Autoregressive Processes on Riemannian Manifolds

Abstract

This paper introduces a Riemannian autoregressive (R-AR) model of order one for manifold-valued time series. The model is specified through an autoregressive process in the tangent space at a reference point, which is mapped to the manifold via the exponential map. It is characterised by two parameters: a reference point $\mu$, representing the intrinsic central tendency through the Fr\'echet mean, and an autoregressive parameter $\phi$, governing the dependence structure and stationarity properties. When these parameters are unknown, their estimation introduces geometric and probabilistic challenges arising from the intrinsic estimation of $\mu$ and from estimating $\phi$ as the base point varies over the manifold and the associated tangent spaces change. The estimation theory is developed using a strong law of large numbers for the sample Fr\'echet mean set of ergodic Markov chains, together with ergodic arguments for the strong consistency of the autoregressive parameter estimator. The framework is validated through numerical simulations in the hyperbolic plane and an application to aerosol size distributions on the Fisher-Rao manifold, demonstrating how the proposed model can characterise mean-reverting dynamics in nonlinear geometries.