Topological Data Analysis & Equation Free Approach

In topological data analysis (TDA), techniques from algebraic topology are combined with statistical methods to reveal, analyze and characterize internal geometric structures of data sets. Recently, topological data analysis has also been applied to predict critical structural changes (e.g., phase transitions) and quantify long-term macroscopic behaviour of complex dynamical systems.

The topological features of topological data analysis have been predicted to render the statistical methods less sensitive to noise. An interesting question is whether this robustness property can be turned into an effective noise-filtering tool and whether topological data processing can be used to reduce the amount of data needed to draw relevant conclusions about the macroscopic behavior of dynamical systems.

TDA unveils the intrinsic topological characteristics of data, discerning features like voids and recesses, thereby opening new avenues in machine learning methodologies for identifying features integral to the data's overall structure. Leveraging persistent homology \cite{Ada09} and associated methodologies allows the transformation of shape information, encapsulating both the local geometry and the global topology of the data, into a vectorised representation suitable for integration into a machine learning framework

The TDA group in Karlstad University focuses on the following projects:

  1. Statistical analysis of topological parameters in combination with equation free methods for the prediction, detection, and characterization of phase transitions in complex networks
  2.  Investigation of the main features of Turing patterns in various biological systems using TDA.

Research Group Members: Associate Professor Nikos I. Kavallaris, Ole Sönnerborn

Matematisk modell
Matematisk modell