Prediction of Fuzzy Nonparametric Regression Function: A Comparative Study of a New Hybrid Method and Smoothing Methods

Document Type : Research Paper

Authors

1 Imam Khomeini International University – Buin Zahra Higher Education Center of Engineering and Technology, Buein Zahra, Ghazvin, Iran

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

3 Department of statistics, North Tehran Branch, Islamic Azad University, Tehran, Iran

4 Department of Statistics, West Tehran Branch, Islamic Azad University, Tehran, Iran

Abstract

In this paper, the fuzzy regression model is considered with crisp inputs and symmetric triangular fuzzy output. This study aims to formulate the fuzzy inference system based on the Sugeno inference model for the fuzzy regression function prediction by the fuzzy least-squares problem-based on Diamond's distance. In this study, the fuzzy least-squares problem is used to optimize consequent parameters, and the results are derived based on the V-fold cross-validation, so that the validity and quality of the proposed method can be guaranteed. The proposed method is used to reduce the bias and the boundary effect of the estimated underlying regression function. Also, a comparative study of the fuzzy nonparametric regression function prediction is carried out between the proposed model and smoothing methods, such as k-nearest neighbor (k-NN), kernel smoothing (KS), and local linear smoothing (LLS). Different approaches are illustrated by some examples and the results are compared. Comparing the results indicates that, among the various prediction models, the proposed model is the best, decreasing the boundary effect significantly. Also, in comparison with different methods, in both one-dimensional and two-dimensional inputs, it may be considered the best candidate for the prediction.

