A 'basic form'-focused modeling and a modified parameter estimation technique for grey prediction models

Document Type : Article

Authors

Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, 424 Hafez Ave, Tehran, Iran

Abstract

Grey modeling is an alternative approach to time series forecasting with growing popularity. There is no theoretical limitation for grey prediction models to adapt to almost every process by taking the appropriate order. However, deficiencies of traditional higher-order models have made researchers overlook such flexibility and make use of first-order models by default. In order to bridge the mentioned gap, this paper makes two contributions. First, a novel discrete modeling is developed with the basic form equation at its heart, which reconciles estimation and prediction processes. Second, the traditional least-squares estimation technique is modified by shifting the focus from nominal parameters to parameters practically employed in the prediction process. The new approach named ‘Basic Form’-focused Grey Model (BFGM) is applied to first-order, second-order, and Verhulst grey models. Then, it is validated through comparing its performance with the traditional approach. Results show that in most cases BFGM makes considerable improvements in simulation and prediction accuracy, while it has reasonable computational complexity. Improvements are especially dramatic when BFGM is applied to GM (2, 1). The resultant BFGM (2, 1) is superior in simulation and short-term prediction and, therefore, can be regarded as the basis for developing efficient higher-order grey formulations.

Keywords

Main Subjects


References
1. Liu, S.-F. and Lin, Y., Grey Systems: Theory and
Applications, Springer, Berlin, Germany (2010).
2. Kayacan, E., Ulutas, B., and Kaynak, O. Grey system
theory-based models in time series prediction", Expert
Syst. Appl., 37(2), pp. 1784-1789 (2010).
3. Xia, M. and Wong, W.K. A seasonal discrete grey
forecasting model for fashion retailing", Knowl. Based
Syst., 57, pp. 119-126 (2014).
4. Jin, M., Zhou, X., Zhang, Z.M., and Tentzeris, M.M.
Short-term power load forecasting using grey correlation
contest modeling", Expert Syst. Appl., 39(1), pp.
773-779 (2012).
5. Bezuglov, A. and Comert, G. Short-term freeway traf-
c parameter prediction: Application of grey system
theory models", Expert Syst. Appl., 62, pp. 284-292
(2016).
6. Chang, B.-R. A tunable epsilon-tube in support
vector regression for re ning parameters of GM(1,1j )
prediction model - SVRGM(1,1j ) approach", IEEE
Int. Conf. on Systems, Man and Cybernetics (SMC),
Washington D.C., USA, pp. 4700-4704 (2003).
7. Chuang, C.-W. and Kao, C.-C. PC-based pseudomodel
following discrete integral variable structure
control of positions in slider-crank mechanisms", J.
Sound Vib., 301(3-5), pp. 510-520 (2007).
8. Hsu, L.-C. Using improved grey forecasting models
to forecast the output of opto-electronics industry",
Expert Syst. Appl., 38(11), pp. 13879-13885 (2011).
9. Li, D.-C., Chang, C.-J., Chen, W.-C., and Chen, C.-
C. An extended grey forecasting model for omnidirectional
forecasting considering data gap di erence",
Appl. Math. Model., 35(10), pp. 5051-5058 (2011).
10. Li, G.-D., Wang, C.-H., Masuda, S., and Nagai, M. A
research on short term load forecasting problem applying
improved grey dynamic model", Int. J. Electr.
