# Adaptive spline fitting with particle swarm optimization

## TL;DR

This paper introduces an adaptive spline fitting method that uses particle swarm optimization and regularization to effectively place knots, improving fit quality especially on noisy data.

## Contribution

It presents a novel PSO-based approach for free knot placement in spline fitting, addressing overfitting and high-dimensional optimization challenges.

## Key findings

- Improved fit quality on noisy data
- Effective avoidance of overfitting through regularization
- Versatile fitting of smooth and non-smooth functions

## Abstract

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.