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ě2ƌƎƢ

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Ǣ?ďǣ

2

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åőě2ƑT­#Oş

R= 1 r

r 1

⎛ ⎝⎜

⎞ ⎠⎟

R−

λ

_{I =}0⇔₍₁−

λ

)2

−r2=0

T­£ì€ƒ

∴

λ

₁₌1_+r,

λ

₂₌1−_r

Ra

1=

λ

1a1

T­ǏƱǂǛƒŨ ƦÊƄž

y1=

1

2(x1+x2), y2= 1

2(x1−x2) žƎƥƆ

ƦÊƄž ∴a

11=a12= 1

2

ĂuŨ”0ƸƴƨƒŨ

a₂ƜH¹ƏÂƛƠƣƢ

a₁₁₊ra₁₂₌(1+r)a₁₁ ra

ĖƑĕ»¹

”00²ƑĂ³

-2s.d. 平均 +2s.d.

PC1

PC2

PC3

PC4

*** 24.05

*** 57.28

*** 119.71

*** 48.27

品種効果

自由度174, 875

主成分

寄与率

(43.5%)

(15.0%)

(9.3%)

(4.5%)

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ƑĂ³ nÖǧ

F”0Ƒ ^9Ƒ Ĭ¦7

ǢsżúǣƍƑƟűƏżƊÝ%ƌżƊ”0Ƒ‘LƦêžźƌŵƋŶƢƑƋżƞűǩ

”0ˆÍƌƜƌƑ^¡Ƒ

ő!Ʀ4ÛžƢ

ǠǠǠ

z

iq

=

x

i T

u

q

(

z

_i1

,...,

z

_iq

)

₌

x

i T

(

u

1

,...,

u

q

)

°ƐƢƌ

z

i

T

=

x

_iT

U

⇔

z

_i

₌

U

T

x

_i

z

_i 1

=

x

i

T

u

1

UT_U= u

1 T_u

1  u1 T_u

q

  

u

q

T_u

1  uq

T_u

q

⎡

⎣ ⎢ ⎢ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎥ ⎥

=I ƌźƤƋŨ

ƎƑƋ

U

T

=

U

−1

⇔

UU

T

₌

I

∴

x

_i

₌

Uz

_i

=^8_H\ tW]% Se'Gu %

x_{i =}U k λ1

0 

0

⎛

⎝ ⎜ ⎜ ⎜ ⎜

⎞

⎠ ⎟ ⎟ ⎟ ⎟

?Y"% 18_H'jsCZ

kAxB'w

_%

T­Š

eigenfaces

Murphy KP (2012) "Machine Learning: A Probabilistic Perspective" The MIT Press.

ĖƑĕ»¹Ƒ

ūŐƑ’ģŬƦĮƗƢ

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102

Ə

30

NíƑĖƑ.

çƦū‘Ƒ¡Ƒ

ƲǛǡǎŬƏ0Ź

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30

Ní

•

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ŷŬ ƠƓƢƟű

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ƂźƋƒŨůƠŴŽƛĖƑÝ %ŐƋĕ»¹ƏZƉŷĶŗƦ ÂƛƊųŶŨFĖÝ%ŴƠ« ƜĽŰ3ƑĖÝ%ƘƋƑĶŗ ƑĆMƦ«a:žƢƟűƏ ū«ń:ŬŵěƥƣƘżƄũ

’ģƦ¡#ƋĜž

Ǣš~ǣ

•

Ní

A

ƌ

B

ƏƈŰƊ

…

–

100

100

ŵHŽƲǛǡǎƏ0ŹƄ\GŨ

Ní

A

ƌ

B

ƒ

100/100 = 1

šżƊŰƢ

ƌžƢ

–

100

0

ŵHŽƲǛǡǎƏ0ŹƄ\GŨ

Ní

A

ƌ

B

ƒ

0/100 = 0

šżƊŰƢ

ƌžƢ

–

100

65

ŵHŽƲǛǡǎƏ0ŹƄ\GŨ

Ní

A

ƌ

B

ƒ

65/100 = 0.65

šżƊŰƢ

ƌžƢ

º0²

Principal co-ordinate analysis (PCO)

•

š~ŴƠŨFpIJƑðŐâxČƦÂƛƢ–Ã

æ}•UVŐƑǘǡƱǚƾǃĶŗŴƠš~ƦĦôżŨº0²żƄĂ³

º0²Ƒj€:

