Mechanical models in nonparametric regression

A common motif in the expositions of spline-based methods in statistical smoothing or numerical interpolation is an allusion to mechanical analogies—motivated perhaps by a desire to provide some explanation why the resulting shapes ought to be regarded as “natural”. The univariate case has its Oxford English Dictionary reference to draftsman spline as “a flexible strip of wood or hard rubber used by draftsmen in laying out broad sweeping curves”, which suggests (amiss!) that the eponymous mathematical object shares exactly the same properties. The introduction of “thin-plate spline” in the bivariate domain usually comes with a more distinctive story about the deformation of an elastic flat thin plate—for instance, page 139 of Green and Silverman [14] or page 108 of Small [32]: if the plate is deformed to the shape of the function f , and is small, then the bending energy is (up to the first order) proportional to the smoothing penalty. The importance attached by the scientific community to such trivia varies: while some consider it a signal from Nature, indicating the righteous path in the potentially endless forest of possibilities—see especially Bookstein [3, 4, 5], but also Bookstein and Green [6], Small [32], Dryden and Mardia [10]—for others it is a marginal curiosity, not deserving to stand in the path of the appreciation of computational and theoretical properties. In our case, the desire of the second author to comprehend the connection between thin-plate splines and total variation penalties led him to cross-questioning of the first author, theoretical physicist with principal interests in gravitation and cosmology; the latter reluctantly, but eventually cooperatively descended into the caverns of “engineering”—theory of elastic and plastic behavior of solid bodies. The unveiled connection not only turned out to be interesting, but yielded also a practical return for the second author: the elucidated mechanical models hinted Koenker and Mizera [19] where—that is, in which community—to look for relevant algorithmic solutions for their proposals. Which could be the end of the story were it not for queries that started to come occassionally thereafter, about a text documenting the apocryphal knowledge. So, here finally an attempt to produce one. The available space allows only for an overview of physical facts relevant to nonparametric regression; for somewhat nonstandard physical derivations (albeit perhaps unsurprising for an expert in solid mechanics), we refer to Balek [2].