Distinguishing between fractional Brownian motion with random and constant Hurst exponent using sample autocovariance-based statistics.

Fractional Brownian motion (FBM) is a canonical model for describing dynamics in various complex systems. It is characterized by the Hurst exponent, which is responsible for the correlation between FBM increments, its self-similarity property, and anomalous diffusion behavior. However, recent research indicates that the classical model may be insufficient in describing experimental observations when the anomalous diffusion exponent varies from trajectory to trajectory. As a result, modifications of the classical FBM have been considered in the literature, with a natural extension being the FBM with a random Hurst exponent. In this paper, we discuss the problem of distinguishing between two models: (i) FBM with the constant Hurst exponent and (ii) FBM with random Hurst exponent, by analyzing the probabilistic properties of statistics represented by the quadratic forms. These statistics have recently found application in Gaussian processes and have proven to serve as efficient tools for hypothesis testing. Here, we examine two statistics-the sample autocovariance function and the empirical anomaly measure-utilizing the correlation properties of the considered models. Based on these statistics, we introduce a testing procedure to differentiate between the two models. We present analytical and simulation results considering the two-point and beta distributions as exemplary distributions of the random Hurst exponent. Finally, to demonstrate the utility of the presented methodology, we analyze real-world datasets from the financial market and single particle tracking experiment in biological gels.

Erratum: "Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions" [Chaos 32, 093114 (2022)].

We point out a minor mistake in Fig. 10 in the published version of our paper [M. Balcerek et al., Chaos 32, 093114 (2022)]. The conclusions drawn from the illustration remain the same.

Testing of two-dimensional Gaussian processes by sample cross-covariance function.

In this paper, we address the issue of testing two-dimensional Gaussian processes with a defined cross-dependency structure. Multivariate Gaussian processes are widely used in various applications; therefore, it is essential to identify the theoretical model that accurately describes the data. While it is relatively straightforward to do so in a one-dimensional case, analyzing multi-dimensional vectors requires considering the dependency between the components, which can significantly affect the efficiency of statistical methods. The testing methodology presented in this paper is based on the sample cross-covariance function and can be considered a natural generalization of the approach recently proposed for testing one-dimensional Gaussian processes based on the sample autocovariance function. We verify the efficiency of this procedure on three classes of two-dimensional Gaussian processes: Brownian motion, fractional Brownian motion, and two-dimensional autoregressive discrete-time process. The simulation results clearly demonstrate the effectiveness of the testing methodology, even for small sample sizes. The theoretical and simulation results are supported by analyzing two-dimensional real-time series that describe the main risk factors of a mining company, namely, copper price and exchange rates (USDPLN). We believe that the introduced methodology is intuitive and relatively simple to implement, and thus, it can be applied in many real-world scenarios where multi-dimensional data are examined.

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.

Multifractional Brownian motion characterization based on Hurst exponent estimation and statistical learning.

This paper proposes an approach for the estimation of a time-varying Hurst exponent to allow accurate identification of multifractional Brownian motion (MFBM). The contribution provides a prescription for how to deal with the MFBM measurement data to solve regression and classification problems. Theoretical studies are supplemented with computer simulations and real-world examples. Those prove that the procedure proposed in this paper outperforms the best-in-class algorithm.

Divergence-Based Segmentation Algorithm for Heavy-Tailed Acoustic Signals with Time-Varying Characteristics.

Many real-world systems change their parameters during the operation. Thus, before the analysis of the data, there is a need to divide the raw signal into parts that can be considered as homogeneous segments. In this paper, we propose a segmentation procedure that can be applied for the signal with time-varying characteristics. Moreover, we assume that the examined signal exhibits impulsive behavior, thus it corresponds to the so-called heavy-tailed class of distributions. Due to the specific behavior of the data, classical algorithms known from the literature cannot be used directly in the segmentation procedure. In the considered case, the transition between parts corresponding to homogeneous segments is smooth and non-linear. This causes that the segmentation algorithm is more complex than in the classical case. We propose to apply the divergence measures that are based on the distance between the probability density functions for the two examined distributions. The novel segmentation algorithm is applied to real acoustic signals acquired during coffee grinding. Justification of the methodology has been performed experimentally and using Monte-Carlo simulations for data from the model with heavy-tailed distribution (here the stable distribution) with time-varying parameters. Although the methodology is demonstrated for a specific case, it can be extended to any process with time-changing characteristics.

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.

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.

Local Defect Detection in Bearings in the Presence of Heavy-Tailed Noise and Spectral Overlapping of Informative and Non-Informative Impulses.

