Assessing dissimilarity of random sets through convex compact approximations and support functions

doctoral thesis

• Title (croatian): Ispitivanje sličnosti slučajnih skupova aproksimacijama konveksnim tijelima i njihovim potpornim funkcijama
• Author: Vesna Gotovac
• Mentor: Nikola Koceić Bilan (mentor)
• Mentor: Kateřina Helisová https://orcid.org/0000-0002-0586-5949 (komentor)
• Committee member: Zbyněk Pawlas https://orcid.org/0000-0002-1042-8073 (predsjednik povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Kateřina Helisová https://orcid.org/0000-0002-0586-5949 (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-04-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - 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: approximations; dissimilarity; envelope tests; kernel tests; random sets; stochastic geometry; support functions
• Keywords (croatian): aproksimacija; jezgreni test; ovojnični test rangova; potporne funkcije; sličnost; slučajni skupovi; stohastička geometrija
• Language: english
• URN:NBN: urn:nbn:hr:217:126277
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: 91 str.
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2019-03-08 11:26:28