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This paper concerns a method of testing the equality of distributions of random convex compact sets. The main theoretical result involves a construction of a metric on the space of distributions of random convex compact sets. We obtain it by using the theory of N - distances and the redefined characteristic function of random convex compact set. We propose an approximation of the metric through its finite- dimensional counterparts. This result leads to a new statistical test for testing the equality of distributions of two random convex compact sets. Consequently, we show a heuristic approach how to determine whether two realisations of random sets that can be approximated by a union of identically distributed random convex compact sets come from the same underlying process using the constructed test. Each procedure is justified by an extensive simulation study and the heuristic method for comparing random sets using their convex compact counterparts is moreover applied to real data concerning histological images of two different types of mammary tissue.

Cilj ovog rada je bio otkrivanje županija u Hrvatskoj visokog rizika od zaraze koronavirusom u prvoj polovini 2021. godine uz pomoć metoda prostorne statistike. U svrhu procjene rizika od zaraze koristimo standardizirani omjer smrtnosti, te tri različita empirijska Bayesova procjenitelja. Za procjenu smrtnost nam je poslužila stopa smrtnosti. Takoder, promatramo utjecaj raznih kovarijabli na mortaitet. Dolazimo do sljedećeg zaključka. Što se tiče morbiditeta, najveći rizik ima Primorsko-Goranska županija, a slijede je Međimurska i Varaždinska. Što se tiče mortaliteta, najveći rizik ima Karlovačka županija, a slijede Varaždinska, Grad Zagreb, Ličko-Senjeska i Osječko-Baranjska. Vodeći se dobnom podjelom stanovništva, pokušali smo pokazati da je populacija starijih od 60 statistički značajna varijabla za povećan mortalitet od korone, ali nismo uspjeli postići konvergenciju Markovljevog lanca. Nakon toga smo prikazali prostornu zagađenost zraka u prvoj polovici 2021. godine. Model kojeg smo proveli ukazao je na statističku značajnost uz konvergenciju Markovljevog lanca. U županijama gdje je zagađenost zraka bila najveća, ponalazimo istog glavnog zagadivača, PM2.5. Očito, kako te sitne čestice PM2.5 uzrokuju probleme u dišnom sustavu, uz kombinaciju sa koronavirusom dolazi do više smrtnih slučajeva. The aim of this work was to detect counties in Croatia with a high risk of infection with the coronavirus in the first half of 2021 with the help of spatial statistics methods. For the purpose of assessing the risk of infection, we use the standardized mortality ratio and three different empirical Bayesian estimators. We used the death rate to estimate mortality. We also observe the influence of various covariables on mortality. We come to the following conclusion. In terms of morbidity, Primorsko-Goranska County has the highest risk, followed by Međimurje and Varaždin. As for mortality, Karlovac County has the highest risk, followed by Varaždin County, the City of Zagreb, Ličko-Senjeska and Osječko-Baranjska. Guided by the age distribution of the population, we tried to show that the population over 60 is a statistically significant variable for increased mortality from corona, but we failed to achieve the convergence of the Markov chain. After that, we presented spatial air pollution in the first half of 2021. The model we implemented indicated statistical significance with the convergence of the Markov chain. In counties where air pollution was the highest, we find the same main pollutant, PM2.5. Obviously, as these tiny PM2.5 particles cause problems in the respiratory system, in combination with the coronavirus, more deaths occur.

Teorija ekstremnih vrijednosti elegantna je i matematički fascinantna teorija koja omogućuje raznoliku primjenu. U prvom dijelu rada cilj nam je pronaći graničnu distribuciju za maksimum nezavisnih i jednako distribuiranih slučajnih varijabli. Definiran je pojam maksimalne domene privlačnosti te klasa funkcija distribucije ekstremnih vrijednosti. Potom je dokazan Fisher i Tippett, Gnedenko teorem koji nam govori kako granične funkcije distribucije čine jednostavnu eksplicitnu jednoparametarsku familiju. U središnjem poglavlju rada predstavljen je prvi pristup domeni privlačnosti te je naveden von Misesov uvjet koji govori o dovoljnom uvjetu za pripadnost domeni privlačnosti. U završnom dijelu rada ustanovili smo nužne i dovoljne uvjete da bi funckija distribucije F pripadala domeni privlačnosti od Gγ., Extreme value theory is an elegant and mathematically fascinating theory which ensures wide application. In the first part of the paper our interest is in finding possible limit distributions for sample maxima of independent and identically distributed random variables. Then the maximum domain of attraction and extreme value distributions were defined. Next Fisher and Tippett, Gnedenko theorem is proved and it tells us that the limit distribution functions form a simple explicit one-parameter family. In the central chapter of the paper the first approach to the domain of attraction is presented and von Mises condition was defined which state a sufficient condition for belonging to a domain of attraction. In the final part of the thesis we established necessary and sufficient conditions for a distribution function F to belong to the domain of attraction of Gγ.

