Nonparametric Regression in Nonstandard Spaces

Abstract

One approach to tackle regression in nonstandard spaces is Fréchet regression, where the value of the regression function at each point is estimated via a Fréchet mean calculated from an estimated objective function. A second approach is geodesic regression, which builds upon fitting geodesics to observations by a least squares method. We compare these two approaches by using them to transform three of the most important regression estimators in statistics -linear regression, local linear regression, and trigonometric projection estimator -to settings where responses live in a metric space. The resulting procedures consist of known estimators as well as new methods. We investigate their rates of convergence in general settings and compare their performance in a simulation study on the sphere.

Dynamic Network Regression

“…Such "object-oriented data" (Wang and Marron 2007;Marron and Alonso 2014) or "random objects" (Müller 2016) can be viewed as random variables lying in a metric space where no Euclidean structure is available and only pairwise distances between the observed data can be defined. Recently, various kinds of non-Euclidean data, for example situated in a Riemannian manifold, such as in the space of covariance matrices (Dryden et al 2009), or in more general metric spaces (Lyons 2013;Schötz 2020) have received increasing attention.…”