2013 2 Vol. 16, No. 1, February 2013

http://cmr.ba.ouhk.edu.hk

1

GM(1,1)

K-Means

(1)GM(1,N)

(2)

GM(1,1)

(3)

(4)(5)

GM(1,N)

______________________________________

2

(Data) (Information)

(Ratio Analysis)

(correlation)

(2006)

5 (2007)

Beaver (1966)

////

/Altman (1968) 22

(Multiple discriminant analysis, MDA) 5

Ohlson (1980) 105

2048 9 Logic

Collins Green (1982) Logistic

(Logistic)

(2006)

3

2000 2004 2005 2007

(1995)

41

Logic -

(2007)

(Cascaded Logistic Model) 43 104

10

(Logit model)

(Cascaded Logistic Model)

1982 (1986)

(1999) 1993

1996 15

(2006)

B/B

GM(1,1)

(2009) 50

GM(1,1)

4

GM(1,1) GARCH(1,1)

(2011)

GM(1,1) SSVR

GM(1,1) GARCH(1,1)

GM(1,1)

K-Means

1.

2.

3.

4.

2009 20

5

2006 2009

2008 2009

2006 2007

Deng (1993) 4

4 5 6

5

1

1

4 5 6

2008 3Q

2008 4Q

2009 1Q

2009 2Q

2009 3Q

2008 2Q

2008 3Q

2008 4Q

2009 1Q

2009 2Q

2009 3Q

2008 1Q

2008 2Q

2008 3Q

2008 4Q

2009 1Q

2009 2Q

2009 3Q

2007 3Q

2008 3Q

2008 4Q

2009 1Q

2009 2Q

2009 3Q

2006 4Q

2007 1Q

2007 2Q

2007 3Q

2007 4Q

2006 3Q

2006 4Q

2007 1Q

2007 2Q

2007 3Q

2007 4Q

6

20

GM(1,N)

GM(1,1)

1

1

20

20

GM(1,N)

GM(1,1)

K-Means

7

GM(1,N) 20

60% 12

()

(1996)

1

))(),...,1(()( 1 mxxkx ii

2

|)()(|minmin 0min kxkx jkij

|)()(|maxmax 0max kxkx jkij

3 |)()(|)( 00 kxkxk jj

4 () 0.5

8

max0

maxmin0

)())(),((

kkxkxr

j

j

5

n

k

jj kxkxrn

xxr1

00 ))(),((1

),( j=1,,m , k=1,,n

6

7

()

(1) ()

)(min)(max

)(min)()(

kxkx

kxkxkx

ii

iii

)(max kxi )(min kxi ()

(2) ()

)(min)(max

)()(min)(

kxkx

kxkxkx

ii

iii

)(max kxi )(min kxi ()

(3) (

OB)

OBkx

OBkxkx

i

ii

)(max

|)(|)(

)(max kxi )(min kxi ()

20 20 MATLAB

9

12

() GM(1,N)

3 (1) GM(1,1)

(2) GM(1,N) N

(3) GM(0,N) N

GM(1,N)

GM(1,N) GM(1,N)

N

i

ii kxbaxdt

dx

2

)1()1(

1

)1(

)(

i. a ib

)()1(1 kx )()1( kxi

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

)0(1

1

2

1

)0()0()1(

n

kk k

kxkxkxx

)()0( kxi Ni ,...,3,2,1 )()0(

1 kx

)(),...(),(),( )0()0(4)0(

3

)0(

2 kxkxkxkx N GM(1,N)

1

)}(),...,2(),1({

...

)}(),...,2(),1({

)}(),...,2(),1({

)}(),...,2(),1({

)0()0()0()0(

)0(

3

)0(

3

)0(

3

)0(

3

)0(

2

)0(

2

)0(

2

)0(

2

)0(

1

)0(

1

)0(

1

)0(

1

kxxxx

kxxxx

kxxxx

kxxxx

NNNN

nk ,...,3,2,1

10

2 (1-AGO)

)}(),...,2(),1({

...

)}(),...,2(),1({

)}(),...,2(),1({

)}(),...,2(),1({

)1()1()1()1(

)1(

3

)1(

3

)1(

3

)1(

3

)1(

2

)1(

2

)1(

2

)1(

2

)1(

1

)1(

1

)1(

1

)1(

1

kxxxx

kxxxx

kxxxx

kxxxx

NNNN

nk ,...,3,2,1

3

GM(1,N)

N

i

ii kxbkazkx2

)1()1(

1

)0(

1 )()()(

2),1(5.0)(5.0)( )1(1)1(

1

)1(

1 kkxkxkz

4

)(...)()()(

...

)3(...)3()3()3(

)2(...)2()2()2(

)1()1(

22

)1(

1

)0(

1

)1()1(

22

)1(

1

)0(

1

)1()1(

22

)1(

1

)0(

1

nxbnxbnaznx

xbxbazx

xbxbazx

NN

NN

NN

N

2

(1)

N

(1)

2

(1)

1

(1)

N

(1)

2

(1)

1

(1)

N

(1)

2

(1)

1

)0(

1

)0(

1

)0(

1

b

...

b

a

(n) x... (n) x(n)z-

....

(3) x... (3) x(3)z-

(2) x... (2) x(2)z-

)n(

...

)3(

)2(

x

x

x

N

T1-T YBB)(Ba

11

N

2

(1)

N

(1)

2

(1)

1

(1)

N

(1)

2

(1)

1

(1)

N

(1)

2

(1)

1

)0(

1

)0(

1

)0(

1

b

...

b

a

(n) x... (n) x(n)z-

....

(3) x... (3) x(3)z-

(2) x... (2) x(2)z-

)n(

...

)3(

)2(

aB

x

x

x

YN

20 20 MATLAB

12

() GM(1,1)

GM(1,1)

( GM )

1 0x kx 0

n,...,x,xxx 0000 21

n ,...,n,k 21

2 (1-AGO)

n,...,x,xxx 1111 21

k

m

mxkx1

01

,...,km 1 ,...,n,k 21

3

12

bax

dt

dx 1

1

bkazkx 10 ,...,n,k 21

b

aa a b

11 111 kxkkxkkz ,...,n,k 32

z k

0.5

4

1-

13-

12-

3

2

0

0

0

nz

z

z

nx

x

x

nx

x

x

YN

0

0

0

3

2

1-

13-

12-

nz

z

z

B

a b

b

aYBBBa N

TT 1-

5 a, b

a

be

a

bxkx ak

-01 11 ,...,n,k 21

13

11 01 xx

6 kx 0

kxkxkx 110 11

()

(1996) GM(1,1)

( FGM )

GM GM

1 GM

)(, ... ),2(),1( 0000 nxxxx

)(, ... ),2(),1( 0000 nxxxx

2Er

GM Er

n,...,Er,ErErEr 32

14

kxkxkEr 00 ,...,n,k 32

3

ak

i

ii kT

ibk

T

iaakrE

1

0

2sin

2cos

2

1

nknT/n-ka ,......3,2,1121

ak

rECP aa

T

*n*k

T

*n*k

T

*n*

T

*n*

T

*n

T

*n

T

**k

T

**k

T

**

T

**

T

*

T

*

T

**k

T

**k

T

**

T

**

T

*

T

*

P

aa

aa

aa

a

2sin

2cos

22sin

22cos

2sin

2cos

2

1

32sin

32cos

322sin

322cos

32sin

32cos

2

1

22sin

22cos

222sin

222cos

22sin

22cos

2

1

aa kka

,b,...,a,b,a,b,aaC 22110

r1 EPPPC TaaTaa

ak

i

ii kT

ibk

T

iaakrE

1

0

2sin

2cos

2

1

,....4,3,2 00 kkrEkxkxa

15

)(

0kxa GM(1,1)

(Cluster Analysis)

(natural grouping)

(metric space)

(2000) (hierarchical

clustering) (nonhierarchical clustering)

K

K-Means

K-Means MacQueen 1967 K

(centroid or seed)

K-Means

1 K (initial cluster)K

2 (mean) (

)

3

16

K-means

20

GM(1,N) 12 2

2

GM(1,N)

0.7093 () 8.1308

0.3078 () 2.7574

() 0.2921 () 2.2374

0.2795 0.6112

0.2588 () 0.4044

0.2292 () 0.1537

0.2157 0.1086

0.188 () 0.0813

0.1597 0.0343

() 0.0768 0.0298

0.0659 0.0274

() 0.0360 0.0254

-

20

2008 2008 2009

2009 MATLAB

MATLAB GM(1,1) 2009

2009 Weka K-Means

4 3 (1996)

FGM 5 FGM

17

3 4 2009

GM(1,N)

98 3Q GM(1,1) 98 3Q GM(1,1)

