arXiv subjects

Meshal Abuqrais

Publications and source records attributed to Meshal Abuqrais.

3 recordsLinked to original sources

arXiv2026

Autoregressive Processes on Riemannian Manifolds

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 $μ$, representing the intrinsic central tendency through the Fréchet mean, and an autoregressive parameter $ϕ$, 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 $μ$ and from estimating $ϕ$ 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 Fréchet 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.

math.ST↗

arXiv2026

Central Limit Theorems for Sample Fréchet Means of Manifold-Valued Markov Chains

In this article, we establish central limit theorems for sample Fréchet means of stationary ergodic Markov chains taking values in manifolds, extending the asymptotic theory previously developed for independent observations to a class of dependent manifold-valued processes. Our results derive the asymptotic normality of the sample Fréchet mean from a central limit condition at the population Fréchet mean, under suitable local regularity conditions. We further provide sufficient geometric and probabilistic conditions under which these assumptions hold, formulated in terms of curvature bounds and a Wasserstein mixing condition. As an application, we establish a central limit theorem for sample Fréchet means for a class of random dynamical systems generated by contractive random maps.

math.PR↗

arXiv2024

A Riemannian covariance for manifold-valued data

The extension of bivariate measures of dependence to non-Euclidean spaces is a challenging problem. The non-linear nature of these spaces makes the generalisation of classical measures of linear dependence (such as the covariance) not trivial. In this paper, we propose a novel approach to measure stochastic dependence between two random variables taking values in a Riemannian manifold, with the aim of both generalising the classical concepts of covariance and correlation and building a connection to Fréchet moments of random variables on manifolds. We introduce generalised local measures of covariance and correlation and we show that the latter is a natural extension of Pearson correlation. We then propose suitable estimators for these quantities and we prove strong consistency results. Finally, we demonstrate their effectiveness through simulated examples and a real-world application.

math.ST↗