A novel kernel regularized nonhomogeneous grey model and its applications. (English) Zbl 1510.62391
Summary: The nonhomogeneous grey model (NGM) is a novel tool for time series forecasting, which has attracted considerable interest of research. However, the existing nonhomogeneous grey models may be inefficient to predict the complex nonlinear time series sometimes due to the linearity of the differential or difference equations based on which these models are developed. In order to enhance the accuracy and applicability of the NGM model, the kernel method in the statistical learning theory has been utilized to build a novel kernel regularized nonhomogeneous grey model, which is abbreviated as the KRNGM model. The KRNGM model is represented by a differential equation which contains a nonlinear function of \(t\). By constructing the regularized problem and using the kernel function which satisfies the Mercer’s condition, the parameters estimation of KRNGM model only involves in solving a set of linear equations, and the nonlinear function in the KRNGM model can be expressed as a linear combination of the Lagrangian multipliers and the selected kernel function, and then the KRNGM model can be solved numerically. Two case studies of petroleum production forecasting are carried to illustrate the effectiveness of the KRNGM model, comparing to the existing nonhomogeneous models. The results show that the KRNGM model outperforms the existing NGM, ONGM, NDGM model significantly.
MSC:
| 62M99 | Inference from stochastic processes |
Software:
GPMLReferences:
| [1] | Ju-Long, D., Control problems of grey systems, Syst Control Lett, 1, 5, 288-294 (1982) · Zbl 0482.93003 |
| [2] | Liu, S.; Forrest, J.; Yang, Y., A brief introduction to grey systems theory, Grey Syst, 2, 2, 89-104 (2012) |
| [3] | Ma, X.; Luo, D.; Ding, X.-F.; Zhou, J.-J., An algorithm based on the GM(1, 1) model on increasing oil production of measures operation for a single well, Proceedings of 2013 IEEE international conference on grey systems and intelligent services (GSIS) (2013) |
| [4] | Wu, L.; Liu, S.; Yao, L.; Yan, S., The effect of sample size on the grey system model, Appl Math Model, 37, 9, 6577-6583 (2013) · Zbl 1426.62279 |
| [5] | Tien, T.-L., A new grey prediction model FGM(1, 1), Math Comput Model, 49, 7, 1416-1426 (2009) · Zbl 1165.62343 |
| [6] | Chen, C.-K.; Tien, T.-L., The indirect measurement of tensile strength by the deterministic grey dynamic model DGDM(1, 1, 1), Int J Syst Sci, 28, 7, 683-690 (1997) · Zbl 0875.93031 |
| [7] | Truong, D. Q.; Ahn, K. K., An accurate signal estimator using a novel smart adaptive grey model SAGM(1, 1), Expert Syst Appl, 39, 9, 7611-7620 (2012) |
| [8] | Xie, N.-m.; Liu, S.-f., Discrete grey forecasting model and its optimization, Appl Math Model, 33, 2, 1173-1186 (2009) · Zbl 1168.62380 |
| [9] | Wang, Z.-X.; Hipel, K. W.; Wang, Q.; He, S.-W., An optimized NGBM(1, 1) model for forecasting the qualified discharge rate of industrial wastewater in China, Appl Math Model, 35, 12, 5524-5532 (2011) |
| [10] | Cui, J.; Dang, Y.; Liu, S., Novel grey forecasting model and its modeling mechanism, Control Decis, 24, 11, 1702-1706 (2009) · Zbl 1212.93009 |
| [11] | Cui, J.; Liu, S.-f.; Zeng, B.; Xie, N.-m., A novel grey forecasting model and its optimization, Appl Math Model, 37, 6, 4399-4406 (2013) · Zbl 1272.93014 |
| [12] | Xie, N.-M.; Liu, S.-F.; Yang, Y.-J.; Yuan, C.-Q., On novel grey forecasting model based on non-homogeneous index sequence, Appl Math Model, 37, 7, 5059-5068 (2013) · Zbl 1426.62280 |
| [13] | Chen, P.-Y.; Yu, H.-M., Foundation settlement prediction based on a novel NGM model, Math Problems Eng, 2014, 1-8 (2014) |
| [14] | Ma, X.; Liu, Z.-b., Predicting the cumulative oil field production using the novel grey ENGM model, J Comput Theor Nanosci, 13, 1, 89-95 (2013) |
| [15] | Vapnik, V. N., Statistical learning theory (1998), Wiley-Interscience · Zbl 0935.62007 |
| [16] | Smola, A. J.; Schölkopf, B., A tutorial on support vector regression, Stat Comput, 14, 3, 199-222 (2004) |
| [17] | Schölkopf, B.; Smola, A.; Müller, K.-R., Kernel principal component analysis, Artificial neural networksłICANN’97, 583-588 (1997), Springer |
| [18] | Schölkopf, B.; Müller, K.-R., Fisher discriminant analysis with kernels, Neural Netw Signal Process IX, 1, 1 (1999) |
| [19] | Rasmussen, C. E.; Nickisch, H., Gaussian processes for machine learning (GPML) toolbox, J Mach Learn Res, 11, 3011-3015 (2010) · Zbl 1242.68242 |
| [20] | Tipping, M. E., Sparse Bayesian learning and the relevance vector machine, J Mach Learn Res, 1, 211-244 (2001) · Zbl 0997.68109 |
| [21] | Suykens, J. A.; Vandewalle, J., Least squares support vector machine classifiers, Neural Process Lett, 9, 3, 293-300 (1999) |
| [22] | Van Gestel, T.; Suykens, J. A.; Lanckriet, G.; Lambrechts, A.; De Moor, B.; Vandewalle, J., Bayesian framework for least-squares support vector machine classifiers, Gaussian processes, and kernel Fisher discriminant analysis, Neural Comput, 14, 5, 1115-1147 (2002) · Zbl 1003.68146 |
| [23] | Suykens, J. A.; Van Gestel, T.; Vandewalle, J.; De Moor, B., A support vector machine formulation to PCA analysis and its kernel version, Neural Netw IEEE Trans, 14, 2, 447-450 (2003) |
| [24] | Mehrkanoon, S.; Falck, T.; Suykens, J. A.K., Approximate solutions to ordinary differential equations using least squares support vector machines, IEEE Trans Neural Netw Learn Syst, 23, 9, 1356-1367 (2012) |
| [25] | Espinoza, M.; Suykens, J. A.; De Moor, B., Partially linear models and least squares support vector machines, Decision and control, 2004. CDC. 43rd IEEE conference on, 4, 3388-3393 (2004), IEEE |
| [26] | Espinoza, M.; Suykens, J. A.; De Moor, B., Kernel based partially linear models and nonlinear identification, IEEE Trans Automat Control, 50, 10, 1602-1606 (2005) · Zbl 1365.93521 |
| [27] | Schölkopf, B.; Smola, A. J., Learning with kernels (2002), MIT Press |
| [28] | Ma, X.; Y, Z.; Y, W., Performance evaluation of kernel functions based on grid search for support vector regression, 2015 IEEE 7th International conference on cybernetics and intelligent systems (CIS) and IEEE conference on robotics, automation andmechatronics (RAM), 283-288 (2015), IEEE |
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