Approximation Errors in Truncated Dimensional Decompositions

Abstract: The main theme of this paper is error analysis for approximations derived from two variants of dimensional decomposition of a multivariate function: the referential dimensional decomposition (RDD) and analysis-of-variance dimensional decomposition (ADD). New formulae are presented for the lower and upper bounds of the expected errors committed by bivariately and arbitrarily truncated RDD approximations when the reference point is selected randomly, thereby facilitating a means for weighing RDD against ADD approximations. The formulae reveal that the expected error from the S-variate RDD approximation of a function of N variables, where $0 \le S < N < \infty$, is at least $2^{S+1}$ times greater than the error from the S-variate ADD approximation. Consequently, ADD approximations are exceedingly more precise than RDD approximations. The analysis also finds the RDD approximation to be sub-optimal for an arbitrarily selected reference point, whereas the ADD approximation always results in minimum error. Therefore, the RDD approximation should be used with caution.