Smooth supersaturated models

Abstract

In areas such as kernel smoothing and non-parametric regression, there is emphasis on smooth interpolation. We concentrate on pure interpolation and build smooth polynomial interpolators by first extending the monomial (polynomial) basis and then minimizing a measure of roughness with respect to the extra parameters in the extended basis. Algebraic methods can help in choosing the extended basis. We get arbitrarily close to optimal smoothing for any dimension over an arbitrary region, giving simple models close to splines. We show in examples that smooth interpolators perform much better than straight polynomial fits and for small sample size, better than kriging-type methods, used, for example in computer experiments.

Keywords:

• regression
• splines
• kernel smoothing
• non-parametric regression
• computer experiments
• algebraic statistics

Acknowledgements

The first and third authors acknowledge the EPSRC grant GR/S63502/01, while the second and third authors acknowledge the EPSRC grant EP/D048893/1 (MUCM project). The authors acknowledge Peter Curtis’ work as part of EPSRC funded PhD studentship ‘Metamodels in design and analysis of computer experiments’.