semi-partial correlation of fuzzy sets - wseas · semi-partial correlation of fuzzy sets nancy p....

Semi-Partial Correlation of Fuzzy Sets

NANCY P. LIN, HAO-EN CHUEH Department of Computer Science and Information Engineering,

Tamkang University, 151 Ying-chuan Road, Tamsui, Taipei,

TAIWAN, R.O.C.

Abstract: - In fuzzy data analysis, a simple correlation coefficient provides us with a good sense of linear relationship between two fuzzy attributes, while a partial correlation coefficient shows the relationship between two fuzzy attributes when the influences of other fuzzy attributes are partialed out from both of the two attributes. But, in some practical applications, we need to use the correlation between two fuzzy attributes when the influences of other fuzzy attributes are removed from only one of the two fuzzy attribute, which is called the semi-partial correlation. The simple and partial correlation analyses on Zadeh’s fuzzy sets have been discussed in previous works [3, 4]. Here, we turn to the analysis of semi-partial correlation between two fuzzy attributes. In this paper, the semi-partial correlation coefficient among fuzzy attributes is analyzed and derived using the membership grades of the fuzzy attributes. Data from previous experiments [3, 4] are used to discuss the semi-partial correlation on fuzzy data set. Key-Words: - Fuzzy set, Linear relationship, Simple correlation, Partial correlation, Semi-partial correlation 1 Preliminary When we deal with crisp data, it is very common to find the correlation coefficient between attributes. The correlation coefficients defined on ordinary crisp sets have been well discussed in the conventional statistics [1, 2, 5]. With data volumes get larger so rapidly, many data may be fuzzy [6, 7, 8, 9] but useful, waiting to be explored, and methods to investigate these fuzzy data are certainly needed. In fuzzy data analysis, when researchers are concerned with the way which two fuzzy attributes are related to each other, the correlation analyses defined on fuzzy sets are must. A simple correlation coefficient [3] provides us with a very good sense of linear relationship between two fuzzy attributes. It can show not only the strength of the linear relationship between two fuzzy attributes but also the direction of the relationship. The simple correlation coefficient shows a positive score if two fuzzy attributes are directly related, and shows a negative score if the attributes bear an inverse relationship to each other. It will be close to zero if two attributes have no systematic linear relationship to each other. In real applications, attributes other than the two under consideration are also responsible for the observed relationship, and the effects of these fuzzy attributes may influence the relationship between the observed fuzzy attributes. The question then might be asked, “If we can hold other fuzzy attributes constant, what will be the value of the relationship between our

interested fuzzy attributes?” Therefore, the analysis of fuzzy partial correlation [4] is thus developed to show the relationship between two fuzzy attributes when the influences of other fuzzy attributes are partialed out from the observed fuzzy attributes. Now we turn to another important variant of correlation analysis which is called the semi-partial correlation. Such a correlation has different interpretation from the pervious ones we have presented. The coefficient of semi-partial correlation can show the relationship between two fuzzy attributes when the influences of other fuzzy attributes are not removed from both of the two interested fuzzy attributes. A semi-partial correlation is more complicated and has larger variety than a partial correlation can explain, which will be analyzed in this paper. Thus, semi-partial correlation on fuzzy sets is considered here, but first, the simple and partial correlation coefficients on fuzzy sets [3, 4] are reminded in Section 2. Definitions for the semi- partial correlation coefficients on fuzzy sets are developed in Section 3. In Section 4, data from previous experiments [3, 4] are used to compute the semi-partial correlation coefficients as we defined. The conclusions are then given in Section 5. 2 Simple and Partial Correlations of Fuzzy Sets

Proceedings of the 6th WSEAS International Conference on Applied Informatics and Communications, Elounda, Greece, August 18-20, 2006 (pp208-213)

Suppose fuzzy sets A , B and FC ⊆ , where F is a fuzzy space. The fuzzy sets A , B andC are defined on the domain of a crisp universal set X with membership functions Aµ , Bµ and Cµ , then A , B andC can be expressed as:

