Mean-squared-displacement statistical test for fractional Brownian motion

Fractional Langevin equation far from equilibrium: Riemann-Liouville fractional Brownian motion, spurious nonergodicity, and aging

We consider the fractional Langevin equation far from equilibrium (FLEFE) to describe stochastic dynamics which do not obey the fluctuation-dissipation theorem, unlike the conventional fractional Langevin equation (FLE). The solution of this equation is Riemann-Liouville fractional Brownian motion (RL-FBM), also known in the literature as FBM II. Spurious nonergodicity, stationarity, and aging properties of the solution are explored for all admissible values α>1/2 of the order α of the time-fractional Caputo derivative in the FLEFE. The increments of the process are asymptotically stationary. However when 1/2<α<3/2, the time-averaged mean-squared displacement (TAMSD) does not converge to the mean-squared displacement (MSD). Instead, it converges to the mean-squared increment (MSI) or structure function, leading to the phenomenon of spurious nonergodicity. When α≥3/2, the increments of FLEFE motion are nonergodic, however the higher order increments are asymptotically ergodic. We also discuss the aging effect in the FLEFE by investigating the influence of an aging time t_{a} on the MSD, TAMSD and autocovariance function of the increments. We find that under strong aging conditions the process becomes ergodic, and the increments become stationary in the domain 1/2<α<3/2.

Statistical Methods for Microrheology of Airway Mucus with Extreme Heterogeneity

The mucus lining of the human airway epithelium contains two gel-forming mucins, MUC5B and MUC5AC. During progression of cystic fibrosis (CF), mucus hyper-concentrates as its mucin ratio changes, coinciding with formation of insoluble, dense mucus flakes. We explore rheological heterogeneity of this pathology with reconstituted mucus matching three stages of CF progression and particle-tracking of 200 nm and 1 micron diameter beads. We introduce statistical data analysis methods specific to low signal-to-noise data within flakes. Each bead time series is decomposed into: (i) a fractional Brownian motion (fBm) classifier of the pure time-series signal; (ii) high-frequency static and dynamic noise; and (iii) low-frequency deterministic drift. Subsequent analysis focuses on the denoised fBm classifier ensemble from each mucus sample and bead diameter. Every ensemble fails a homogeneity test, compelling clustering methods to assess levels of heterogeneity. The first binary level detects beads within vs. outside flakes. A second binary level detects within-flake bead signals that can vs. cannot be disentangled from the experimental noise floor. We show all denoised ensembles, within- and outside-flakes, fail a homogeneity test, compelling additional clustering; next, all clusters with sufficient data fail a homogeneity test. These levels of heterogeneity are consistent with outcomes from a stochastic phase-separation process, and dictate applying the generalized Stokes-Einstein relation to each bead per cluster per sample, then frequency-domain averaging to assess rheological heterogeneity. Flakes exhibit a spectrum of gel-like and sol-like domains, outside-flake solutions a spectrum of sol-like domains, painting a rheological signature of the phase-separation process underlying flake-burdened mucus.

Trap-and-Track for Characterizing Surfactants at Interfaces

Understanding the behavior of surfactants at interfaces is crucial for many applications in materials science and chemistry. Optical tweezers combined with trajectory analysis can become a powerful tool for investigating surfactant characteristics. In this study, we perform trap-and-track analysis to compare the behavior of cetyltrimethylammonium bromide (CTAB) and cetyltrimethylammonium chloride (CTAC) at water-glass interfaces. We use optical tweezers to trap a gold nanoparticle and statistically analyze the particle's movement in response to various surfactant concentrations, evidencing the rearrangement of surfactants adsorbed on glass surfaces. Our results show that counterions have a significant effect on surfactant behavior at the interface. The greater binding affinity of bromide ions to CTA+ micelle surfaces reduces the repulsion among surfactant head groups and enhances the mobility of micelles adsorbed on the interface. Our study provides valuable insights into the behavior of surfactants at interfaces and highlights the potential of optical tweezers for surfactant research. The development of this trap-and-track approach can have important implications for various applications, including drug delivery and nanomaterials.

