3. Grey Modeling

1. 3. Grey Modeling GREY MODEL

2. Grey Modeling • In grey system theory, a dynamic model with a group of differential equations called grey differential model is developed. • A stochastic process whose amplitudes vary with time is referred to as a grey process; • The grey modeling is based on the generating series rather than on the raw one; • The grey derivative and grey differential equation are defined and proposed in order to build a grey model; • To build a grey model, only a few data (as few as four) are needed.

3. Grey Model • Grey model, denoted by GM(n,h) model, is a dynamic model which consists of a group of grey differential equations, where n is the order of grey differential equations and h is the number of considered variables. • Grey models play an important role for the sequence (series) forecasting problem in the grey system theory. • Among all GM(n,h) models, the most commonly utilized grey model is the GM(1,1) model.

4. 3.1: GM(1,1) Model • Let x(0) = {x(0)(1), x(0)(2),…, x(0)(n)} be a raw series and x(1) = AGO x(0), then x(0)(k) + az(1)(k) = b, k = 2,3,…,n. is a grey differential model. • This model is called GM(1,1) model since it consists only one variable. • z(1)(k) = 0.5x(1)(k) + 0.5x(1)(k-1), k = 2,3,…,n • a is the development coefficient. • b is the grey input.

5. GM(1,1) Model • Since x(0) = {x(0)(1), x(0)(2),…, x(0)(n)} and x(1) = {x(1)(1), x(1)(2),…, x(1)(n)} satisfy the GM(1,1) model, the following equations are held. • Error: , Cost function: • B is called a data matrix, yn is the data vector.

6. Solution of GM(1,1) Model • According to the least square method, we have • Another solution of a and b:

7. Whitened Equation • The whitened differential equations: • Initial value: x(0)(1) • Complete solution: • Let t = k + 1  • Predicted value:

8. Modeling Process • Take 1-AGO to original sequence x(0) • Construct the data matrix B and the data vector yn • Identify the development coefficient a and the grey input bby • Forecast the original sequence by

9. Exponential Law & Class Ratio • Let x(t) be a continuous function and c and a are constant, if x(t) = ceat, then x(t) satisfies the continuousexponential law. • Let x(t) be a continuous function and c, a and are constant, if x(t) = ceat+, then x(t) satisfies non-homogeneousexponential law. • Let x={x(1), x(2),…, x(n)}, the class ratio of series x at point k is defined as (k) = x(k-1)/x(k)

10. Class Ratio Let x={x(1), x(2),…, x(n)} • White class ratio: (k) = x(k-1)/x(k) = const, k • Non-homogeneous class ratio at point k: (k) = [x(k-1)x(k-2)]/[x(k)x(k-1)] If (k) = const, then the series x satisfies the non-homogeneouswhiteexponential law.

11. Class Ratio • Class ratio of r-AGO series x(r)  (r)(k) = x(r)(k-1)/x(r)(k), k = 2,3,…,n; r = 0,1,2,… • If a series x(0) can be used to build a GM(1,1) model, the its class ratio must satisfy that

12. Example 3.1 • Let x(0)={79.8, 74, 61, 51} 1-AGO: x(1)={79.8, 153.8, 241.8, 265.8} z(1)={z(1)(2), z(1)(3), z(1)(4)}={116.8, 184.3, 240.3}

13. Example of GM(1,1)

14. Equivalent Model 1 • x(0)(k) + az(1)(k) = b, z(1)(k) = 0.5x(1)(k) + 0.5x(1)(k-1), k = 2,3,…,n. • x(0)(k) =    x(1)(k 1),k = 2,3,…,n. Proof: x(0)(k) + 0.5a[x(1)(k) +x(1)(k1)] = b  x(1)(k) = x(1)(k1) + x(0)(k) [1+0.5a] x(0)(k) + a x(1)(k1) = b  [1+0.5a] x(0)(k) = b  a x(1)(k1)

15. Equivalent Model 2 • x(0)(k) + az(1)(k) = b, z(1)(k) = 0.5x(1)(k) + 0.5x(1)(k-1), k = 2,3,…,n. Proof: Formx(0)(k) =    x(1)(k 1), k = 2,3,…,n, we have k = 2: x(0)(2) =    x(1)(1) k = 3: x(0)(3) =    x(1)(2) =    [x(1)(1) + x(0)(2)] = (1   ) x(0)(2) 

16. Equivalent Model 3 • x(0)(k) + az(1)(k) = b, z(1)(k) = 0.5x(1)(k) + 0.5x(1)(k-1), k = 2,3,…,n. • The forbidden region for a is (,2)(+2,). • If a = 2, then  GM(1,1) model disappears. • If a = 2, then  GM(1,1) model is meaningless.

17. 3.2: Grey Series GM(1,1) • The first datum (1) is the grey number. • A GM(1,1) model built by above grey series has the following characteristic: 1. The developing coefficienta is independent of the first datum (1). 2. The predicted value is independent of (1). 3. The grey inputb is crucially dependent on (1). 4. The generating series is dependent on the grey number (1).

18. Grey Series GM(1,1) • To build a GM(1,1) model, the series must consist of at least four data. • If only three past data are available , then x(0)cannot be modeled. • However, then x(0)can be modeled and .

19. Example 3.2

20. 3.3: GM(1,N) Model • A grey differential equation having N variables is called GM(1,N) whose expression can be written as follows: where bi is said to be the ith influence coefficient which means that xi exercises influence on x1(the behavior variable).

21. GM(1,N) Model • Based on the least squared method, we have

22. GM(1,N) Model • The GM(1,N) whitened differential equation: • From the whitened differential eq., we have where

23. Example 3.3 • Original series: x1={134.8,148.2,145.3,146.6,154.4,153.7} x2={141.6,168.5,176.1,169.8,169.3,176.0} x3={152.8,173.2,203.0,214.2,221.1,244.1} x4={172.8,229.9,277.3,332.6,383.8,423.6} • Initializing  ={1,1.0994,1.0778,1.0875,1.1454,1.1402} ={1,1.1899,1.2436,1.1991,1.1956,1.2429} ={1,1.1335,1.3285,1.4018,1.4469,1.5975} ={1,1.3304,1.6047,1.9247,2.2210,2.4513}

24. Example of GM(1,N)

25. Example of GM(1,N) • By the Least Square Method, we have • From GM(1,N) model

26. Example of GM(1,N)

27. GM(1,1) v.s. GM(1,N) • GM(1,1) model plays an important role in grey forecasting, grey programming and grey control. • GM(1,N) model has laid an important foundation for regional economic programming and grey multivariable control.CANNOT use to predict the considered sequences.