Prof. Xie Nai-ming Nanjing University of Astronautics and Aeronautics, China Grey Information and Mechanism of Grey system Modelling ” August 09, 2016 Leicester

2 / Contents Part Ⅰ : Thinking about grey information Part Ⅱ : Grey models based on limited data Part Ⅲ : Grey models based on poor information Part Ⅳ : Further thinking about grey structure

3 / Contents Part Ⅰ : Thinking about grey information Part Ⅱ : Grey models based on limited data Part Ⅲ : Grey models based on poor information Part Ⅳ : Further thinking about grey structure

4 Part Ⅰ : Thinking about grey information ” In the process of system analysis, forecasting, evaluation, decision- making and optimization, computational results are totally relied on what kinds of information that we could collect. Probability and Random information Statistic Theory Vague information Fuzzy Logical Grey information Grey System Theory Certainty information A lot of Traditional system models and theories

5 Part Ⅰ : Thinking about grey information Black Information Unknown Black system White Information known completely White system Grey Information known partially Grey system Input Information Input Information

6 Part Ⅰ : Thinking about grey information Grey Number covered set can be known, the real value is unknown Number sequence is very short, could not satisfy other theory assumption Limited data Poor information System structure is not clearly known Grey structure Grey structure

7 / Contents Part Ⅰ : Thinking about grey information Part Ⅱ : Grey models based on limited data Part Ⅲ : Grey models based on poor information Part Ⅳ : Further thinking about grey structure

8 General thinking of Forecasting Modelling Data Test DataForecasting Data Part Ⅱ : Grey models based on limited data

9 Collected Sequence X Generated sequence XD D t Stage 1Stage 2Stage 3 B C Z shock influence endpoint Z shock influence factor Forecasting model is constructed on the basis of assumption that the trend should be keep. Like in Statistic theory, assumed that  1 +  2 +  +  n =0 Under shock influence  1 +  2 +  +  n ≠0

10 Definition 1 Assume that X=(x(1), x(2),…, x( n)) is a sequence of raw data, D an buffer operator, and XD=( x(1)d, x(2)d,…, x( n)d) a D’s buffer sequence.  Weakening operator.  Strengthening operator k X X=X (0) k X X=X (1) X XD D Part Ⅱ : Grey models based on limited data

11 Modelling variables selection Modelling sequence generating Model form choice Background value generating Forecasting/ simulating sequence Parameter solving of main variable Error analysis Properties analysis Grey action coefficients Part Ⅱ : Grey models based on limited data Real Applications Information collected  To construct valuable novel grey forecast models  To collect and express grey information?  To apply in valuable real applications?

12 Part Ⅱ : Grey models based on limited data Grey incidence model

13 The fundamental idea of grey incidence analysis is that the closeness of a relationship is judged based on the similarity level of the geometric patterns of sequence curves. The more similar the curves are, the higher degree of grey incidence between sequences; and via versa. X X X X2,X3,which is more similar with X1 ？ Part Ⅱ : Grey models based on limited data

14 Part Ⅱ : Grey models based on limited data On distance Measure of area Panel matrix information

15 Part Ⅱ : Grey models based on limited data To summarize  Limited data modelling is mainly focused on grey forecasting models.  Grey incidence models were firstly used to test accuracy of grey forecasting models.  Grey incidence models could be solely used for system evaluation, index selection or decision-making according to different relationships of sequences, matrix, etc.

16 / Contents Part Ⅰ : Thinking about grey information Part Ⅱ : Grey models based on small sample Part Ⅲ : Grey models based on poor information Part Ⅳ : Further thinking about grey structure

17 Part Ⅲ : Grey models based on poor information 3.1 Introduction The Grey Number Different with the real number system, grey number operation should be solved to overcome some new problems. So we will find that operation + and - are not invertible operations, operation × and ÷ are not invertible operations in existing algorithms of grey numbers.This kind of problem is not only laid in grey system theory, but also laid in fuzzy theory, interval number, etc.

Definitions of grey numbers Definition 3.1 Assume in a real system, the variables are expressed incompletely or people are difficult to catch the exact information of them, the true value is unknown because of the limited information while the boundary or possible value set can be known. The set is defined as the information background of a grey number. is the true value of the grey number. Then (1) is a grey number under the information background. (2) is the value-covered set of. (3) is the true value of grey number. Generally, we marked the grey number as Part Ⅲ : Grey models based on poor information

Definitions of grey numbers Definition 3.2 Let is a value-covered set of grey number, if (1) is a continuous set, i.e. a interval number, we call as the continuous covered set of grey number and as a continuous grey number. Marked as or abbreviate as. (2) is a discrete set, we call as the discrete covered set of grey number and as a discrete grey number. Marked as. (3) is a union set of continuous sets and discrete sets, we call as the mixed covered set of grey number and the grey number is called a mixed grey number. is marked as or abbreviate as. Part Ⅲ : Grey models based on poor information

