Vectors Of Arbitrary Dimensions Research Articles

Scalar measure of the relationship between several random vectors

The problem of estimating the closeness of co-relation (interdependence) between several random vectors of arbitrary dimension is considered. These random vectors can have arbitrary multidimensional continuous distribution laws. Earlier, within the framework of the entropy approach, indicators were obtained for estimating the closeness of the correlation relationship between the components of one random vector and between two random vectors. The goal of the study is to generalize the previously obtained results to the case of several random vectors. The analytical expression for the coefficient of the closeness of co-relation between several random vectors is obtained. This coefficient is expressed through the indices of determination of conditional regressions between the components of random vectors. For the introduced scalar measure of the relationship, a number of particular results were obtained, which turned out to be known correlation coefficients. A rather simple formula expressed through the determinants of each random vector and the determinant of their combination is derived for the case of Gaussian random vectors. The proposed coefficient can be used to study the network structures consisting of many subsystems. In particular, the interpretation of the correlation coefficient between the elements of the network structure and the other elements can be introduced as the correlation coefficient of the system at the vertex. The introduced measure is quite simply interpretable and allows an unambiguous assessing of the closeness of co-relation between several random vectors of arbitrary dimensions and can be used on real data samples. An example of calculating the closeness of the co-relation between three Gaussian random vectors is presented.

General Extremal Dependence Concepts

The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges; recent developments in extremal dependence concepts have drawn a lot of attention in probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in the literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is actually focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements and this not only for historical reasons. Rearrangement functions describe the relationship between random variables in a completely deterministic way and this implies several advantages on the approximation of solutions in a broad class of optimization problems and allows a deeper understanding of dependence itself.