Flexible Regression Methods for Curve and Shape Data

The overarching goal of this project is the extension of the methodological toolbox for flexible regression for curve and shape data developed in the first funding period in several directions. In particular, we identified several areas, where there are still gaps in the literature, see also our discussion of these open problems in St?cker et al. (2024), which - while related - weren’t yet addressed in the first funding period. These concern in particular the challenging case of re-parametrization invariance (elastic analysis), where only a quotient metric space structure instead of a manifold structure is available, and which is also necessary to tackle for the case of both types of invariances simultaneously. They include in particular:

1. Error-prone observations: While we addressed the realistic sparsely observed case in the first funding period - a novel development compared to previously existing elastic functional and functional shape data methods - we did not explicitly consider additional error. While error in curve observations is likely common in applications due to imprecise measurements (similarly to the sparse functional data case), it also poses considerable additional challenges, in particular for elastic methods based on derivatives in the square-root- velocity framework.

2. Elastic functional and functional shape covariates: While we focused on the elastic functional and functional shape outcome case in the first funding period, the case of corresponding covariates is not yet solved satisfactorily. We discuss in St?cker et al. (2024) why an existing approach (Ahn et al., 2020) for elastic covariates has several shortcomings.

3. As proven again in Volkmann et al. (2023a) and Volkmann et al. (2023b) for the case of multivariate functional data, principal component analysis (PCA) is a key tool for visualisation of variability in complex data and serves as a building block in other analysis methods including for parsimonious basis representations in regression modelling. Multivariate functional PCA (MFPCA) would be similarly useful in the elastic case.