Document Type : Research Paper


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


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.


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.
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.
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.
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.
Lai, M. J., & Wang, L. (2013). Bivariate penalized splines for regression. Statistica sinica, 1399-1417.
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.
Schwarz, K., & Krivobokova, T. (2016). A unified framework for spline estimators. Biometrika103(1), 121-131.
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.
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.
Papp, D., & Alizadeh, F. (2014). Shape-constrained estimation using nonnegative splines. Journal of computational and graphical statistics23(1), 211-231.
Ma, S., Racine, J. S., & Yang, L. (2015). Spline regression in the presence of categorical predictors. Journal of applied econometrics30(5), 705-717.
Zhou, J., Chen, Z., & Peng, Q. (2016). Polynomial spline estimation for partial functional linear regression models. Computational statistics, 31(3), 1-23.
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.
Powell, J. D. (1970). Curve fitting by splines in one variable. Numerical approximation to functions and data, 12(1), 65-83.
Jupp, D. L. (1978). Approximation to data by splines with free knots. SIAM journal on numerical analysis15(2), 328-343.
Ma, S. (2014). A plug-in the number of knots selector for polynomial spline regression. Journal of nonparametric statistics26(3), 489-507.
Wang, X. (2008). Bayesian free-knot monotone cubic spline regression. Journal of computational and graphical statistics, 17(2), 373-387.
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.
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.
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.
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.
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.
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.
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.
Holland, J. H. (1975). Adaptation in natural and artificial systems. University of Michigan Press.
Lee, T. (2002). On algorithms for ordinary least squares regression spline: a comparative study. Journal of statistical computation and simulation, 72(8), 647-663.
Yoshimoto, F., Harada, T., & Yoshimoto, Y. (2003). Data with a spline using a real-coded Genetic Algorithm. Computer-aided design35(8), 751-760.
Pittman, J. (2002). Adaptive splines and Genetic Algorithms. Journal of computational and graphical statistics11(3), 615-638.
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.
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.
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.
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.
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.
Karadede, Y., & Özdemir, G. (2018). A hierarchical soft computing model for parameter estimation of curve fitting problems. Soft computing22(20), 6937-6964.
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.
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.
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.
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.
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.