Smooth supersaturated models

In areas such as kernel smoothing and non-parametric regression there is emphasis on smooth interpolation and smooth statistical models. Splines are known to have optimal smoothness properties in one and higher dimensions. It is shown, with special attention to polynomial models, that smooth interpolators can be constructed by first extending the monomial basis and then minimising a measure of smoothness with respect to the free parameters in the extended basis. Algebraic methods are a help in choosing the extended basis which can also be found as a saturated basis for an extended experimental design with dummy design points. One can get arbitrarily close to optimal smoothing for any dimension and over any region, giving a simple alternative models of spline type. The relationship to splines is shown in one and two dimensions. A case study is given which includes benchmarking against kriging methods.

💡 Research Summary

The paper addresses the long‑standing problem of constructing smooth interpolators in non‑parametric regression and kernel smoothing, where splines are traditionally regarded as the gold standard. The authors propose a novel “oversaturated” modeling framework that builds on polynomial regression but augments the monomial basis with additional, deliberately redundant terms. These extra basis functions—referred to as dummy or filler terms—do not affect the fit to the observed data directly; instead, they provide a set of free parameters that can be tuned to minimise a chosen smoothness criterion, typically the integrated squared second‑derivative (or curvature) of the fitted surface.

Mathematically, the model takes the form

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