scholarly journals Detection of Noise Signal in the Additive Mixture Based on the Second-Order Cumulant

2021 ◽  
Vol 4 (2) ◽  
Author(s):  
Anton Iehorovych Bereznytskyi
Keyword(s):  
Sample Size ◽  
Relative Error ◽  
Acoustic Signal ◽  
Second Order ◽  
Independent Random Variables ◽  
Threshold Values ◽  
Noise Interference ◽  
Cumulant Method ◽  
Correct Determination ◽  
Noise Diagnostics

In order to determine the technical condition of energetic objects with the objective of ensuring their operational reliability, durability and safety, systems of noise diagnostics, which are based on the analysis of acoustic diagnostic signals. A promising area of noise diagnostics are cumulant methods, based on cumulant analysis, which involves the use of cumulants and cumulant coefficients. In known literature no characteristics of detection of a signal within an interference-containing additive mixture with the use of a second-order cumulant (variance) can be found. That is why the objective of the paper is to study the use of cumulant method on the basis of point estimations of variance for a sample of momentary values for detection of an acoustic signal against the background of noise interference. The research was carried out by way of modeling the additive mixture of signal and interference using the MATLAB® software package. Interference is a model of a noise acoustic signal, which accompanies the operation of properly functional equipment. Signal is a model of an acoustic signal which is created with the occurrence of a malfunction. Signal and interference are independent random variables, so the property of additivity of cumulants was used – the variance of a mixture equals the sum of variances of signal and interference. The decision about the presence of a signal was made on the basis of testing two statistical hypotheses. The null hypothesis – the signal is absent, variance equals to the variance of the interference. The first hypothesis – the signal is present, variance equals to the variance of the mixture. Additional parameters: probability of a Type I error 0,01, probability of correct determination 0,99. The relative error of estimation determined the minimal sample size. These values allowed for the calculation of the threshold value, upon the exceeding of which by the variance estimation, the decision on the presence of signal is made. For each sample, assessments of variance were made. Experimental probability of correct determination is calculated as a total number of decisions taken regarding the presence of a signal, divided by the number of realizations, and corresponds to the value of the specified probability of correct determination. Its relative error was calculated in order to control the validity of the results. Also, kernel density estimation of the probability of the variance assessment for the case of a signal with normal distribution. As shown by the graphs, the assessments have a distribution that is close to normal. The conducted study proves that a variance -based cumulant method allows to detect a signal against the background of noise interference. The necessary sample size, which shows the number of the necessary momentary values, is given in the paper. That is to say that with the help of the frequency of an analogue digital converter the needed duration of the recording of a real for assessment of its variance can be obtained, and the decision on the presence or absence of a signal is to be made on the basis of the specified threshold values. The results of the study can be added to the known sample sizes and threshold values for the coefficients of asymmetry and excess with different distributions. Application of the described method requires additional testing on real acoustic signals and has the areas of use in systems of noise diagnostics.

scholarly journals Sample Sizes Needed for Specified Margins of Relative Error in the Estimates of the Repeatability and Reproducibility Standard Deviations

2005 ◽  
Vol 88 (5) ◽  
pp. 1503-1510 ◽  
Author(s):  
Foster D McClure ◽  
Jung K Lee
Keyword(s):  
Standard Deviation ◽  
Sample Size ◽  
Relative Error ◽  
Analytical Method ◽  
Actual Error ◽  
Repeatability Standard Deviation ◽  
Repeatability And Reproducibility ◽  
Standard Deviations ◽  
True Values ◽  
Addition Formulas

Abstract Sample size formulas are developed to estimate the repeatability and reproducibility standard deviations (sr and sR) such that the actual error in (sr and sR) relative to their respective true values, σr and σR, are at predefined levels. The statistical consequences associated with AOAC INTERNATIONAL required sample size to validate an analytical method are discussed. In addition, formulas to estimate the uncertainties of (sr and sR) were derived and are provided as supporting documentation.Formula for the Number of Replicates Required for a Specified Margin of Relative Error in the Estimate of the Repeatability Standard Deviation