Goodness-of-fit test for stochastic processes using even empirical moments statistic

In this paper, we introduce a novel framework that allows efficient stochastic process discrimination. The underlying test statistic is based on even empirical moments and generalizes the time-averaged mean-squared displacement framework; the test is designed to allow goodness-of-fit statistical testing of processes with stationary increments and a finite-moment distribution. In particular, while our test statistic is based on a simple and intuitive idea, it enables efficient discrimination between finite- and infinite-moment processes even if the underlying laws are relatively close to each other. This claim is illustrated via an extensive simulation study, e.g., where we confront α-stable processes with stability index close to 2 with their standard Gaussian equivalents. For completeness, we also show how to embed our methodology into the real data analysis by studying the real metal price data.

Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. However, recent single-particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that cannot be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process that preserves the properties of the fractional Brownian motion at a single trajectory level; however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical, and statistical analysis of the fractional Brownian motion with the random Hurst exponent. The explicit formulas for probability density function, mean-squared displacement, and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely, two-point, uniform, and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition, which we demonstrate analytically and numerically.

Time-averaged mean squared displacement ratio test for Gaussian processes with unknown diffusion coefficient

The time-averaged mean squared displacement (TAMSD) is one of the most common statistics used for the analysis of anomalous diffusion processes. Anomalous diffusion is manifested by non-linear (mostly power-law) characteristics of the process in contrast to normal diffusion where linear characteristics are expected. One can distinguish between sub- and super-diffusive processes. We consider Gaussian anomalous diffusion models and propose a new approach used for their testing. This approach is based on the TAMSD ratio statistic for different time lags. Similar to the TAMSD, this statistic exhibits a specific behavior in the anomalous diffusion regime. Through its structure, it is independent of the diffusion coefficient, which, in general, does not influence anomalous diffusion behavior. Thus, the TAMSD ratio-based approach does not require preliminary knowledge of the diffusion coefficient's value, in contrast to the TAMSD-approach, where this value is crucial in the testing procedure. Based on the quadratic form representation of the TAMSD ratio, we calculate its main characteristics and propose a step-by-step testing procedure that can be applied for any Gaussian process. For the anomalous diffusion model used here, namely, the fractional Brownian motion, we demonstrate the effectiveness of the proposed methodology. We show that the new approach outperforms the TAMSD-based one, especially for small sample sizes. Finally, the methodology is applied to the real data from the financial market.

Discriminating Gaussian processes via quadratic form statistics

Gaussian processes are powerful tools for modeling and predicting various numerical data. Hence, checking their quality of fit becomes a vital issue. In this article, we introduce a testing methodology for general Gaussian processes based on a quadratic form statistic. We illustrate the methodology on three statistical tests recently introduced in the literature, which are based on the sample autocovariance function, time average mean-squared displacement, and detrended moving average statistics. We compare the usefulness of the tests by taking into consideration three very important Gaussian processes: the fractional Brownian motion, which is self-similar with stationary increments (SSSIs), scaled Brownian motion, which is self-similar with independent increments (SSIIs), and the Ornstein-Uhlenbeck (OU) process, which is stationary. We show that the considered statistics' ability to distinguish between these Gaussian processes is high, and we identify the best performing tests for different scenarios. We also find that there is no omnibus quadratic form test; however, the detrended moving average test seems to be the first choice in distinguishing between same processes with different parameters. We also show that the detrended moving average method outperforms the Cholesky method. Based on the previous findings, we introduce a novel procedure of discriminating between Gaussian SSSI, SSII, and stationary processes. Finally, we illustrate the proposed procedure by applying it to real-world data, namely, the daily EURUSD currency exchange rates, and show that the data can be modeled by the OU process.

Testing of Multifractional Brownian Motion

Fractional Brownian motion (FBM) is a generalization of the classical Brownian motion. Most of its statistical properties are characterized by the self-similarity (Hurst) index 0<H<1. In nature one often observes changes in the dynamics of a system over time. For example, this is true in single-particle tracking experiments where a transient behavior is revealed. The stationarity of increments of FBM restricts substantially its applicability to model such phenomena. Several generalizations of FBM have been proposed in the literature. One of these is called multifractional Brownian motion (MFBM) where the Hurst index becomes a function of time. In this paper, we introduce a rigorous statistical test on MFBM based on its covariance function. We consider three examples of the functions of the Hurst parameter: linear, logistic, and periodic. We study the power of the test for alternatives being MFBMs with different linear, logistic, and periodic Hurst exponent functions by utilizing Monte Carlo simulations. We also analyze mean-squared displacement (MSD) for the three cases of MFBM by comparing the ensemble average MSD and ensemble average time average MSD, which is related to the notion of ergodicity breaking. We believe that the presented results will be helpful in the analysis of various anomalous diffusion phenomena.

