. 1

1.1 { xt : t = 0,1, 2,...} , xt t (stochastic system). : (i) (ii) , , xt t = 1, 2,.... xt t t [0, ].

(iii) xt [0,t] t 0. (iv) , xt yt , t = 1, 2,.... (v) , xt yt , t = 1, 2,....

(vi) , , xt yt , (l , a, h) t. t t = (l , a, h, t ). , xt t, (i),

xt t, (ii) (iii), xt

. . .

. 1.

t, (iv) (v) xt t, (vi). , (i)-(iii), , (iv)-(vi), . S, () R, R d , t , R k , t t .. (l , a, h) - (spatial time series), . (vi) . . 1.2. , { xt : t = 0,1, 2,...} , , (deterministic) , . , (..), , { X n : n N} , () { X t : t T } , ( , F , P ) . ,

{x

t

= X t ( ) : t T }

t (sample path, realization) .. { X t : t T } . { xt = X t ( ) : t T }.

{X t : t T },

T {ti T } ti < ti +1 (i = 1,..., n 1). X (t ( n ) )

2

. . .

. 1.

n- .. ( X t1 ,..., X tn ) Ft ( n ) ... X (t ( n ) ),

D = {Ft : t ( n ) = (t1 ,..., tn ) t i < t i+1 , i=1,...,n-1 n N},(n)

Ft ( n ) ( x ( n ) ) = P[ X t1 x1 ,..., X tn xn ], x ( n ) = ( x1 ,..., xn ) R n n N,

.. N n ( x | , ), n N. ( . 1.5 1.7). m- .. X (t ( m ) ) = ( X t1 ,..., X tm ) ', m < n, m .. X (t ( n ) ), ... Ft ( m ) D . (-) . :Ft ( m ) ( x ( m ) ) = P[ X t1 x1 ,..., X tm xm ] = P[ X t1 x1 ,..., X tm xm , X tm+1 < ,..., X tn < ] = Ft ( n ) ( x(m)

(1.2.1)

, ,..., ), x

(m)

R m, n N m < n,m

. t ( m ) ( u( m ) ) = t ( n ) ( u( m ) , 0,..., 0), u( m ) R m m, n N m < n.

(1.2.2)

(1.2.1), (1.2.2), (Kolmogorov compatibility conditions). , , .. { X t : t T } . .

3

. . .

. 1.

1.2.1 (Kolmogorov). (n) n D = {Ft ( n ) : t R n N}, (1.2.1), (1.2.2), .. { X t : t T } D . 1.2.1 ( ).

{ X t : t 0}

. .. Brown. 1.3.

{X t : t T }

t = 0, , t = s, { xt : 0 t s} , X s + h , h > 0, X s + h

{ xt : t [0, s]}

.

, , . . . 1.3.1 ( ). n N , ti T

{X t : t T }

(i = 1,..., n) h T

(X

t1

,..., X tn X t1 + h ,..., X tn + h .

) (

)

(1.3.1)

, . ( ).

4

. . .

. 1.

, = E[ X ], 2 = V [ X ] , , 2 , xy = Cov ( X , Y ). ,

{ X t : t T } (t ) = E[ X t ], t T , 2 (t ) = V [ X t ] = E[( X t t ) 2 ], t T ,

(1.3.2)

(1.3.3)

(ACVF)

(t , h) = Cov( X t , X t + h ) = E[( X t t )( X t + h t + h )], t , h T . 2 (t ) = (t , 0) = Cov ( X t , X t ), t T .

(1.3.4)

(1.3.5)

, , , . E[| X |2 ], E[| Y |2 ] < , , , Cauchy-Schwarz

E[| X || Y |] {E[ X 2 ]E[Y 2 ]}1/ 2 ,:

| E[ X ] | E[| X |] {E[ X 2 ]}1/ 2 < , | E[Y ] | E[| Y |] {E[Y 2 ]}1/ 2 < ,

| Cov( X , Y ) | {V [ X ]V [Y ]}1/ 2 < ., { X t : t T } :

= E[ X t ], t T , (h) = Cov( X t , X t + h ), t , h T , 2 = V [ X t ] = (0) | ( h) | h T .

