Nonlinear Wavelet Tools for Noisy and Indirect Data

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The wavelet transform is an important new tool which has made a huge impact on a variety of important problems. For example, wavelets have been successfully applied to data compression problems. What makes wavelets good for data compression is also what makes wavelets good for statistical estimation. Wavelets and associated ideas have the potential to make a huge contribution to problems in involving noisy and indirect observations. StatSci Division, MathSoft, Inc. proposes to develop a collection of statistical techniques based on the principle of wavelet shrinkage. Wavelet shrinkage refers to estimations obtained by taking the wavelet transformation, followed by shrinking the empirical wavelet coefficients towards zero, followed by inverse transformation. These techniques will be bundled in an object-oriented software toolkit. The research plan is divided into four components: (1) a wavelet shrinkage advisor to help sift through the myriad of techniques, (2) a suite of diagnostics and remedies to diagnose and fix common problems, (3) new wavelet shrinkage algorithms, (4) new statistical theory behind wavelet shrinkage. The tools developed in their research will apply to a wide variety of statistical and scientific problems, including signal extraction, nonparametric regression, density estimation, inverse problems, image reconstruction and enhancement, change-point analysis, and analysis of massive datasets.