Computational Statistics

, Volume 31, Issue 3, pp 1079–1105

# Geometrically designed, variable knot regression splines

• Dimitrina S. Dimitrova
• Steven Haberman
• Richard J. Verrall
Original Paper

## Abstract

A new method of Geometrically Designed least squares (LS) splines with variable knots, named GeDS, is proposed. It is based on the property that the spline regression function, viewed as a parametric curve, has a control polygon and, due to the shape preserving and convex hull properties, it closely follows the shape of this control polygon. The latter has vertices whose x-coordinates are certain knot averages and whose y-coordinates are the regression coefficients. Thus, manipulation of the position of the control polygon may be interpreted as estimation of the spline curve knots and coefficients. These geometric ideas are implemented in the two stages of the GeDS estimation method. In stage A, a linear LS spline fit to the data is constructed, and viewed as the initial position of the control polygon of a higher order ($$n>2$$) smooth spline curve. In stage B, the optimal set of knots of this higher order spline curve is found, so that its control polygon is as close to the initial polygon of stage A as possible and finally, the LS estimates of the regression coefficients of this curve are found. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline coefficients. Numerical examples are provided and further supplemental materials are available online.

## Keywords

Spline regression B-splines Greville abscissae Variable knot splines Control polygon

C13 C14

## Notes

### Acknowledgments

The authors would like to acknowledge support received through a research grant from the UK Institute of Actuaries. The authors would also like to thank Simon Kimber for providing them with the $$\hbox {BaFe}_2\hbox {As}_2$$ dataset and the results from the Rietveld fit given in Kimber et al. (2009). The sincere encouragement received by David van Dyk, and his help in discussing and providing invaluable advice on ways to improve the paper are greatly appreciated.

## Supplementary material

180_2015_621_MOESM1_ESM.pdf (2 mb)
Supplementary material 1 (pdf 2031 KB)

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