A Fuzzy Linear Regression Model Based on Dissemblance Index
Fuzzy regression models are useful for investigating
the relationship between explanatory variables and responses in fuzzy
environments. To overcome the deficiencies of previous models and
increase the explanatory power of fuzzy data, the graded mean
integration (GMI) representation is applied to determine
representative crisp regression coefficients. A fuzzy regression model
is constructed based on the modified dissemblance index (MDI),
which can precisely measure the actual total error. Compared with
previous studies based on the proposed MDI and distance criterion, the
results from commonly used test examples show that the proposed
fuzzy linear regression model has higher explanatory power and
forecasting accuracy.
[1] J. Neter, W. Wasserman, and M. Kutner, Applied Linear Regression
Models. Homewood, IL: Richard D. Irwin Inc., 1983.
[2] N.R. Draper and H. Smith, Applied Regression Analysis, New York: John
Wiley & Sons, 1998.
[3] L.A. Zadeh, “Fuzzy sets,” Inf. and Control, vol. 8, pp. 338–356, 1965.
[4] H. Tanaka, T. Okuda, and K. Asai, “On fuzzy-mathematical
programming,” J. Cybern., vol. 3, pp. 37–46, 1974.
[5] J. Kacprzyk and M. Fedrizzi, Fuzzy Regression Analysis, Warsaw:
Omnitech Press, 1992.
[6] S. Ferson and L. R. Ginzburg, “Different methods are needed to propagate
ignorance and variability,” Rel. Eng. Syst. Safety, vol. 54, pp. 133-144,
1996.
[7] D. Dubois and H. Prade, “Gradualness, uncertainty and bipolarity:
Making sense of fuzzy sets,” Fuzzy Sets Syst., vol. 192, pp. 3-24, 2012.
[8] K.Y. Cai, “System failure engineering and fuzzy methodology: an
introductory overview,” Fuzzy Sets Syst., vol. 83, pp. 113–133, 1996.
[9] S. Fruehwirth-Schnatter, “On statistical inference for fuzzy data with
applications to descriptive statistics,” Fuzzy Sets Syst., vol. 50, pp. 143–
165, 1992.
[10] H. Tanaka, S. Uejima, and K. Asai, “Fuzzy linear regression model,”
IEEE Trans. Syst., Man, Cybern., vol. 10, pp. 2933-2938, 1980.
[11] H. Tanaka, S. Uegima, and K. Asai, “Linear regression analysis with
fuzzy model,” IEEE Trans. Syst., Man, Cybern., vol. SMC-12, pp.
903-907, 1982.
[12] B.Wu and N.F. Tseng, “A new approach to fuzzy regression models with
application to business cycle analysis,” Fuzzy Sets Syst., vol. 130, pp. 33–
42, 2002.
[13] Y.Q. He, L.K. Chan, and M.L. Wu, “Balancing productivity and
consumer satisfaction for profitability: Statistical and fuzzy regression
analysis,” Eur. J. Oper. Res., vol. 176, pp. 252-263, 2007.
[14] C.Y. Huang, and G.H. Tzeng, “Multiple generation product life cycle
predictions using a novel two-stage fuzzy piecewise regression analysis
method,” Technol. Forecasting & Social Change, vol. 75, 12-31, 2008.
[15] S. Imoto, Y. Yabuuchi, and J.Watada, “Fuzzy regression model of R&D
project evaluation,” Applied Soft Comput., vol. 8, pp. 1266-1273, 2008.
[16] S.J. Mousavia, K. Ponnambalamb, and F. Karray, “Inferring operating
rules for reservoir operations using fuzzy regression and ANFIS,” Fuzzy
Sets Syst., vol. 158, pp. 1064-1082, 2007.
[17] L.H. Chen and C.C. Hsueh, “Fuzzy regression models using the
least-squares method based on the concept of distance,” IEEE Trans.
Fuzzy Syst., vol. 17, pp. 1259-1272, 2009.
[18] P. Diamond, “Fuzzy least squares,” Inf. Sci., vol. 46, pp. 141-157, 1988.
