Multiple and Partial Correlation Coefficients of Fuzzy Sets

Transcription

1 Quality & Quantity (2007) 41: Springer 2006 DOI /s Multiple and Partial Correlation Coefficients of Fuzzy Sets WEN-LIANG HUNG 1 and JONG-WUU WU 2, 1 Department of Mathematics Education, National Hsinchu Teachers College, Hsin-Chu, Taiwan, R.O.C.; 2 Department of Applied Mathematics, National Chiayi University, Chiayi City 60004, Taiwan, R.O.C. Abstract. In many applications, multiple correlation and partial correlation for three or more fuzzy sets are very important, but Chiang and Lin (1999, Fuzzy Sets and Systems 102: ) do not solve this problem. Here, we propose a method to calculate the multiple correlation and partial correlation for fuzzy data, by adopting the concepts from the multivariate correlation model. In order to fit into normal framework, we use empirical logit transform (see, Agresti, [1990, Categorical Data Analysis. New York: Wiley]; Johnson and Wichern, [1992, Applied Multivariate Statistical Analysis 3rd edn. Engelwood Cliffs; Prentice- Hall.]) for membership function grades to achieve this. Key words: correlation model, empirical logit transform, fuzzy sets, multiple correlation, partial correlation. 1. Introduction In the study of multivariate correlation models one is naturally very interested in relationships among the variables. One set of measures useful to this end consists of the coefficients of multiple correlation and the coefficients of partial correlation. All partial correlation coefficients measure the correlation between two variables. What we are interested in, finding the coefficients of multiple correlation and the coefficients of partial correlation between fuzzy sets (FSs), which can tell us the relationships among the FSs. Chiang and Lin (1999) discussed the correlation coefficient between two FSs, by adopting the concepts from the conventional statistics. Similar works have been done by Bustince and Burillo (1995), Gerstenkorn and Manko (1991), Hong and Hwang (1995) and Yu (1993). In many applications, partial correlation and multiple correlation for three or more FSs Author for correspondence: Jong-Wuu-wu, Department of Applied Mathematics, National Chiayi University, Chiayi City 60004, Taiwan, R.O.C., jwwu@mail.ncyu. edu.tw

2 334 WEN-LIANG HUNG AND JONG-WUU WU are very important, but they does not solve this problem. In this paper, we solve this problem via multivariate correlation model. In order to fit into normal framework, we use empirical logit transform (see, Agresti, 1990; Johnson and Wichern, 1992) for membership function grades to achieve this. The outline of the paper is as follows. The multiple and partial correlation coefficients of crisp variables are discussed in Section 2. Formulas of multiple and partial correlation coefficients over fuzzy data are derived in Section 3. We give some examples to illustrate the our proposed formulas in Section 4. Some conclusions are summarized in Section Multiple and Partial Correlation Coefficients of Crisp Sets Correlation models are employed to study the nature of the relations between the variables, and also may be used for making inferences about any one of the variables on the basis of the others. The correlation model most widely employed is the normal correlation model. It is well known that the conditional multivariate correlation models are equivalent to the normal error multiple regression model. Hence, in the case where the variables are jointly normally distributed, all inferences on one variable conditional on the other variables being fixed are carried out by the usual multiple regression techniques (see Neter et al., 1999). Suppose that Y 1,Y 2,...,Y p are included in a multivariate normal correlation model. The coefficient of correlation between Y i and Y j (i j,i,j = 1, 2,...,p) is denoted by ρ ij and its square, ρij 2, is called the coefficient of determination and ρij 2 is defined as follows: ρij 2 = σ i 2 σi j 2, σi 2 where σi j 2 is the variance of conditional distributions of Y i when Y j are fixed, and σi 2 is the variance of Y i. To estimate ρij 2, we regress Y i on Y j and obtain the total sum of squares SSTO(Y i ) and the regression sum of squares SSR(Y j ). The estimator, denoted by rij 2 is: rij 2 = SSR(Y j) SSTO(Y i ). (1) The square-root of r 2 ij, denoted by r ij, is the estimated coefficient of correlation between Y i and Y j. A plus or minus sign is attached to this measure according to whether the slope of the fitted regression line is positive or negative. As for, a coefficient of multiple determination associated with each variable. Suppose that Y 1,Y 2,...,Y p are included in the correlation model. The

