# ПОЛУРЕКУРРЕНТНАЯ НЕПАРАМЕТРИЧЕСКАЯ ИДЕНТИФИКАЦИЯ УСЛОВНЫХ ФУНКЦИОНАЛОВ СЛАБОЗАВИСИМЫХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ

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• 2008 , 1(2)

621.391.1:519.25

.. , ..

, -

.

. -

.

: , ,

.

, -

, .

, , .. -

Y X,

,

X = x.

, ,

, ..

,

, -

.

1- , -

. -

.

-

-

.

1.

, l -

1

( , , )l

X X X= Y , lX R , 1Y R ,

Rl l- .

• 58 .. , ..

Y -

F(X) X, .. Y=F(X). -

F . -

, -

, ,

() :

( ) ( )

( )

( )

( )

( )

1

1

,

| R

R

yf x y dya x

r x yf y x dyp x p x

= = = ,

( ),f x y (l + 1)-

( ),Z X Y= 1lR + , ( )( )

( )

,|

f x yf y x

p x=

1Y R , X x= , ( )p x -

X , ( ) ( )1

,

R

a x yf x y dy= .

, -

, -

. -

() :

( )

( )

( )( )

1

2

2

,

R

y f x y dy

D Y x r xp x

= .

-

, .

( ) kg y y= , k = 1, 2, 3, 4.

-

( ) ( ) ( )

( ) ( )

( )

( )

( )

1

1

,

R

R

g y f x y dyG x

J x g y f y x dyp x p x

= = = , (1.1)

( )g . , -

( ) ( ) ( )21 2D Y x J x J x= , ( )2

1g y y= , ( )2g y y= .

(1.1) x

[ ] ( )[ ] ( )

[ ] ( )

n

n

n

G xJ x

p x= , (1.2)

[ ] ( ) [ ] ( ) [ ] ( )( )

[ ][ ]

1 1

1

1 n nn n n l

nn k

k

g Y x XG x G x G x K

n hh

=

=

(1.3)

• 59

[ ] ( ) [ ] ( ) [ ] ( )

[ ][ ]

1 1

1

1 1 nn n n l

nn k

k

x Xp x p x p x K

n hh

=

=

( )G x ( )p x ;

( ),i i i

Z X Y= , ( ),1 ,

, ,i i i l

X X X= , 1,i n= , (l+1)- , -

( ) ( ),f z f x y= ; ( )K u ; [ ] 0n kh > -

, 1,k l= .

[1], , -

(1.2) ,

(1.2) .

, -

: , , ,

, -

, -

, .

2.

:

( ) ( )1

( ) ,rr

R

a x g y f x y dy= , ( ) ( )1

( ) ,rr

R

a x g y f x y dy+ = ,

r = 0, 1. ( ) ( )0p x a x=

( )1a x

[ ] ( ) [ ] ( )1n r n ra x a x= [ ] ( )

[ ]

( )[ ]

1

1

1 1 r nnn r l

nn k

k

x Xa x g Y K

n hh

=

, (2.1)

[ ] [ ] [ ]

1 ,1 ,

1

, ,

i l i li

n i i l

x X x Xx XK K

h h h

=

, r = 0, 1.

(2.1) .

n, n i i 1 n. -

[ ] ( )

[ ]

( )[ ]1

1

1 1n r iin r l

i ii k

k

x Xa x g Y K

n hh=

=

=

. (2.2)

(2.2)

( 1), ( 2), -

( 1) [ ] ( )n ra x [ ] ( )n sa x (-

3).

• 60 .. , ..

1. ( )ra x ,

( ) 1l

R

K u du = , ( ) 0K u , sup ( )l

r

x R

a x+

< ,

[ ] 0n kh 1,k l= ,

[ ]lim ( ) ( ) 0.rn rn

Ea x a x

= (2.3)

, [ ]nh

:

[ ] [ ] [ ]( )1

1 n q q qqki k n k n k

i

h S h o hn =

= + ; (2.4)

[ ] [ ] [ ]( )1

lq ql ql

qsi k i s i sk

h C h o h=

= + , 1,s l= , (2.5)

q ; qkS qsC , n.

2. ( )ra x l

R ,

( ) 1l

R

K u du = , ( )2

1l

k

R

u K u du = , 1, 2, ,k l= , ( ) ( )K u K u= , ( )2

21

supl

lr

kx R k

a x

x=

• 61

[ ]( )2

*( )

n ru a x ( )

( )( )

4

42 4

2 42 2

1 l

llr l

rs Rs

a xA a x K u du n

x

+

+

=

, (2.8)

( ) ( )( )

( )

4 22

44 2 4 34

44 4

4 2

4

44 2 4

4 2

ll

lll l

ll

l ll

l

l lA l l l

l

+

++ ++

++ +

+

+

+

= + + +

+

,

[ ]

( )

( )[ ]

2

2

2

2

r

l

i k i l

r

k

a x

xh h

a x

x

=

, 1,i n= , 1,k l= ; (2.9)

[ ]

( ) ( )( )

( )( )

( )

1

2 4

2 121 2 4

2 2

2

2 2

1

4 l

llr

rso s l

n k

Rr

k

a xl a x

xh K u du n

l l a x

x

+

=+

+

=

+

,

1,k l= . (2.10)

1 3 1 2 , -

[3].

