Theory and Applications of Sharp Nonparametric Estimation and Learning

建议编号：0243606这项研究的主要重点是开发数据驱动的统计估计和学习的通用方法，这些方法受到环境、医学和生物应用的激励和测试。主要的智力目标有三个：(A)在具有已知尖锐渐近性的设置(如删失或有偏数据集)的情况下，发展关于小数据集的尖锐最优性的开始以及统计模型之间的等价性的理论；(B)在具有间接观测和滋扰函数的模型的情况下(如异方差非参数回归中的误差密度估计或恢复时间序列中的隐藏分量)，发展固定精度的尖锐估计和抽样理论；(C)在具有未知算子的逆问题中，发展隐含尖锐估计的数据驱动的学习机。实际问题包括塞维莱塔国家野生动物保护区植物时空结构的统计建模，阿尔伯克基流域砷浓度的建模，城市污水处理厂的研究，汉坦病毒传播的统计建模，以及恢复磁共振图像的学习机。这项研究的主要重点是与桑迪亚国家实验室和北卡罗来纳大学医学院合作，开发自适应统计估计和学习的算法和软件，这些算法和软件受到下列环境、医疗和生物应用的激励和测试：塞维莱塔国家野生动物保护区植物时空结构的统计建模；阿尔伯克基流域砷浓度的建模；市政污水处理厂的研究；汉坦病毒传播的统计建模；用于恢复磁共振图像的学习机。这项研究的更广泛影响是通过众所周知的应用程序来定义的，这些应用程序可以鼓励学生学习数学，并可以帮助更广泛的受众理解统计学的重要性。其影响基于以下活动：(1)开发一门关于自适应统计估计的新课程，通过联合国大学网络方案授课；(2)每周为本科生和研究生举办科学研讨会(部分由私人补助金支持)，并在联合国大学数学意识周期间为高中生举办讲座；(3)定期在巴伦西亚州立大学校区举办的外联研讨会上介绍情况，以扩大代表不足群体的参与；(4)在调查员的网页上张贴所开发的软件、数据库和实际调查结果，使更多的受众感兴趣；(4)在非技术性期刊上发表有益于社会的医学、环境和生物学研究成果。

abstractPI: SAM EFROMOVICHproposal number: 0243606The primary focus of this research is to develop general methods of data-driven statistical estimation and learning motivated by and tested on environmental, medical and biological applications. The main intellectual objectives are threefold:(A) In the case of settings with known sharp asymptotics (like censored or biased datasets), develop the theory of the onset of the sharp optimality and the equivalence between statistical models for small datasets; (B) In the case of models with indirect observations and nuisance functions (like error density estimation in heteroscedastic nonparametric regression or recovery of a hidden component in time series), develop the theory of sharp estimation and sampling with fixed accuracy; (C) In the case of inverse problems with unknown operator, develop data-driven learning machines implying sharp estimation. Practical problems include statistical modeling of temporal and spatial structures of plants in Sevilleta National Wildlife Refuge, modeling of arsenic concentration in Albuquerque water basin, the study of municipal wastewater treatment plants, statistical modeling of spreading hantavirus, and learning machines for recovery magnetic resonance images.The primary focus of this research is to develop, in collaboration with Sandia National Laboratories and the UNM Medical School, algorithms and software for adaptive statistical estimation and learning motivated by and tested on the following environmental, medical and biological applications: Statistical modeling of temporal and spatial structures of plants in Sevilleta National Wildlife Refuge; Modeling of arsenic concentration in Albuquerque water basin; Study of municipal wastewater treatment plants; Statistical modeling of spreading hantavirus; Learning machines for recovery magnetic resonance images. The broader impact of the research is defined by the well-understood applications that can encourage students to study mathematics and can help a broader audience to understand the importance of statistics. The impact is based on the following activities:(i) Developing a new course on adaptive statistical estimation taught via the UNM web-based program;(ii) Weekly scientific seminars (supported in part by private grants) held for undergraduate and graduate students, and talks during the UNM mathematical awareness weeks for high-school students;(iii) Regular presentations at outreach seminars conducted by the UNM Valencia campus to broaden participation of under-represented groups;(iv) Posting the developed software, databases, and practical findings, that can be of interest to a broader audience, on the investigator's webpage;(iv) Publishing of medical, environmental and biological findings, benefiting the society, in non-technical journals.