A Cluster Target Similarity Based g y Principal Component Analysis - - PowerPoint PPT Presentation

A Cluster‐Target Similarity Based g y Principal Component Analysis for Interval‐Valued Data

University of Tsukuba School of Systems and Information Engineering Mika Sato‐Ilic

Energy Evaluation Data

[51,71] [81,91] … [60,70] [60,90]

• 1. Oil

UK2 UK1 … JP2 JP1 Energy … … [65,75] [50,60] … [20,40] [60,100]

• 3. Coal with

CCS [80,91] [80,91] … [80,95] [90,120]

• 2. Coal

[ , ] [ , ] [ , ] [ , ]

• 1. Oil

… … [0,20] [0,20] … [30,45] [60,80]

• 5. Geothermal

[60,80] [45,65] … [50,85] [70,120]

• 4. Nuclear

CCS … … [50 70] [60 72] [50 60] [70 100]

• 8. On. Wind,

[60,70] [60,70] … [20,35] [40,100]

• 7. Biomass

[10,40] [0,10] … [30,40] [30,70]

• 6. Solar PV

… … [60,70] [65,75] … [40,60] [70,100]

• 10. Hydro

[80,90] [60,80] … [50,65] [83,111]

• 9. Mun/Ind

Waste [50,70] [60,72] … [50,60] [70,100] large … … …

J i t R h ESRC f d d S E G t SPRU (S i d T h l P li R h U i it f S )

[87,97] [87,97] … [65,85] [80,120]

• 11. Gas

y

…

Joint Research: ESRC‐funded Sussex Energy Group at SPRU (Science and Technology Policy Research, University of Sussex) Sustainable Energy/Environment & Public Policy (SEPP), University of Tokyo (ESRC: Economic and Social Research Council)

Principal Component Analysis based on Classification Structure by Fuzzy Clustering y y g

Multi‐dimensional Space

１０

(p dimensional space) p: Number of Variables

１ ３ ７ ８ ６ ５ ２ ９ １１ ４ Adaptable Classification Structure Adaptable Number of Clusters

Principal Component Analysis for Metric Projection Principal Component Analysis for Metric Projection L X P

Metric Projection

L X P

L

 :

y x x x x y x y x       

L L

L d X L d L P inf ) , ( , )} , ( : { ) (

yL

X :

: Inner Product Space

L :

: Nonempty Subset of X

Principal Component Analysis for Metric Projection

L X P

L

 : Metric Projection P

L

) ( ) ( ve nonexpansi is X P P

L L

    y x y x y x , ) ( ) (

L X i i t t l t t i t th h F

L : Convex Chebyshev Set

: Convex Chebyshev Set

L X in point nearest

• ne

least at exists there , each For  x

Analysis Component Principal Objects

• f

ity Dissimilar : ) ( ) ( ) , ( y x y x y x y x

L L

P P C      Space Projected

• n

Objects

• f

ity Dissimilar : ) ( ) ( j y y x y

L L

P P 

Principal Component Analysis Principal Component Analysis

         n i x x X

1

1 ) ( ~ ~ x x                     

a ip i i n

p a x X n i x x X

1 1

1 ) ( , , 1 ), , , ( , ~ x x x x x           

na a p

p n p a x X

1

Variables

• f

Number : Objects

• f

Number : , , 1 , ), , , ( x x x   



  

a p a

X X F p n

1 1

) ( )' ( Minimize Variables

• f

Number : Objects

• f

Number : l x l x



   

   

a a a p a a a

X X X X X X X X F X X X

1 1 * 1 1 1

) ' ) ' ( ( )' ' ) ' ( ( ' ) ' ( x x x x x l

  p p a

X X X X

1 1 1

, , by Spanned Subspace to Projection : ' ) ' ( x x 

  



   

     

p b a b a b a b a b a

X X X X X X X X

1 1 ' 1

) ( ' ) ' ( ) ( ) ( ' ) ' ( ) ( x x x x x x x x

Principal Component Analysis

 



X p

X X X X P X X X X

1 1 1

' ) ' ( , , by Spanned Subspace to Projection : ' ) ' ( x x   

X X X X X X

Symmetry P P Idempotent P P P : ' :

 

        

p a X a a a p a X a a X a n b a n b a

P P P F V V

*

) ' ' ( ) ( )' ( , , x x x x x x x x x x x x

  



  p a a

P P

' 1 1

) ( ) ( ) ( ) ( x x x x x x x x

  

