第8章 ガウス過程回帰による異常検知
TRANSCRIPT
-
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