ПОЛУРЕКУРРЕНТНАЯ НЕПАРАМЕТРИЧЕСКАЯ ИДЕНТИФИКАЦИЯ УСЛОВНЫХ ФУНКЦИОНАЛОВ СЛАБОЗАВИСИМЫХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ

  • Published on
    05-Apr-2017

  • View
    216

  • Download
    4

Embed Size (px)

Transcript

  • 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 .

Recommended

View more >