Power Energy Syst., 33(4), pp. 809-816 (2011).
11. Li, H. and Xiao, T.-Y. Improved Generalized Energy
Index method for comprehensive evaluation and prediction
of track irregularity", J. Stat. Comput. Simul.,
84(6), pp. 1213-1231 (2014).
12. Wu, L.-F., Liu, S.-F., Yao, L.-G., Xu, R.-T., and Lei,
X.-P. Using fractional order accumulation to reduce
M. Hashem-Nazari et al./Scientia Iranica, Transactions E: Industrial Engineering 25 (2018) 2867{2880 2879
errors from inverse accumulated generating operator of
grey model", Soft Comput., 19(2), pp. 483-488 (2015).
13. Wu, L.-F., Liu, S.-F., and Yang, Y.-J. A gray model
with a time varying weighted generating operator",
IEEE Trans. Syst. Man Cybern. Syst., 46(3), pp. 427-
433 (2016).
14. Zhou, W. and He, J.-M. Generalized GM (1,1) model
and its application in forecasting of fuel production",
Appl. Math. Model., 37(9), pp. 6234-6243 (2013).
15. Wang, J.-Z., Zhu, S.-L., Zhao, W.-G., and Zhu, W.-
J. Optimal parameters estimation and input subset
for grey model based on chaotic particle swarm optimization
algorithm", Expert Syst. Appl., 38(7), pp.
8151-8158 (2011).
16. Chang, C.-J., Li, D.-C., Chen, C.-C., and Chen, C.-S.
A forecasting model for small non-equigap data sets
considering data weights and occurrence possibilities",
Comput. Ind. Eng., 67, pp. 139-145 (2014).
17. Chiang, H.-K. and Tseng, C.-H. The grey GM(2,1)
integral variable structure controller of synchronous
reluctance motor drive", IEEE Int. Conf. on Control
Applications, Taipei, Taiwan, pp. 278-283 (2004).
18. Cui, J., Liu, S.-F., Zeng, B., and Xie, N.-M. A novel
grey forecasting model and its optimization", Appl.
Math. Model., 37(6), pp. 4399-4406 (2013).
19. Kayacan, E. and Kaynak, O. Single-step ahead prediction
based on the principle of concatenation using
grey predictors", Expert Syst. Appl., 38(8), pp. 9499-
9505 (2011).
20. Li, G.-D., Masuda, S., and Nagai, M. Predictor design
using an improved grey model in control systems", Int.
J. Comput. Integr. Manuf., 28(3), pp. 297-306 (2015).
21. Li, K.-T. and Chen, Y.-P. Predictor design of a
novel grey model PGM21 using pseudo second-order
information", JSME Int. J. Ser. C: Mech. Syst. Mach.
Elem. Manuf., 49(2), pp. 395-400 (2006).
22. Lin, C.-B., Su, S.-F., and Hsu, Y.-T. High-precision
forecast using grey models", Int. J. Syst. Sci., 32(5),
pp. 609-619 (2001).
23. Luo, X. and Chang, X.-H. A novel data fusion scheme
using grey model and extreme learning machine in
wireless sensor networks", Int. J. Control Autom.
Syst., 13(3), pp. 539-546 (2015).
24. Evans, M. An alternative approach to estimating the
parameters of a generalised Grey Verhulst model: An
application to steel intensity of use in the UK", Expert
Syst. Appl., 41(4, Part 1), pp. 1236-1244 (2014).
25. Wang, Y., Song, Q.-B., Macdonell, S., Shepperd, M.,
and Shen, J.-Y. Integrate the GM(1,1) and Verhulst
models to predict software stage e ort", IEEE Trans.
Syst. Man Cybern. C Appl. Rev., 39(6), pp. 647-658
(2009).
26. Xiao, X.-P. and Qin, L.-F. A new type solution and
bifurcation of grey Verhulst model", J. Grey Syst.,
24(2), pp. 165-174 (2012).
27. Xu, J., Tan, T., Tu, M., and Qi, L. Improvement
of grey models by least squares", Expert Syst. Appl.,
38(11), pp. 13961-13966 (2011).
28. Zhu, X.-L. Application of composite grey BP neural
