Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories

Abstract

Extensive time-series encoding the position of particles such as viruses, vesicles, or individual proteins are routinely garnered in single-particle tracking experiments or supercomputing studies. They contain vital clues on how viruses spread or drugs may be delivered in biological cells. Similar time-series are being recorded of stock values in financial markets and of climate data. Such time-series are most typically evaluated in terms of time-averaged mean-squared displacements (TAMSDs), which remain random variables for finite measurement times. Their statistical properties are different for different physical stochastic processes, thus allowing us to extract valuable information on the stochastic process itself. To exploit the full potential of the statistical information encoded in measured time-series we here propose an easy-to-implement and computationally inexpensive new methodology, based on deviations of the TAMSD from its ensemble average counterpart. Specifically, we use the upper bound of these deviations for Brownian motion (BM) to check the applicability of this approach to simulated and real data sets. By comparing the probability of deviations for different data sets, we demonstrate how the theoretical bound for BM reveals additional information about observed stochastic processes. We apply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracer beads measured in mucin hydrogels, and of geographic surface temperature anomalies. Our analysis shows how the large-deviation properties can be efficiently used as a simple yet effective routine test to reject the BM hypothesis and unveil relevant information on statistical properties such as ergodicity breaking and short-time correlations.

Objective comparison of methods to decode anomalous diffusion

“…The best accuracy is obtained for CTRW and LW, for which the method of team E is able to identify their markedly different features. However, it shows a higher level of error when discriminating between Gaussian processes, such as FBM and SBM 39 . The worst performance is obtained for ATTM, whose trajectories display a large heterogeneity in diffusion coefficients and lack a characteristic timescale.…”

“…In recent years, advances in fluorescence techniques have greatly increased the availability of high-precision trajectories of single molecules in living systems 35 , producing an increasing drive to develop methods for quantifying anomalous diffusion 16,25,32,[36][37][38][39] . Furthermore, the recent blossoming of machine learning has promoted the accessibility of new powerful tools for data analysis 40 and further widened the palette of available methods 33,[41][42][43] .…”

Characterization of anomalous diffusion classical statistics powered by deep learning (CONDOR)

“…Various methods have been introduced over time to characterize anomalous diffusion in real data. Traditional methods are based on the direct analysis of trajectory statistics [15] via the examination of features such as the MSD [16][17][18][19], the velocity autocorrelation function [20] and the power spectral density [21,22] among others [15,[23][24][25][26][27][28][29][30][31][32][33][34]. More recently, the blossoming of machine learning approaches has expanded the toolbox of available methods for the characterization of anomalous diffusion with algorithms based on random forests [35,36], gradient boosting techniques [35], recurrent neural networks [37] and convolutional neural networks [38,39].…”

Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions

“…We note here that non-Gaussian shapes can be determined not only by the kurtosis. For the analysis of data deviations from a Gaussian can be determined from large-deviation analyses [83] or via the codifference [84].…”