Document Type : Research Paper

Author

University of Torbat heydarieh, Torbat Heydarieh, Khorasan Razavi, Iran.

Abstract

Curve fitting is a computational problem in which we look for a base objective function with a set of data points. Recently, nonparametric regression has received a lot of attention from researchers. Usually, spline functions are used due to the difficulty of the curve fitting. In this regard, the choice of the number and location of knots for regression is a major issue. Therefore, in this study, a Genetic algorithm simultaneously determines the number and location of the knots based on two criteria comprise of least square error and capability process index. The proposed algorithm performance has been evaluated by some numerical examples. Simulation results and comparisons reveal that the proposed approach in curve fitting has satisfactory performance. Also, a sensitivity analysis on the number of knots has been illustrated by an example. Finally, simulation results from a real case in statistical process control show that the proposed Genetic algorithm works well in practice.

Keywords

Main Subjects

Dierckx, P. (1995). Curve and surface with splines. Oxford University Press.
DiMatteo, I., Genovese, C. R., & Kass, R. E. (2001). Bayesian curve‐fitting with free‐knot splines. Biometrika88(4), 1055-1071. https://doi.org/10.1093/biomet/88.4.1055
Ahmed, S. M., Biswas, T. K., & Nundy, C. K. (2019). An optimization model for aggregate production planning and control: a genetic algorithm approach. International journal of research in industrial engineering8(3), 203-224. http://dx.doi.org/10.22105/riej.2019.192936.1090
Zhao, X., Zhang, C., Yang, B., & Li, P. (2011). Adaptive knot placement using a GMM-based continuous optimization algorithm in B-Spline curve approximation. Computer-aided design43(6), 598-604. https://doi.org/10.1016/j.cad.2011.01.015
Gazioglu, S., Wei, J., Jennings, E. M., & Carroll, R. J. (2013). A note on penalized regression spline estimation in the secondary analysis of case-control data. Statistics in biosciences5(2), 250-260. https://doi.org/10.1007/s12561-013-9094-9
Lai, M. J., & Wang, L. (2013). Bivariate penalized splines for regression. Statistica sinica, 1399-1417. http://dx.doi.org/10.5705/ss.2010.278
Seo, H. S., Song, J. E., & Yoon, M. (2013). An outlier detection method in penalized spline regression models. Korean journal of applied statistics26(4), 687-696. https://doi.org/10.5351/KJAS.2013.26.4.687
Schwarz, K., & Krivobokova, T. (2016). A unified framework for spline estimators. Biometrika103(1), 121-131. https://doi.org/10.1093/biomet/asv070
Montoril, M. H., Morettin, P. A., & Chiann, C. (2014). Spline estimation of functional coefficient regression models for    time series with correlated errors. Statistics & probability petters92(1), 226-231. https://doi.org/10.1016/j.spl.2014.05.021
Yang, Y., & Song, Q. (2014). Jump detection in time series nonparametric regression models: a polynomial spline approach. Annals of the institute of statistical mathematics66(2), 325-344. https://doi.org/10.1007/s10463-013-0411-3
Papp, D., & Alizadeh, F. (2014). Shape-constrained estimation using nonnegative splines. Journal of computational and graphical statistics23(1), 211-231. https://doi.org/10.1080/10618600.2012.707343
Ma, S., Racine, J. S., & Yang, L. (2015). Spline regression in the presence of categorical predictors. Journal of applied econometrics30(5), 705-717. https://doi.org/10.1002/jae.2410
Zhou, J., Chen, Z., & Peng, Q. (2016). Polynomial spline estimation for partial functional linear regression models. Computational statistics, 31(3), 1-23. https://doi.org/10.1007/s00180-015-0636-0
Daouia, A., Noh, H., & Park, B. U. (2016). Data envelope with constrained polynomial splines. Journal of the royal statistical society: series b (statistical methodology)78(1), 3-30. https://doi.org/10.1111/rssb.12098
Powell, J. D. (1970). Curve fitting by splines in one variable. Numerical approximation to functions and data, 12(1), 65-83. https://doi.org/10.2307/2316601
Jupp, D. L. (1978). Approximation to data by splines with free knots. SIAM journal on numerical analysis15(2), 328-343. https://doi.org/10.1137/0715022
Ma, S. (2014). A plug-in the number of knots selector for polynomial spline regression. Journal of nonparametric statistics26(3), 489-507. https://doi.org/10.1080/10485252.2014.930143
Wang, X. (2008). Bayesian free-knot monotone cubic spline regression. Journal of computational and graphical statistics, 17(2), 373-387. https://doi.org/10.1198/106186008X321077
Engin, O., & İşler, M. (2021). An efficient parallel greedy algorithm for fuzzy hybrid flow shop scheduling with setup time and lot size: a case study in apparel process. Journal of fuzzy extension and applications, 3(3), 249-262. http://dx.doi.org/10.22105/jfea.2021.314312.1169
Goli, A., Zare, H. K., Moghaddam, R., & Sadeghieh, A. (2018). A comprehensive model of demand prediction based on hybrid artificial intelligence and metaheuristic algorithms: a case study in dairy industry. Journal of industrial and systems engineering, 11(1), 190-203. https://doi.org/10.1080/10485252.2014.930143
