## Topological Data Analysis - Topološka analiza podatkov

Basic topological concepts and models and their use in data analysis will be introduced.

Predstavili bomo osnovne topološke pojme in metode ter prikazali, kako si z njimi pomagamo pri analizi podatkov.

What is topology?

Topology is a mathematical discipline which deals with understanding and comparing shapes. Topological properties of objects are, for example, the number of connected pieces, the number of holes and tunels in it, and so on, while geometric quantities like distances, volumes, angles are typically not interesting from the topological point of view.

Kaj pa je topologija?

Topologija je veja matematike, ki se ukvarja z razumevanjem in primerjanjem oblik. Pri topologiji nas zanima iz koliko kosov (ali komponent) je sestavljen objekt, koliko lukenj in tunelov je v njem... Ne zanimajo pa nas geometrijske mere kot so ploščine, volumni, koti in dolžine, saj so odvisni od tega, kako smo podatke o objektu zajeli - od oddaljenosti, lege ...

How is topology used in data anaysis?

We are exposed to huge quantities of digital data, that is, numbers, that are continuously measured, compared, analyized,... One of the big problems today is what to do with them and how to extract useful information from them. In addition to a number of existing methods and approaches from various fields, topological methods are very suitable for dealing with certain problems, for example:

• data can be ordered using simple and transparent geometric structures which are used for modeling the domain
• topological properties of these structures help understand their shape
• the shapes obtained can be compared giving new relationships and patterns linking different data domains
• ...

Zakaj je topologija uporabna pri analizi podatkov?

Zasuti smo z množico digitalnih podatkov, torej številk, ki se neprestano zajemajo, merijo, primerjajo, analizirajo ... Za razumevanje in uporabo teh podatkov so potreba dobra orodja in metode.

S topološkimi metodami lahko:

• podatke uredimo v pregledne geometrijske strukture, iz katerih lahko rekonstruiramo, od kje so bili zajeti;
• izračunamo njihove "topološke" lastnosti, ki pomagajo razumeti oblike;
• primerjamo tako dobljene oblike in poskušamo najti nove vzorce in povezave med različnimi podatki;
• ...

Course contents

1. Topological models: triangulations and simplicial complexes, cell complexes
2. Finding holes and tunels: homology groups
3. Distinguishing between the details and the big picture: persistent homology

Vsebina predmeta

1. Topološki modeli: triangulacije in simplicialni kompleksi.
2. Iskanje lukenj in tunelov: homološke grupe.
3. Razlika med podrobnostmi in veliko sliko: vztrajna homologija.

Last modified: Tuesday, 19 December 2023, 4:45 PM