Keywords

References

  1. A. Celmins, Multidimensional least-squares fitting of fuzzy models, Fuzzy sets and systems, 22, 245 253 (1997).
  2. C. B. Cheng and E. S. Lee, Nonparametric fuzzy regression k-NN and kernel smoothing techniques, Computers and Mathematics with Applications, 38, 239-251 (1999).
  3. C. B. Cheng, E. S. Lee, Applying Fuzzy Adoptive Network to Fuzzy Regression Analysis, Computers and Mathematics with Applications, 38, 123-140 (1999).
  4. C. B. Cheng, E.S. Lee, Fuzzy regression with radial basis function networks, Fuzzy Sets and Systems, 119, 291-301 (2001).
  5. T. E. Dalkilic and T. Apaydin, A fuzzy adaptive network approach to parameter estimation in cases where independent variables come from an exponential distribution, Journal of Computational and Applied Mathematics, 233, 36-45 (2009).
  6. T. E. Dalkilic, T. Apaydin, Parameter Estimation by ANFIS in Cases Where Outputs are Non-Symmetric Fuzzy Numbers, International Journal of Applied Science and Technology, 92-103 (2014).
  7. M. Danesh, S. Danesh, A Combinatorial Algorithm for Fuzzy Parameter Estimation with Application to Uncertain Measurements, Journal of AI and Data Mining, 8, 525-533 (2020).
  8. M. Danesh, S. Danesh, K. khalili, Multi-Sensory Data Fusion System for Tool Condition Monitoring Using Optimized Artificial Fuzzy Inference System, Aerospace Mechanics Journal, 15, 103-118 ( 2020).
  9. P. Diamond, Fuzzy least squares, Information Sciences, 46, 141-157 (1988).
  10. V.A. Epanechnikov, non-parametric estimation of a multivariate probability density, Theory Probab. Appl, 14, 153-158 (1969).
  11. J. Fan, I. Gijbels, Local Polynomial Modelling and Its Applications, Chapman & Hall, London (1996).
  12. R. Farnoosh, J. Ghasemian, O.S fard, A modification on ridge estimation for fuzzy nonparametric regression, Iranian Journal of Fuzzy System, 9, 75-88 (2012).
  13. L.V. Fausett, Fundamentals of neural networks: architectures, algorithms, and applications. Prentice Hall (1994).
  14. M.H. Fazel Zarandi, I.B. Turksen, J. Sobhani, A.A. Ramezanianpour, Fuzzy polynomial neural networks for approximation of the compressive strength of concrete, Applied Soft Computing, 8, 488-498 (2008).
  15. J.D. Hart, Nonparametric Smoothing and Lack-of-Fit Tests, Springer-Verlag, New York (1997).
  16. 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, 307-319 (2004).
  17. H. Ishibuchi, K. Kwon and H. Tanaka, A learning algorithm of fuzzy neural networks with triangular fuzzy weights, Fuzzy Sets and Systems, 71, 277-293 (1995).
  18. H. Ishibushi, H. Tanaka, Fuzzy neural networks with interval weights and its application to fuzzy regression analysis, Fuzzy Sets and Systems, 57, 27-39 (1993).
  19. H. Ishibushi, H. Tanaka, Fuzzy regression analysis using neural networks, Fuzzy Sets and systems, 50, 257-265 (1992).
  20. J.S.R. Jang. ANFIS: adaptive-network-based fuzzy inference system, IEEE Trans Syst. Man Cyber, 23, 665-685 (1993).
  21. J.S.R. Jang, Self-learning fuzzy controllers based on temporal back-propagation, IEEE Transactions on Neural Network , 3, 714-723 (1992).
  22. A. Jayadevsa, S.A. Rahman, A neural network with O(N) neurons for ranking N numbers in O(1/N) time. IEEE Transactions on Circuits and Systems I: Regular Papers, 51, 2044-2051 (2004).
  23. A. Kenar Koohi, H. Soleimanjahi, Sh. Falahi, The application of the new intelligent system (ANFIS) in prediction of human papilloma virus oncogenicity potency, arak medical university Journal (Amuj, 13, 95-105 (2010).
  24. B. Kim, R. R. Bishu, Evaluation of fuzzy linear regression models by comparing membership functions, Fuzzy Sets and Systems, 100, 343-351 (1998).
  25. K.J. Kim, H.R. Chen, a comparison of fuzzy and non-parametric linear regression, Computers Ops Res, 24, 505-519 (1997).
  26. H. Lee, H. Tanaka, Fuzzy approximations with non-symmetric fuzzy parameters in fuzzy regression analysis, Journal of the Operations Research Society of Japan, 42, 98-112 (1999).
  27. D.O. Loftsgaarden and G.P. Quesenberry, A nonparametric estimate of a multivariate density function, Annals of Mathematical Statistics, 36, 1049-1051 (1965).
  28. M. Ming, M. Friedman and A. Kandel„ General fuzzy least squares, Fuzzy Sets and Systems, 88, 107-118 (1997).
  29. M. Modarres, E. Nasrabadi and MM. Nasrabadi, A mathematical-programming approach to fuzzy linear regression analysis, Applied Mathematics and Computation, 163, 977-989 (2005).
  30. M. Mosleh, M. Otadi, S. abbasbandy, Evaluation of fuzzy regression model by fuzzy neural networks, Journal of Computational and Applied Mathematics, 234, 825-834 (2010).
  31. E. A. Nadaraya, on estimating Regression, Theory Probab. Appl, 9, 141-142 (1964).
  32. E. A. Nadaraya, Estimation of a bivariate probability density, Soobshch. Akad. Nauk Gruz. SSR, 36, 267-268 (1964).
  33. M.M. Nasrabadi, E. Nasrabadi, A mathematical-programming approach to fuzzy linear regression analysis, Applied Mathematics and Computation, 155, 873-881 (2004).
  34. M. Otadi, Fully fuzzy polynomial regression by fuzzy neural networks, Neurocomputing, 142, 486-493 (2014).
  35. E. Pasha, T. Razzaghnia, T. Allahviranloo, Gh. Yari. and H. R. Mostafaei, A new mathematical programming approach in fuzzy linear regression models, Applied Mathematical Sciences, 35, 1715-1721 (2007).
  36. W. Pedrycz, F. Gomide and D.A. Savic, An introduction to fuzzy sets: analysis and design, Fuzzy sets and systems, 39, 51-63 (1991).
  37. T. Razzaghnia, Regression parameters prediction in data set with outliers using neural network, Hacettepe journal of mathematics and statistics, 48, 1170-1184 (2019).
  38. T. Razzaghnia, S. Danesh & A. Maleki, Hybrid fuzzy regression with trapezoidal fuzzy data, Proceeding of the SPIE, 834901-834921 (2011).
  39. T. Razzaghnia, S. Danesh, Nonparametric Regression with Trapezoidal Fuzzy Data”, International Journal on Recent and Innovation Trends in Computing and Communication (IJRITCC), 3, 3826-3831 (2015).
  40. M. Rezal, J. A. Ghani, M. Z. Nuawi and C. H. C. Haeon, Online tool wear prediction system in the turning process using an adaptive neuro-fuzzy inference system, Applied Soft Computing, 13, 1960- 1968 (2013).
  41. C. L. Sabharwal, An implementation of hybrid approach to indexing image data bases, ACM Symp. On Applied Computing, 2, 421-426 (1999).
  42. A. Shapiro, The merge of neural networks, fuzzy logic, and genetic algorithms, Insurance: Mathematic, 31, 115-131 (2002).
  43. M. Stone, Cross validation choice and assessment of statistical predictions, Journal of the Royal Statistical Society, 36 (series B), 111-147 (1974).
  44. T. Takagi, M. Sugeno, Fuzzy identification of systems and its application to modelling and control, IEEE Transactions on Systems, Man and Cybernetics , 15, 16-132 (1985).
  45. A. Talei, L. Chua, and T. Wong, ”Evaluation of rainfall and discharge inputs used by Adaptive Network based on Fuzzy Inference Systems (ANFIS) in rainfall-runoff modeling”, Journal of Hydrology, 391, 248-262 (2010).
  46. S. Tamer, M.A. Kamel, M. Hassan, ANFIS For Fault Classification in the Transmission Lines, The Online Journal on Electronics and Electrical Engineering (OJEEE), 2 (2008).
  47. H. Tanaka, Fuzzy data analysis by possibilistic linear models, Fuzzy Sets and Systems, 24, 363-375 (1987).
  48. H. Tanaka, S. Uejima, K. Asia, Linear regression analysis with fuzzy model, IEEE Transactions on Systems, Man and Cybernetics, 12, 903-907 (1982).
  49. H. Tanaka, I. Hayashi, J. Watada, Possibility linear regression analysis for fuzzy data, European Journal of Operational Research , 40, 389-396 (1989).
  50. H. Tanaka, J. Watada, Possibilistic linear systems and their application to the linear regression mode, Fuzzy Sets and System, 27, 275-289 (1988).
  51. N. Wang, W. X. Zhang, C. L. Mei, Fuzzy nonparametric regression based on local linear smoothing technique, Information Sciences, 177, 3882-3900 (2007).
  52. P. Zhang, Model selection via multifold cross validation, Annals of Statistics, 21, 299-313 (1993).
Global Analysis and Discrete Mathematics
Volume 6, Issue 1
August 2021
Pages 143-177

Files

History

  • Receive Date: 07 January 2021
  • Revise Date: 23 June 2021
  • Accept Date: 22 August 2021

Share

How to cite

Statistics