(i) (ii)

5

. . .

. 1.

(1.3.1). , 2 , , , . . 1.3.2 ( ).

{ X t : t T }

, , (i) (ii). , , .. . H (ACF) :

( h ) = Corr ( X 0 , X h ) =

Cov ( X 0 , X h )

V [ X 0 ]V [ X h ]

=

( h) , h T . ( 0)

(1.3.6)

ACF | ( h ) | 1, Cauchy-Schwarz. . . . , . 1.5 1.7 , Kolmogorov. 1: X t = U cos ( t ) + V sin t , ( , ] U , V

.. , 2 2 U = V = 0, U = V = 1 ( X , Y ) = 0 . { X t : t ()

} ( ). :

E [ X t ] = E [U ] cos ( t ) + E [V ] sin ( t ) = 0, t .

6

. . .

. 1.

()

Cov ( X t , X t + h ) = E [ X t X t + h ] E [ X t ] E [ X t + h ] = E [ X t X t + h ]

= E U 2 cos ( t ) cos ( (t + h) ) + UV cos ( t ) sin ( (t + h) ) + UV sin ( t ) cos ( (t + h) ) + V 2 sin ( t ) sin ( (t + h) ) = E U 2 cos ( t ) cos ( (t + h) ) + E [UV ] cos [t ] sin ( (t + h) ) = cos ( t ) cos ( (t + h) ) + sin ( t ) sin ( (t + h) )

+ E [UV ] sin ( t ) cos ( (t + h) ) + E V 2 sin ( t ) sin ( (t + h) )

= cos ( t ( t + h ) ) = cos ( h ) , t.

2: X t = t + t 1 , t N ( 0, ) , t ,

. { X t : t () ()

}

. :

E [ X t ] = E [ t ] + E [ t 1 ] = 0.

Cov ( X t , X t + h ) = Cov ( t + t 1 , t + h + t + h 1 )

= Cov ( t , t + h ) + Cov ( t , t + h 1 ) + Cov ( t 1 , t + h ) + Cov ( t 1 , t + h 1 ) = Cov ( t , t + h ) + Cov ( t , t + h 1 ) + Cov ( t 1 , t + h ) + 2Cov ( t 1 , t + h 1 ) (1 + 2 ) 2 , h = 0, = 2 , h = 1, 0, h > 1. 1 2 ( ) t. . 3: { X n : n

} . ( n = 1, 2,...)

X n = Z1 + Z 2 + ... + Z n

Z v2

( v = 1, 2,...)

.. = 0

. { X n : n

}

.

7

. . .

. 1.

, E [ X n ] = nE [ Z ] = 0, n n+k Cov ( X n , X n + k ) = Cov X n , X n + Z v v = n +1 n = Cov ( X n , X n ) + Cov Z v , v =1 = V [ X n ] = nV [ Z ] = n 2 .

v = n +1

Z

n+k

v

1.4.

, , / , . . , , . . , (outliers), . . ,

X t = mt + st + Yt , t ,

(1.4.1)

mt t , st Yt . mt

8

. . .

. 1.

mt = + t , mt = exp{ t}, tmt = 0 + 1t + 2t 2 + ... + k t k .

..mt

mt = 0 + 1t + 2t

2

0 st :st = sin 2t = sin

t

2 t 1 , , d = , , d v

: st = k sin ( 2k t + k ).k =1

st

0

t

, X t = mt + st + Yt .

9

. . .

. 1.

Xt

0

t

t () ( )

.

st ,

X t = mt + Yt , t = 1,..., n. mt .(1)

(1.4.2)

{ X t } n, {xt , t = 1,..., n}, mt t , . , mt = + t , :

(1.4.3)

{ x t}t =1 t

n

2

,

10

. . .