Shahsavari, N., Abolhasani, M. H., Sheikhi, H., Mohammadi Andargoli, H., & Abolhasani, H. (2014). A novel Genetic algorithm for a flow shop scheduling problem with fuzzy processing time. International journal of research in industrial engineering3(4), 1-12. https://doi.org/10.1007/s10463-013-0411-3
Sanagooy Aghdam, A., Kazemi, M. A. A., & Eshlaghy, A. T. (2021). A hybrid GA–SA multiobjective optimization and simulation for RFID network planning problem. Journal of applied research on industrial engineering, 8(Spec. Issue), 1-25. http://dx.doi.org/10.22105/jarie.2021.295762.1357
Khalili, S. & Mosadegh Khah, M. (2020). A new queuing-based mathematical model for hotel capacity planning: a Genetic algorithm solution. Journal of applied research on industrial engineering7(3), 203-220. http://dx.doi.org/10.22105/jarie.2020.244708.1187
Rezaee, F., & Pilevari, N. (In Press). Mathematical model of sustainable multilevel supply chain with meta-heuristic algorithm approach (case study: atmosphere group: industrial and manufacturing power plant). Journal of decisions and operations research. DOI: 10.22105/dmor.2021.270853.1310
Alizadeh Firozi, M., Kiani, V., & Karimi, H. (2022). Improved Genetic algorithm with diversity and local search for uncapacitated single allocation hub location problem. Journal of decisions and operations research6(4), 536-552. http://dx.doi.org/10.22105/DMOR.2021.272989.1325
Irshad, M., Khalid, S., Hussain, M. Z., & Sarfraz, M. (2016). Outline capturing using rational functions with the help of GA. Applied mathematics and computation274(1), 661-678. https://doi.org/10.1016/j.amc.2015.10.014
Holland, J. H. (1975). Adaptation in natural and artificial systems. University of Michigan Press. https://doi.org/10.7551/mitpress/1090.003.0007
Lee, T. (2002). On algorithms for ordinary least squares regression spline: a comparative study. Journal of statistical computation and simulation, 72(8), 647-663. https://doi.org/10.1080/00949650213743
Yoshimoto, F., Harada, T., & Yoshimoto, Y. (2003). Data with a spline using a real-coded Genetic Algorithm. Computer-aided design35(8), 751-760. https://doi.org/10.1016/S0010-4485(03)00006-X
Pittman, J. (2002). Adaptive splines and Genetic Algorithms. Journal of computational and graphical statistics11(3), 615-638. https://doi.org/10.1198/106186002448
Tongur, V., & Ülker, E. (2016). B-Spline curve knot estimation by using niched Pareto genetic algorithm (npga). In Intelligent and evolutionary systems (pp. 305-316). Springer, Cham.
Garcia, C. H., Cuevas, F. J., Trejo-Caballero, G., & Rostro-Gonzalez, H. (2015). A hierarchical GA approach for curve fittingwith B-splines. Genetic programming and evolvable machines16(2), 151-166. https://doi.org/10.1007/s10710-014-9231-3
Gálvez, A., Iglesias, A., Avila, A., Otero, C., Arias, R., & Manchado, C. (2015). Elitist clonal selection algorithm for optimal choice of free knots in B-Spline data. Applied soft computing26(1), 90-106. https://doi.org/10.1016/j.asoc.2014.09.030
Fengler, M. R., & Hin, L. Y. (2015). A simple and general approach to fitting the discount curve under no-arbitrage constraints. Finance research letters, 15(1), 78-84. https://doi.org/10.1016/j.frl.2015.08.006
Liu, G. X., Wang, M. M., Du, X. L., Lin, J. G., & Gao, Q. B. (2018). Jump-detection and curve estimation methods for discontinuous regression functions based on the piecewise B-Spline function. Communications in statistics-theory and methods47(23), 5729-5749. https://doi.org/10.1080/03610926.2017.1400061
Wu, Z., Wang, X., Fu, Y., Shen, J., Jiang, Q., Zhu, Y., & Zhou, M. (2018). Fitting scattered data points with ball B-Spline curves using particle swarm optimization. Computers & graphics, 72(1), 1-11. https://doi.org/10.1016/j.cag.2018.01.006
Karadede, Y., & Özdemir, G. (2018). A hierarchical soft computing model for parameter estimation of curve fitting problems. Soft computing22(20), 6937-6964. https://doi.org/10.1007/s00500-018-3413-5
Ramirez, L., Edgar, J., Capulin, C. H., Estudillo-Ayala, M., Avina-Cervantes, J. G., Sanchez-Yanez, R. E., & Gonzalez, H. R. (2019). Parallel hierarchical Genetic algorithm for scattered data fitting through B-Splines. Applied sciences9(11), 2336. https://doi.org/10.3390/app9112336
Li, M., & Lily, D. L. (2020). A novel method of curve fitting based on optimized extreme learning machine. Applied artificial intelligence, 34(12). 849-865. https://doi.org/10.1080/08839514.2020.1787677
Yeh, R., Youssef, S. N., Peterka, T., & Tricoche, X. (2020). Fast automatic knot placement method for accurate B-Spline curve fitting. Computer-aided design, 128(1). 102905. https://doi.org/10.1016/j.cad.2020.102905
Haupt, R. L., & Haupt, S. E. (2004). Practical genetic algorithms. John Wiley & Sons, New York.
Bureick, J., Alkhatib, H., & Neumann, I. (2019). Fast converging elitist genetic algorithm for knot adjustment in B-Spline curve approximation. Journal of applied geodesy13(4), 317-328. https://doi.org/10.1515/jag-2018-0015
Toutounji, H., & Durstewitz, D. (2018). Detecting multiple step changes using adaptive regression splines with application to neural recordings. Frontiers in neuroinformatics, 12(1), 210-231. https://doi.org/10.3389/fninf.2018.00067