Understanding cumulative sum operator in grey prediction model with integral matching. (English) Zbl 1451.93211
Summary: Grey prediction models have been widely used in various fields and disciplines. Cumulative sum operator, also called accumulative generation operator, is an essential step in grey modelling, but until now relatively limited attention has been paid to its mechanism of action. In this paper, we introduce the integral matching to explain it. By using the integral transformation, the grey prediction model whose nature is modelling the cumulative sum series with a differential equation proves to be equivalent to that modelling the original series with a reduced differential equation. The cumulative sum operator is the discretization and approximation of the definite integral terms by using the piecewise constant integral and thus can be improved by using the piecewise linear integral. Simulation studies detail the advantages in terms of the stability and robustness to noise.
References:
| [1] | Ferbar Tratar, L.; Mojškerc, B.; Toman, A., Demand forecasting with four-parameter exponential smoothing, Int J Prod Econ, 181, 162-173 (2016) |
| [2] | Young, P. C., Refined instrumental variable estimation: maximum likelihood optimization of a unified Box-Jenkins model, Automatica, 52, 35-46 (2015) · Zbl 1309.93178 |
| [3] | Lau, KW; Wu, QH, Local prediction of non-linear time series using support vector regression, Pattern Recognit, 41, 5, 1539-1547 (2008) · Zbl 1132.62080 |
| [4] | Barrow, DK; Crone, SF, Cross-validation aggregation for combining autoregressive neural network forecasts, Int J Forecast, 32, 4, 1120-1137 (2016) |
| [5] | Cheng, C.; Sa-Ngasoongsong, A.; Beyca, O.; Le, T.; Yang, H.; Kong, ZJ, Time series forecasting for nonlinear and non-stationary processes: a review and comparative study, IIE Trans, 47, 10, 1053-1071 (2015) |
| [6] | Liu, SF; Yang, YJ; Forrest, J., Grey data analysis: methods, models and applications (2017), Springer-Verlag, Singapore · Zbl 1406.93002 |
| [7] | Deng, JL, Grey theory and methods for socioeconomic system, Soc Sci China, 6, 47-60 (1984) |
| [8] | Chang, C-J; Li, D-C; Huang, Y-H; Chen, C-C, A novel gray forecasting model based on the box plot for small manufacturing data sets, Appl Math Comput, 265, 400-408 (2015) · Zbl 1410.62166 |
| [9] | Li, KW; Liu, L.; Zhai, JN; Khoshgoftaar, TM; Li, TM, The improved grey model based on particle swarm optimization algorithm for time series prediction, Eng Appl Artif Intell, 55, 285-291 (2016) |
| [10] | Chang, C.-J.; Yu, L. P.; Jin, P., A mega-trend-diffusion grey forecasting model for short-term manufacturing demand, J Oper Res Soc, 67, 12, 1439-1445 (2016) |
| [11] | Dang, Y. G.; Liu, S. F.; Jia, C. K., The GM models that x(n) be taken as initial value, Kybernetes, 33, 2, 247-254 (2004) · Zbl 1083.93509 |
| [12] | Tien, T-L, A new grey prediction model FGM(1, 1), Math Comput Model, 49, 7-8, 1416-1426 (2009) · Zbl 1165.62343 |
| [13] | Xu, J.; Tan, T.; Tu, M.; Qi, L., Improvement of grey models by least squares, Expert Syst Appl, 38, 11, 13961-13966 (2011) |
| [14] | Li, D-C; Chang, C-J; Chen, C-C; Chen, W-C, Forecasting short-term electricity consumption using the adaptive grey-based approach—an asian case, Omega, 40, 6, 767-773 (2012) |
| [15] | Zhao, Z.; Wang, JZ; Zhao, J.; Su, ZY, Using a grey model optimized by differential evolution algorithm to forecast the per capita annual net income of rural households in china, Omega, 40, 5, 525-532 (2012) |
| [16] | Liu, L.; Wang, QR; Wang, JZ; Liu, M., A rolling Grey model optimized by particle swarm optimization in economic prediction, Comput Intell, 32, 3, 391-419 (2016) |
| [17] | Tabaszewski, M.; Cempel, C., Using a set of GM(1,1) models to predict values of diagnostic symptoms, Mech Syst Signal Process, 52-53, 416-425 (2015) |
| [18] | Bezuglov, A.; Comert, G., Short-term freeway traffic parameter prediction: application of grey system theory models, Expert Syst Appl, 62, 284-292 (2016) |
| [19] | Hu, Y.-C.; Jiang, P.; Lee, P.-C., Forecasting tourism demand by incorporating neural networks into Grey-Markov models, J Oper Res Soc, 70, 1, 12-20 (2019) |
| [20] | Cui, J.; Liu, SF; Zeng, B.; Xie, NM, A novel grey forecasting model and its optimization, Appl Math Model, 37, 6, 4399-4406 (2013) · Zbl 1272.93014 |
| [21] | Luo, D.; Wei, B. L., Grey forecasting model with polynomial term and its optimization, J Grey Syst, 29, 3, 58-69 (2017) |
| [22] | Wei, BL; Xie, NM; Hu, AQ, Optimal solution for novel grey polynomial prediction model, Appl Math Model, 62, 717-727 (2018) · Zbl 1462.62571 |
| [23] | Ma, X.; Hu, YS; Liu, ZB, A novel kernel regularized nonhomogeneous grey model and its applications, Commun Nonlinear Sci Numer Simul, 48, 51-62 (2017) · Zbl 1510.62391 |
| [24] | Xie, N. M.; Wang, R. Z., A historic review of grey forecasting models, J Grey Syst, 29, 4, 1-29 (2017) |
| [25] | Li, D-C; Chang, C-J; Chen, W-C; Chen, C-C, An extended grey forecasting model for omnidirectional forecasting considering data gap difference, Appl Math Model, 35, 10, 5051-5058 (2011) · Zbl 1228.62162 |
| [26] | Chang, C.-J.; Li, D.-C.; Chen, C.-C.; Chen, C.-S., A forecasting model for small non-equigap data sets considering data weights and occurrence possibilities, Comput Ind Eng, 67, 139-145 (2014) |
| [27] | Xiao, XP; Guo, H.; Mao, SH, The modeling mechanism, extension and optimization of grey GM(1, 1) model, Appl Math Model, 38, 5-6, 1896-1910 (2014) · Zbl 1428.62417 |
| [28] | Dattner, I., A model-based initial guess for estimating parameters in systems of ordinary differential equations, Biometrics, 71, 4, 1176-1184 (2015) · Zbl 1419.62083 |
| [29] | Gene, H. G.; Charles, F. V.L., Matrix computations (2013), The Johns Hopkins University Press · Zbl 1268.65037 |
| [30] | Carroll, RJ; Ruppert, D.; Stefanski, LA, Measurement error in nonlinear models: a modern perspective (2006), Chapman & Hall/CRC · Zbl 1119.62063 |
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