Regression and curve fitting for networks and shapes

Complex object data such as networks and shapes are becoming increasingly available, and so there is a need to develop suitable methodology for statistical analysis. Networks can be represented as graph Laplacian matrices, which are a type of manifold-valued data. Shapes of 3D objects are also a type ofmanifold-valued data, invariant to translation, rotation and scale. Our main objective is to estimate a regression curve from a sample of graph Laplacian matrices or 3D shapes conditional on a set of Euclidean covariates, for example in dynamic objects where the covariate is time. We develop an adapted Nadaraya-Watson estimator which has uniform weak consistency for estimation using Euclidean and power Euclidean metrics, and we also explore splines on shape spaces.