Document Type : Research Paper
Authors
1 Department of Mathematics, University of Birjand, Birjand, Iran
2 Faculty of Basic Sciences, Shiraz University of Technology, Shiraz, Iran
3 Department of Statistics, Shahid Bahonar University of Kerman, Kerman, Iran
Abstract
The fuzzy linear regression model with fuzzy input-output data and
crisp coefficients is studied in this paper. A linear programming
model based on goal programming is proposed to calculate the
regression coefficients. In contrast with most of the previous works, the
proposed model takes into account the centers of fuzzy data as an
important feature as well as their spreads in the procedure of
constructing the regression model. Furthermore, the model can deal
with both symmetric and non-symmetric triangular fuzzy data as well
as trapezoidal fuzzy data which have rarely been considered in the
previous works. To show the efficiency of the proposed model, some
numerical examples are solved and a simulation study is performed.
The computational results are compared with some earlier methods.
Keywords
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regression model}, Iranian Journal of Fuzzy Systems, {\bf 5}\textbf{(2)} (2008), 1-19.
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programming and network flows}, Jhon Wiley and Sons, New York, 1990.
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and methods}, Iranian Journal of Fuzzy Systems, {\bf 3}\textbf{(1)} (2006), 1-22.
\bibitem{Celmins:LSMFFVD} A. Celmins, {\it Least squares model fitting to fuzzy vector
data}, Fuzzy Sets and Systems, {\bf 22} (1987), 245-269.
\bibitem{ChBa:FRLADE} S. H. Choi and J. J. Buckley, {\it Fuzzy regression using least
absolute deviation estimators}, Soft Computing, {\bf 12}\textbf{(3)} (2008), 257-263.
\bibitem{ChKiPa:NRQE} S. H. Choi, H. K. Kim and K. O. Park, {\it Nonlinear regression quantiles estimation}, Journal of the Korean Statistical Society, {\bf 29} (2000), 187-199.
\bibitem{CDuGiSa:LSELRMFR} R. Coppi, P. D'Urso, P. Giordani and A. Santoro,
{\it Least squares estimation of a linear regression model with LR fuzzy response},
Computational Statistics \& Data Analysis, {\bf 51} (2006), 267-286.
\bibitem{Diam:FLS} P. Diamond, {\it Fuzzy least squares}, Information Sciences,
{\bf 46} (1988), 141-157.
\bibitem{Diam:LSFSV} P. Diamond, {\it Least squares fitting of several variables},
Preprint of Second IFSA Congress, Tokyo, July (1987), 20-25.
\bibitem{DiKl:MSFS} P. Diamond and P. Kloeden, {\it Metrics spaces of fuzzy sets:
theory and applications}, World Scientific, 1994.
\bibitem{D'Urso:LRFCIFCOD} P. D'Urso, {\it Linear regression analysis for fuzzy/crisp input and fuzzy/crisp
output data}, Computational Statistics \& Data Analysis, {\bf 42} (2003), 47-72.
\bibitem{HaMaYa:AGPAFLR} H. Hassanpour, H. R. Maleki and M. A. Yaghoobi, {\it A goal programming
approach to fuzzy linear regression with non-fuzzy input and fuzzy
output data}, APJOR, {\bf 26}\textbf{(5)} (2009), 587-604.
\bibitem{HaMaYa:ANEFLRMCMF} H. Hassanpour, H. R. Maleki and M. A. Yaghoobi, {\it A note on evaluation
of fuzzy linear regression models by comparing membership functions}, Iranian Journal of Fuzzy Systems, {\bf 6}\textbf{(2)} (2009), 1-6.
\bibitem{HoBeSm:ASMCFLR} M. Hojati, C. R. Bector and K. Smimou, {\it A simple method for computation
of fuzzy linear regression}, EJOR, {\bf 166} (2005), 172-184.
\bibitem{HuYa:AOADOFRM} W. L. Hung and M. S. Yang, {\it An omission approach for detecting outliers in
fuzzy regression models}, Fuzzy Sets and Systems, {\bf 157} (2006), 3109-3122.
\bibitem{JOZ:OELDTPFLR} S. Jozsef, {\it On the effect of linear data transformations in
possibilistic fuzzy linear regression}, Fuzzy Sets and Systems,
{\bf 45} (1992), 185-188.
