A nonlinear modeling with linear fuzzy numbers for replicated response measures. (English) Zbl 07552614

Summary: In this study, it is aimed to present a flexible modeling approach for replicated response measured data set via linear fuzzy numbers (LFNs), e.g. pentagonal linear fuzzy numbers (PLFNs), trapezoidal linear fuzzy numbers (TrLFNs), and triangular linear fuzzy numbers (TLFNs). For this purpose, a fuzzification formula of replicated measures was proposed with calculating quartile based descriptive statistics. The model parameters were also assumed as LFNs whereas the input variable was crisp. In order to obtain the predicted fuzzy nonlinear regression model, least squares (LS) approach and hybrid optimization algorithm, hybrid of Genetic Algorithm and Quasi-Newton algorithm (GA-QN), were used as estimation and optimization methods, respectively. Two data sets were chosen from the literature for application purpose. It is seen from the obtained results that the predicted fuzzy nonlinear functions, obtained with PLFNs, TrLFNs, TLFNs, have equal performance according to the nonparametric statistical tests. However, it is seen from the medians of prediction errors that the predicted fuzzy nonlinear model with PLFNs is the most preferable one.


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[1] Bardossy, A.; Bogaroli, I.; Duckstein, L., Fuzzy nonlinear regression analysis of dose-response relationships, European Journal of Operational Research, 66, 1, 36-51 (1993) · Zbl 0765.62063 · doi:10.1016/0377-2217(93)90204-Z
[2] Bates, D. M.; Watts, D. G., Nonlinear regression analysis and its applications (1988), New York: John Wiley and Sons, New York · Zbl 0728.62062
[3] Buckley, J. J.; Feuring, T., Linear and non-linear fuzzy regression: Evolutionary algorithm solutions, Fuzzy Set and Systems, 112, 3, 381-94 (2000) · Zbl 0948.62048 · doi:10.1016/S0165-0114(98)00154-7
[4] Celmins, A., Practical approach to nonlinear fuzzy regression, SIAM Journal of Science and Statistical Computing, 12, 3, 521-46 (1991) · Zbl 0725.65153
[5] Chan, K. Y.; Kwong, C. K.; Fogarty, T. C., Modeling manufacturing processes using a genetic programming-based fuzzy regression with detection of outliers, Information Science, 180, 4, 506-18 (2010) · doi:10.1016/j.ins.2009.10.007
[6] Chan, K. Y.; Kwong, C. K.; Dillon, T. S.; Fung, K. Y., An Intelligent fuzzy regression approach for affective product design that captures nonlinearity and fuzziness, Journal of Engineering Design, 22, 8, 523-42 (2011) · doi:10.1080/09544820903550924
[7] Chan, K. Y.; Kwong, C. K., Modeling of epoxy dispensing process using a hybrid fuzzy regression approach, The International Journal of Advanced Manufacturing Technology, 65, 1-4, 589-600 (2013) · doi:10.1007/s00170-012-4202-4
[8] Chang, S. S. L.; Zadeh, L. A., On fuzzy mappings and control, IEEE Transactions on Systems, Man, and Cybernetics, 2, 1, 30-4 (1972) · Zbl 0305.94001 · doi:10.1109/TSMC.1972.5408553
[9] Coppi, R.; D’Urso, P.; Giordania, P.; Santoro, A., Least squares estimation of a linear regression model with LR fuzzy response, Computational Statistics & Data Analysis, 51, 267-86 (2006) · Zbl 1157.62460 · doi:10.1016/j.csda.2006.04.036
[10] Dubois, D.; Prade, H., Operations on fuzzy numbers, International Journal of Systems Science, 9, 6, 613-26 (1978) · Zbl 0383.94045 · doi:10.1080/00207727808941724
[11] Dunyak, J. P.; Wunsch, D., Fuzzy regression by fuzzy number neural networks. Fuzzy., Set and Systems, 112, 3, 371-80 (2000) · Zbl 0948.62049 · doi:10.1016/S0165-0114(97)00393-X
[12] D’Urso, P., Linear regression analysis for fuzzy/crisp input and fuzzy/crisp output data, Computational Statistics & Data Analysis, 42, 47-72 (2003) · Zbl 1429.62337
[13] D’Urso, P.; Gastaldi, T., A least-squares approach to fuzzy linear regression analysis, Computational Statistics & Data Analysis, 34, 427-40 (2000) · Zbl 1046.62066 · doi:10.1016/S0167-9473(99)00109-7
[14] D’Urso, P.; Gastaldi, T., An “orderwise” polynomial regression procedure for fuzzy data, Fuzzy Sets and Systems, 130, 1-19 (2002) · Zbl 1033.62070 · doi:10.1016/S0165-0114(02)00055-6
[15] D’Urso, P.; Massari, R.; Santoro, A., A class of fuzzy clusterwise regression models, Information Sciences, 180, 4737-62 (2010) · Zbl 1204.62112 · doi:10.1016/j.ins.2010.08.018