[19] P.T. Chang and E.S. Lee, “A generalized fuzzy weighted least-squares
regression,” Fuzzy Sets Syst., vol. 82, pp. 289-298, 1996.
[20] M. Ma, M. Friedman, and A. Kandel, “General fuzzy least squares,”
Fuzzy Sets Syst., vol. 88, 107-118, 1997.
[21] C. Kao and C.L. Chyu, “Least-squares estimates in fuzzy regression
analysis, Eur. J. Oper. Res., vol. 148, pp. 426-435, 2003.
[22] H. Tanaka, “Fuzzy data analysis by possibilistic linear models,” Fuzzy
Sets Syst., vol. 24, 363-375, 1987.
[23] H. Tanaka and J. Watada, “Possibilistic linear systems and their
application to the linear regression model,” Fuzzy Sets Syst., vol. 27, pp.
275-289, 1988.
[24] H. Tanaka, I. Hayashi, and J. Watada, “Possibilistic linear regression
analysis for fuzzy data,” Eur. J. Oper. Res., vol. 40, 389-396, 1989.
[25] D. Savic andW. Pedrycz, “Evaluation of fuzzy linear regression models,”
Fuzzy Sets Syst., vol. 39, 51–63, 1991.
[26] C. Kao and C.L. Chyu, “A fuzzy linear regression model with better
explanatory power,” Fuzzy Sets Syst., vol. 126, 401-409, 2002.
[27] L.H. Chen and C.C. Hsueh, “A mathematical programming method for
formulating a fuzzy regression model based on distance criterion,” IEEE
Trans. Syst., Man, Cybern. B, Cybern., vol. 37, 705-711, 2007.
[28] S.P. Chen and J.F. Dang, “A variable spread fuzzy linear regression
model with higher explanatory power and forecasting accuracy,” Inf. Sci.
vol. 178, 3973–3988, 2008.
[29] D.H. Hong and C. Hwang, “Interval regression analysis using quadratic
loss support vector machine,” IEEE Trans. Fuzzy Syst., vol. 13, 229–237,
2005.
[30] P.Y. Hao and J.H. Chiang, “Fuzzy regression analysis by support vector
learning approach,” IEEE Trans. Fuzzy Syst., vol. 16, 428–441, 2008.
[31] C. Kao and P.H. Lin, “Entropy for fuzzy regression analysis,” Int. J. Syst.
Sci., vol. 36, 869-876, 2005.
[32] D.T. Redden and W.H. Woodall, “Properties of certain fuzzy linear
regression methods,” Fuzzy Sets Syst., vol. 64, 361-375, 1994.
[33] B. Kim and R.R. Bishu, “Evaluation of fuzzy linear regression models by
comparing membership functions,” Fuzzy Sets Syst., vol. 100, 343-352,
1998.
[34] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and
Applications. International Edition, Taiwan: Prentice-Hall, 2002.
[35] S.H. Chen and C.H. Hsieh, “Graded mean integration representation of
generalized fuzzy numbers,” J. Chinese Fuzzy Syst. Association, vol. 5,
1–7, 1991.
[36] S.H. Chen and C.H. Hsieh, “Representation, ranking, distance, and
similarity of L-R type fuzzy number and application,” Australian J Intell.
Process. Syst., vol. 6, 217–229, 2000.
[37] A. Kaufmann and M.M. Gupta, Introduction to Fuzzy Arithmetic Theory
and Applications. Boston: International Thomson Computer Press, 1991.
[38] M.S. Bazaraa, H.D. Sherali, and C.M. Shetty, Nonlinear Programming:
Theory and Algorithms, Second ed. Singapore: JohnWiley& Sons, 1993.
[39] LINGO User’s Guide, LINDO Syst. Inc., Chicago, IL, 2003.
[40] L.A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets
Syst., vol. 1, 3-28, 1978.
[1] J. Neter, W. Wasserman, and M. Kutner, Applied Linear Regression
Models. Homewood, IL: Richard D. Irwin Inc., 1983.
