Theory and Applications of Sharp Nonparametric Estimation and Learning

摘要：SAM efromovich提案号：0243606本研究的主要重点是开发数据驱动的统计估计和学习的一般方法，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方面：(A)在已知尖锐渐近的情况下（如审查或有偏差的数据集），发展尖锐最优性开始的理论和小数据集统计模型之间的等效性；(B)对于具有间接观测和干扰函数的模型（如异方差非参数回归中的误差密度估计或时间序列中隐藏分量的恢复），发展具有固定精度的尖锐估计和抽样理论；(C)对于未知算子的逆问题，开发数据驱动的学习机。实际问题包括塞维利亚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用于恢复磁共振图像的学习机器。这项研究的主要重点是与桑迪亚国家实验室和新墨西哥大学医学院合作，开发用于自适应统计估计和学习的算法和软件，这些算法和软件的动机是在以下环境、医学和生物应用中进行测试：塞维利亚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模拟研究城市污水处理厂的研究；汉坦病毒传播的统计模型；磁共振图像恢复学习机。该研究的广泛影响是通过易于理解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了解统计的重要性。其影响基于以下活动：(i)通过新墨西哥大学网络方案开设一门关于适应性统计估计的新课程；每周为本科生和研究生举办科学研讨会（部分由私人资助），并在新墨西哥大学数学认识周期间为高中生举办讲座；定期在新墨西哥大学瓦伦西亚校区举办的外联研讨会上发言，以扩大代表性不足群体的参与；（四）在研究者的网页上公布广大读者可能感兴趣的开发的软件、数据库和实际发现；（四）在非技术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.