Teorija slučajnih skupova

master's thesis

• Title (english): Random set theory
• Author: Daniela Ivanković
• Mentor: Bojan Basrak (mentor)
• Committee member: Bojan Basrak (predsjednik povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog diplomskog rada bio je dati motivaciju za uvođenje pojma zatvorenog slučajnog skupa. Budući da je zatvoreni slučajni skup prirodno poopćenje pojma slučajnog vektora, na početku smo dali pregled već poznatih rezultata. Nakon definicije pojma zatvorenog slučajnog skupa pokazali smo neke jednostavne primjere. Uveli smo funkcije distribucije i kapaciteta i pokazali njihova osnovna svojstva. Iskazali smo Choquetov teorem koji daje nužne i dovoljne uvjete da bi funkcija bila funkcija kapaciteta nekog zatvorenog slučajnog skupa. Definirali smo Choquetov integral koji omogućuje integriranje neaditivnih funkcija. Funkcije kapaciteta zatvorenih slučajnih skupova su neaditivne pa Choquetov integral koristimo pri definiranju tri tipa konvergencije zatvorenih slučajnih skupova. Naposljetku, promotrili smo što se dogada s konvergencijom ME plotova, odnosno grafova funkcije očekivanog viška. ME plotovi se koriste u analizi i modeliranju opaženih ekstremnih vrijednosti. Na simuliranim podacima iz generalizirane Pareto razdiobe i F razdiobe te na stvarnim podacima potvrdili smo teorijski rezultat o konvergenciji ME plotova prema linearnoj funkciji.

Abstract (english)

The aim of this master thesis was to provide the motivation for the introduction of a random closed set. Since a random closed set is a natural generalization of the concept of a random vector, at the beginning we provided an overview of some well known results. After defining the concept of a random closed set, we presented some simple examples. We have introduced the distribution and capacity functions and demonstrated their basic properties. We have presented the Choquet’s theorem which gives the necessary and sufficient conditions for a function to be a capacity function of a random closed set. We have defined the Choquet integral which allows the integration of the nonadditive functions. Capacity functionals are nonadditive so we use Choquet integral to define the three modes of convergence of a series of random closed sets. Lastly, we observed what happens with the convergence of the ME plots, that is, the mean excess function plots. ME plots are used in the analysis and modeling of extreme value observations. On the simulated data from the generalized Pareto distribution and the F distribution and on the real–world data, we illustrated the theorem of the convergence of ME plots to a linear function.

• Keywords: zatvoreni slučajni skup; Choquetov teorem; Choquetov integral; ME plotovi; Pareto razdioba; F razdioba
• Keywords (english): random closed set; Choquet’s theorem; Choquet integral; ME plots; Pareto distribution; the F distribution
• Language: croatian
• URN:NBN: urn:nbn:hr:217:730289
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-10-27 11:12:00