Pointwise Bias Error Bounds for Response Surface Approximations and Min-Max Bias Design

Two approaches addressing response surface approximation errors due to model inadequacy (bias error) are presented, and a design of experiments minimizing the maximal bias error is proposed. Both approaches assume that the functional form of the true model is known and seek, at each point in design space, worst case bounds on the absolute error. The first approach is implemented prior to data generation. This data independent error bound can identify locations in the design space where the accuracy of the approximation fitted on a given design of experiments may be poor. The data independent error bound can easily be implemented in a search for a design of experiments that minimize the bias error bound as it requires very little computation. The second approach is to be used posterior to the data generation and provides tightened error bound consistent with the data. This data dependent error bound requires the solution of two linear programming problems at each point. The paper demonstrates the data independent error bound for design of experiments of two-variable examples. Randomly generated polynomials in two variables are then used to validate the data dependent bias-error bound distribution.