In this paper, the statistical dependence between random variables is quantified by mutual information and estimated using a k nearest neighbor based approximation.
In structural reliability the dependence structure between random variables is almost exclusively modeled by Gauss (normal or Gaussian) copula; however, this implicit assumption is typically not corroborated.
Normally, the variation between random variables is estimated by the distance variation of each random variable from their mean in units of standard deviation.
Analyzing and estimating covariances between random variables is an important and interesting problem with manifold applications to Economics, Finance, Actuarial Science, Engineering, Statistics, and other areas (see, e.g., Egozcue et al. [1], Furman and Zitikis [2 5], Zitikis [6], and references therein).
Since the relationship between random variables is not based on the distribution, but rather constructed from mathematical structures between random variables, dependence can be evaluated using non-parametric correlation statistics (Spearman's ρs and Kendall's τ), which are independent on marginal distributions.
Also, the dependencies between different random variables are often vaguely known and, thus, not included in the modeling.
Due to the curse of dimensionality, the underlying dependence relationships between these random variables are difficult to capture.
The stochastic structural responses, which establish the relationship between structural responses and random variables, are achieved using a stochastic multi-scale finite element method, which integrates computational homogenisation with the stochastic finite element method.
The relations between different concepts of measurability for fuzzy random variables are contained in the papers of Colubi et al. [45], Terán Agraz [46], López-Díaz and Ralescu [47].