The problem of the informative frequency band (IFB) selection for local fault detection is considered in the paper. There are various approaches that are very effective in this issue. Most of the techniques are vibration-based and they are related to the cyclic impulses detection (associated with the local fault) in the background noise. However, when the background noise in the vibration signal has non-Gaussian impulsive behavior, the classical methods seem to be insufficient. Recently, new techniques were proposed by several authors and interesting approaches were tested for different non-Gaussian signals. We demonstrate the comparative analysis related to the results for three selected techniques proposed in recent years, namely the Alpha selector, Conditional Variance-based selector, and Spearman selector. The techniques seem to be effective for the IFB selection for the non-Gaussian distributed vibration signals. The main purpose of this article is to investigate how spectral overlapping of informative and non-informative impulsive components will affect diagnostic procedures. According to our knowledge, this problem was not considered in the literature for the non-Gaussian signals. Nevertheless, as we demonstrated by the simulations, the level of overlapping and the location of a center frequency of the mentioned frequency bands have a significant influence on the behavior of the considered selectors. The discussion about the effectiveness of each analyzed method is conducted. The considered problem is supported by real-world examples.

Identification, Decomposition and Segmentation of Impulsive Vibration Signals with Deterministic Components-A Sieving Screen Case Study.

Condition monitoring is a well-established field of research; however, for industrial applications, one may find some challenges. They are mostly related to complex design, a specific process performed by the machine, time-varying load/speed conditions, and the presence of non-Gaussian noise. A procedure for vibration analysis from the sieving screen used in the raw material industry is proposed in the paper. It is more for pre-processing than the damage detection procedure. The idea presented here is related to identification and extraction of two main types of components: (i) deterministic (D)-related to the unbalanced shaft(s) and (ii) high amplitude, impulsive component randomly (R) appeared in the vibration due to pieces of ore falling down of moving along the deck. If we could identify these components, then we will be able to perform classical diagnostic procedures for local damage detection in rolling element bearing. As deterministic component may be AM/FM modulated and each impulse may appear with different amplitude and damping, there is a need for an automatic procedure. We propose a method for signal processing that covers two main steps: (a) related to R/D decomposition and including signal segmentation to neglect AM/FM modulations, iterative sine wave fitting using the least square method (for each segment), signal filtering technique by subtraction fitted sine from the raw signal, the definition of the criterion to stop iteration by residuals analysis, (b) impulse segmentation and description (beginning, end, max amplitude) that contains: detection of the number of impulses in a decomposed random part of the raw signal, detection of the max value of each impulse, statistical analysis (probability density function) of max value to find regime-switching), modeling of the envelope of each impulse for samples that protrude from the signal, extrapolation (forecasting) envelope shape for samples hidden in the signal. The procedure is explained using simulated and real data. Each step is very easy to implement and interpret thus the method may be used in practice in a commercial system.

Omnibus test for normality based on the Edgeworth expansion.

Statistical inference in the form of hypothesis tests and confidence intervals often assumes that the underlying distribution is normal. Similarly, many signal processing techniques rely on the assumption that a stationary time series is normal. As a result, a number of tests have been proposed in the literature for detecting departures from normality. In this article we develop a novel approach to the problem of testing normality by constructing a statistical test based on the Edgeworth expansion, which approximates a probability distribution in terms of its cumulants. By modifying one term of the expansion, we define a test statistic which includes information on the first four moments. We perform a comparison of the proposed test with existing tests for normality by analyzing different platykurtic and leptokurtic distributions including generalized Gaussian, mixed Gaussian, α-stable and Student's t distributions. We show for some considered sample sizes that the proposed test is superior in terms of power for the platykurtic distributions whereas for the leptokurtic ones it is close to the best tests like those of D'Agostino-Pearson, Jarque-Bera and Shapiro-Wilk. Finally, we study two real data examples which illustrate the efficacy of the proposed test.

Probabilistic properties of detrended fluctuation analysis for Gaussian processes.

Detrended fluctuation analysis (DFA) is one of the most widely used tools for the detection of long-range dependence in time series. Although DFA has found many interesting applications and has been shown to be one of the best performing detrending methods, its probabilistic foundations are still unclear. In this paper, we study probabilistic properties of DFA for Gaussian processes. Our main attention is paid to the distribution of the squared error sum of the detrended process. We use a probabilistic approach to derive general formulas for the expected value and the variance of the squared fluctuation function of DFA for Gaussian processes. We also get analytical results for the expected value of the squared fluctuation function for particular examples of Gaussian processes, such as Gaussian white noise, fractional Gaussian noise, ordinary Brownian motion, and fractional Brownian motion. Our analytical formulas are supported by numerical simulations. The results obtained can serve as a starting point for analyzing the statistical properties of DFA-based estimators for the fluctuation function and long-memory parameter.

