Statistical inference for manifolds attracts much attention because of its power of working with more general forms of data or geometric objects. We study regression and stochastic differential equations on manifolds from the intrinsic point of view.

Firstly, we are able to provide alternative parametrizations for data that lie on Lie group in the problem of fitting a regression model, by mapping this space intrinsically onto its Lie algebra, while we explore the behaviour of fitted values when this base point is chosen differently. Due to the nature of our data in the application of soft tissue artefacts, we employ two correlation structures, namely Matern and quasi-periodic correlation functions when using the generalized least squares, and show that some patterns of the residuals are removed.

Secondly, we construct a generalization of the Ornstein-Uhlenbeck process on the cone of covariance matrices SP(n) endowed with two popular Riemannian metrics, namely Log-Euclidean (LE) and Affine-Invariant (AI) metrics. We show that the Riemannian Brownian motion on SP(n) has infinite explosion time as on the Euclidean space and establish the calculation for the horizontal lifts of smooth curves. Moreover, we provide Bayesian inference for discretely observed diffusion processes of covariance matrices associated with either the LE or the AI metrics, and present a novel diffusion bridge sampling method using guided proposals when equipping SP(n) with the AI metric. The estimation algorithms are illustrated with an application in finance, together with a goodness-of-fit test comparing models associated with different metrics. Furthermore, we explore the multivariate volatility models via simulation study, in which covariance matrices in the models are assumed to be unobservable.