Pointwise RMS bias error estimates for design of experiments

Response surface approximations offer an effective way to solve complex problems. However limitations on computational expense pose restrictions to generate ample data and a simple model is typically used for approximation leaving the possibility of errors due to insufficient model known as bias errors. This paper presents a method to estimate pointwise RMS bias errors in response surface models. Prior to generation of data, RMS bias error estimates can be used to construct design of experiments (DOEs) to minimize the maximal RMS bias errors or compare different DOEs. It is demonstrated that for high dimensional design spaces, central composite design gives a reasonable trade-off between variance and bias errors under conventional assumption of quadratic model and cubic true function. The results indicate that compared to bounds on bias error, RMS estimates are less affected by increase in the dimensionality of the problem. Latin Hypercube Sampling (LHS), though very popular, may yield large unsampled regions which can be successfully detected by implementing an alternate criterion like maximum standard error or maximum RMS bias error in design space. We further demonstrate that poor LHS designs can be eliminated by using more than one DOE. With the help of a non-polynomial example problem, it is demonstrated that the proposed method is correctly able to identify the regions of high errors even when assumed true model is not accurate.