Consistency of Ridge Function Fields for Varying Nonparametric Regression

A nonparametric regression model proposed in [Pelletier and Frouin, Applied Optics, 2006] as a solution to the geophysical problem of ocean color remote sensing is studied. The model, called ridge function field, combines a regression estimate in the form of a superposition of ridge functions, or equivalently a neural network, with the idea pertaining to varyingcoefficients models, where the parameters of a parametric family are allowed to vary with other variables. Under mild assumptions on the underlying distribution of the data, the strong universal consistency of the least-squares ridge function fields estimate is established. Index Terms — Varying coefficients model, ridge function approximation, nonparametric regression, universal consistency, least squares. AMS 2000 Classification: 62G08, 62G05. ∗Corresponding author.