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Assessing dissimilarity of random sets through convex compact approximations and support functions
Vesna Gotovac (2017)
University of Zagreb
Faculty of Science
Department of Mathematics
Cite this document...

Gotovac, V. (2017). Assessing dissimilarity of random sets through convex compact approximations and support functions (Doctoral thesis). Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:126277

Gotovac, Vesna. "Assessing dissimilarity of random sets through convex compact approximations and support functions." Doctoral thesis, University of Zagreb, Faculty of Science, 2017. https://urn.nsk.hr/urn:nbn:hr:217:126277

Gotovac, Vesna. "Assessing dissimilarity of random sets through convex compact approximations and support functions." Doctoral thesis, University of Zagreb, Faculty of Science, 2017. https://urn.nsk.hr/urn:nbn:hr:217:126277

Gotovac, V. (2017). 'Assessing dissimilarity of random sets through convex compact approximations and support functions', Doctoral thesis, University of Zagreb, Faculty of Science, accessed 25 March 2019, https://urn.nsk.hr/urn:nbn:hr:217:126277

Gotovac V. Assessing dissimilarity of random sets through convex compact approximations and support functions [Doctoral thesis]. Zagreb: University of Zagreb, Faculty of Science; 2017 [cited 2019 March 25] Available at: https://urn.nsk.hr/urn:nbn:hr:217:126277

V. Gotovac, "Assessing dissimilarity of random sets through convex compact approximations and support functions", Doctoral thesis, University of Zagreb, Faculty of Science, Zagreb, 2017. Available at: https://urn.nsk.hr/urn:nbn:hr:217:126277

Metadata
TitleAssessing dissimilarity of random sets through convex compact approximations and support functions
Title (croatian)Ispitivanje sličnosti slučajnih skupova aproksimacijama konveksnim tijelima i njihovim potpornim funkcijama
AuthorVesna Gotovac
MentorNikola Koceić Bilan (mentor)
MentorKateřina Helisová (komentor)
ORCID: 0000-0002-0586-5949
Committee memberZbyněk Pawlas (predsjednik povjerenstva)
ORCID: 0000-0002-1042-8073
Committee memberBojan Basrak (član povjerenstva)
Committee memberKateřina Helisová (član povjerenstva)
ORCID: 0000-0002-0586-5949
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2017-04-28, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Universal decimal classification
(UDC)		51
NATURAL SCIENCES
Mathematics
Abstract
In recent years random sets were recognized as a valuable tool in modelling different processes in fields like biology, biomedicine or material sciences. Nevertheless, the full potential of these applications has not yet been reached and one of the main problems in their advancement is the usual inability to correctly differentiate between underlying processes which generate real-world realisations. This research introduces a measure of dissimilarity of random sets through a heuristic based on convex compact approximations and support functions. For assessing dissimilarity two statistical approaches were used, namely the envelope test and the test of equality of distributions of two random convex compact sets based on \(\mathfrak{N}\)-distances. In the latter, the vector space of hedgehogs which are defined as differences of convex compact sets, is employed. The methodology is justified through simulation studies of common random models like Boolean and Quermass-interaction processes. Also an example of the application of proposed methodology to real data is presented concerning the distinction between sample images of mastophatic breast tissue and mammary cancer.
Abstract (croatian)
Posljednjih godina slučajni skupovi su prepoznati kao koristan alat kod modeliranja raznih procesa u područjima znanosti kao što su biologija, biomedicina i nauka o materijalima. Pun potencijal ove primjene nije još dosegnut, a jedna od glavnih prepreka je najčešće nemogućnost razlikovanja pozadinskih procesa koji generiraju opažene realizacije. U ovoj disertaciji se uvodi mjera različitosti slučajnih skupova putem heuristike bazirane na aproksimaciji konveksnim tijelima i njihovim potpornim funkcijama. Za ocjenu različitosti korištena su dva statistička pristupa, ovojnični test rangova te test jednakosti po distribuciji dva slučajna konveksna tijela baziran na \(\mathfrak{N}\)--udaljenostima. Pri konstrukciji drugo spomenutog testa neki pojmovi su poopćeni na vektorski prostor ježeva kojeg tvore formalne razlike konveksinih tijela u \(\mathbb{R}^d\). Simulacijska studija koja opravdava korištenu metodologiju je provedena na standardnim modelima slučajnih skupova kao što su Boolean i Quermass-interaction procesi. Također je prikazan primjer primjene predložene metodologije na ispitivanje sličnosti uzoraka slika mastopatičnog tkiva dojke i raka mliječne žlijezde.
Keywords
Keywords (croatian)
Languageenglish
URN:NBNurn:nbn:hr:217:126277
Study programmeTitle: Matematika
Study programme type: sveučilišni
Study level: poslijediplomski doktorski
Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
Type of resourcetext
Extent91 str.
File originBorn digital
Access conditionsRestricted access
Terms of usehttp://rightsstatements.org/vocab/InC/1.0/
RepositoryRepository of Faculty of Science
Created on2019-03-08 11:26:28

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