3.

, 0,1, 2, ,lt

X R t = , -

( ), ,xF P , { },

,

x

s t uF X s u t=

- , ,u

X s u t . -

, 1, ,

nX X -

( ) ,

( ) ( ) ( ) ( )0, ,;

sup sup 0

t tt A F B F

P AB P A P B

+

= , 0 > .

( ) ( )1i i

X S .

1 2 ,

i

Z , 1,i n= , , , -

.

, ,

n -

, 1, ,

nZ Z .

• 62 .. , ..

4. ( ) ( )1i i

Z S , [ ]

0(

qm dm

< , 1

02

q< < ,

( ) 1l

R

K u du = , ( )2

1l

k

R

u K u du = , 1, 2, ,k l= , ( )2

lR

K u du < , ( ) ( )K u K u= ,

( )ra x lR , ( )sup

lr

x R

a x

< , ( ) ( )2supl

r

x R

a x+

+

< , 0 > ,

1 ,sup ( , )l

m rs

x R

a x x+

< 1m > . n

[ ] ( ) [ ] ( )( )cov ,n r n sa x a x =

( )

[ ]

( )

[ ]

( )

1 2

2 1

12

1

l

r s j

llRj n j

n j

a x SK u du o

C nhnh

+

+

+

= +

, 1,j l= . (3.1)

(. 1 1).

4.

:

( )( ) ( ) ( )4 4

11, , , , , ,

l

i i j i j ki i j i j kf z s u w P Z z Z s Z u Z wz s u w

+

+ + ++ + +

= <

4 4l + -

( )1, , ,i i j i j kZ Z Z Z+ + + , ( )( ) ( ) ( ) ( ) ( ) ( )1 1

1 ,, , ,

r r r ri i j i j k r

R R

a x y x y g v g s g v g s+

+ + + =

( ) ( )( ) ( )2

1, , , , , , ,

ri i j i j kg s f x v y s x v y s dvdsdv ds

+

+ + + ,

( )( ) ( ) ( ) ( ) ( ) ( )2

1 1 1 , 23 3

, ,

r r rj j k r

l

a x y x g v g s g v

++

+ + + +

+

=

( )( ) ( )1 1 1 , , , , ,j j kf x v y s x v dvdsdv+ + + .

[ ] ( )nJ x 3.

2. :

1) ( ) ( )1i i

Z S , [ ]

0(

qm dm

< , 1

02

q< < , ( ) 1l

R

K u du = ,

( )2 1l

k

R

u K u du = , 1, 2, ,k l= , ( )2

lR

K u du < , ( ) ( )K u K u= , ( )ra x -

lR , ( )supl

r

x R

a x

< , ( )( ) ( )1 ,sup , , ,l

i i j i j k r

x R

a x x x x+

+ + +

< , , , 1i j k , i j k n+ + ,

( )( ) ( ) ( )1 1 1 , 2sup , ,l

j j k rx R

a x x x+

+ + + +

• 63

2) [ ] ( )

[ ] ( )

1

0

n

n

a x

a x

-

[ ]( )nCd , [ ]nd n , 0 < , .

[ ] ( )nJ x

[ ] ( )

[ ] ( ) [ ]

( )

( )

( )

( )( )

21 12 22 1

2 3

0 1 0 0l

n j

ln Rj n j

a x S a x a xu K u du

a x C nh a x a x

= +

( ) ( )[ ]

22 2

21 0

24 2 21

0 0

1

4 4

l

k n kk

k

a x a xS h

a x a=

+ +

( )[ ]

( ) ( )[ ]

22 2

2 21 1 0

2 22 3 21 1

02

l l

k kn k n kk k

k k

a x a x a xS h S h

x a x= =

( )[ ]

[ ][ ]

22

2 21

2 21 1

1l l

k n k n jlk jk n j

a xS h o h

x nh= =

+ +

, 1,j l= . (4.1)

4 2

[2].

5.

p

{ }, 1,0,1t

X t = -

p:

( )1, , ,t t t p tX X X a = + , (5.1)

{ }t -

-

, , , , px y R ,

( ) ( ) 1 1 p py x y a x a x + + + , 10 1pa a< + + C < ( ) ( )x x dx C + ,

t

X -

( ) ( ) ( )aC a e , 0 > ( )C a < , ( ) 0a > ,

( )1, , pa a a= , ia (5.2).

1

, ,p n

X X

, (5.1).

(5.1)

• 64 .. , ..

[ ] ( )[ ]

[ ] [ ]

[ ][ ] [ ]

1 1

1 1

1

1 1

1 1

1

1, ,

1, ,

n p i piip

i i i pi k

kn p

n p i pi

pi i i p

i kk

x Xx XX K

h hh

xx Xx X

Kh h

h

=

=

=

=

=

. (5.3)

6.

1 ,

, [ ]o

nh [ ] ( )( )

2

n ru a x

( )2ra x , ( )ra x ( )2

2

r

k

a x

x

.

