時系列解析

B L12(2016-07-11 Mon) : Time-stamp: 2016-07-11 Mon 17:25 JST hig,, Excel,. http://hig3.net ( ) L12 B(2016) 1 / 24

L11-Q1 Quiz : 1 E[R] = 1 2, V[R] = 9 12 = 3 4. R(t), E[X(30)] = E[X(0)] + 30 1 2 = 115, V[X(30)] = 3 90 4 30 = 4. 2 x = X(30),T = 30, f(x; 115, 90 4 ) = 1 2π 90 4 e (x 115) 2 2 (90/4).,, z = x 115 90 4, P = 125 120 10/ 90/4 f(x; 115, 90 4 ) dx = 1 e z2 /2 dz. 5/ 90/4 2π ( ) L12 B(2016) 2 / 24

, 10 P = Q( sqrt90/4 ) Q( 5 ) = Q(1.05) Q(2.11) = 0.1469 0.0174 90/4 L11-Q2 Quiz : 1 s = r 4, s = 0. E[R] = 2 3, E[R2 ] = 11 25 6, V[R] = 18., E[X(30)] = 100 + 2 3 30 = 120, V[X(30)] = 25 18 30 = ( 5 3 15) 2. ( ) L12 B(2016) 3 / 24

2 130 P = 120 = 1 2π( 5 3 15) 2 5 10/( 3 15) 0 (x 120)2 2( e 5 3 15) 2 dx 1 2π e z2 2 dz = Q(0) Q(10/( 5 3 15)). ( ) L12 B(2016) 4 / 24

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Time Series Analysis t x(0), x(1), x(2),..., x(t),....,. x(t) t = 0, 1, 2, 3,.... 1 1 t X(t).. t T t > T. ( ) ( ) L12 B(2016) 6 / 24

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: Moving Average x(t) (smoothing) y(t) 2l + 1. y(t) = 1 2l + 1 t+l t =t l x(t ) 2l y(t) = 1 2l ( 1 2 x(t l+1)+x(t l+2)+ +x(t)+ +x(t+l 2)+ 1 2 x(t+l 1)) ( ) L12 B(2016) 8 / 24

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: : (covariance) x, y C xy = 1 n Y (,+) (+,+) n (x i x) (y i y) i=1 Y の平均値 (, ) X の平均値 I(2015)L04 (+, ) X (+, ) = (x i x, y i y ). ( ) L12 B(2016) 12 / 24

Y 0 2 4 6 8 10 X Y 0 2 4 6 8 10 X Y : 0 2 4 6 8 10 X Y 0 2 4 6 8 10 X Y 0 2 4 6 8 10 X r = C xy s x s y s x, s y : 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 r = 0.99 r = 0.55 r = 0 r = 0.55 r = 0.99 : x y : x y / : / 2 I(2015)L04 ( ) L12 B(2016) 13 / 24

:, k, t, k y(t), y(t k) 2,, k = 1 x y y(1) y(2) y(2 k) y(3) y(3 k).. y(t 1) y(t 2) y(t ) y(t 1) y(t ) ( ) L12 B(2016) 14 / 24

: C(k) = 1 T k T t=k+1 y = 1 T k (y(t) y)(y(t k) y) T t=k+1 y(t) r(k) = = C(k) C(0) C(0) y(t) T. ( ) L12 B(2016) 15 / 24

: k, k. ( ), k r(k). ( )..,, y. y(t) (t = 1, 2,..., T ) T ( ). Excel, ( ) L12 B(2016) 16 / 24

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m =AR Autoregression m AR(m) Y (t):, t = 0, 1, 2, 3,... Y (t) = m a k Y (t k) + R(t) k=1, R. E[R(t)] =0, E[R(t)Y (s)] =0 (t > s), E[R(t)R(s)] =σ 2 δ t,s = σ 2 { 1 (t = s) 0 ( ) R(t),. ( ) L12 B(2016) 18 / 24

E[Y (t)], E[Y (t)y (s)] t s, t,, ( ) AR(m) a k. ( ) L12 B(2016) 19 / 24

AR(1) a 1 = 1. E[R(t)] = 0, V[R(t)] = σ 2. E[R(t)] = µ 0, (R(t) µ) µ. 1 f o r ( t ){ / / 2 x=x+getrandom ( g e t u n i f o r m ( ) ) ; 3 } AR(1) a 1 = ϕ. E[R(t)] = 0, V[R(t)] = σ 2. 1 f o r ( t ){ / AR( 1 ) / 2 x=p h i x+getrandom ( g e t u n i f o r m ( ) ) ; 3 } ( ) L12 B(2016) 20 / 24

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AR(1), AR(1) r(k) = ϕ k. Y (t), a 1, σ 2.. ϕ = 0.9, σ = 1 ϕ = 0.9, σ = 1 ϕ = 0.2, σ = 1 ϕ = 0.2, σ = 3 ( ) L12 B(2016) 22 / 24

( ) 2016-07-27 3 ( ) 14. ( 28 /100),.. (2016-07-20 )., debugger1,,., (cont15,inverse01) (contrwsim01 ) ( ) L12 B(2016) 23 / 24

manaba., 2016-07-29,, ( ),. (1-502) /Math 1-614 https://manaba.ryukoku.ac.jp manaba ( ) L12 B(2016) 24 / 24