)|))(,{( XxxxA A ∈= µ ,

)|))(,{( XxxxB B ∈= µ ,

)|))(,{( XxxxC C ∈= µ ,

where Aµ , Bµ , ]1,0[: →XCµ . In previous papers [3, 4], the simple correlation coefficient and the partial correlation coefficient of two fuzzy sets A and B are defined as follows. 2.1 Simple correlation coefficient of fuzzy sets Assume that (( )(),(, 111 xxx BA µµ ), ( ),(, 22 xx Aµ

)( 2xBµ ), … … , ( )(),(, nBnAn xxx µµ )) is a random sample drawn from a crisp universal set X , where Aµ and Bµ are the grades of the membership functions of the fuzzy sets A and B . Then the simple correlation coefficient between fuzzy sets A and B is defined by

22,

,BA

BABA

sssr

⋅= (1)

, where

1))(())((1

, −∑ −⋅−

= =

nxx

sni BiBAiA

BAµµµµ

(2)

is the covariance of fuzzy sets A and B . AAA ss ,2 = ,

BBB ss ,2 = are the variances of fuzzy attributes A and

B , respectively. Properties of the simple correlation coefficient can be referenced in [3]. If we know that the effect of fuzzy attribute C also accounts for the observed relationship between A and B . What would happen to the relationship

between A and B , if the fuzzy attribute C was held constant? This is explained in the next subsection [4]. 2.2 Partial correlation coefficient of fuzzy sets Assume that simple correlation coefficients are computed between pairs of the fuzzy attributes A , B and C according to formula (1), namely, BAr , , CBr ,

and CAr , . Then the partial correlation coefficient between the fuzzy attributes A and B , when the effects of fuzzy attribute C on fuzzy sets A and B are partialed out, is defined as

2,

2,

,,,,

)(1)(1 CBCA

CBCABACBA

rr

rrrr

−⋅−

⋅−=• (3).

Properties of the partial correlation coefficient can be referenced in [4]. A simple correlation coefficient provides us with a good sense of linear relationship between two fuzzy attributes, while a partial correlation coefficient of two fuzzy attributes shows the relationship when the influences of other fuzzy attributes are removed from these two attributes. 3 Semi-Partial Correlation of Fuzzy Sets In this section, we turn to the discussion of the semi-partial correlation. Assume that (( ),(, 11 xx Aµ

),( 1xBµ )( 1xCµ ),( ),(, 22 xx Aµ )(),( 22 xx CB µµ ),

… … , ( ),(, nAn xx µ ),( nB xµ )( nC xµ )) is a random sample drawn from a crisp universal set X , where

Aµ , Bµ and Cµ are the corresponding membership

grades of the fuzzy attributes A , B and C . Here, we are interested in finding the relationship between two fuzzy attributes A and B with the influence of fuzzy attribute C removed from fuzzy attribute B but not fuzzy attribute A . This semi-partial correlation,

)(, CBAr • , is defined in the following subsection. 3.1 First-order semi-partial correlation coefficient of fuzzy sets Assume that the simple correlation coefficients, BAr , ,

CBr , and CAr , of fuzzy attributes A , B and C have been calculated according to formula (1). Then the semi-partial correlation coefficient between the fuzzy attributes A and B with the influence of fuzzy attribute C removed from the fuzzy attribute B can be calculated as:

2,

,,,)(,

)(1 CB

CBCABACBA

r

rrrr

−

⋅−=• (4).

Similarly, the semi-partial correlation =• )(, CABr

2,

,,,

)(1 CA

CBCABA

r

rrr

−

⋅− (5) shows the relationship between


the fuzzy attributes A and B with the influence of the fuzzy attribute C removed from fuzzy attribute A . Properties of the semi-partial correlation coefficient are stated here. First, the semi-partial correlation coefficient, )(, CABr • , is different from

)(, CBAr • because of their different meanings. )(, CBAr • means the relationship between the fuzzy attributes A and B with the influence of fuzzy attribute C

removed from the fuzzy attribute B , but comparatively, )(, CABr • means the relationship

between the fuzzy attributes A and B with the influence of fuzzy attribute C removed from the other fuzzy attribute, A . We can also see the difference between them from their expressions, (4) and (5). Therefore, there are six different semi-partial correlation coefficients among three fuzzy attributes A , B and C , say )(, CBAr • , )(, CABr • , )(, ACBr • , )(, ABCr • ,

)(, BCAr • and )(, BABr • , although there are only three different partial correlation coefficients among the same attributes, say BAr , , CBr , and CAr , .

