This three-year project advances the statistical field of functional data analysis by rigorously developing a new compound estimation paradigm for simultaneously estimating a mean response and several of its derivatives from noisy data. A hybrid between local modeling and global modeling, compound estimation circumvents the difficulties associated with local averaging (e.g., kernel), local modeling (e.g., local regression), and global modeling (e.g., spline) approaches. These difficulties include: the empirical disparity between asymptotic theory and finite-sample performance in local averaging; the pointwise character of local modeling, along with the incompatibilities between mean response estimates and derivative estimates in that setting; and, the unrealistic assumptions tacitly imposed in global modeling, such as there being three nonzero derivatives with discontinuities in the third derivative. Compound estimation is particularly promising for problems in which physical phenomena are described by differential equations, problems in which modeling velocities and accelerations is of scientific importance, and pattern recognition problems in which features of higher-order derivatives can be exploited for classification. One such pattern recognition problem has remained an outstanding challenge in nanoscale engineering: can the configuration of nanosize metallic particles, agglomerates, and structures on or near a surface be inferred from surface wave scattering data? If so, then a major hurdle to building real-time diagnostic tools in nanoscale engineering can be overcome, thereby advancing future nanomanufacturing efforts. Compound estimation offers a solution to this pattern recognition problem, as mean response estimates and derivative estimates from surface wave scattering data can be employed to objectively determine the most plausible configurations of nanosize particles. Although there is a wealth of underlying mathematics, the principal motivation for compound estimation is practical: to provide tractable yet realistic descriptions of the possibly complicated relationships among variables of scientific interest. Many applications are envisioned, including studies of: human growth and development patterns in biology; temporal trajectories for infectious disease incidence in public health; dose-response relationships for pharmacological treatments in medicine; stock market and gross domestic product trends in economics; and, the behavior of nanosize particles in engineering. This three-year project gives particular attention to the last application, both to guide the development of compound estimation and to address an important scientific problem in its own right. There is a crucial need for advanced instrumentation allowing real-time on-line diagnostics of chemical and physical processes in nanoscale engineering; compound estimation, through its solution to a pattern recognition problem, can help to create the ""eyes"" and ""brain"" for such diagnostics.
Theoretical and methodological developments in compound estimation, along with empirical findings, will be published in appropriate venues and presented to local, regional, and national audiences. User-friendly software will be made freely available online. This project will positively impact graduate and undergraduate education at the University of Kentucky, through both the direct involvement of five students during its three-year course and its visibility to incoming or prospective students as an example of exciting multidisciplinary research. Finally, the investigative team's connections with the Appalachian Math and Science Program and the Umbrella Program on Nanoscale Engineering will be utilized to raise undergraduate and high school student awareness of and appreciation for statistics and applications.
