Αυτο-συσχέτιση ( auto-correlation )

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Αυτο-συσχέτιση ( auto-correlation ). covariance («συνδιασπορά») και συντελεστής συσχέτισης (correlation coefficient) αυτο-συσχέτιση (auto-correlation) βασικά παραδείγματα. Covariance («συνδιασπορά»). παράδειγμα : - PowerPoint PPT Presentation

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  • - (auto-correlation)covariance () (correlation coefficient)- (auto-correlation)

  • Covariance (): i i , . (i,i), i = 1,2,3, , N : i i, .. i, i ? : iiii 1: 2: ,i i ?

  • : covariance ()

    : - Cov(,) > 0: ( ) ( ), - Cov(,) < 0: ( ) ( ), - Cov(,) = 0: , Cov ( ) , Cov: Cov ?

    ( i)( Yi)

  • r (correlation coefficient) ( 1/(-1) ) ) -1 r 1( i)( Yi)

  • :r = 1 r = -1: r > 0: ( , ), r 1r < 0: , - ( , ), - r -1r = 0:

  • : X(ti) (X(t1),X(t1+k)), (X(t2),X(t2+k)), (X(t3),X(t3+k)), .. (X(tN-k),X(tN)) . k - tX(ti)X(ti+k)

  • -: = 0,1,2,3, ., N-1 rk () -() [auto-correlation (function), acf]r-k = rkr0 = (-1)2 / (-1)2 = 1: k ) rk k ) rk /4 /2 -1 rk 1, k

  • -: {rk} (correlation) / {rk} , . .. (ti) (ti+k) , , {rk} (: ), . ,

  • -, : : , (AR-1, a1=0.7, u2 [-1,1])-(acf), /4

    acf, k = 20

    1/e acf , rk > 0 ) (characteristic time) =

  • 3 c c:= acf (c 10.5)c:= acf (c 11)c:= acf 1/e (e Euler, 1/e 0.37) (c 2.5) , 1/e time : acf acf, k = 20acf, k = 10log-linear

    log-lin, rk exp[-a k]1/e1/e

  • , 40 c 10.5 ( acf ) c 11 ( acf )c 2.5 ( acf 1/e)

    ) 10 2.5

  • (ti)(ti)X(ti+1)X(ti+20) (X(ti), X(ti+k)), i = 1,2,3, , N-kk = 1k = 20 , ,

  • : ( )=++x (1o )(2 )

  • -, a: ,X(ti) = 10 sin(2 ti/39.5)-(acf), ,. acf acf ,

  • ? k, y ,rk (biased, underestimated),

    ) rk /4 /2

    - (decay) (correlation), . k rk 0

  • -, a: , + acf acf) , acf ,X(ti) = 10 sin(2 ti/39.5)

    -(acf), N/4

    acf ( )

  • : acf (ti) = a sin( ti) X = 0 ) rk i sin( ti) sin( ti+k)sin(A) sin(B) = [ cos(B-A) - cos(A+B)] ) rk (1/2) i [ cos( (ti+k-ti)) - cos( (ti+k+ti)) ] cos( k) - i cos( (ti+k+ti)) k= 0 ( !)) acf , , 1 (r0 = 1)

  • 5: X(ti) = 10 sin(2 ti / 39.5) + 50.0 i = 1, 2, 3, , N, N = 512 - k = 0,1,2,3, ... , /4 ( 0 = 0 !)

  • p = 0

  • -, a: , [-2,2]-(acf)

    r0 = 1, rk 0, k =1,2,3, ) (completely random) (white noise)

    : , rk = (k)= - (uncorrelated)

  • rk 0 ? , , 95% rk (95% confidence interval) 5% rk , ! :

    ) : (1) -, (2) 95% rk acf

  • 6: X(ti) = G(ti), i=1,2,3, , N, N = 512 G(ti) Gauss ( = 5 = 2)

    X(ti) X(ti), Gauss -, , (confidence interval)

  • Gauss Mathematica: