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On a new grey clustering evaluation model
Sifeng Liu et al
Present by Naiming Xie
E-mail:sifeng.liu@dmu.ac.uk; sfliu@nuaa.edu.cn
Institute for Grey System Studies
This work was supported by a project of Marie
Curie International Incoming Fellowship within the
7th European Community Framework Programme
entitled “Grey Systems and Its Application to Data
Mining and Decision Support”（Grant No. FP7PIIF-GA-2013-629051）, and
a project of the Leverhulme Trust International
Network entitled “Grey Systems and Its
Applications” (IN-2014-020)
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Outlines
1 Introduction
2 The three kind of whitenization functions
3 The mixed center-point triangular
whitenization functions
4 Steps of the grey clustering evaluation
model using mixed center-point triangular
whitenization functions
5 An Example
1 Introduction
Grey clustering evaluation models, a
kind of uncertain clustering evaluation
model which using whitenization
function is more suitable to be used to
solve the problem of clustering
evaluation with poor information, are
used widely for uncertain systems
analysis.
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1 Introduction
Professor Deng Julong proposed the grey variable
clustering model in 1986 using whitenization
functions which is similar to the membership
function (L.A. Zadeh, 1965) or the probability
density function.
Xiao Xinping(1997),Dong Fenyi(2010) and others
are improved and optimized grey cluster evaluation
models from different perspectives. Zhang Qishan
studied measurement problem of Grey
Characteristics of Grey Clustering Result(Q.S.
Zhang, 2002).
1 Introduction
The most used grey clustering model
1)The grey variable weight clustering model(J.L.
Deng, 1986) ;
2)The grey fixed weight clustering evaluation
model(S.F. Liu, 1993);
3)The grey cluster evaluation model using endpoint triangular whitenization functions(S.F. Liu et
al, 1993),;
4)The grey cluster evaluation model using centerpoint triangular whitenization functions(S.F. Liu,
N.M. Xie, 2011) etc.
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1 Introduction
Grey variable weight clustering model is
applicable to the problems with such criteria
that have the same meanings and dimensions.
When the criteria for clustering with different
meanings, dimensions, grey fixed weight
clustering evaluation model and grey cluster
evaluation model using triangular
whitenization function are suitable.
1 Introduction
Compared with grey variable weight clustering model and
grey fixed weight clustering model, Grey clustering
evaluation model using end-point triangular whitenization
functions suitable for the situation that all grey boundary is
clear, but the most likely points belonging to each grey
class are unknown; grey clustering evaluation model using
center-point triangular whitenization functionsis suitable
for those problems that is easier to judge the most likely
points belonging to each grey class, but the grey boundary
is not clear.
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1 Introduction
S.F.Liu, B.J.Li et al presented a mixed whitenization
function in 1998, but the mixed model hadn’t drew
into the center-point triangular whitenization
functions for 17 years.
2 Three kind of whitenization functions
1) The lower measure whitenization function
f jk [  , , x jk ( 3 ), x jk ( 4 )]
0



f jk (x) 1
 k
 xj (4) x
xk (4) xk (3)
j
j
x[0, xkj (4)]
x[0, xkj (3)]
x[xkj (3), xkj (4)]
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2 Three kind of whitenization functions
2) The triangular (moderate measure) whitenization
function
f
k
j
[ x kj (1), x kj ( 2 ),  , x kj ( 4 )]
0

k
 x  xj (1)
 xk (2)  xk (1)

j
k
f j (x)   j

 xk (4)  x
 j
 xkj (4)  xkj (2)
x [xkj (1), xkj (4)]
x [xkj (1), xkj (2)]
x [xkj (2), xkj (4)]
2 Three kind of whitenization functions
3) The upper measure whitenization function
f
k
j
[ x kj (1) , x kj ( 2 ) ,  ,  ]
 0,
xxkj(1)
 k
 xxj (1)
k
fj (x) k
, x[xkj(1),xkj(2)]
k
xj (2)xj (1)
k
 1,
x
x

j (2)

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3 The mixed center-point triangular
whitenization functions
Assume that the turning point of grey class 1, grey
1
s
class s are j，j and the center-point

