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MARKOV CHAIN A, B and C are three towns. Each year: 10% of the residents of A move to B 30% of the residents of A move to C 20% of the residents of B move.

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Presentation on theme: "MARKOV CHAIN A, B and C are three towns. Each year: 10% of the residents of A move to B 30% of the residents of A move to C 20% of the residents of B move."— Presentation transcript:

1 MARKOV CHAIN A, B and C are three towns. Each year: 10% of the residents of A move to B 30% of the residents of A move to C 20% of the residents of B move to A 30% of the residents of B move to C 20% of the residents of C move to A 20% of the residents of C move to B Everybody else stays put. No one dies. No one is born.

2 A B C A, B, and C are three towns. 10 people live in A 15 people live in C 20 people live in B

3 A B C Each year 10% of the residents of A move to B.10 and 20% of the residents of A move to C.20

4 A B C Each year 20% of the residents of B move to A.20 and 30% of the residents of B move to C.30

5 A B C Each year 20% of the residents of C move to A.20 and 20% of the residents of C move to B.20

6 A B C

7 A B C

8 A B C

9 B A C A B C This year Next year Where will everyone be in 10 years?

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11 A B C

12 A B C A n+1 =.7A n

13 A C A n+1 =.7A n +.2B n +.2C n B

14 A C A n+1 =.7A n +.2B n +.2C n B n+1 =.1A n +.5B n +.2C n B

15 A C A n+1 =.7A n +.2B n +.2C n B n+1 =.1A n +.5B n +.2C n C n+1 =.2A n +.3B n +.6C n B

16 A C A n+1 =.7A n +.2B n +.2C n B n+1 =.1A n +.5B n +.2C n C n+1 =.2A n +.3B n +.6C n B

17 A n+1 =.7A n +.2B n +.2C n B n+1 =.1A n +.5B n +.2C n C n+1 =.2A n +.3B n +.6C n Note: = 1. This accounts for 100% of those in town A in year n C n+1 =.2A n +.3B n +.6C n Note: = 1. This accounts for 100% of those in town B in year nNote: = 1. This accounts for 100% of those in town C in year n

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20 Because the sum of the numbers in each column of M is 1, 1 is an eigenvalue for M, and An eigenvector belonging to 1 will describe stable proportions. The null space of 1I – M = the null space of -=

21 The null space of = Suppose there are this year 140 people in A, 80 people in B, and 130 people in C. How many will be in each town next year? ?


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