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

提案编号：dms-0204552和dms-0203901 pis：李蒙和Thomas Chun Man Lee标题：合作研究：具有不规则设计的自洽和小波回归。该奖项是为一个综合研究项目而设，该研究项目将由哈佛大学(牵头机构)的Pi Meng和科罗拉多州立大学的Pi Lee联合开展，研究如何将自洽原理用于具有不规则设计的小波回归。小波估计器具有优良的理论性质，能够适应非常复杂的空间和频率不均匀。此外，当回归设计点为正则时，它们的计算速度非常快。然而，当设计点不规则时，就像统计应用中的典型情况一样，标准的小波方法不再适用。这项合作研究建议从缺失数据的角度来解决这一问题，方法是将不规则设计问题视为常规设计问题，但包含缺失数据。这种新的视角允许研究人员在自洽原理的指导下，应用成熟的缺失数据方法来构造高效的非规则设计小波估计器，以及计算这种估计量的快速算法。小波回归是一种强大的曲线曲面拟合方法，引起了不同领域的研究人员，特别是应用数学家、计算机科学家、工程师和统计学家的极大关注。在许多不完全数据问题中，自相容是构造最有效的统计估计量的基本统计原理。这项合作研究工作将这两种强大的方法结合在一起，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，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when the design points are not regular, as is typical in statistical applications, standard wavelet methods are no longer applicable.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.