Second, when CBr , =1, the calculation of )(, CBAr • will cause an error according to formula (4). Thus, the semi-partial correlation coefficient between the fuzzy attributes A and B with the influence of fuzzy attribute C removed from the fuzzy attribute B is not defined, when there is a linear relationship between B andC . Similarly, when CAr , =1, )(, CABr • is also not

defined. Comparatively, when CAr , =0,

=• )(, CBAr2

,

,,,

)(1 CB

CBCABA

r

rrr

−

⋅−

2,

2

,,,

)(1)0(1 CB

CBCABA

r

rrr

−⋅−

⋅−=

2,

2,

,,,

)(1)(1 CBCA

CBCABA

rr

rrr

−⋅−

⋅−=

CBAr •= , . That means, if there is no linear relationship between A andC , then the relationship between the fuzzy attributes A and B with the influence of fuzzy attribute C removed from the fuzzy attribute B is the same as the relationship between the fuzzy attributes A and B with the influence of fuzzy attribute C

removed from both of the two fuzzy attributes A and B . Similarly, when CBr , =0, then )(, CABr • is the same

as CABr •, .

Third, when CAr , =0 and CBr , =0, according to

formulas (4) and (5), then BACBA rr ,)(, =• )(, CABr •= . That is, the two semi-partial correlation coefficients will be the same as the simple correlation coefficient in this situation. Notice that if BAr , =0, that means there is no linear

relationship between A and B , then whether the influence of fuzzy attribute C is removed from fuzzy attribute A or B , it is not necessarily to discuss the semi-partial relationship between fuzzy attributes A and B , in spite that the calculation of )(, CBAr • and

)(, CABr • may not cause any error. Next, we contiune to discuss the semi-partial relationship among four or more fuzzy attributes, and then derive the generalized semi-partial correlation coefficient for fuzzy sets in the following subsection. 3.2 Generalized semi-partial correlation coefficient of fuzzy sets Suppose there are qp + fuzzy sets ,,,, 21 pAAA L

qpp AA ++ ,,1 L defined on the domain of a crisp

universal set X with membership functions ,1Aµ

qpAA +µµ ,,

2L , and then each fuzzy attribute iA can

be expressed as: ))(,{( xxAiAi µ= )}| Xx ∈ , where

]1,0[: →XiAµ , for qpi += L1 .

Assume that ,),(),(,{(21

LkAkAk xxx µµqpA +

µ

|))( kx kx X∈ , }1 nk L= is a random sample of size n from a crisp universal set X with the corresponding membership grades of the qp + fuzzy attributes. Also, the correlation coefficients between pairs of the fuzzy attributes ,,,, 21 pAAA L

,,1 L+pA qpA + , defined on their membership grades have been calculated, and expressed as a matrix form of size )( qp + )( qp +× , R .

=

++++

+

+

qPqpjqpqp

qpijii

qpj

rrr

rrr

rrr

R

,,1,

,,1,

,1,11,1

KKMOMNM

LLMNMOM

LL

, where jir , is the simple correlation coefficient of

the fuzzy attributes iA and jA [3, 4]. Notice that the


simple correlation matrix R is symmetric, and all its diagonal elements, iir , , for qpi += L1 are 1. Next, the simple correlation matrix R need to be partitioned into four parts as follows:

=

2221

1211

RRRR

R

, where 11R is the pp × correlation matrix of fuzzy

attributes iA and jA , ,1 pi L= pj L1= . 12R is

the qp × correlation matrix of fuzzy attributes

iA and jA , ,1 pi L= ,1+= pj ,L qp + . 21R is

the pq × correlation matrix of iA and jA , =i

,,1 L+p ,qp + pj L1= . 22R is the qq ×

correlation matrix of iA and jA , ,1+= pi ,L

,qp + qppj ++= ,,1 L . Here, our objective is to find the relationship between two fuzzy attributes iA and jA , ∈ji ,

},,2,1{ pL , with the influences of other q fuzzy

attributes qpp AA ++ ,,1 L removed from the fuzzy

attribute jA . This semi-partial correlation coefficient,

),,1(, qppjir ++• L , is defined as:

jj

jiqppji

r

rr

,

,),,1(,

'

'=++• L (6)

, where jir ,' is the ji, th element of the pp × matrix

'R . 211

221211' RRRRR ⋅⋅−= − . According to the above definition, if 2=p and

1=q , then we have a similar case as subsection 3.1.

