High Dimensional Model Representation With Principal Component Analysis

Abstract

In engineering design, spending excessive amount of time on physical experiments or expensive simulations makes the design costly and lengthy. This issue exacerbates when the design problem has a large number of inputs, or of high dimension. High dimensional model representation (HDMR) is one powerful method in approximating high dimensional, expensive, blackbox (HEB) problems. One existing HDMR implementation, random sampling HDMR (RSHDMR), can build an HDMR model from random sample points with a linear combination of basis functions. The most critical issue in RSHDMR is that calculating the coefficients for the basis functions includes integrals that are approximated by Monte Carlo summations, which are error prone with limited samples and especially with nonuniform sampling. In this paper, a new approach based on principal component analysis (PCA), called PCAHDMR, is proposed for finding the coefficients that provide the best linear combination of the bases with minimum error and without using any integral. Several benchmark problems of different dimensionalities and one engineering problem are modeled using the method and the results are compared with RSHDMR results. In all problems with both uniform and nonuniform sampling, PCAHDMR built more accurate models than RSHDMR for a given set of sample points.