0 0 3 0

1 1 1 1

1 1 1 1

3 1 1 1

1 1 1 1

3 1 3 1

3 1 3 1

1 1 1 1

3 1 3 1

0 3 3 3

3 1 3 1

0 1 3 1

3 1 3 1

0 0 0 0

1 0 3 0

2 2 2 2

1 1 3 1

0 0 3 0

0 1 3 1

2 0 3 0

60% 54%

K-Means

20 202C 190

2020

GM(1,1) 2009

4 2009

1 0

5 GM(1,1) ( 4)

javascript:setCode('2357');javascript:setCode('2324');javascript:setCode('2382');javascript:setCode('2356');javascript:setCode('2353');javascript:setCode('2301');javascript:setCode('2352');javascript:setCode('3231');javascript:setCode('2315');javascript:setCode('2331');javascript:setCode('2377');javascript:setCode('8008');javascript:setCode('1604');javascript:setCode('2362');javascript:setCode('2376');javascript:setCode('2474');javascript:setCode('2385');javascript:setCode('3005');javascript:setCode('2395');javascript:setCode('2387');

18

( 5)

4 5

114 114/190=0.6 60%(

)

4 2009

0 1 1 3 1 3 3 1 3 0 3 0 3 0 1 2 1 0 0 2

0 1 1 0 1 1 0 1 1 3 0 0 0 1 1 0 1 1 0 1 3 0 0 0 1 0 1 3 0 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 3 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 3 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 3 0 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1

javascript:setCode('2324');javascript:setCode('2324');javascript:setCode('2324');javascript:setCode('2324');javascript:setCode('2382');javascript:setCode('2382');javascript:setCode('2356');javascript:setCode('2356');javascript:setCode('2356');javascript:setCode('2353');javascript:setCode('2353');javascript:setCode('2301');javascript:setCode('2301');javascript:setCode('2301');javascript:setCode('2301');javascript:setCode('2352');javascript:setCode('2352');javascript:setCode('2352');javascript:setCode('3231');javascript:setCode('3231');javascript:setCode('2315');javascript:setCode('2315');javascript:setCode('2315');javascript:setCode('2315');javascript:setCode('2331');javascript:setCode('2331');javascript:setCode('2377');javascript:setCode('2377');javascript:setCode('2377');javascript:setCode('2377');javascript:setCode('8008');javascript:setCode('8008');javascript:setCode('8008');javascript:setCode('1604');javascript:setCode('1604');javascript:setCode('2362');javascript:setCode('2362');javascript:setCode('2376');javascript:setCode('2376');javascript:setCode('2474');javascript:setCode('2474');javascript:setCode('2474');javascript:setCode('2474');javascript:setCode('2385');javascript:setCode('2385');javascript:setCode('2385');javascript:setCode('2385');javascript:setCode('3005');javascript:setCode('3005');javascript:setCode('3005');javascript:setCode('3005');javascript:setCode('2395');javascript:setCode('2395');javascript:setCode('2387');javascript:setCode('2387');javascript:setCode('2387');javascript:setCode('2387');javascript:setCode('2357');javascript:setCode('2324');javascript:setCode('2382');javascript:setCode('2356');javascript:setCode('2353');javascript:setCode('2301');javascript:setCode('2352');javascript:setCode('3231');javascript:setCode('2315');javascript:setCode('2331');javascript:setCode('2377');javascript:setCode('8008');javascript:setCode('1604');javascript:setCode('2362');javascript:setCode('2376');javascript:setCode('2474');javascript:setCode('2385');javascript:setCode('3005');javascript:setCode('2395');javascript:setCode('2387');

19

5 GM(1,1) 98 ()

0 1 1 1 1 1 1 1 1 3 1 1 1 0 0 2 1 0 1 0

0 1 1 0 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 3 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1

GM(1,1) 60%GM(1,N)

GM(1,1) 54% 4

-

5 GM(1,N)

GM(1,1) FGM 2008

2008 2008 2009 2009

2009 6

5 FGM

GM(1,1) GM(1,N)

5

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20

GM(1,N) FGM

6 5 2009

GM(1,N)

98 3Q

GM(1,1)

FGM

98 3Q

GM(1,1)

FGM

0 1 1 3 1 1

1 2 2 1 1 1

1 1 1 1 0 0

3 1 1 1 0 0

1 1 1 1 0 0

3 1 1 3 1 1

3 2 1 3 0 0

1 1 1 1 0 0

3 1 1 3 1 1

0 3 3 3 3 3

3 1 1 3 1 1

0 1 1 3 1 1

3 1 1 3 1 1

0 0 0 0 1 1

1 2 2 3 0 1

2 2 2 2 2 2

1 1 1 3 1 1

0 1 1 3 1 1

0 1 1 3 1 1

2 2 2 3 1 1

54% 55% 66% 72%

-

GM(1,N)