 

 

           

p b a X b a b a b a b a b a X b a b a X b a

P P P

1

) ( )' ( ) ( )' ( ) ( ) ( ) ( ) ( x x x x x x x x x x x x x x x x

  

    

   

p b a b X a b a p a a X a a a b a b a X b a b a b a

P P p

1 1 1

) ' ' ( 2 ) ' ' ( 2 x x x x x x x x

   b a a 1 1

Covariance between Variables

*

F

Fuzzy Cluster based Principal Component Analysis

  





     

p b a b a X b a b a X b a

P P

1 '

) ( ) ( ) ( ) ( x x x x x x x x



   

     

p b a b a X b a b a b a b a

P

1 1

) ( )' ( ) ( )' ( x x x x x x x x

 

    

   

p b a b X a b a p a a X a a a b a

P P p

1 1 1

) ' ' ( 2 ) ' ' ( 2 x x x x x x x x

Covariance between Variables

*

F Adaptable Classification Structure based on an Appropriate Number of Clusters Dissimilarity Structure of Objects in Higher Dimensional Space

Selection of an Appropriate Number of Clusters Selection of an Appropriate Number of Clusters

~ ~

) ( ) (

S U a K

a a 

 

 S

) ( ~

) ( ) (

b a S U b K

b b

   

 S

Clusters

• f

Number : K

) ( b a 

) (

i Cl t f N b th h f St t ti Cl ifi Data Similarity Observed :

l

l S U S

) (

, , 1 Clustering Fuzzy

• f

Result A is Clusters

• f

Number the when for Structure tion Classifica :

l

K l l S U ） （  

) ( ) (

Using by Similarity Restored ~

l l

U S ：

S S S S

K l

} ~ ~ { t ~ Cl t S l t

) ( ) 2 ( ) (

S S S S

K l

Data Original for Structure tion Classifica e Explainabl Most Select } , , { among to Closest Select

) ( ) 2 ( ) (

 l is Clusters

• f

Number e Appropriat

Asymmetric Similarity of Interval Asymmetric Similarity of Interval‐Valued Data Valued Data

, , 1 , , , 1 ]), , ([ ) ( p a n i y y y Y

ia ia ia

     

   

| ) ( inf ) ( | ) ( sup ) , , ( ) , , (

1 1

y y y x d y x d y x y x d d y y and y y between ity Dissimilar

p jp j j ip i i

     



  y y

     

 

| ) ( inf ) ( | ) ( sup | ) , ( inf ) , ( , | ) , ( sup

1

y x y x d y y d y y y y d d y y y x d y x d y x y x d d

p ja ja a ia ja ij

       

 



 

 

| ) , ( inf ) , ( , | ) , ( sup

1

y x y x d y y d y y y y d d

ia ia a ja ia ji

    



) ( j i d d

ji ij

  ) ( , , 1 , }, { max / 1

,

j i s s n j i d d s

ji ij ij j i ij ij

      ) ( j

ji ij

Asymmetric Fuzzy Clustering Model Asymmetric Fuzzy Clustering Model

( ) ( ) (Sato and Sato, 1995) (Sato and Sato, 1995)

Asymmetric Similarity Data Asymmetric Similarity Data

 

n j i j i s s s S

ji ij ij

,..., 1 , ), ( , ,     n j i u u w s

ij jl K K ik kl ij

,..., 1 , ,      j and i Objects Between Similarity Asymmetric s n j i u u w s

ij ij jl k l ik kl ij

: ,..., 1 , ,

1 1





 

 l and k Clusters Between Similarity Asymmetric w k Cluster a to i Ojbect an

• f

ess Belongingn

• f

Degree u

kl ik

: : s s w w Error

ji ij lk kl ij :

  

) , 1 ( , 1 ], 1 , [

1

   



m u u

K k ik ik

Clusters

• f

Number K Objects

• f

Number n : :

1  k

Asymmetric Similarity of Clusters Asymmetric Similarity of Clusters

K l k w K 1 1 1

) (

  K l k w

K kl

w kl

, , 1 , , exp 1 1

) (

~

    



 

| | | | log ) ( 2 1 ~

) ( ) , ( ) , ( 1 ) , ( ) , ( ) , ( ) (

1 ) , (

                        

