ظ • ֬ม͕࿈ଓͷͱ͖ʢ֬ີؔʣ µi = E { xi } = Z dxi xip ( xi) µi = E { xi } = X k x (k) i P ⇣ x (k) i ⌘ • ֬ม͕ࢄͷͱ͖ʢ࣭֬ྔؔʣ µ = ¯ x = 1 N N X i=1 xi • ؍ଌσʔλ͕ N ݸ༩͑ΒΕ͍ͯΔͱ͖
पล֬ • ֬ม͕ࢄͷͱ͖ʢ࣭֬ྔؔʣ • ͍ͯ͠ΔಛྔͰͳ͍ͷͥΜͿʢੵʣ ΛͱΔ ϋϯόʔά͕͖͔ʁ yes no sum ΤϏϑϥΠ ͕͖͔ʁ yes 60 40 100 no 30 20 50 sum 90 60 p ( xi) = Z dx1 Z dx2 · · · Z dxi 1 Z dxi+1 · · · Z dxd p ( x1, x2, . . . , xd)
ϕΫτϧతͳղऍ • N ݸͷଌఆ͕͋Δͱ͖ɺ ij = 1 N N X n=1 ( xni µi)( xnj µj) = 1 N N X n=1 dnidnj = 1 N di · dj di = ( x1i µi, x2i µi, . . . , xNi µi)T = ( d1i, d2i, . . . , dNi)T ͱ͓͘ͱɺڞࢄੵʢͷఆഒʣʹͳΔ ϕΫτϧۭؒͱͯ͠ ѻ͏ͨΊʹ ֤ಛྔͰ୯Ґ͕ ἧ͍ͬͯΔඞཁ͕͋Δ
2 i = 1 N di · di = 1 N |di |2 i = = 1 p N |di | ϕΫτϧతͳղऍ ⇢ij = ij i j = (1 /N ) di · dj (1 / p N ) |di | (1 / p N ) |dj | = di · dj |di | |dj | = cos ✓ij
ฏۉɾࢄͱઢܗม • ઢܗมΛߟ͑Δ y = ax + b E { y } = E { ax + b } = a E { x } + b = aµ + b • ฏۉͱࢄҎԼͷΑ͏ʹԠ Var { y } = E ( y Ey )2 = E [ ax + b ( aµ + b )]2 = E a 2( x µ )2 = a 2E ( x µ )2 = a 2Var { x } = a 2 2
ແ૬ؔԽ y = ST x E { y } = E ST x = ST E { x } = ST µ Var { y } = E ( y E { y })( y E { y })T = E (ST x ST µ )(ST x ST µ )T = E ST ( x µ )[ST ( x µ )]T = E ST ( x µ )( x µ )T S = ST E ( x µ )( x µ )T S = ST ⌃S
ന৭Խ E { u } = E n⇤ 1/2ST ( x µ )o = ⇤ 1/2ST (E { x } µ ) = 0 Var { u } = E n⇤ 1/2ST ( x µ )( x µ )T S(⇤ 1/2)T o = ⇤ 1/2ST E ( x µ )( x µ )T S(⇤ 1/2)T = ⇤ 1/2ST ⌃S(⇤ 1/2)T = ⇤ 1/2⇤(⇤ 1/2)T = I • ඪ४Խ͞Εɺ͔ͭແ૬ؔԽ͞Εͨʂ
ਖ਼ن • ૬ؔͷ͚ͩճస͠ɺඪ४ภࠩͷ͚ͩҾ͖৳͞Ε͍ͯΔ • ന৭Խͷٯ ( x µ )T ⌃ 1( x µ ) = ( x µ )T [S⇤S 1] 1( x µ ) = ( x µ )T S⇤ 1S( x µ ) = [ST ( x µ )]T ⇤ 1[ST ( x µ )] = y T ⇤ 1 y (* y ⌘ ST ( x µ )) = y T (⇤1/2)T ⇤1/2 y = (⇤1/2 y )T (⇤1/2 y ) = z T z (* z ⌘ ⇤ 1/2 y )