B

=

XX

T

• q¼'ðŐ,Ə0xžƢn"ƑƶǟǎǛƑňČƦūƶǟǎǛŐƑ

ǘǡƱǚƾǃĶŗŬŴƠĿƏÂƛƢũ

• n"ƑƶǟǎǛƑq¼'ðŐƋƑºƦnŧqƑě2XƌžƢƌŨiÞ ãƑƶǟǎǛƌjÞãƑƶǟǎǛƑ,îƦijÞãƑğûƌžƢě2 ƒ

b

_ij

₌

x

_ik

x

_jk

k=1

q

∑

• ƎųŨxƑŊ‹ƒ>ÍƏůƢƌžƢũżƄŵƇƊŨbijƦiƝjƏƈŰƊMƦ ƌƢƌ

d

_ij2

=

(

x

_ik

−

x

_jk

)

2

k=1

q

∑

=

x

_ik2

k=1

q

∑

+

x

_ik2

k=1

q

∑

−

2

x

_ik

x

_jk

k=1

q

∑

=

b

_ii

₊

b

_jj

−

2b

_ij

• źƑƌŶiÞãƌjÞãƑƶǟǎǛŐƑĶŗdijƒ

b

_ij

i=1

n

∑

=

x

_ik

x

_jk

k=1 q

∑

=

i=1

n

∑

x

_jk

x

_ik

i=1 k

∑

=

0

k₌₁

q

∑

†IfJh|

º0²Ƒj€:

ǢĄŶǣ

• żƄŵƇƊŨŵ”ơñƈũ

d_ij2

i=1

n

∑

= (b_ii+b_jj−2b_ij)

i=1

n

∑

= b_ii

i=1

n

∑

+nb_jj

d_ij2

j=1

n

∑

= (b_ii+b_jj−2b_ij)

j=1

n

∑

= b_jj

j=1

n

∑

+nb_ii= b_ii

i=1

n

∑

+nb_ii

d_ij2

j=1

n

∑

i=1

n

∑

= (b_ii+b_jj−2b_ij)

j=1

n

∑

i=1

n

∑

=n b_ii

i=1

n

∑

+nn b_jj

j=1

n

∑

=2n b_ii

i=1

n

∑

• żƄŵƇƊŨŵ”ơñƈũ

di. 2

=1

n dij

2

j=1

n

∑

=1

n i=1bii

n

∑

+bii

di. 2

=1

n dij

2

i=1

n

∑

=1

n i=1bii

n

∑

+bjj

d.. 2

= 1

n2 dij

2 j=1 n

∑

i=1 n

∑

=2

n i=1bii

n

∑

bij=−

1 2(dij

2

−di. 2

−d.j

2

+d.. 2 )

źźƋŨ

Œ†I 7#&&

#Ž†IB'Š

€% %

†IB Š€&

B=XXT'qX

'n&"

A

(

u

₁

,...,

u

q

)

=

(

u

1

,...,

u

q

)

λ

₁

0



0

λ

q

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

ƸǐƱǂǛ0Ĥ

Au

1

=

λ

1

u

1 ǠǠǠ

Au

q

=

λ

q

u

q

°ƐƢƌ

AU

₌

U

_Λ

ƎųŨ

u

i T

u

j

=

0

u

i

T

u

i

=

1,

U

T

_U

=

I

⇔

U

T

₌

U

−1

⇔

UU

T

₌

I

∴

A

₌

U

_Λ

U

T

u

_i*

₌

λ

_i

u

_i ƌžƢƌŨ

A

=

U

*

U

*T źźƋŨ

U

*

=

(

u

1 *

,...,

u

q *

)

ƎƑƋ

B

=

XX

T ƑXƒŨ

X

=

U

Λ

1/2

Λ

1/2

=

diag(

λ

1 1/2

,

…

_,

λ

q

1/2

)

żƄŵƇƊ

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š~ƑŤŰƜƑH]ƒŨ

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感覚距離

PC2

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主成分得点の差

0

1

2

3

0.2 0.4 0.6 0.8 1.0

0

1

2

3

4

5

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http://konicaminolta.jp/instruments/knowledge/color/part2/06.html

II. ŏàĊ: ŕŻƣƄ»¹ƦĦƢǧŐƑãƏƒĠŲƎŰƨǍǙDŽƑĖƑ`¹Ž

(m)

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1

10-2

10-4

10-6

10-8

10-10

10-12

10-14

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780

700

600

500

400 380

(nm)

780

700

600

520

440

360

(nm)

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ý_ĈKAŞYƑ

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Syafaruddin et al. (2006) Breed Sci 56:75-79

