Skip to main content
Springer Nature Link
Log in

A fuzzy functional linear regression model with functional predictors and fuzzy responses

  • Application of soft computing
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

A novel functional regression model was introduced in this research in which, the predictor is a curve linked to a scalar fuzzy response variable. An absolute error-based penalized method with SCAD loss function was also proposed to evaluate the unknown components of the model. For this purpose, a concept of fuzzy-valued function was developed and discussed. Then, a fuzzy large number notion was proposed to estimate the fuzzy-valued function. The performance of the proposed method was examined by some common goodness-of-fit criteria. The efficiency of the proposed method was then evaluated through two numerical examples; a simulation study and an applied example in the scope of watershed management. The proposed method was also compared with several common fuzzy regression models in cases where the functional data were converted to scalar ones.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Akbari MG, Hesamian G (2019a) Elastic net oriented to fuzzy semi-parametric regression model with fuzzy explanatory variables and fuzzy responses. IEEE Trans Fuzzy Syst 17:2433–2442

    Google Scholar 

  • Akbari MG, Hesamian G (2019b) A partial-robust-ridge-based regression model with fuzzy predictors-responses. J Comput Appl Math 351:290–301

    MathSciNet  MATH  Google Scholar 

  • Amiri MJ, Zarei AR, Abedi-Koupai J, Eslamian S (2019) The performance of fuzzy regression method for estimating of reference evapotranspiration under controlled environment. Int J Hydrol Sci Technol 9:28–38

    Google Scholar 

  • Arqub OA (2017) Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integro differential equations. Neural Comput Appl 28:1591–1610

    Google Scholar 

  • Arqub OA, Al-Smadi M, Momani S, Haya T (2016) Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method. Soft Comput 20:3283–3302

    MATH  Google Scholar 

  • Arqub OA, Al-Smadi MM (2020) Fuzzy conformable fractional differential equations: novel extended approach and new numerical solutions. Soft Comput 24:12501–12522

    Google Scholar 

  • Atalay KD, Eraslan E, Cinar MO (2015) A hybrid algorithm based on fuzzy linear regression analysis by quadratic programming for time estimation: an experimental study in manufacturing industry. J Manuf Syst 36:182–188

    Google Scholar 

  • Azadeh A, Alajdad SMHM, Bioki TA (2014) A neuro-fuzzy regression approach for estimation and optimization of gasoline consumption. Int J Serv Oper Manag 17:221–256

    Google Scholar 

  • Buckley JJ, Hayashi Y (1994) Fuzzy genetic algorithm and applications. Fuzzy Sets Syst 61:129–136

    MathSciNet  Google Scholar 

  • Celmins A (1987) Least squares model fitting to fuzzy vector data. Fuzzy Sets Syst 22:245–269

    MathSciNet  MATH  Google Scholar 

  • Chachi J (2019) A weighted least-squares fuzzy regression for crisp input-fuzzy output data. IEEE Trans Fuzzy Syst 17:739–748

    Google Scholar 

  • Chan KY, Engelke U (2015) Fuzzy regression for perceptual image quality assessment. Eng Appl Artif Intell 43:102–110

    Google Scholar 

  • Chan KY, Kwong CK (2013) Modeling of epoxy dispensing process using a hybrid fuzzy regression approach. Int J Adv Manuf Technol 65:589–600

    Google Scholar 

  • Chan KY, Lam HK, Yiu CKF, Dillon TS (2017) A flexible fuzzy regression method for addressing non-linear uncertainty on aesthetic quality assessments. IEEE Trans Syst Man Cybern Syst 47:2363–2377

    Google Scholar 

  • Cheng CB, Lee ES (1999) Non-parametric fuzzy regression K-NN and kernel smoothing techniques. Comput Math Appl 38:239–251

    MathSciNet  MATH  Google Scholar 

  • Chiou JM, Müller HG, Wang JL (2004) Functional response models. Stat Sin 14:675–693

    MathSciNet  MATH  Google Scholar 

  • Chiou JM, Yang YF, Chen YT (2016) Multivariate functional linear regression and prediction. J Multivar Anal 146:301–312

    MathSciNet  MATH  Google Scholar 

  • Choi SH, Buckley JJ (2008) Fuzzy regression using least absolute deviation estimators. Soft Comput 12:257–263

    Google Scholar 

  • Choi SH, Yoon JH (2010) General fuzzy regression using least squares method. Int J Syst Sci 41:477–485

