root of (k)]].sup.T], (8)

and set w = [[0.3, 1.sub.]T], then we have

g(k) = [w.sup.T][phi](k). (9)

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 [phi] [30]. Firstly, we consider the regularized problem as follows:

min J ([alpha], w, e) = 1/2 [[alpha].sup.2] + 1/2 [[parallel]w[parallel].sup.2] + [gamma]/2 [n.summation over (k=2)] [e.sup.2.sub.k]

s.t. x (k) = [alpha]x (k - 1) + [w.sup.T][phi](k) + [beta] + [e.sub.k], (10)

where [gamma] is the regularized parameter to balance the flatness of the fitting curve and the error bound, and [parallel]x[parallel] 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

L ([alpha], w, e; [lambda])

= J ([alpha], w, e)

- [n.summation over (k=2)] [[lambda].sub.k] {[alpha]x(k - 1) + [w.sup.T][phi](k) + [beta] + [e.sub.k] - x(k)}, (11)

where [[lambda].sub.k] is the Lagrangian multiplier. Then the KKT conditions can be given as

[mathematical expression not reproducible]. (12)

By eliminating the [alpha], w and [e.sub.k], the KKT conditions can be converted to the following linear system:

[mathematical expression not reproducible], (13)

where

[mathematical expression not reproducible], (14)

and [I.sub.n-1] and n -1 dimensional identity matrix with all the diagonal elements are 1 and others are zero. The inner product [[OMEGA].sub.ij] can be expressed using a kernel function which satisfies the Mercer's condition, that is,

[[OMEGA].sub.ij] = [phi] (i +1) x [phi] (j +1) = K (1 + 1, j +1). (15)

The Gaussian kernel is often employed, which is defined as

K(i, j) = exp {-[(i - j).sup.2]/2[[sigma].sup.2]}, (16)

where [sigma] 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 [beta] and the Lagrangian multipliers [[lambda].sub.k] can be obtained. The parameter [alpha] 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:

[mathematical expression not reproducible]. (17)

Noticing the second equation in the KKT conditions (12), which can be rewritten as w = [[summation].sup.n.sub.j=2] [[lambda].sub.[phi]](j), we have

[w.sup.T] [phi]([tau]) = [n.summation over (j=2)][[lambda].sub.j][phi](j) x [phi]([tau]). (18)

Using the inner product (15), we can rewrite the nonlinear function (18) as follows:

[w.sup.T] [phi]([tau]) = [n.summation over (j=2)][[lambda].sub.j]K (j, [tau]). (19)

For convenience, we note the following discrete function:

[phi]([tau]) = [w.sup.T] [phi]([tau]) + [beta] = [n.summation over (j=2)] [[lambda].sub.j]K (j, [tau]) + [beta]. (20)

Then the solution (17) can be rewritten as

x(k) = [[alpha].sup.k-1]x(1) + [k.summation over ([tau]=2)] [a.sup.k-[tau]] [phi]([tau]). (21)

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 mapping [phi], because all the computational procedures of the KARGM(1, 1) model only involve in the inner product [phi](i) x [phi](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 known a priori, and the "Black" part is the linear combination [w.sup.T][phi](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 function K(x,x) and the regularized parameter [gamma] in (10).

Step 2. Compute the parameters [beta] and [[lambda].sub.2], [[lambda].sub.3], ..., [[lambda].sub.n] by solving the linear system (13), and then compute the parameter a 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

MAPE = 1/n [n.summation over (k=1)] [absolute value of ([[??](k) - [??](k)]/x(k))] x 100(%), (22)

where x(k) is the real value and [??](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 ([10.sup.4] [m.sup.3]) 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.

The kernel parameter [sigma] and the regularized parameter y 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 [sigma] and [gamma] are plotted in Figure 1. It can be seen that the minimum MAPE is found at the point [sigma] = 0.9 and [gamma] = 14; 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 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.

It can be seen in Table 2 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 ([10.sup.4] [m.sup.3]) 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.

In this case, the kernel parameter [sigma] and the regularized parameter [gamma] are tuned using 5-fold cross validation, and the results of the MAPE for each pair of [sigma] and [gamma] are plotted in Figure 4. The minimum MAPE is found at the point [sigma] = 2.0 and [gamma] = 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.

It can be seen in Table 4 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 [w.sup.T][phi], 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.

