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常検知と変化検知 8 章 ガウス過程回帰による常検知 @progranate

第8章 ガウス過程回帰による異常検知

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  • 8 @progranate

  • x, yD D={(x(1), y(1)), , (x(n), y(n))} * xMy

    x: y:

    2

  • x: y:

    x

    y

    0.7 v

    3

  • x: y:

    x

    y

    0.7 v

    f(x):

    y

    f

    4

  • 5

  • f(x)p(f(x)|D)

    2 1. p(y|x,2) 2. f(x) p(fN)

    6

  • 1: p(y | x,2) y:

    y f(x) 2

    p(y x, 2 ) = N y f (x), 2( ) (8.1)

    7

  • 2: f(x) p(fN) (1/2) x, x2

    x, xf(x), f(x)f(x)f(x)

    pf (x)f (x ')

    !

    "##

    $

    %&&= N 0,

    K(x, x) K(x, x ')K(x ', x) K(x ', x ')

    '

    ())

    *

    +,,

    !

    "

    ##

    $

    %

    &&

    K(x, x): xx

    (8.3)

    8

  • 2: f(x) p(fN) (2/2) x(1), , x(N)N

    x(1), , x(N)f(x(1)), , f(x(N))fN

    p( fN ) = N( fN | 0,K ) fN = ( f (x(1) ),, f (x (N ) ))T

    K: (i, j)K(x(i), x(j))

    *

    (8.5) (8.4)

    9

  • N(fN | 0,K)f(x)

    8.2 xN50[-5, 5]f(x)

    fN

    10

  • 11

  • 2D

    p(y | x, D, 2)

    p(y | x, D, 2)

    p(y | x,D, 2 ) = dfN(y | f (x), 2 )

    p( f (x) |D)f(x) 2

    1: DfNp(fN | D) 2: p(fN | D)p(f(x) | D)

    (8.2)

    12

  • 1: p(fN | D) (1/4)

    p(D|fN,2) {y(1), y(N)}

    fN

    p(fN)

    p( fN D) =p(D fN ,

    2 )p( fN )d f 'N p(D fN ,

    2 )p( fN ')

    p(D fN ,2 ) = N(y(n) f (n), 2 ) = N(yN fN ,

    2IN )n=1

    N

    p( fN ) = N( fN 0,K )

    yN y(1),, y(n){ }

    (8.6)

    (8.7)

    (8.5)

    13

  • 1: p(fN | D) (2/4) p(fN | D)

    2

    p(x|y)p(y)

    p(y | x) = N(y | Ax + b,D)p(x) = N(x |,)

    p(x | y) = N(x | M ATD1(y b)+1{ },M)p(y) = N(y | A+b,D+ AAT )

    M ATD1A+1( )1

    (8.8)

    (8.9)

    (8.10)

    (8.11)

    (8.12)

    14

  • 1: p(fN | D) (3/4)

    p(y | x) = N(y | Ax + b,D)p(x) = N(x |,)

    p(x | y)

    = N(x | M ATD1(y b)+1{ },M)

    M ATD1A+1( )1

    p(D | fN ,2 ) = N(yN | fN ,

    2IN )p( fN ) = N( fN | 0,K )

    p( fN |D,2 )

    = N fN MIN 2IN( )

    1yN( ),M( )

    = N fN1 2MyN ,M

    "

    #$

    %

    &'

    M 1 2IN +K

    1#

    $%

    &

    '(1

    y yN ,A IN ,b 0,D 2IN , 0, K

    p( fN D) = p(D fN ,2 )p( fN )

    (8.13)

    15

  • 1: p(fN | D) (4/4) MM 1

    2IN +K

    1#

    $%

    &

    '(1

    A+BDC[ ]1 = A1 A1B D1 +CA1B!" #$

    1CA1 (8.14)

    M 1 2IN

    !

    "#

    $

    %&1

    1 2IN

    !

    "#

    $

    %&1

    IN K + IN1 2IN

    !

    "#

    $

    %&1

    IN!

    "##

    $

    %&&IN

    1 2IN

    !

    "#

    $

    %&1

    = 2 IN 2 K + 2IN( )

    1( ) (8.16)

    (8.17)M 2K K + 2IN( )1

    (K+2IN)

    16

  • 1fNp(fN | D)

    2fNyN p(fN)

    D=(yN)

    p( fN |D,2 ) = N fN

    1 2MyN ,M

    !

    "#

    $

    %&

    M 2K K + 2IN( )1

    (8.13)

    (8.17)

    17

  • 2: p(f(x) | D) (1/5) p(fN | D)p(f(x) | D) p(fN | D): N p(f(x) | D): xf(x)

    p(f(x) | D)

    p( f (x) |D) = d fN p( f (x) | fN )p( fN |D)fN f(x)

    1

    (8.18)

    p(f(x) | fN)p(f(x), fN)

    18

  • 2: p(f(x) | D) (2/5) f(x)fN (8.5)

    pf (x)fN

    !

    "##

    $

    %&&= N 0,

    Ko kT

    k K

    '

    ())

    *

    +,,

    !

    "

    ##

    $

    %

    && (8.19)

    k = K x, x (1)( ),,K x, x (N )( )( )T

    Ko = K x, x( )

    19

  • 2: p(f(x) | D) (3/5)

    x

    xN(x| , ) xbxaN(xa|a|b, a|b)

    x =xaxb

    !

    "

    ##

    $

    %

    &&

    =ab

    !

