SF2956 Topological Data Analysis 7.5 credits

Topologisk dataanalys

Show course information based on the chosen semester and course offering:

Offering and execution

No offering selected

Select the semester and course offering above to get information from the correct course syllabus and course offering.

Course information

Content and learning outcomes

Course contents *

The course contains the following topics:

  • Kleinberg theorem about impossibility of clustering,
  • metric spaces and dendrograms,
  • classical hierarchical clustering schemes (single, complete, average, and Haussdorff linkage),
  • elements of simplicial complexes,
  • transforming data into simplicial complexes via Chech and Vietoris -Rips constructions
  • extracting homology out of data based simplicial complexes
  • persistence modules, barcoding , and feature visualization

Intended learning outcomes *

In this course we present topology-based constructions of some stable signatures (called bar codes) of data sets and how to use these signatures to extract information out of outcomes of various hierarchical clustering schemes. We also present how to use these methods as a preprocessing step transforming possibly complex data into objects, which might be more suitable for statistical analysis. We also illustrate how these methods can help in visualizing certain aspects of the data. Part of the course is to learn how to extract homology out of data based simplicial complexes. The course addresses both theoretical and practical aspects of topological data analysis techniques. Computer-aided project work will be part of the learning activity.

To pass the course, the student should:

  • explain the concepts of a metric space and an ultrametric space

  • be able to visualise ultra metrics as dendrograms

  • extract bar codes from dendrograms

  • be able to calculate bottle neck distances between bar codes

  • use bar codes to look for homological features in data sets

  • be familiar with basic methods of extracting homology of data based simplical complexes

  • be familiar with stability properties of classical hierarchical clustering schemes

To receive the highest grade, the student should in addition be able to:

  • apply the discussed methods such as hierarchical clustering and bar coding of various homology groups to visualise properties of different type of data

Course Disposition

Lectures, presentations, work with computer-aided data analysis.

Literature and preparations

Specific prerequisites *

Courses in liner algebra, calculus in one and several variables, numerical methods, modern methods in statistical learning, computer intensive methods.

Recommended prerequisites

No information inserted

Equipment

No information inserted

Literature

Various books and lecture notes presented on the course web page.

Examination and completion

Grading scale *

A, B, C, D, E, FX, F

Examination *

    Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

    The examiner may apply another examination format when re-examining individual students.

    Written exam.

    Other requirements for final grade *

    Passed final exam.

    Opportunity to complete the requirements via supplementary examination

    No information inserted

    Opportunity to raise an approved grade via renewed examination

    No information inserted

    Examiner

    Wojciech Chachólski

    Further information

    Course web

    Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

    Course web SF2956

    Offered by

    SCI/Mathematics

    Main field of study *

    Mathematics

    Education cycle *

    Second cycle

    Add-on studies

    No information inserted

    Contact

    Wojtek Chacholski (wojtek@kth.se)

    Ethical approach *

    • All members of a group are responsible for the group's work.
    • In any assessment, every student shall honestly disclose any help received and sources used.
    • In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.