例）多項式曲線フィッティング

2. Key word
p linear model
p (error function)
p (maximum likehood)
p over fitting
p (generalization)
p (regularization)

6. !" ∶ 1
$" ∶ )&'((2+!" + 0.3 )( 0,
$"
)&'((2+!

7. !, # $ %, &
)⇒ )*+(2.%
$ %, & /% ̂1

8. !(#, %) # %
, %
% '
!(#, %) = )* + ),# + )-#-
+ ⋯ + )/#/
= 0
12*
/
)1#1

9. !

10. ) (
!(#) -
#
!(#) =
1
2
(
)*+
,
-(.), #) − 1)
2

11. !

12. . ! = {$%}
0
M
i j j i
j
A w T
=
=å
1
( ) ,
N
i j
ij n
n
A x +
=
= å 1
( )
N
i
i n n
n
T x t
=
= å

13. !(#, %) = (
)*+
,
-)#) , .(%) =
1
2
(
1*2
3
!(#1, %) − 51
6
1 0 0
( )
{ }( ) 0
N M M
j j
j n n j n
n j ji
E w
w x t w x
w = = =
¶
¢= - =
¶
å å å
1 0
{ } 0
N M
j i
j n n n
n j
w x t x
= =
- =å å
1 0 1
0
N M N
j j i
j n n n
n j n
w x t x+
= = =
- =åå å
1
( ) ,
N
i j
ij n
n
A x +
=
= å 1
( )
N
i
i n n
n
T x t
=
= å
0
M
i j j i
j
A w T
=
=å
0 000 0
0
M
N NM M N
w TA A
A A w T
æ ö æ öæ ö
ç ÷ ç ÷ç ÷
=ç ÷ ç ÷ç ÷
ç ÷ç ÷ ç ÷
è øè ø è ø
!
" # " " "
$
7 8

14. ! ", $ = &
'()
*
+'"'
$∗
-
., / ! ", $

15. !(#, %∗
) = )* !(#, %∗
) = )* + ),#
-.
)sin(23#

16. !(#, %∗
) = )* + ),# + )-#-
+ ).#. !(#, %∗
) = /
01*
2
)0#0
)sin(27#
89

17. !
"(!∗
) = 0

18. !(#, %)

19. !"#$ = 2!((∗ ⁄) ,
!((∗
(∗ :
:
=
1
N
/
012
3
4(50, (∗) − 80
9
)

22. 3 4
.
4
!"( . 3
3 )

23. ‖"‖ ≡ "$" = "&
'
+ ")
'
+ ⋯ + "+
'
,-(/) =
1
2
3
45)
6
7(84, /) − ;4
'
+
<
2
‖/‖'

24. !
! = e$%&
'((*) =
1
2
.
/0%
1
2(3/, *) − 6/
7 +
!
2
‖*‖7
M = 9

25. !
M = 9
! = e$ = 1
&'()) =
1
2
,
-./
0
1(2-, )) − 5-
6 +
1
2
‖)‖6

26. !∗
#
$%(') =
1
2
,
-./
0
1(2-, ') − 5-
6 +
#
2
‖'‖6

27. !"#$ %
%

28. - )(
) )

31. ( ) ( ) ( )f ax y af x f y+ = +

32. !(#, %) #
)!('x + *, %) = '!(#, %) + !(*, %
= '! #, % + ! *, %
= ',- + ',.# + ',/#/ + ⋯ + ',1#1 + ,- + ,.* + ,/*/ + ⋯ + ,1*1
= ! '# + *, %
= ,- + ,. '# + * + ,/('# + *)/+ ⋯ + ,1('# + *)1
) .!(#, %) #

33. ! ", $% + '
= ($*+ + ,+) + ($*. + ,.)" + ($*/ + ,/)"/ + ⋯ + ($*1 + ,1)"1
)!(", $% + ') = $!(", %) + !(", '
$! ", % + ! ", '
= $*+ + $*." + $*/"/ + ⋯ + $*1"1 + ,+ + ,." + ,/"/ + ⋯ + ,1"1
!(", %) % .
!(", %) %

34. 1 0
( )
{ } 0
N M
j i
j n n n i
n ji
E w
w x t x w
w
l
= =
¶
= - + =
¶
å å
!
!"($) =
1
2
)
*+,
-
.(/*, $) − 2*
3 +
5
2
‖$‖3
0
M
i j j i i
j
A w w Tl
=
+ =å
1
( ) ,
N
i j
ij n
n
A x +
=
= å 1
( )
N
i
i n n
n
T x t
=
= å
ij ij ijA A Il= +!
0
M
ij j i
j
A w T
=
=å !
0 000 0
0
M
N NM M N
w TA A
A A w T
l
l
+ æ ö æ öæ ö
ç ÷ ç ÷ç ÷
=ç ÷ ç ÷ç ÷
ç ÷ç ÷ ç ÷+è øè ø è ø
!
" # " " "
$