Nonlinear function approximation over high-dimensional domains

摘要

An algorithm is disclosed for constructing nonlinear models from high-dimensional scattered data. The algorithm progresses iteratively adding a new basis function at each step to refine the model. The placement of the basis functions is driven by a statistical hypothesis test that reveals geometric structure when it fails. At each step the added function is fit to data contained in a spatio-temporally defined local region to determine the parameters, in particular, the scale of the local model. The proposed method requires no ad hoc parameters. Thus, the number of basis functions required for an accurate fit is determined automatically by the algorithm. The approach may be applied to problems including modeling data on manifolds and the prediction of financial time-series. The algorithm is presented in the context of radial basis functions but in principle can be employed with other methods for function approximation such as multi-layer perceptrons.