Assume that there are three fuzzy attributes 1A , 2A

and 3A , and a simple correlation matrix of them, say R . First, we divide the matrix R into four parts as we need.

=

3,32,31,3

3,22,21,2

3,12,11,1

rrrrrrrrr

R

[ ] [ ]

=

=

3,3

3,2

3,1

2,31,3

2,21,2

2,11,1

2221

1211

r

rr

rr

rrrr

RRRR

.

We can see that, in this case, 11R is the 22 ×

correlation matrix of 1A and 2A . 12R is the 12 ×

correlation matrix of 1A and 3A , and of 2A and 3A .

21R is the transposed matrix of 12R . 22R is the

correlation of fuzzy attribute 3A and itself, 3,3r . Then, the semi-partial correlation coefficient between fuzzy attributes 1A and 2A with the influence

of the fuzzy attribute 3A removed from 2A will be

2,2

2,1)32(,1

'

'

r

rr =• (7)

, where jir ,' is the ji, th element of the 22 × matrix 'R . Let’s simplify the matrix 'R .

211

221211' RRRRR ⋅⋅−= −

−

=

2,21,2

2,11,1

rrrr [ ]2,31,3

3,33,2

3,1 1rr

rrr

×

×

−

=

2,21,2

2,11,1

rrrr [ ]2,31,3

3,3

3,2

3,3

3,1

rr

r

r

r

r

×

⋅⋅

⋅⋅

−

=

3,3

2,33,2

3,3

1,33,2

3,3

2,33,1

3,3

1,33,1

2,21,2

2,11,1

rrr

rrr

rrr

rrr

rrrr

⋅−

⋅−

⋅−

⋅−

=

3,3

2,33,22,2

3,3

1,33,21,2

3,3

2,33,12,1

3,3

1,33,11,1

r

rrr

r

rrr

r

rrr

r

rrr

.

The semi-partial correlation coefficient, )32(,1 •r , can be obtained as:

2,2

2,1)32(,1

'

'

r

rr =•

3,3

2,33,22,2

3,3

2,33,12,1

rrr

r

rrr

r

⋅−

⋅−

=

11

1

2,33,2

2,33,12,1

rr

rrr

⋅−

⋅−

=

2,33,2

2,33,12,1

1 rr

rrr

⋅−

⋅−=

23,2

2,33,12,1

1 r

rrr

−

⋅−= .


Notice that the all diagonal elements of R are 1, that is 13,32,21,1 === rrr . Clearly, we obtain a same result as subsection 3.1. 4 Experiment Data from previous experiments [3, 4] are used here to compute and discuss the semi-partial correlation coefficients as we defined. First, the main points of the previous experiments are given here again. A random sample has been obtained with a questionnaire which is designed for evaluating university students’ feeling toward their objective accomplishment by their membership in university activities. There are four parts, 33 questions in this questionnaire. Part I includes 8 questions on objective accomplishment. Part II includes 10 questions on group atmosphere. Part III includes 10 questions on characteristics of a group leader. Part IV includes 5 questions on environmental conditions. For each question, we provide 7 possible choices for them to choose from to show their agreement with our statement. All responses to the questionnaire are fuzzified by the following 1Aµ (8), showing the level of agreement to these questions. But, some responses are transformed by the following 2Aµ (9), when the questions are asked in a reversed way.

<≤≤≤

<≤

−+

−

=.75,53

,31

2.02.01.01.0

4.02.0

)(1

xforxfor

xfor

xx

x

xAµ (8)

<≤≤≤<≤

−−−

=.75,53,31

2.04.11.09.02.02.1

)(2

xforxforxfor

xxxx

xAµ (9)

To simplify, only four fuzzy attributes are mainly considered to represent the four parts in the questionnaire. Namely, O represents the objective accomplishment, A represents the group atmosphere, C represents the comments on leader's characteristic, and E represents the group environment. In each part, the mean value of the membership grades which corresponds to the answers to the questions is treated as the value of each main attribute. In previous experiments, the simple correlation matrix of fuzzy attributesO , A , C and E , say R , has been obtained as the following one.