  K l K k K l K k K l K k K kl

I tr w

K k

μ μ ) ( ) ( | |

) , ( ) , ( 1 ) , ( ' ) , ( ) , ( ) , ( ) , ( ) , (

1 ) , (

        

  K l K k K k K l K k K l K k K l

K k

μ μ μ μ μ μ k Cl f M i C i V i k Cluster in Data

• f

Value Expected :

) , (



K k

μ ] 1 , [ ~ ), ( , ~ ~ k Cluster for Matrix Covariance Variance :

) ( ) ( ) ( ) , (

    

K kl K lk K kl K k

w l k w w

Clusters

• f

Number K :

Criterion for Selection of Number of Clusters Criterion for Selection of Number of Clusters



n K ij ijs

s

) (

~

  

 



n K ij n ij j i j j

s s K C

) ( 2 1

2

~ ) (

 

    j i ij j i ij

s s

1 1

Alignment Degree of Agreement Alignment Degree of Agreement

K K

Alignment Degree of Agreement Alignment Degree of Agreement

n j i u u w s

K jl k l K ik K kl K ij

,..., 1 , , ~

) ( 1 1 ) ( ) ( ) (

  

 

Restored Asymmetric Similarity

Clusters

• f

Number K :

K k ,..., 1 

Concentration around Expected Value Concentration around Expected Value f h iff f h iff for the Different w for the Different wkl

kl



n K ) (

~



 



n n j i K ij ijs

s K C

1 ) (

2

) (

 

    j i K ij j i ij

s s

1 ) ( 1 2

2

~        a m C E C P

2 4 2

2 ) | )] ˆ ( ˆ [ ) ˆ ( ˆ (|         c w s s C E w s s C P exp 2 ) | )] , , ( [ ) , , ( (|  ) ˆ , ˆ , ( ˆ ) ˆ , ˆ , ( ˆ w s s C w s s C

kl ij ij



Concentration around Expected Value Concentration around Expected Value f h Diff f h Diff

) 1 ( 6 | ) ~ ~ ( ˆ ) ˆ ˆ ( ˆ |  n n m w s s C w s s C

for the Different w for the Different wkl

kl

) 1 ( , | ) , , ( ) , , ( |

2

    n n m ma w s s C w s s C

ˆ ˆ ) ˆ , ~ ~ , ( ˆ ) ~ , ~ , ( ˆ   w s d s s C w s s C , ~ ~ ) ~ ~ ~ ( ~ ~ ) ~ ~ ~ ( ~ ~ , ~ 2 ) ˆ , ~ , ( ˆ ) ˆ , ~ ~ , ( ˆ

2

         s s s s d s d s s s s d s d s s s s d s w s s C w s d s s C ) 1 ( , 2 ~ , ~    n n m m s s d

) ~ , ~ , ( ˆ ) ~ , ~ , ( ˆ ), ˆ , ˆ , ( ˆ ) ˆ , ˆ , ( ˆ w s s C w s s C w s s C w s s C

kl ij ij kl ij ij

 

Clusters

• f

Number K :

K k ,..., 1 

Theorem: (C. McDiarmid)

| ) , , , ˆ , , , ( ) , , ( | sup

1 1 1 1 ˆ , , ,

1

   

 

c x x x x x f x x f

i n i i i n x x x

i n

  



, 2 exp 2 ) | ) , , ( ) , , ( (|

2 1 1

                all for X X Ef X X f P   , exp 2 ) | ) , , ( ) , , ( (|

1 2 1 1

        





  all for c X X Ef X X f P

n i i n n

n i for satisfies R A f A Set a in Values Taking Variables Random t Independen X X

n n

   1 : : , ,

1 

n i for satisfies R A f    1 :

) 1 ( , 6 | ) ~ , ~ , ( ˆ ) ˆ , ˆ , ( ˆ |

2

    n n m w s s C w s s C ) 1 ( , | ) , , ( ) , , ( |

2

 n n m ma w s s C w s s C      a m

2 4 2

ˆ ˆ           c a m w s s C E w s s C P exp 2 ) | )] , ˆ , ( [ ) , ˆ , ( (|  

Fuzzy Cluster based Principal Component Analysis

  





     

p b a b a X b a b a X b a

P P

1 '

) ( ) ( ) ( ) ( x x x x x x x x



   

     

p b a b a X b a b a b a b a

P

1 1

) ( )' ( ) ( )' ( x x x x x x x x

 

