Mathematical Sciences: Adaptive Estimation: New Tools, New Settings

提案：DMS 95- 05151 PI： 伊恩·约翰斯通- 1981年，大卫·多诺霍- 1984年 院校：斯坦福大学 职务名称： 自适应估计，新工具，新设置 摘要： 该研究开发了理论统计学的基本工具，以帮助理解这种自适应程序，并展示了如何调整它们 所以它们是噪声识别和稳定的。这种方法的基础是a）神谕的思想，它完全知道如何适应表征 理想情况下，B）在存在噪声数据的情况下自适应的目标是 量化可实现的程序（不具有特权） 关于对象的信息）可以模仿预言机，以及c） 尽可能接近Oracle。 该项目还通过比较不同种类的预言器，例如时间频率预言器和时间尺度预言器，开发了比较不同适应方案的方法。 这是我们早期 结果小波，这种方法被用来表明，小波具有接近理想的空间自适应的属性。 此外，正在开发一个计算环境， 实施并系统地测试这些方法。 作为提议者早期关于小波的工作的进一步成果，该项目研究了小波收缩的一些改进和扩展，例如在分类、置信带、相关数据和正交基的选择等方面。 本研究旨在发展统计理论和计算工具，在一般领域的自适应方法表示和分析信号，图像和其他对象。 一方面，这个项目是由在信号处理，图像处理， 时频分析，语音处理分析。 在这些领域中，几乎每天都会出现新的信号表示，沿着选择表示的新原则。 这样的表示对于数据压缩是有用的，这不是这里的主要兴趣;但是它们对于噪声去除也是有用的， 信号解释，这是统计人员需要考虑的重要领域。 另一方面，这个项目是由发展统计理论的目标，可以给这种自适应方案的清晰的理解。在信号处理中提出的一些最高度自适应的方案对传统的统计思想提出了真实的挑战。 例如，一些适应计划 搜索（显式或隐式）通过数千或数百万的信号表示，以达到其最终结果。 一个统计学家，当考虑将这种方法应用于噪声信号时， 不得不问结果在多大程度上仅仅反映了通过噪声窥探、检测实际上由于噪声和由于有力搜索的伪结构的效果。 该研究旨在开发理论统计学中的基本工具，以帮助理解这种自适应程序，并展示如何调整它们，使它们能够识别噪声和稳定。 这个项目可能有两个副产品。 首先，一些结果可能是刺激和/或有用的社区的“自适应程序的发明者”在信号，图像，语音和时间/频率，和相关的社区。 第二，理论工作可能会刺激统计人员更有兴趣在这些方向上作出进一步贡献。

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.