Theory and Applications of Sharp Nonparametric Estimation and Learning

University Of New Mexico, Albuquerque NM

Investigators

Abstract

abstract PI: SAM EFROMOVICH proposal number: 0243606 The 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.