A direct prediction of the shape parameter—A purely scattered data approach

Abstract

In this paper we present an approach which predicts directly without search the optimal choice of the shape parameter c contained in the multiquadrics (−1)⌈β2⌉(c2+∥x∥2)β2,β>0, and the inverse multiquadrics (c2+∥x∥2)β2,β<0. Unlike the simplex scheme where the data points are required to be evenly spaced, as in a recent paper of the author, here we allow them to be arbitrarily scattered in the simplex, making it much more useful. Besides this, we aim at helping non-mathematicians use our approach and hence remove some complicated requirements for the domain. The drawback is that its theoretical ground is not so strong as in the evenly spaced data setting. However, experiments show that it works well. The experimentally optimal value of c coincides with the theoretically predicted one. Since the fill distance, which reflects the amount of data points needed, involved is always of reasonable size, this approach is supposed to be practically useful.