時系列解析と自己回帰モデル
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1 B L11( Mon) : Time-stamp: Mon 11:04 JST hig,,,.,. ( ) L11 B(2017) 1 / 28
2 L10-Q1 Quiz : , x[]={1,1,3,3,3,8}; (. ) 2 x = 0, 1, 2,..., 9 10, 10. u[]={0,2,0,3,0,0,0,0,1,0}; ( ) L10-Q3 Quiz : ( ) L11 B(2017) 2 / 28
3 i n t sum=0, count =0; f o r ( k=0;k<samplesize ; k++){ sum+=x [ k ] ; i f ( x [ k]<=5) count++; } ex =( double ) sum/samplesize ; px=(double ) count /SAMPLESIZE ; L10-Q4 Quiz : double ex =0.0, px =0.0; f o r ( x =0;x<XMAX; x++){ ex+=p [ x ] x ; i f ( x<=5) px+=p [ x ] ; } ( ) L11 B(2017) 3 / 28
4 L10-Q5 Quiz( ) B R,, 1 cm 2cm, cm. 1 30? cm 125cm., Φ(z) = 1 z 2π e u2 /2 du.. t X(t), X(t + 1) = X(t) + R(t + 1), X(0) = 100. ( ) L11 B(2017) 4 / 28
5 , R(t), { 1/3 ( 1 r < 2) f(r) = 0 ( ).. ( ) L11 B(2017) 5 / 28
6 ( ) L11 B(2017) 6 / 28
7 : : ( ) L11 B(2017) 7 / 28
8 : Time Series Analysis t x(0), x(1), x(2),..., x(t),....,. x(t) t = 0, 1, 2, 3, t X(t).. t T t > T. ( ) R f R (t),. ( ) L11 B(2017) 8 / 28
9 : Moving Average x(t) (smoothing) y(t) (2l + 1). y 2l+1 (t) = 1 2l + 1 t+l t =t l x(t ) x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7) x(8) x(9) x(10) x 2l y 2l (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)) x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7) x(8) x(9) x(10) x ( ) L11 B(2017) 9 / 28
10 : 3 3 x(t) = + + ( ).,,, ( ). ( ) ( ), ( ) ( ). ( ) L11 B(2017) 10 / 28
11 :. ( ) L11 B(2017) 11 / 28
12 : L11-Q1 Quiz( ), 3,4. t x y 3 y 4 ( ) L11 B(2017) 12 / 28
13 : : (covariance) x, y C xy = 1 n Y (,+) (+,+) n (x i x) (y i y) i=1 Y の平均値 (, ) 1.8, 3.6 X の平均値 (+, ) X (+, ) = (x i x, y i y ). ( ) L11 B(2017) 13 / 28
14 Y X Y X Y : X Y X Y X r = C xy s x s y s x, s y : r = 0.99 r = 0.55 r = 0 r = 0.55 r = 0.99 : x y : x y / : / 2 1.8, 3.6 ( ) L11 B(2017) 14 / 28
15 :, k, t, k x(t), x(t k) 2,, k = 1 x y x(1) x(2) x(2 1) x(3) x(3 1).. x(t 1) x(t 1 1) x(t ) x(t 1) x(t ) k x y x(1). x(k + 1) x(1).. x(t ) x(t k). x(t ) ( ) L11 B(2017) 15 / 28
16 : kj autocovariance C(k) = 1 T k T t=k+1 x = 1 T k kj autocorrelation r(k) = (x(t) x)(x(t k) x) T t=k+1 = C(k) C(0) C(0) x(t) T. x(t) ( ) L11 B(2017) 16 / 28
17 : correlogram k, k. ( ), k ( ), r(k)., r(k) =. ( ) L11 B(2017) 17 / 28
18 : L11-Q2 Quiz( ),. ( ) L11 B(2017) 18 / 28
19 10 11 : ( ) L11 B(2017) 19 / 28
20 m =AR Autoregression m AR(m) X(t):, t = 0, 1, 2, 3,... X(t) = m a k X(t k) + R(t) k=1, R(t). E[R(t)] =0, E[R(t)X(s)] =0 (t > s), E[R(t)R(s)] =σ 2 δ t,s = σ 2 { 1 (t = s) 0 ( ) R(t),. ( ) L11 B(2017) 20 / 28
21 AR(1) E[R] = 0, AR(1). a 1 = 1. E[R(t)] = 0, V[R(t)] = σ 2. 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 } ( ) L11 B(2017) 21 / 28
22 L11-Q3 Quiz(AR(1) ) AR(1) X(t + 1) = ϕ X(t) + R(t + 1), R(t). ϕ = 1. 1 X(2) X(0), R(1), R(2). 2 X(0) = a,, P (X(0) = a) = 1., R(t) N(0, σ 2 ) (AR(1) )., X(2). 3 (2), X(t) (t 1). ( ) L11 B(2017) 22 / 28
23 ( ) L11 B(2017) 23 / 28 X(t) E[X(t)] t E[X(t)X(s)] t s, t,., t.
24 ,, µ = E[X(t)]. k C(k) = E[(X(t) µ)(x(t + k) µ)] k = 0 C(0) = E[(X(t) µ) 2 ]=. k r(k) = C(k) C(0). ( ) L11 B(2017) 24 / 28
25 , 1,. :t :. Excel ( ) L11 B(2017) 25 / 28
26 AR(1), X(t) =ϕx(t 1) + R(t) =ϕ(ϕx(t 2) + R(t 1)) + R(t) = = ϕ t X(0) + ϕ t 1 R(1) + ϕr(t 1) + R(t). E[X(t)] =ϕ t E[X(0)]. E[X(t)] = 0. E[X(t)X(t + k)] =E[(ϕ t X(0) + ϕ t 1 R(1) + ϕr(t 1) + R(t)) (ϕ t+k X(0) + ϕ t+k 1 R(1) + + ϕ k+1 R(t 1) + ϕ k R(t) + + R(t + k))] =ϕ 2t+k E[X(0)X(0)] + (ϕ 2t+k 2 + ϕ 2t+k 4 + ϕ k )σ 2 E[X(t)X(t)] =ϕ 2t E[X(0)X(0)] + (ϕ 2t 2 + ϕ 2t ϕ 0 )σ 2. r(k) = E[X(t)X(t + k)] E[X(t)X(t)] = ϕ k. ( ) L11 B(2017) 26 / 28
27 AR(1), AR(1) r(k) = a k 1 = ϕ k. X(t), a 1 = ϕ, σ 2.. ϕ = 0.9, σ = 1 ϕ = 0.9, σ = 1 ϕ = 0.2, σ = 1 ϕ = 0.2, σ = 3 ( ) L11 B(2017) 27 / 28
28 (1-502) /Math Learn Math Moodle B (3-B105) ( ) (3-B105) (3-B105) ( ) ( ) ( ) L11 B(2017) 28 / 28
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