Nonlinear methods for Functional Data Analysis lead to flexible and versatile statistical models, inference and analysis methods for data that include samples of random functions. Such data accrue in the study of time-dynamic phenomena such as electricity consumption curves or biological trajectories and also in a large number of longitudinal studies across the sciences. Methods for functional data analysis have been rapidly evolving over the past few years and are increasingly viewed as essential for the analysis of time-dynamic phenomena. To date, linear models for functional data have been relatively well investigated, both in terms of theoretical and practical aspects, and statistical tools that are based on these methods are available for data analysis. However, the class of linear functional models is quite narrow and often not adequate in practical data analysis. In contrast, relatively little is known about more general and more flexible nonlinear approaches. This research seeks to remedy this situation by developing a class of nonlinear functional methods. The potential value of such models for applications is high, especially in scenarios where one observes repeated functions in time or space, or where one wishes to study regression relations that include functional components as predictors or responses. Nonlinear functional methodology includes representations of samples of trajectories by means of nonlinear components or through mixture models. These approaches are useful for applications where random time warping plays a role, and for the construction of quantiles in functional regression settings. The proposed methodology and associated software provides a sensible balance between increased flexibility and structural constraints.
The investigator and his research group develop new methods aimed at the statistical analysis of repeatedly observed trends and trajectories. Such data are increasingly common due to new sophisticated sensors, measurement systems and the widening recognition that a deep understanding and interpretation of time trends and their patterns is often key to better individual and societal decision making. This new methodology is useful to gain insights into the dynamics of time-dependent processes such as human growth, characteristics of freeway traffic patterns, or the comparison of lifetables across countries and calendar years. The proposed nonlinear approaches to such functional data lead to improved and more compact descriptions and to better predictions of outcomes that are related to observed time trends.
