Mathematical Sciences: Adaptive Estimation: New Tools, New Settings

建议：DMS 95-05151 PI(S)：Iain Johnstone-1981年，David Donoho-1984年机构：斯坦福大学标题：自适应估计，新工具，新设置摘要：这项研究开发了理论统计中的基本工具，以帮助理解这种自适应过程，并展示了如何调整它们，使它们能够识别噪声和稳定。这种方法的基础是a)Oracle的思想，它非常清楚如何在理想情况下调整表示，b)在有噪声的数据存在的情况下进行适应的目标是量化可实现的过程(没有关于对象的特权信息)可以模仿Oracle的程度，以及c)尽可能接近Oracle的过程的设计。该项目还开发了通过比较不同类型的先知来比较不同适应方案的方法，例如时频先知和时间尺度先知。这是我们早期关于小波的结果的推广，其中该方法被用来表明小波具有近理想的空间自适应的性质。此外，正在开发一个计算环境，用于执行和系统地测试这些方法。作为作者早期小波工作的进一步发展，该项目研究了小波收缩的一些改进和扩展，例如在分类、置信度、相关数据和正交基的选择方面。这项研究试图在表示和分析信号、图像和其他对象的自适应方法的一般领域发展统计理论和计算工具。一方面，这个项目是由信号处理、图像处理、时频分析、语音处理分析等领域的工作人员对改编的极高兴趣推动的。这些领域几乎每天都会出现新的信号表示法，以及选择表示法的新原则。这些表示对于数据压缩是有用的，这不是这里的主要兴趣；但它们也对噪声去除和信号解释有用，这是统计学家要考虑的重要领域。另一方面，这个项目是由发展统计理论的目标推动的，该理论可以清楚地理解这种适应性方案。信号处理中提出的一些适应性最强的方案对传统的统计思维构成了真正的挑战。例如，一些自适应方案(显式或隐式地)搜索数千或数百万个信号的表示，以便得到它们的最终结果。统计学家在考虑将这些方法应用于有噪声的信号时，通过培训被迫询问结果在多大程度上仅仅反映了通过噪声进行窥探、检测实际上由于噪声以及由于大力搜索而产生的伪结构的效果。这项研究试图开发理论统计中的基本工具，以帮助理解这种适应过程，并展示如何调整它们，使它们能够识别噪音和稳定。这个项目可能有两个衍生产品。首先，一些结果可能对信号、图像、语音和时间/频率方面的“自适应程序发明者”社区以及相关社区具有刺激性和/或有用的作用。其次，理论工作可能会激发统计学家对在这些方向上做出进一步贡献的更大兴趣。

Proposal: DMS 95- 05151 PI(s): Iain Johnstone - 1981, David Donoho - 1984 Institution: Stanford Title: Adaptive Estimation, New Tools, New Settings Abstract: The research develops fundamental tools in theoretical statistics to aid understanding of such adaptive procedures, and shows how to tune them so they are noise-cognizant and stable. Underlying the approach are a) the idea of oracles, which know perfectly well how to adapt representations ideally, b) the idea that the goal of adaptation in the presence of noisy data is to quantify how closely realizable procedures (which do not have privileged information about the object) can mimic an oracle, and c) the design of procedures coming as close as possible to the oracle. The project also develops methods for comparing different adaptation schemes by comparing oracles of different kinds, for example time-frequency oracles and time-scale oracles. This is an outgrowth of our earlier results on wavelets, where this approach was used to show that wavelets have a property of being nearly-ideally spatially adaptive. In addition a computational environment is being developed for implementing and systematically testing such approaches. As a further outgrowth of the proposers' earlier work on wavelets, the project studies a number of improvements and extensions of wavelet shrinkage, for example in the directions of classification, confidence bands, correlated data and selection of orthogonal bases. This research seeks to develop statistical theory and computational tools in the general area of adaptive methods of representing and analyzing signals, images and other objects. On the one hand, this project is prompted by the extremely high interest in adaptation on the part of people working in fields of signal processing, image processing, time-frequency analysis, speech processing analysis. New signal representations appear in these fields almost daily, along with new principles fo r selecting representations. Such representations are useful for data compression, which is not the main interest here; but they are also useful for noise removal and signal interpretation, which are important areas for statisticians to consider. On the other hand, this project is prompted by the goal of developing statistical theory which can give clear understanding of such adaptive schemes. Some of the most highly adaptive schemes being suggested in signal processing pose a real challenge to traditional statistical thinking. For example, some adaptation schemes search (either explicitly or implicitly) through thousands or millions of representations of a signal in order to arrive at their final result. A statistician, when thinking about applying such methods to a noisy signal, is by training forced to ask to what extent the result merely reflects the effects of snooping through noise, detecting pseudo-structure which actually due to noise, and due to the vigorous search. The research seeks to develop fundamental tools in theoretical statistics to aid understanding of such adaptive procedures, and shows how to tune them so they are noise-cognizant and stable. This project may have two spin-offs. First, some of the results may be stimulating and/or useful to the community of ``inventors of adaptive procedures'' in signal, image, speech, and time/frequency, and related communities. Second, the theoretical work may stimulate statisticians to take more interest in making further contributions in such directions.