Osnovni cilj ovog rada je aksiomatski pristup teoriji integracije. U prvom poglavlju su izneseni osnovni rezultati Riemannovog i Lebesgueovog pristupa pojmu integrala. U drugom poglavlju uvodimo Daniellov integral, pojmove donjih i gornjih funkcija te pojam sumabilne funkcije. U trećem poglavlju dajemo vezu Daniellovog integrala i teorije mjere. Na samom kraju iznosimo Stoneov aksiom i objašnjavamo njegovu važnost., Primary goal of this paper is axiomatic approach to theory of integration. In the first chapter we give basic results of Riemann and Lebesgue integral. In the second chapter we introduce definition of Daniell integral, definitions of lower and upper functions and definition of summable function. In the third chapter we give a link between Daniell integral and measure theory. At the very end we discuss Stone’s axiom and it’s importance.

Glavni cilj ovog rada je predstaviti najefikasnije načine rješavanja zadanih algoritama., The main objective of this master’s thesis is to present the most efficient ways of solving algorithms.

U ovom radu obrađeni su problemi prilagodbe jednog i K, K ≥ 2 uzorka. Kod problema jednog uzorka, statistički testovi prilagodbe su testovi koji nam govore predstavljaju li određeni podaci uzorka podatke koje biste očekivali pronaći u stvarnoj populaciji. Statistički testovi prilagodbe za problem K uzorka daju odgovore o jednakosti distribuciji dva ili više nezavisnih uzoraka. Svi obrađeni testovi razjašnjeni su kroz svoje testne statistike, osnovne komponente dekompozicije statistike te nulte distribucije. Kod problema jednog uzorka obrađeni su testovi prilagodbe redom: Kolmogorov-Smirnovljev, Anderson-Darlingov, Cramér-von-Misesov i Watsonov. Kao primjer ispitujemo kvalitetu uniformnog pseudo-slučajnog generatora u softveru R. Skup 100,000 slučajno izgeneriranih brojeva može se naći pod nazivom PRG. U podacima o koncetraciji PCB (poliklorirani bifenil) 65 Anacapa ptica je od interesa testiranje normalnosti. U poglavlju o problemu K uzoraka, bavimo se Kolmogorov-Smirnovljevim testom i Anderson-Darlingovim. Sličnosti i razlike testova za probleme jednog i K uzorka su objašnjene. Rad završavamo problemom usporedbe distribucija dva uzorka slučajnih brojeva i K-uzorka iz skupa podataka "Iris". Ovaj poznati skup podataka Iris o perunici daje mjere u centimetrima varijabli duljine i širine čašice i duljine i širine latice za 50 cvjetova svake od 3 vrste perunike. Vrste su Iris setosa, versicolor i virginica., In this paper, the goodness-of-fit problems of one and K, K ≥ 2 samples are dealt with. In one-sample problems, statistical goodness-of-fit tests are tests that tell us whether certain sample data represent the data you would expect to find in the real population. Statistical goodness-of-fit tests for the K sample problem provide answers about the equality of the distribution of two or more independent samples. All processed tests are explained through their test statistics, the basic components of the decomposition of statistics and the null distribution. In the case of one-sample problems, the Kolmogorov-Smirnov, Anderson-Darling, Cramér-von-Mises and Watson tests were processed in order. We analyze the distributions of the data sets PCB and PRG. In the chapter on the K samples problem, we deal with the Kolmogorov-Smirnov test and the Anderson-Darling test. The similarities and differences of the tests for the one-sample and K-sample problems are explained. We end the paper with the problem of comparing the distributions of two samples of random numbers and the K-sample from the "Iris" data set.

Ovaj rad pokriva ključne ideje za dizajniranje algoritama. Vidimo kako upotreba određenih struktura podataka pri dizajniranju algoritma može biti efikasnija od upotrebe drugih struktura podataka. Proučili smo standardne probleme poput sortiranja, dodavanja elemenata u određenu strukturu podataka kao i brisanje elemenata. Prošli smo kroz ključne strukture podataka poput listi, stogova, redova i binarnih stabala te pogledali njihov utjecaj na složenost prilikom dizajniranja algoritma za standardne probleme. Pokazali smo implementaciju navedenih algoritama u Java programskom jeziku te pokazali primjere u kojima odabir struktura podataka ne utječe na složenost algoritma i primjere kada odabir ispravne strukture podataka za dani problem smanjuje složenost algoritma za dani problem., This paper covers the key ideas for designing algorithms. We see how the use of certain data structures when designing an algorithm may be more efficient than the use of other data structures. We studied standard problems such as sorting, adding and deleting elements to a certain data structure. We went through key data structures such as lists, stacks, queues and binary trees and looked at their impact on complexity when designing algorithms for standard problems. We showed the implementation of the mentioned algorithms in the Java programming language and showed examples in which the selection of data structures does not affect the complexity of the algorithm and examples when the selection of the correct data structure for a given problem reduces the complexity of algorithm for the given problem.

