Bayesian inference of L\'evy walks via hidden Markov models

TL;DR

This paper develops a Bayesian inference method using hidden Markov models to analyze Le9vy walks, enabling accurate parameter estimation and model classification even with noisy or limited data.

Contribution

It introduces a novel Bayesian inference framework for Le9vy walks based on hidden Markov models, including a Markovian decomposition scheme for power-law waiting times.

Findings

Successfully extracts power-law exponents from noisy data

Accurately classifies Le9vy walk trajectories

Performs well with moderate noise levels

Abstract

The L\'evy walk is a non-Brownian random walk model that has been found to describe anomalous dynamic phenomena in diverse fields ranging from biology over quantum physics to ecology. Recurrently occurring problems are to examine whether observed data are successfully quantified by a model classified as L\'evy walks or not and extract the best model parameters in accordance with the data. Motivated by such needs, we propose a hidden Markov model for L\'evy walks and computationally realize and test the corresponding Bayesian inference method. We introduce a Markovian decomposition scheme to approximate a renewal process governed by a power-law waiting time distribution. Using this, we construct the likelihood function of L\'evy walks based on a hidden Markov model and the forward algorithm. With the L\'evy walk trajectories simulated at various conditions, we perform the Bayesian inference for parameter estimation and model classification. We show that the power-law exponent of the flight-time distribution can be successfully extracted even at the condition that the mean-squared displacement does not display the expected scaling exponent due to the noise or insufficient trajectory length. It is also demonstrated that the Bayesian method performs remarkably inferring the L\'evy walk trajectories from given unclassified trajectory data set if the noise level is moderate.