Ekstremni volumni kvazi-kopul

Belšak, Matej (author), Zalar, Aljaž (mentor), Vuk, Martin (comentor)

Diplomska naloga obravnava problem iskanja ekstremnih volumnov kvazi-kopul, ki so posplošene oblike kopul in se uporabljajo za modeliranje odvisnosti med večdimenzionalnimi slučajnimi spremenljivkami. Ekstremni volumni kvazi-kopul so kvantitativno merilo, koliko se lahko kvazi-kopule razlikujejo od kopul. Osredotočimo se na iskanje ekstremnih volumnov za višje dimenzije s pristopom linearnega programiranja. Izpeljani linearni program rešujemo z metodo simpleksov in metodo notranje točke v programu Mathematica in Matlab ter primerjamo dobljene rezultate. Izkaže se, da domneva za izračun spodnje meje volumna v članku iz 2023 ne drži. Ekstremni volumni, pridobljeni z metodo simpleksov v Mathematici, so v obliki racionalnih števil in lahko prispevajo k možnosti za določitev točne formule za ekstremne volumne kvazi-kopul.

• Language: Slovenian
• Keywords: kopula, kvazi-kopula, ekstremni volumni, linearno programiranje, metoda simpleksov, metoda notranje točke
• Work type: Bachelor thesis/paper
• Organization: FRI - Faculty of Computer and Information Science
• Year: 2024
• PID: 20.500.12556/RUL-165325
• Publication date in RUL: 02.12.2024

Secondary language

• Language: English
• Title: Extreme volumes of quasi-copulas

Abstract

This thesis addresses the problem of finding the extreme volumes of quasi-copulas, which are generalized forms of copulas used to model dependencies between multidimensional random variables. The extreme volumes of quasi-copulas provide a quantitative measure of how much quasi-copulas can differ from copulas. We focus on finding extreme volumes for higher dimensions using the approach of linear programming. The derived linear program is solved using the simplex method and the interior-point method in Mathematica and Matlab, and the obtained results are compared. It turns out that the conjecture for calculating the lower bound of the volume in the 2023 article does not hold. The extreme volumes, obtained using the simplex method in Mathematica, are in the form of rational numbers and may contribute to the possibility of determining an exact formula for the extreme volumes of quasi-copulas.

• Keywords: copula, quasi-copula, extreme volumes, linear programming, simplex method, interior-point method