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

提案ID：DMS - 0204552和DMS -0203901 PI：Xiao-Li Meng和托马斯Chun Man Lee标题：合作研究：自洽性与非正则小波回归摘要本奖项是为了奖励哈佛大学的皮萌教授的一项综合性研究项目（牵头机构）和科罗拉多州立大学的PI Lee，关于在不规则设计的小波回归中使用自洽原理。小波估计器具有优良的理论性能，能够适应非常复杂的空间和频率不均匀性。此外，当回归设计点是规则的时，它们的计算非常快。 然而，当设计点是不规则的，是典型的统计应用，标准小波方法不再适用。这项合作研究提出从丢失数据的角度来解决这个问题，将不规则设计问题视为规则设计问题，但缺少数据。这一新的视角使得研究者可以应用自洽原理指导下的缺失数据方法来构造有效的非规则设计小波估计，以及计算此类估计的快速算法。小波回归是一种强大的曲线和曲面拟合方法，吸引了不同领域研究人员的极大关注，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的统计估计量的基本统计原则。这项合作研究工作结合了这两种强大的方法，旨在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，其中不规则性是规则而不是例外。

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when the design points are not regular, as is typical in statistical applications, standard wavelet methods are no longer applicable.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.