The problem of the Dedekind-MacNeille completion of the class of n–copulas originated in 2005 and is well-known to all the experts in the field. Unlike in the bivariate case, where the solution is ...just the class of n-quasi-copulas, in case that n>2 this class is simply too big. We solve the problem by identifying an appropriate concrete subclass together with concrete meet and join operations so that the requirements for the completion are satisfied. Of course, the construction is made so that the induced order coincides with the starting pointwise order on n-quasi-copulas. Additionally, we want to enhance and promote the technique used to solve this problem obtained by upgrading a method developed by the authors of this paper to solve some other questions. We call the version of the method we present here the Al_gebraic Obstacles in the Ge_ometry of N_egative Volumes (ALGEN for short).
The geometrical interpretation of symmetry of a function on the unit square is that it takes the same value in mirror images w.r.t. the main diagonal. Here, this concept is generalized by considering ...the reflection w.r.t. a curve representing the graph of an automorphism of the unit interval. A function on the unit square that takes the same value in mirror images w.r.t. such a curve is called symmetric w.r.t. that curve. Moreover, a measure is proposed for quantifying to what extent a (quasi-)copula can be regarded as being asymmetric w.r.t. a given curve. The major part of the paper is concerned with establishing lower and upper bounds on this degree of asymmetry. Finally, it is shown that these bounds are sharp within the class of copulas.
Zero-sets of copulas de Amo, Enrique; Fernández-Sánchez, Juan; Úbeda-Flores, Manuel
Fuzzy sets and systems,
08/2020, Letnik:
393
Journal Article
Recenzirano
We study conditions on sets in order to be zero-sets of semi-copulas, quasi-copulas, and with special attention, of copulas. We find necessary and sufficient conditions for characterizing the ...zero-sets of absolutely continuous copulas and copulas whose support coincides with the closure of the complementary of the zero-set. Moreover, we study several topological properties and the lattice-theoretic structure, and characterize the zero-sets of the class of Archimedean copulas.
In this study, we focus on bivariate transformations of trivariate quasi-copulas. We characterize necessary and sufficient conditions of transformations that can be written in the form of ...compositions between two quasi-copulas and a quadratic polynomial function. The conditions only depend on the coefficients of the quadratic polynomial. The set of these coefficients is convex with linear-section boundary and it lies on the seven-dimensional Euclidean space. All extreme points of this set have been characterized via CAS and can be used to construct quasi-copulas. Construction examples are also given.
We study the relationship between the poset of supermodular n-quasi-copulas and the posets of n-quasi-copulas and n-copulas. It is known that, for n⩾3, the poset of n-quasi-copulas is not ...order-isomorphic to the Dedekind–MacNeille completion of the poset of n-copulas. We complement the latter result by showing that, for n⩾3, the structure of the poset of n-quasi-copulas is more closely related to that of the poset of supermodular n-quasi-copulas than that of the poset of n-copulas.
In this paper we study the smallest and the greatest M-Lipschitz continuous n-ary aggregation functions with a given diagonal section. We show that several properties that were studied for the ...smallest and the greatest 1-Lipschitz continuous binary aggregation functions with a given diagonal section extend naturally to higher dimensions while considering different Lipschitz constants. Just as in the binary case, we show that the smallest n-quasi-copula with a given diagonal section coincides with the smallest 1-Lipschitz n-ary aggregation function with that diagonal section. Additionally, we show that the smallest n-quasi-copula with a given diagonal section, called the Bertino n-quasi-copula, is supermodular for any n⩾2.
•We study the class of supermodular n-quasi-copulas.•We show that some properties of 2-copulas extend to supermodular n-quasi-copulas.•We introduce the class of k-dimensionally-increasing ...n-quasi-copulas.•We solve an open problem regarding a characterization of n-quasi-copulas.
We show that as the dimensionality increases, more and more interesting classes of operations can be identified between the class of n-quasi-copulas and the class of n-copulas. One such class is the class of supermodular n-quasi-copulas. We observe that some properties of 2-copulas that cannot be generalized to higher-dimensional copulas, hold true for supermodular n-quasi-copulas. Additionally, we show that trying to generalize a volume-based characterization of bivariate copulas to higher dimensions, results in a restrictive class of n-quasi-copulas.
Copulas are functions that link an
n
-dimensional distribution function with its one-dimensional margins. In this contribution we show how
n
-variate copulas with given values at two arbitrary points ...can be constructed. Thereby, we also answer a so far open question whether lower and upper bounds for
n
-variate copulas with given value at a single arbitrary point are achieved. We also introduce and discuss the concept of an
F
-copula which is needed for proving our results.
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