High Dimensional Model Representation Supported by Products of Univariate Functions

Abstract

This work extends the High Dimensional Model Representation (HDMR) proposed by Sobol and many contributions to the development have been made by some other authors including Rabitz and his group and the author and his group. The additivity measurers developed and used by the author and his group enabled us to measure the truncation approximation quality of HDMR. Now it is very well known that the multiplicativity in the target function of HDMR prevents to truncate it at univariate or at most bivariate level. It requires all expansion. In HDMR an ascending multivariance structure is used. Hence the leading constant component is followed by the univariate, bivariate terms and so on. This work extends the representation by feeding univariate factors to support each such term in missing variables. The result is a new representation which is enable us to exactly have the target function at the first component even if it is purely multiplicative.