Recognition of stable distribution with Lévy index α close to 2

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.

Application of Machine Learning Tools for Long-Term Diagnostic Feature Data Segmentation

In this paper, a novel method for long-term data segmentation in the context of machine health prognosis is presented. The purpose of the method is to find borders between three data segments. It is assumed that each segment contains the data that represent different statistical properties, that is, a different model. It is proposed to use a moving window approach, statistical parametrization of the data in the window, and simple clustering techniques. Moreover, it is found that features are highly correlated, so principal component analysis is exploited. We find that the probability density function of the first principal component may be sufficient to find borders between classes. We consider two cases of data distributions, Gaussian and α-stable, belonging to the class of non-Gaussian heavy-tailed distributions. It is shown that for random components with Gaussian distribution, the proposed methodology is very effective, while for the non-Gaussian case, both features and the concept of moving window should be re-considered. Finally, the procedure is tested for real data sets. The results provided here may be helpful in understanding some specific cases of machine health prognosis in the presence of non-Gaussian noise. The proposed approach is model free, and thus it is universal. The methodology can be applied for any long-term data where segmentation is crucial for the data processing.

Goodness-of-fit test for $$\alpha$$-stable distribution based on the quantile conditional variance statistics

The class of $$\alpha$$ α -stable distributions is ubiquitous in many areas including signal processing, finance, biology, physics, and condition monitoring. In particular, it allows efficient noise modeling and incorporates distributional properties such as asymmetry and heavy-tails. Despite the popularity of this modeling choice, most statistical goodness-of-fit tests designed for $$\alpha$$ α -stable distributions are based on a generic distance measurement methods. To be efficient, those methods require large sample sizes and often do not efficiently discriminate distributions when the corresponding $$\alpha$$ α -stable parameters are close to each other. In this paper, we propose a novel goodness-of-fit method based on quantile (trimmed) conditional variances that is designed to overcome these deficiencies and outperforms many benchmark testing procedures. The effectiveness of the proposed approach is illustrated using extensive simulation study with focus set on the symmetric case. For completeness, an empirical example linked to plasma physics is provided.

How a Nonequilibrium Bath and a Potential Well Lead to Broken Time-Reversal Symmetry—First-Order Corrections on Fluctuation–Dissipation Relations

The noise that is associated with nonequilibrium processes commonly features more outliers and is therefore often taken to be Lévy noise. For a Langevin particle that is subjected to Lévy noise, the kicksizes are drawn not from a Gaussian distribution, but from an α-stable distribution. For a Gaussian-noise-subjected particle in a potential well, microscopic reversibility applies. However, it appears that the time-reversal-symmetry is broken for a Lévy-noise-subjected particle in a potential well. Major obstacles in the analysis of Langevin equations with Lévy noise are the lack of simple analytic formulae and the infinite variance of the α-stable distribution. We propose a measure for the violation of time-reversal symmetry, and we present a procedure in which this measure is central to a controlled imposing of time-reversal asymmetry. The procedure leads to behavior that mimics much of the effects of Lévy noise. Our imposing of such nonequilibrium leads to concise analytic formulae and does not yield any divergent variances. Most importantly, the theory leads to simple corrections on the Fluctuation–Dissipation Relation.

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.

Tempered linear and non-linear time series models and their application to heavy-tailed solar flare data

In this paper, we introduce two tempered linear and non-linear time series models, namely, an autoregressive tempered fractionally integrated moving average (ARTFIMA) with α-stable noise and ARTFIMA with generalized autoregressive conditional heteroskedasticity (GARCH) noise (ARTFIMA-GARCH). We provide estimation procedures for the processes and explain the connection between ARTFIMA and their tempered continuous-time counterparts. Next, we demonstrate an application of the processes to modeling of heavy-tailed data from solar flare soft x-ray emissions. To this end, we study the solar flare data during a period of solar minimum, which occurred most recently in July, August, and September 2017. We use a two-state hidden Markov model to classify the data into two states (lower and higher activity) and to extract stationary trajectories. We do an end-to-end analysis and modeling of the solar flare data using both ARTFIMA and ARTFIMA-GARCH models and their non-tempered counterparts. We show through visual inspection and statistical tests that the ARTFIMA and ARTFIMA-GARCH models describe the data better than the ARFIMA and ARFIMA-GARCH, especially in the second state, which justifies that tempered processes can serve as the state-of-the-art approach to model signals originating from a power-law source with long memory effects.

