マルコフ連鎖の時間発展の数値計算
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- ことこ はぎにわ
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1 B L07( Mon : Time-stamp: Mon 8:4 JST hig ( L07 B(206 / 20
2 L05-Q TA Prob and Sol:, {, 2}. M = ( p(0 = ( 0 p(t. p(0 = 2 ( p(t. ( L07 B(206 2 / 20
3 M λ, λ 2, u, u 2, λ = λ 2 = 6, u = ( 4 s, u 2 = ( s (s 0. λ = 7 ( 4,. 2 u s = 7. u 2 s =. p(t. p(t =M t p(0 =(UDU t p(0 = ( u u 2 ( λ t 0 0 λ t 2 ( u u 2 p(0 ( L07 B(206 / 20
4 U p(0, ( a b. ( p(t = u λ t u 2λ t 2 ( a b =a u λ t + b u 2 λ t 2. p(0, p(0 = a u + b u 2 a, b a =, b = 4 7. p(t = 7 ( ( 7 ( 6 t. t + p(t, a =.. 6 <, p(t 7 ( 4 (t + ( L07 B(206 4 / 20
5 0.75 p(,t p(2,t p(x,t 4/7 0.5 / t p(t = 7 ( ( ( 6 t ( L07 B(206 5 / 20
6 L06-Q2 Quiz : ( λ = (, 0 ( 0 s + k (s 0 k 0., 0 ( ( 0 (, u =, u 2 = 0 2, s u + ( s u 2 (0 s 0. ( λ =, 0 + s(s 0., ( ( 0+, u =. p(0 p(t = a u t + b u 2 t + c u ( t. ( L07 B(206 6 / 20
7 ( p(0 = 2 a, b, c, 0 p(t = 2 u + 2 u u ( t 2 p( = 4. ( 4 2 p(0 = a, b, c, p( = p(t = u + 2 u u ( t. {} {2, }.,. ( L07 B(206 7 / 20
8 L06-Q Quiz : T λ,, ( λ =, ω, ω 2 ( ω ( s, s, ω 2 ω 2 s (s 0 ω λ =, ( s, u =. 2 λ = ω, ω 2, ω = ω 2 =.,., p(0 = u = p(t. ( L07 B(206 8 / 20
9 L06-Q4 Quiz : λ =, ( s (s 0. s, u = 2 (., ( s (s 0. s, u 2 = (, p(t = a u t + b u 2 ( t. ( 2 2 p(0 = a, b a =, b = 0 p(t = u. 2 p( = u. ( p(0 = 2 a, b p(t = u + 6 u 2 ( t.., x =, 2. ( L07 B(206 9 / 20
10 4 ( L07 B(206 0 / 20
11 I x = 0,..., m m. C. p(t, p(x, t double p [m] = {. 0, 0. 0,...., 0. 0 } ; /. m. / / {p ( 0, t, p (, t, p ( 2, t,... p (m, t } / M = ( p p 2 p 2 p 22 double M[ ] [ m]= { { 0., 0. }, { 0. 9, 0. 7 } } ; / 2 / {{p, p 2 }, {p 2, p 22 }} q = M p q y = M yx p x. x ( L07 B(206 / 20
12 p [ ] p ( x, 0 ; 2 p ; f o r ( t { 4 pn=m p ; / / 5 p=pn ; 6 p ; 7 } : / 2 M a r k o v Time stamp : Mon 06:28 JST hig 4 / 5 #d e f i n e CRT SECURE NO WARNINGS / V i s u a l C++ 6 #i n c l u d e <s t d i o. h> 7 8 / m / 9 #d e f i n e NS 0 i n t m u l t i p l y t r a n s ( double pn, double p ; 2 i n t p r i n t d i s t ( double p, i n t t, i n t m ; 4 i n t main ( { 5 i n t t, tmax ; 6 i n t x ; 7 double p [ NS ] ; / p ( t / 8 double pn [ NS ] ; / p ( t + / 9 i n t m=ns ; / / s c a n f ( %d, &tmax ; 2 p r i n t f ( #T=%d\n, tmax ; / / 26 t =0; p [ 0 ] =. 0 ; p [ ] = 0. 0 ; p [ 2 ] = 0. 0 ; 27 p r i n t d i s t ( p, t,m ; ( L07 B(206 2 / 20
13 2: f o r ( t=;t<=tmax ; t++{ m u l t i p l y t r a n s ( pn, p ; / / 2 f o r ( x =0;x<m; x++{ p [ x]=pn [ x ] ; 4 } 5 p r i n t d i s t ( p, t,m ; 6 } 7 r e t u r n 0 ; 8 } 9 40 / p M q=m p. / 4 i n t multiply trans ( double q, double p { 42 i n t x, y ; 4 i n t m=ns ; 44 / / 45 double M[ ] [ NS] = {{0.5, 0. 5, 0. 0 }, 46 { 0. 5, 0. 5, 0. 0 }, 47 { 0. 0, 0. 0,. 0 } } ; 48 f o r ( y =0;y<m; y++{ 49 q [ y ]=0; 50 f o r ( x =0;x<m; x++{ 5 q [ y]+=m[ y ] [ x ] p [ x ] ; 52 } 5 } 54 r e t u r n ; 55 } / t p / 58 i n t print dist ( double p, i n t t, i n t m{ 59 i n t x ; 60 p r i n t f ( %d,, t ; 6 f o r ( x =0;x<m; x++{ 62 p r i n t f ( %f,, p [ x ] ; 6 } 64 p r i n t f ( \n ; 65 r e t u r n 0 ; 66 } ( L07 B(206 / 20
14 4 ( L07 B(206 4 / 20
15 L07-Q Quiz(, {x} = {0,, 2,..., 99} M = double p [ 0 0 ], q [ 0 0 ] ; p, q, p q = M p i n t m u l t i p l y t r a n s ( double q [ ], double p [ ] ;. M 2, M. ( L07 B(206 5 / 20
16 00.? ( ( 0 0, + 0 m = 99?? ( L07 B(206 6 / 20
17 4 ( L07 B(206 7 / 20
18 p(x, t = P (X(t = x, E[ϕ(X(t] = x ϕ(xf(x = = m ϕ(xp (X(t = x x= m ϕ(xp(x, t x= ( L07 B(206 8 / 20
19 L07-Q2 Quiz(, {x} = {, 2}. ( M = 6. p(0 = ( 0 E[(X(t 2 ] p(, log p(t p(.. 2 ( L07 B(206 9 / 20
20 (-502 /Math I? manaba ( L07 B( / 20
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