1,...,

nZ Z

[ ] 1( ,..., )o

nnh Z Z

[ ] ( )n ra x .

-

[ ] 1( ,..., )o

nnh Z Z [ ] ( )

[ ] ( )

[ ] ( )

1

0

n

n

n

a x

J xa x

= ( )J x

[ ] ( ) ( )[1

1 0

1

,..., argmin

so

n hs isi

h Z Z g Y

>

=

=

( )

[ ][ ]

( )

[ ][ ]

2

1

1 1

1

1 1

1 1

s j i j s i s

l lj i j

kj kk k

s i j i s

l lj i j

kj kk k

g Y X X g Y X XK K

h hsh hs

X X X XK K

h hsh hs

= =

= =

+ +

, 2,s n= . (6.1)

[ ] [ ] [ ]1 2, , ,o o o

nh h h

:

1) : [ ]1 2

1

11

4 2

X Xh h

= =

;

2) [ ] [ ] [ ]1 2, , ,o o o

nh h h (6.1) 2,3, ,s n= ;

n [ ] [ ] [ ]1 2, , ,o o o

nh h h .

[ ] ( )n x

p (6.2).

[ ] ( )n x

• 65

[ ] ( )[ ]

[ ] [ ]

[ ][ ] [ ]

11 1

1 11

1

11

1 11

1

, ,

1, ,

n p i pi i

pi i i p

i kk

n pn p i pi

pi i i p

i kk

x XX x XK

h hh

xx Xx X

Kh h

h

++

= +

+

=

+

= +

+

=

=

. (6.3)

[ ]o

sh (6.2) -

[ ] ( )1 1,...,o

ssh X X

+=

[ ][ ] [ ]

1 11 1 1

0 1

1 11

1

argmin , ,

s s j i j i p j p

hs i pi j i j j p

j kk

X X X X XX K

h hh

+ + +

> +

= +

+

=

= +

[ ][ ]

1 11

11

2

, , /

i p s ps i s

p

s ps k

k

X XX X XK

hs hhs h

+ ++

+

+

=

+

[ ][ ] [ ]

11 1

11

1

1/ , ,

s i j i p j p

pj i j j p

j kk

X X X XK

h hh

+ +

+

+

=

+

[ ][ ]

2

1 1

11

2

1, ,

i p s pi s

p

s ps k

k

X XX XK

hs hhs h

+ +

+

+

=

+

, 2,s n= .

7. 1-

{ }, 1, 2, ,100t

X t = -

( )1t t tX X = + , (7.1)

{ }t -

, ( ) / 25 tt e = + .

, (5.2) -

t

X .

(7.1) 6 -

{t} 0,0001, 0,1, 1, 10. -

.

• 66 .. , ..

. {t} - :

6 100

1 2

1 1

6 99t t

i t

R Xi Xi

= =

=

,

Xi , i- , Xi

, -

100 . Xi

6 100

i 1 1

1 1

6 100t

t

X Xi

= =

=

,

cp

100%R

X = .

. 1, -

, {t} -

.

1

D{t}

D{t} R 0,0001 0,011 0,218

0,1 0,2 3,95

1 0,522 10,442

10 1,293 24,813

.

6 100

1 81

1 1100%

6 20

t t

i t t

X X

X= =

=

. (7.1)

2

D{t}

D{t} 0,0001 0,1 1 10, % 0,2 5,6 20,7 146

. 2 : D{t}, .

t

Xi , 80,100t = ,

, D{t} 0,0001,

85%, 0,1 76,7%, 73,3%,

66,7%. , , -

.

• 67

8. -

(6.3) -

100 -

- 15

2006 . 17 2007 .

-

. :

1) 20 (6.3);

2) 21- ;

3) ,

, , ;

4) 21- ;

5) , 17 .

, 18 2007 . 17 2007 .

12,7%,

1,3%, 6,4%. 8,3,

50% , , -

1- .

, -

1- ,

2- 3- .

. 3,

t

X t, t

X

t, (t 1)- ,

t t

t

X X

X

t.

3

T Xt, .

tX , .

t t

t

X X

X

21 3705,1 3631 0,02

30 3834,53 3796,31 0,01

40 3852,67 3839,77 0,003

50 3398,42 3631,84 0,069

60 3771,8 3740,17 0,008

70 3976,24 3919,44 0,014

80 4125,75 4127,9 0,001

90 3955,54 3972,77 0,004

100 3930,47 3919,52 0,003

,

, .

, . -

. 1.

• 68 .. , ..

0,05

0,1

0 0,2 0,4 0,6 0,8

. 1.

, ,

.

:

1) -

, -

. , -

, .

2)

,

;

.

3)

.

4)

- -

50%.

1. Gyrfi L., Kohler M., and Walk H. Week and strong universal consistency of semi-recursivekernel and partitioning regression estimates // Statist. Decisions. 1998. V. 16. P. 1 18.

2. .., .. . .: ,, 1997. 336 .

3. .., .. // . 2003.

289. . 187 200.

4. .., .. //. . . . 1986. . 2(153). . 118 122.

5. .., .., .. - . .: , 2004. 508 .

-

, -

17 2007 .