Finite element thin plate splines for surface fitting

Abstract

Surface fitting and smoothing splines techniques are widely used in practice to fit data arising from many different application areas such as meteorology, insurance and stock exchange. A common problem is the approximation of functions of many variables for given values of the function at various points. What makes the problem even more complicated is that the given or observed values may contain noise. We are particularly interested in data mining applications which deal with very large databases. These problems arise in many different areas. One basic aim in data mining is to model functional relationships of high dimensional data sets which introduce the “curse of dimensionality". In order to overcome this curse, additive and interaction splines have been used. Generalised additive models, if interaction terms are limited to order two interactions, lead to the determination of coupled surfaces and curves. Thus, an important part of any data analysis algorithm for these problems is the determination of an approximating surface for extremely large data sets. A variational characterisation of the thin plate smoothing splines was proposed by Duchon. He also proposed to use a radial basis function approach to solve this problem. However, this leads to symmetric indefinite dense linear system of equations. It was seen that this system can be reduced to a positive definite system of equations which can be solved by a conjugate gradient method. Further improvements using ideas from multipole expansions and Lagrange functions [1, 2, 3] lead to methods which are of O(n) or O(n log(n)) in complexity where n is the number of observations. In this work, a new smoothing method is proposed which can be viewed as a discrete thin plate spline. This new approach combines the favourable properties of finite element surface fitting with the ones of thin plate splines. In Section 2, the method is introduced which is based on first order techniques similar to mixed finite element techniques for the biharmonic equation. The numerical solution of the linear system of equations is discussed in Section 3. In Section 4, the technique is illustrated with an example. Conclusions are given in Section 5.