Functional additive models on manifolds of planar shapes and forms
This work addresses the problem of statistical modeling for shape and form data in fields like biology and biophysics, representing an incremental extension of existing methods to handle geometric constraints.
The authors tackled the problem of regression modeling for planar shapes and forms by extending generalized additive models to respect quotient geometry, using squared geodesic distance as loss and a geodesic response function. They proposed a Riemannian L2-Boosting algorithm for fitting and demonstrated its usefulness in analyzing astragalus shapes, cell forms, and bottle outlines.
The "shape" of a planar curve and/or landmark configuration is considered its equivalence class under translation, rotation and scaling, its "form" its equivalence class under translation and rotation while scale is preserved. We extend generalized additive regression to models for such shapes/forms as responses respecting the resulting quotient geometry by employing the squared geodesic distance as loss function and a geodesic response function to map the additive predictor to the shape/form space. For fitting the model, we propose a Riemannian $L_2$-Boosting algorithm well suited for a potentially large number of possibly parameter-intensive model terms, which also yields automated model selection. We provide novel intuitively interpretable visualizations for (even non-linear) covariate effects in the shape/form space via suitable tensor-product factorization. The usefulness of the proposed framework is illustrated in an analysis of 1) astragalus shapes of wild and domesticated sheep and 2) cell forms generated in a biophysical model, as well as 3) in a realistic simulation study with response shapes and forms motivated from a dataset on bottle outlines.