    MATH  Google Scholar 

  • Chukhrova N, Johannssen A (2019) Fuzzy regression analysis: systematic review and bibliography. Appl Soft Comput. https://doi.org/10.1016/j.asoc.2019.105708

    Article  MATH  Google Scholar 

  • Chung W (2012) Using the fuzzy linear regression method to benchmark the energy efficiency of commercial buildings. Appl Energy 95:45–49

    Google Scholar 

  • De Andres-Sanchez J (2016) Fuzzy regression analysis: an actuarial perspective. In: Kahraman C, Kabak O (eds) Fuzzy statistical decision-making: theory and applications, studies in fuzziness and soft computing, vol 343. Springer, Berlin, pp 175–201

    Google Scholar 

  • D’Urso P, Gastaldi T (2002) An order-wise polynomial regression procedure for fuzzy data. Fuzzy Sets Syst 130:1–19

    MATH  Google Scholar 

  • D’Urso P, Massari R (2013) Weighted least squares and least median squares estimation for the fuzzy linear regression analysis. Metron 71:279–306

    MathSciNet  MATH  Google Scholar 

  • Fan YY, Foutz N, James GM, Jank W (2014) Functional response additive model estimation with online virtual stock markets. Ann Appl Stat 8:2435–2460

    MathSciNet  MATH  Google Scholar 

  • Faraway J (1997) Regression analysis for a functional response. Technometrics 39:254–261

    MathSciNet  MATH  Google Scholar 

  • Febrero-Bande M, Oviedo de la Fuente ML (2012) Statistical computing in functional data analysis: the \(R\) package. J Stat Softw 51:1–28

    Google Scholar 

  • Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer Science & Business Media, Berlin

    MATH  Google Scholar 

  • Gholizadeh H, Javadian N, Fazlollahtabar H (2018) Fuzzy regression integrated with genetic-tabu algorithm for prediction and optimization of a turning process. Int J Adv Manuf Technol 96:2781–2790

    Google Scholar 

  • Goldsmith J, Scheipl F (2014) Estimator selection and combination in scalar-on-function regression. Comput Stat Data Anal 70:362–372

    MathSciNet  MATH  Google Scholar 

  • Hall P, Horowitz JL (2007) Methodology and convergence rates for functional linear regression. Ann Stat 35:70–91

    MathSciNet  MATH  Google Scholar 

  • Hao PY, Chiang JH (2008) Fuzzy regression analysis by support vector learning approach. IEEE Trans Fuzzy Syst 16:428–441

    Google Scholar 

  • Hassanpour H, Maleki HR, Yaghoobi MA (2011) A goal programming approach to fuzzy linear regression with fuzzy input–output data. Soft Comput 15:1569–1580

    MATH  Google Scholar 

  • Hastie T, Mallows C (1993) A statistical view of some chemometrics regression tools. Technometrics 35:140–143

    Google Scholar 

  • He YL, Wei CH, Long H, Ashfaq RAR, Huang JZ (2017) Random weight network-based fuzzy non-linear regression for trapezoidal fuzzy number data. Appl Soft Comput 70:959–979

    Google Scholar 

  • Hesamian G, Shams M (2016) Parametric testing statistical hypotheses for fuzzy random variables. Soft Comput 20:1537–1548

    MATH  Google Scholar 

  • Hesamian G, Akbari MG, Asadollahi M (2017) Fuzzy semi-parametric partially linear model with fuzzy inputs and fuzzy outputs. Expert Syst Appl 71:230–239

    Google Scholar 

  • Hong T, Wang P (2014) Fuzzy interaction regression for short term Load forecasting. Fuzzy Optim Decis Mak 13(1):91–103

    MATH  Google Scholar 

  • Hu YC (2014) Multilayer perceptron for robust non-linear interval regression analysis using genetic algorithms. Sci World J 970931:1–8

    Google Scholar 

  • Huang CH (2012) A reduced support vector machine approach for interval regression analysis. Inf Sci 217:56–64

    MathSciNet  MATH  Google Scholar 

  • Icen D, Demirhan H (2016) Error measures for fuzzy linear regression: Monte Carlo simulation approach. Appl Soft Comput 46:104–114

    Google Scholar 

  • James G (2002) Generalized linear models with functional predictors. J R Stat Assoc Ser B 64:411–432

