A MODIFICATION ON RIDGE ESTIMATION FOR FUZZY NONPARAMETRIC REGRESSION

Document Type: Research Paper

Authors

1 School of Mathematics, Iran University of Science and Tech- nology, Narmak, Tehran-16846, Iran

2 School of Mathematics, Iran University of Science and Technol- ogy, Narmak, Tehran-16846, Iran

3 School of Mathematics and Computer Science, Damghan Uni- versity, Damghan, Iran

Abstract

This paper deals with ridge estimation of fuzzy nonparametric
regression models using triangular fuzzy numbers. This estimation method
is obtained by implementing ridge regression learning algorithm in the La-
grangian dual space. The distance measure for fuzzy numbers that suggested
by Diamond is used and the local linear smoothing technique with the cross-
validation procedure for selecting the optimal value of the smoothing param-
eter is fuzzi ed to t the presented model. Some simulation experiments are
then presented which indicate the performance of the proposed method.

Keywords


[1] C. B. Cheng and E. S. Lee, Fuzzy regression with radial basis function networks, Fuzzy Sets
and Systems, 119 (2001), 291-301.
[2] P. Diamond, Fuzzy least squares, Information Sciences, 46 (1988), 141-157.
[3] N. R. Draper and H. Smith, Applied Regression Analysis, Wiley, New York, 1980.
[4] H. Drucker, C. Burges, L. Kaufman, A. Smola and V. N. Vapnik, Support vector regression
machines, in: M. C. Mozer, M. I. Jordan, T. Petsche, Eds., Advances in Neural Information
Processing Systems, MIT Press, Cambridge, MA, 9 (1996), 155-162.
[5] O. S. Fard and A. V. Kamyad, Modi ed k-step method for solving fuzzy initial value problems,
Iranian Journal of Fuzzy Systems, 8(1) (2011), 49-63.
[6] J. Fan and I. Gijbels, Local Polynomial Modeling and Its Applications, Chapman & Hall,
London, 1996.
[7] W. Hardle, Applied Nonparametric Regression, Cambridge University Press, New York, 1990.
[8] J. D. Hart, Nonparametric Smoothing and Lack-of- t Tests, Springer-Verlag, New York, 1997.
[9] T. J. Hastie and R. J. Tibshirani, Generalized Additive Models, Chapman & Hall, London,
1990.
[10] D. H. Hong and C. Hwang, Support vector fuzzy regression machines, Fuzzy Sets and Systems,
138 (2003), 271-281.
[11] D. H. Hong, C. Hwang and C. Ahn, Ridge estimation for regression models with crisp inputs
and Gaussian fuzzy output, Fuzzy Sets and Systems, 142 (2004), 307-319.

[12] A. E. Hoerl and R. W. Kennard, Ridge regression: biased estimates for nonorthogonal prob-
lems, Technometrics, 12 (1970), 55-67.
[13] H. Ishibuchi and H. Tanaka, Fuzzy regression analysis using neural networks, Fuzzy Sets and
Systems, 50 (1992), 257-265.
[14] H. Ishibuchi and H. Tanaka, Fuzzy neural networks with interval weights and its application
to fuzzy regression analysis, Fuzzy Sets and Systems, 57 (1993), 27-39.
[15] B. Kim and R. R. Bishu, Evaluation of fuzzy linear regression models by comparing mem-
bership functions, Fuzzy Sets and Systems, 100 (1998), 343-352.
[16] R. X. Liu, J. Kuang, Q. Gong and X. L. Hou, Principal component regression analysis with
SPSS, Computer Methods and Programs in Biomedicine, 71 (2003), 141-147.
[17] S. Pourahmad, S. M. T. Ayatollahi and S. M. Taheri, Fuzzy logistic regression: a new possi-
bilistic model and its application in clinical vague status, Iranian Journal of Fuzzy Systems,
8 (2011), 1-17.
[18] H. Shakouri G and R. Nadimi, A novel fuzzy linear regression model based on a non-equality
possibility index and optimum uncertainty, Applied Soft Computing, 9 (2009), 590-598.
[19] C. Saunders, A. Gammerman and V. Vork, Ridge regression learning algorithm in dual vari-
able, Proceedings of the 15th International Conference on Machine Learning, (1998), 515-521.
[20] N. Wang, W. X. Zhang and C. L. Mei, Fuzzy nonparametric regression based on local linear
smoothing technique, Information Sciences, 177 (2007), 3882-3900.
[21] M. S. Yang and C. H. Ko, On a class of fuzzy c-numbers clustering procedures for fuzzy data,
Fuzzy Sets and Systems, 84 (1996), 49-60.