Partial orders and hierarchical clustering

Gručenje podatkov velja za eno najpomembnejših metod podatkovnega rudarjenja, ki se kot nova informacijska tehnologija dnevno razvija. Razvrščanja objektov v gruče so se tekom let raziskovalci lotevali na več načinov, kar s seboj prinese obilico različnih metod in postopkov. V diplomski nalogi se podrobneje seznanimo z merili za podobnost objektov znotraj posamezne gruče. Predstavljenih je več metod, od tega so tri hierarhične metode implementirane, predstavljene pa so tudi razlike med njimi. Vsaka razvrstitev objektov v gruče je matematično gledano ekvivalenčna relacija. Dva podatka sta ekvivalentna, če sta v isti gruči. V prvem delu je razvito matematično orodje, s katerim kasneje raziskujemo lastnosti podatkovne hierarhije, ki nastane med izvajanjem algoritmov gručenja. Končna ugotovitev kaže na to, da je reducirani graf podatkovne hierarhije, ki ga dobimo tekom razvrščanja hierarhičnih algoritmov gručenja, enak poti, za nehiearhično metodo $K$-voditeljev pa je to graf brez povezav.Data clustering is the task of organizing a set of object into groups (clusters) according to some similarity measure of the objects. As such, data clustering represents one of the most important methods of data mining. Over the years, the researchers have classified objects into clusters in several ways, which brings up plenty of different methods and procedures. In this graduation thesis, we implemented three hierarchical clustering methods and studied differences between them. Each classification of objects is from mathematical point of view an equivalence relation. Two data objects are equivalent if they are in the same cluster. In the first part, a mathematical framework of data hierarchies is developed. It enables us to study characteristics of data hierarchy, which we obtain during the execution of clustering algorithms. We show that reduced graph of data hierarchy, which results from execution of a hierarchical clustering algorithm, is a path. During nonhierarchical algorithm of K-means we obtain a graph without edges