Response Surface Modeling By Local Kernel Partial Least Squares

Response surface modeling (RSM) is often used to analyze the relationshipbetween multiple variables. RSM samples the data through desigh of experiment, andestimates parameters by least squares. However, small sized samples and multiplecorrelations could lead to multi-collinearity problem. Because of this problem, theaccuracy and the reliability of model would not be guaranteed. Recenly, Partial LeastSquares (PLS) has been introduced to build the response surface for this problem.However, PLS is a linear method, which is not capable to deal with the non-linearmodel. Therefore, the purpose of this paper is to establish the nonlinear model. Insummary, this paper has the following main contributions:To solve this problem, we propose local modeling. In local modeling, we alsodefine a new performance criterion, which is used to estimate the parameters of eachsub-model. To adjust the influence of experimental point on each sub-space, weintroduce a weight function matrix in the new performance criterion.Based on local modeling, we propose local partial least squares (LPLS). Thisapproach aim to achieve the global nonlinear by local linear, and select the model byusing the cross-validation. LPLS can successfully solve multi-collinearity problems.However, the local space presents linear characteristics in this method. In addition,LPLS, KPLS may lead to "extreme value missing phenomenon".To avoid the "extreme value missing phenomenon", we propose local kernel partialleast squares (LKPLS). This approach maps the data in the original space into a featurekernel space by using the kernel function in the reproducing kernel Hilbert space.LKPLS builds a non-linear model in the local area. Except building a nonlinear PLSmodel directly, this approach also builds a nonlinear transformation for original data.At last, we examine the new approaches in several experiments to verify theproposed method. Besides, the latter experiments also verify the stability of the newalgorithm on small data sets. Moreover, the results show the proposed method workswell when occurring to"extreme value missing phenomenon." Finally, the new methodshave been applied in the research project of Tsinghua&Mitsubishi Heavy Industries.