In fitting data with a spline, finding the optimal placement of knots can significantly improve the quality of the fit. However, the challenging high-dimensional and non-convex optimization problem associated with completely free knot placement has been a major roadblock in using this approach. We present a method that uses particle swarm optimization (PSO) combined with model selection to address this challenge. The problem of overfitting due to knot clustering that accompanies free knot placement is mitigated in this method by explicit regularization, resulting in a significantly improved performance on highly noisy data. The principal design choices available in the method are delineated and a statistically rigorous study of their effect on performance is carried out using simulated data and a wide variety of benchmark functions. Our results demonstrate that PSO-based free knot placement leads to a viable and flexible adaptive spline fitting approach that allows the fitting of both smooth and non-smooth functions.

在用样条曲线拟合数据时，找到最佳结点位置可以显着提高拟合质量。但是，与完全自由打结放置有关的具有挑战性的高维和非凸优化问题一直是使用此方法的主要障碍。我们提出一种结合粒子群优化（PSO）和模型选择的方法来解决这一挑战。在此方法中，通过显式正则化解决了因自由结放置而导致的结聚类而导致的过度拟合问题，从而显着提高了对高噪声数据的性能。描述了该方法中可用的主要设计选择，并使用模拟数据和各种基准功能对它们对性能的影响进行了统计严格的研究。