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Halftone Visibility
Published in Daniel L. Lau, Gonzalo R. Arce, Modern Digital Halftoning, 2018
Daniel L. Lau, Gonzalo R. Arce
Stable distributions describe a rich class of processes that allow heavy tails and skewness in their functions [7]. The class was characterized by Lévy in 1925 [77] and is described by four parameters: an index of stability, α ∈ (0, 2]; a dispersion parameter, γ > 0; a skewness parameter, δ ∈ [-1, 1]; and a location parameter, β ∈ R. The stability parameter α measures the thickness of the tails of the distribution and provides this model with the flexibility needed to characterize a wide range of impulsive processes. The dispersion γ is similar to the variance of the Gaussian distribution. When the skewness parameter is set to δ = 0, the stable distribution is symmetric about the location parameter β. Symmetric stable processes are also referred to as symmetric alpha-stable or simply SαS.
Simulation Tools
Published in Harold Klee, Randal Allen, Simulation of Dynamic Systems with MATLAB® and Simulink®, 2018
The log-stable distribution is frequently used to model investment returns. Returns are expressed in decimal form, where negative returns represent losses and positive returns represent profit. We then normalize the returns by adding one and taking the natural log of the result. Once in this form, the returns conform to a stable distribution. The probability density function (pdf) for a (fat-tail) stable distribution is f(x:α,β,γ,δ)=1γg(x−δγ;α,β)
Correlation and functions of random variables
Published in Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke, Statistics in Engineering, 2019
Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke
The results for the mean and variance of a linear combination of random variables apply for any distributions for which means and variances are defined, but the form of the distribution is not specified. If the random variables have a normal distributions then any linear combination is also normally distributed, and this holds whether or not the variables are correlated. Probability distributions with this property are known as stable distributions, and the only stable distribution with a finite variance is the normal distribution7.
The impact of node arrival process and stochastic edge growth on scale-free distribution in complex networks
Published in International Journal of Modelling and Simulation, 2020
F. Safaei, H. Yeganloo, M. Moudi
It is noteworthy to mention that Cauchy, Levy, and normal distributions belong to a kind of distribution known as stable distribution. Taking the advantage of large probability mass in the tail of density function, stable distributions are counted appropriate for many types of phenomena in engineering, economy, physics, and hydrology [53]. Since there is no any analytical form of the stable distribution for the inverse CDF of random variable except Cauchy distribution, we cannot apply the inverse transfer function to generate random numbers of these kinds of distributions. Regarding this fact, to generate the random numbers of the density function of random stable variables (α is the index of stability, β is the skewness parameter, γ is the scale parameter, and δ is the location parameter), a computational program that has been proposed in [53] is utilized.
Improved Classification Accuracy for Diagnosing the Early Stage of Parkinson’s Disease Using Alpha Stable Distribution
Published in IETE Journal of Research, 2023
The 12 VRI slices are selected and the intensity levels are getting normalized using alpha-stable distribution to achieve uniform intensity outside the striatum (reference region) throughout images in the dataset. The normalization technique is adjusting the parameters in such a way that the reference region of images must have the same α stable distribution. Because, the analysis of the intensity distribution of the reference region is better as it has the uniform distribution in terms of shape and level than in the striatum regions which have worst distribution. The stable distribution is highly influenced by four important parameters: δ, γ, α, and β. Calculation of the parameter is done for all images using the MLE method.