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Elastic and plastic splines some experimental comparisons Roger Koenker and Ivan Mizera Abstract We give some empirical comparisons between two nonparametric regression methods based on regularization the elastic or thin plate splines and plastic splines based on total variation penalties 1 Nonparametric regression with elastic and plastic splines We seek f to fit the dependence of response points zi R on two dimensional covariates xi yi living in the domain To this end we employ regularization f is obtained as a minimizer of 1 n X zi f xi yi J f i 1 The first term is traditionally called in fidelity since it measures the overall lack of fit of f xi yi to zi The second penalty term shrinks the solution towards a more plausible or desirable alternative the extent of this shrinkage being controlled by the regularization parameter See Green and Silverman 1994 Eubank 1999 or Wahba 1990 1 1 Elastic splines We coin the term elastic splines for what are usually called thin plate splines defined on the idealized domain R2 They arise as solutions of 1 with the penalty ZZ 2 2 2 2 J2 f fxx x y 2fxy x y fyy x y dx dy R2 This penalty can be considered a natural extension of the easily interpretable oneR dimensional prototype f 00 2 The only unnatural feature is the fact that the penalty is evaluated over all of R2 instead of over a more realistic bounded domain for instance the convex hull of the xi yi points The latter alternative was considered among others by Green and Silverman 1994 who coined the name finite window thin plate splines However the algorithm for the finite window alternative is more involved though not necessarily slower and mostly important 2 Roger Koenker and Ivan Mizera not available to us at the present moment therefore this version of elastic splines will not be considered further here The name elastic splines comes from the quite well known physical model underlying the whole setting in which the penalty is interpreted as the potential energy of a

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