Bound analysis in univariately truncated Generalized High Dimensional Model Representation for random-data partitioning: Interval GHDMR

Abstract

Generalized High Dimensional Model Representation (GHDMR) method is based on a divide-and-conquer philosophy and it approximately reduces the N-dimensional interpolation to N number of one-dimensional interpolations. It is an exact representation which uses Dirac delta functions as the weight functions. GHDMR's components with multivariances higher than univariance are quite complicated and we prefer to use only its constant and univariate terms. In this work, we try to construct a new high dimensional model representation based method, Interval GHDMR, to be used when a random discrete multivariate data set which has construction errors coming from incapabilities or limited capabilities of the measurement devices or tools is given and an analytical structure for a multivariate function that passes through the nodes of this data set is to be determined. The errors coming from the construction of the data set that describes the sought multivariate function result in a band structure for the sought function instead of an analytical structure. The main purpose, here, is to estimate the total error formed through the given method in the given multivariate interpolation problem and, to this end, to partition the given multivariate data into at most univariate data and to determine an approximate band structure (or interval) for the sought function. By this way, the computational complexity and CPU time needed for computer based applications of the problem decrease.