[2] N.R. Draper and H. Smith, Applied Regression Analysis, New York: John
Wiley & Sons, 1998.
[3] L.A. Zadeh, “Fuzzy sets,” Inf. and Control, vol. 8, pp. 338–356, 1965.
[4] H. Tanaka, T. Okuda, and K. Asai, “On fuzzy-mathematical
programming,” J. Cybern., vol. 3, pp. 37–46, 1974.
[5] J. Kacprzyk and M. Fedrizzi, Fuzzy Regression Analysis, Warsaw:
Omnitech Press, 1992.
[6] S. Ferson and L. R. Ginzburg, “Different methods are needed to propagate
ignorance and variability,” Rel. Eng. Syst. Safety, vol. 54, pp. 133-144,
1996.
[7] D. Dubois and H. Prade, “Gradualness, uncertainty and bipolarity:
Making sense of fuzzy sets,” Fuzzy Sets Syst., vol. 192, pp. 3-24, 2012.
[8] K.Y. Cai, “System failure engineering and fuzzy methodology: an
introductory overview,” Fuzzy Sets Syst., vol. 83, pp. 113–133, 1996.
[9] S. Fruehwirth-Schnatter, “On statistical inference for fuzzy data with
applications to descriptive statistics,” Fuzzy Sets Syst., vol. 50, pp. 143–
165, 1992.
[10] H. Tanaka, S. Uejima, and K. Asai, “Fuzzy linear regression model,”
IEEE Trans. Syst., Man, Cybern., vol. 10, pp. 2933-2938, 1980.
[11] H. Tanaka, S. Uegima, and K. Asai, “Linear regression analysis with
fuzzy model,” IEEE Trans. Syst., Man, Cybern., vol. SMC-12, pp.
903-907, 1982.
[12] B.Wu and N.F. Tseng, “A new approach to fuzzy regression models with
application to business cycle analysis,” Fuzzy Sets Syst., vol. 130, pp. 33–
42, 2002.
[13] Y.Q. He, L.K. Chan, and M.L. Wu, “Balancing productivity and
consumer satisfaction for profitability: Statistical and fuzzy regression
analysis,” Eur. J. Oper. Res., vol. 176, pp. 252-263, 2007.
[14] C.Y. Huang, and G.H. Tzeng, “Multiple generation product life cycle
predictions using a novel two-stage fuzzy piecewise regression analysis
method,” Technol. Forecasting & Social Change, vol. 75, 12-31, 2008.
[15] S. Imoto, Y. Yabuuchi, and J.Watada, “Fuzzy regression model of R&D
project evaluation,” Applied Soft Comput., vol. 8, pp. 1266-1273, 2008.
[16] S.J. Mousavia, K. Ponnambalamb, and F. Karray, “Inferring operating
rules for reservoir operations using fuzzy regression and ANFIS,” Fuzzy
Sets Syst., vol. 158, pp. 1064-1082, 2007.
[17] L.H. Chen and C.C. Hsueh, “Fuzzy regression models using the
least-squares method based on the concept of distance,” IEEE Trans.
Fuzzy Syst., vol. 17, pp. 1259-1272, 2009.
[18] P. Diamond, “Fuzzy least squares,” Inf. Sci., vol. 46, pp. 141-157, 1988.
[19] P.T. Chang and E.S. Lee, “A generalized fuzzy weighted least-squares
regression,” Fuzzy Sets Syst., vol. 82, pp. 289-298, 1996.
[20] M. Ma, M. Friedman, and A. Kandel, “General fuzzy least squares,”
Fuzzy Sets Syst., vol. 88, 107-118, 1997.
[21] C. Kao and C.L. Chyu, “Least-squares estimates in fuzzy regression
analysis, Eur. J. Oper. Res., vol. 148, pp. 426-435, 2003.
[22] H. Tanaka, “Fuzzy data analysis by possibilistic linear models,” Fuzzy
Sets Syst., vol. 24, 363-375, 1987.