    "

    ##

    $

    %

    && =

    aa abba bb

    "

    #

    $$

    %

    &

    ''

    a|b = a +abbb1 xb b( )

    (8.20)

    a|b = aa abbb1ba

    (8.21)

    (8.23)

    20

  • 2: p(f(x) | D) (4/5) fNf(x)

    (8.21)(8.23)

    x =xaxb

    !

    "

    ##

    $

    %

    && =

    ab

    !

    "

    ##

    $

    %

    && =

    aa abba bb

    "

    #

    $$

    %

    &

    '' f =

    f (x)fN

    !

    "##

    $

    %&& = 0

    0

    !

    "#

    $

    %& =

    Ko kT

    k K

    "

    #$$

    %

    &''

    a|b = kTK1( fN 0) = k

    TK1 fNa|b = Ko k

    TK1k

    p f (x) fN( ) = N f (x) kTK1 fN ,Ko kTK1k( ) (8.27)

    21

  • 2: p(f(x) | D) (5/5) (8.18)p( f (x) |D) = d fN p( f (x) | fN )p( fN |D) (8.18)

    N f (x) kTK1 fN ,Ko kTK1k( ) N fN 1 2 MyN ,M

    !

    "#

    $

    %&

    p f (x) D( ) = N f (x) f (x ), 2f (x )( ) f (x) = k

    T K + 2IN( )1yN

    2f (x) = Ko kT K + 2IN( )

    1k

    (8.28)

    (8.29)

    p(y | x) = N(y | Ax + b,D) p(x) = N(x |,)p(y) = N(y | A+b,D+ AAT )

    22

  • p(y | x) = N(y | Ax + b,D) p(x) = N(x |,)p(y) = N(y | A+b,D+ AAT )

    p(y | x, D, 2)p(y | x,D, 2 ) = dfN(y | f (x), 2 )

    p( f (x) |D)N f (x) f (x ),

    2f (x )( )

    f (x ) = kT K + 2IN( )

    1yN

    2f (x ) = Ko kT K + 2IN( )

    1k

    p y x,D, 2( ) = N y y (x), 2y (x)( )y (x) = k

    T K + 2IN( )1yN

    2f (x) =2 +Ko k

    T K + 2IN( )1k

    (8.31)

    (8.32)

    (8.30)

    y(x)x

    23

  • 24

  • T2

    T2 = a(x ') = (x ' )T 1(x ' )

    =1N

    x (n)n=1

    N

    (2.9)

    =1N

    (x (n) )(x (n) )Tn=1

    N

    a(y ', x ') = log p y ' x ',D, 2( )=12log 2 y

    2 (x '){ }+ 12 y2 (x ')y 'y (x '){ }

    2

    (8.33)

    25

  • T2

    T2 =

    a(y ', x ') = log p y ' x ',D, 2( )=12log 2 y

    2 (x '){ }+ 12 y2 (x ')y 'y (x '){ }

    2

    a(x ') = (x ' )T 1(x ' )

    =1N

    x (n)n=1

    N

    (8.33)

    (2.9)

    =1N

    (x (n) )(x (n) )Tn=1

    N

    x

    26

  • 8.3 8.250

    50 : (x, y)={(-4, -2), (-2.8, 0), (-1, 1), (0, 2), (2.2, -1)}

    27

  • 28

  • 2 2

    2 E(2|D)2

    E( 2 D) d fN p D fN ,2( ) p( fN )

    (8.11)

    E( 2 D) N yN 0,2IN +K( )

    (8.36)

    (8.37)

    29

  • 2 2

    -2

    K = 2 !K

    logE( 2 D) N2log(2 2 ) 1

    2log IN + !K

    2

    2yNT IN + !K( )

    1yN

    2 1NyNT IN + !K( )

    1yN

    K p103

    (8.38)

    (8.39)

    30

  • 31

  • wikipedia

    :x : y N x

    D = (x (1), y(1) ),, (x (N ), y(N ) ){ }

    32

  • y ymin: DN []+0

    J(x) = dyp(y | x,D, 2 )

    ymin y[ ]+ (8.42)

    33

  • J(x) = dyN(y |y (x), y2 (x))

    ymin

    (ymin y)

    = duN(u | 0,1)(ymin u y (x)y (x))yminy y

    = y (x) z(z)+ N(z | 0,1)[ ]

    z =ymin y (x) y (x)

    (v) = du

    u N(u | 0,1)

    ddu

    N(u | 0,1) = uN(u | 0,1)

    J(x) = dyp(y | x,D, 2 )

    ymin y[ ]+

    (8.43)

    (8.44)

    34

  • z[]z

    yD8.3

    x z

    J(x) = y (x) z(z)+ N(z | 0,1)[ ]

    J(x) y (x) z(x)[ ]+

    (8.43)

    (8.45)

    35

  • 36

  • (1/2) 2

    y = xT = XXT + 2IM( )XyNX x (1),, x (N )"# $%

    yN X( )T yN X( )+ 2T

    2

    (8.46)

    37

  • (2/2) (8.14)

    , y

    = 2IN 4X IN +

    2XTX( )1XT{ }XyN

    k = XTx K = XTXy = 2kT IN

    2 2K + IN( )1K{ } yN

    = 2kT 2K + IN( )1

    2K + IN( ) 22K{ } yN= kT K + 2IN( )

    1yN y(x)

    38

  • 8

    39

  • N

    p y x,D, 2( ) = N y y (x), 2y (x)( )y (x) = k

    T K + 2IN( )1yN

    2f (x) =2 +Ko k

    T K + 2IN( )1k

    a(y ', x ') = log p y ' x ',D, 2( )=12log 2 y

    2 (x '){ }+ 12 y2 (x ')y 'y (x '){ }

    2

    40