FGM 2008 2008

2008 2008 2009 2009

2009 ( 7)

64% 72% 6

5

(2011)

javascript:setCode('2357');javascript:setCode('2324');javascript:setCode('2382');javascript:setCode('2356');javascript:setCode('2353');javascript:setCode('2301');javascript:setCode('2352');javascript:setCode('3231');javascript:setCode('2315');javascript:setCode('2331');javascript:setCode('2377');javascript:setCode('8008');javascript:setCode('1604');javascript:setCode('2362');javascript:setCode('2376');javascript:setCode('2474');javascript:setCode('2385');javascript:setCode('3005');javascript:setCode('2395');javascript:setCode('2387');

21

7 GM(1,N) FGM 6 98

98 3Q FGM 98 3Q FGM

0 0 3 1

1 1 0 1

1 1 3 3

3 1 0 1

1 1 1 1

3 0 2 2

3 0 1 0

1 1 0 0

3 0 0 0

0 3 2 0

64%

-

5

2007 2008 2008

2009 2009 GM(1,N)

GM(1,1) FGM K-Means

8 2009

5 5

GM(1,N) GM(1,N)

GM(1,1) 54% 66% 54% 60%

FGM 55% 72% 56% 70%

8

FGM GM(1,1)

-

2009 2008

2008 9

javascript:setCode('2357');javascript:setCode('2377');javascript:setCode('2324');javascript:setCode('8008');javascript:setCode('2382');javascript:setCode('1604');javascript:setCode('2356');javascript:setCode('2362');javascript:setCode('2353');javascript:setCode('2376');javascript:setCode('2301');javascript:setCode('2474');javascript:setCode('2352');javascript:setCode('2385');javascript:setCode('3231');javascript:setCode('3005');javascript:setCode('2315');javascript:setCode('2395');javascript:setCode('2331');javascript:setCode('2387');

22

2008 9 14

2008 9 9 15 9 17

2007

4 5 2006

2006 2007 2007 2007

2007 GM(1,N)

GM(1,1) FGM

9 FGM 5

4 FGM

2009

GM(1,N) 5

FGM 7

9 2007

4 5

GM(1,N) GM(1,N)

GM(1,1) 64% 54% 65% 56%

FGM x x 59% 70%

http://zh.wikipedia.org/zh-tw/%E9%9B%B7%E6%9B%BC%E5%85%84%E5%BC%9Fhttp://zh.wikipedia.org/zh-tw/%E7%BE%8E%E6%9E%97%E8%AD%89%E5%88%B8http://zh.wikipedia.org/zh-tw/%E7%BE%8E%E5%9C%8B%E9%8A%80%E8%A1%8Chttp://zh.wikipedia.org/w/index.php?title=2008%E5%B9%B49%E6%9C%88%E5%85%A8%E7%90%83%E8%82%A1%E5%B8%82%E5%A4%A7%E5%B4%A9%E7%9B%A4&action=edit&redlink=1

23

1.

GM(1,N)

2. GM(1,N)

3. GM(1,1)

4. 5

4

6 5

5.

6. GM(1,N)

1.

2.

3.

24

1. 20

2.

3.

1.

2.

3.

4.

25

1996GM(1,1)1996

91-101

1999

2011

14

1 1-27

2009 50

2007

2007 Cascaded Logistic Model

10 2 1-15

2000

2006

1995

5 2 205-219

2006 B/B

1986

1996

2006-

26

Altman, E. I. (1968). Financial ratio, discriminant analysis and the predication of

corporate bankruptcy, Journal of Finance, 23(4), 589609.

Beaver, W. H. (1966). Financial ratio as predictors of failure, Empirical research in

accounting: selected study, Journal of Accounting Research, 4, 71111.

Collins, R. A., & Green, R. D. (1982). Statistical methods for bankruptcy

forecasting, Journal of Economics and Business, 34(4), 349354.

Deng, J. L. (1993). Grey differential equation, Journal of Grey System, 5(1), 114.

MacQueen, J. B. (1967). Some methods for classification and analysis of

multivariate observations. Proceedings of 5th Berkeley Symposium on

Mathematical Statistics and Probability, Berkeley, University of California Press, 1,

281-297.

Ohlson, J. A. (1980). Financial ratios and the probabilistic prediction of bankruptcy,

Journal of Accounting Research, 18(1), 109131.