Abstract

Recent research has focused on statistical methods for data defined on non-Euclidean spaces, particularly within the framework of Riemannian manifolds. In Air Traffic Management, when a moving object is highly maneuverable, an alternative to the Euclidean framework is to switch to the formalism of Riemannian geometry. Instead of representing trajectory points as positions in a Euclidean state space, they can be treated as points on a Lie group or, more generally on a Riemannian manifold. In this work, we will examine issues related to the Nadaraya-Watson regression estimator—as defined in Davis et al(2010)—in the context where the response variable is a random variable taking values ​​in a Riemannian manifold, while the explanatory variables (the predictors) are Euclidean random variables. We will discuss the asymptotic bias and variance, as well as the mean squared error, of the intrinsic Nadaraya-Watson regression estimator in the Riemannian manifold ramework. We present a study using simulated data, covering both classical models—such as linear regression or models on SE(3)—and models for which an explicit expression for the exponential map is unknown.