Flexible regression methods for curve and shape data

• Applicant: Professorin Dr. Sonja Greven
• Subject Area: Statistics and Econometrics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 431707411

Using modern imaging and tracking devices, researchers in a wide range of areas collect more and more data, where each observation corresponds to a two- or higher-dimensional curve. Examples are movement patterns and bone outlines. In some settings, these can be viewed as multivariate functional data. In others, the functional shape is primarily of interest, i.e. the equivalence class of the curve accounting for invariance to translation, rotation, scaling and/or re-parameterization along the curve. This induces a non-Euclidean geometry on the resulting quotient spaces (shape spaces). In the first funding period, we developed additive regression frameworks for multivariate trajectory-valued and shape-valued responses under different combinations of invariances, focusing on realistic settings of potentially sparsely observed curves as well as flexible yet interpretable additive covariate effects, while respecting the intrinsic geometries of the respective quotient spaces. In this continuation, we aim to extend this flexible additive regression framework for curve and shape data in several directions. First, we will extend our methods to handle not only the realistic sparsely observed setting, but also additional measurement error. Second, we will extend multivariate functional principal component analysis, a key tool for visualization of variability in such complex data and a useful building block for parsimonious regression modeling, to the elastic re-parametrization invariant case. Third, while we focused on settings with elastic functional and functional shape outcomes in the first funding period, we will now consider the case of such covariates. The goal of this project is to advance the field of functional shape analysis both in terms of the theory and in terms of usefully applicable methods for real data analysis problems. All developments will be implemented in the open-source software R and applied in collaborative projects. Overall, the developed framework will thus greatly extend the availability and flexibility of regression models for curve and shape data analysis.

• DFG Programme: Research Grants