# American Institute of Mathematical Sciences

September  2017, 9(3): 391-410. doi: 10.3934/jgm.2017015

## Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry

 1 University of Copenhagen Universitetsparken 5, DK-2100 Copenhagen E, Denmark 2 Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark

Received  January 2016 Revised  June 2016 Published  June 2017

We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed in the frame bundle lead to a family of probability distributions on the manifold. We explain how data mean and covariance can be interpreted as points in the frame bundle or, more precisely, in the bundle of symmetric positive definite 2-tensors analogously to the parameters describing Euclidean normal distributions. We discuss a factorization of the frame bundle projection map through this bundle, the natural sub-Riemannian structure of the frame bundle, the effect of holonomy, and the existence of subbundles where the Hörmander condition is satisfied such that the Brownian motions have smooth transition densities. We identify the most probable paths for the underlying Euclidean Brownian motion and discuss small time asymptotics of the transition densities on the manifold. The geometric setup yields an intrinsic approach to the estimation of mean and covariance in non-linear spaces.

Citation: Stefan Sommer, Anne Marie Svane. Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 391-410. doi: 10.3934/jgm.2017015
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Sampled data on an ellipsoid realized as endpoints of sample paths of the process $X_t$ (black lines and points). Mean $x_0=\pi(u_0)$ (green point) and covariance $\sigma^{-1}$ (ellipsis over mean) are estimated by minimizing (15). The most probable paths for the driving process connects $x_0$ and the sample paths (gray lines) and minimize the distances $d_{{\text{Sym}} ^+ M}\left(\sigma, q^{-1}(x_i)\right)$.
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