Točkovni procesi i njihovi momenti

master's thesis

• Author: Bernard Blagajčević
• Mentor: Bojan Basrak (mentor)
• Committee member: Bojan Basrak (predsjednik povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog diplomskog rada bio je upoznati točkovne procese i mjere njihovih momenata. Nakon prvog poglavlja upoznati smo s osnovnim konceptima potrebnim za razumijevanje ostatka rada. Nakon definicije točkovne mjere slijedi precizna definicija točkovnog procesa. Prikazani su matematički alati koji se koriste za karakterizaciju distribucije točkovnih procesa. Dani su najvažniji primjeri točkovnih procesa koji se koriste za ilustraciju važnih pojmova. Dodatno, upoznajemo se sa stacionarnim točkovnim procesima. Navodimo definiciju kontaktne udaljenosti procesa od točke te njenu distribuciju dovodimo u vezu s funkcionalom kapaciteta. Nastavljamo s definicijom mjere intenziteta kao očekivanja slučajne varijable koja predstavlja broj točaka u određenom području. Lebesgueovom indukcijom dokazana je Campbellova formula koja se koristi za računanje očekivanja slučajne varijable \(\sum_{x \in X}f(x)\). Zatim su definirane mjere drugog momenta te funkcija gustoće drugog momenta. U nastavku se upoznajemo s mjerama drugog momenta stacionarnih točkovnih procesa te s \(K\)-funkcijom koja služi za opisivanje interakcije između točaka. Ilustriramo primjenu \(K\)-funkcije na stvarnim podacima. Na kraju je dan prikaz Janossyjevih mjera koje dovodimo u vezu s distribucijom točkovnih procesa te njihovim mjerama momenta.

Abstract (english)

The aim of this thesis was to get acquainted with point processes and their moment measures. In the first chapter, we introduce the basic concepts needed to understand the rest of the work. After the definition of point measure, a precise definition of point process is given. We have presented the mathematical tools which are used for characterization of the point processes distribution. The most important examples of point processes that illustrate the main concepts are given. In addition, we introduce with stationary point processes. There is the definition of contact distance of the process from the given point and its distribution is brought into connection with the capacity functional. We continue with the definition of the intensity measure as the expectation of a random variable that represents the number of points in a given area. Campbell’s formula is used to calculate expected value of random variable \(\sum_{x \in X}f(x)\), which is proved by Lebesgue induction. Second moment measures and second moment density are defined as well. We present the second moment measures of stationary point processes and the \(K\)-function that describes the interaction between points. We illustrate the application of \(K\)-function on real data. Finally, we define Janossy measures that we connect with the distribution of point processes and their moment measures.

• Keywords: točkovni proces; točkovna mjera; stacionarni točkovni procesi; Lebesgueova indukcija; Campbellova formula; Janossyjeve mjere
• Keywords (english): point process; point measure; stationary point process; Lebesgue induction; Campbell’s formula; Janossy measures
• Language: croatian
• URN:NBN: urn:nbn:hr:217:859049
• Study programme: Title: Finance and Business Mathematics; 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: 2017-12-18 13:26:11