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System identification of autoregressive processes on Stiefel and Grassmann manifolds are presented and studied. We define the system parameters as elements in the orthogonal group and we show that the system can be estimated by averaging over observations. Then we propose an algorithm on how to compute these system parameters using conjugate gradient descent on Stiefel and Grassmann manifolds, respectively.

The projected normal distribution, with isotropic variance, on the 2-sphere is considered using intrinsic statistics. It is shown that in this case, the expectation commutes with the projection, and that the covariance of the normal variable has a 1-1 correspondence with the intrinsic covariance of the projected normal distribution. This allows us to estimate, after the model identification, the parameters of the underlying normal distribution that generates the data.

A generalization of the extended Kalman filter for Stiefel manifold-valued measurements is proposed. The underlying mathematics in order to explain this generalization is thouroughly presented. We also give some simulations for the 2-sphere and the space of orthogonal 4-by-2 matrices which show significant improvement of the Extended Kalman Filter compared to only relying on raw measurements.

In this paper we first introduce the setting of filtering on Stiefel manifolds. Then, assuming the underlying system process is constant, the convergence of the extended Kalman filter with Stiefel manifold-valued observations is proved. Finally, some simulations are presented for a select few Stiefel manifolds and the speed of convergence is studied.