# ГЕНЕРИРОВАНИЕ ГАУССОВЫХ МАРКОВСКИХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ

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• 4, 2006

4

519.2

. . . . . . . . . . , . , . . . . . . . . . . ,

. 2n1 n . . n .

Correlation matrices and precision matrices of Markov Gauss sequences are analyzed. We provethat the precision matrix of an order n Markov sequence has the 2n1diagonals property. NonMarkovprocesses are approximated by Markov sequences via retaining the principal diagonals of the precisionmatrix. Also, we develop some procedures to generate order n Markov sequences.

[1]. , ' . ' . ' (') n' , ' ' ' MATLAB' '.

,X XX M B (p 1) ' . MX ' BX [2, 3]

1 1| , ..., | , ..., ;

k k p

T T TX k p k x x x xm m m mM M M

11 12

21 22

,XB B

BB B (1)

B11, B22 k pk , ' M '

B | :p k kf X X

;p k k kM M C X M (2)

22 12;B B CB (3)

121 11.C B B (4)

BX ' X, t ' ,

2 1

2

2 2 3

1 2

1 ...

1 ...

,1 ...

... ... ... ... ...

... 1

p

p

pX

p

b b b

b b b

b b b

b b b

B (5)

b=k( t) . , (5) ' (1), B22 = b22 =

2, , B11:

• 4, 2006 5

2

2 21

11 2 2

1 0 0 ... 0

1 0 ... 0

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1 0 0 1 0

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b

b b b

b b b

b b b b

b

B (6)

.

2 1 221 ... ,

p pb b bB

(4), ' (2):

121 11 0 0 ... 0 bC B B

, b, p2 . ' ' [1].

xp1, xp ' x1, , xp2

2 32

21 1 2 2

...,

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222

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2

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b

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(7), , ' xp .

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(5):2 = 1. |p k kX X

(Xpk Xp)

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2 4 5 3 3

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,...

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1,

1

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(7). ' m < pk B11,. . (8) ' .

(3) ' ('

• 4, 20066

, [2, 3]). ' 1XD B ' (6).

11 12

221

,XB B

BB

11 12

21 22d

D DD

D

( )

11 11 12 21 11 12 12 22

2 221 11 21 21 12 22

11 12

21

,1

X

d

d

B D B D B D BB D

B D D B D

I II

I

I11 (p 1) (p 1)'; I210 0 ... 0 0 ; 12 21.

TI I '

221 12 22 1mdB D

2 21 12

22

1,

d

B D

(3) xp

2 2 2 11 1 21 11 12| , ..., p px x B B B

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22 22

1.

d

d d

B D B B B (9)

11 12 22 12 12 0dB D B I

112 22 11 12.dD B B (10)

(10) (9)

1 12 22 21 11 12 22 21 11 12

22 22 22

1 1,p

d d

d d d

B B B B B B(11)

'.

n

n' ' ' [1]

1 1

1

, ..., | , ...,

, ..., | , ..., ,

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k p k n k

f x x x x

f x x x x (12)

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11 1

22 1

1

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0 ... 0 ....

... ... ... ... ... ...

0 ... 0 ...

kk n

kk n

p k k n p k k

c c

c c

c c

C (13)

, ' ' (2)

7 1 6 4 4 5 5 6 6| , ..., ,x x x c x c x c xCX

(13)

4 5 60 0 0 ,c c cC 4 0,c 5 0,c 6 0.c

. ' n' 2n+1 () , ' .

, n' ' (13) , ' 2n 1 :

11 12 1 1

21 22 2 2

1 1 1 2

2 2

1

... 0 ... 0

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0 ... ... ... ...

0 ... ... ... ... ... ...

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n

n

n n

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. , ' p n', k , ' (4) (3) pk

122 21 21,C D D BD (15)

122,B D (16)

D21, D22 p k k p k p k D. , ' DB=I

21 12 22 22 ,D B D B I

I p k p k .

• 4, 2006 7

122 22 21 12 ;B D I D B

1 122 22 22 21 12.D B D D B (17)

(16) (3) (15) (17):

122 22 21 12 22 12.B B D D B B CB

(15) (14), ' (14) 2n+1 . ' 122D ' (14), D21 ' . , ' (' ) :

0 0 0

0 0

0

,

0

0 0

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D

21

0 0.

