Multiscale representation for irregularly spaced data

In this paper, we propose a multiscale method for representing inhomogeneous functions (surfaces) from irregularly spaced noisy observations that have inherent multiscale structure. The proposed multiscale method is based on a novel combination of the standard discrete wavelet transform with newly defined pseudo data. The pseudo data, which can be considered as a preprocessing of the original data, play a crucial role in deriving the proposed method and motivating a practical algorithm. The proposed algorithm using the empirical pseudo data is computationally fast, simple to describe and easy to implement. Moreover, results from numerical examples and real data analysis demonstrate the promising empirical properties of the proposed method.