1. A nonlinear hybrid fuzzy least- squares regression model Olga Poleshchuk, Evgeniy Komarov Moscow State Forest University, Russia

2. 2 The approaches under the heading of “Fuzzy Regression”: (a) Methods proposed by H.Tanaka and investigated by H.Tanaka, A.Celmins, D.Savic, W.Pedrycz, Y.-H.O.Chang, B.M.Ayyub, H.Ishibuchi. The coefficients of input variables are assumed to be fuzzy numbers. (a) Methods proposed by H.Tanaka and investigated by H.Tanaka, A.Celmins, D.Savic, W.Pedrycz, Y.-H.O.Chang, B.M.Ayyub, H.Ishibuchi. The coefficients of input variables are assumed to be fuzzy numbers. (b) Method proposed by R.J. Hathaway and J.C. Bezdek, where first the fuzzy clusters determined by fuzzy clustering define how many ordinary regressions are to be constructed, one for each cluster. Next each fuzzy cluster is used to determine the most appropriate ordinary regression that is to be applied for a new input from the ordinary regressions determined in the first place. (b) Method proposed by R.J. Hathaway and J.C. Bezdek, where first the fuzzy clusters determined by fuzzy clustering define how many ordinary regressions are to be constructed, one for each cluster. Next each fuzzy cluster is used to determine the most appropriate ordinary regression that is to be applied for a new input from the ordinary regressions determined in the first place. (c) Methods proposed by I.B.Turksen, D.H.Hong, C.H.Hwang, where the fuzzy functions approach to system modeling was developed. These methods are based on a fuzzy clustering together with the least squares estimation techniques and approach that identifies the fuzzy functions using support vector machines. (c) Methods proposed by I.B.Turksen, D.H.Hong, C.H.Hwang, where the fuzzy functions approach to system modeling was developed. These methods are based on a fuzzy clustering together with the least squares estimation techniques and approach that identifies the fuzzy functions using support vector machines.

3. 3 A quadratic hybrid fuzzy least- squares regression

4. 4 The method for formalization the meanings of qualitative characteristic

5. 5 A quadratic hybrid fuzzy least- squares regression

6. 6 A weighted interval

7. 7 Distance between fuzzy numbers

8. 8 Weighted intervals and distances between initial output fuzzy numbers and model fuzzy numbers

9. 9 Optimization problem

10. 10 Identifying a model fuzzy number with meanings of qualitative characteristic

11. 11 Hybrid standard deviation, hybrid correlation coefficient, hybrid standard error of estimate

12. 12 Numerical example TABLE I Students’ grades

13. 13 Numerical example TABLE II Membership functions of grades

14. 14 Numerical example TABLE III Membership functions of grades

15. 15 Numerical example Linear hybrid fuzzy least-squares regression (1)

16. 16 Numerical example Quadratic hybrid fuzzy least-squares regression (2)

17. 17 Numerical example Ordinary regression (3)

18. 18 Numerical example TABLE IV Predicted and observed data

19. 19 Conclusions  Quadratic hybrid fuzzy least-squares regression based on weighted intervals was developed.  The method for formalization qualitative characteristics’ meanings was developed.  The numerical example has demonstrated that developed hybrid regression model can be used for analysis relations among linguistic variables with success.