Mathematical Sciences: Adaptive Estimation: New Tools, New Settings

提案：DMS 95- 05151 PI(s): Iain Johnstone - 1981， David Donoho - 1984机构：Stanford标题：自适应估计，新工具，新设置摘要：该研究开发了理论统计学中的基本工具，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并保持稳定。这种方法的基础是：a) oracle的思想，它非常清楚如何理想地适应表征；b)在嘈杂数据存在的情况下，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）能够模仿oracle的程度；c)过程的设计尽可能接近oracle。本项目还通过比较不同类型的神谕，如时频神谕和时标神谕，发展了比较不同适应方案的方法。这是我们对小波的早期结果的一个结果，其中这种方法被用来表明小波具有近乎理想的空间自适应特性。此外，正在为执行和系统地测试这种方法开发一个计算环境。作为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、置信带、相关数据和正交基选择的方向上。本研究旨在发展统计理论和计算工具，用于表示和分析信号、图像和其他对象的自适应方法的一般领域。一方面，由于信号处理、图像处理、时频分析、语音处理分析等领域的工作人员对自适应有着极高的兴趣，促使了本项目的开展。几乎每天都有新的信号表示出现在这些领域，同时也有新的选择表示的原则。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；但它们对噪声去除和信号解释也很有用，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，这可以给这种适应性方案提供清晰的理解。在信号处理中提出的一些最具适应性的方案对传统的统计思维提出了真正的挑战。例如，一些适应方案搜索（或显式或隐式）通过数千或数百万个信号的表示，以达到他们的最终结果。一个统计学家，当考虑将这种方法应用于一个有噪声的信号时，通过训练被迫问，结果在多大程度上仅仅反映了通过噪声窥探的效果，检测到实际上是由噪声引起的伪结构，以及由于剧烈的搜索。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过程，并展示如何调整它们，使它们能够识别噪声并保持稳定。这个项目可能有两个副产品。首先，某些结果可能对信号、图像、语音、时间/频率以及相关领域的“自适应程序发明者”群体具有启发性和/或有用性。其次，理论工作可能会刺激统计学家更有兴趣在这些方向上做出进一步的贡献。

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.