    

   

p b a b X a b a p a a X a a a b a

P P p

1 1 1

) ' ' ( 2 ) ' ' ( 2 x x x x x x x x

Covariance between Variables

*

F Adaptable Classification Structure based on an Appropriate Number of Clusters Dissimilarity Structure of Objects in Higher Dimensional Space

Fuzzy Cluster based Covariance Matrix Fuzzy Cluster based Covariance Matrix with respect to Variables with respect to Variables with respect to Variables with respect to Variables

u

n i n K i t i m ik

 

  ) ( ) ( x x x x x n u C

i n i K k m ik i k

  

  

 

1 1 1 1 1

, x

i k   1 1

) , 1 ( , 1 ], 1 , [

1

   





m u u

K k ik ik

Fuzzy Clustering

n i t i



  ) ( ) ( x x x x Special case n C

i i i







1

) ( ) ( x x x x





 

K k ik ik

u u

1

1 }, 1 , {

Hard Clustering n : Number of Objects K : Number of Clusters

Fuzzy Cluster based Covariance Matrix Fuzzy Cluster based Covariance Matrix with respect to Variables with respect to Variables with respect to Variables with respect to Variables

u

n i n K i t i m ik

 

  ) ( ) ( x x x x x n u C

i n i K k m ik i k

  

  

 

1 1 1 1 1

, x

i k   1 1

) , 1 ( , 1 ], 1 , [

1

   





m u u

K k ik ik



    

n b ib a ia i ab ab

p b a x x x x w c c C , , 1 , ), )( ( ), ( 

1  k

  

 K m k n i b ib a ia i ab ab

u x p

1

, , , ), )( ( ), (

   

 

 

n K m ik k ik i i ia a

u u w n x x

1 1

,

 

  i k ik

u

1 1

n : Number of Objects K : Number of Clusters

Fuzzy Cluster based Covariance Matrix Fuzzy Cluster based Covariance Matrix with respect to Variables with respect to Variables with respect to Variables with respect to Variables



    

n b ib a ia i ab ab

p b a x x x x w c c C , , 1 , ), )( ( ), ( 

 

 K m ik i b ib a ia i ab ab

u p

1

) )( ( ) (





 

K k ik ik

u u

1

1 ], 1 , [

  





n K m ik k ik i

u w

1

) , 1 (   m

 

  i k ik 1 1

Crisp Classification of an Object i l f f b Becoms Larger

i

w

Uncertainty Classification of an Object i Becomes Smaller

i

w

n n : Number : Number of

• f Objects

Objects K K : N : Number mber of Clusters usters

Fuzzy Cluster based Covariance Matrix with Fuzzy Cluster based Covariance Matrix with respect to Variables respect to Variables respect to Variables respect to Variables



    

n b ib a ia i ab ab

p b a x x x x w c c C , , 1 , ), )( ( ), ( 

  

 K m ik n ia i b ib a ia i ab ab

u x

1

   

 

 

n K m ik k ik i i ia a

u w n x

1 1

, ) , 1 (   m

n K

 

  i k ik 1 1

Fuzzy Clustering

w w u u

i i i k ik ik

    

 

 

1 , 1 ], 1 , [

1 1

y g

i w u u

i K ik ik

    



, 1 1 }, 1 , {

Hard Clustering

n

i k ik ik



1

n n : Number : Number of

• f Objects

Objects K K : N : Number mber of Clusters usters

Fuzzy Cluster based Covariance Matrix Fuzzy Cluster based Covariance Matrix for for Interval Interval‐Valued Data Valued Data for for Interval Interval Valued Data Valued Data



    

n b ib a ia i ab ab

p b a x x x x w c c C , , 1 , ), )( ( ), ( 

  

 K m n i b ib a ia i ab ab

u x p b a x x x x w c c C

1

, , 1 , ), )( ( ), (

   

 

 

n K m ik k ik i i ia a

u u w n x x

1 1

,

 

  i k ik

u

1 1

n : Number of Objects K : Number of Clusters

Fuzzy Cluster based Covariance When xi are Interval Valued Data ? When xia are Interval‐Valued Data ?

