  In this paper, we address the problem of adaptive learning for autoregressive
moving average (ARMA) model in the quaternion domain. By transforming the
original learning problem into a full information optimization task without
explicit noise terms, and then solving the optimization problem using the
gradient descent and the Newton analogues, we obtain two online learning
algorithms for the quaternion ARMA. Furthermore, regret bound analysis
accounting for the specific properties of quaternion algebra is presented,
which proves that the performance of the online algorithms asymptotically
approaches that of the best quaternion ARMA model in hindsight.
