Soft Computing

, Volume 23, Issue 23, pp 12189–12198 | Cite as

Fuzzy ridge regression with fuzzy input and output

  • Mohammad Reza RabieiEmail author
  • Mohammad Arashi
  • Masoumeh Farrokhi


In regression modeling, existence of multicollinearity may result in linear combination of the parameters, leading to produce estimates with wrong signs. In this paper, a fuzzy ridge regression model with fuzzy input–output data and crisp coefficients is studied. We introduce a generalized variance inflation factor, as a method to identify existence of multicollinearity for fuzzy data. Hence, we propose a new objective function to combat multicollinearity in fuzzy regression modeling. To evaluate the fuzzy ridge regression estimator, we use the mean squared prediction error and a fuzzy distance measure. A Monte Carlo simulation study is conducted to assess the performance of the proposed ridge technique in the presence of multicollinear data. The fuzzy coefficient determination of the fuzzy ridge regression model is higher compared to the fuzzy regression model, when there exists sever multicollinearity. To further ascertain the veracity of the proposed ridge technique, two different data sets are analyzed. Numerical studies demonstrated the fuzzy ridge regression model has lesser mean squared prediction error and fuzzy distance compared to the fuzzy regression model.


Fuzzy arithmetic Generalized variance inflation factor Goodness of fit Fuzzy ridge regression 



The authors would like to thank two anonymous reviewers for their constructive comments which led to put many details in the paper and significantly improved the presentation.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does no contain any studies with human participants or animals performed by any of the authors.


