An algorithm for selecting a good value for the parameter c in radial basis function interpolation

The accuracy of many schemes for interpolating scattered data with radial basis functions depends on a shape parameter c of the radial basis function. In this paper we study the effect of c on the quality of fit of the multiquadric, inverse multiquadric and Gaussian interpolants. We show, numerically, that the value of the optimal c (the value of c that minimizes the interpolation error) depends on the number and distribution of data points, on the data vector, and on the precision of the computation. We present an algorithm for selecting a good value for c that implicitly takes all the above considerations into account. The algorithm selects c by minimizing a cost function that imitates the error between the radial interpolant and the (unknown) function from which the data vector was sampled. The cost function is defined by taking some norm of the error vector E = (E-1, ..., E-N)(T) where E-k = E-k = f(k) - S-(k)(x(k)) and S-(k) is the interpolant to a reduced data set obtained by removing the point x(k) and the corresponding data value f(k) from the original data set. The cost function can be defined for any radial basis function and any dimension. We present the results of many numerical experiments involving interpolation of two dimensional data sets by the multiquadric, inverse multiquadric and Gaussian interpolants and we show that our algorithm consistently produces good values for the parameter c.