scholarly journals Measures of Dependence for α-Stable Distributed Processes and Its Application to Diagnostics of Local Damage in Presence of Impulsive Noise

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Grzegorz Żak ◽  
Marek Teuerle ◽  
Agnieszka Wyłomańska ◽  
Radosław Zimroz
Keyword(s):  
Impulsive Noise ◽  
Complex Nature ◽  
Local Damage ◽  
Second Moment ◽  
Measures Of Dependence ◽  
Heavy Tailed Distributions ◽  
Noisy Observation ◽  
Alternative Measures ◽  
Heavy Tailed ◽  
A New Technique

Local damage detection in rotating machinery is simply searching for cyclic impulsive signal in noisy observation. Such raw signal is mixture of various components with specific properties (deterministic, random, cyclic, impulsive, etc.). The problem appears when the investigated process is based on one of the heavy-tailed distributions. In this case the classical measure can not be considered. Therefore, alternative measures of dependence adequate for such processes should be considered. In this paper we examine the structure of dependence of alpha-stable based systems expressed by means of two measures, namely, codifference and covariation. The reason for using alpha-stable distribution is simple and intuitive: signal of interest is impulsive so its distribution is heavy-tailed. The main goal is to introduce a new technique for estimation of covariation. Due to the complex nature of such vibration signals applying novel methods instead of classical ones is recommended. Classical algorithms usually are based on the assumption that theoretical second moment is finite, which is not true in case of the data acquired on the faulty components. Main advantage of our proposed algorithm is independence from second moment assumption.

scholarly journals Alternative Measures of Dependence for Cyclic Behaviour Identification in the Signal with Impulsive Noise—Application to the Local Damage Detection

Electronics ◽  
2021 ◽  
Vol 10 (15) ◽  
pp. 1863
Author(s):  
Justyna Hebda-Sobkowicz ◽  
Jakub Nowicki ◽  
Radosław Zimroz ◽  
Agnieszka Wyłomańska
Keyword(s):  
Damage Detection ◽  
Impulsive Noise ◽  
Complex Case ◽  
Local Damage ◽  
Measures Of Dependence ◽  
Cyclic Behaviour ◽  
Dependence Measure ◽  
Alternative Measures ◽  
Heavy Tailed ◽  
Non Gaussian

The local damage detection procedures in rotating machinery are based on the analysis of the impulsiveness and/or the periodicity of disturbances corresponding to the failure. Recent findings related to non-Gaussian vibration signals showed some drawbacks of the classical methods. If the signal is noisy and it is strongly non-Gaussian (heavy-tailed), searching for impulsive behvaior is pointless as both informative and non-informative components are transients. The classical dependence measure (autocorrelation) is not suitable for non-Gaussian signals. Thus, there is a need for new methods for hidden periodicity detection. In this paper, an attempt will be made to use alternative measures of dependence used in time series analysis that are less known in the condition monitoring (CM) community. They are proposed as alternatives for the classical autocovariance function used in the cyclostationary analysis. The methodology of the auto-similarity map calculation is presented as well as a procedure for a “quality” or “informativeness” assessment of the map is proposed. In the most complex case, the most resistant to heavy-tailed noise turned out the proposed techniques based on Kendall, Spearman and Quadrant autocorrelations. Whereas in the case of the local fault disturbed by the Gaussian noise, the most efficient proved to be a commonly-known approach based on Pearson autocorrelation. The ideas proposed in the paper are supported by simulation signals and real vibrations from heavy-duty machines.

Probability and Random Processes

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Random Processes ◽  
Distribution Functions ◽  
Basic Logic ◽  
Heavy Tailed Distributions ◽  
Random Components ◽  
Classical Distribution ◽  
Heavy Tailed ◽  
Basic Facts ◽  
Stochastic Events ◽  
Stochastic Event

Phenomena, systems, and processes are rarely purely deterministic, but contain stochastic,probabilistic, or random components. For that reason, a probabilistic descriptionof most phenomena is necessary. Probability theory provides us with the tools for thistask. Here, we provide a crash course on the most important notions of probabilityand random processes, such as odds, probability, expectation, variance, and so on. Wedescribe the most elementary stochastic event—the trial—and develop the notion of urnmodels. We discuss basic facts about random variables and the elementary operationsthat can be performed on them. We learn how to compose simple stochastic processesfrom elementary stochastic events, and discuss random processes as temporal sequencesof trials, such as Bernoulli and Markov processes. We touch upon the basic logic ofBayesian reasoning. We discuss a number of classical distribution functions, includingpower laws and other fat- or heavy-tailed distributions.

scholarly journals A Method for Confidence Intervals of High Quantiles

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 70
Author(s):  
Mei Ling Huang ◽  
Xiang Raney-Yan
Keyword(s):  
Confidence Intervals ◽  
Computational Algorithm ◽  
Geometric Mean ◽  
Asymptotic Properties ◽  
Quantile Estimation ◽  
Heavy Tailed Distributions ◽  
High Quantiles ◽  
High Quantile ◽  
Heavy Tailed ◽  
Infinite Moments

The high quantile estimation of heavy tailed distributions has many important applications. There are theoretical difficulties in studying heavy tailed distributions since they often have infinite moments. There are also bias issues with the existing methods of confidence intervals (CIs) of high quantiles. This paper proposes a new estimator for high quantiles based on the geometric mean. The new estimator has good asymptotic properties as well as it provides a computational algorithm for estimating confidence intervals of high quantiles. The new estimator avoids difficulties, improves efficiency and reduces bias. Comparisons of efficiencies and biases of the new estimator relative to existing estimators are studied. The theoretical are confirmed through Monte Carlo simulations. Finally, the applications on two real-world examples are provided.

scholarly journals Optimal Randomness for Stochastic Configuration Network (SCN) with Heavy-Tailed Distributions

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 56
Author(s):  
Haoyu Niu ◽  
Jiamin Wei ◽  
YangQuan Chen
Keyword(s):  
Research Work ◽  
Cauchy Distribution ◽  
Classification Model ◽  
Test Accuracy ◽  
Functional Link ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Improved Performance ◽  
Hidden Layer ◽  
Hidden Nodes

Stochastic Configuration Network (SCN) has a powerful capability for regression and classification analysis. Traditionally, it is quite challenging to correctly determine an appropriate architecture for a neural network so that the trained model can achieve excellent performance for both learning and generalization. Compared with the known randomized learning algorithms for single hidden layer feed-forward neural networks, such as Randomized Radial Basis Function (RBF) Networks and Random Vector Functional-link (RVFL), the SCN randomly assigns the input weights and biases of the hidden nodes in a supervisory mechanism. Since the parameters in the hidden layers are randomly generated in uniform distribution, hypothetically, there is optimal randomness. Heavy-tailed distribution has shown optimal randomness in an unknown environment for finding some targets. Therefore, in this research, the authors used heavy-tailed distributions to randomly initialize weights and biases to see if the new SCN models can achieve better performance than the original SCN. Heavy-tailed distributions, such as Lévy distribution, Cauchy distribution, and Weibull distribution, have been used. Since some mixed distributions show heavy-tailed properties, the mixed Gaussian and Laplace distributions were also studied in this research work. Experimental results showed improved performance for SCN with heavy-tailed distributions. For the regression model, SCN-Lévy, SCN-Mixture, SCN-Cauchy, and SCN-Weibull used less hidden nodes to achieve similar performance with SCN. For the classification model, SCN-Mixture, SCN-Lévy, and SCN-Cauchy have higher test accuracy of 91.5%, 91.7% and 92.4%, respectively. Both are higher than the test accuracy of the original SCN.

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