. 1.

mt = + t ,

(1.4.4)

= x t , = C /C ,tx tt

(1.4.5)

x = n 1 t xt , t = n 1 t t = (n + 1) / 2

Ctx = t (t t )( xt x ), Ctt = t (t t ) = n(n 2 1) /12.2

(1.4.6)

. , :{ j : j = 0,..., k }

min

{x t =1 t

n

0

1t 2t 2 ... k t k } .2

(1.4.7)

(2)

mt

() (moving average smoothing method). 1 q 1). {xt : t = 1, 2,...} t . d = 12. st : st = st + d

d

l =1 t + l

s

= 0,

t .

. d , xk ,l xt

t = (k 1)d + l , (k = 1,..., K l = 1,..., d ). l = t mod d . ( n {xt }, d n = Kd ).(1)

mt , , mt . xt . k, mk () , mk =d

x dl =1

1

k ,l

, k = 1,..., K ,

(1.4.17)

K . sl (l = 1,..., d ). . l

15

. . .

. 1.

sl =

1 K

(xk =1

K

k ,l

mk ), l = 1,..., d ,

(1.4.18)

d l =1 sl = 0. xk ,l = mk + sl + Yk ,l , k = 1,..., K , l = 1,..., d ,

(1.4.19)

Yk ,l = xk ,l mk sl , k = 1,..., K , l =1,...,12.

(1.4.20)

.

. sl , l = 1,..., d , mt , st sl , l = t mod d (t = 1,..., n), mt .(2)

{ X t } mt . . . (deseasonalization). :1 ().

d = 2q, {xt , t = 1,..., n}, mt = {0,5 xt q + xt q +1 + ... + xt + q 1 + 0,5 xt + q } / d , t = 1, 2,...,

(1.4.21)

d = 2q + 1,

16

. . .

. 1.

mt = { xt q + xt q +1 + ... + xt + q 1 + xt + q } / d , t = 1, 2,....

(1.4.22)

mt =

j = q

a j xt + j

q

j = q

a

q

j

= 1.

mt , t = 1, 2,..., st , 2

q j = q

a j st + j = 0, t.

wl = 1 K

{xk =1

K

( k 1) d + l

m( k 1)d +l } , l = 1,..., d .

d l =1 wl = 0.

wl w, w wl , l = 1,..., d . :1 d wl d l =1

sl = wl w = wl

l = 1,..., d ,

(1.4.23)

st = sl

l = t mod d , t = 1,..., n.

(1.4.24)

3

st

17

. . .

. 1.

dt = xt st ,

t = 1,..., n.

(1.4.25)

mt . Yt = xt mt sl , t = 1,..., n,

(1.4.26)

.(3) d

st = st + d , t = 1, 2,..., X t X t d , t = d , d + 1,..., n, st . , d (difference operator at lag d) d = I B d

X t = mt + st + Yt , t = 1, 2,..., d X t = X t X t d = d (mt + st + Yt ) = mt mt d + Yt Yt d , t = d + 1, d + 2,.... { d X t : t = d + 1, 2,...},

t = d mt = mt mt d , t = d + 1, 2,..., , k, k = ( I B)k {t , t = d + 1,...}, k.

18

. . .

. 1.

1.5.

. 1: g : (, ) n = (1 ,..., n ) ' n

i , j =1

g (i j ) i

n

j

0 (, = 0).

g :

.

i , j =1

g (t t )i i j

n

j

0 (, = 0)

n , = (1 ,..., n ) '

ti

(i = 1,..., n).

1: ( ).

g .:

() g : { X n : n } . g ( h) = Cov ( X n , X n + h ) = Cov ( X n h , X n ) = g ( h), h .

(1.5.1)

= (1 ,..., n ) '

n

t = (t1 ,..., tn ) ' Z n

n 0 V i X ti = V [ ' X ] = ' n , i =1

19

. . .

. 1.

n = ij i , j =1 (nxn)- ij = Cov( X ti , X t j ) = g (| ti t j |), i, j = 1,..., n,

n

(1.5.2)

{ X n : n ()

}.

g : , . 0 (

) n = ij , ij = g (| ti t j |), i , j =1 (i, j = 1,..., n) .. X ti (i = 1,..., n), n N t = (t1 ,..., tn ) ' R n . n

n

.. { X t : t R}. Kolmogorov. . , n .. X t ( n ) = ( X t1 ,..., X tn ) ' n (0, n ), n = ij , i , j =1 ij = g (| ti t j |), i, j = 1,..., n, (. 1.7 ) .. 1 t (n) ( u ) = exp u ' n u , u 2 n n

.