\bibitem{KaCh:AFLRMBEP} C. Kao and C. L. Chyu, {\it A fuzzy linear regression model with better
explanatory power}, Fuzzy Sets and Systems, {\bf 126} (2002), 401-409.
\bibitem{KaCh:LSEFRA} C. Kao and C. L. Chyu, {\it Least-squares
estimates in fuzzy regression analysis}, EJOR, {\bf 148} (2003), 426-435.
\bibitem{KiBi:EFLRMCMF} B. Kim and R. R. Bishu, {\it Evaluation of fuzzy linear regression models by
comparing membership functions}, Fuzzy Sets and Systems, {\bf 100} (1998), 343-352.
\bibitem{KiYoLi:APLSEFO} H. K. Kim, J. H. Yoon and Y. Li, {\it Asymptotic properties of least squares
estimation with fuzzy observations}, Information Sciences,
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analysis from the point of view risk}, International Journal of Uncertainty,
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a general framework}, Computers \& Operations Research, {\bf 27} (2000), 635-652.
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intervals}, Fuzzy Sets and Systems, {\bf 63} (1994), 45-55.
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regression models}, Fuzzy Sets and Systems, {\bf 64} (1994), 361-375.
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method for solving fuzzy linear programming problems}, Iranian Journal of
Fuzzy Systems, {\bf 4}\textbf{(2)} (2007), 31-46.
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Plenum Press, New York, 1993.
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\bibitem{SaYa:MFLRAFIOD} M. Sakawa and H. Yano, {\it Multiobjective fuzzy linear
regression analysis for fuzzy input-output data}, Fuzzy Sets and
Systems, {\bf 47} (1992), 173-181.
\bibitem{SaPe:EFLRM} D. A. Savic and W. Pedrycz, {\it Evaluation of fuzzy linear
regression models}, Fuzzy Sets and Systems, {\bf 39} (1991), 51-63.
\bibitem{ShKhGh:OLOFSFRICMC} E. Shivanian, E. Khorram and A. Ghodousian, {\it Optimization of linear
objective function subject to fuzzy relation inequalities constraints with max-average composition}, Iranian Journal of Fuzzy Systems, {\bf 4}\textbf{(2)} (2007), 15-30.
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Fuzzy Sets and Systems, {\bf 24} (1987), 363-375.
\bibitem{TaHaWa:PLRAFD}H. Tanaka, I. Hayashi and J. Watada, {\it Possibilistic linear
regression analysis for fuzzy data}, EJOR, {\bf 40} (1989), 389-396.
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model}, IEEE Transactions on Systems, Man and Cybernetics,
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spaces}, Iranian Journal of Fuzzy Systems, {\bf 5}\textbf{(2)} (2008),
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\bibitem{AbMa:pcPFNRC} S. M. Abu Nayeem and M. Pal, {\it The p-center problem on fuzzy
networks and reduction of cost}, Iranian Journal of Fuzzy Systems,
{\bf 5}\textbf{(1)} (2008), 1-26.
\bibitem{ArTa:EPFLRM} A. R. Arabpour and M. Tata, {\it Estimating the parameters of a fuzzy linear
regression model}, Iranian Journal of Fuzzy Systems, {\bf 5}\textbf{(2)} (2008), 1-19.
\bibitem{BaJaSh:LPandNF} M. S. Bazaraa, J. J. Jarvis and H. D. Sherali, {\it Linear
programming and network flows}, Jhon Wiley and Sons, New York, 1990.
\bibitem{CaVe:APFOMM} J. M. Cadenas and J. L. Verdegay, {\it A primer on fuzzy optimization models
and methods}, Iranian Journal of Fuzzy Systems, {\bf 3}\textbf{(1)} (2006), 1-22.
\bibitem{Celmins:LSMFFVD} A. Celmins, {\it Least squares model fitting to fuzzy vector
data}, Fuzzy Sets and Systems, {\bf 22} (1987), 245-269.
\bibitem{ChBa:FRLADE} S. H. Choi and J. J. Buckley, {\it Fuzzy regression using least
absolute deviation estimators}, Soft Computing, {\bf 12}\textbf{(3)} (2008), 257-263.
