A Comparison of Several Nonparametric Fuzzy Regressions with Trapezoidal Data

Document Type : Original Article


1 Department of Statistics, North Tehran Branch, Islamic Azad University, Tehran, Iran

2 Young Researchers and Elite Club, East Tehran Branch, Islamic Azad university, Tehran, Iran.

3 Department of Statistics, West tehran Branch, Islamic Azad University, Tehran, Iran


In this paper, three methods of nonparametric fuzzy regression with crisp input and asymmetric trapezoidal fuzzy output, are compared. It analyzes the three nonparametric techniques in statistics, namely local linear smoothing (L-L-S), K- nearest neighbor Smoothing (K-NN) and kernel smoothing (K-S) with trapezoidal fuzzy data to obtain the best smoothing parameters. In addition, it makes an analysis on three real-world datasets and calculates the goodness of fit to illustrate the application of the proposed method.
In this paper, we propose to analyze the three nonparametric regression techniques in statistical regression, namely local linear smoothing (L-L-S), the K- nearest neighbor smoothing (K-NN) and the kernel smoothing techniques (K-S) with trapezoidal fuzzy data.
This article is organized as follows: In section 2, we have some preliminaries about fuzzy nonparametric regression and trapezoidal fuzzy data. In section 3, smoothing methods for trapezoidal fuzzy numbers are proposed and in section 4, two numerical examples are solved.


[1] H. Tanaka, S. Uejima, K. Asia, “Linear regression analysis with fuzzy model”, IEEE Transactions on Systems, Man, and Cybernetics,Vol12,P 903-907, 1982.
 [2] Danesh, S., Farnoosh, R., Razzaghnia, T. (2015). “Fuzzy nonparametric regression based on adaptive neuro fuzzy inference system”, Neurocomputing, Vol173, P1450-1460, 2015.
[3] Naderkhani, R., Behzad, M.H., Razzaghnia, T. et al. “Fuzzy Regression Analysis Based on Fuzzy Neural Networks Using Trapezoidal Data”. Int. J. Fuzzy Syst. Vol23, P1267–1280, 2021.
[4]Cheng, C. B., Lee, E. S ,. “Nonparametric fuzzy regression K-NN and Kernel Smoothing techniques”, Computers and Mathematics with Applications, Vo 38, P 239-251, 1999.
[5] R. Farnoosh, J. Ghasemian and o. SolaymaniFard, “A modification on ridge estimation for fuzzy nonparametric regression”, Iraninan Journal of Fuzzy systems,Vol 9,P 75-88, 2012.
[6]  H. Ishibushi, H. Tanaka, “Fuzzy regression analysis using neural networks”, Fuzzy Sets and Systems, Vol50,P257-265,1992.
[7] H. Ishibushi, H. Tanaka, “Fuzzy neural networks with interval weights and its application to fuzzy regression analysis”, Fuzzy Sets and Systems, Vol 57,P 27-39,1993.
[8] J. Fan, I. Gijbels, Local polynomial modeling and its applications, Chapman & Hall, London, 1996.
[9] W. Hardle, Applied Nonparametric Regression, Cambridge University Press, New York, 1990.
[10] M.Danesh ,S.Danesh , T.Razzaghnia, A.Maleki, “Prediction of fuzzy nonparametric regression function: a comparative study of a new hybrid method and smoothing methods”, Global Analysis and Discrete Mathematics, Vol
6, Issue 1, P 143–177,2021.
 [11] N.Wang, W.X. Zhang and C.L Mei, “Fuzzy nonparametric regression based on local linear smoothing
technique”, Information Sciences, Vol177, P 3882-3900,
[12] P. Diamond, “Fuzzy least squares”, Information Sciences,Vol46, P 141-157, 1988.
[13] T. Razzaghnia, E. Pasha, E. Khorram, A. Razzaghnia, “Fuzzy linear regression analysis with trapezoidal coefficients”, First Joint Congress On Fuzzy And Intelligent Systems, 2007, Aug. 29-31, Mashhad, Iran.
[14] D. O. Loftsgaarden and G.P. Quesenberry, “A nonparametric estimate of a multivariate density function”, Annals of Mathematical Statistics,Vol36, P1049-1051, 1965.
[15]  R. Coppi, P.D'Urso, P. Giordani, A Santoro, “Least squares estimation of a linear regression model with LR fuzzy response”, Computational Statistics and Data Analysis, Vol 51,P 267-286,2006.
[16] M. Stone, “Cross-validatory choice and assessment of statistical predictions”, Journal of the Royal Statistical Society, VOl 36 (Series B), P111-147, 1994.
[17] Chang, P. T., Lee, E. S., “A generalized fuzzy weighted least-squares regression”, Fuzzy Sets and Systems,Vol82, P289-298, 1996.
 [18] C.-B. Cheng, E. S. Lee, “Applying Fuzzy Adoptive Network to Fuzzy Regression Analysis”,Computers and Mathematics with Applications,Vol38, P 123-140, 1999.