(1.5.3)

, m < n, m- .. X t ( m ) = ( X t1 ,..., X tm ) ' m (0, m ), m = ij , ij = g (| ti t j |), i, j = 1,..., m, .. i , j =1 1 t (m) ( u ) = exp u ' m u , u 2 m m

.

(1.5.4)

Kolmogorov . :

20

. . .

. 1.

t ( m ) ( u) = t ( n ) ( u, 0), u

m

m, n m

m < n,

(1.5.5)

u

:0 = u ' m u. m

u (u ', 0 ') n = (u ', 0 ') m 0 0

.. 1.6. , X i (i = 1,..., n) i = E[ X i ] , ii i2 = V [ X i ], i = 1,..., n, , ij = Cov[ X i , X j ]

(i j = 1,..., n). . () (..) X = ( X 1 ,..., X n ) ' E [ X1 ] = E[X ] = . E [ X n ]

(1.6.1)

() (dispersion matrix) .. X = ( X 1 ,..., X n ) ' ( n n) - = D[ X ] = Cov( X i , X j ) .

(1.6.2)

(n m) X = [ X i , j ],M = E[ X] = E ( X ij ) .

(1.6.3)

X = ( X 1 ,..., X n ) ' Y = (Y1 ,..., Ym ) ', .

21

. . .

. 1.

Cov ( X i , Y j ) i j. (n m) XY = [Cov ( X i , Y j )],

.. X Y . XY = Cov(X,Y) = E[( X - E[ X ])(Y - E[Y ]) '].

(1.6.4)

.. X .. X , D[ X ] = Cov( X , X ) = XX . : XY = E[ XY' ] E[ X ]E[Y ]'

(1.6.5)

' YX = Cov (Y , X ) = [Cov (Y j , X i )] = [Cov ( X i , Y j )]' = Cov( X , Y ) ' = XY .

(1.6.6)

. 1. X n .. . Y = + BX , = ( 1 ,..., m ) ' m B (m n) -. () () : () , E[Y ] = + B, D[Y ] = B B'.

(1.6.7)

E[Y ] = E[ + BX ] = + BE[ X ] = + B.()

D[Y ] = E[(Y - E[Y ])(Y - E[Y ]) '] = E[ B ( X - )( X - ) ' B '] = BE[( X - )( X - ) ']B ' = BB '.

22

. . .

. 1.

2.

Cov(a + BX , c + DY ) = BCov[ X , Y ]D ',: Z = + BX W = c + DY () :

(1.6.8)

Cov(a + BX , c + DY ) = E[( E[ Z ])(W E[W ]) '] = E[ B ( X E[ X ])(Y E[Y ]) ' D '] = BE[( X E[ X ])(Y E[Y ]) ']D ' = BCov[ X , Y ]D '. . Z = + bX W = c + dY .. Cov( Z ,W ) = bdCov( X , Y ). 3: = D [ X ] . .. ,

Cov( X , Y ) = E[( X X )(X Y )]= Cov(Y , X ). () 1, = 0 B = b ' = ( b1 ,..., bn ) ' 0 '. .. Y = b ' X :

0 Var[Y ] = Var[b ' X ] = b ' b. 4: ( n n )-

= PP ', P ( n n )- = diag ( 1 ,...., n ) j 0

(1.6.9)

( j = 1,..., n )

. j > 0

( j = 1,..., n ) .

23

. . .

. 1.

: . j P j ( j = 1,..., n ) 1.7. . , X = ( X 1 ,..., X n ) ' ...f ( x) = 1 (2 )n/2 1/ 2

||

1 exp Q ( x ) 2

(1.7.1)

Q( x ) = 1 ( x - ) ' 1 ( x - ), 2

x

n

.

(1.7.2)

( | | 0). : 1.

