Detection of Gravitational Waves Using Topological Data Analysis

Non-spinning Binary Black Holes Coalescing Event

Overview

This project focuses on detecting gravitational waves caused by non-spinning binary black hole mergers through a combination of simulation, topological data analysis (TDA), and machine learning. Gravitational waves, one of Einstein’s most profound predictions, are emitted by massive events like black hole mergers. In this project, we simulate time series data for gravitational waves, apply topological methods to extract features, and use a machine learning model to classify and detect gravitational wave signals.

Objective:

The goal is to develop a systematic approach for detecting gravitational waves from non-spinning binary black holes coalescing, using topological data analysis to process time series data and a Support Vector Machine (SVM) classifier for signal detection.

Methodology

1. Data Simulation with PyCBC

We used the PyCBC package to simulate gravitational waveforms from non-spinning binary black hole mergers. PyCBC is an open-source software package designed for the analysis of gravitational waves, and it provides tools for:

• Generating gravitational wave signals from astrophysical sources (like binary black holes).
• Filtering noisy data to search for gravitational wave signals.
• Performing parameter estimation and waveform modeling.

Using PyCBC, we simulated 2000 samples of gravitational wave time-series data, representing binary black hole coalescence, and generated an equal number of noise samples to balance the dataset.

2. Topological Data Analysis (TDA)

• Sliding Window Embedding: We employed the sliding window method to convert the 1-dimensional time series into higher-dimensional feature spaces.
• Gaussian Kernel: Applied to smooth out the time series data before embedding.
• Persistent Homology: Calculated to measure the shape of the data by computing Betti curves. Betti curves represent the topological features of the embedded time series and have been shown to be resistant to noise.

3. Feature Extraction

• The Betti Curves are used as feature vectors for the next step of classification. These curves capture the essential topological features from the gravitational wave signals and allow for effective classification.

4. Machine Learning Model

• Support Vector Machine (SVM): The extracted feature vectors (Betti curves) were used as input for the SVM model. The model was trained to distinguish between true gravitational wave signals and noise.
• Accuracy: We achieved a maximum accuracy of 97.3% by varying the signal-to-noise ratio, with the best performance obtained for an embedding dimension of 8 and a time delay of 1.

Data Processing Workflow

1. Data Loading: Simulated time-series data for gravitational wave signals and noise.
2. Gaussian Smoothing: Applied Gaussian kernels to reduce noise and enhance signal clarity.
3. Sliding Window Embedding: The time series data was embedded into a higher-dimensional space using sliding window techniques.
4. Persistent Homology: Computed Betti curves to represent the topological features of the data.
5. Feature Extraction: Betti curves served as feature vectors for machine learning.
6. Classification: A Support Vector Machine (SVM) was trained and tested to classify gravitational wave signals with high accuracy.

Notebooks

• sample_DataSet-TDA.ipynb: Contains the code for data simulation (using PyCBC) and the process of generating Betti curves from the gravitational wave and noise data.
• ML_v5.ipynb: Includes the machine learning model (SVM) used to classify the signals based on the topological features.

Results

The project demonstrates that topological data analysis, when combined with machine learning, can be highly effective in detecting gravitational waves. The results show that Betti curves, as feature vectors, are robust against noise, and the SVM classifier can distinguish gravitational wave signals from noise with high accuracy (97.3%).

Key Achievements:

• High accuracy in detecting non-spinning binary black hole coalescing events from noisy data.
• Robust feature extraction using Betti curves that are resistant to noise.
• Efficient classification of gravitational wave signals using an SVM model.

How to Use

Prerequisites:

• Python 3.x
• Libraries: pandas, numpy, scikit-learn, gudhi (for topological data analysis), matplotlib
• PyCBC: Install PyCBC by running:

  pip install pycbc