ࣝผڥք fij( x ) = gi( x ) gj( x ) = ( x µi)T ⌃ 1 i ( x µi) + ln |⌃i | 2 ln P(Ci) ( x µj)T ⌃ 1 j ( x µj) ln |⌃j | + 2 ln P(Cj) = x ⌃ 1 i x x ⌃ 1 i µi µi⌃ 1 i x + µi⌃ 1 i µi x ⌃ 1 j x x ⌃ 1 j µj µj⌃ 1 j x + µj⌃ 1 j µj + ln |⌃i | ⌃j 2 ln P(Ci) P(Cj) = x (⌃ 1 i ⌃ 1 j ) x + 2( µ T j ⌃ 1 j µ T i ⌃ 1 i ) x + µT i ⌃ 1 i µiµT j ⌃ 1 j µj + ln |⌃i | ⌃j 2 ln P(Ci) P(Cj) ) x T S x + 2 c T x + F = 0 ʢ2 ࣍ࣝผؔʣ +µT i ⌃ 1 i µi µT j ⌃ 1 j µj + ln |⌃i | |⌃j | 2 ln P(Ci) P(Cj) fij( x ) = gi( x ) gj( x ) = ( x µi)T ⌃ 1 i ( x µi) + ln |⌃i | 2 ln P(Ci) ( x µj)T ⌃ 1 j ( x µj) ln |⌃j | + 2 ln P(Cj) = x ⌃ 1 i x x ⌃ 1 i µi µi⌃ 1 i x + µi⌃ 1 i µi x ⌃ 1 j x x ⌃ 1 j µj µj⌃ 1 j x + µj⌃ 1 j µj + ln |⌃i | ⌃j 2 ln P(Ci) P(Cj) = x (⌃ 1 i ⌃ 1 j ) x + 2( µ T j ⌃ 1 j µ T i ⌃ 1 i ) x + µT i ⌃ 1 i µiµT j ⌃ 1 j µj + ln |⌃i | ⌃j 2 ln P(Ci) P(Cj) x ⌃ 1 j x + x ⌃ 1 j µj + µj⌃ 1 j x µj⌃ 1 j µj
ࣝผڥք ⌃i = ⌃j = I P(Ci) = P(Cj) • ͔ͭ ͷͱ͖ fij( x ) = 2( µ T j ⌃ 1 j µ T j ⌃ 1 i ) x + µ T i ⌃ 1 i µi µ T i ⌃ 1 i µi = 2 ( µ T j µ T i ) x + µ T i µi µ T j µj = 0 x T x + 2 µ T i x + µ T i µi x T x + 2 µ T j x µ T j µj = 0 ( x µi)T ( x µi) ( x µj)T ( x µj) = 0 ) ( x µi)T ( x µi) = ( x µj)T ( x µj) x T x 2 µ T i x + µ T i µi x T x + 2 µ T j x µ T j µj = 0 • ྆ล σ ͰׂΓɺ x^T x Λ͠Ҿ͖ ʢ࠷ۙ๏……ͱຊʹॻ͍ͯ͋Δ͕ઢܗผੳʢLDAʣͰʁ ʢฏۉ͔ΒͷڑΛൺ͍ͯΔͷͰʣʣ +µT i ⌃ 1 i µi µT j ⌃ 1 j µj fij( x ) = 2( µ T j ⌃ 1 j µ T j ⌃ 1 i ) x + µ T i ⌃ 1 i µi µ T i ⌃ 1 i µi = 2 ( µ T j µ T i ) x + µ T i µi µ T j µj = 0
1 มਖ਼نͷ߹ @Lln(ˆ µ, 2) @µ = @ @µ " 1 2 2 N X i=1 ( xi µ )2 # µ=ˆ µ = 0 1 2 2 N X i=1 2( xi ˆ µ )( 1) = 0 N X i=1 xi N X i=1 ˆ µ = 0 ) ˆ µ = 1 N N X i=1 xi
1 มਖ਼نͷ߹ @Lln( µ, ˆ2) @ 2 = @ @ 2 " N 2 ln 2 1 2 2 N X i=1 ( xi µ )2 # 2=ˆ2 = 0 N 2 1 ˆ2 1 2 1 (ˆ2)2 ( 1) N X i=1 ( xi µ )2 = 0 N ˆ2 + 1 (ˆ2)2 N X i=1 ( xi µ )2 = 0 ) ˆ2 = 1 N N X i=1 ( xi µ )2
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