3 MULTIPLE AND PARTIAL CORRELATION COEFFICIENTS OF FUZZY SETS 335 coefficient of multiple determination associated with, say, Y 1 is denoted by ρ p 2, and defined as follows: ρ p 2 = σ 1 2 σ p 2, σ1 2 where σ p 2 is the variance of conditional distributions of Y 1 when the other variables are fixed, and σ1 2 is the variance of Y 1. Moreover, the positive square-root of a coefficient of multiple determination is the coefficient of multiple correlation. Thus, the coefficient of multiple correlation for Y 1 is ρ p = ρ p 2. To estimate ρ p, we need the total sum of squares for Y 1, denoted by SSTO(Y 1 ). Then we require the regression sum of squares when Y 1 is regressed on Y 2,Y 3,...,Y p, denoted by SSR(Y 2,Y 3,...,Y p ). The estimator, denoted by R p, is: R p 2 = SSR(Y 2,Y 3,...,Y p ). (2) SSTO(Y 1 ) The positive square-root of R p 2, denoted by R p, is the estimated coefficient of multiple correlation for Y 1. Next, we consider the correlation of two variables when other variables are fixed. Such as the correlation between Y 1 and Y 2 in the conditional joint distribution when Y 3,Y 4,...,Y p are fixed at given levels. When all variables are jointly normally distributed, this correlation, say, ρ p does not depend on the levels where Y 3,Y 4,...,Y p are fixed. ρ p is called the coefficient of partial correlation between Y 1 and Y 2 when Y 3,Y 4,...,Y p are fixed. The square of ρ p is called the coefficient of partial determination is denoted by ρ p 2. Thus, to estimate ρ p, we regress Y 1 on Y 3,Y 4,...,Y p and obtain the error sum of squares SSE(Y 3,Y 4,...,Y p ). Next, we regress Y 1 on Y 2,Y 3,...,Y p and obtain the error sum of squares SSE(Y 2,Y 3,...,Y p ). The estimator of ρ p 2 then is: r p 2 = SSE(Y 3,Y 4,...,Y p ) SSE(Y 2,Y 3,...,Y p ). (3) SSE(Y 3,Y 4,...,Y p ) Remark: The coefficient of partial determination can be expressed in terms of simple correlation coefficients. For example: r = (r 12 r 23 r 13 ) 2 (1 r 2 23 )(1 r2 13 ),

4 336 WEN-LIANG HUNG AND JONG-WUU WU r = (r 12 4 r 23 4 r 13 4 ) 2 (1 r )(1 r ), where r 12 denotes the coefficient of simple correlation between Y 1 and Y 2,r 23 denotes the coefficient of simple correlation between Y 2 and Y 3, and so on. Extensions are straightforward. 3. Multiple and Partial Correlation Coefficients of Fuzzy Sets Suppose there is a fuzzy set A F, where F is a fuzzy space. The FS A is defined on a crisp set X with a membership function µ A, it can be expressed as Zimmermann (1991) A ={(x, µ A (x)) x X}, where µ A : X [0, 1]. Assume that there is a random sample (x 1,...,x n ) X, along with a sequence of p-ple data, ((µ A1 (x 1 ), µ A2 (x 1 ),...,µ Ap (x 1 )),...,(µ A1 (x n ), µ A2 (x n ),...,µ Ap (x n ))), which correspond to the grades of the membership functions of FSs A 1,A 2,...,A p defined on X. We will try to develop formulas for both multiple and partial correlation cofficients between FSs A 1,A 2,...,A p by means of normal correlation models. Therefore, the sequence of data has to be adapted in order to fit into normal framework. We use logit transform to achive this (see Johnson and Wichern, 1992). Let µ Aj (x i ) Z iaj = log, i= 1, 2,...,n, j= 1, 2,...,p. 1 µ Aj (x i ) In order to avoid log(0) or log( ), therefore, we approximate Z iaj the empirical logit transform (see Agresti, 1990) as following by using Y iaj = log µ Aj (x i ) + 1 2n, i= 1, 2,...,n, j= 1, 2,...,p. 1 µ Aj (x i ) + 1 2n Ordinarily, we are interested in the relationship between the two FSs, for example, A i and A j. By Equation (1) in Section 2, we define the coeficient of determination: ra 2 i A j = SSR(Y A j ), i j, i, j = 1, 2,...,p. (4) SSTO(Y Ai )