0 0 0D

(15)

0 0

0 0C

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. ' (15) ,

11 11 12 21 1 1... 0;p k p kb d b d b d

21 11 22 21 2 1... 0;p k p kb d b d b d (18)

. . . . . . . . . . . . . . . . . . . . . . . . . . .

11 211 2 1... 0,p k p k p k p k p kb d b d b d

' di1. ' , ' B , .

(18) , ' . . , . , (13) ' , ' D21. 2n+1' ' .

' n' ' k ' (12): k = n. k

• 4, 20068

n' ) Xi m=1, 2,..., p n , 1

niX '

n . ' Bi (3) (2) Mi. m=1 (, . .) m = 2, 3,... . Bi Mi ' ' (2), (3), (11),(16), (17) (15), (19), (20) ' .

', Z ' ,i iX B . RANDN[4] Y, Y :

,dZ A Y 1/2 ,d Y YYA U U

UY ; Y ,YB ' COV [4]. I J ' Y Z , J , I n. ', I=10 J. '

X AZ , 1/2 TX XXA U U , (21)

UX ; X Bi (16) m . m ' n (2), (4),(15), (19) (20). m ' m.

' [6]. ' ' X ' BX DX D , ' 2n+1 . D 1nB D ' n' ( ').

0ijd , , ,

1 .n XB D B

1. ' R( ) = exp( /2)cos ,

= 0,45, '

1,0000 0,1249 0,6064 0,2312 0,3289

0,1249 1,0000 0,1249 0,6064 0,2312

0,6064 0,1249 1,0000 0,1249 0,6064

0,2312 0,6064 0,1249 1,0000 0,1249

0,3289 0,2312 0,6064 0,1249 1,0000

XB

1,6950 0,2934 1,0583 0,0809 0,0063

0,2934 1,7458 0,4769 1,0403 0,0809

.1,0583 0,4769 2,4027 0,4769 1,0583

0,0809 1,0403 0,4769 1,7458 0,2934

0,0063 0,0809 1,0583 0,2934 1,6950

XD

d14 d25 d41 d52 0,0809,d15 d51 0,0063 , D

2

0,9788 0,0999 0,5950 0,1644 0,3430

0,0999 0,9578 0,1055 0,5696 0,1644

.0,5950 0,1055 0,9822 0,1055 0,5950

0,1644 0,5696 0,1055 0,9578 0,0999

0,3430 0,1644 0,5950 0,0999 0,9788

B

2. X '

1

exp cos sin ,R

1/2. X ' ,

2

4,3 6,0 4,4 0 0 0 0

6,0 12,8 12,2 6,1 0 0 0

4,4 12,2 17,2 13,4 6,4 0 0

0 6,1 13,4 17,5 13,4 6,1 0

0 0 6,4 13,4 17,2 12,8 4,4

0 0 0 6,1 12,2 12,8 6,0

0 0 0 0 4,4 6,0 4,3

D

' 12 2 ;B D B2 : 4 0,8761, 5 0,7047.

• 4, 2006 9

3. X ' . 1.

, , .

(19) .

n C

3 0 0 0 0682.0 4200,1 8683,1

4 0 0 3180,0 0682.0 4200,1 8683,1

5 0 9120,0 3180,0 0682.0 4200,1 8683,1

' ' '.

4. exp 2 cos ,R ' 1/2, ' X . X ' , ( 7 10 000 ). (15) '

0 0,0188 0,0666 0,1897 0,5053.

0 0,0095 0,0572 0,1627 0,0656C

. 2 .

' . 2n+1'' n' ' ' . ' , ' . ' .

1. . ., . . '. .: . , 1977. 488 .

2. . ..: , 1974. 491 .

3. . . . ' . .: , 1985.640 .

4. . . MATLAB: . ..: , 1997. 350 .

5. . . ' : . . // . . . II. . / . ., 2006.. 257260.

6. . . . .:. , 1966. 679 .

. 1. !:1 ;24 ! , ! , 10000

. 2. :1 , 2 , 3 4 !