Power spectral density of a single Brownian trajectory: what one can and cannot learn from it

Abstract

The power spectral density (PSD) of any time-dependent stochastic process X t is a meaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of X t over an infinitely large observation time T, that is, it is defined as an ensemble-averaged property taken in the limit  ¥ T . A legitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation time T. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is a fluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.

Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems

“…The admissible coefficients f i and θ j are chosen such that the sequence V k is stationary, resembling the physical velocity process (hence the choice of notation 'V') or highly confined motion 8 .…”

“…In most of the literature the model (8) was not meant to explain the behaviour of the system. Rather, the philosophy was concentrated on controlling the data.…”

“…Given the estimates of the f i and θ j the future behaviour can be predicted (in the sense of the least square error or similar), which is a crucial element, for instance, in financial forecasting. Additionally, by inverting relation (8), the Z k can be estimated from the V k , therefore various white noise tests can be used to verify the goodness of fit of the model [31]. 5 Of course, other forms may also emerge from such an approach, for instance, Gamma [34] or stable [28] distributions.…”

“…in time of the mean-squared displacement (MSD) with the diffusion coefficient D, and the Gaussian probability density function (PDF) Alternatively to the MSD (1) Brownian motion is characterised by a 1/f 2 frequency dependence of the associated ensemble and single trajectory power spectra [7,8].…”

Single trajectory characterization via machine learning

“…In this scenario, determining whether a single-molecule trajectory (or a short segment of it) displays normal or anomalous behavior by its tMSD scaling exponent and associating the motion to a specific physical model are elements of paramount importance to gain insight about the biophysical mechanism underlying anomalous diffusion, thus providing a detailed picture of a variety of phenomena. Recent works in this direction have focused on classification schemes based on optimization procedures [8], power spectral density [25], or Bayesian approaches [26,27].…”