5 MULTIPLE AND PARTIAL CORRELATION COEFFICIENTS OF FUZZY SETS 337 The square-root of ra 2 i A j, denoted by r Ai A j, is the correlation coefficient between A i and A j. The r Ai A j is given the same sign as that of the corresponding regression coefficient in the fitted regression line. By Equation (2), we define the coefficient of multiple determination associated with FS A 1 is: RA 2 1 A 2 A 3...A p = SSR(Y A 2,Y A3,...,Y Ap ). (5) SSTO(Y A1 ) The positive square-root of RA 2 1 A 2 A 3...A p, denoted by R A1 A 2 A 3...A p, is the coefficient of multiple correlation for FS A 1. On the other hand, we use Equation (3) to define the coefficient of partial determination: ra 2 1 A 2 A 3 A 4...A p = SSE(Y A 3,Y A4,...,Y Ap ) SSE(Y A2,Y A3,...,Y Ap ). (6) SSE(Y A3,Y A4,...,Y Ap ) The square-root of r 2 A 1 A 2 A 3 A 4...A p, denoted by r A1 A 2 A 3 A 4,...A p, is the coefficient of partial correlation between FSs A 1 and A 2 when A 3,A 4,...,A p are fixed. Usually the partial correlation coefficient is given the same sign as that of the corresponding regression coefficient in the fitted regression function. 4. Examples 4.1. example 1 Let X ={All the girls in the university}, our interests are the estimates on the degree of prettiness, the degree of elegance and the degree of smart of the girls in this university, we cannot measure over all the girls in the campus, therefore, a sample of six girls was taken at random from the campus (x 1,...,x 6 ) = (Mary, Judy, Linda, Susan, Betty, Julia). Now, let us define three FSs (i.e. p = 3) over the crisp set X, A 1 = {Pretty girl},a 2 ={Elegant girl},a 3 ={Smart girl}, then we collect information and obtain the membership grades of these six girls concerning FSs A 1,A 2 and A 3 as Name Mary Judy Linda Susan Betty Julia µ A µ A µ A Our initial interest was in the multiple correlation coefficient R A1 A 2 A 3 and the partial correlation coefficient r A1 A 2 A 3, respectively. We ran two separate regressions, Y A1 on Y A3 and Y A1 on Y A2 and Y A3. We obtain in turn SSE(Y A3 ) = ,

6 338 WEN-LIANG HUNG AND JONG-WUU WU and SSTO(Y A1 )=2.6772, SSR(Y A2,Y A3 )=1.3510, SSE(Y A2,Y A3 )= The R 2 A 1 A 2 A 3 and r 2 A 1 A 2 A 3 then were calculated corresponding to (5) and (6): R 2 A 1 A 2 A 3 = SSR(Y A 2,Y A3 ) SSTO(Y A1 ) = = , r 2 A 1 A 2 A 3 = SSE(Y A 3 ) SSE(Y A2,Y A3 ) SSE(Y A3 ) = = Hence R A1 A 2 A 3 = and r A1 A 2 A 3 = (The sign of r A1 A 2 A 3 here is positive because the regression coefficient for Y A2 when Y A1 is regressed on Y A2 and Y A3 is positive.) Next we ascertained from a printout of the regression of Y A1 on Y A2 that r A1 A 2 = By the same argument, we can obtain and and r A1 A 3 = 0.028, r A2 A 3 = R A2 A 1 A 3 = , R A3 A 1 A 2 = r A1 A 3 A 2 = , r A2 A 3 A 1 = example 2 De et al. (2000) defined some operations on intuitionistic FSs (cf. Atanassov, 1986), and presented some examples to illustrate these definitions will be useful while dealing with various linguistic hedges like very, more or less, very very, etc. involved in the problems under intuitionistic fuzzy environment. Here, we consider FSs in X ={6, 7, 8, 9, 10} as follows: LARGE ={(6, 0.1), (7, 0.3), (8, 0.6), (9, 0.9), (10, 1.0)}, Very LARGE ={(6, 0.01), (7, 0.09), (8, 0.36), (9, 0.81), (10, 1.0)}, Not very LARGE ={(6, 0.99), (7, 0.91), (8, 0.64), (9, 0.19), (10, 0.0)}, Very very LARGE ={(6, 0.0), (7, 0.01), (8, 0.13), (9, 0.66), (10, 1.0)}, More or less LARGE={(6, 0.32), (7, 0.55), (8, 0.77), (9, 0.95), (10, 1.0)}.

