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Many modern data sets are sampled with error from complex high-dimensionalsurfaces. Methods such as tensor product splines or Gaussian processes areeffective/well suited for characterizing a surface in two or three dimensionsbut may suffer from difficulties when representing higher dimensional surfaces.Motivated by high throughput toxicity testing where observed dose-responsecurves are cross sections of a surface defined by a chemical's structuralproperties, a model is developed to characterize this surface to predictuntested chemicals' dose-responses. This manuscript proposes a novel approachthat models the multidimensional surface as a sum of learned basis functionsformed as the tensor product of lower dimensional functions, which arethemselves representable by a basis expansion learned from the data. The modelis described, a Gibbs sampling algorithm proposed, and is investigated in asimulation study as well as data taken from the US EPA's ToxCast highthroughput toxicity testing platform.

Bayesian Additive Adaptive Basis Tensor Product Models for Modeling High Dimensional Surfaces: An application to high-throughput toxicity testing