arXiv subjects

Damjan Škulj

Publications and source records attributed to Damjan Škulj.

11 recordsLinked to original sources

arXiv2025

A complete characterization of maximal copulas with a given track section

Bivariate copulas with prescribed diagonal section were first studied by Bertino. Their maximality was studied so far only from the point of view of upper bounds which brings quasi-copulas into the picture and limits the resulting set substantially. We propose to study maximality of these families in the order theoretic sense. A copula C with given diagonal section δ is called undominated if there is no copula C' {\neq} C with the same diagonal section δ such that C {\leq} C'. The main contribution of this paper is a new method that provides copulas of the kind. This method generates a much wider class that contains the known upper bounds as a very small subclass. There was a recent call for the study of asymmetry which is addressed by our class better than by the known ones. Corresponding quasi-copulas can be obtained from our copulas via splicing techniques. Most results are given on the level of tracks.

math.ST↗

arXiv2025

Reversible Imprecise Markov Chains

Reversible Markov chains play a central role in stochastic modelling and in algorithms such as Markov chain Monte Carlo (MCMC). Motivated by the fundamental importance of reversibility in classical settings, this paper develops a theoretical framework for reversible imprecise Markov chains. We focus on their structural properties and their representation through joint distribution matrices. Adopting the strong independence interpretation, we reverse every precise chain compatible with a given imprecise Markov chain specification. Since the reversed ensemble generally cannot be encoded by the usual forward model defined by an imprecise initial distribution and a set of transition matrices, we introduce a symmetric representation based on credal sets of two-step joint distribution (or edge measure) matrices. This strictly more expressive framework naturally admits the reversal operation and reduces reversibility to simple matrix symmetry. Moreover, forward and reverse dynamics can be described simultaneously within a single closed convex set, providing a unified structural basis for the analysis of expectations of path-dependent functionals. We illustrate the theory with random walks on graphs and outline methods for computing lower and upper expectations of such functionals.

math.PR↗

arXiv2022

Normal cones corresponding to credal sets of lower probabilities

Credal sets are one of the most important models for describing probabilistic uncertainty. They usually arise as convex sets of probabilistic models compatible with judgments provided in terms of coherent lower previsions or more specific models such as coherent lower probabilities or probability intervals. In finite spaces, credal sets usually take the form of convex polytopes. Many properties of convex polytopes can be derived from their normal cones, which form polyhedral complexes called normal fans. We analyze the properties of normal cones corresponding to credal sets of coherent lower probabilities. For two important classes of coherent lower probabilities, 2-monotone lower probabilities and probability intervals, we provide a detailed description of the normal fan structure. These structures are related to the structure of the extreme points of the credal sets. To arrive at our main results, we provide some general results on triangulated normal fans of convex polyhedra and their adjacency structure.

math.PR↗

arXiv2022

A complete characterization of normal cones and extreme points for $p$-boxes

Probability boxes, also known as $p$-boxes, correspond to sets of probability distributions bounded by a pair of distribution functions. They fall into the class of models known as imprecise probabilities. One of the central questions related to imprecise probabilities are the intervals of values corresponding to expectations of random variables, and especially the interval bounds. In general, those are attained in extremal points of credal sets, which denote convex sets of compatible probabilistic models. The aim of this paper is a characterization and identification of extreme points corresponding to $p$-boxes on finite domains. To accomplish this, we utilize the concept of normal cones. In the settings of imprecise probabilities, those correspond to sets of random variables whose extremal expectations are attained in a common extreme point. Our main results include a characterization all possible normal cones of $p$-boxes, their relation with extreme points, and the identification of adjacency structure on the collection of normal cones, closely related to the adjacency structure in the set of extreme points.

math.PR↗

arXiv2020

Computing bounds for imprecise continuous-time Markov chains using normal cones

The theory of imprecise Markov chains has achieved significant progress in recent years. Its applicability, however, is still very much limited, due in large part to the lack of efficient computational methods for calculating higher-dimensional models. The high computational complexity shows itself especially in the calculation of the imprecise version of the Kolmogorov backward equation. The equation is represented at every point of an interval in the form of a minimization problem, solvable merely with linear programming techniques. Consequently, finding an exact solution on an entire interval is infeasible, whence approximation approaches have been developed. To achieve sufficient accuracy, in general, the linear programming optimization methods need to be used in a large number of time points. The principal goal of this paper is to provide a new, more efficient approach for solving the imprecise Kolmogorov backward equation. It is based on the Lipschitz continuity of the solutions of the equation with respect to time, causing the linear programming problems appearing in proximate points of the time interval to have similar optimal solutions. This property is exploited by utilizing the theory of normal cones of convex sets. The present article is primarily devoted to providing the theoretical basis for the novel technique, yet, the initial testing shows that in most cases it decisively outperforms the existing methods.

