Evolutionary computation for optimal knots allocation in smoothing splines

Abstract In this paper, a novel methodology is presented for optimal placement and selections of knots, for approximating or fitting curves to data, using smoothing splines. It is well-known that the placement of the knots in smoothing spline approximation has an important and considerable effect on the behavior of the final approximation [1] . However, as pointed out in [2] , although spline for approximation is well understood, the knot placement problem has not been dealt with adequately. In the specialized bibliography, several methodologies have been presented for selection and optimization of parameters within B-spline, using techniques based on selecting knots called dominant points , adaptive knots placement, by data selection process, optimal control over the knots, and recently, by using paradigms from computational intelligent, and Bayesian model for automatically determining knot placement in spline modeling. However, a common two-step knot selection strategy, frequently used in the bibliography, is an homogeneous distribution of the knots or equally spaced approach [3] . In order to optimize the placement and numbers of knots required for approximation using smoothing splines, an Evolutionary Computation Paradigms ( ECP ) based on a Multi-Objective Genetic Algorithm has been developed, with the main purpose of avoiding the large number of local minima (in terms of approximation error for different system complexity or number of knots) existing in the problem of knots placement. The accuracy, computationally efficient and robustness of the algorithm presented will be compared by different experimental results, with other approaches presented in the bibliography, showing the main advantages of the proposed methodology.