Collaborative Research: Self-Consistency and Wavelet Regressions with Irregular Designs

提案id: DMS - 0204552和DMS - 0203901pi: Xiao-Li孟和Thomas Chun Man Lee题目：合作研究：不规则设计的自一致性和小波回归摘要本奖项是由哈佛大学PI孟（领导单位）和科罗拉多州立大学PI Lee共同开展的关于不规则设计的小波回归自一致性原理应用的综合研究项目。小波估计具有良好的理论性质，能够适应非常复杂的空间和频率非均匀性。此外，当回归设计点是规则点时，它们的计算速度非常快。然而，当设计点不规则时，如统计应用中典型的情况，标准小波方法就不再适用了。这项合作研究建议从缺失数据的角度来解决这个问题，通过将不规则设计问题视为缺失数据的规则设计问题。这种新的视角允许研究人员应用完善的缺失数据方法，在自一致性原则的指导下，构建有效的不规则设计小波估计器，以及快速算法来计算这些估计器。小波回归是一种强大的曲线和曲面拟合方法，引起了不同领域研究者的极大关注，特别是应用数学家、计算机科学家、工程师和统计学家。自洽性是在许多不完全数据问题中构造最有效统计估计的基本统计原理。这项合作研究工作结合了这两种强大的方法，目的是使小波方法更适用于现实生活中的问题，从医学成像到渔业经济再到全球变暖，在这些问题中，不规则是规则而不是例外。

Proposal IDs: DMS - 0204552 and DMS - 0203901PIs: Xiao-Li Meng and Thomas Chun Man LeeTitle: COLLABORATIVE RESEARCH: SELF-CONSISTENCY AND WAVELET REGRESSIONS WITH IRREGULAR DESIGNSAbstractThis award is for a comprehensive research project for a joint investigation, to be conducted by PI Meng of Harvard University (the lead institution) and PI Lee of Colorado State University, on the use of the self-consistency principle for wavelet regressions with irregular designs. Wavelet estimators enjoy excellent theoretical properties and they are capable of adapting to very complex spatial and frequency inhomogeneities. In addition, their computation is very fast when the regression design points are regular. However, when the design points are not regular, as is typical in statistical applications, standard wavelet methods are no longer applicable. This collaborative research proposes to attack this problem from a missing-data perspective by viewing an irregular-design problem as a regular-design one but with missing data. This new perspective allows the investigators to apply well-established missing-data methods, guided by the self-consistency principle, to construct efficient irregular-design wavelet estimators, as well as fast algorithms to compute such estimators.Wavelet regression is a powerful curve and surface fitting method that has attracted enormous attention from researchers across different fields, in particular applied mathematicians, computer scientists, engineers, and statisticians. Self-consistency is a fundamental statistical principle for constructing the most efficient statistical estimators in many incomplete data problems. This collaborate research effort combines these two powerful methods with the aim to make wavelet methods much more applicable to real-life problems, varying from medical imaging to fishery economy to global warming, where irregularities are rules rather than exceptions.