The research objective of this award is to develop a control theoretic framework and efficient numerical schemes for nonparametric estimation of shape and dynamics constrained functions, with applications to emerging fields such as systems biology. The project will focus on three important and interrelated components: (i) smoothing spline estimation of functions subject to constraints; (ii) computation and analysis of penalized polynomial spline estimators for constrained functions; and (iii) applications to genetic regulatory networks, degradation analysis in reliability engineering, and joint-drug treatment in biomedical research. The underlying theoretical foundation is based on constrained optimal control, complementarity theory, and asymptotic statistics. The obtained estimation algorithms will be implemented on various biological, biomedical and engineering systems subject to constraints.
Constraints are pervasive in engineering and science. Efficient estimation methods to be developed in this research will yield better understanding of complex biological systems, and provide accurate predication of products' lifetime and joint-drug effects on patients. The proposed research goes beyond traditional areas of control technology with many novel applications in statistics, systems biology, reliability engineering, and biomedical practice. The research findings of this project will be disseminated through PIs' close collaboration with engineers and an epidemiologist. This project also intends to integrate research with education and outreach activities at University of Maryland Baltimore County and Purdue University. Examples include developing new curricula and recruiting students to gain hands-on research experience. Undergraduate summer research and K-12 training will be supported with particular focus on minority and women participants.
Various static or dynamic models of biological, engineering, and economic systems contain functions subject to inequality shape constraints; typical examples include monotone and convex functions. Efficient estimation of shape constrained functions from error polluted data is critical to numerous applied fields, and incorporation of (given) shape constraints into an estimation process will improve accuracy and efficiency of estimation. Shape constrained estimation problems give rise to a number of numerical and analytical challenges in estimator characterization and performance analysis due to inequality constraints. This project studies characterization, computation, and statistical analysis of shape constrained estimation schemes using techniques from constrained optimal control/optimization and nonparametric statistics. Intellectual Merit: 1. Computation of general shape constrained smoothing splines subject to control and other constraints. Shape constrained smoothing splines find a wide range of applications in many applied fields, and we formulate these splines as a constrained optimal control problem subject to control constraints. Because of inequality constraint induced nonsmoothness, new numerical tools are needed to compute the constrained smoothing splines. In this project, a modified nonsmooth Newton's algorithm is developed to compute optimal solutions of the smoothing splines, and its numerical convergence is established. Moreover, several other algorithms are developed to handle additional initial state constraints. Numerical results illustrate their effectiveness. 2. Statistical performance analysis of shape constrained estimators. Shape constrained estimators are developed for monotone and convex functions using B-splines. In order to evaluate the performance of these estimators, we exploit several novel tools from constrained optimization and nonparametric statistics. Specifically, since the optimality conditions of the constrained spline estimators yield a family of size-varying piecewise linear functions, a critical uniform Lipschitz property is established. This property leads to uniform convergence, with which we show optimal convergence rates and design several estimators adaptive to parameter variations in function classes. These results are extended to general shape constrained problems. Moreover, we also show the attained estimators achieve minimax upper and lower bounds, and thus completely characterize optimal performance of these estimators. 3. Applications. The proposed shape constrained estimators are applied to several applied problems. (3.1) Estimation of periodic boundary functions subject to monotone temporal constraint with application to estimation of range expansion of imported fire ants using data from US Department of Agriculture Animal and Plant Health Inspection Service. (3.2) A P-spline based convex estimator is developed to estimate the relation between age and eye lens weight of rabbits in Australia. (3.3) Estimation of monotone components in genetic regulation networks and Heaviside-type functions using EEG data in biomedical applications. Broader Impacts: 1. The research of this project extends control techniques to a novel and nontraditional field, namely, shape constrained nonparametric estimation, and applies several new tools from complementarity theory and constrained optimal control to address challenging problems in statistics, e.g. computation of general constrained smoothing splines and establishment of uniform Lipschitz property of constrained spline estimators. The research outcomes and findings of this project broaden applications of control and optimization techniques, and promote awareness of these techniques in statistics. Hence they open a door for further interdisciplinary collaboration between control engineering and statistics. Funded by this grant, the PI has given many talks at international conferences to broaden and deepen interests of this interdisciplinary area. 2. This grant supported Teresa Lebair (Ph.D. student of the PI) as a research assistant to carry out research on computation and analysis for shape constrained estimation. Teresa was supported by this grant to travel to Purdue University for research collaboration during PI's sabbatical at Purdue in 2012-2013. She was also supported to attend the SIAM conferences in 2013 and 1014, where she spoke about her research. This gave Teresa excellent opportunities to learn state-of-art of research topics and other relevant problems in this field. 3. This grant supported Peter Davis for an undergraduate summer research at UMBC. Peter studied nonsmooth dynamical systems and uncertainty analysis subject to parameter perturbations. 4. The PI has been a faculty advisor of the SIAM Student Chapter at UMBC during 2011-2012 and a faculty advisor of PME (honorary national math society) in 2009-211. He has led a number of student activities at these chapters to promote interests in STEM areas. Moreover, he has been a co-organizer of the 3rd SIAM Mid-Atlantic Regional Applied Math Student Conference in 2012.
