# Generalization of Auto-Regressive Hidden Markov Models to Non-Linear Dynamics and Unit Quaternion Observation Space

## TL;DR

This paper extends Auto-Regressive Hidden Markov Models to handle non-linear dynamics and orientation data using unit quaternions, enabling more accurate modeling of complex time series in various fields.

## Contribution

It introduces two novel generalizations of ARHMM: non-linear AR dynamics in Cartesian space and linear quaternion-based orientation dynamics.

## Key findings

- Enhanced modeling of complex dynamics
- Better orientation representation with quaternions
- Applicable to various latent variable models

## Abstract

Latent variable models are widely used to perform unsupervised segmentation of time series in different context such as robotics, speech recognition, and economics. One of the most widely used latent variable model is the Auto-Regressive Hidden Markov Model (ARHMM), which combines a latent mode governed by a Markov chain dynamics with a linear Auto-Regressive dynamics of the observed state. In this work, we propose two generalizations of the ARHMM. First, we propose a more general AR dynamics in Cartesian space, described as a linear combination of non-linear basis functions. Second, we propose a linear dynamics in unit quaternion space, in order to properly describe orientations. These extensions allow to describe more complex dynamics of the observed state. Although this extension is proposed for the ARHMM, it can be easily extended to other latent variable models with AR dynamics in the observed space, such as Auto-Regressive Hidden semi-Markov Models.