Max-stabilni procesi

master's thesis

• Author: Bruno Hrastić
• Mentor: Bojan Basrak (mentor)
• Committee member: Bojan Basrak (predsjednik povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-11-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog rada bio je prikazati osnovna teorijska svojstva max-stabilnih procesa. Osim fundamentalnog de Haanovog teorema uveden je i pojam ekstremalnih funkcija. Predstavljene su i neke poznatije parametarske familije (u jednodimenzionalnom i dvodimenzionalnom okruženju): Smithov, Schlatherov, Brown-Resnickov, ekstremalni \(t\) i Tukeyev proces. Od tih procesa izabrali smo Brown-Resnickov proces za detaljnije proučavanje zbog nekih njegovih zanimljivih svojstava. Uvedeni su pojmovi Brown-Resnick stacionarnosti te Brown-Resnick slučajnog polja. Osim toga objašnjena je veza s variogramom, miješanim moving maxima procesima, a i dokazan je teorem koji omogućuje prvi algoritam simulacije. Uz taj algoritam objašnjena su još dva, simulacija spektralnom mjerom koji se pokazao najefikasnijim i uvjetna simulacija koji se može koristiti u primjenama.

Abstract (english)

The aim of this thesis was to present basic theoretical properties of max-stable processes. Apart from the fundamental de Haan theorem we also introduced the term extremal functions. Also featured are some of the more familiar parametric families (in one-dimensional and two-dimensional environment): Smith, Schlather, Brown-Resnick, Extremal \(t\) and Tukey process. From these processes we chose Brown-Resnick’s process for a more detailed study because of his interesting features. The terms Brown-Resnick stationarity and Brown-Resnick random field were also introduced. In addition, we explained the relation with the variogram and mixed moving maxima processes. We proved a theorem which enables the first simulation algorithm. Along this algorithm, two more simulation algorithms were demonstrated. One that used spectral measure, which proved to be the most efficient one, and conditional simulation that can be used in applications.

• Keywords: de Haanov teorem; ekstremalne funkcije; Smithov proces; Schlatherov proces; Brown-Resnickov proces; ekstremalni \(t\) proces; Tukeyev proces
• Keywords (english): de Haan theorem; extremal functions; Smith process; Schlather process; Brown-Resnick process; Extremal \(t\) process; Tukey process
• Language: croatian
• URN:NBN: urn:nbn:hr:217:336281
• 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: 2018-05-24 11:16:41