PRML 多項式曲線フィッティング 1.1
- 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 ! " # " " " $