[23] H. Tanaka and J. Watada, “Possibilistic linear systems and their
application to the linear regression model,” Fuzzy Sets Syst., vol. 27, pp.
275-289, 1988.
[24] H. Tanaka, I. Hayashi, and J. Watada, “Possibilistic linear regression
analysis for fuzzy data,” Eur. J. Oper. Res., vol. 40, 389-396, 1989.
[25] D. Savic andW. Pedrycz, “Evaluation of fuzzy linear regression models,”
Fuzzy Sets Syst., vol. 39, 51–63, 1991.
[26] C. Kao and C.L. Chyu, “A fuzzy linear regression model with better
explanatory power,” Fuzzy Sets Syst., vol. 126, 401-409, 2002.
[27] L.H. Chen and C.C. Hsueh, “A mathematical programming method for
formulating a fuzzy regression model based on distance criterion,” IEEE
Trans. Syst., Man, Cybern. B, Cybern., vol. 37, 705-711, 2007.
[28] S.P. Chen and J.F. Dang, “A variable spread fuzzy linear regression
model with higher explanatory power and forecasting accuracy,” Inf. Sci.
vol. 178, 3973–3988, 2008.
[29] D.H. Hong and C. Hwang, “Interval regression analysis using quadratic
loss support vector machine,” IEEE Trans. Fuzzy Syst., vol. 13, 229–237,
2005.
[30] P.Y. Hao and J.H. Chiang, “Fuzzy regression analysis by support vector
learning approach,” IEEE Trans. Fuzzy Syst., vol. 16, 428–441, 2008.
[31] C. Kao and P.H. Lin, “Entropy for fuzzy regression analysis,” Int. J. Syst.
Sci., vol. 36, 869-876, 2005.
[32] D.T. Redden and W.H. Woodall, “Properties of certain fuzzy linear
regression methods,” Fuzzy Sets Syst., vol. 64, 361-375, 1994.
[33] B. Kim and R.R. Bishu, “Evaluation of fuzzy linear regression models by
comparing membership functions,” Fuzzy Sets Syst., vol. 100, 343-352,
1998.
[34] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and
Applications. International Edition, Taiwan: Prentice-Hall, 2002.
[35] S.H. Chen and C.H. Hsieh, “Graded mean integration representation of
generalized fuzzy numbers,” J. Chinese Fuzzy Syst. Association, vol. 5,
1–7, 1991.
[36] S.H. Chen and C.H. Hsieh, “Representation, ranking, distance, and
similarity of L-R type fuzzy number and application,” Australian J Intell.
Process. Syst., vol. 6, 217–229, 2000.
[37] A. Kaufmann and M.M. Gupta, Introduction to Fuzzy Arithmetic Theory
and Applications. Boston: International Thomson Computer Press, 1991.
[38] M.S. Bazaraa, H.D. Sherali, and C.M. Shetty, Nonlinear Programming:
Theory and Algorithms, Second ed. Singapore: JohnWiley& Sons, 1993.
[39] LINGO User’s Guide, LINDO Syst. Inc., Chicago, IL, 2003.
[40] L.A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets
Syst., vol. 1, 3-28, 1978.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:72423", author = "Shih-Pin Chen and Shih-Syuan You", title = "A Fuzzy Linear Regression Model Based on Dissemblance Index", abstract = "Fuzzy regression models are useful for investigating
the relationship between explanatory variables and responses in fuzzy
environments. To overcome the deficiencies of previous models and
increase the explanatory power of fuzzy data, the graded mean
integration (GMI) representation is applied to determine
representative crisp regression coefficients. A fuzzy regression model
is constructed based on the modified dissemblance index (MDI),
which can precisely measure the actual total error. Compared with
previous studies based on the proposed MDI and distance criterion, the
results from commonly used test examples show that the proposed
fuzzy linear regression model has higher explanatory power and
forecasting accuracy.", keywords = "Dissemblance index, fuzzy linear regression, graded
mean integration, mathematical programming.", volume = "8", number = "9", pages = "1278-6", }
