997 2002 5 Copyright c 997-2002 by Manabu Kano. All rights reserved.
3 2 4 2.... 4 2.2... 4 2.3 m... 7 2.4... 8 3 8 3.... 8 3.2... 9 4 0 4.... 0 4.2... 2 4.3... 3 2
2 0 Fig. z z 0 z z 2 z 2 PCA; Principal Component Analysis z z 2 z z z Fig. Ax,x 2 z A z AB z OB O z OB OA 2 OB 2 + AB 2 z 2 x, x 2 z x 2 Fig. x 3
OA 0 + 2 P {x p }p, 2,,P {x p } MM P {z m } z m w pm x p m, 2,,M 2 p z m m {w pm }p, 2,,P; m, 2,,M z {x p }p, 2,,P m {z m }m 2,,M {z m }m, 2,,m wpm 2 m, 2,,M 3 p 2 2. Sec. P N {x np}n, 2,,N; p, 2,,P { x p }p, 2,,P {x np } x np x np x p n, 2,,N; p, 2,,P 4 X x x 2 x P x 2 x 22 x 2P X...... x N x N2 x NP 5 2.2 z Eq.2 w w 2 w. w P 6 4
n x n x n x n2 x np 7 z t n t n w p x np p x n w 8 z t n N t t 2 t 9. t z σ 2 z t N N N N t N t Xw 0 N n t n N x n w n N P w p x np n p N w p x np V i, j v ij v ij p n 0 σz 2 N tt t N Xw T Xw w T Vw 2 0 V N N N XT X 3 N x ni x nj 4 n N n x ni x i x nj x j 5 5
v ij v ji V V T z Eq.3 σz 2 Lagrange Lagrange λ J w T Vw λ w T w 6 J w J w 0 J w J J w w 2. J w P 2 v p w p p 2 v 2p w p p 2λ. 2 v Pp w p p 2Vw 2λw w w 2. w P 0 7 V λi w 0 8 Lagrange λ det V λi 0 9 Lagrange λ z w V V P λ P z σz 2 w z σz 2 Eq.2 w Eq.8 Eq.2 Eq.8 w T w σz 2 w T Vw w T λw λ 20 z σz 2 V λ z σz 2 V w 6
2.3 m 2 {w m }m 2, 3,,M m z m m 2,,M σz 2 m {z m }m, 2,,m w m Eq.3 m w m m {w i }i, 2,,m V λ i I w i 0 2 { w T i w i j j 22 0 i j m w m Lagrange J m w T mvw m λ m w T m w m m µ i w T mw i 23 J m w m m z m m 2,,M {z m }m, 2,,m J m w m 0 i m J m 2Vw m 2λ m w m µ i w i w m i 0 24 w T j j, 2,,m Eq.22 w T j Vw m µ j 0 j, 2,,m 25 V V T Eq.2 w T j Vw m w T mvw j w T m λ jw j 0 j, 2,,m 26 Eq.25 µ j 0 j, 2,,m 27 Eq.24 V λ m I w m 0 28 Eq.2 m z m σz 2 m V m m m z m σz 2 m V m w m 7
2.4 V P N x npn, 2,,N; p, 2,,P x np x np x p σ xp 29 x p,σ xp p x p X R N X T X 30 m z m σ 2 z m R m m z m w m 0 x npn, 2,,N; p, 2,,P x np x np x p σ xp σ xp σ xpe σ xp 3 σ xpe p x p 3 3. P V V p, p v pp x p σx 2 p p p v pp trv 32 8
m σz 2 m V m λ m P σz 2 p p p trv 33 m m λ p C m λ m λ p p λ m trv 34 C m proportion % m m P m m C i i i λ i trv 35 accumulated proportion, cumulative proportion V R 80% 80% 3.2 factor loading m z m p x p r zmx p σ2 z mx p σ zm σ xp 36 9
σ zm,σ xp z m,x p σz 2 mx p N z m,x p z m : t m t 2ṃ.. Xw m 37 x p : t Nm x p x 2p. Xcp 38 x Np c p p 0. c p p. 0 P 39 c p z m x p σ 2 z m λ m σz 2 mx p N Xw m T Xc p N wt m XT Xc p Vw m T c p λ m w T mc p λ m w pm 40 r zmx p λ mw pm λm σ xp λm w pm σ xp 4 r zmx p λ m w pm 42 4 4. P N X λ i i, 2,,P 0
λ i { > 0, for i, 2,,r 0, for i r +,r+2,,p 43 r + r + P {x n }n, 2,,N r r {w i }i, 2,,r r i w i i i z i z i w pi x p 44 p i t i t i Xw i 45 MM P score matrix T t t 2 t M 46 loading matrix P w w 2 w M 47 T XP 48. loading matrix P 2. Eq.48 3. x new t new t new x new P 49 0 Sec.
4.2 P x n n, 2,,N z x n ˆx n ˆx n t n w T x n w w T 50 t n x n w x n M ˆx n ˆx n M t nm w T m 5 m P 3,M 2 Fig.2 Eq.5 loading matrix P ˆx n t n t n2 t nm P T N x n PP T 52 ˆX XPP T 53 M ˆX X ˆX X E E X ˆX X TP T X XPP T XI PP T 54 x n z 3 t n2 w 2 T z 2 e n x n - x ˆ n T T x ˆ n t n w + t n2 w 2 t n w T z Fig. 2 P 3,M 2 2
4.3 P N X i λ i w i N XT Xw i λ i w i i, 2,,P 55 N λ i λ i X T Xw i λ i w i i, 2,,P 56 λ i X T X i w i loading matrix P w w 2 w P 57 P Eq.58 X XPP T 58 σ i λ i i, 2,,P 59 σ 0 σ 2 Σ... 60 0 σ P u i Xw i i, 2,,P 6 σ i U u u 2 u P 62 XP UΣ 63 X UΣP T 64 Σ U P Eq.64 X P Eq.22 U U T U U T U i,j U T U ij U T U ij Xw i T Xw j σ i σ j w T i X T Xw j σ i σ j 3
λ j w T i σ i σ w j j { i j 0 i j 65 U X Eq.64 r X r σ i u i w T i 66 i u T i u j w T i w j { { i j 0 i j i j 0 i j 67 68 X T X XX T r λ i w i w T i 69 i r λ i u i u T i 70 i X T X XX T X T X XX T X T X P Σ 2 P T 7 XX T UΣ 2 U T 72. i w i X i σ i 2. i w i X T X i λ i 4