Appendix

A. The KKT Conditions for Equality Constrained Convex Minimization

Consider the equality constrained optimization problem

min f (x)

s.t. [h.sub.j](x) = 0, j = 1, 2, ..., m, (A.1)

where the objective function f and the constraint functions [h.sub.j](x) = 0 (j = 1, 2, ..., m) are supposed to be continuously differentiable at point x*. One defines the Lagrangian function as

L (x; [lambda]) = f(x) - [m.summation over (i=1)][[lambda].sub.j][h.sub.j](x), (A.2)

where the [lambda] = [[[[lambda].sub.1], ..., [[lambda].sub.m]].sup.T] are the Lagrangian multipliers. The point x* and the Lagrangian multipliers must satisfy the KKT conditions if the point x* is a local minimum. In this case, the KarushCKuhnCTucker (KKT) conditions can be interpreted using the Lagrangian function as

[[partial derivative]L (x; [lambda])]/[partial derivative]x = 0

[[partial derivative]L (x; [lambda])]/[partial derivative][[lambda].sub.j] = 0 j = 1, 2, ..., m. (A.3)

The KKT conditions are sufficient and necessary to the optimization problem (A.1) if the objective function f(x) is convex and the constrained functions = 0 (j = 1, 2, ..., m) are linear functions. In this case the local minimum x* which satisfies the KKT conditions is a global minimum; that is, the solution of the minimization problem (A.1) is equivalent to the solution of the KKT conditions (A.3). (More details of the KKT conditions can be seen in [32], pages: 243-245.)

B. A Brief Summary of the 5-Fold Cross Validation Used in the Case Studies

With a given time series {x(1), x(2), ..., x(n)}, the 5-fold cross validation can be described as follows.

Step 1. Divide the original time series into 5 subsets randomly, and mark them as [S.sub.1], [S.sub.2], ..., [S.sub.5].

Step 2. Build the KARGM(1, 1) model using four subsets and the rest subset is used for validation. (For example, the first time, the KARGM(1, 1) model is built on the subset [S.sub.1], [S.sub.2], [S.sub.3], [S.sub.4], and the subset [S.sub.5] is used for validation. This procedure will be repeated for 5 times, and all the subset will be used for validation.) Compute the MAPE for all the validation procedures.

Step 3. For [sigma] = 0.1, 0.2, ..., 2 and [sigma] = 1, 2, ..., 20, repeat Step 2 and store the values of the MAPE with all the pairs of a and [gamma].

Step 4. Output the values of optimal a and y corresponding to the minimum MAPE.

http://dx.doi.org/10.1155/2016/5471748

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Doctoral Research Foundation of Southwest University of Science and Technology (no. 16zx7140).

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School of Science, Southwest University of Science and Technology, Mianyang, China Correspondence should be addressed to Xin Ma; cauchy7203@gmail.com Received 2 July 2016; Revised 23 October 2016; Accepted 14 November 2016 Academic Editor: Michele Betti

Caption: Figure 1: The MAPE of cross validation of KARGM(1, 1) with different [sigma] and [gamma] using the 5-fold cross validation in Case Study 1.

Caption: Figure 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).

Caption: Figure 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).

Caption: Figure 4: The MAPE of cross validation of KARGM(1, 1) with different [sigma] and [gamma] using the 5-fold cross validation in Case Study 2.
Table 1: Monthly gas production data of the B51 in Sichuan, China.

Month     Gas production

1            90.2837
2            72.1744
3            54.7787
4            54.1369
5            64.6249
6            63.9314
7            30.8003
8            60.5955
9            58.2767
10            7.5851
11           34.2398
12           44.0861
13           49.6398
14           51.7017
15           44.5744
16           53.2518
17           50.3282
18           37.8982
19           36.4632
20           48.0127

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).

           Gas
Month   production   ARGM(1, 1)   Error    NDGM(1, 1, k)   Error

1        90.2837      90.2837       0         90.2837        0
2        72.1744      59.1583     18.03       73.2911       1.55
3        54.7787      51.0645      6.78       64.7782      18.25
4        54.1369      48.9598      9.56       57.9739       7.09
5        64.6249      48.4125     25.09       52.6425      18.54
6        63.9314      48.2702     24.50       48.5808      24.01
7        30.8003      48.2332     56.60       45.6134      48.09
8        60.5955      48.2236     20.42       43.5893      28.07
9        58.2767      48.2211     17.25       42.3785      27.28
10        7.5851      48.2204     535.73      41.8686      451.98
11       34.2398      48.2202     40.83       41.9629      22.56
12       44.0861      48.2202      9.38       42.5780       3.42
13       49.6398      48.2202      2.86       43.6422      12.08
14       51.7017      48.2202      6.73       45.0933      12.78
15       44.5744      48.2202      8.18       46.8780       5.17

MAPE                              52.13                    45.39

16       53.2518      48.2202      9.45       48.9502       8.08
17       50.3282      48.2202      4.19       51.2703       1.87
18       37.8982      48.2202     27.24       53.8041      41.97
19       36.4632      48.2202     32.24       56.5221      55.01
20       48.0127      48.2202      0.43       59.3988      23.71

