バイオメトリクス授業資料 iwatawiki lec02 s

060310391

0560565

2

2016/10/3 13:00-14:45

@1 - 4

• 

• 

• 

– 

– 

– 

– 

• 

h/p://konicaminolta.jp/instruments/knowledge/color/part2/07.html

R 151

G 109

B 121

182 × 167 × 3

= 91182

pixel

256 0-255 3 256^3=16 777 216

•  RGB: Red, Green, Blue

–  ³

•  CMY: Cyan, Magenta, Yellow

–  ³

•  HSV: Hue ^{SaturaRon Value}

– 

•  HSL: Hue SaturaRon Luminance

–  ^100%

100% 0%

h/p://homepage2.niUy.com/studio_AURK/ccconv/Color/hue.html

•  RGB ^CMY

C = 255 – R

M = 255 – G

Y = 255 – B

•  RGB ^HSV

^{max,min RGB}

V = max

S = 255 × (max – min) / max

R max

H = 60 × (B – G) / (max – min)

G max

H = 60 × (R – B) / (max – min)

B max

H = 60 × (G – R) / (max – min)

H H ← H + 360

• 

^50×50

50 × 50 × 3 = 7500

0

255

1

(R, G, B) = (23, 14, 22)

(R, G, B) = (200, 198, 168)

(R, G, B) = (201, 200, 173)

(R, G, B) = (200, 199, 172)

50 × 50 × 3 = 7500

7500

Principal component analysis (PCA)

• 

50 × 50 × 3 = 7500

7500

• 

• 

2 1

u

1

x

₁

, x

₂

1 u

₁

u

₁

u

₁

u

1

x

₁

x

₂

u

1

u

1

= u

₁

u

₁

x

₁

x

₂

ε

_i

= y

_i

− (α + βx

_i

)

( 2

SS_E

= ε

_i²

i n

∑ ^{= (y}

i

^{− α − βx}

i

⁾

2 i

n

∑

:

∂SS_E

∂β ⁼⁻² _i ^{(yi−α − βxi)xi}

n

∑

^{= 0}

∂SS_E

∂α ⁼⁻² _i ^{(yi−α − βxi)}

n

∑

^{= 0}

a = y_i

i

∑

n ^{n− b} _i^xi

∑

n n = y − bx

b =

x_iy_i−

i

∑

n _i^xi

∑

n _i^yi

∑

n ⁿ

x_i²− x_i

i

∑

n

( )

^{2 n}

i

∑

n

= ^{i(xi− x )(yi− y )}

∑

n

(x_i− x )²

i

∑

n

n x_i

i

∑

n

x_i

i

∑

n ^xi 2 i

∑

n

#

$

%

%

&

' ( ( a b

#

$ % ^& ' ( = ^iyi

∑

n

x_iy_i

i

∑

n

#

$

%

%

&

' (

-3 _{-2 -1} 0 _{1 2} 3 (

455055

x

y

y_i ˆ

y _i=

α

+

β

x_i x_i

ε

i

15

(a)  ^y

(b)

1

x,y

16

2 1

u

1 T

_u

1

^{= u}

11 2

_{+ u}

12 2

= 1

1

n ₋₁ ^z

¹ⁱ

2

i₌₁ n

∑

z1i

^{= x}

ⁱ T_u

1

^{= x} (

1i ^x2i

)

^u_u¹¹

12

"

# $ ^%

&

' = u

₁₁^x_1i

^{+ u}

₁₂^x_2i

(u₁

x 0

u

1

_x

i
Z_1i ^{= u}₁ ^x_i

Z_1i

1 n₋₁

x_1i²

i₌₁ n

∑

^x1i^x2i i₌₁

n

∑

x1i^x2i i₌₁

n

∑

^x2i 2 i₌₁

n

∑

$

%

&

&

&

&

'

( ) ) ) )

u11

u12

$

% & ^' ( ) = λ

u11

u12

$

% & ^' ( ) L_(u

1^{,λ) =}

1 n₋₁ ^(u11

x1i^{+ u}12^x2i⁾

2−λ(u₁₁²+ u₁₂² −1)

i=1 n

∑

∂L

∂u₁₁⁼ 1

n_−1i ^{2(u11x1i+ u12x2i)x1i− 2λu11= 0}

=1 n

∑

∂L

∂u₁₂⁼ 1

n₋₁ ^{2(u11x1i+ u12x2i)x2i− 2λu12= 0}

i₌₁ n

∑

1 n₋₁^{u11 x1i}

2 i₌₁

n

∑

^{+ u}12 ^x1i^x2i i₌₁

n

$

∑

% & ^'