\bibitem{ChKiPa:NRQE} S. H. Choi, H. K. Kim and K. O. Park, {\it Nonlinear regression quantiles estimation}, Journal of the Korean Statistical Society, {\bf 29} (2000), 187-199.
\bibitem{CDuGiSa:LSELRMFR} R. Coppi, P. D'Urso, P. Giordani and A. Santoro,
{\it Least squares estimation of a linear regression model with LR fuzzy response},
Computational Statistics \& Data Analysis, {\bf 51} (2006), 267-286.
\bibitem{Diam:FLS} P. Diamond, {\it Fuzzy least squares}, Information Sciences,
{\bf 46} (1988), 141-157.
\bibitem{Diam:LSFSV} P. Diamond, {\it Least squares fitting of several variables},
Preprint of Second IFSA Congress, Tokyo, July (1987), 20-25.
\bibitem{DiKl:MSFS} P. Diamond and P. Kloeden, {\it Metrics spaces of fuzzy sets:
theory and applications}, World Scientific, 1994.
\bibitem{D'Urso:LRFCIFCOD} P. D'Urso, {\it Linear regression analysis for fuzzy/crisp input and fuzzy/crisp
output data}, Computational Statistics \& Data Analysis, {\bf 42} (2003), 47-72.
\bibitem{HaMaYa:AGPAFLR} H. Hassanpour, H. R. Maleki and M. A. Yaghoobi, {\it A goal programming
approach to fuzzy linear regression with non-fuzzy input and fuzzy
output data}, APJOR, {\bf 26}\textbf{(5)} (2009), 587-604.
\bibitem{HaMaYa:ANEFLRMCMF} H. Hassanpour, H. R. Maleki and M. A. Yaghoobi, {\it A note on evaluation
of fuzzy linear regression models by comparing membership functions}, Iranian Journal of Fuzzy Systems, {\bf 6}\textbf{(2)} (2009), 1-6.
\bibitem{HoBeSm:ASMCFLR} M. Hojati, C. R. Bector and K. Smimou, {\it A simple method for computation
of fuzzy linear regression}, EJOR, {\bf 166} (2005), 172-184.
\bibitem{HuYa:AOADOFRM} W. L. Hung and M. S. Yang, {\it An omission approach for detecting outliers in
fuzzy regression models}, Fuzzy Sets and Systems, {\bf 157} (2006), 3109-3122.
\bibitem{JOZ:OELDTPFLR} S. Jozsef, {\it On the effect of linear data transformations in
possibilistic fuzzy linear regression}, Fuzzy Sets and Systems,
{\bf 45} (1992), 185-188.
\bibitem{KaCh:AFLRMBEP} C. Kao and C. L. Chyu, {\it A fuzzy linear regression model with better
explanatory power}, Fuzzy Sets and Systems, {\bf 126} (2002), 401-409.
\bibitem{KaCh:LSEFRA} C. Kao and C. L. Chyu, {\it Least-squares
estimates in fuzzy regression analysis}, EJOR, {\bf 148} (2003), 426-435.
\bibitem{KiBi:EFLRMCMF} B. Kim and R. R. Bishu, {\it Evaluation of fuzzy linear regression models by
comparing membership functions}, Fuzzy Sets and Systems, {\bf 100} (1998), 343-352.
\bibitem{KiYoLi:APLSEFO} H. K. Kim, J. H. Yoon and Y. Li, {\it Asymptotic properties of least squares
estimation with fuzzy observations}, Information Sciences,
{\bf 178} (2008), 439-451.
\bibitem{KoBa} R. Koenker and G. Bassett, {\it Regression quantiles}, Econometrica,
{\bf 46} (1978), 33-50.
\bibitem{LINGO} L. Schage, {\it LINGO release 6.0}, LINDO System, Inc., 1999.
\bibitem{MATLAB} {\it MATLAB 7.0}, The Mathworks, (www.mathworks.com).
\bibitem{MPL} Maximal software inc., (www.maximalsoftware.com).
