C 조 박재무 윤동일 장석민

1. Mathematical Modeling Land use change C조 박재무 윤동일 장석민

2. Index 1.Preliminary 2.Problem 3.Summary 4.Examples

3. 1.Preliminary

4. Markov process What is Markov process? A Markov process is a stochastic model that has the Markov property. What is Markov property? one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process's full history. I.e., conditional on the present state of the system, its future and past are independent

5. Example Forest and deforest A = = X == = = =

6. 2.Problem

7. Suppose that the 2013 state of land use in a city of 50 square miles of area is I Residentially used 30% II Commercially used 20% III Educationally used 15% IV Recreationally used 10% V Industrially used 25%. Assume that the transition probabilities for 5-year intervals are given by the matrix from I from II from III from IV from V 0.4 0.2 0.2 0.1 0.1 to I 0.2 0.3 0.1 0.1 0.3 to II A= 0.2 0.1 0.2 0.2 0.2 to III 0.1 0.1 0.3 0.6 0.1 to IV 0.1 0.3 0.2 0.0 0.3 to V Determine the limit state of the land-use succession in the above example. Problem

8. Matlab Code And result A=[ 0.4 0.2 0.2 0.1 0.1; 0.2 0.3 0.1 0.1 0.3; 0.2 0.1 0.2 0.2 0.2; 0.1 0.1 0.3 0.6 0.1; 0.1 0.3 0.2 0.0 0.3]; X=[30 ; 20; 15; 10; 25]; TOL=10^(-5); for i=1:1000 new_X=A*X; if norm(new_X-X)<TOL break else X=new_X; end end • 19.582247678672204 • 18.973026017712101 • X= 18.102696833411947 • 27.241063785982242 • 16.100965684221556

9. Change A A1=[ 0.4 0.4 0.3 0.0 0.1; 0.2 0.2 0.3 0.1 0.2; 0.2 0.2 0.1 0.2 0.3; 0.1 0.1 0.1 0.3 0.2; 0.1 0.1 0.2 0.4 0.2]; ( singular matrix ) A2=[ 0.3 0.4 0.1 0.6 0.1; 0.2 0.0 0.2 0.1 0.3; 0.1 0.2 0.1 0.1 0.2; 0.2 0.2 0.2 0.1 0.1; 0.2 0.2 0.4 0.1 0.3];

10. Change =[30 ; 20; 15; 10; 25]; =[20 ; 30 ; 10 ; 0 ; 40]; =[20 ; 20 ; 20 ; 20 ; 20];

11. Property of A ∙ Each column sum is 1 ∙ A-I is singular ⇒B is singular ⇒A-I is singular ∙ 1 is a eigenvalue of A

12. Property of A ∙ For all eigenvalues of A, ∙ As k→(c is constant)

13. Property of A

14. Converge Why does converge? Let be a eigenvalues of A with an associated collection of linearly independent eigenvectors { A has precisely one eigenvalue, that is largest in magnitude, so that 1=

15. is not affected by Since k is constant and is determined by A ⇒is not affected by

16. e-vector Since 1 is a eigenvalue of A

17. 3.Summary

18. In land use change problem, ∙is not affected by initial ∙ is affected by transition matrix A ∙ is a eigenvector of A which is corre- sponding to eigenvalue of A

19. 4.Examples

20. Example 1 Campus Master Plan - Land use change between 1997 and 2004 in campus - Increase green space, student support and academic instruction - Decrease parking space

21. Example 2 Biogeochemical modeling - carbon sequestration and reduction of greenhouse gas

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