Volume 17, pp. 168-180, 2004.

Multidimensional smoothing using hyperbolic interpolatory wavelets

Markus Hegland, Ole M. Nielsen, and Zuowei Shen

Abstract

We propose the application of hyperbolic interpolatory wavelets for large-scale $d$-dimensional data fitting. In particular, we show how wavelets can be used as a highly efficient tool for multidimensional smoothing. The grid underlying these wavelets is a sparse grid. The hyperbolic interpolatory wavelet space of level $j$ uses $O(j^{d-1}{2}^j)$ basis functions and it is shown that under sufficient smoothness an approximation error of order $O\left(\left(\begin{array}{c}j+d-1d-1\end{array}\right)2^{-2j}\right)$ can be achieved. The implementation uses the fast wavelet transform and an efficient indexing method to access the wavelet coefficients. A practical example demonstrates the efficiency of the approach.

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Key words

sparse grids, predictive modelling, wavelets, smoothing, data mining.

AMS subject classifications

65C60, 65D10, 65T60.