Parameter optimization in multiquadric response surface approximations

Multiquadric (MQ) response surface approximation uses the function SigmaC(i)root|X-X-i|(2)+h(1/2) to interpolate a given set of data. The performance of MQ approximation depends on the shift parameter h. Efficient methods of computing the optimal shift parameter based on the leave-one-out cross validation technique are presented in this paper. We also proved that the condition number of the MQ coefficient matrix is an increasing function of the shift parameter h. Two numerical examples are included to illustrate the proposed formulation.