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Evolutionary computation for optimal knots allocation in smoothing splines

机译：平滑样条曲线中最优节结分配的进化计算

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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.
机译：在本文中，提出了一种新颖的方法，可以使用平滑样条线对结的最佳放置和选择，曲线的近似或拟合进行拟合。众所周知，在平滑样条曲线逼近中，结的位置对最终逼近的行为具有重要且相当大的影响[1]。然而，正如[2]中指出的那样，尽管已经很好地理解了近似样条，但结结问题尚未得到充分的解决。在专门的书目中，已经提出了几种用于选择和优化B样条内参数的方法，这些方法使用以下技术：通过选择称为优势点的结，自适应结放置，通过数据选择过程，对结的最佳控制，以及最近通过使用计算智能范式和贝叶斯模型自动确定样条线建模中的结位置。然而，在书目中经常使用的常见的两步结选择策略是结的均匀分布或等距方法[3]。为了优化使用平滑样条进行逼近所需的结的位置和节数，已开发了一种基于多目标遗传算法的进化计算范例（ECP），其主要目的是避免大量局部极小值（在存在于打结位置问题中的不同系统复杂性或打结数量的近似误差项。所提出算法的准确性，计算效率和鲁棒性将通过不同的实验结果进行比较，并与参考书目中提出的其他方法进行比较，从而显示出所提出方法的主要优势。

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