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 following proposed 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.