X = ( X 1 ,..., X n ) '

n (n m) - X = + BZ , Z = ( Z1 ,..., Z m ) ' Zi , i = 1,..., n, (0,1). .. Z ... : 1 m f Z ( z ) = (2 ) m / 2 exp zk2 , z 2 k =1 m

,

(1.7.3)

[ ] = 0 D[ ] = I n , (n n) -.

24

. . .

. 1.

, .. X :

=E[X ]= = D[ X ] = BB '.

(1.7.4)

1. n > m. =D[ X ] , k m < n. X n k k ( ). 2. (..) (t ) .. X = ( X 1 ,..., X n ) ' (t ) = E ei t ' X , t n

i = 1.

(1.7.5)

.. (t ), t

n

, ( t ) ( 0 ) = 1.

3. . ' ' X = ( X 1 , X 2 )' , .. X k nk , k = 1, 2, n1 + n2 = n, .. X1 :1 ( t1 ) = E ei t '1 X1 = E ei ( t ' X1 + 0 ' X 2 ) = ( t1 , 0 ) , t1 n1

.

(1.7.6)

4. . :

X k , j = 1,..., n, .. .. .. X = ( X 1 ,..., X n ) ' .. n n n ( t ) = E ei t'X = E exp i t j X j = E exp {it j X j } = j (t j ), t j =1 j =1 i =1

n

.

, ..

25

. . .

. 1.

.. .. .. .. .. . .. .. : 1 ( t ) = exp t 2 , t . 2

(1.7.7)

:

1. .. .. Z = ( Z1 ,..., Z m ) ' Z j , j = 1,..., m, :m m 1 m 1 (t ) = j (t j ) = exp t 2 = exp t 2 , t j j 2 j =1 j =1 2 j =1 m

.

(1.7.8)

, B (n m) - X .. m- .. X = ( X 1 ,..., X m ) '. .. n- .. Y = + BX 2. n

Y ( t ) = ei t ' X ( B ' t ),

t

n

.

(1.7.9)

: .. (. 1.7.5). 1: .. = ( 1 ,..., n ) ' :1 X (t ) = exp i t' - t' t , t n . (1.7.10) 2 > 0, .. X

f X ( x ) = (2 ) n / 2 1/ 2

1 exp ( x - ) ' 1 ( x - ) , 2

x

n

.

(1.7.11)

26

. . .

. 1.

: .. = ( 1 ,..., n ) ' X = + BZ = ,

BB ' = , Z = ( Z1 ,..., Z m ) ' Z j , j = 1,..., n, (0,1). 1 2 .. .. : 1 X (t ) = E ei t'X = ei t' Z ( B ' t ) = ei t' exp t ' BB ' t 2 1 = exp i t ' - t ' BB ' t , t n . 2

> 0,

= PP ' PP ' = I n , (n n) -, (n n) - j , j = 1,..., n. = PP ', = diag ( 1 ,..., n ) j > 0, j = 1,..., n, PP ' = I n .

1 : 1 = P1 P ' = Pdiag ( 1 1 ,..., 1 ) P ' n

(1.7.12) (1.7.13)

1/ 2 = P1/ 2 P ' = Pdiag ( 1 1/ 2 ,..., 1/ 2 ) P '. n

2, .. = 1/ 2 ( ) : (t ) = exp{t ' t / 2} = exp{ t 2 / 2} = exp{t 2 / 2}, t j jj =1 j =1 n n n

.

27

. . .

. 1.

.. E[ ] = 0 V [ ] = I n . .. Z j , j = 1,..., n, . .. f Z ( z ) = (2 ) n / 2 exp{ 1 n 2 1 z j } = (2 ) n / 2 exp{ 2 z ' z}, z 2 j =1n

.

x = + 1/ 2 z z = 1/ 2 ( x - ) J (z ) =| |1/ 2 0, .. X = + 1/ 2 1 f X ( x ) = f Z ( z ( x )) J ( x ) = (2 ) n / 2 | |1/ 2 exp{ ( x ) ' 1 ( x )}, 2

x

n

.

