[14, 17, 18] , . - [4, 7], , [27, 28].
p - [29,30]:
, (3.7)
,
, ,
(); .
(3.7) :
, t =2,.., M (3.8)
p - , , [29, 30]:
, (3.9)
, ; .
, (3.9) , p =1,2, .. . , .
, , (3.8) (3.9):
(3.10)
(3.11)
(3.10), (3.11) , , [31]:
, (3.12)
. (3.13)
[14, 17, 18, 29], ( .. .. [30]), (3.12) (3.13) :
, (3.14)
. (3.15)
(3.12) (3.13) , . (3.14), (3.15) , (3.12), (3.13) :
.
(3.14) (3.15) - .
(3.12), (3.13) . [31]:
,
:
,
.
.
, :
|
|
- ;
- , , ;
- , 100.
, ( ARX, ARIMA ..) .
:
- (, , ARX-);
- (, ) - .
[76].
, -, . , [14, 16 18, 32].
3.3. ( , , )
, , , [3, 23].
, , , , , .., .
, () , , V 1, V 2,, V N .
[12, 26], j - () V j =[ V 1 j, V 2 j,, Vnj ] T.
, [12, 26] : V 1, V 2,, V N; L n, , .. V ×=[ V 1×, V 2×,, Vn ×]T F × =[ f 1×, f 2×,, fm ×]T , m < n.
, () [11,12, 26]:
|
|
, (3.16)
V; . n - , ( ) . (3.16) [11, 25], , n - n - .
V, F , :
. (3.17)
(3.17) . , F , V .
, , [10, 33]. V V 1, V 2,, V N
[10,33,34]:
(3.18)
V; U i .
(3.18) V j K V F j . K V [35 37], , [10, 33]. F n, (3.18) m = n [34].
, , [38 43]. ξ:
(3.19)
(3.19) (3.18). , K V , K X, . , ξ , K :
.
, , , ξ . , , V
.
, K V, K V - . (3.19) V j [38 40].
. () , V. m < n ( ) (3.19), n m , , ξ. , ξ . m (3.19) , , ..:
|
|
, .
, , () [38 42], () [34, 43, 44]. [77], .. , ( ).
, , ( V) [38 42]. m (3.19) [45]:
(3.19) , F. F m (m < n) , , .
, , V F, . , , .
(, ) () (3.19) V ( ) [45].
, . [34, 40]. , [12, 41, 42]. , .
(3.19), , .
(), , (). () , .
, , .. , [21, 34].
|
|
- , [17, 41, 42].
() , [1], [46, 47], [1, 4].
, - , () [48 52].
, , , . , .. [12, 25, 26, 38 42]. , , [78].
3.4. ( )
[53 55], ( ) [56].
[53, 54, 57]:
, (3.20)
- x (t) y (t) ;
x (t). TR , T () k (t).
x (t) y (t) , . (3.20).
, , (3.20) [57, 58]:
, . (3.21)
[9, 14, 57 59] .
[60]. , (3.20) (3.21) , (). , [59].
, x (t) y (t) , , (3.20) (3.21).
[55, 56]: , [55]:
(3.22) |
: , . T , .
, . , , :
, (3.23)
λ ; .
l , l+ 2 [55, 56]:
(3.24)
(3.24) , (3.23) [55, 57, 58, 61]. : , .
|
|
, : (3.22) ; . [55].
- [57]. , , (3.22), (3.24) . , . [57, 62]. .
, [6], ARMA-, , , , . .
3.5. ( )
, , , [1, 2, 4, 15, 19, 63].
() , ( -). . [1, 5, 9, 14, 15] :
(3.25)
, , (3.26)
() j 1 ; i - j - ; , , . .
(3.26) j - i - [1], j - i - [2, 4, 5, 8].
(3.25) [1, 5, 9, 14, 15]:
(3.27)
,
; , .
, (3.25), (3.26) [9, 64, 65]. , , , , .. . .
, (3.26), . , , , .
[1, 2, 5]. . , , ( ). [5, 64, 65], . , , .
(3.25) (3.27) , . , j - i - , , () . j - i - , , i - .
[66] ARIMA- ( (3.5) , , . , [79], .
. ( [2, 5], .. [19, 65]), [65] , .