Grey Information and Mechanism of Grey system Modelling

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

/ 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 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

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

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

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

/ 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 7

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

Part Ⅱ : Grey models based on limited data 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 B C Under shock influence 1 + 2+ + n≠ 0 Collected Sequence X Z shock influence endpoint D Generated sequence XD t Stage 1 Stage 2 Z shock influence factor Stage 3 9

Part Ⅱ : Grey models based on limited data X X 3 2 1 6 X=X(0) X=X(1) 4 2 0 1 2 3 4 k 0 • Definition 1 Assume that • X=(x(1), x(2), …, x( n)) • is a sequence of raw data, D an buffer operator, and • 2 1 3 4 k D X XD=( x(1)d, x(2)d, …, x( n)d) • a D’s buffer sequence. ØWeakening operator. ØStrengthening operator XD 10

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

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

Part Ⅱ : Grey models based on limited data X 2, X 3 , which is more similar with X 1 ？ 25 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. 20 15 X 1 X 2 10 X 3 5 0 1 2 3 4 5 X 1 6 3 4 9 15 X 2 9 4. 5 6 13. 5 22. 5 X 3 11 8 9 14 20 13

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

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

/ 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 16

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. 17

Part Ⅲ: Grey models based on poor information 3. 2 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 18

Part Ⅲ: Grey models based on poor information 3. 2 Definitions of grey numbers Definition 3. 2 Let (1) is a value-covered set of grey number is a continuous set, i. e. a interval number, we call covered set of grey number as (2) and as the continuous as a continuous grey number. Marked or abbreviate as is a discrete set, we call , if . as the discrete covered set of grey number as a discrete grey number. Marked as . (3) is a union set of continuous sets and discrete sets, we call mixed covered set of grey number. abbreviate as and the grey number is marked as as the is called a mixed or . 19

Part Ⅲ: Grey models based on poor information 3. 3 The algorithms of grey numbers 1. Self-minus and self-divide of grey number 2. Covered operation of simple grey number 3. Covered operation of complex grey number 4. Covered operation of multiple grey number Definition 3. 3 Let number . and as the value-covered set and true value of grey as an operation. Let the general and as the result of and as the value-covered of the grey number on the. Then we have operation formula. abbreviated as. 20

Part Ⅲ: Grey models based on poor information 3. 3 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 , , , and . Calculate the value-covered sets of. The results 21

Part Ⅲ: Grey models based on poor information 3. 3 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 , , , and . Calculate the value-covered sets of. 22

Part Ⅲ: Grey models based on poor information 3. 3 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 valuecovered set of complex number can be calculated by two optimized model: Example 3. 3 Suppose value-covered of , , and . Calculate the. According to the Definition 2. 6 we can get the results: Let , corresponding value-covered set . Then the. 23

Part Ⅲ: Grey models based on poor information 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: Let Then the corresponding value-covered set 24

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

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

Part Ⅲ: Grey models based on poor information Case 2: Grey forecasting model IN-DGM model the lower boundary sequence Assume we collect the original interval grey number sequence is the upper boundary sequence the mean value sequence The parameters’ values Table 1 Simulate value of IN-DGM model No. 1 2 3 4 5 MPSVSC (%) Simulate value of Actual value IN-NDGM model d. L d. U Mean PSVSC (%) APEM (%) d. L 19. 3 24. 5 33. 3 48. 7 d. U 21. 1 26. 5 36. 1 52. 3 Mean 20. 2 25. 5 34. 7 50. 5 24. 45 33. 39 48. 63 26. 46 36. 17 52. 24 25. 46 34. 78 50. 43 1. 93 3. 19 1. 69 0. 17 0. 24 0. 13 74. 6 78. 8 76. 7 74. 60 78. 81 76. 71 0. 12 1. 73 27 0. 01

Part Ⅲ: Grey models based on poor information 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 28

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

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

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

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

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

/ 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 34

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

Conclusions u. 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 theory perfect. u. 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. 36

Thank you! ” Prof. Xie Nai-ming For more info please contact us: Email: xienaiming@nuaa. edu. cn Web: http: //igss. nuaa. edu. cn/ August 09, 2016 Leicester