MAPE                              14.71                    26.13

Month   DGM(1, 1)   Error    NGBMOP    Error    KAGM(1, 1)   Error

1        90.2837      0      90.2837    0.00     90.2837     0.00
2        63.5896    11.89    72.8510    0.94     70.0022     3.01
3        61.0010    11.36    65.2404   19.10     55.9178     2.08
4        58.5177     8.09    61.4602   13.53     53.5025     1.17
5        56.1356    13.14    58.5062    9.47     65.0060     0.59
6        53.8504    15.77    55.9700   12.45     61.2468     4.20
7        51.6582    67.72    53.6995   74.35     34.3068     11.38
8        49.5553    18.22    51.6200   14.81     59.2332     2.25
9        47.5380    18.43    49.6887   14.74     55.7955     4.26
10       45.6028    501.22   47.8783   531.21    12.5928     66.02
11       43.7464    27.76    46.1701   34.84     32.4912     5.11
12       41.9656     4.81    44.5509    1.05     45.1528     2.42
13       40.2572    18.90    43.0106   13.35     49.4121     0.46
14       38.6184    25.31    41.5412   19.65     51.3527     0.68
15       37.0463    16.89    40.1364    9.96     45.2044     1.41

MAPE                50.63              51.30                 7.00

16       35.5382    33.26    38.7911   27.16     47.2950     11.19
17       34.0915    32.26    37.5008   25.49     50.3266     0.00
18       32.7037    13.71    36.2619    4.32     50.5842     33.47
19       31.3724    13.96    35.0710    3.82     50.5825     38.72
20       30.0953    37.32    33.9254   29.34     50.5832     5.35

MAPE                26.10              18.02                 17.75

Table 3: Monthly gas production data of the B51 in Sichuan, China.

Month     Gas production

1              12.8
2               9.2
3               8.8
4               9.2
5               8.5
6               9.3
7              10.3
8               9.4
9               9.3
10             11.2
11             12.2
12              11
13              9
14              8.2
15              6.4
16              7.5
17              6.8
18              7.5
19              7.7
20              8.5

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).

           Gas
Month   production   ARGM(1, 1)   Error   NDGM(1, 1, k)   Error

1          12.8         12.8        0         12.8          0
2          9.2         10.83      17.75       7.95        13.60
3          8.8          9.89      12.33       8.56         2.70
4          9.2          9.43      2.48        9.11         0.99
5          8.5          9.21      8.33        9.58        12.72
6          9.3          9.10      2.13        9.97         7.20
7          10.3         9.05      12.13       10.26        0.38
8          9.4          9.03      3.98        10.44       11.11
9          9.3          9.01      3.07        10.50       12.96
10         11.2         9.01      19.57       10.43        6.91
11         12.2         9.01      26.18       10.19       16.48
12          11          9.00      18.14       9.77        11.14
13          9           9.00      0.04        9.16         1.74
14         8.2          9.00      9.80        8.31         1.34
15         6.4          9.00      40.68       7.20        12.55

MAPE                              11.77                    7.45

16         7.5          9.00      20.05       5.80        22.63
17         6.8          9.00      32.40       4.07        40.17
18         7.5          9.00      20.05       1.96        73.90
19         7.7          9.00      16.93       -0.58       107.55
20         8.5          9.00      5.92        -3.60       142.40

MAPE                              19.07                   77.33

Month   DGM(1, 1)   Error   NGBMOP   Error   KAGM(1, 1)   Error

1         12.8        0      12.8      0        12.8      0.00
2         9.60      4.35     9.09    1.19       9.22      0.21
3         9.57      8.79     9.41    6.91       8.88      0.96
4         9.55      3.77     9.50    3.23       8.80      4.33
5         9.52      12.00    9.53    12.15      9.00      5.89
6         9.49      2.08     9.54    2.61       9.41      1.22
7         9.47      8.08     9.54    7.40       9.60      6.81
8         9.44      0.44     9.52    1.31       9.54      1.52
9         9.41      1.23     9.50    2.17       9.93      6.76
10        9.39      16.17    9.47    15.41     10.95      2.21
11        9.36      23.26    9.44    22.59     11.64      4.60
12        9.34      15.12    9.41    14.46     11.04      0.35
13        9.31      3.45     9.37    4.14       9.41      4.59
14        9.28      13.22    9.33    13.83      7.76      5.42
15        9.26      44.67    9.29    45.21      6.78      5.94

MAPE                10.44            10.17                3.39

16        9.23      23.10    9.25    23.36      6.65      11.34
17        9.21      35.40    9.21    35.42      7.14      5.07
18        9.18      22.42    9.16    22.20      7.86      4.78
19        9.16      18.91    9.12    18.44      8.44      9.61
20        9.13      7.42     9.07    6.76       8.77      3.14

MAPE                21.45            21.24                6.79
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Title Annotation:Research Article
Author:Ma, Xin
Publication:Mathematical Problems in Engineering
Date:Jan 1, 2016
Words:5850
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