7 MULTIPLE AND PARTIAL CORRELATION COEFFICIENTS OF FUZZY SETS 339 Table I. The partial correlation coefficients computed from Equation (3) when the fuzzy set LARGE is fixed VL NVL VVL MLL VL (1.000) ( 1.000) (0.982) (0.985) NVL ( 1.000) (1.000) ( 0.982) ( 0.985) VVL (0.982) ( 0.982) (1.000) (0.937) MLL (0.985) ( 0.985) (0.937) (1.000) Note. The values in parenthese are correlation coefficients computed from Equation (1). Our objective is to find the coefficients of partial correlation when the FS LARGE is fixed. The result is summarized in Table I. The following abbreviated notations are used in Table I. L LARGE, VL Very LARGE, VVL Very very LARGE, MLL More or less LARGE, NVL Not very LARGE. Based on Table I, we obtain that r MLL,VL L = 0.930, r MLL,VL = 0.985, (7) r MLL,NVL L = 0.930, r MLL,NVL = 0.985, (8) r MLL,VVL L = 0.939, r MLL,VVL = (9) Thus (7) (9) suggested to the analyst that the FS More or less LARGE tends to be heavily influenced by the FS LARGE. For example, when the FS LARGE is not considered, the relation between More or less LARGE and Very very LARGE is positive and indicating that the degree of linear association is extremely high. But, the relation between More or less LARGE and Very very LARGE is negative and indicating that the degree of linear association is extremely high with LARGE is fixed. 5. Concluding Remarks In this paper, we develop formulas for both multiple and partial correlation coefficients between FSs by means of the empirical transform for membership function grades. Computer multiple regression packages can be used to compute the multiple and partial correlation coefficients between FSs.

8 340 WEN-LIANG HUNG AND JONG-WUU WU References Agresti, A. (1990). Categorical Data Analysis. New York: Wiley. Atanassov, K. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20: Bustince, H. & Burillo, P. (1995). Correlation of interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems 74: Chiang, D. A. & Lin, P. N. (1999). Correlation of fuzzy sets. Fuzzy Sets and Systems 102: De, S. K., Biswas, R. & Roy, A. R. (2000). Some operations on intuitionistic fuzzy sets. Fuzzy Sets and Systems 114: Gerstenkorn, T. & Manko, J. (1991). Correlation of intuitionistic fuzzy sets. Fuzzy Sets and Systems 44: Hong, D. H. & Hwang, S. Y. (1995). Correlation of intuitionistic fuzzy sets in probability spaces. Fuzzy Sets and Systems 75: Johnson, R. A. & Wichern, D. W. (1992). Applied Multivariate Statistical Analysis, 3rd edn. Englewood Cliffs, NJ: Prentice-Hall. Neter, J., Kutner, M. H., Nachtsheim, C. & Wasserman, W. (1999). Applied Linear Regression Models, 3rd edn. New York: McGraw-Hill. Yu, C. (1993). Correlation of fuzzy numbers. Fuzzy Sets and Systems 55: Zimmermann, H. J. (1991). Fuzzy Set Theorey and Its Applications, 2nd edn. Taiwan: Kluwer Academic Publishers, Published and Reprinted by Maw Chang Book Company.