1. Effectiveness of GM(1,1) Model: Evidence from Random Simulations
Dr Chaoqing Yuan
Institute for Grey System Studies
Nanjing University of Aeronautics and Astronautics

2. Contents GM(1,1) model and the issues of its effectiveness Method
Results
Comparisons with ARIMA
Conclusion
Future works

3. GM(1,1) model and the issues of its effectiveness
2)the whitenization function of GM(1,1) model:
3)the restored values :

4. GM(1,1) model and the issues of its effectiveness
It can reveal the development trend of the sequence by its Accumulating Generation Operational (AGO)
(Sifeng Liu et al, 2013)
Sifeng Liu, Forrest J., Yingjie Yang. Advances in Grey Systems Research. Journal of Grey System, 2013; 25: 1-18. 

5. GM(1,1) model and the issues of its effectiveness
GM(1,1) model is a compatible model group consisted of differential, difference and exponential models
(Julong Deng, 1987)
Julong Deng. Three properties of Grey Forecasting Model GM (1,1)- the issue on the optimization structure and optimization information volume of grey predictive control.  
Journal of Huazhong University of Science and Technology, 1987; 05:1-6.(In Chinese)

6. GM(1,1) model and the issues of its effectiveness
The issues of effectiveness
Reliability
Feasibility
Fitment

7. GM(1,1) model and the issues of its effectiveness
Reliability
The length of the original data, at least four, is enough or not?
The results obtained from small data sequence are reliable and convincible?

8. GM(1,1) model and the issues of its effectiveness
Feasibility
It can only be used for exponential sequences?
Can it be used for Linear growth sequences ?

9. GM(1,1) model and the issues of its effectiveness
Fitment
GM(1,1),Rolling GM(1,1) and Metabolism GM(1,1) model, which one is better?

10. Method The Random Experiments Step 1: Generating the random sequences
Step 2: GM(1,1) Modeling
Step 3:The outputs of the Experiments

11. The frame of Random Experiments
Method
The frame of Random Experiments

12. Method Step 1: Generating the random sequences
X(0)(1): starting Point of the sequence
a,b or : growth rate of the sequence
: random disturbance

13. Method Step 2: GM(1,1) Modeling Class ratio test and smooth ratio test

14. Method Step 3:The outputs of the Experiments
MAPEs and percentage errors
Comparisons

15. Results
average fitting MAPEs of combination sequences

16. Results
average predicted percentage errors of combination sequences

17. Results
The random disturbances are not conducive to grey modeling. A rather large random disturbance will reduce forecast accuracy of GM(1,1) model.
GM(1,1) is also suitable for Linear Growth Sequence
When the random disturbances are rather small, four data are usually good; with the growth of random disturbances, five data are the best.
Metabolism GM(1,1) model is slightly better when <0.15 , otherwise GM(1,1) model is better according to the percentage errors.
It is noteworthy that grey models should almost not be used when >0.2.

18. Comparisons with ARIMA

19. Comparisons with ARIMA
Through comparing, it is found that the fitted values of ARIMA model respond less to the fluctuations because they are bounded by its long-term trend while those of GM(1,1) model respond more due to the usage of the latest four data.
the residues of the two models are opposite in a statistical sense, according to Wilcoxon signed rank test. So a hybrid model is constructed with these two models, and its Mean Absolute Percent Error (MAPE) is smaller than ARIMA model and GM(1,1) model.

20. Comparisons with ARIMA

21. Comparisons with ARIMA
And the prediction percentage errors of GM(1,1) model are quite small with an average of 0.15%.
ARIMA describes the linear trend perfectly with a average of prediction percentage errors of 0.25%, which is larger than that of GM(1,1) model.
GM(1,1) model can give almost perfect prediction while ARIMA accurately reveals the long-term liner trend.

22. Future works Comparison with ARIMA model with the given data sets
Standard data sets for testing GM(1,1) model and the extended models

23. Thanks a lot!