Ǣ?ďǣ

2*24,1hU9eo

f

(

_x

)

Ʀ

g(

x

)

=

0

ƑƜƌƋ«aǢrǣ:žƢ

Ǣ5ø±ǣ

L(

_x

,

λ

)

≡

f

(

_x

)

+

λ

g(

_x

)

Ƒ¸#ƦŲƢxƦ™üžƢ

ǙƲǙǟƷǗő¡ (Lagrangian)

ǙƱǙǟƷǗ ¡ (Lagrange multiplier)

∇L

=

∇f

+

λ

∇g

=

0

ƑĤƦÂƛƢ

Ǣ?ďǣ

ǙƲǙǟƷǗƑ®j ¡ÃƑSĤ

http://www.isigas.com/LagrangeMP.html Ə0Ŵơ§ŰĬ¦ŵůơƘž

g

(

x

₁

,

x

₂

)

=

0

f

(

x

₁

,

x

₂

)

∇

f

∇

g

g

(x)=0

ƌ

f

(x)

ƑóŤĈƒ

:a

ƊŰ

&

#

∇

f

−

λ∇

g

=

0

Ǣ?ďǣ

ǙƲǙǟƷǗ®j ¡ÃƑ

g(x

₁

,

x

₂

)

=

x

₁

+

x

₂

−

1

=

0

ƑƋŨ

f

(

x

1

,

x

2

)

=

1

−

x

1 2

−

x

2

2

Ʀ«a:žƢ

L

₍

_x

_,

_λ

₎

₌

₁

₋

x

1 2

−

x

2 2

+

λ

(

x

1

+

x

2

−

1)

Ƒ¸#ƦŲƢxƦ™üžƢ

∂

L

∂

x

1

=

−

2

x

1

+

λ

=

0

∂

L

∂

x

2

=

−

2

x

₂

₊

_λ

₌

0

∂

L

∂λ

=

x

1

+

x

2

−

1

₌

0

'‰

x

1

=

1

2

,

x

2

=

1

2

,

λ

=

1

Ǣ?ďǣ0 Ƒ«a:ǧ

3

¼'Ƒ\G

z

1i

=

u

11

x

1i

+



+

u

1m

x

mi

=

u

1 T

x

i

V

(

z

1

)

=

1

n

−

1

(

z

1i

−

z

1

)

2

i=1

n

∑

=

s

_jk

u

1j

u

1k

=

u

1 T

Vu

1

k=1

m

∑

j=1

m

∑

sjkƒx_jƌx_kƑ*0 

Ʀ

u

T

u

=

u

11

2

₊

_

₊

_u

1M

2

₌

1

5ø± ƑƋ«a:žƢ

z₁Ƒ0 

ƌžƢũ

L

(

u

1

,

λ

)

=

u

1 T

Vu

1

−

λ

(

u

1 T

u

1

−

1)

Ƒ«a:

∂

_L

∂

u

₁

=

2

Vu

₁

₋

2

_λ

u

₁

₌

0

₍

V

₋

_λ

_I

₎

u

1

=

0

T­#Oş

V(z₂)=u 2

T Vu

2

L

(

u

2

,

λ

,

ν

)

=

u

2 T

_Vu

2

−

λ

(

u

2 T

_u

2

−

1)

−

ν

(

u

2 T

_u

1

)

Ǣ?ďǣ

3

¼'Ƒ\GƒŨŻƠƏ

...

Ǒƪǟǂ

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ò1”0ƅŹƋǁǡƼƑLjǙƿƯŵ;0ƏĬ¦ƋŶƎŰ\GŨ¼Ƒ1¼ĂG

z

2i

=

u

₂₁

x

₁i

+



+

u

₂m

x

mi

=

u

₂

T

x

_i

u

2 T

_u

2

=

u

21

2

₊

_

₊

_u

2M 2

₌

1

ƦďŲƢũ

ųƟƔ

u

2 T

u

1

=

u

21

u

11

+



+

u

2M

u

1M

=

0

ǢǏƱǂǛƑ,î=0 → äǣ

ƑƋ«a:žƢũ Ʀ

Ƒ«a:

∂

_L

∂

u

₂

=

2

Vu

₂

₋

2

_λ

u

₂

₋

_ν

u

₁

₌

0

_u

2 T

_u

1

=

u

1 T

_u

2

=

0,

u

1 T

_u

1

=

1

Ɵơ

ν

=

0

(

V

₋

_λ

_I

₎

u

₌

₀

ƝƒơT­#OşƏ

Y#u

1T'`%

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