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

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when the design points are not regular, as is typical in statistical applications, standard wavelet methods are no longer applicable.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