TDA to Dynamics

Topological Data Analysis Application to Dynamical System Time Series and Images

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Repository Overview
This repository accompanies the publication, "Topological Data Analysis Application to Dynamical System Time Series and Images," providing implementations of two scenarios that correspond to the research presented in the paper. Each scenario offers tools and instructions for analyzing dynamical systems using Topological Data Analysis (TDA) methods.

Scenarios and Setup Instructions

Scenario 1: MATLAB Implementation

• Purpose: To generate the time series data and phase space images of dynamical systems.
• Instructions:
  • Run the like_rossler.m file in MATLAB to generate the time series and phase space images.

Scenario 2: Python Implementation

The Python folder contains Jupyter notebooks that analyze topological features of the data generated by MATLAB and extract machine learning features.

1. Simplex_creation.ipynb

   • Purpose: Introduces simplexes, Betti numbers, and Euler numbers, providing foundational topological calculations.

2. point_cloud_to_persistent_diagram.ipynb

   • Purpose: Walks through an example using point clouds, constructing Rips filtrations, and calculating homology.

3. simplexes_and_homology_calculation.ipynb

   • Purpose: Implements Rips complex creation on the time series generated by MATLAB, offering an introduction to Rips complex.

4. cubical_homology_grayscaleimage.ipynb

   • Purpose: Computes V and T construction homology of grayscale images of dynamical systems.

5. machine_learning_features.ipynb

   • Purpose: Converts persistence diagrams into machine learning features, including persistent landscapes, persistence images, and Betti curves.

Citation

If you use any files from this repository, please cite the associated publication:

Shah, W. H. "Topological Data Analysis Application to Dynamical System Time Series and Images."