A novel hybrid high-dimensional model representation (HDMR) based on the combination of plain and logarithmic high-dimensional model representations

Abstract

This chapter focuses on a new version of hybrid high- dimensional model representation for multivariate functions. High- dimensional model representation (HDMR) was proposed to approximate the multivariate functions by the functions having less number of independent variables. Toward this end, HDMR disintegrates a multivariate function to components which are, respectively, constant, univariate, bivariate, and so on in an ascending order of multivariance. HDMR method is a scheme truncating the representation at a prescribed multivariance. If the given multivariate function is purely additive then HDMR method spontaneously truncates at univariance, otherwise the highly multivariant terms are required. On the other hand, if the given function is dominantly multiplicative then the logarithmic HDMR method which truncates the scheme at a prescribed multivariance of the HDMR of the logarithm of the given function is taken into consideration. In most cases the given multivariate function has both additive and multiplicative natures. If so then a new method is needed. Hybrid high-dimensional model representation method is used for these types of problems. This new representation method joins both plain high-dimensional model representation and logarithmic high-dimensional model representation components via an hybridity parameter. This work focuses on the construction and certain details of this novel method.