2: n .. X = ( X 1 ,..., X n ) ' c = ( c1 ,..., cn ) ' n .. Y = c ' X N (c ' , c ' c ) . :() 3 = 0 B = c '. () c n .. Y = c ' X N (c ' , c ' c ). .. .. :Y (t ) = E[exp{i tY }] = E[exp{i tc ' X }] = exp{i tc ' c ' c t 2 }, t , c 2 1n

.

t = 1 :Y (1) = E[exp{iY }] = E[exp{i c ' X }] = exp{i c ' c ' c}, c 2 1n

.

X (c ) = E[exp{i c ' X }] = exp{i c ' c ' c}, c 2 1n

,

28

. . .

. 1.

.. X N (c ' , c ' c ). ( ) . :

2.

X = ( X 1 ,..., X n ) 'n

Y = ' X , .

,

5. , X = ( X 1 ,..., X n ) ' N ( , ).

X = + 1/ 2

Z = ( Z1 ,..., Z n ) ', ..

Zi , i = 1,..., n, . 3: .. X = ( X 1 ,..., X n ) ' N ( , ) . .. X X n X = 1 1, X 2 n2 (n1 + n2 = n)

, n = 1 1 = 11 21 2 n2 k = E [ X k ] , k = Cov ( X k , X ) , 12 n1 22 n2

(k,

= 1, 2 ) . :

() .. X1 X 2 12 = 0. () 22 > 0, X 1 X 2 : 1 N 1 + 12 221 ( X 2 2 ) , 11 12 22 21 .

(1.7.14)

: () X1 , X 2

29

. . .

. 1.

Cov ( X 1 , X 2 ) = 0 12 = 0.

21 = 0, = 11 0 0 . 22

.. X :1 X ( t ) = exp i t' t' t 2 1 T T = exp i ( t1 , t 2 ) 1 ( t1T , t 2 ) 11 0 1 2 0 t1 22 t 2

1 T T = exp i ( t1T 1 + t 2 2 ) ( t1T 11t1 + t 2 22 t 2 ) 2 2 1 = exp i t T j t T jj t j , t1 n1 t 2 j j 2 j =1

n2

.

.. , 4, .. X1 , X 2 . () | 22 | 0 .. Y = X 1 1 12 221 ( X 2 2 ) ,

(1.7.15)

E [Y ] = 0.

30

. . .

. 1.

YY = D [Y ] = Cov (Y , Y ) =

= Cov ( X1 1 12 221 ( X 2 2 ) , X1 1 12 1 ( X 2 2 ) ) 22 1 = Cov ( X1 12 22 X 2 , X1 12 221 X 2 )

= Cov ( X1 , X 1 ) Cov ( X 1 , X 2 ) 221 21 12 221Cov ( X 2 , X1 ) + 12 221Cov ( X 2 , X 2 ) 221 21 = 11 12 221 21 ,

Cov (Y , X 2 ) = Cov ( X1 1 12 221 ( X 2 2 ) , X 2 ) = = Cov ( X1 12 221 X 2 , X 2 ) = Cov ( X1 , X 2 ) 12 221 Cov ( X 2 , X 2 ) = 12 12 221 22 = 12 12 = 0 .

, 0 11 12 221 21 Y X N , 2 0 2

0 , 22

(1.7.16)

.. Y X 2 . X 2 , .. X 1 E exp {i t ' X 1} | X 2 , : E exp {i t ' X 1} | X 2 = E exp i t ' Y + 1 + 12 221 ( X 2 2 ) | X 2

{

}

= exp i t ' 1 + 12 221 ( X 2 2 ) E exp {i t 'Y } | X 2 1 12 221 ( X 2 2

{ = exp {i t ' +

= E exp {i t 'Y } exp i t ' 1 + 12 221 ( X 2 2 ) | X 2

{

}

} ) } E exp {i t 'Y } ,

31

. . .

. 1.

Y X 2 . .. Y (1.7.16), 1 -1 E exp {i t 'Y } = exp t ' 11 12 22 21 t , 2

1 E exp {i t ' X1} X 2 = exp i t ' 1 + 12 221 ( X 2 2 ) t ' 11 12 221 21 t , 2 .. N 1 + 12 221 ( X 2 2 ) , 11 12 221 21 .

32