Breaking of time-reversal symmetry for a particle in a parabolic potential that is subjected to Lévy noise: Theory and an application to solar flare data

The noise in nonequilibrium systems commonly contains more outliers as compared to equilibrium systems and is often best described with a Lévy distribution. Many systems in which there are fluctuations around a steady-state throughput can be modeled as a Lévy-noise-subjected particle in a parabolic potential. We consider an overdamped particle in a parabolic potential that is subjected to noise. Microscopic reversibility and time-reversal symmetry apply if the particle is subject to Gaussian distributed noise, but are violated if the noise is Lévy. A parameter to detect this violation is formulated. We, furthermore, develop an understanding for how the time-reversal asymmetry depends on the time Δt between the sample points and on the stability index, α, of the Lévy noise. With solar flare data it is shown how the time-reversal asymmetry parameter of a signal can be used to obtain the α of the underlying noise.

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.

Glucocorticoids rapidly inhibit cell migration through a novel, non-transcriptional HDAC6 pathway

Glucocorticoids (GCs) act through the glucocorticoid receptor (GR, also known as NR3C1) to regulate immunity, energy metabolism and tissue repair. Upon ligand binding, activated GR mediates cellular effects by regulating gene expression, but some GR effects can occur rapidly without new transcription. Here, we show that GCs rapidly inhibit cell migration, in response to both GR agonist and antagonist ligand binding. The inhibitory effect on migration is prevented by GR knockdown with siRNA, confirming GR specificity, but not by actinomycin D treatment, suggesting a non-transcriptional mechanism. We identified a rapid onset increase in microtubule polymerisation following GC treatment, identifying cytoskeletal stabilisation as the likely mechanism of action. HDAC6 overexpression, but not knockdown of αTAT1, rescued the GC effect, implicating HDAC6 as the GR effector. Consistent with this hypothesis, ligand-dependent cytoplasmic interaction between GR and HDAC6 was demonstrated by quantitative imaging. Taken together, we propose that activated GR inhibits HDAC6 function, and thereby increases the stability of the microtubule network to reduce cell motility. We therefore report a novel, non-transcriptional mechanism whereby GCs impair cell motility through inhibition of HDAC6 and rapid reorganization of the cell architecture.This article has an associated First Person interview with the first author of the paper.

Boltzmann-distribution-equivalent for Lévy noise and how it leads to thermodynamically consistent epicatalysis

Nonequilibrium systems commonly exhibit Lévy noise. This means that the distribution for the size of the Brownian fluctuations has a "fat" power-law tail. Large Brownian kicks are then more common as compared to the ordinary Gaussian distribution. We consider a two-state system, i.e., two wells and a barrier in between. The barrier is sufficiently high for a barrier crossing to be a rare event. When the noise is Lévy, we do not get a Boltzmann distribution between the two wells. Instead we get a situation where the distribution between the two wells also depends on the height of the barrier that is in between. Ordinarily, a catalyst, by lowering the barrier between two states, speeds up the relaxation to an equilibrium, but does not change the equilibrium distribution. In an environment with Lévy noise, on the other hand, we have the possibility of epicatalysis, i.e., a catalyst effectively altering the distribution between two states through the changing of the barrier height. After deriving formulas to quantitatively describe this effect, we discuss how this idea may apply in nuclear reactors and in the biochemistry of a living cell.

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.

Entanglement Properties and Quantum Phases for a Fermionic Disordered One-Dimensional Wire with Attractive Interactions

A fermionic disordered one-dimensional wire in the presence of attractive interactions is known to have two distinct phases, a localized and superconducting, depending on the strength of interaction and disorder. The localized region may also exhibit a metallic behavior if the system size is shorter than the localization length. Here we show that the superconducting phase has a distribution of the entanglement entropy distinct from the metallic regime. The entanglement entropy distribution is strongly asymmetric with a Lévy α-stable distribution (compared to the Gaussian metallic distribution), as is seen also for the second Rényi entropy distribution. Thus, entanglement properties may reveal properties which cannot be detected by other methods.

