Nonparametric Curve Estimation: Theory and Practice

摘要：本研究旨在发展数据驱动的非参数曲线估计的统计理论、方法论和方法，这些数据驱动的非参数曲线估计受到生物、心理、医学和环境应用的驱动和测试，主要目的是弥合渐近理论和应用之间的桥梁。主要的智力目标是：(i)对于具有间接观测的模型（如删节和偏倚回归，隐藏成分，存在非参数趋势和波动性的时间序列建模），发展急剧极小极大估计理论以及模仿知道直接观测和/或统计学家不可用的讨厌函数的预言机；（ii）在受控实验中，针对（在更普遍的标准下）最优解决方案取决于估计和/或干扰函数的情况，发展顺序抽样和估计的理论、方法学和方法；(3)发展小波估计局部聚合的理论和方法。实际问题包括对市政污水处理厂的研究、对阿尔伯克基水盆中污染物和残留消毒剂水平的建模、对怀孕期间的饮用模式和行为变化开始的分析、对塞维利亚国家野生动物保护区植物时空结构的研究、用于恢复磁共振图像的学习机器以及对卫星数据的分析。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新墨西哥大学医学院和工程学院合作，开发用于统计学习和自适应估计的算法和软件，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，这将有助于研究全球天气变化；城市污水处理厂的研究；阿尔伯克基流域污染物和剩余消毒剂水平变化点的建模和分析及其在国土安全饮用水监测中的应用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；分析怀孕期间饮酒模式和行为改变，减少复发的可能性。研究的更广泛影响是通过易于理解的应用程序来定义的，这些应用程序将有益于社会，并帮助学生和更广泛的受众了解数学科学的重要性。影响将基于以下拟议的活动：(i)研究生将参与研究；（二）为促进学习，将为本科生和研究生举办科学研讨会，并在新墨西哥大学高中生数学认识周期间由提案人和学生发表演讲。为了扩大代表性不足的群体的参与，将定期在新墨西哥大学盖洛普分校和瓦伦西亚分校举办的外联讨论会上作介绍。开发的软件将免费提供。造福社会的医学、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.