Bound Analysis in Univariately Truncated Factorized HDMR for a Multivariate Function Given at the Nodes of an Hypergrid

摘要

High Dimensional Model Representation (HDMR), and its varieties find important positions in the theory and approximation of multivariate functions. The function may be given either in an analytical form or a finite data set. In this work we deal with the case of finite data set which is composed of the positions of the nodes of a hypergrid and the values of the function under consideration given at each node. Since this data set somehow contains uncertainties in the positions of the nodes and the function values because of certain incapabilities in the data sampling devices, it is needed to construct a band structure as the model of that problem. Here, we focus on the univariate HDMR partitioning of this data and the case where univariate Factorized HDMR is at the target in modelling the given data with such uncertainties. These uncertainties cause a band structure in the model and this urges us to get upper and lower bounds for the constant and univariate factors of FHDMR.