Fractional Dynamics Identification via Intelligent Unpacking of the Sample Autocovariance Function by Neural Networks

Many single-particle tracking data related to the motion in crowded environments exhibit anomalous diffusion behavior. This phenomenon can be described by different theoretical models. In this paper, fractional Brownian motion (FBM) was examined as the exemplary Gaussian process with fractional dynamics. The autocovariance function (ACVF) is a function that determines completely the Gaussian process. In the case of experimental data with anomalous dynamics, the main problem is first to recognize the type of anomaly and then to reconstruct properly the physical rules governing such a phenomenon. The challenge is to identify the process from short trajectory inputs. Various approaches to address this problem can be found in the literature, e.g., theoretical properties of the sample ACVF for a given process. This method is effective; however, it does not utilize all of the information contained in the sample ACVF for a given trajectory, i.e., only values of statistics for selected lags are used for identification. An evolution of this approach is proposed in this paper, where the process is determined based on the knowledge extracted from the ACVF. The designed method is intuitive and it uses information directly available in a new fashion. Moreover, the knowledge retrieval from the sample ACVF vector is enhanced with a learning-based scheme operating on the most informative subset of available lags, which is proven to be an effective encoder of the properties inherited in complex data. Finally, the robustness of the proposed algorithm for FBM is demonstrated with the use of Monte Carlo simulations.

Statistical analysis of superstatistical fractional Brownian motion and applications

Recent advances in experimental techniques for complex systems and the corresponding theoretical findings show that in many cases random parametrization of the diffusion coefficients gives adequate descriptions of the observed fractional dynamics. In this paper we introduce two statistical methods which can be effectively applied to analyze and estimate parameters of superstatistical fractional Brownian motion with random scale parameter. The first method is based on the analysis of the increments of the process, the second one takes advantage of the variation of the trajectories of the process. We prove the effectiveness of the methods using simulated data. Also, we apply it to the experimental data describing random motion of individual molecules inside the cell of E.coli. We show that fractional Brownian motion with Weibull-distributed diffusion coefficient gives adequate description of this experimental data.

Cholesterol modulates acetylcholine receptor diffusion by tuning confinement sojourns and nanocluster stability

Translational motion of neurotransmitter receptors is key for determining receptor number at the synapse and hence, synaptic efficacy. We combine live-cell STORM superresolution microscopy of nicotinic acetylcholine receptor (nAChR) with single-particle tracking, mean-squared displacement (MSD), turning angle, ergodicity, and clustering analyses to characterize the lateral motion of individual molecules and their collective behaviour. nAChR diffusion is highly heterogeneous: subdiffusive, Brownian and, less frequently, superdiffusive. At the single-track level, free walks are transiently interrupted by ms-long confinement sojourns occurring in nanodomains of ~36 nm radius. Cholesterol modulates the time and the area spent in confinement. Turning angle analysis reveals anticorrelated steps with time-lag dependence, in good agreement with the permeable fence model. At the ensemble level, nanocluster assembly occurs in second-long bursts separated by periods of cluster disassembly. Thus, millisecond-long confinement sojourns and second-long reversible nanoclustering with similar cholesterol sensitivities affect all trajectories; the proportion of the two regimes determines the resulting macroscopic motional mode and breadth of heterogeneity in the ensemble population.

Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement

The most common way of estimating the anomalous scaling exponent from single-particle trajectories consists of a linear fit of the dependence of the time-averaged mean-square displacement on the lag time at the log-log scale. We investigate the statistical properties of this estimator in the case of fractional Brownian motion (FBM). We determine the mean value, the variance, and the distribution of the estimator. Our theoretical results are confirmed by Monte Carlo simulations. In the limit of long trajectories, the estimator is shown to be asymptotically unbiased, consistent, and with vanishing variance. These properties ensure an accurate estimation of the scaling exponent even from a single (long enough) trajectory. As a consequence, we prove that the usual way to estimate the diffusion exponent of FBM is correct from the statistical point of view. Moreover, the knowledge of the estimator distribution is the first step toward new statistical tests of FBM and toward a more reliable interpretation of the experimental histograms of scaling exponents in microbiology.