Approximation Errors in Truncated Dimensional Decompositions

TL;DR

This paper analyzes the approximation errors of two multivariate function decomposition methods, revealing that ADD is significantly more accurate than RDD and highlighting the importance of choosing the right method for error minimization.

Contribution

The paper introduces new bounds for expected errors in RDD and ADD, showing ADD's superior accuracy and optimality over RDD for arbitrary reference points.

Findings

ADD approximations are more accurate than RDD by a factor of at least 2^{S+1}.

RDD is sub-optimal for arbitrary reference points, while ADD minimizes error.

The paper provides formulas for expected error bounds in both RDD and ADD.

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.