Empirical Joint Function for Interval Empirical Joint Function for Interval‐Valued Data Valued Data

(Bertrand and (Bertrand and Goupil Goupil, 2000) , 2000)

) ( ]) [ ] ([ 1 , ) ( ) , ( 1 ) , (

1 n i b a i b a

i Z y y I n y y f   



] [ I t l E h f Di t ib ti U if , , , , , ) ( ]) , [ ], , ([ , 1 ) , (

ib ib ia ia ib ib ia ia b a i

y y y y Otherwise i Z y y y y y y I      

2  p

] , [ Interval Each for

• n

Distributi Uniform

ia ia y

y

) (i Z

Xb

Object k

2 p

ib

y

) (i Z

Object i Object k

]) , [ ], , ([

ib ib ia ia

y y y y

ib

y

ib

y

Object i Object j

) (i Z

ia

y

ia

y

Xa

Fuzzy Cluster based Covariance Matrix Fuzzy Cluster based Covariance Matrix for for Interval Interval‐Valued Data Valued Data





    

K n i b ib a ia i ab ab

p b a x x x x w c c C

1

, , 1 , ), )( ( ), ( 

   

 

 

n K m K k m ik i n i ia a

u w n x x

1 1

,

 

  i k m ik

u

1 1

When xia are Interval‐Valued Data ?

) , ( ~ ) )( ( ˆ ), ˆ ( ˆ

 

 

   

b a b a b b a a ab ab

dy dy y y f y y y y c c C Mean Empirical Symbolic : ) ( 2 1 ) ( ) )( ( ) (

  

   

 

n ia ia a b a b a b b a a ab ab

y y n y y y y y f y y y y Function Joint Empirical Weighted : ) ( ) , ( 1 ) , ( ~ 2

1







n b a i i b a i

i Z y y I w n y y f n ) (

1  i

i Z n

Fuzzy Cluster based Covariance Matrix Fuzzy Cluster based Covariance Matrix f I l I l V l d D V l d D for for Interval Interval‐Valued Data Valued Data

) ˆ ( ˆ

ab

c C 

n n n b a b a b b a a ab

y y w y y w dy dy y y f y y y y c

 

     

     1 ) ( 1 ) ( 1 1 ) , ( ~ ) )( ( ˆ

b a i ib ib i a i ia ia i b i ib ib ia ia i

y y n y y w y n y y w y n y y y y w n

  

  

       

1 1 1

1 2 ) ( 1 2 ) ( 1 ) )( ( 4 1 ) , 1 ( ,

1

  





m u w

n K K k m ik i

) , ( ,

1 1

 

 

u

n i K k m ik i

PCA based on Fuzzy Covariance for PCA based on Fuzzy Covariance for Interval Interval‐Valued Data Valued Data Interval Interval‐Valued Data Valued Data

) (

'

l l l l ) , , , (

1 12 11 1 p

l l l   l

ˆ

Corresponding Eigen‐Vector

C

Corresponding Eigen Vector For the Maximum Eigen‐Value of

:

1 1

l z Y 

First Principal Component

y y y y Y

ia ia ia ia),

(    p a n i y y

ia ia

1 1 2 ), (     p a n i , , 1 , , , 1

Conventional PCA for Interval Conventional PCA for Interval‐Valued Data Valued Data

Centers Method Centers Method

i

y y  ~

1 11

   

c p c

y y  , 2

ia ia c ia

y y y  

] , [

ia ia ia

y y y 

, ~

1

        

c np c n

y y Y    

p a n i   , 1 , , , 1  

p

C Y ~ } ~ cov{ 

Covariance Matrix for Interval‐Valued Data

b a ib ib n i ia ia ab ab

y y y y y y n c c C     



) ( ) ( 4 1 ~ ), ~ ( ~

1

(Billard and Diday, 2000)

i

n



4

1





n b a i b a

i Z y y I n y y f ) ( ) , ( 1 ) , (

(Bertrand and (Bertrand and Goupil Goupil, 2000) , 2000)

Principal Component Analysis：Centers Method

 i

i Z n

1

) (

( p , ) , )

Comparison between Proposed PCA and Conventional PCA for Interval‐Valued Data

) , ( ~ ) )( ( ˆ ), ˆ ( ˆ

 

     

   

b a b a b b a a ab ab

dy dy y y f y y y y c c C Function Joint Empirical Weighted : ) ( ) , ( 1 ) , ( ~

1



  



n i b a i i b a

i Z y y I w n y y f ) (

1 i

) , 1 ( ,

1

  

 



m u w

n K m K k m ik i

Fuzzy Clustering

1 1



 

u

i k m ik

) ( ) )( ( ~ ) ~ ( ~

 

