Mathematical Problems in Engineering

Volume 2016, Article ID 5471748, 9 pages

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

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

School of Science, Southwest University of Science and Technology, Mianyang, China

Received 2 July 2016; Revised 23 October 2016; Accepted 14 November 2016

Academic Editor: Michele Betti

Copyright © 2016 Xin Ma. 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

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 model, abbreviated as ARGM, is a novel discrete grey model which is easy to use and accurate in prediction of approximate nonhomogeneous exponential time series. However, the ARGM is essentially a linear model; thus, its applicability is still limited. In this paper a novel kernel based ARGM model is proposed, abbreviated as KARGM. The KARGM 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 model are compared to the existing discrete univariate grey prediction models, including ARGM, NDGM, DGM, and NGBMOP, and it is shown that the KARGM outperforms the other four models.

#### 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 [6], DGMD [7], NGM [8], and SAGM [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 model [10], which is based on the basic GM model. This novel technique has also been used to build the NDGM model based on the NGM [11]. And in our previous works, this technique has been extended to build the discrete GM 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 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 model [19] and is also called the autoregressive GM model (ARGM) [20] as it is essentially in the autoregressive formulation. It is unnecessary to use the 1-AGO when building the ARGM model; thus it is very easy to use. The ARGM 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 model in some applications [19, 20]. However, the ARGM model is essentially a linear model; thus its applicability is limited.

In order to improve the applicability of the ARGM model, we use the kernel method to build a novel kernel based ARGM model, abbreviated as KARGM. 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 model.

The rest of this paper is organized as follows. Section 2 presents a brief overview of the existing ARGM model; Section 3 presents the modelling procedures of the KARGM; 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 Model

With a given time series , the autoregressive GM (ARGM) model is represented as the following linear difference equation [20]:

The parameters and can be obtained using the least squares method as follows:where

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

The discrete function (4) is used to compute the predicted values.

#### 3. The Proposed KARGM Model

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

##### 3.1. Representation of the KARGM Model

With a given time series , the KARGM model is represented as the following difference equation:where is a nonlinear mapping, which is defined asand is a higher dimensional feature space; is a vector in .

The linear combination is a nonlinear function of . For example, if we consider a nonlinear functionand define the nonlinear mapping asand set , then we have

##### 3.2. Parameters Estimation of the KARGM Model

Being different from the ARGM model, we cannot simply use the least squares method to estimate the parameters of the KARGM model, because it is not always computationally feasible to find the formulation of the nonlinear mapping [30]. Firstly, we consider the regularized problem as follows:where 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 aswhere is the Lagrangian multiplier. Then the KKT conditions can be given as

By eliminating the , and , the KKT conditions can be converted to the following linear system:whereand and dimensional identity matrix with all the diagonal elements are 1 and others are zero. The inner product can be expressed using a kernel function which satisfies the Mercer’s condition, that is,The Gaussian kernel is often employed, which is defined aswhere 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 and the Lagrangian multipliers can be obtained. The parameter can then be computed using the first equation in the KKT conditions (12).

##### 3.3. The Solution of the KARGM Model

The KARGM model (5) can be easily solved using the recursive method as follows:

Noticing the second equation in the KKT conditions (12), which can be rewritten as , we have

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

For convenience, we note the following discrete function:

Then the solution (17) can be rewritten as

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 , because all the computational procedures of the KARGM model only involve in the inner product , and it can be expressed by a proper kernel function.

Actually, the KARGM 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 , which is expressed by the kernel function and finally determined by the raw data. So we can say that the KARGM model is a real “Grey” model.

##### 3.4. Summary of Computational Steps

The computational steps of the KARGM model are summarized as follows.

*Step 1. *Select an appropriate kernel function and the regularized parameter in (10).

*Step 2. *Compute the parameters and by solving the linear system (13), and then compute the parameter 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 model. The monthly production data are collected from two gas wells in Sichuan, China. In order to compare the performance of the KARGM model and other existing discrete grey models, the ARGM, NDGM [11], DGM [10], and the nonlinear grey Bernoulli model with optimal parameter (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 aswhere is the real value and 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 () 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 model.