( ) = λu₁₁ 1

n₋₁ ^{u11 x1ix2i}

i₌₁ n

∑

^{+ u}12 ^x2i 2 i₌₁

n

$

∑

% & ^'

( ) = λu₁₂

(V)

{

Vu

1

^{= λu}

1

}

u

1

^{= 0}

[ ]

var(x) = ¹

n₋₁ ^{(xi− x}

i=1 n

∑

⁾²

cov(x, y) = ¹

n−1_i=1^{(xi− x}

n

∑

^)(yi^{− y ) 2}

r = ^{cov(x, y)} var(x) var(y)⁼

(x_i− x

i=1 n

∑

^)(yi^{− y )}

(x_i− x

i=1 n

∑

^{)2 (y}i^{− y} i=1

n

∑

⁾²

-3 _{-2 -1} 0 _{1 2} 3

-2-10123

length

width

(x_i− x )(y_i− y )

x y

2 4

×(-1)

2

V

₌

6.00 2.03

2.03 2.98

"

# $ ^%

&

' =

6 2

2 3

"

# $ ^%

&

'

7, 2 ^u1=

2 / 5 1/ 5

"

#

$ ^%

& ' u2=

1/ 5

−2 / 5

#

$

% ^& ' (

1 2.5 -0.2

2 3.4 3.4

3 1.0 1.9

4 -2.4 -0.8

5 -3.0 -2.4

6 0.1 -1.6

7 -2.5 -0.5

8 -2.9 1.2

9 1.8 0.1

10 1.6 -1.0

-3 _{-2 -1} 0 _{1 2} 3

-2-10123

length

width

Au _{= λ} u

A 0 u A _λ

A _n×n A _{n λ1, λ2, …, λn}

u_{1, u2, …, un}

{u₁, u₂, …, u_n}

(A − λI)u = 0

u₌₀ 0

0

A − λI = 0

V

₌

6 2

2 3

"

# $ ^%

&

'

6 2

2 3

"

# $ ^%

&

'

u1

u2

"

# $ ^%

&

' = λ

u1

u2

"

# $ ^%

&

'

6 _{− λ} 2

2 3 _{− λ}

$

% & ^'

( )

u1

u2

$

% & ^'

( ) =

0

0

$

% & ^'

( )

6 _{− λ} 2

2 3 _{− λ} ^{= 0}

^{(6 −λ)(3 −λ) − 4 = (λ− 2)(λ− 7) = 0}

λ=2 ^2u21+ u22= 0 ^u₂₁² + u₂₂² _{= 1} ^u21 u₂₂

"

# $ ^%

& ' = ^{1/ 5}

−2 / 5

"

# $ ^%

& ' λ=7 ^u11− 2u12= 0 ^u₁₁²+ u₁₂² _{= 1} ^u11

u₁₂

"

# $ ^%

& ' = ^{2 / 5}

1/ 5

"

# $ ^%

& '

2, 7

u₁₌ u₁₁ u₁₂

"

# $ ^%

& ' =

2 / 5 1/ 5

"

# $ ^%

&

' ^u2=

u₁₁ u₁₂

"

# $ ^%

& ' =

1/ 5

−2 / 5

"

# $ ^%

& '

k 2 B

50 × 50 × 3 = 7500

7500

z₁, z₂, …

^•

• 

z

_ij

_{= u}

_j1

x

_1i

_{++ u}

_jM

x

_Mi

_{= x}

_i^T

^u

_j

j i principal component score

1. 

2.  ^-3 -2 -1 ⁰ 1 2 ³

-2-10123

length

width

z_{11= 2.5}( −0.2)^{2 / 5}

1/ 5

#

$

% ^& ' ( = 2.2

z₁₂=

(

^2.5 −0.2

)

^{1 / 5}

−2 / 5

⎛

⎝⎜

⎜

⎞

⎠⎟

⎟^{= 1.3}

PC1 PC2

1 2.5 -0.2 2.2 1.3

2 3.4 3.4 4.6 -1.5

3 1.0 1.9 1.8 -1.2

4 -2.4 -0.8 -2.5 -0.3

5 -3.0 -2.4 -3.7 0.8

6 0.1 -1.6 -0.6 1.5

7 -2.5 -0.5 -2.4 -0.6

8 -2.9 1.2 -2.0 -2.3

9 1.8 0.1 1.7 0.7

10 1.6 -1.0 1.0 1.6

0

1 N − 1 ^zji

2 i N

∑

= ¹ N − 1

zj Tz

j= ¹

N − 1^(Xuj)

T(Xu_j)