Model of the Vibration Signal of the Vibrating Sieving Screen Suspension for Condition Monitoring Purposes.

Diagnostics of industrial machinery is a topic related to the need for damage detection, but it also allows to understand the process itself. Proper knowledge about the operational process of the machine, as well as identification of the underlying components, is critical for its diagnostics. In this paper, we present a model of the signal, which describes vibrations of the sieving screen. This particular type is used in the mining industry for the classification of ore pieces in the material stream by size. The model describes the real vibration signal measured on the spring set being the suspension of this machine. This way, it is expected to help in better understanding how the overall motion of the machine can impact the efforts of diagnostics. The analysis of real vibration signals measured on the screen allowed to identify and parameterize the key signal components, which carry valuable information for the following stages of diagnostic process of that machine. In the proposed model we take into consideration deterministic components related to shaft rotation, stochastic Gaussian component related to external noise, stochastic α-stable component as a model of excitations caused by falling rocks pieces, and identified machine response to unitary excitations.

Identification and Statistical Analysis of Impulse-Like Patterns of Carbon Monoxide Variation in Deep Underground Mines Associated with the Blasting Procedure.

The quality of the air in underground mines is a challenging issue due to many factors, such as technological processes related to the work of miners (blasting, air conditioning, and ventilation), gas release by the rock mass and geometry of mine corridors. However, to allow miners to start their work, it is crucial to determine the quality of the air. One of the most critical parameters of the air quality is the carbon monoxide (CO) concentration. Thus, in this paper, we analyze the time series describing CO concentration. Firstly, the signal segmentation is proposed, then segmented data (daily patterns) is visualized and statistically analyzed. The method for blasting moment localization, with no prior knowledge, has been presented. It has been found that daily patterns differ and CO concentration values reach a safe level at a different time after blasting. The waiting time to achieve the safe level after blasting moment (with a certain probability) has been calculated based on the historical data. The knowledge about the nature of the CO variability and sources of a high CO concentration can be helpful in creating forecasting models, as well as while planning mining activities.

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.

Mean-squared-displacement statistical test for fractional Brownian motion.

Anomalous diffusion in crowded fluids, e.g., in cytoplasm of living cells, is a frequent phenomenon. A common tool by which the anomalous diffusion of a single particle can be classified is the time-averaged mean square displacement (TAMSD). A classical mechanism leading to the anomalous diffusion is the fractional Brownian motion (FBM). A validation of such process for single-particle tracking data is of great interest for experimentalists. In this paper we propose a rigorous statistical test for FBM based on TAMSD. To this end we analyze the distribution of the TAMSD statistic, which is given by the generalized chi-squared distribution. Next, we study the power of the test by means of Monte Carlo simulations. We show that the test is very sensitive for changes of the Hurst parameter. Moreover, it can easily distinguish between two models of subdiffusion: FBM and continuous-time random walk.

Elucidating distinct ion channel populations on the surface of hippocampal neurons via single-particle tracking recurrence analysis.

Protein and lipid nanodomains are prevalent on the surface of mammalian cells. In particular, it has been recently recognized that ion channels assemble into surface nanoclusters in the soma of cultured neurons. However, the interactions of these molecules with surface nanodomains display a considerable degree of heterogeneity. Here, we investigate this heterogeneity and develop statistical tools based on the recurrence of individual trajectories to identify subpopulations within ion channels in the neuronal surface. We specifically study the dynamics of the K^{+} channel Kv1.4 and the Na^{+} channel Nav1.6 on the surface of cultured hippocampal neurons at the single-molecule level. We find that both these molecules are expressed in two different forms with distinct kinetics with regards to surface interactions, emphasizing the complex proteomic landscape of the neuronal surface. Further, the tools presented in this work provide new methods for the analysis of membrane nanodomains, transient confinement, and identification of populations within single-particle trajectories.

Discriminating between Light- and Heavy-Tailed Distributions with Limit Theorem.

In this paper we propose an algorithm to distinguish between light- and heavy-tailed probability laws underlying random datasets. The idea of the algorithm, which is visual and easy to implement, is to check whether the underlying law belongs to the domain of attraction of the Gaussian or non-Gaussian stable distribution by examining its rate of convergence. The method allows to discriminate between stable and various non-stable distributions. The test allows to differentiate between distributions, which appear the same according to standard Kolmogorov-Smirnov test. In particular, it helps to distinguish between stable and Student's t probability laws as well as between the stable and tempered stable, the cases which are considered in the literature as very cumbersome. Finally, we illustrate the procedure on plasma data to identify cases with so-called L-H transition.