, , . , ARMA . : . , ARIMA (p, d, q), , d .
, , . , . , w (t) - W (t), wn = w (tn), n =0,1,, N. W (t) g (t, a) s (t), ..
W (t) = g (t, a) + s (t). (3.10)
t a. , , , .
- AR (p), MA (q), ARMA (p, q). g (t, a), . g (t, a) . TS - (trend stationary series - ).
, (3.10) , (. . 3.5, 3.7, 3.9, 3.11). d . , , DS (difference stationary series - ).
. .
: TS -. ,
Xt = α + β t + at, t = 1,..., T, (3.11)
.. .
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, Δ Xt = Xt - Xt 1 = β + (at - at - 1 ). at - , , ( Δ Xt) = β.
: DS . ,
Xt = α + Xt 1 + at, t = 1,..., T. (3.12)
Δ Xt = Xt - Xt 1= = α + at , ( Δ Xt) = α. Xt
Xt = α + Xt 1 + at = α + (α + Xt 2 + at - 1) + at = 2α + Xt 2 + at 1 + at =
=... = at + . (3.13)
, Xt , .
(3.11) (3.12) : , . - at, - .
( ) .
(3.11) , Xt 0 = Xt (α + β t) = at, .. .
(3.13) : Xt 0 = Xt - at = , . , : .
( ) . Δ Xt - . Δ Xt = β + (at - at - 1 ) (1) , β. Δ Xt = α + at (0) , α.
, . , TS - , DS , .
TS DS, , , (3.11) (3.12). TS -: TSt = 1 +0,5 t + at, at ~ N (0,1), DS : DSt = 0,1 + DSt 1 + at, at ~ N (0;0,25). .3.13.
.3.13. TS () DS ()
, .. , . (.3.1 3.2) .
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TS -: Xt = 0,78 + 0,50 t; DS : Xt = -0,075 + 0,093 t.
3.1 TS -
3.2 DS -
, DS = 0,05.
.3.14.
.3.14. ( - TS -; - DS )
.3.14, TS - . DS . [5,6]: , . , , ( ) .
. .3.15 , TS -. .3.16 . , - : , , AR MA . , , .
.3.15. TS -
) )
.2.16. () () TS -
DS .3.17, 3.18. , , DS - .
.3.17. DS -
) )
.3.18. () () DS -