The algorithms of grey numbers 2. Covered operation of simple grey number 3. Covered operation of complex grey number Definition 3.3 Let and as the value-covered set and true value of grey number. Let and as the value-covered set and true value of grey number. as an operation. Let as the result of and on the operation. Let as the value-covered of the grey number.Then we have the general operation formula. abbreviated as. 1. Self-minus and self-divide of grey number 4. Covered operation of multiple grey number Part Ⅲ : Grey models based on poor information

The algorithms of grey numbers Covered operation of simple grey number Definition 3.4 Suppose grey numbers and have the corresponding discrete value- covered sets and. If, then. So the value-covered set of complex grey number is, where Example 3.1,.Calculate the value-covered sets of,, and. The results Part Ⅲ : Grey models based on poor information

The algorithms of grey numbers Covered operation of simple grey number Definition 3.5 Suppose grey numbers and have the corresponding continuous value- covered sets and. If,, then. So the value-covered set of complex grey number is, where Example 3.2,.Calculate the value-covered sets of,, and. Part Ⅲ : Grey models based on poor information

The algorithms of grey numbers Covered operation of multiple grey number Definition 3.6 Suppose grey numbers have the corresponding continuous covered sets, if is the result under two or more operations.. Then the corresponding value- covered set of complex number can be calculated by two optimized model: Let,. Then the corresponding value-covered set. Example 3.3 Suppose, and.Calculate the value-covered of, and. According to the Definition 2.6 we can get the results: Part Ⅲ : Grey models based on poor information

24 Definition 3.7 Suppose grey numbers is composed with two parts and. have the corresponding discrete covered sets and have the corresponding continuous covered sets, if is the result under two or more operations. Where. Then the corresponding value-covered set of complex number can be calculated by two optimized model: Then the corresponding value-covered set Let Part Ⅲ : Grey models based on poor information

25 Curve 1 Curve 2 Curve m Fig. Simulative and predicative value of grey number sequence Predicativ e value original sequence Definition of grey numbers Operations of grey numbers Simulative and predicative value based on optimized methods Part Ⅲ : Grey models based on poor information Grey number forecasting

26 Percent of simulating value set covered (PSVSC(%)) A B (PSVSC(%))=1-A÷B Part Ⅲ : Grey models based on poor information

27 Case2 : Grey forecasting model Assume we collect the original interval grey number sequence is No. Actual value Simulate value of IN-NDGM model PSVSC (%) APEM (%) dLdUMeandLdUMean MPSVSC (%) 1.73 MAPEM (%) 0.14 Table 1 Simulate value of IN-DGM model The parameters’ values the lower boundary sequence the upper boundary sequence the mean value sequence IN-DGM model Part Ⅲ : Grey models based on poor information

28 Table 2 Parameters’ values of DGM (1, 1) model and GM (1, 1) model Table 3 Simulate value of DGM (1, 1) model and GM (1, 1) model Part Ⅲ : Grey models based on poor information

29 Part Ⅲ : Grey models based on poor information Grey Cluster model  Whitenization weight function

30 Criterion m Criterion 2 Criterion 1 … The fundamental idea of Grey cluster model

31 Part Ⅲ : Grey models based on poor information Grey Decision making model

32 Part Ⅲ : Grey models based on poor information Grey Decision making model

33 To summarize  Grey number is linked with information background.  Grey numbers operation could be connected with optimization.  Rely on grey number information, Grey cluster, grey decision- making,g rey game, grey control, etc could be constructed. Of course, they could be utilized in different real applications. Part Ⅲ : Grey models based on poor information

34 / Contents Part Ⅰ : Thinking about grey theory Part Ⅱ : Grey models based on small sample Part Ⅲ : Grey models based on poor information Part Ⅳ : Further thinking about grey structure

35 Part Ⅳ : Further thinking about grey structure Similar with limited data forecasting, grey structure is generated by shock influence of events.  Finicial Crisis  EU debt crisis  UK exit EU  China’s “One belt and One Road” strategy  … Similar with limited data forecasting, grey structure is generated by shock influence of events.  Finicial Crisis  EU debt crisis  UK exit EU  China’s “One belt and One Road” strategy  …

36  Grey system theory is just 35 years old, it is a developing uncertainty theory, the framework and mechanism still should be improved and developed, and novel models, new applications should be further studied. We expect more and more scholars join us to make the theory perfect.  Whatever grey system theory, fuzzy logical, statistic theory are studied and reflected the real world from different sides. In our viewpoint, real world is full of uncertainty information rather than certainty information. Any of uncertainty theory is like a blind people feels an elephant and it just could describe the real world from one side. Maybe we can make progress on each theory and combine these different uncertainty information so as to character real world better in the near future. Conclusions

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