Adaptive free-knot splines

Satoshi Miyata, Xiaotong Shen

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


This article proposes a function estimation procedure using free-knot splines as well as an associated algorithm for implementation in nonparametric regression. In contrast to conventional splines with knots confined to distinct design points, the splines allow selection of knot numbers and replacement of knots at any location and repeated knots at the same location. This flexibility leads to an adaptive spline estimator that adapts any function with inhomogeneous smoothness, including discontinuity, which substantially improves the representation power of splines. Due to uses of a large class of spline functions, knot selection becomes extremely important. The existing knot selection schemes - such as stepwise selection - suffer the difficulty of knot confounding and are unsuitable for our purpose. A new knot selection scheme is proposed using an evolutionary Monte Carlo algorithm and an adaptive model selection criterion. The evolutionary algorithm locates the optimal knots accurately, whereas the adaptive model selection strategy guards against the selection error in searching through a large candidate knot space. The performance of the procedure is examined and illustrated via simulations. The procedure provides a significant improvement in performance over the other competing adaptive methods proposed in the literature. Finally, usefulness of the procedure is illustrated by an application to actual dataset.

Original languageEnglish (US)
Pages (from-to)197-213
Number of pages17
JournalJournal of Computational and Graphical Statistics
Issue number1
StatePublished - Mar 2003


  • Adaptive model selection
  • Discontinuity
  • Evolutionary algorithms
  • Inhomogeneous
  • Nonparametric regression
  • Signal processing
  • Smoothness of unknown degree and type
  • Spatial adaptation
  • Variable multiple knots


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