network forecasting model to motor vehicle fatality
risk", 2nd Int. Conf. on Computer Modeling and
Simulation, Sanya, China, pp. 236-240 (2010).
29. Wang, J.-Z., Dong, Y., Wu, J., Mu, R., and Jiang, H.
Coal production forecast and low carbon policies in
China", Energy Policy, 39(10), pp. 5970-5979 (2011).
30. Wang, K., Wu, L.-Y., and Liu, Y.-W. Condition
prediction of power transformer based on discrete
gray model", IEEE Conf. on Power Engineering and
Automation (PEAM), Wuhan, China, pp. 280-283
(2011).
31. Wang, Z.-X., Dang, Y.-G., and Liu, S.-F. Unbiased
grey Verhulst model and its application", Syst. Eng.
Theory Pract., 29(10), pp. 138-144 (2009).
32. Xie, N.-M. and Liu, S.-F. Discrete grey forecasting
model and its optimization", Appl. Math. Model.,
33(2), pp. 1173-1186 (2009).
33. Xie, N.-M., Liu, S.-F., Yang, Y.-J., and Yuan, C.-
Q. On novel grey forecasting model based on nonhomogeneous
index sequence", Appl. Math. Model.,
37(7), pp. 5059-5068 (2013).
34. Guo, X.-J., Liu, S.-F., and Fang, Z.-G. Study on
a grey Verhulst self-memory model and application",
IEEE Int. Conf. on Grey Systems and Intelligent
Services (GSIS), Macao, Macao, pp. 118-122 (2013).
35. Li, G.-D., Masuda, S., Yamaguchi, D., Nagai, M., and
Wang, C.-H. An improved grey dynamic GM(2, 1)
model", Int. J. Comput. Math., 87(7), pp. 1617-1629
(2010).
36. Choi, T.-M., Hui, C.-L., Liu, N., Ng, S.-F., and Yu, Y.
Fast fashion sales forecasting with limited data and
time", Decis. Support Syst., 59, pp. 84-92 (2014).
37. Liu, L., Wang, Q.-R., Wang, J.-Z., and Liu, M.
A rolling grey model optimized by particle swarm
optimization in economic prediction", Comput. Intell.,
32(3), pp. 391-419 (2014).
38. Shih, C.-S., Hsu, Y.-T., Yeh, J., and Lee, P.-C.
Grey number prediction using the grey modi cation
model with progression technique", Appl. Math.
Model., 35(3), pp. 1314-1321 (2011).
39. Stewart, J. Multivariable Calculus: Concepts and Contexts,
Thomson Brooks/Cole, Belmont, USA (2009).
40. Tsai, C.-F. Dynamic grey platform for ecient forecasting
management", J. Comput. Syst. Sci., 81(6),
pp. 966-980 (2015).
41. Wu, L.-F., Liu, S.-F., Yang, Y.-J., Ma, L.-H., and Liu,
H.-X. Multi-variable weakening bu er operator and
its application", Inf. Sci., 339, pp. 98-107 (2016).
42. Tsaur, R.-C. Forecasting analysis by using fuzzy grey
regression model for solving limited time series data",
Soft Comput., 12(11), pp. 1105-1113 (2008).
2880 M. Hashem-Nazari et al./Scientia Iranica, Transactions E: Industrial Engineering 25 (2018) 2867{2880
43. Tien, T.-L. The deterministic grey dynamic model
with convolution integral DGDMC(1,n)", Appl. Math.
Model., 33(8), pp. 3498-3510 (2009).
44. Wu, L.-F., Liu, S.-F., Cui, W., Liu, D.-L., and
Yao, T.-X. Non-homogenous discrete grey model with
fractional-order accumulation", Neural Comput. Appl.,
25(5), pp. 1215-1221 (2014).
45. Xia, T.-B., Jin, X.-N., Xi, L.-F., Zhang, Y.-J., and
Ni, J. Operating load based real-time rolling grey
forecasting for machine health prognosis in dynamic
maintenance schedule", J. Intell. Manuf., 26(2), pp.
269-280 (2015).
46. Hashem-Nazari, M., Esfahanipour, A., and Fatemi
Ghomi, S.M.T. Non-equidistant 'Basic Form'-
focused Grey Verhulst Models (NBFGVMs) for illstructured
socio-economic forecasting problems", J.
Bus. Econ. Manag, 18(4), pp. 676-694 (2017).
http://dx.doi.org/10.3846/16111699.2017.1337045.