The analysis of samples of curves is a field of growing relevance in statistics. Samples of curves arise when a time-dependent process is observed on a group of individuals. Such curves present both time and amplitude variability ("horizontal" and "vertical" variability), and both types of variation have to be statistically modeled in order to draw valid inference from the data. The investigator is developing dynamic regression models, that is, models for prediction of response curves based on explanatory curves, that explicitly take into account time variability. The properties of these methods are being studied theoretically, by simulation, and by the analysis of real-life data sets. Computer software implementing these methods is also being developed.
Data consisting of samples of curves include, among many others, human growth curves, time-dependent gene expression profiles, daily air pollutant concentrations, and stock prices. The investigator's work will help scientists work on a broad range of applications and these statistical techniques will help provide new insights into scientific areas as diverse as medicine, genetics, environmental studies, and economics.
The primary goal of this project was to develop new statistical methods for the analysis of samples of curves. Samples of curves arise, for example, in longitudinal human growth studies where the entire growth trajectory is observed for each individual, or in medical image analysis where angiographic images of veins and arteries are taken. In this project we specifically developed methods to predict certain features of the curves on the basis of other features; for example, to predict the location of an aneurysm on the basis of geometric features of the angiographic images. The statistical methods developed during this project, like all statistical methods, can be applied to a wide variety of data and therefore have the potential of making an impact in different fields of application. For example, the methods developed in this project were applied to: neuromotor data, where the lip trajectory of a speaker was predicted in terms of muscular neural activity; environmental data, where ozone concentration trajectories were predicted in terms of concentrations of oxides of nitrogen; beetle growth data, where genetic and environmental components of the growth curves were estimated in order to assess heritability of the traits, which is important from an evolutionary perspective; and angiographic image analysis, where the severity of aneurysms was classified in terms of geometric characteristics of the upper carotid artery. These are just a few examples that illustrate the breadth of application of the statistical methods developed during this project. In addition to the specific impact on science and technology, this project can have a broader impact in society, since the statistical tools developed here can be used for the analysis of data arising in strategic areas such as public health and economy, and therefore contributing to the improvement of well-being of individuals in society. For example, applying these methods to the analysis of environmental data like ground-level ozone formation can lead to new discoveries that can inform environmental public policy; applying these methods to the analysis of medical images can lead to implementation of less invasive diagnostic procedures, thus improving public health. This project also contributed to the development of human resources by providing opportunities for research, teaching and mentoring in mathematical sciences, specifically in Statistics, since the PI supervised a number of PhD theses in the area and taught graduate-level courses using data, algorithms and software that arose from this project.