= ¹

N − 1 uj

T_XT_Xu j= uj

T_Vu j= λj^uj

T_u j= λj

Vu_j=λ_ju_j

V eigenvalue λ1≥λ2≥  ≥λ_M≥ 0 ^ui

z_j j

λ

_j

^/ λ

_i

i=1 M

∑

λ_i

i=1 j

∑

^{/ λi}

i=1 M

∑

j contribuRon cumulaRve

1 M −1^X

T_{X= V}

x 0

i λ_i

x

0.77 0.77

0.33 1.00

var(z

₁

) = 7.0

…

PC1 PC2

1 2.5 -0.2 2.2 1.3

2 3.4 3.4 4.6 -1.5

3 1.0 1.9 1.8 -1.2

4 -2.4 -0.8 -2.5 -0.3

5 -3.0 -2.4 -3.7 0.8

6 0.1 -1.6 -0.6 1.5

7 -2.5 -0.5 -2.4 -0.6

8 -2.9 1.2 -2.0 -2.3

9 1.8 0.1 1.7 0.7

10 1.6 -1.0 1.0 1.6

var(l) = 6.0

V

₌

6 2

2 3

"

# $ ^%

&

'

var(w) = 3.0

var(z

₂

) = 2.0

• 

• 

1

•  ¹

• 

2

R =

^{1 r}

r 1

⎛

⎝⎜

⎞

⎠⎟

R − λI = 0 ⇔ (1− λ)²

− r

²

= 0

∴ λ

1

= 1+ r, λ

2

= 1− r

Ra1

= λ

1^a1

y₁₌ ¹

2^{(x1+ x2), y2=} 1

2^{(x1− x2)}

∴a

₁₁

= a

₁₂

= ¹

2

a2

a_{11+ ra12= (1+ r)a11} ra11+ a12= (1+ r)a12

-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%)

z

_iq

_{= x}

_i^T

^u

_q

(z

_i1

,..., z

_iq

_{) = x}

_i^T

(u

₁

,..., u

_q

) ^z

_i^T

_{= x}

_i^T

^{U ⇔ z}

_i

_{= U}

^T

^x

_i

z

_i

1

= x

^iT

^u

1

U^T_{U =} u1

T_u

1 ^{ u}1

T_u q

  

u_q^Tu

1 ^{ u}q

T_u q

⎡

⎣

⎢

⎢⎢

⎢

⎤

⎦

⎥

⎥⎥

⎥

= I

^U

^T

= U

⁻¹

^{⇔ UU}

^T

= I

∴x

i

= Uz

i

= 2 x T 2

x_{i = U} k λ₁

0

 0

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟

2

2

eigenfaces

Murphy KP (2012) "Machine Learning: A ProbabilisRc PerspecRve" The MIT Press.

102

30

30

• 

•  ^{A B …}

–  100 ¹⁰⁰

A B 100/100 = 1

–  100 ⁰

A B 0/100 = 0

–  100 ⁶⁵

A B 65/100 = 0.65

Principal co-ordinate analysis (PCO)

• 

h/p://aoki2.si.gunma-u.ac.jp/lecture/misc/princo-ex.html

B = XX

^T

•  q ⁿ

•  n ^{q n×q X i}

j ij

b

_ij

₌ x

_ik

x

_jk

k=1

∑

q

•  ^{x bij i j}

d

_ij²

₌ (x

_ik

− x

_jk

)

²

k=1

∑

q

= ^x

ik

2 k=1

∑

q

+ ^x

ik 2 k=1

∑

q

^{− 2}

_k=1

^x

ik

^x

jk

∑

q

= b

ii

+ b

jj

^{− 2b}

ij

•  ^{i j dij}

b

_ij

i=1

∑

n

=

_k=1

^x

ik

^x

jk

∑

q

=

i=1

∑

n

^x

jk _i=1

^x

ik

∑

k

= 0

k =1

∑

q

{

•  dij

2 i=1

∑

n = _i=1(bii+ bjj− 2bij⁾

∑

n = _i=1bii

∑

n + nbjj

dij 2 j=1

∑

n = _j=1(bii+ bjj− 2bij⁾

∑

n = _j=1bjj

∑

n + nbii= _i=1bii

∑

n + nbii

d_ij²

j=1

∑

n i=1

∑

n = _j=1^(bii+ bjj− 2bij⁾

∑

n i=1

∑

n = n _i=1^bii

∑

n + nn _j=1^bjj

∑

n = 2n _i=1^bii

∑

n

• 

d_i.²₌¹ n ^dij

2 j=1

∑

n =¹ n ^i=1bii

∑

n + bii

d_i.²₌¹ n ^dij

2 i=1

∑

n =¹ n ^i=1bii

∑

n + bjj

d.. 2= ¹

n^{2 dij}

2 j=1

∑

n i=1

∑

n =² n ^i=1bii

∑

n

b_{ij= −}¹ 2^(dij

2− di. 2− d.j

2+ d.. 2)

