Generalized partially linear models on Riemannian manifolds

Abstract

We introduce generalized partially linear models with covariates on Riemannian manifolds. These models, like ordinary generalized linear models, are a generalization of partially linear models on Riemannian manifolds that allow for scalar response variables with error distribution models other than a normal distribution. Partially linear models are particularly useful when some of the covariates of the model are elements of a Riemannian manifold, because the curvature of these spaces makes it difficult to define parametric models. The model was developed to address an interesting application: the prediction of children's garment fit based on three‐dimensional scanning of their bodies. For this reason, we focus on logistic and ordinal models and on the important and difficult case where the Riemannian manifold is the three‐dimensional case of Kendall's shape space. An experimental study with a well‐known three‐dimensional database is carried out to check the goodness of the procedure. Finally, it is applied to a three‐dimensional database obtained from an anthropometric survey of the Spanish child population. A comparative study with related techniques is carried out.