The II Method for Estimating Multivariate Functions From Noisy Data

Abstract

The Π method for estimating an underlying smooth function of M variables, (x _l , …, x_m ), using noisy data is based on approximating it by a sum of products of the form Π _m φ _m (x _m ). The problem is then reduced to estimating the univariate functions in the products. A convergent algorithm is described. The method keeps tight control on the degrees of freedom used in the fit. Many examples are given. The quality of fit given by the Π method is excellent. Usually, only a few products are enough to fit even fairly complicated functions. The coding into products of univariate functions allows a relatively understandable interpretation of the multivariate fit.

KEY WORDS:

• Function estimation
• Knot deletion
• Nonparametric regression
• Regression splines