Finite Element Thin Plate

Thin plate splines have been used successfully to model curves and surfaces. A new application is in data mining where they are used to model interaction terms. These interaction splines break the \curse of di-mensionality" by reducing the high-dimensional nonparametric regression problem to the determination of a set of interdependent surfaces. However , the determination of the corresponding thin plate splines requires the solution of a dense linear system of equations of order n where n is the number of observations. For data mining applications n can be in the millions , and so standard thin plate splines, even using fast algorithms may not be practical. A nite element approximation of the thin plate splines will be described. The method uses H 1 elements in a formulation which only needs rst order derivatives. The resolution of the method is chosen independently of the number of observations which only need to be read from secondary storage once and do not need to be stored in memory. The formulation leads to a saddle point problem. Convergence and solution of the method and its relationship to the standard thin plate splines will be discussed. x1. Introduction Recent years have seen a considerable eeort to develop reliable and eecient data mining tools to discover hidden knowledge in very large data bases. Data mining methods have been applied to a number of application areas, for instance the calculation of proot according to customer, product and acquisition channel; trend and response prediction and identiication; the development of tools to examine corporate intelligence reports, tax/insurance frauds and patterns in airline crashes. The interested reader can refer to 5] for detailed information. The fundamental issue is the development and adaptation of algorithms to extract some useful information from very large databases. Often the number of observations can be of the order of millions, where there may be thousands of variables recorded. All rights of reproduction in any form reserved.