Title Statističko modeliranje ekstremnih vrijednosti - metoda maksimuma blokova Title (english) Statistical modeling of extreme values - maximum block method Author Anđela Spajić Mentor Hrvoje Planinić (mentor)Committee member Hrvoje Planinić (predsjednik povjerenstva)Committee member Snježana Lubura Strunjak (član povjerenstva)Committee member Pavle Goldstein (član povjerenstva)Committee member Siniša Slijepčević (član povjerenstva)Granter University of Zagreb Defense date and country 2022-09-28, Croatia Scientific / art field, NATURAL SCIENCES Abstract Cilj ovoga rada predstaviti je teorijske rezultate potrebne za razumijevanje statističkoga modeliranja ekstremnih vrijednosti korištenjem metode maksimuma blokova te ilustrirati primjenu metode maksimuma blokova na primjerima iz stvarnoga svijeta. Na samom je početku rada opisana reprezentacija dostupnih podataka kao nezavisnih i jednako distribuiranih slučajnih varijabli, kao i postupak pronalaska maksimalnih vrijednosti unutar blokova. Glavni cilj metode maksimuma blokova određivanje je distribucije tako dobivenih maksimuma, o čemu govori i Fisher-Tippett-Gnedenkov teorem, čiji iskaz i dokaz predstavljaju središnji rezultat ovoga rada. Važnost Fisher-Tippett-Gnedenkova teorema ogleda se u tome što navodi sve moguće familije funkcija distribucije kojima može pripadati funkcija distribucije promatranih maksimuma, a to su Gumbelova, Frechetova i Weibullova familija, i to bez obzira na distribuciju početnih podataka. Dan je i kratak osvrt na Gumbelovu, Frechetovu i Weibullovu domenu atrakcije zajedno s primjerima poznatih distribucija koje pripadaju pojedinoj domeni. Nadalje, predstavljen je pojam GEV familije funkcija distribucije kao parametrizacije koja ujedinjuje Gumbelovu, Frechetovu i Weibullovu familiju te opisan način prilagodbe optimalnoga GEV modela dostupnim podacima. U samoj procjeni parametara modela, kao i donošenju zaključaka o njihovoj preciznosti, ključna je bila metoda maksimalne vjerodostojnosti sa svojim asimptotskim svojsvima. U ekstrapolaciji budućih ekstremnih vrijednosti iz GEV modela ključni pojmovi bili su razina povrata te povratni period. Razina povrata je vrijednost za koju se očekuje da bi u svakome bloku mogla biti nadmašena s vjerojatnošću p, gdje je p vrijednost recipročna željenom povratnom periodu. Primjena metode maksimuma blokova ilustrirana je na primjerima iz stvarnoga svijeta, i to u procjeni maksimalnih morskih razina, procjeni minimalne izdržljivosti staklenih vlakana te analizi maksimalnih ljetnih temperatura u Zagrebu. Naposljetku je dan i pregled osnovnih pojmova klasične matematičke statistike koji su ključni za razumijevanje rezultata iznesenih u radu.
Abstract (english) The prime objective of this thesis is to give an introduction into the statistical theory that is crucial for understanding extreme value modelling via block maxima method, as well as to try and illustrate the use of said block maxima method on real-life data. At first, it is described how to represent the available data as a sequence of independent, identically distributed random variables and how block maxima are actually found. The main objective of block maxima method consists of determining the distribution of block maxima, which can sometimes be achieved by applying Fisher-Tippett-Gnedenko theorem. Fisher-Tippett-Gnedenko theorem, along with its proof, represents the central theoretical result of this thesis. The importance of Fisher-Tippet-Gnedenko theorem lies in the fact that it provides all distribution families that can appear as limits for the block maxima distributions, regardless of the distribution of the initial data. Said distribution families are widely known as Gumbel, Frechet and Weibull families. This thesis also contains a brief review of maximum domains of attraction of Gumbel’s, Frechet’s and Weibull’s distributions, along with examples of some famous distribution functions that belong to one of said maximum domains of attraction. Next, GEV family is introduced as an unification of Gumbel’s, Frechet’s and Weibull’s family and a description is given on how to adapt the right GEV model to the available data. The key to estimating model parameters, as well as to determining the confidence of obtained estimates, surely was maximum likelihood estimation with its asymptotic properties. Moreover, the terms return level and return period are proposed as fundamental in understanding the extrapolated extreme values. Return level denotes a value that might be exceeded in any particular block with probability p, which corresponds to the reciprocal value of the observed return period. The application of block maxima method is illustrated with the analysis of three real-life datasets: the first one contains data about annual maximum sea-levels, the second dataset consists of measurements of glass fiber breaking strengths, and the third one includes measurements of maximum daily temperatures in Zagreb, Croatia. At last, this thesis also contains a short overview of classical statistical results that are crucial for understanding the presented extreme value theory results.
Keywords
Fisher-Tippett-Gnedenkov teorem
Gumbelova
Frechetova i Weibullova familija
GEV familije
razina povrata
povratni period
matematička statistika
Keywords (english)
Fisher-Tippett-Gnedenko theorem
Gumbel
Frechet and Weibull families
GEV family
return level
return period
mathematical statistics
Language croatian URN:NBN urn:nbn:hr:217:044328 Study programme Title: Mathematical Statistics Type of resource Text File origin Born digital Access conditions Open access Terms of use Created on 2022-11-08 11:47:21
