Efficient particle swarm optimization approach for data fitting with free knot B -splines

Data fitting through B -splines improves dramatically if the knots are treated as free variables. However, in that case the approximation problem becomes a very difficult continuous multimodal and multivariate nonlinear optimization problem. In a previous paper, Yoshimoto et al. (2003) [18] solved this problem for explicit curves by using a real-code genetic algorithm. However, the method does not really deal with true multiple knots, so the cases of data with underlying functions having discontinuities and cusps are not fully addressed. In this paper, we present a new method to overcome such a limitation. The method applies the particle swarm optimization (PSO) paradigm to compute an appropriate location of knots automatically. Our scheme yields very accurate results even for curves with singularities and/or cusps. Several experiments show that our proposal is very efficient and improves previous results (including those by Yoshimoto et al. (2003) in [18]) significantly in terms of data points error, AIC and BIC criteria. Furthermore, the important case of true multiple knots is now satisfactorily solved. ► A new metaheuristic approach for knot placement in data fitting with B -splines is presented. ► It does not assume any condition (continuity, differentiability, etc.) on the underlying function of data. ► It can automatically obtain truly identical multiple knots when needed. ► The method is very fast, easy to implement and requires neither human intervention nor further pre-/post-processing. ► It shows excellent performance and outperforms previous approaches in terms of accuracy and flexibility.