In this paper, we deal with a problem of local damage detection in bearings in the presence of a high-energy impulsive noise. Such a problem was identified during diagnostics of bearings in raw materials crusher. Unfortunately, classical approaches cannot be applied due to the impulsive character of the noise. In this paper we propose, a procedure that cancels out impulsive noise rather than extracts signal of interest. The methodology is based on the regime switching model with two regimes: first corresponding to high-energy noncyclic impulses and second to the rest of the signal. We apply the proposed technique to a simulated signal as well as to the real one. Effectiveness of the method is presented graphically using time series, time-frequency spectrogram, and classical envelope analysis. The obtained results indicate efficiency of the method in impulsive noise cancellation and improve the ability to detect a damage.

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.

The vibration signals acquired on machines usually have complex spectral structure. As the signal of interest (SOI) is weak (especially at an early stage of damage) and covers some frequency range (around structural resonance), it requires its extraction from a raw observation. Until now, most of the techniques assumed the presence of Gaussian noise. Unfortunately, there are cases when the non-informative part of the signal (considered as the noise) is non-Gaussian due to the random disturbances or nature of the process executed by the machine. Thus, the problem can be formulated as the extraction of the SOI from the non-Gaussian noise. Recently this problem has been recognized by several authors and some new ideas have been developed. In this paper, we would like to compare these techniques for benchmark signals (Gaussian noise, cyclic impulsive signals, non-cyclic impulsive signals with random amplitudes and locations of impulses and a mixture of all of them). Our analysis will cover spectral kurtosis, kurtogram, stability index (Alpha selector), conditional variance-based selector, spectral Gini index, spectral smoothness index and infogram. Finally, a discussion on the efficiency of each method is provided.

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.

We propose a generalization of the widely used fractional Brownian motion (FBM), memory-multi-FBM (MMFBM), to describe viscoelastic or persistent anomalous diffusion with time-dependent memory exponent $\alpha \left(t\right)$ in a changing environment. In MMFBM the built-in, long-range memory is continuously modulated by $\alpha \left(t\right)$. We derive the essential statistical properties of MMFBM such as its response function, mean-squared displacement (MSD), autocovariance function, and Gaussian distribution. In contrast to existing forms of FBM with time-varying memory exponents but a reset memory structure, the instantaneous dynamic of MMFBM is influenced by the process history, e.g., we show that after a steplike change of $\alpha \left(t\right)$ the scaling exponent of the MSD after the $\alpha$ step may be determined by the value of $\alpha \left(t\right)$ before the change. MMFBM is a versatile and useful process for correlated physical systems with nonequilibrium initial conditions in a changing environment.