    MathSciNet  MATH  Google Scholar 

  • Jiang H, Kwong CK, Park WY (2017) Probabilistic fuzzy regression approach for preference modeling. Eng Appl Artif Intell 64:286–294

    Google Scholar 

  • Khan UT, Valeo C (2015) A new fuzzy linear regression approach for dissolved oxygen prediction. Hydrol Sci J 60:1096–1119

    Google Scholar 

  • Kim IK, Lee WJ, Yoon JH, Choi SH (2016) Fuzzy regression model using trapezoidal fuzzy numbers for re-auction data. Int J Fuzzy Log Intell Syst 16:72–80

    Google Scholar 

  • Kocadagli O (2013) A novel non-linear programming approach for estimating CAPM beta of an asset using fuzzy regression. Expert Syst Appl 40:858–865

    Google Scholar 

  • Kula K, Apaydin A (2008) Fuzzy robust regression analysis based on the ranking of fuzzy sets. Int J Uncertain Fuzziness Knowl Based Syst 16:663–681

    MathSciNet  MATH  Google Scholar 

  • Lee KH (2005) First course on fuzzy theory and applications. Springer-Verlag, Berlin

    MATH  Google Scholar 

  • Liu Y, Chen Y, Zhou J, Zhong S (2015) Fuzzy linear regression models for QFD using optimized h values. Eng Appl Artif Intell 39:45–54

    Google Scholar 

  • Liu HT, Wang J, He YL, Ashfaq RAR (2017) Extreme learning machine with fuzzy input and fuzzy output for fuzzy regression. Neural Comput Appl 28:3465–3476

    Google Scholar 

  • Luciano S, Barnabás B (2009) Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal Theory Methods Appl 71:1311–1328

    MathSciNet  MATH  Google Scholar 

  • McLean MW, Hooker G, Staicu AM, Scheipl F, Ruppert D (2014) Functional generalized additive models. J Comput Graph Stat 23:249–269

    MathSciNet  Google Scholar 

  • Megri F, Megri AC, Djabri R (2016) An integrated fuzzy support vector regression and the particle swarm optimization algorithm to predict indoor thermal comfort. Indoor Built Environ 25:1248–1258

    Google Scholar 

  • Modarres M, Nasrabadi E, Nasrabadi MM (2005) Fuzzy linear regression models with least square errors. Appl Math Comput 163:977–989

  • Muzzioli S, Ruggieri A, De Baets B (2015) A comparison of fuzzy regression methods for the estimation of the implied volatility smile function. Fuzzy Sets Syst 266:131–143

    MathSciNet  MATH  Google Scholar 

  • Pehlivan NY, Apaydin A (2016) Fuzzy radial basis function network for fuzzy regression with fuzzy input and fuzzy output. Complex Intell Syst 2:61–73

    Google Scholar 

  • Radchenko P, Qiao X, James GM (2015) Index models for sparsely sampled functional data. J Am Stat Assoc 110:824–836

    MathSciNet  MATH  Google Scholar 

  • Ramsay JO, Dalzell CJ (1991) Some tools for functional data analysis. J Roy Stat Soc B 53:539–572

    MathSciNet  MATH  Google Scholar 

  • Ramsay JO, Silverman BW (2006) Functional data analysis. Wiley, New York

    MATH  Google Scholar 

  • Reiss PT, Goldsmith J, Shang HL, Ogden RT (2017) Methods for scalar-on-function regression. Int Stat Rev 85:228–249

    MathSciNet  Google Scholar 

  • Roh SK, Ahn TC, Pedrycz W (2012) Fuzzy linear regression based on polynomial neural networks. Expert Syst Appl 39:8909–8928

    Google Scholar 

  • Roldan L, de Hierro AF, Martinez-Moreno J, Aguilar-Pena C, Lopez R, de Hierro C (2016a) A fuzzy regression approach using Bernstein polynomials for the spreads: computational aspects and applications to economic models. Math Comput Simul 128:13–25

    MathSciNet  MATH  Google Scholar 

  • Roldan L, de Hierro AF, Martinez-Moreno J, Aguilar-Pena C, Lopez R, de Hierro C (2016b) Estimation of a fuzzy regression model using fuzzy distances. IEEE Trans Fuzzy Syst 24:344–359

    MATH  Google Scholar 

  • Ryan BF, Joiner BL, Cryer JD (2012) MINITAB handbook: update for release 16. Brooks/Cole Publishing Co., Pacific Grove

