Veri bilimi`nde hiyerarşik yapılar

In recent years, the increase of studies analyzing data as complex systems lead clustering to play key role. Hierarchical clustering is one of the most popular clustering method in data science. It is a useful method with its comprehensible application, graphical analysis and with its resulting hierarchical tree. This thesis aims to study the mathematical background of the hierarchical clustering structures of a particular data by using metric and ultrametric spaces' features as well as graph theoretical tools. First of all, we study metric spaces, normed spaces and ultrametric spaces. Besides some examples, including the remarkable p-adic spaces, the topological properties of these spaces are studied. Then, we study how to interpret a particular data by means of a metric and ultrametric space. Ultrametric tree models of similarity and association are used to produce the representation of the data. We gave the equivalence of agglomerative hierarchical clustering model using single linkage and the graph theoretical model using minimal spanning tree. We tackled here some notions of Graph Theory which helps us to visualize the data and mainly the question how to obtain a Minimum Spanning Tree (MST) from a graph which represents the optimization process. Finally, we analyze the data obtained from PISA-mathematical and PISA-reading performance evolution over $4$ years for $10$ OECD countries. We analyze these particular data by using minimum spanning tree model which are obtained by using certain algorithms (Prim\& Kruskal) and programs (Python\& Sage). The results of our data analysis allow us to make a meaningful conclusion about the evolution of mathematics and reading performance in the considered $10$ OECD countries.