Ovaj se rad bavi osnovama podržanog učenja iznoseći sve poznate metode korištene u PU poput Bellmanovih jednadžbi, Dinamičkog programiranja, teorema poboljšanja strategije,Monte-Carlo učenja, učenja s vremenskom razlikom itd. Osnovni problem je bio nalaženje optimalne strategije uz pomoć optimalne vrijednosne funkcije, i to u okolini bez modela. Kombinacijom navedenih metoda, posebno MC iteracije strategija, napisan je kod koji je uspješno nalazio optimalne strategije na primjeru igre Blackjack., This paper deals with the basics of reinforcement learning, presenting all known methods used in PU such as Bellman equations, Dynamic programming, strategy improvement theorem, MC learning, temporal difference learning, etc. The basic problem was finding an optimal strategy that converges to an optimal value function, and in a model-free environment. By combining the above methods, especially MC strategy iteration, a code was written that successfully found optimal strategies on the example of the Blackjack game.

Slučajni skupovi igraju ključnu ulogu u modeliranju brojnih procesa u biologiji, medicini i znanosti o materijalima. U prvom dijelu rada koji se odnosi na teroijske rezultate definiran slučajni zatvoreni skup u \(\mathbb{R}^{d}\) kao funkcija nad vjerojatnosnim prostorom te su navedeni ilustrativni primjeri. Potom je definiran pojam funkcionala kapaciteta te su istražena njegova svojstva. U središnjem poglavlju rada dokazan je Choquetov teorem koji nam govori o nužnim i dovoljnim uvjetima da bi funkcional bio funkcional kapaciteta jedinstvenog slučajnog zatvorenog skupa, a time i da na jedinstven način odreduje distribuciju pripadajućeg slučajnog zatvorenog skupa. U završnom dijelu rada napravljena je simulacijska studija čiji cilj je bio istražiti koliko dobro funkcional kapaciteta opisuje realizaciju slučajnog skupa te može li se mjera sličnosti bazirana na funkcionalu kapaciteta iskoristiti za klasteriranje realizacija različitih modela. Rezultati istraživanja pokazali su da funkcional kapaciteta vrlo dobro razlikuje odgovarajuće realizacije pa se može koristiti kao mjera sličnosti za klasteriranje., Random sets play an essential role in modelling several phenomena in biology, medicine and material science. The first part of the paper, which refers to the theoretical results, a random closed set in \(\mathbb{R}^{d}\) is defined as a function over the probability space and illustrative examples are given. Then the concept of capacity functional was defined and its properties were investigated. In the central chapter of the paper Choquet’s theorem is proved. It tells us what are the necessary and sufficient conditions for a functional to be a capacity functional of a unique random closed set, and thus to uniquely determine the distribution of the corresponding random closed set. In the final part of the paper a simulation study was made the goal of which was to investigate how well the capacity functional describes the realization of a random set and whether the similarity measure based on the capacity functional can be used for clustering the realizations of different models. The results of the research showed that the capacity functional distinguishes the corresponding realizations very well, so it can be used as a measure of similarity for clustering.

U ovom radu upoznajemo se s teorijom Hilbertovih prostora reproducirajućih jezgri te objašnjavamo matematičku pozadinu primjene jezgrenih funkcija u umjetnoj inteligenciji. Istražujemo vezu između pozitivno semidefinitnih funkcija i jezgrenih funkcija te svojstva različitih pozitivno semidefinitnih funkcija na proizvoljnom skupu. U prvom poglavlju uvodimo pojam potpunih metričkih, normiranih te unitarnih prostora i istražujemo njihova svojstva. U drugom poglavlju gradimo teoriju Hilbertovih prostora reproducirajućih jezgri, a u zadnjem poglavlju pokazujemo kako se navedena teorija primjenjuje u umjetnoj inteligenciji., In this thesis, we introduce the theory of reproducing kernel Hilbert spaces and explain the mathematical background of the application of kernel functions in artificial intelligence. We investigate the relationship between positive semidefinite functions and kernel functions and the properties of different positive semidefinite functions on an arbitrary set. In the first chapter, we introduce the concept of complete metric, normed and pre-Hilbert spaces and explore their properties. In the second chapter we build a theory of reproducing kernel Hilbert spaces and in the last chapter we show how this theory is applied in artificial intelligence.

