Maximum likelihood, neural networks and posterior mean for inferring fractional Brownian motion parameters
Abstract
Fractional Brownian motion (fBM) is a model of anomalous diffusion parameterised by a Hurst exponent H ∈ (0,1) and a generalised diffusion coefficient DH > 0. We jointly estimate H and DH using maximum likelihood (MLE) from density measurements with Poisson counting noise and single-particle trajectory with Gaussian localisation error. We show that one density measurement in time cannot infer both H and DH and that several time points are needed, which matters most for inference precision. For trajectories, we show that a neural network trained with a squared error loss approximates the posterior mean of H, which can be computed exactly at negligible cost. Paired with the profile-likelihood estimate of DH, the posterior mean has the lowest root mean squared error in both H and DH, except near H=0 and H=1 where MLE estimates H better. In vesicle trajectories from two cell types, H and DH form a continuum rather than distinct modes of motion. The posterior probability that each window within a track is subdiffusive or superdiffusive shows that 26% of MRC5 fibroblast and 21% of HeLa vesicle trajectories contain both subdiffusive and superdiffusive windows, against 2% and 7% expected from estimation uncertainty alone. Intracellular cargoes therefore switch between subdiffusion and superdiffusion over time rather than forming populations with fixed H.
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BibTeX
@misc{han2026maximum,
title = {{Maximum likelihood, neural networks and posterior mean for inferring fractional Brownian motion parameters}},
author = {Daniel S. Han and Vaishnavi Ananthanarayanan and Matthew J. Simpson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19888},
eprint = {2610.19888},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19888},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Maximum likelihood, neural networks and posterior mean for inferring fractional Brownian motion parameters
AU - Daniel S. Han
AU - Vaishnavi Ananthanarayanan
AU - Matthew J. Simpson
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19888
UR - https://arxiv.org/abs/2610.19888
AB - Fractional Brownian motion (fBM) is a model of anomalous diffusion parameterised by a Hurst exponent H ∈ (0,1) and a generalised diffusion coefficient D_H > 0. We jointly estimate H and D_H using maximum likelihood (MLE) from density measurements with Poisson counting noise and single-particle trajectory with Gaussian localisation error. We show that one density measurement in time cannot infer both H and D_H and that several time points are needed, which matters most for inference precision. For trajectories, we show that a neural network trained with a squared error loss approximates the posterior mean of H, which can be computed exactly at negligible cost. Paired with the profile-likelihood estimate of D_H, the posterior mean has the lowest root mean squared error in both H and D_H, except near H=0 and H=1 where MLE estimates H better. In vesicle trajectories from two cell types, H and D_H form a continuum rather than distinct modes of motion. The posterior probability that each window within a track is subdiffusive or superdiffusive shows that 26% of MRC5 fibroblast and 21% of HeLa vesicle trajectories contain both subdiffusive and superdiffusive windows, against 2% and 7% expected from estimation uncertainty alone. Intracellular cargoes therefore switch between subdiffusion and superdiffusion over time rather than forming populations with fixed H.
ER -