2
j
,
3
j
, , 
s  1
j
, s 1})
of grey class k(k {2,3，
respectively.
3 The mixed center-point triangular
whitenization functions
For grey class 1 and grey class s, we take lower
1
1
2
measure whitenization function f j [  ,  ,  j ,  j ] , and
s s1 s
upper measure whitenization function f j [j ,j ,,]
as corresponding whitenization function, and for
 , s  1}) , we take triangular
grey class k ( k  {2,3，
whitenization function fjk(kj 1,kj ,,kj 1 )
as
corresponding whitenization function.
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3 The mixed center-point triangular
whitenization functions
y
y  f j1(x) y  f j 2 (x)
y  f j k (x)
aj j1 j 2 j3
jk1j k jk1
y  f j s1(x) y  f j (x)
s
1
o
j s2 j s1 js bj x
图 6.4.2 中心点混合三角白化权函数示意图
4 Steps of the new grey clustering
evaluation model
Steps of the new grey clustering evaluation model
using mixed center-point triangular whitenization
functions as follows
Step 1: Determine the turning point of grey class
1, grey class s 1j，sj and the center-point

2
j
,
3
j
, , 
s  1
j
, s 1})
of grey class k(k {2,3，
respectively according to the evaluation
requirements and the grey class number s.
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4 Steps of the new grey clustering
evaluation model
Step 2: Setting the whitenization function of grey
class 1, 2, …,s
f j1[  ,  , 1j ,  2j ]
 , s  1})
fjk(kj 1,kj ,,kj 1 ) k ( k  {2,3，
f js[sj1,sj ,,]
Step 3: Determine the clustering weight for
each index w , j  1, 2 ,  , m
j
4 Steps of the new grey clustering
evaluation model
Step 4: Compute clustering coefficient  i of object i
regarding to grey class k
k

k
i

m

j 1
f
k
j
( x ij ) w
j

ik }  ik , determine the
Step 5: According to max{
1ks
object i belonging to grey class k *
Step 6:If there are several objects belonging to the
same grey class, then sort them according grey
clustering coefficient or decision coefficient with
synthetic measure and principle of maximum value.
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5 An Example
The evaluation of project for discipline construction
of an university
There are 6 primary indicators to reflect the
performance of a project for discipline construction
based on extensive surveys, 1)faculty, 2)scientific
research, 3)students cultivation, 4)disciplines
platform construction, 5)conditions for construction
and 6)academic communication.
The corresponding weights are 0.21, 0.24, 0.23,
0.14, 0.1,and 0.08 respectively.
5 An Example
Disciplines construction evaluation
Academic communication
Construction conditions Disciplines platform Students cultivation Scientific research
Faculty Fig.3 Evaluation indicator system of project for discipline construction
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5 An Example
All the evaluation scores of the 6 indexes of 41
projects for discipline construction are laid in the
interval of [40, 100]
The evaluation results are divided into four grey
class of 1, 2, 3, 4 corresponding to class poor,
moderate, good and excellent, respectively.
Step 1: Determine the turning point of grey class 1
and class 4, then the center-point of class 2, class 3
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1j  60 j2  70,3j  80  j  90
5 An Example
Step 2: Setting the whitenization
function of grey class 1,2,3,4
 0
x [40,70]

1
x [40,60]
f（
x

)
1

j
 70  x x [60,70]
70  60


0

 x  70
3
f（
j x)  
 80  70
 90  x
 90  80

x  [ 70 ,90 ]
x  [ 70 ,80 ]
x  [80 ,90 ]


0

 x  60
2
f（
j x)  
 70  60
 80  x
 80  70

x  [ 60 ,80 ]
x  [ 60 , 70 ]
x  [ 70 ,80 ]
 0
x [80,100]
 x  80
f（x)  
x [80,90]
9080 x [90,100]
 1
4
j
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5 An Example
Step 3: The weight for each index are 0.21, 0.24,
0.23, 0.14, 0.1,and 0.08 respectively.
Step 4: Compute clustering coefficient
Indicator
name
Faculty
value
81
Academic
Scientific
Students Disciplines Construction
conditions communication
research cultivation platform
87
92
78
74
53
5 An Example
grey class
x1
x2
x3
x4
x5
x6
i
excellent
good
medium
poor
0.1
0.9
0
0
0.7
0.3
0
0
1.0
0
0
0
0
0.8
0.2
0
0
0.4
0.6
0
0
0
0
1.0
0.419
0.413
0.088
0.080
 k    i4  0.419  i3  0 . 413
Step 5: max
1 k  4 i
It also shows that the execution effect of the
project is situated between grey class "excellent"
and "good“.
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Thanks for Attention!
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