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The $\prod $ Method for Estimating Multivariate Functions from Noisy Data

Leo Breiman
Technometrics
Vol. 33, No. 2 (May, 1991), pp. 125-143
DOI: 10.2307/1269038
Stable URL: http://www.jstor.org/stable/1269038
Page Count: 19
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The $\prod $ Method for Estimating Multivariate Functions from Noisy Data
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Abstract

The $\prod $ method for estimating an underlying smooth function of M variables, (x1,... ,xM), using noisy data is based on approximating it by a sum of products of the form $\prod_{m}\phi _{m}(x_{m})$. 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 $\prod $ 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.

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