math.OC↗

arXiv2020

Some multivariate imprecise shock model copulas

Bivariate imprecise copulas have recently attracted substantial attention. However, the multivariate case seems still to be a "blank slate". It is then natural that this idea be tested first on shock model induced copulas, a family which might be the most useful in various applications. We investigate a model in which some of the shocks are assumed imprecise and develop the corresponding set of copulas. In the Marshall's case we get a coherent set of distributions and a coherent set of copulas, where the bounds are naturally corresponding to each other. The situation with the other two groups of multivariate imprecise shock model induced copulas, i.e., the maxmin and the the reflected maxmin (RMM) copulas, is substantially more involved, but we are still able to produce their properties. These are the main results of the paper that serves as the first step into a theory that should develop in this direction. In addition, we unfold the theory of bivariate imprecise RMM copulas that has not yet been done before.

math.PR↗

arXiv2019

Constructing copulas from shock models with imprecise distributions

The omnipotence of copulas when modeling dependence given marg\-inal distributions in a multivariate stochastic situation is assured by the Sklar's theorem. Montes et al.\ (2015) suggest the notion of what they call an \emph{imprecise copula} that brings some of its power in bivariate case to the imprecise setting. When there is imprecision about the marginals, one can model the available information by means of $p$-boxes, that are pairs of ordered distribution functions. By analogy they introduce pairs of bivariate functions satisfying certain conditions. In this paper we introduce the imprecise versions of some classes of copulas emerging from shock models that are important in applications. The so obtained pairs of functions are not only imprecise copulas but satisfy an even stronger condition. The fact that this condition really is stronger is shown in Omladič and Stopar (2019) thus raising the importance of our results. The main technical difficulty in developing our imprecise copulas lies in introducing an appropriate stochastic order on these bivariate objects.

math.PR↗

arXiv2019

Errors bounds for finite approximations of coherent lower previsions on finite probability spaces

Coherent lower previsions are general probabilistic models allowing incompletely specified probability distributions. However, for complete description of a coherent lower prevision -- even on finite underlying sample spaces -- an infinite number of assessments is needed in general. Therefore, they are often only described approximately by some less general models, such as coherent lower probabilities or in terms of some other finite set of constraints. The magnitude of error induced by the approximations has often been neglected in the literature, despite the fact that it can be significant, with substantial impact on consequent decisions. An apparent reason is that no widely used general method for estimating the error seems to be available at the moment. This paper provides a practically applicable method that allows calculating an upper bound for the maximal error induced by approximating a coherent lower probability with its values on a finite set of gambles. An algorithm is also provided with an estimation of its computational complexity.

math.PR↗

arXiv2016

Perturbation bounds and degree of imprecision for uniquely convergent imprecise Markov chains

The effect of perturbations of parameters for uniquely convergent imprecise Markov chains is studied. We provide the maximal distance between the distributions of original and perturbed chain and maximal degree of imprecision, given the imprecision of the initial distribution. The bounds on the errors and degrees of imprecision are found for the distributions at finite time steps, and for the stationary distributions as well.

math.PR↗

arXiv2015

Random walks on graphs with interval weights and precise marginals

We propose a model of random walks on weighted graphs where the weights are interval valued, and connect it to reversible imprecise Markov chains. While the theory of imprecise Markov chains is now well established, this is a first attempt to model reversible chains. In contrast with the existing theory, the probability models that have to be considered are now non-convex. This presents a difficulty in computational sense, since convexity is critical for the existence of efficient optimization algorithms used in the existing models. The second part of the paper therefore addresses the computational issues of the model. The goal is finding sets of weights which maximize or minimize expectations corresponding to multiple steps transition probabilities. In particular, we present a local optimization algorithm and numerically test its efficiency. We show that its application allows finding close approximations of the globally best solutions in reasonable time.

math.OC↗

arXiv2012

Interval matrix differential equations

The matrix differential equation $x'(t) = Q(t)x(t), x(0) = x_0$ is considered in the case where $Q(t)$ is an unspecified matrix function of time, with the only constraint that $Q(t)\in \mset$ for every $t$, where $\mset$ is a prescribed closed and convex set of matrices. We provide the solution of the generalised equation by defining an exponential of a set of matrices. Although the defintion is not directly applicable to calculate the solutions, we provide an approximate method along with the estimation of maximal possible error. In particular, the method allows estimating continuous time imprecise Markov chains.

math.CA↗