New formulation of minimum-bias central composite experimental design and Gauss quadrature

Response surface methods provide a powerful tool for constructing approximations to complex response functions. Statistical design of experiments is usually used to select optimal points that minimize the error in the resulting response surface approximation. Traditionally, data points are selected using minimum-variance designs, for example the D-optimal design, which may result in large bias errors for low-order approximation. Minimum-bias criteria have been developed for selecting data points to minimize the bias error of a response surface approximation. The present work developed a minimum-bias counterpart to the popular minimum-variance central composite designs. In addition, a new formulation of the minimum-bias design that assigns unequal weights to the design points, based on Gauss quadrature, is explored. Example problems are evaluated and the results obtained from D-optimal, the traditional minimum-bias, and the new Gauss-quadrature-based minimum-bias designs are compared. It is shown that the Gauss-quadrature-based minimum-bias design criterion results in the most accurate approximations and provides analytical solutions to a wider range of approximation domains than the traditional minimum-bias design. Response surface approximations based on minimum-bias central composite designs are more accurate than those constructed from traditional central composite design. Moreover, it is shown that using weights in regression has little influence on the accuracy of the response surface approximation in Gauss-quadrature minimum-bias designs.