A RECURSIVE LOCAL POLYNOMIAL APPROXIMATION METHOD USING DIRICHLET CLOUDS AND RADIAL BASIS FUNCTIONS

We present a recursive function approximation technique that does not require the storage of the arrival data stream. Our work is motivated by algorithms in stochastic optimization which require approximating functions in a recursive setting such as a stochastic approximation algorithm. The unique collection of these features in this technique is essential for nonlinear modeling of large data sets where the storage of the data becomes prohibitively expensive and in circumstances where our knowledge about a given query point increases as new information arrives. The algorithm presented here employs radial basis functions (RBFs) to provide locally adaptive parametric models (such as linear models). The local models are updated using recursive least squares and only store the statistical representative of the local approximations. The resulting scheme is very fast and memory efficient without compromising accuracy in comparison to methods well accepted as the standard and some advanced techniques used for functional data analysis in the literature. We motivate the algorithm using synthetic data and illustrate the algorithm on several real data sets.