Agnieszka Wyłomańska

Recognition of stable distribution with Lévy index α close to 2.

We address the problem of recognizing α-stable Lévy distribution with Lévy index close to 2 from experimental data. We are interested in the case when the sample size of available data is not large, thus the power law asymptotics of the distribution is not clearly detectable, and the shape of the empirical probability density function is close to a Gaussian. We propose a testing procedure combining a simple visual test based on empirical fourth moment with the Anderson-Darling and Jarque-Bera statistical tests and we check the efficiency of the method on simulated data. Furthermore, we apply our method to the analysis of turbulent plasma density and potential fluctuations measured in the stellarator-type fusion device and demonstrate that the phenomenon of the L-H transition from low confinement, L mode, to a high confinement, H mode, which occurs in this device is accompanied by the transition from Lévy to Gaussian fluctuation statistics.

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.

REGIME VARIANCE TESTING — A QUANTILE APPROACH

In this paper, we examine time series that exhibit behavior related to two or more regimes with dierent statistical properties. The motivation of our study are two real data sets from plasma physics with an observable two-regimes structure. In this paper, we develop a procedure to estimate the critical point of the division in a structural change in a time series. Moreover, we propose three tests to recognize such specific behavior. The presented methodology is based on the empirical second moment and its main advantage is the assumption of a lack of distribution. Moreover, the examined statistical properties are expressed in the language of empirical quantiles of the squared data, therefore, the methodology is an extension of the approach known from the literature. Theoretical results are confirmed by simulations and analysis of real data of turbulent laboratory plasma.

Codifference as a practical tool to measure interdependence

Correlation and spectral analysis represent the standard tools to study interdependence in statistical data. However, for the stochastic processes with heavy-tailed distributions such that the variance diverges, these tools are inadequate. The heavy-tailed processes are ubiquitous in nature and finance. We here discuss codifference as a convenient measure to study statistical interdependence, and we aim to give a short introductory review of its properties. By taking different known stochastic processes as generic examples, we present explicit formulas for their codifferences. We show that for the Gaussian processes codifference is equivalent to covariance. For processes with finite variance these two measures behave similarly with time. For the processes with infinite variance the covariance does not exist, however, the codifference is relevant. We demonstrate the practical importance of the codifference by extracting this function from simulated as well as real data taken from turbulent plasma of fusion device and financial market. We conclude that the codifference serves as a convenient practical tool to study interdependence for stochastic processes with both infinite and finite variances as well.

Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes

In the last decade the subordinated processes have become popular and have found many practical applications. Therefore in this paper we examine two processes related to time-changed (subordinated) classical Brownian motion with drift (called arithmetic Brownian motion). The first one, so called normal tempered stable, is related to the tempered stable subordinator, while the second one–to the inverse tempered stable process. We compare the main properties (such as probability density functions, Laplace transforms, ensemble averaged mean squared displacements) of such two subordinated processes and propose the parameters’ estimation procedures. Moreover we calibrate the analyzed systems to real data related to indoor air quality.

Impulsive Noise Cancellation Method for Copper Ore Crusher Vibration Signals Enhancement

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.

Procedures for Decision Thresholds Finding in Maintenance Management of Belt Conveyor System – Statistical Modeling of Diagnostic Data

Belt conveyors are a key component in material transportation system in both opencast lignite mining and underground copper mines in Poland. Regardless of the structure of the mine, the problem of maintenance of belt conveyors is important (from the entire mining process point of view) for many reasons, such as: (a) conveyors are spatially distributed over a large area, (b) they create logically structured form of complex and heavy components, (c) they are operating in harsh mining environmental conditions, (d) failure of any belt conveyor might result in downtime of the entire production line or its major part. The paper discusses the issue of maintenance of gearboxes used in the conveyor drive systems. The authors have developed a CMMS-class system using GIS technology to support management of conveyors’ network. Its fundamental role is to make right decisions for the exchange of components of the drive systems or allow them to continue their work. Such defined problem requires determination of complex decision rules and the definition of appropriate thresholds of diagnostic parameters. The article presents the procedures for determining decision thresholds, based on statistical modeling of diagnostic data and multidimensional data clustering. By selection of suitable distribution of the data and appropriate statistical parameters, multidimensional data analysis has been performed to determine threshold values for the effective identification of the condition of machines and their components.

Recent Developments in Vibration Based Diagnostics of Gear and Bearings Used in Belt Conveyors

Local damage detection in bearings/gearboxes is one of the most intensively explored problems in condition monitoring literature. Also for mechanical systems used in mining industry this issue might be critical due to short time local overloading of surfaces in contact in gear-pair or bearings that often happens during operation. In general, the problem of local damage detection is well defined in literature, however, specific factors related to the mining industry, require adaptation of existing methods or even developing new approaches. In the paper, some of the most promising techniques with mining machinery context are briefly re-called. The key problems identified for mining machines are: operation under time-varying load/speed conditions, presence of time varying signal to noise ratio and non-Gaussian noise (impulses that appear incidentally, randomly, not with expected cycle or cyclically, however with different cycle related to another damage). All these situations motivated us to find novel solution. The paper might be considered as brief review of recent achievements in the field rather that comprehensive, holistic description of the problem.

Periodic Autoregressive Modeling of Vibration Time Series From Planetary Gearbox Used in Bucket Wheel Excavator

Vibration signals acquired from machines operating under non-stationary operations are difficult to process due to their time varying spectral content, statistical properties, signal to noise ratio etc. In case of damaged machine vibration analysis, the classical damage detection approach might be defined as informative and non-informative contents separation. It can be done in many ways, including model based approaches. One of the most known solutions for constant load/speed operations exploits autoregressive (AR) modeling of the deterministic high energy components that often mask the weak impulsive and stochastic part of the signal. After establishing the model, the residual signal is extracted and further analyzed. In the case presented here, AR modeling is considered inappropriate because of the variation of speed/load conditions. To illustrate importance of the problem and novelty of our approach, a planetary gearbox vibration will be analyzed. The gearbox operates in a bucket wheel excavator (heavy duty mining machine) subjected to cyclic load/speed variation due to the digging/excavating process. Due to periodicity of the excavation process, it seems appropriate to assume a periodic autoregressive (PAR) model for the deterministic high energy components. In the chapter several topics will be discussed: real data inspired motivation for PAR modeling, estimation details, simulations and PAR based inverse filtering for extraction of the informative stochastic part of the signal. Finally, we present some comparison of PAR and AR for modeling the deterministic high energy part.

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.