Nonlinear Wavelet Tools for Noisy and Indirect Data

小波变换是一种重要的新工具，对各种重要问题产生了巨大的影响。例如，小波已经成功地应用于数据压缩问题。小波在数据压缩方面的优势也是小波在统计估计方面的优势。波和相关的想法有潜力作出巨大贡献的问题，涉及噪音和间接的观察。MathSoft，Inc.建议开发一个基于小波收缩原理的统计技术集合。小波收缩是指通过进行小波变换，然后将经验小波系数向零收缩，然后进行逆变换而获得的估计。这些技术将被捆绑在一个面向对象的软件工具包。该研究计划分为四个部分：（1）小波收缩顾问，以帮助筛选无数的技术，（2）一套诊断和补救措施，以诊断和解决常见问题，（3）新的小波收缩算法，（4）小波收缩背后的新统计理论。在他们的研究中开发的工具将适用于各种各样的统计和科学问题，包括信号提取，非参数回归，密度估计，逆问题，图像重建和增强，变点分析和大规模数据集的分析。

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.