U ovom radu su opisana tri paradoksa iz vjerojatnosti. U svakom paradoksu je opisan njegov problem, rješenje tog problema te njegova primjena u nastavi matematike. Prvi opisani je paradoks Montyja Halla. Kod njega smo zaključili da promjena izbora vrata donosi veću vjerojatnost za osvajanje nagrade. Nadalje, kod paradoksa dviju omotnica, kod igre TEP, smo došli do zaključka da nije najbolje rješenje mijenjati izbor omotnice. I na samom kraju, opisali smo rođendanski paradoks kod kojeg je glavni problem bio odrediti vjerojatnost da dvoje ljudi ima rođendan na isti dan., There are three paradoxes described in this thesis. For each paradox, the problem of the paradox is described, its solution and the implementation in teaching mathematics are presented. Monty Hall paradox is the first one described. The conclusion is that a change in the choice of doors yields a higher probability of winning the award. Furthermore, in the paradox of two envelopes, regarding the game TEP, we derive a conclusion that it is not a very good solution to change the envelope. In the end, we have described the birthday paradox where the main problem is to determine the probability of two people having a birthday on the same day.

U Hrvatskom osnovnoškolskom obrazovanju vjerojatnost i statistika imaju neopravdani nedostatak prostora. Naime, one su upravo te grane matematike s čijim se rezultatima učenici susreću svakodnevno putem medija i s kojima će se susretati u kojem god smjeru usmjerili svoje daljnje obrazovanje. Rad opisuje i prati odličnu ideju NOK-a, podržanu i razrađenu u CKR-u, gdje se vjerojatnost i statistika uvodi od prvog razreda osnovne škole. U radu je korišten neformalni, konceptualni pristup, gdje se raznovrsnim primjerima i aktivnostima pokušava učenicima približiti osnovne statističke metode prikupljanja, obrade i analize podataka i apstraktni pojam vjerojatnosti slučajnog događaja. Spomenute aktivnosti potiču učenje iz znatiželje i učenje na temelju iskustva. Provođenjem stvarnih statističkih istraživanja na nastavi, učenicima se olakšava učenje statističkih pojmova, rasvjetljuje potreba izradbe grafičkih prikaza i njihove analize. Također, iako je vjerojatnost „a posteriori“ ostavljena za viši obrazovni stupanj, ovdje se koristimo eksperimentima pri utvrđivanju vjerojatnosti, te u svrhu usporedbe s teorijskom vjerojatnošću., Within the Croatian primary school education, more space should be given to probability and statistics as they are unjustly neglected. In fact, it is the branch of mathematics with whose results students encounter every day through the media and which they will continue to encounter in whatever direction they focus their further education. This thesis describes and follows the great NOK's idea, supported and elaborated in CKR, where probability and statistics are introduced from the first grade. The informal conceptual approach in the thesis uses various examples and activities to bring basic statistical methods, such as collecting, processing, and analysis of data, as well as the abstract concept of the probability of random events, closer to students. These activities encourage students to learn out of curiosity and based on experience. The implementation of real statistical research in class facilitates the apprehension of statistical concepts and clarifies the purposes of graphs and their analysis. Also, although the probability "a posteriori" is left for higher education, here we use experiments to determine probabilities, with the purpose of comparing it with the theoretical probability.

Razni događaji i pojave u svijetu se mogu prikazati pomoću točkovnih procesa, a motivacija za njihovu analizu se krije u želji za otkrivanjem veza izmedu tih točaka. Analiziranjem konkretnih točkovnih uzoraka se može doći do važnih zaključaka koji su presudni za razumijevanje pojedinih dijelova raznih znanosti. Na jednostavan način su definirani točkovni procesi, njihova distribucija i primjena. Nadalje, definirani su binomni i Poissonov točkovni proces kao dva osnovna primjera. Kroz simulacije su prikazane razlike između dva spomenuta primjera, te par njihovih različitih grafičkih prikaza. Karakteristike točkovnih procesa kao što su intenzitet i K-funkcija daju bolji uvid u podatke opisane točkovnihm procesom. U ovom radu su objašnjeni najosnovniji pojmovi i primjeri primjene točkovnih procesa, kao što je primjena kod testiranja Poissonove hipoteze., Various events and phenomena in the world can be represented by point processes, and the motivation for their analysis lies in the desire to discover the connections between these points. By analyzing specific point patterns, important conclusions can be reached that are crucial for understanding individual parts of various sciences. Point processes, their distribution and application are defined in a simple way. Furthermore, the binomial and Poisson point processes are defined as two basic examples. Through the simulations, the differences between the two mentioned examples are shown, and a couple of their different graphical representations as well. The characteristics of point processes such as intensity and K-function give a better insight into the data described by the point process. This thesis explains the most basic concepts and examples of the application of point processes, such as the application in testing the Poisson hypothesis.