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 timeseries are most typically evaluated in terms of time-average mean-squared displacements, 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 time-averaged mean-squared displacement from its ensemble average counterpart. Specifically, we use the upper bound of these deviations for Brownian motion 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 Brownian motion 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 Brownian motion hypothesis and unveil crucial information on statistical properties such as ergodicity breaking and short-time correlations. III. DATA SETS FOR LARGE-DEVIATION ANALYSISWe here describe the data used in our analysis below. These contain both BM and non-Brownian processes.

Empirical anomaly measure for finite-variance processes

“…The length of the trajectory is N = 300. The similar trajectory lengths we observe in real-life data, see for instance [19]. As one can see the values of the statistic for the simulated trajectory are less than zero for all considered τ s. Moreover, we have also zoomed the beginning and the end of the plot to see the specific behavior of the EAM statistic for small and large values of the arguments.…”

“…The most popular is the time-changed BM driven by the so-called inverse to the strictly increasing Lévy stable subordinator [14][15][16][17] for which the PDF is described by the fractional diffusion equation [8]. The family of anomalous diffusion models contains also the processes with time-or position-dependent diffusion coefficients such as scaled BM [18,19] or heterogeneous diffusion models [20]. It is worth to mention also the superstatistical process where the diffusion coefficient is a random variable [21] or diffusive diffusivity model, called also Brownian yet non-Gaussian diffusion process, where the diffusion coefficient is described by another process (like the Ornstein-Uhlenbeck one) [22].…”

Unravelling the origins of anomalous diffusion: from molecules to migrating storks

“…Indeed, this characterization is a major challenge in various fields including single particle tracking and movement ecology (17)(18)(19), and much effort is made to develop techniques to tackle it; see, e.g., Refs. (20)(21)(22)(23).…”

Moses, Noah and Joseph Effects in Coupled Lévy Processes

“…These days, there are many studies which use techniques such as machine learning [48,49], Bayesian statistics [50] and more, e.g. [51], to try to distinguish between various known models such as continuous-time random walk, fractional Brownian motion and others, which lead to anomalous scaling of the MSD, based only on data obtained from single trajectories generated in a simulation. The characterization of anomalous diffusion using three additional exponents M, L and J, in addition to the Hurst, and the various time-series-analysis based methods to obtain them that we presented in this paper, may add additional tools to be used for such purpose.…”