Generalized partially linear models on Riemannian manifolds

TL;DR

This paper introduces generalized partially linear models on Riemannian manifolds, enabling flexible modeling of responses with various error distributions, especially useful for shape data like children's body scans.

Contribution

It extends partially linear models to Riemannian manifolds, addressing the challenge of modeling covariates on curved spaces, with applications to 3D shape data.

Findings

Effective prediction of children's garment fit from 3D scans.

Successful application to Kendall's shape space.

Comparison shows advantages over existing techniques.

Abstract

The generalized partially linear models on Riemannian manifolds are introduced. These models, like ordinary generalized linear models, are a generalization of partially linear models on Riemannian manifolds that allow for 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 3D scanning of their body. 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 3D database is carried out to check the goodness of the procedure. Finally it is applied to a 3D database obtained from an anthropometric survey of the Spanish child population. A comparative study with related techniques is carried out.