Semiparametric Analysis of Variance with Tensor Product Thin Plate Splines

SUMMARY We consider fitting multivariate models to response data based on analysis-of-variance (ANOVA)-like decompositions of functions of several variables, f(t 1,..., td) = C + σα f α(ta + σα β f αβ(t α, t β) + · · · · A theory for fitting (some components of) these models with polynomial smoothing splines exists when each tα is in a subset of the real line. In this case the various estimated components turn out to be certain tensor sums and products of polynomial splines. This approach may not be natural when one or more of the 'variables' are geographic, in particular where nature does not know north from east. In this case splines of radial structure, such as thin plate splines, for the geographic component, are more natural. In this paper we extend this theory of polynomial smoothing spline ANOVA models to include variables which take values in Euclidean k-space, and fits which turn out to include sums and products of both polynomial and thin plate smoothing splines. The cases of most interest would be k = 2 and k = 3. The formulation, interpretation and calculation of the models are discussed, and an application of the technique is illustrated. This work can be used to build predictive ANOVA-like models which describe a response as a function of spatial, temporal and other variables and to explore their interactions. The models can be fitted by using the existing publicly available code RKPACK.