Application of spectral decomposition of ²²²Rn activity concentration signal series measured in Niedźwiedzia Cave to identification of mechanisms responsible for different time-period variations

The authors present an application of spectral decomposition of (222)Rn activity concentration signal series as a mathematical tool used for distinguishing processes determining temporal changes of radon concentration in cave air. The authors demonstrate that decomposition of monitored signal such as (222)Rn activity concentration in cave air facilitates characterizing the processes affecting changes in the measured concentration of this gas. Thanks to this, one can better correlate and characterize the influence of various processes on radon behaviour in cave air. Distinguishing and characterising these processes enables the understanding of radon behaviour in cave environment and it may also enable and facilitate using radon as a precursor of geodynamic phenomena in the lithosphere. Thanks to the conducted analyses, the authors confirmed the unquestionable influence of convective air exchange between the cave and the atmosphere on seasonal and short-term (diurnal) changes in (222)Rn activity concentration in cave air. Thanks to the applied methodology of signal analysis and decomposition, the authors also identified a third process affecting (222)Rn activity concentration changes in cave air. This is a deterministic process causing changes in radon concentration, with a distribution different from the Gaussian one. The authors consider these changes to be the effect of turbulent air movements caused by the movement of visitors in caves. This movement is heterogeneous in terms of the number of visitors per group and the number of groups visiting a cave per day and per year. Such a process perfectly elucidates the observed character of the registered changes in (222)Rn activity concentration in one of the decomposed components of the analysed signal. The obtained results encourage further research into precise relationships between the registered (222)Rn activity concentration changes and factors causing them, as well as into using radon as a precursor of geodynamic phenomena in the lithosphere.

Comment on "probability distribution of power fluctuations in turbulence"

Bandi et al. [Phys. Rev. E 79, 016309 (2009)] reported a closed form expression for the probability density function of the product of two correlated normal random variables and proposed an approximation for the distribution of its mean. Here, we question the closed form expression and derive exact expressions for the distribution of the mean. The proposed approximation is shown to be poor.

Stochastic Modeling of Time Series with Application to Local Damage Detection in Rotating Machinery

Raw vibration signals measured on the machine housing in industrial conditions are complex and can be modeled as an additive mixture of several processes (with different statistical properties) related to normal operation of machine, damage related to one (or more) of its part, some noise, etc. In the case of local damage in rotating machines, contribution of informative process related to damage is hidden in the raw signal so its detection is difficult. In this paper we propose to use the statistical modeling of vibration time series to identify these components. Building the model of raw signal may be ineffective. It is proposed to decompose signal into set of narrowband sub-signals using time-frequency representation. Next, it is proposed to model each sub-signal in the given frequency range and classify all signals using their statistical properties. We have used several parameters (called selectors because they will be used for selection of sub-signals from time-frequency map for further processing) for analysis of sub-signals. They have base in statistics theory and can be useful for example in testing of normality of data set (vibration time series from machine in good condition is close to Gaussian, damaged not). Results of such modeling will be used in the sub-signals classification procedure but also in defects detection. We illustrate effectiveness of novel technique using real data from heavy machinery system.

Fractional process as a unified model for subdiffusive dynamics in experimental data

We show how to use a fractional autoregressive integrated moving average (FARIMA) model to a statistical analysis of the subdiffusive dynamics. The discrete time FARIMA(1,d,1) model is applied in this paper to the random motion of an individual fluorescently labeled mRNA molecule inside live E. coli cells in the experiment described in detail by Golding and Cox [Phys. Rev. Lett. 96, 098102 (2006)] as well as to the motion of fluorescently labeled telomeres in the nucleus of live human cells (U2OS cancer) in the experiment performed by Bronstein et al. [Phys. Rev. Lett. 103, 018102 (2009)]. It is found that only the memory parameter d of the FARIMA model completely detects an anomalous dynamics of the experimental data in both cases independently of the observed distribution of random noises.