[16] D’Urso, P.; Massari, R.; Santoro, A., Robust fuzzy regression analysis, Information Sciences, 181, 4154-74 (2011) · Zbl 1242.62073 · doi:10.1016/j.ins.2011.04.031
[17] D’Urso, P.; Massari, R., Weighted least squares and least median squares estimation for the fuzzy linear regression analysis, Metron, 71, 279-306 (2013) · Zbl 1302.62170 · doi:10.1007/s40300-013-0025-9
[18] D’Urso, P.; Santoro, A., Fuzzy clusterwise linear regression analysis with symmetrical fuzzy output variable, Computational Statistics & Data Analysis, 51, 287-313 (2006) · Zbl 1157.62461 · doi:10.1016/j.csda.2006.06.001
[19] Gaeta, M.; Loia, V.; Tomasiello, S., A Fuzzy Functional Network for nonlinear regression problems, International Journal of Knowledge Engineering and Soft Data Paradigms, 4, 4, 351 (2014) · doi:10.1504/IJKESDP.2014.069290
[20] Gill, P. E.; Murry, W.; Wright, M. H., Practical optimization (1981), New York: Academic Press, New York · Zbl 0503.90062
[21] He, Y. L.; Wang, X. Z.; Huang, J. Z., Fuzzy nonlinear regression analysis using a random weight network, Information Sciences, 364, 222-40 (2016) · Zbl 1427.68265 · doi:10.1016/j.ins.2016.01.037
[22] Holland, J. H., Adaptation in natural and artificial systems (1975), Ann Arbor: The University of Michigan Press, Ann Arbor · Zbl 0317.68006
[23] Hong, D. H.; Hwang, C., Support fuzzy regression machines, Fuzzy Set and Systems, 138, 2, 271-81 (2003) · Zbl 1026.62076 · doi:10.1016/S0165-0114(02)00514-6
[24] Ishibuchi, H.; Nii, M., Fuzzy regression using asymmetric fuzzy coefficients and fuzzied neural networks, Fuzzy Sets and Systems, 119, 2, 273-90 (2001) · Zbl 0964.62051 · doi:10.1016/S0165-0114(98)00370-4
[25] Kamble, A. J., Some notes on pentagonal fuzzy numbers, International Journal of Fuzzy Mathematical Archive, 13, 2, 113-21 (2017)
[26] Kennedy, F. C.; Dhanalakshmi, V., Cone properties of linear fuzzy numbers, Global and Stochastic Analysis, 4, 1, 95-109 (2017)
[27] Kumar, R.; Pathinathan, T., Sieving out the poor using fuzzy decision making tools with reference to Nalanda District, Bihar, India, International Conference on Convergence Technology, 5, 1, 890-1 (2015)
[28] Mondal, S. P.; Mandal, M., Pentagonal fuzzy number, its properties and application in fuzzy equation, Future Computing and Informatics Journal, 2, 2, 110-7 (2017) · doi:10.1016/j.fcij.2017.09.001
[29] Nasrabadi, E.; Hashemi, S. M., Robust fuzzy regression analysis using neural networks, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 16, 4, 579-98 (2008) · Zbl 1151.62337 · doi:10.1142/S021848850800542X
[30] Pandit, P. K., Fuzzy nonlinear regression using artificial neural networks, International Journal of Advances in Applied Mathematics and Mechanics, 2, 1, 53-63 (2014) · Zbl 1359.62313
[31] Pathinathan, T.; Minj, A., Interval-valued pentagonal fuzzy numbers, International Journal of Pure and Applied Mathematics, 119, 9, 177-87 (2018)
[32] Seng, K. Y.; Nesterov, I.; Gini, P. V., Fuzzy least squares for identification of individual pharmacokinetic parameters, IEEE Transactions Biomedical Engineering, 56, 12, 2796-805 (2009)
[33] Türkşen, Ö.; Apaydın, A., A modeling approach based on fuzzy least squares method for multi-response experiments with replicated measures, 153-158 (2014)
[34] Türkşen, Ö.; Güler, N., Comparison of fuzzy logic based models for the multi-response surface problems with replicated response measures, Applied Soft Computing, 37, 887-96 (2015) · doi:10.1016/j.asoc.2015.09.028
[35] Türkşen, Ö., Analysis of response surface model parameters with Bayesian approach and fuzzy approach, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 24, 1, 109-22 (2016) · Zbl 1378.62035 · doi:10.1142/S0218488516500069
[36] Türkşen, Ö., A fuzzy modeling approach for replicated response measures based on fuzzification of replications with descriptive statistics and golden ratio, Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22, 1, 153-9 (2018) · doi:10.19113/sdufbed.89217
[37] Yao, C. C.; Yu, P. T., Fuzzy regression based on asymmetric support vector machines, Applied Mathematics and Computation, 182, 1, 175-93 (2006) · Zbl 1113.62084 · doi:10.1016/j.amc.2006.03.046
[38] Zadeh, L. A., The concept of a Linguistic variable and applications to approximate reasoning-part-I, II, III, Information Science, 8, 3, 199-249 (1975) · Zbl 0397.68071 · doi:10.1016/0020-0255(75)90036-5
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