Flexible regression methods for curve and shape data

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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 re-parameterization along the curve. This induces a non-Euclidean geometry on the resulting quotient spaces (shape spaces). 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. In particular, a general and flexible regression framework for curves and shapes in two (or potentially higher) dimensions will be developed and implemented.Successively generalizing from additive models for functional data to those for multivariate functional data and for functional shape data, this framework will offer unprecedented flexibility in the following respects: it will allow for modeling curve and shape responses modulo re-parameterization; intrinsically and modularly account for different combinations of invariances with respect to re-parameterization, translation, rotation and/or scaling according to the needs in a specific data scenario; allow for curves and shapes to be irregularly or sparsely sampled, or observed as an ensemble of shapes; and include various additive covariate effect types including linear, non-linear and random effects. Additionally, appropriate effects for scalar-on-curve and -shape regression will be developed. While building on interpretable linear and additive predictors, the framework will conform to the intrinsic geometries of the spaces arising from the respective invariances. 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 analysis.