On the generalization of AR processes to riemannian manifolds

The autoregressive (AR) process is fundamental to linear signal processing and is commonly used to model the behaviour of an object evolving on Euclidean space. In real life, there are myriad examples of objects evolving not on flat spaces but on curved spaces such as the surface of a sphere. For instance, wind-direction studies in meteorology and the estimation of relative rotations of tectonic plates based on observations on the Earth's surface deal with spherical data, while subspace tracking in signal processing is actually inference on the Grassmann manifold. This paper considers how to extend the AR process to one evolving on a curved space, or in a general, a manifold. Doing so is non-trivial, and in fact, several different extensions are proposed, along with their advantages and disadvantages. Algorithms for estimating the parameters of these generalized AR processes are derived.