# Online Learning Algorithms for Quaternion ARMA Model

## TL;DR

This paper introduces online learning algorithms for quaternion ARMA models, utilizing gradient and Newton methods, with regret bounds showing asymptotic optimality in adaptive signal processing.

## Contribution

It presents novel online algorithms for quaternion ARMA models and provides regret analysis tailored to quaternion algebra properties.

## Key findings

- Algorithms achieve asymptotic optimality
- Regret bounds are established for the algorithms
- Effective adaptive learning in quaternion signal processing

## Abstract

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.