 

  dy dy y y f y y y y c c C Function Joint Empirical : ) , ( 1 ) , ( ) , ( ) )( ( ), (

  

   

    

n b a i b b a b a b b a a ab ab

y y I y y f dy dy y y f y y y y c c C

i  1

Function Joint Empirical : ) ( ) , (

1



 i b a

i Z n y y f

d l i Centers Method

i wi   , 1

Hard Clustering

Fuzzy Cluster based Covariance Matrix Fuzzy Cluster based Covariance Matrix with respect to Variables with respect to Variables with respect to Variables with respect to Variables



    

n b ib a ia i ab ab

p b a x x x x w c c C , , 1 , ), )( ( ), ( 

  

 K m ik n ia i b ib a ia i ab ab

u x

1

   

 

 

n K m ik k ik i i ia a

u w n x

1 1

, ) , 1 (   m

n K

 

  i k ik 1 1

w w u u

i i i k ik ik

    

 

 

1 , 1 ], 1 , [

1 1

i w u u

i K ik ik

    



, 1 1 }, 1 , { n

i k ik ik



1

n : Number of Objects K : Number of Clusters

Energy Evaluation Data

[51,71] [81,91] … [60,70] [60,90]

• 1. Oil

UK2 UK1 … JP2 JP1 Energy … … [65,75] [50,60] … [20,40] [60,100]

• 3. Coal with

CCS [80,91] [80,91] … [80,95] [90,120]

• 2. Coal

[ , ] [ , ] [ , ] [ , ]

• 1. Oil

… … [0,20] [0,20] … [30,45] [60,80]

• 5. Geothermal

[60,80] [45,65] … [50,85] [70,120]

• 4. Nuclear

CCS … … [50 70] [60 72] [50 60] [70 100]

• 8. On. Wind,

[60,70] [60,70] … [20,35] [40,100]

• 7. Biomass

[10,40] [0,10] … [30,40] [30,70]

• 6. Solar PV

… … [60,70] [65,75] … [40,60] [70,100]

• 10. Hydro

[80,90] [60,80] … [50,65] [83,111]

• 9. Mun/Ind

Waste [50,70] [60,72] … [50,60] [70,100] large … … …

J i t R h ESRC f d d S E G t SPRU (S i d T h l P li R h U i it f S )

[87,97] [87,97] … [65,85] [80,120]

• 11. Gas

y

…

Joint Research: ESRC‐funded Sussex Energy Group at SPRU (Science and Technology Policy Research, University of Sussex) Sustainable Energy/Environment & Public Policy (SEPP), University of Tokyo (ESRC: Economic and Social Research Council)

Asymmetric Dissimilarity

Objects

1 2 3 4 5 6 7 8 9 10 11 1 1.00 0.70 0.78 0.83 0.38 0.36 0.66 0.78 0.78 0.70 0.77 2 0 67 1 00 0 51 0 81 0 05 0 03 0 37 0 50 0 63 0 43 0 80 2 0.67 1.00 0.51 0.81 0.05 0.03 0.37 0.50 0.63 0.43 0.80 3 0.73 0.49 1.00 0.77 0.45 0.50 0.77 0.73 0.75 0.70 0.57 4 0 70 0 70 0 70 1 00 0 22 0 22 0 51 0 62 0 67 0 54 0 67 4 0.70 0.70 0.70 1.00 0.22 0.22 0.51 0.62 0.67 0.54 0.67 5 0.44 0.14 0.58 0.42 1.00 0.78 0.64 0.59 0.46 0.62 0.26 6 0.29 0.00 0.49 0.30 0.64 1.00 0.43 0.36 0.30 0.41 0.12 6 0.29 0.00 0.49 0.30 0.64 1.00 0.43 0.36 0.30 0.41 0.12 7 0.67 0.39 0.81 0.63 0.63 0.50 1.00 0.82 0.66 0.81 0.51 8 0.80 0.55 0.81 0.78 0.55 0.46 0.85 1.00 0.80 0.81 0.66 9 0.80 0.69 0.84 0.82 0.42 0.40 0.70 0.81 1.00 0.79 0.76 10 0.76 0.53 0.84 0.75 0.62 0.55 0.86 0.86 0.83 1.00 0.64 11 0.75 0.82 0.61 0.80 0.19 0.17 0.50 0.64 0.73 0.56 1.00