Characterization of a class of second-order density functions

1967 ◽  
Vol 4 (1) ◽  
pp. 123-129 ◽  
Author(s):  
C. B. Mehr
Keyword(s):  
Exponential Distribution ◽  
Gamma Distribution ◽  
Gaussian Distribution ◽  
Random Variables ◽  
Second Order ◽  
Independent Random Variables ◽  
Density Functions ◽  
Linear Filtering ◽  
Non Linear

Distributions of some random variables have been characterized by independence of certain functions of these random variables. For example, let X and Y be two independent and identically distributed random variables having the gamma distribution. Laha showed that U = X + Y and V = X | Y are also independent random variables. Lukacs showed that U and V are independently distributed if, and only if, X and Y have the gamma distribution. Ferguson characterized the exponential distribution in terms of the independence of X – Y and min (X, Y). The best-known of these characterizations is that first proved by Kac which states that if random variables X and Y are independent, then X + Y and X – Y are independent if, and only if, X and Y are jointly Gaussian with the same variance. In this paper, Kac's hypotheses have been somewhat modified. In so doing, we obtain a larger class of distributions which we shall call class λ1. A subclass λ0 of λ1 enjoys many nice properties of the Gaussian distribution, in particular, in non-linear filtering.

scholarly journals Second Order Expansions for High-Dimension Low-Sample-Size Data Statistics in Random Setting

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1151
Author(s):  
Gerd Christoph ◽  
Vladimir V. Ulyanov
Keyword(s):  
Sample Size ◽  
High Dimension ◽  
Random Variable ◽  
Second Order ◽  
Multivariate Normal ◽  
Edgeworth Expansions ◽  
Sample Correlation ◽  
Data Statistics ◽  
Student’S T ◽  
Size Data

We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error bounds. Depending on the type of normalization, we get three different limit distributions: Normal, Student’s t-, or Laplace distributions. The paper continues studies of the authors on approximation of statistics for random size samples.

IMPRECISE SECOND-ORDER MODEL FOR A SYSTEM OF INDEPENDENT RANDOM VARIABLES

2005 ◽  
Vol 13 (02) ◽  
pp. 177-193 ◽  
Author(s):  
LEV V. UTKIN
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Second Order ◽  
Independent Random Variables ◽  
Order Model ◽  
First Order ◽  
Probability Bounds ◽  
Uncertainty Model ◽  
Lower And Upper Probabilities ◽  
Interval Valued

A new hierarchical uncertainty model for combining different evidence about a system of statistically independent random variable is studied in the paper. It is assumed that the first-order level of the model is represented by sets of lower and upper previsions (expectations) of random variables and the second-order level is represented by sets of lower and upper probabilities which can be viewed as confidence weights for interval-valued expectations of the first-order level. The model is rather general and allows us to compute probability bounds and "average" bounds for previsions of a function of random variables. A numerical example illustrates this model.

scholarly journals Higher-Order Approximate Periodic Solutions of a Nonlinear Oscillator with Discontinuity by Variational Approach

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
M. Orhan Kaya ◽  
S. Altay Demirbağ ◽  
F. Özen Zengin
Keyword(s):  
Approximate Solution ◽  
Periodic Solutions ◽  
Relative Error ◽  
Variational Approach ◽  
Nonlinear Oscillator ◽  
Higher Order ◽  
Second Order ◽  
Elastic Force ◽  
Force Term ◽  
The Third

He's variational approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). Three levels of approximation have been used. We obtained 1.6% relative error for the first approximate period, 0.3% relative error for the second-order approximate period. The third approximate solution has the accuracy as high as 0.1%.

scholarly journals The Law of the Iterated Logarithm for Linear Processes Generated by a Sequence of Stationary Independent Random Variables under the Sub-Linear Expectation

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1313
Author(s):  
Wei Liu ◽  
Yong Zhang
Keyword(s):  
Independent Random Variable ◽  
Random Variables ◽  
Random Variable ◽  
Second Order ◽  
Law Of Iterated Logarithm ◽  
Independent Random Variables ◽  
Linear Processes ◽  
Iterated Logarithm ◽  
The Law

In this paper, we obtain the law of iterated logarithm for linear processes in sub-linear expectation space. It is established for strictly stationary independent random variable sequences with finite second-order moments in the sense of non-additive capacity.

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