Modular Analysis of Complex Systems with Numerically Described Multidimensional Probability Distributions

2017 ◽  
pp. 171-176
Author(s):  
J. Stefan Bald
Keyword(s):  
Complex Systems ◽  
Probability Distributions ◽  
Modular Analysis ◽  
Multidimensional Probability

scholarly journals SOLVING NOT COMPLETELY INTEGRABLE QUANTILE PFAFFIAN DIFFERENTIAL EQUATIONS

2017 ◽  
Vol 18 (3.1) ◽  
pp. 20-39
Author(s):  
L.E. Melkumova ◽  
S. Ya. Shatskikh
Keyword(s):  
Differential Equations ◽  
Integral Manifold ◽  
Probability Distributions ◽  
Maximum Dimension ◽  
Conditional Quantile ◽  
Completely Integrable ◽  
Conditional Quantiles ◽  
Integral Manifolds ◽  
Pfaffian Equations ◽  
Multidimensional Probability

The present work deals with quantile Pfaffian differential equations which are constructed using two-dimensional conditional quantiles of multidimensional probability distributions. As it was shown in [3] in case when the initial probability distributions have reproducible conditional quantiles this kind of Pfaan equations is completely integrable and the integral manifold is the conditional quantile of maximum dimension. In this paper we discuss properties of integral manifolds of maximum possible dimension for quantile Pfaffian equations which are not completely integrable. Manifolds of this type are described in terms of conditional quantiles of intermediate dimensions.

Star-shaped order for distributions characterized by several parameters and some applications

2021 ◽  
pp. 1-11
Author(s):  
Idir Arab ◽  
Milto Hadjikyriakou ◽  
Paulo Eduardo Oliveira ◽  
Beatriz Santos
Keyword(s):  
Complex Systems ◽  
Probability Distributions ◽  
Real Parameter ◽  
Applied Probability ◽  
Aging Properties ◽  
Multidimensional Parameter ◽  
Complex Models

Abstract The star-shaped ordering between probability distributions is a common way to express aging properties. A well-known criterion was proposed by Saunders and Moran [(1978). On the quantiles of the gamma and F distributions. Journal of Applied Probability 15(2): 426–432], to order families of distributions depending on one real parameter. However, the lifetime of complex systems usually depends on several parameters, especially when considering heterogeneous components. We extend the Saunders and Moran criterion characterizing the star-shaped order when the multidimensional parameter moves along a given direction. A few applications to the lifetime of complex models, namely parallel and series models assuming different individual components behavior, are discussed.

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