Nonparametric Curve Estimation: Theory and Practice

该提案的重点是发展数据驱动的非参数曲线估计的统计理论、方法和方法，并在生物、心理、医学和环境应用上进行测试，主要目的是在渐近理论和应用之间架起桥梁。主要的智力目标是：(I)对于有间接观测的模型(如删失和有偏回归、隐藏分量、在存在非参数趋势和波动性的情况下的时间序列建模)，发展尖锐极小极大估计以及模仿已知直接观测和/或干扰函数的先知的理论；(Ii)对于对照实验，发展最优(在更常见的标准下)解取决于估计和/或干扰函数的情况下的序贯抽样和估计的理论、方法和方法；(Iii)发展小波估计的局部聚集的理论和方法。实际问题包括对城市污水处理厂的研究、阿尔伯克基流域污染物和残留消毒剂水平的建模、分析饮用模式和孕期开始的行为变化、研究塞维莱塔国家野生动物保护区植物的时空结构、恢复核磁共振图像的学习机和卫星数据分析。这项研究的主要重点是与桑迪亚和洛斯阿拉莫斯国家实验室以及UNM医学和工程学院合作，开发统计学习和适应性估计的算法和软件，并在以下环境、医疗和生物应用方面进行测试：对塞维莱塔国家野生动物保护区植物的时空结构进行统计建模，这将使人们能够研究全球气候变化；城市污水处理厂的研究；阿尔伯克基流域污染物和残留消毒剂水平变化点的建模和分析，以及应用于国土安全饮用水监测；用于恢复磁共振图像和分析包括温度和湿度在内的环境卫星数据的学习机；分析怀孕期间的饮酒模式和行为改变，以降低复发的可能性。这项研究的更广泛的影响被定义为广为人知的应用程序，这些应用程序将造福社会，帮助学生和更广泛的受众理解数学科学的重要性。影响将基于以下拟议活动：(I)研究生将参与研究；(Ii)为促进学习，将为本科生和研究生举办科学研讨会，并将在新墨西哥大学高中生数学意识周期间介绍提出者和学生的演讲。(3)为扩大任职人数不足群体的参与，将在新墨西哥州大学盖洛普校区和巴伦西亚校区举办的外联研讨会上定期介绍情况。(4)开发的软件将免费提供。有益于社会的医学、环境和生物学研究成果将发表在非技术性期刊上。

ABSTRACT The proposal focuses on developing statistical theory, methodology and methods of data-driven nonparametric curve estimation motivated by and tested on biological, psychological, medical, and environmental applications, with the main aim to bridge the asymptotic theory and applications. The main intellectual objectives are: (i) For models with indirect observations (like censored and biased regression, hidden components, modeling of time series in the presence of nonparametric trend and volatility) develop the theory of sharp minimax estimation as well as of mimicking of oracles that know direct observations and/or nuisance functions unavailable to the statistician; (ii) For controlled experiments, develop the theory, methodology and methods of sequential sampling and estimation for the case where optimal (under more common criteria) solution depends on estimated and/or nuisance functions; (iii) Develop the theory and methods of a local aggregation of wavelet estimates. Practical problems include the study of municipal wastewater treatment plants, modeling of levels of contaminants and residual disinfectants in Albuquerque water basin, analysis of drinking patterns and behavior change initiation during pregnancy, study of temporal and spatial structures of plants in Sevilleta National Wildlife Refuge, learning machines for recovery magnetic resonance images and the analysis of satellite data. The primary focus of this research is to develop, in collaboration with Sandia and Los Alamos National Laboratories as well as with the UNM Medical and Engineering Schools, algorithms and software for statistical learning and adaptive estimation 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 which will allow to study global weather changing; The study of municipal wastewater treatment plants; Modeling and analysis of change points in levels of contaminants and residual disinfectants in Albuquerque water basin with applications to a homeland security drinking water monitoring; Learning machines for recovery magnetic resonance images and the analysis of environmental satellite data including temperature and humidity; Analysis of drinking patterns and behavior change initiation during pregnancy which reduces likelihood of relapses. The broader impact of the research is defined by well--understood applications that will benefit the society and help students and a broader audience to understand the importance of mathematical sciences. The impact will be based on the followingproposed activities: (i) Graduate students will participate in the research; (ii) To promote learning, scientific seminars will be held for undergraduate and graduate students, and talks by the proposer and the students will be presented during the UNM mathematical awareness weeks for high-school students. (iii) To broaden participation of under-represented groups, regular presentations will be held at outreach seminars conducted by the UNM Gallup and Valencia campuses. (iv) The developed software will be freely available. Medical, environmental and biological findings, benefiting the society, will be published in not-technical journals.