=

000.14662.02758.01409.04662.0000.11853.02086.02758.01853.0000.14850.01409.02086.04850.0000.1

ECAO

ECAO

R

Here, what we are interested is to obtain the semi- partial relationships between the fuzzy attributesO , A , C and E . According to formula (4), these

targeted semi-partial correlation coefficients can be obtained, and shown as follows:

,4542.0)(, =•CAOr ,4563.0)(, =•COAr

,1208.0)(, =•ACOr ,1357.0)(, =• AOCr

,4641.0)(, =•EAOr ,4506.0)(, =•EOAr

,0074.0)(, =•AEOr ,0082.0)(, =•AOEr

,1616.0)(, =•ECOr ,1444.0)(, =•EOCr

,0493.0)(, =•CEOr .0446.0)(, =•COEr In [3, 4], the partial correlation coefficients between the fuzzy attributes O , A , C and E have also been obtained. At that time, when the influence of fuzzy attribute A or C is removed from the simple correlation coefficient betweenO and E , the partial correlation coefficients, AEOr •, and CEOr •, , decrease significantly.

,0085.0, =• AEOr 0504.0, =•CEOr .

The reason is that the fuzzy attributes A and C have great influences on the relationship between the fuzzy attributes O and E , thus, when the influence of A or C is removed, the partial correlation coefficients dropped so much. But, between fuzzy attribute O and E , which is influenced more? We can’t obtain any conclusion from the previous experiments. According to our experiment, we can see that removing specific attribute from different observed attribute bring about different results. So we can answer the above question, and get the following conclusions. When the influence of A is removed from fuzzy attribute E , the semi-partial correlation coefficient, ,)(, AEOr • , dropped more significantly.

When the influence of C is removed from fuzzy attribute O , the semi-partial correlation coefficient,

)(, COEr • , decreased more significantly. 5 Conclusions The simple and partial correlation analyses on Zadeh’s fuzzy sets have been discussed in previous works [3, 4]. Here, we turn to the analysis of semi- partial correlation which has different interpretation from the previous two ones. A simple correlation coefficient [3] can show the strength of the linear relationship between two fuzzy attributes and the direction of the relationship. A partial correlation


coefficient [4] can provide us the relationship between two fuzzy attributes with other fuzzy attributes held constant. However, a semi-partial correlation can tell us the different relationships, when the influences of other fuzzy attributes are removed from different observed fuzzy attribute. In previous experiments [3, 4], although the analysis of partial correlation coefficient can show that the fuzzy attributes A and C have the great influences on the relationship between fuzzy attributes O and E , we can’t clearly know the differences between the influences of A and C on different fuzzy attributes O and E . Here, the semi- partial correlation coefficients are computed, and thus, when the fuzzy attribute A or C is removed from fuzzy attribute O and removed from fuzzy attribute E , the relationships between the fuzzy attributesO and E are obtained differently. References: [1] T. W. Anderson, An Introduction to Multivariate

Statistical Analysis, 2nd Ed. John Wiley & Sons, 1984.

[2] S. F. Arnold, Mathematical Statistics, Prentice- Hall, New Jersey, 1990.

[3] D. A. Chiang, N. P. Lin, Correlation of Fuzzy Sets, Fuzzy Sets and Systems, Vol. 102, 1999, pp. 221-226.

[4] D. A. Chiang, N. P. Lin, Partial Correlation of Fuzzy Sets, Fuzzy Sets and Systems, Vol. 110, 2000, pp. 209-215.

[5] S. Dowdy, S. Wearden, Statistics for Research, John Wiley & Sons, 1983.

[6] G. J. Klir, T. A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice-Hall International, Inc., 1988.

[7] G. J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic, Theory and Applications, Prentice-Hall International, Inc., 1995.

[8] L. A. Zadeh, Fuzzy sets, Information and Control, Vol. 8, 1965, pp. 338-353.

[9] H. -J. Zimmermann, Fuzzy Set Theory and Its Applications, 2nd Ed., Kluwer Academic Publishers, 1991.


semi-partial correlation of fuzzy sets - wseas · semi-partial correlation of fuzzy sets nancy p....

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