An alternative procedure for selecting a good value for the parameter c in RBF-interpolation

Abstract

The impact of the scaling parameter c on the accuracy of interpolation schemes using radial basis functions (RBFs) has been pointed out by several authors. Rippa (Adv Comput Math 11:193-210, 1999) proposes an algorithm based on the idea of cross validation for selecting a good such parameter value. In this paper we present an alternative procedure, that can be interpreted as a refinement of Rippa's algorithm for a cost function based on the euclidean norm. We point out how this method is related to the procedure of maximum likelihood estimation, which is used for identifying covariance parameters of stochastic processes in spatial statistics. Using the same test functions as Rippa we show that our algorithm compares favorably with cross validation in many cases and discuss its limitations. Finally we present some computational aspects of our algorithm.

Interpolation of spatial data – A stochastic or a deterministic problem?

“…The PIT histograms in Figure 3 are quite far from uniformity, suggesting that the assumption of a Gaussian distribution is rather questionable. It is quite remarkable that REML, which is based on this assumption, does an excellent job in selecting good parameters, and we found that this is also true for many other test cases (see [71]). …”

“…A major drawback seems to be the strong assumption that Z is Gaussian under which the maximum likelihood estimator is derived. In [71], however, an alternative derivation of REML in the framework of kernel interpretation (where much weaker modelling assumptions are made) is given, and a numerical study with several non-stochastic test cases is presented in which REML often yields very good choices of K.…”

The method of approximate particular solutions for the time-fractional diffusion equation with a non-local boundary condition

“…Many interesting works have been carried out to address this issue for PDE problems, see, e.g. [39][40][41]34,42,33,43] and the references therein. In [34], Rippa proposed a leave-one-out cross validation (LOOCV) algorithm to estimate the interpolation error and use it to select an optimal value of the shape parameter.…”

Analysis, Simulation and Prediction of Multivariate Random Fields with PackageRandomFields

“…The second often used method is the maximum likelihood (ML) method which relies on the additional important assumption of the Gaussianity of the field. Showing that the ML approach is not too far from an established approach in approximation theory, Scheuerer (2011) gives some indication why ML often works well even when the assumption of Gaussianity is violated. The ML method and its variants (geoR, Ribeiro Jr. andDiggle 2001, CompRandFld, Padoan and) is frequently implemented for parametric inference on spatial data.…”