# Mixture of hidden Markov models for accelerometer data

## TL;DR

This paper introduces a finite mixture of hidden Markov models tailored for accelerometer data, enabling characterization of physical activity patterns and subpopulation identification without predefined activity levels.

## Contribution

It presents a novel mixture HMM approach that estimates activity levels from data and accounts for population heterogeneity, with theoretical guarantees on classification accuracy.

## Key findings

- Misclassification probability decreases exponentially with data length.
- Model effectively captures activity transition dynamics.
- Numerical simulations support theoretical results.

## Abstract

Motivated by the analysis of accelerometer data, we introduce a specific finite mixture of hidden Markov models with particular characteristics that adapt well to the specific nature of this type of data. Our model allows for the computation of statistics that characterize the physical activity of a subject (\emph{e.g.}, the mean time spent at different activity levels and the probability of the transition between two activity levels) without specifying the activity levels in advance but by estimating them from the data. In addition, this approach allows the heterogeneity of the population to be taken into account and subpopulations with homogeneous physical activity behavior to be defined. We prove that, under mild assumptions, this model implies that the probability of misclassifying a subject decreases at an exponential decay with the length of its measurement sequence. Model identifiability is also investigated. We also report a comprehensive suite of numerical simulations to support our theoretical findings. Method is motivated by and applied to the PAT study.