  1. Akbari M, Mohammadalizadeh R, Rezaei M (2012) Bootstrap statistical inference about the regression coefficients based on fuzzy data. Int J Fuzzy Syst 14(4):549–556MathSciNetGoogle Scholar
  2. Arabpour A, Tata M (2008) Estimating the parameters of a fuzzy linear regression model. Iran J Fuzzy Syst 5(2):1–19MathSciNetzbMATHGoogle Scholar
  3. Asai HTSUK (1982) Linear regression analysis with fuzzy model. IEEE Trans Syst Man Cybern 12:903–907zbMATHGoogle Scholar
  4. Balasundaram S (2011) Kapil: Weighted fuzzy ridge regression analysis with crisp inputs and triangular fuzzy outputs. Int J Adv Intell Paradig 3(1):67–81MathSciNetGoogle Scholar
  5. Belsley DA, Kuh E, Welsch RE (1980) Detecting and assessing collinearity. In: Regression diagnostics: identifying influential data and sources of collinearity. Wiley, New YorkzbMATHGoogle Scholar
  6. Chachi J, Roozbeh M (2015) A fuzzy robust regression approach applied to bedload transport data. Commun Stat Simul Comput 46(3):1703–1714MathSciNetzbMATHGoogle Scholar
  7. Coppi R, D’Urso P, Giordani P, Santoro A (2006) Least squares estimation of a linear regression model with lr fuzzy response. Comput Stat Data Anal 51(1):267–286MathSciNetzbMATHGoogle Scholar
  8. Diamond P (1988) Fuzzy least squares. Inf Sci 46(3):141–157MathSciNetzbMATHGoogle Scholar
  9. Donoso S, Marín N, Vila M (2007) Fuzzy ridge regression with non symmetric membership functions and quadratic models. In: International conference on intelligent data engineering and automated learning. Springer, pp 135–144Google Scholar
  10. Dubois DJ (1980) Fuzzy sets and systems: theory and applications, vol 144. Academic press, CambridgezbMATHGoogle Scholar
  11. Gibbons DG (1981) A simulation study of some ridge estimators. J Am Stat Assoc 76(373):131–139zbMATHGoogle Scholar
  12. Gildeh BS, Gien D (2002) A goodness of fit index to reliability analysis in fuzzy model. In: 3rd WSEAS international conference on fuzzy sets and fuzzy systems. Interlaken, Switzerland, pp 11–14Google Scholar
  13. Hassanpour H, Maleki H, Yaghoobi M (2010) Fuzzy linear regression model with crisp coefficients: a goal programming approach. Iran J Fuzzy Syst 7(2):1–153MathSciNetzbMATHGoogle Scholar
  14. Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12(1):55–67zbMATHGoogle Scholar
  15. Hojati M, Bector C, Smimou K (2005) A simple method for computation of fuzzy linear regression. Eur J Oper Res 166(1):172–184MathSciNetzbMATHGoogle Scholar
  16. Hong DH, Hwang C (2003) Support vector fuzzy regression machines. Fuzzy Sets Syst 138(2):271–281MathSciNetzbMATHGoogle Scholar
  17. Hong DH, Hwang C, Ahn C (2004) Ridge estimation for regression models with crisp inputs and Gaussian fuzzy output. Fuzzy Sets Syst 142(2):307–319MathSciNetzbMATHGoogle Scholar
  18. Kim HK, Yoon JH, Li Y (2008) Asymptotic properties of least squares estimation with fuzzy observations. Inf Sci 178(2):439–451MathSciNetzbMATHGoogle Scholar
  19. Marquaridt DW (1970) Generalized inverses, ridge regression, biased linear estimation, and nonlinear estimation. Technometrics 12(3):591–612Google Scholar
  20. McDonald GC, Galarneau DI (1975) A Monte Carlo evaluation of some ridge-type estimators. J Am Stat Assoc 70(350):407–416zbMATHGoogle Scholar
  21. Namdari M, Yoon JH, Abadi A, Taheri SM, Choi SH (2015) Fuzzy logistic regression with least absolute deviations estimators. Soft Comput 19(4):909–917Google Scholar
  22. Newhouse JP, Oman SD (1971) An evaluation of ridge estimatorsGoogle Scholar
  23. Redden DT, Woodall WH (1994) Properties of certain fuzzy linear regression methods. Fuzzy Sets Syst 64(3):361–375MathSciNetzbMATHGoogle Scholar
  24. Snee RD, Marquardt DW (1984) Comment: collinearity diagnostics depend on the domain of prediction, the model, and the data. Am Stat 38(2):83–87Google Scholar
  25. Tanaka H (1987) Fuzzy data analysis by possibilistic linear models. Fuzzy Sets Syst 24(3):363–375MathSciNetzbMATHGoogle Scholar
  26. Tanaka H, Watada J (1988) Possibilistic linear systems and their application to the linear regression model. Fuzzy Sets Syst 27(3):275–289MathSciNetzbMATHGoogle Scholar
  27. Tanaka H, Hayashi I, Watada J (1989) Possibilistic linear regression analysis for fuzzy data. Eur J Oper Res 40(3):389–396MathSciNetzbMATHGoogle Scholar
  28. Tihonov AN (1963) Solution of incorrectly formulated problems and the regularization method. Sov Math Dokl 4:1035–1038MathSciNetGoogle Scholar
  29. Woods H, Steinour HH, Starke HR (1932) Effect of composition of Portland cement on heat evolved during hardening. Ind Eng Chem 24(11):1207–1214Google Scholar
  30. Wu HC (2008) Fuzzy linear regression model based on fuzzy scalar product. Soft Comput 12(5):469–477zbMATHGoogle Scholar
  31. Xu R, Li C (2001) Multidimensional least-squares fitting with a fuzzy model. Fuzzy Sets Syst 119(2):215–223MathSciNetzbMATHGoogle Scholar
  32. Zhou J, Zhang H, Gu Y, Pantelous AA (2018) Affordable levels of house prices using fuzzy linear regression analysis: the case of Shanghai. Soft Comput 22(16):5407–5418Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Mohammad Reza Rabiei
    • 1
    Email author
  • Mohammad Arashi
    • 1
  • Masoumeh Farrokhi
    • 1
  1. 1.Department of Statistics, Faculty of Mathematical SciencesShahrood University of TechnologyShahroodIran

Personalised recommendations