    Google Scholar 

  • Shakouri Ganjavi H, Nadimi R, Ghaderi SF (2017) Investigation on objective function and assessment rule in fuzzy regressions based on equality possibility, fuzzy union and intersection concepts. Comput Ind Eng 110:207–215

    Google Scholar 

  • Simonnet M (1996) The strong law of large numbers. In: Measures and probabilities. Universitext. Springer, New York

  • Sugeno M (1985) An introductory survey of fuzzy control. Inf Sci 36:59–83

    MathSciNet  MATH  Google Scholar 

  • Tanaka H, Ishibuchi H (1992) Possibilistic regression analysis based on linear programming. In: Kacprzyk J, Fedrizzi M (eds) Fuzzy regression analysis. Omnitech Press and Physica-Verlag, Warsaw, Heidelberg, pp 47–60

    Google Scholar 

  • Trott M (2004) The mathematica guidebook for programming. Springer-Verlag, New York

    MATH  Google Scholar 

  • Usset J, Staicu AM, Maity A (2016) Interaction models for functional regression. Comput Stat Data Anal 94:317–330

    MathSciNet  MATH  Google Scholar 

  • Wahba G (1990) Spline models for observational data. Society for Industrial and Applied Mathematics, Siam, Philadelphia

    MATH  Google Scholar 

  • Wang N, Zhang WX, Mei CL (2007) Fuzzy non-parametric regression based on local linear smoothing technique. Inf Sci 177:3882–3900

    MATH  Google Scholar 

  • Wieszczy P, Grzegorzewski P (2016) Support vector machines in fuzzy regression. In: de Tre G, Grzegorzewski P, Kacprzyk J, Owsinski JW, Penczek W, Zadrozny S (eds) Challenging problems and solutions in intelligent systems, studies in computational intelligence, vol 634. Springer, Switzerland, pp 103–138

    Google Scholar 

  • Yang Z, Yin Y (2013) Robust fuzzy varying coefficient regression analysis with crisp inputs and Gaussian fuzzy output. J Comput Sci Eng 7:263–271

    Google Scholar 

  • Yoon JH, Choi SH, Grzegorzewski P (2016) On asymptotic properties of the multiple fuzzy least squares estimator. In: Ferraro MB, Giordani P, Vantaggi B, Gagolewski M, Gil MA, Grzegorzewski P, Hryniewicz O (eds) Soft methods for data science, advances in intelligent systems and computing, vol 456. Springer, Berlin, pp 525–532

    Google Scholar 

  • Yuan M, Cai T (2010) A reproducing kernel Hilbert space approach to functional linear regression. Ann Stat 38:3412–3444

    MathSciNet  MATH  Google Scholar 

  • Zeng W, Feng Q, Li J (2017) Fuzzy least absolute linear regression. Appl Soft Comput 52:1009–1019

  • 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 22:5407–5418

  • Zolfaghari ZS, Mohebbi M, Najariyan M (2014) Application of fuzzy linear regression method for sensory evaluation of fried donut. Appl Soft Comput 22:417–423

  • Zuo H, Zhang G, Pedrycz W, Behbood V, Lu J (2018) Granular fuzzy regression domain adaptation in Takagi–Sugeno fuzzy models. IEEE Trans Fuzzy Syst 26:847–858

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and anonymous reviewer for his/her constructive suggestions and comments, which improved the presentation of this.

Author information

Authors and Affiliations

  1. Department of Statistics, Payame Noor University, Tehran, 19395-3697, Iran

    Gholamreza Hesamian

  2. Department of Statistics, University of Birjand, Birjand, 615-97175, Iran

    Mohammad Ghasem Akbari

Authors
  1. Gholamreza Hesamian
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Mohammad Ghasem Akbari
    View author publications

    You can also search for this author inPubMed Google Scholar

Contributions

GH: Conceptualization, Methodology, Writing-review & editing. MGA: Software, Validation, Investigation.

Corresponding author

Correspondence to Mohammad Ghasem Akbari.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical standard

This article does not contain any studies with human participants or animals performed by the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hesamian, G., Akbari, M.G. A fuzzy functional linear regression model with functional predictors and fuzzy responses. Soft Comput 26, 3029–3043 (2022). https://doi.org/10.1007/s00500-021-06435-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-021-06435-4

Keywords