X 2 2

X ,

A(u₁

,..., u

_q

_{) = (u}

₁

,..., u

_q

)

λ

₁

0



0 λ

_q

⎡

⎣

⎢

⎢ ⎢

⎤

⎦

⎥

⎥ ⎥

Au

1

^{= λ}

1

^u

1

^Au

^q

^{= λ}

^q

^u

^q

AU = UΛ

u

i T

u

j

= 0

u

i

T

_u

i

_{= 1,} ^U

T

U = I ⇔ U

^T

= U

⁻¹

^{⇔ UU}

^T

= I

∴A = UΛU

^T

u

^*_i

_{= λ}

_i

u

_i

_{A = U}

^*

_U

^{*T U*}

= (u

1

*

,..., u

^*_q

)

B = XX

^{T X}

_{X = UΛ}

^1/2

_Λ

^1/2

_{= diag(} ^λ

₁^1/2

,…, ^λ

_q^1/2

)

B

感覚距離 PC2

PC4 主成分得点の差 0123

0.2 0.4 0.6 0.8 1.0

012345

…

1

( )

PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10

PCO1 -1.0-0.50.00.51.0

PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10

PCO2 -1.0-0.50.00.51.0

u

2

3

4 1

x x

• 

–  ^{3 4}

• 

– 

h/p://konicaminolta.jp/instruments/knowledge/color/part2/06.html

II. :

_(m) 10 ²

1

10 ^-2

10 ^-4

10 ^-6

10 ^-8

10 ^-10

10 ^-12

10 ^-14

 





 





 FM



780

700

600

500

400

380

(nm)

780

700

600

520

440

360

(nm)

< >

< >

Brassica rapa L.

Brassica napus L.

rapa

< >

Brassica rapa L.

Brassica napus L.

: Nikon D70

: Ultra AchromaRc Takumar

• 

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

f (x)

^g( x) = 0

L(x, ^{λ ) ≡} f (x) + ^λ g(x)

x

(Lagrangian)

(Lagrange mulRplier)

∇L = ∇f + λ ∇g = 0

h/p://www.isigas.com/LagrangeMP.html

g(x

₁

, x

₂

) = 0

f (x

₁

, x

₂

)

∇g ∇f

g(x)=0 f(x)

∇f − λ ∇g = 0

λ

g(x

₁

, x

₂

) = x

₁

+ x

₂

−1 = 0

f (x

₁

, x

₂

) = 1− x

₁²

− x

₂²

L _{(x, λ ) = 1− x}

1

2

− x

₂²

+ λ (x

₁

+ x

₂

−1)

x

∂ ^L

∂ ^x

₁

^{= −2x}

¹

^{+ λ = 0}

∂ ^L

∂ ^x

₂

^{= −2x}

²

^{+ λ = 0}

∂ ^L

∂λ ^{= x}

¹

^{+ x}

²

^{−1 = 0}

x

1

⁼

1

2 ^,

x

2

⁼

1

2 ^{, λ = 1}

3

^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

∑

s_jk x_j x_k

^u

^T

^{u = u}

¹¹²

^{+  + u}

^1M2

^{= 1}

z₁

L(u

1

^{,λ) = u}

1 T

Vu

1

^{− λ(u}

1 T

u

1

⁻¹⁾

∂L

∂u

₁

^{= 2Vu}

¹

^{− 2λu}

¹

^{= 0 (V − λ I)u}

¹

^{= 0}

V_{(z2) = u2}^TVu₂

L(u

2

^{,λ,ν ) = u}

2 T

Vu

2

^{− λ(u}

2 T

u

2

^{−1) − ν (u}

2 T

u

1

⁾

3 ...

x

1 1

^z

²ⁱ

^{= u}

²¹

^x

¹ⁱ

^{+ + u}

^2m

^x

^mi

^{= u}

²

T

x

i

^u

^2T

^u

²

^{= u}

²¹²

^{+  + u}

^2M2

^{= 1}

^u

²

T

u

₁

_{= u}

₂₁

u

11

^{+  + u}

2M

^u

1M

^{= 0}

=0 →

∂L

∂u

₂

^{= 2Vu}

²

^{− 2λu}

²

^{− νu}

¹

^{= 0 u}

²

T

u

1

^{= u}

1 T

u

2

^{= 0, u}

1 T

u

1

^{= 1}

ν = 0

(V − _{λI)u = 0}

2

1,2

Q.

PDF MS-Word

e

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A4 1

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