1: Oil 3: Coal with 5: Geothermal 7: Biomass 9: Mun/Ind 11: Gas 1: Oil 3: CCS 5: Geothermal 7: Biomass 9: Waste 11: Gas 2: Coal 4: Nuclear 6: Solar PV 8:

• On. Wind,

large 10: Hydro

Selection for Number of Clusters Selection for Number of Clusters



 n j i K ij ijs

s

1 ) (

~

n j i u u w s

K jl K K K ik K kl K ij

,..., 1 , , ~

) ( ) ( ) ( ) (

  

 

     



n j i K ij n j i ij j i

s s K C

1 ) ( 1 2 1

2

~ ) (

j k l j 1 1



 

K 2 3 4

(Number of Clusters)

2 3 4 C(K) 0.93 0.90 0.91

R lt f P d PCA Result of Proposed PCA

40 ponent 7 10 20 40 mponent 7 10 11 20 Principal Comp 1 3 5 8 9 11 20 Principal Comp 1 3 5 8 9 11 −20 Second P 2 4 5 6 −20 Second P 2 4 5 6 −250 −200 −150 −100 −50 First Principal Component −250 −200 −150 −100 −50 First Principal Component

Result of Cluster 1 Result of Cluster 2

Results of Weights g

40 nt 7

2 nt 7 8

20 4 ipal Component 1 3 8 9 10 11

1 ipal Component 1 5 8 9 10 11

−20 Second Principa 1 2 4 5 6

−2 −1 Second Principa 1 2 3 4 6

−250 −200 −150 −100 −50 First Principal Component S 4

−4 −2 2 4 −2 Se 6

First Principal Component

First Principal Component

1

 



u w

K k m ik

Proposed PCA Centers Method

1 1





 

u w

n i K k m ik i

Hard Clustering Fuzzy Clustering

) , 1 (   m

Comparison of Cumulative Proportion Comparison of Cumulative Proportion

O di PCA Proposed PCA Ordinary PCA

(Centers Method) (Centers Method)

0 86 0 82 0.86 0.82

Conclusions Conclusions Conclusions Conclusions

(1) Propose a PCA based on Fuzzy Clustering C id i Di i il it St t i Hi h Considering Dissimilarity Structure in Higher Dimensional Space (2) Numerical Examples p

Kushiro‐Marshland

Landsat Data; 1024 X 1024 pixels, 7 Kinds of Lights, July ‐ October, 1993

Result of Proposed PCA for the First Component p p

~ ˆ

 

 

Function Joint Empirical Weighted : ) ( ) , ( 1 ) , ( ~ ) , ( ~ ) )( ( ˆ ), ˆ ( ˆ

1

  

     

    

n i b a i i b a b a b a b b a a ab ab

i Z y y I w n y y f dy dy y y f y y y y c c C ) (

1  i

i Z n

) , 1 ( ,

1

  

 



m u u w

n K m K k m ik i 1 1



 

u

i k ik

Result of Proposed PCA for the Second Component

) , ( ~ ) )( ( ˆ ), ˆ ( ˆ

 

 

   

b b b b b b

dy dy y y f y y y y c c C Function Joint Empirical Weighted : ) ( ) , ( 1 ) , ( ~ ) , ( ) )( ( ), (

1

  

    



n i b a i i b a b a b a b b a a ab ab

i Z y y I w n y y f dy dy y y f y y y y c c C

) , 1 ( ,

1

  

 



m u u w

n K m K k m ik i 1 1



 

u

i k ik

Result of Centers Method for the First Component p

) ( ) )( ( ~ ) ~ ( ~

 

 

d d f C Function Joint Empirical : ) ( ) , ( 1 ) , ( ) , ( ) )( ( ~ ), ~ (

1

  

   

    

n i b a i b a b a b a b b a a ab ab

i Z y y I n y y f dy dy y y f y y y y c c C

i wi   , 1

) (

1  i

i Z n

Result of Centers Method for the Second Component p

) ( ) )( ( ~ ) ~ ( ~

 

 

d d f C Function Joint Empirical : ) ( ) , ( 1 ) , ( ) , ( ) )( ( ~ ), ~ (

1

  

   

    

n i b a i b a b a b a b b a a ab ab

i Z y y I n y y f dy dy y y f y y y y c c C

i wi   , 1

) (

1  i

i Z n