…}…‰…R…tŸA“½‡Ì−î‚b

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1 2012 OR 2 ( ) 2012 OR 2 1 / 29

2 1 2 3 Google PageRank ( ) 2012 OR 2 2 / 29

3 41 1, 2004,, 1, 2 3, 2, 5 5, 6-4,? 42 2, 2, 3, 2,? ( ) 2012 OR 2 3 / 29

4 Exercise 41, 43, or, 70% 30%, 60% 40% ( ) 2012 OR 2 4 / 29

5 44, ,, p 1, 1 p 1 b, N, 0 p = 045, b = 90, N = 100, N 0? Exercise 42, ( ) 2012 OR 2 5 / 29

6 t {X t } # X t, OK t T := {0, 1, 2, } {X t } = {X 0, X 1, X 2, } #, X t S, S #, ( ) 2012 OR 2 6 / 29

7 ,, Def Markov property i 0, i 1,, i, j S, t T, {X t } P { X t+1 = j X 0 = i 0, X 1 = i 1,, X t = i } = P { X t+1 = j X t = i } ( ) 2012 OR 2 7 / 29

8 n Def 2 i 0, i 1,, i, j, k S, t T, {X t } 2 P{X t+1 = j X 0 = i 0,, X t 1 = i, X t = k} = P{X t+1 = j X t 1 = i, X t = k} Def n t T, i 0, i 1,, i t, i t+1 S,, {X t } n P{X t+1 = i t+1 X 0 = i 0,, X t n+1 = i t n+1,, X t = i t } = P{X t+1 = i t+1 X t n+1 = i t n+1,, X t = i t } , 2,, 90%,, 50%,, 70%,, 50%, 2 ( ) 2012 OR 2 8 / 29

9 n n = 0 OCRO HLI RGWR NMIELWIS EU LL NBNESEBYA TH EEI ALHENHTTPA OOBTTVA NAH BRL n = 1 ON IE ANSOUTINYS ARE T INCOTORE ST BE S DEAMY ACHIN DILONASIVE TUCOOWE AT TEASONARE FUSO TIZIN ANDY TOBE SEACE CTISBE 2 n = 2 IN NO IST LAT WHEY CRATICT FROURE GROICD PONDENOME OF DEMONSTURES OF THE REPTAGIN IS REGOACTIONA OF CRE ( ) 2012 OR 2 9 / 29

10 j, k S, t T P{ X t+1 = k X t = j } = P{ X 1 = k X 0 = j } (=: p jk ), {X t } p 11 p 12 p 1N p 21 p 22 p 2N P := p N1 p N2 p NN p jk := P{ X 1 = k X 0 = j }, j, k = 1,, N Exercise ( ) 2012 OR 2 10 / 29

11 Exercise 44, a 1/6 1/3 b 1/4 Exercise 45, d 1/5 C a 1/50 1/50 c 5/6 b d 1/3 e 1/5 f g ( ) 2012 OR 2 11 / 29

12 60 t + 1 1 2 3 4 5 6 7 8 9 10 11 12 1 0 30 2 0 0 0 1 7 8 3 5 4 60 2 2 0 30 0 0 0 8 2 11 0 4 3 60 3 30 2 0 0 0 0 5 8 2 5 3 5 60 4 0 0 0 0 30 2 6 4 1 1 7 9 60 t 5 0 0 0 2 0 30 4 5 2 10 0 7 60 6 0 0

12 12 60 t t Yahoo! ( ) 2012 OR 2 12 / 29

13 p (2) jk := P{ X t+2 = k X t = j } = ( 2 P (2) := p (2) jk ) P (2) = P 2 (, m P (m) := P m S p jh p hk h=1 p (m) jk ) Exercise 46 43, 2, 3 ( ) 2012 OR 2 13 / 29

14 m π (m) k m k π (m) := (π (m),, π (m) ) m 1 S π (m) = π (m 1) P = π (0) P m Exercise 47 43,, m ( ) 2012 OR 2 14 / 29

15 π π = π P, π P π (0) = π π (0) = π π (1) = π (0) P = π P = π π (2) = π (1) P = π P = π π (m) = π (m 1) P = π P = π, m Exercise ( ) 2012 OR 2 15 / 29

16 i j i j, i j i j, i j, i i i j j i i j, j k i k, irreducible set closed set absorbing state transient set, ( ) 2012 OR 2 16 / 29

17 Exercise , ( ) 2012 OR 2 17 / 29

18 j Exercise 49 d j := {k p (k) jj > 0} d j 2 j d j = Exercise 410, a, b, c, d ( ) 2012 OR 2 18 / 29 5/6 d 4/5 1/5 1/4 a C 1/6 1/3 2/3 b 3/4

19 Theorem ( ), lim k p(k) jk = π k for any j 1 2, p (k) jk 3, k or ( ) 2012 OR 2 19 / 29

20 Google PageRank HP H = [h ij ] { 1/li HPi HPj h ij = 0 otherwise, l i :=HPi HP Exercise 411, H? ( ) 2012 OR 2 20 / 29

21 i H i = (0,, 0), 1/N S := H + a ( 1 N e N) { 1 HP i, a i := 0 otherwise S H ( ) 2012 OR 2 21 / 29

22 (cnt d), G := αs + (1 α) 1 N e Ne N PageRank ( ) 2012 OR 2 22 / 29

23 Assignment 41 1, H S 2 G 3 α = 085 b a C e d ( ) 2012 OR 2 23 / 29

24 D Esopo& Lefkowitz(1977) 25, 本塁打 (H) p H 二塁打 (D) p D p D ヒット (H) 0アウト四死球 (W) 0アウト p S +p W p H p S 0 アウト p H p O p O アウト (O) p O p D p H p D p D p S +p W 1 アウト 1 アウト p H p S 1 アウト ( ) 2012 OR 2 24 / 29

25 ( ) 2012 OR 2 25 / 29

26 ( ) 2012 OR 2 26 / 29

27 4 Aaa AAA Aa AA A A Baa BBB Ba BB B B Caa CCC Ca CC C C D D ( ) 2012 OR 2 27 / 29

28 a j := j N a j = pa j+1 + (1 p)a j 1, j = 1,, N 1 a 0 = 1, a N = 0 ( ) 2012 OR 2 28 / 29

29 , 2004, Web, vol54, no12, pp739(25) 743(29), 2009,, vol57, no6, pp308(136) 314(142), 2012 D Esopo DA Lefkowitz B The distribution of runs in the game of baseball, In Optimal Strategies in Sports (Ladany SP, Machol RE eds), North-Holland, 1977 Hillier FS, Lieberman GL Introduction to Operations Research ninth edition (chapter 16), McGraw Hill, 2010 Langville AN, Meyer CD Google PageRank, 2009 Google s PageRank and Beyond, Princeton Univ Press, 2006 ( ) 2012 OR 2 29 / 29

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