\bibitem{MiFrKa:GFLS} M. Ming, M. Friedman and A. Kandel, {\it General
fuzzy least-squares}, Fuzzy Sets and Systems, {\bf 88} (1997), 107-118.
\bibitem{MoNaNa:FLRAPVR} M. Modarres, E. Nasrabadi and M. M. Nasrabadi, {\it Fuzzy linear regression
analysis from the point of view risk}, International Journal of Uncertainty,
Fuzziness and Knowledge-Based Systems, {\bf 12}\textbf{(5)} (2004), 635-649.
\bibitem{MoNaNa:FLRMLSE} M. Modarres, E. Nasrabadi and M. M. Nasrabadi, {\it Fuzzy linear regression models with least square errors}, Applied Mathematics and Computation, {\bf 163} (2005), 977-989.
\bibitem{MoTa:PFFLSR} J. Mohammadi and S. M. Taheri, {\it Pedomodels fitting with fuzzy least
squares regression}, Iranian Journal of Fuzzy Systems, {\bf 1}\textbf{(29)} (2004), 45-62.
\bibitem{OzDu:MOFRGF} E. C. \"{O}zelkan and L. Duckstein, {\it Multi-objective fuzzy regression:
a general framework}, Computers \& Operations Research, {\bf 27} (2000), 635-652.
\bibitem{Pet:FLRFI} G. Peters, {\it Fuzzy linear regression with fuzzy
intervals}, Fuzzy Sets and Systems, {\bf 63} (1994), 45-55.
\bibitem{ReWo:PCFLRM} D. T. Redden and W. H. Woodall, {\it Properties of certain fuzzy linear
regression models}, Fuzzy Sets and Systems, {\bf 64} (1994), 361-375.
\bibitem{SaMaZa:AnZMSFLPp} M. R. Safi, H. R. Maleki and E. Zaeimazad, {\it A note on zimmermann
method for solving fuzzy linear programming problems}, Iranian Journal of
Fuzzy Systems, {\bf 4}\textbf{(2)} (2007), 31-46.
\bibitem{Sak:FSIMO} M. Sakawa, {\it Fuzzy sets and interactive multiobjective optimization},
Plenum Press, New York, 1993.
\bibitem{SaYa:FLRA} M. Sakawa and H. Yano, {\it Fuzzy linear regression and its applications}, In:
Studies in Fuzziness, Fuzzy Regression Analysis,
J. Kacprzyk and M. Fedrizzi, Omnitech Press, Warsaw,
Poland, {\bf1} (1992), 61-80.
\bibitem{SaYa:MFLRAFIOD} M. Sakawa and H. Yano, {\it Multiobjective fuzzy linear
regression analysis for fuzzy input-output data}, Fuzzy Sets and
Systems, {\bf 47} (1992), 173-181.
\bibitem{SaPe:EFLRM} D. A. Savic and W. Pedrycz, {\it Evaluation of fuzzy linear
regression models}, Fuzzy Sets and Systems, {\bf 39} (1991), 51-63.
\bibitem{ShKhGh:OLOFSFRICMC} E. Shivanian, E. Khorram and A. Ghodousian, {\it Optimization of linear
objective function subject to fuzzy relation inequalities constraints with max-average composition}, Iranian Journal of Fuzzy Systems, {\bf 4}\textbf{(2)} (2007), 15-30.
\bibitem{Tan:FDAPLM} H. Tanaka, {\it Fuzzy data analysis by possibilistic linear models},
Fuzzy Sets and Systems, {\bf 24} (1987), 363-375.
\bibitem{TaHaWa:PLRAFD}H. Tanaka, I. Hayashi and J. Watada, {\it Possibilistic linear
regression analysis for fuzzy data}, EJOR, {\bf 40} (1989), 389-396.
\bibitem{TaUeAs:LRAFM} H. Tanaka, S. Uejima and K. Asai, {\it Linear regression analysis with fuzzy
model}, IEEE Transactions on Systems, Man and Cybernetics,
{\bf 12} (1982), 903-907.
\bibitem{VaKa:tBAFNS} S. M. Vaezpour and F. Karimi, {\it T-best approximation in fuzzy normed
spaces}, Iranian Journal of Fuzzy Systems, {\bf 5}\textbf{(2)} (2008),
93-99.
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)