On the selection of the most adequate radial basis function

Radial basis function (RBF) methods can provide excellent interpolants for a large number of poorly distributed data points. For any finite data set in any Euclidean space, one can construct an interpolation of the data by using RBFs. However, RBF interpolant trends between and beyond the data points depend on the RBF used and may exhibit undesirable trends using some RBFs while the trends may be desirable using other RBFs. The fact that a certain RBF is commonly used for the class of problems at hand, previous good behavior in that (or other) class of problems, and bibliography, are just some of the many valid reasons given to justify a priori selection of RBF. Even assuming that the justified choice of the RBF is most likely the correct choice, one should nonetheless confirm numerically that, in fact, the most adequate RBF for the problem at hand is the RBF chosen a priori. The main goal of this paper is to alert the analyst as to the danger of a priori selection of RBF and to present a strategy to numerically choose the most adequate RBF that better captures the trends of the given data set. The wing weight data fitting problem is used to illustrate the benefits of an adequate choice of RBF for each given data set.