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professional thesis
Odabir i regularizacija linearnih modela s primjenom u aktuarstvu
Zagreb: University of Zagreb, Faculty of Science, 2020. urn:nbn:hr:217:054531
Petrunić, Ivan
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Petrunić, I. (2020). Odabir i regularizacija linearnih modela s primjenom u aktuarstvu (Professional thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:054531

Petrunić, Ivan. "Odabir i regularizacija linearnih modela s primjenom u aktuarstvu." Professional thesis, University of Zagreb, Faculty of Science, 2020. https://urn.nsk.hr/urn:nbn:hr:217:054531

Petrunić, Ivan. "Odabir i regularizacija linearnih modela s primjenom u aktuarstvu." Professional thesis, University of Zagreb, Faculty of Science, 2020. https://urn.nsk.hr/urn:nbn:hr:217:054531

Petrunić, I. (2020). 'Odabir i regularizacija linearnih modela s primjenom u aktuarstvu', Professional thesis, University of Zagreb, Faculty of Science, accessed 27 November 2024, https://urn.nsk.hr/urn:nbn:hr:217:054531

Petrunić I. Odabir i regularizacija linearnih modela s primjenom u aktuarstvu [Professional thesis]. Zagreb: University of Zagreb, Faculty of Science; 2020 [cited 2024 November 27] Available at: https://urn.nsk.hr/urn:nbn:hr:217:054531

I. Petrunić, "Odabir i regularizacija linearnih modela s primjenom u aktuarstvu", Professional thesis, University of Zagreb, Faculty of Science, Zagreb, 2020. Available at: https://urn.nsk.hr/urn:nbn:hr:217:054531

Metadata
TitleOdabir i regularizacija linearnih modela s primjenom u aktuarstvu
Title (english)Selection and Regularization of Linear Models With Actuarial Applications
AuthorIvan Petrunić
MentorBojan Basrak (mentor)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2020-07-03, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U kontekstu linearnih modela, postavlja se pitanje koji od prediktora (kojih može biti vrlo mnogo) su najviše povezani s odazivom. Drugo pitanje koje se javlja je: mogu li se koeficijenti linearnog modela tako prilagoditi, da model daje dobra predviđanja za nove, dosada neviđene podatke? Da bismo dali odgovore na ta pitanja, prvo uvodimo osnovne pojmove i ideje iz područja statistike i statističkog učenja kao što su kompromis između preciznosti predviđanja i interpretabilnosti modela, nadzirano i nenadzirano učenje, mjerenje kvalitete prilagodbe modela te kompromis između pristranosti i varijance. Zatim promatramo klasičan linearni regresijski model metodom najmanjih kvadrata, koji će biti temelj za kasnije proširenje. Nakon toga, opisujemo nekoliko metoda pomoću kojih možemo procijeniti pogrešku na testnom skupu, kao što su metoda validacijskog skupa, pojedinačna unakrsna validacija (LOOCV) te \(k\)-struka unakrsna validacija. U glavnom dijelu rada, prvo promatramo metode odabira podskupa prediktora, kao što su metoda odabira najboljeg podskupa, postupni odabir unaprijed te postupni odabir unatrag. Opisujemo kako odabrati najbolji model korištenjem statistika \(C_p\), BIC i prilagođeni \(R^2\), te pomoću unakrsne validacije. Zatim opisujemo dvije metode smanjenja koeficijenata (regularizacije): ridge i lasso regresiju. Konačno, opisanu teoriju praktično primjenjujemo u slučaju osiguranja od profesionalne odgovornosti liječnika.
Abstract (english)
A question that arises in the context of linear models is: which of the predictors (and there could be many of them) is most closely related to the response? Another question is: could the linear model coefficients be adjusted in a way that the model performs well on previously unseen data? In order to answer these questions, we first introduce basic concepts and ideas from statistics and statistical learning, such as the trade-off between prediction accuracy and model interpretability, supervised and unsupervised learning, measurement of the quality of fit and the bias-variance trade-off. We then consider the classical least squares linear regression model, which will be the basis for further extension. After that, we describe several methods for the test error estimation, such as the validation set approach, the leave-one-out cross-validation (LOOCV), and the \(k\)-fold cross-validation. In the central part of the thesis, we first consider subset selection methods, such as the best subset selection, forward stepwise selection and backward stepwise selection. We describe how to select the best model using the \(C_p\), BIC and adjusted \(R^2\) statistics, as well as using cross-validation. We then describe two shrinkage (or regularization) methods: the ridge regression and the lasso. Finally, we apply the theory to the case of medical malpractice insurance.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:054531
Study programmeTitle: Actuarial Mathematics
Study programme type: university
Study level: postgraduate specialist
Academic / professional title: sveučilišni/a specijalist/ specijalistica aktuarske matematike (sveučilišni/a specijalist/ specijalistica aktuarske matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Embargo expiration date: 2022-07-08
Terms of useLicence In copyright icon
Created on2020-07-08 11:31:19