Gravity Wave Identification using Metropolis-Hastings MCMC

This project implements a Markov Chain Monte Carlo method -- specifically the Metropolis-Hastings algorithm to identify Gravitational wave-like signals from noisy time series data. It uses simulated data containing a gravitational wave–like signal with added noise.

Objective

We aim to recover the parameters α, β, and γ of the analytical model that describes the signal:

$$h(t)=\alpha e^{t}\left(1-\tanh\left(2(t-\beta)\right)\right)\sin(\gamma t)$$

• α controls the amplitude of the signal
• β shifts the signal in time
• γ controls the oscillation frequency

The parameters vary within these ranges: $$0<\alpha<2, \quad 1<\beta<10,\quad 1<\gamma<20$$

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Method

We perform a random walk in the 3D parameter space:

$$\theta = (\alpha,\beta,\gamma)$$

using the Metropolis–Hastings algorithm to sample from the posterior probability distribution:

$$P(\theta | \text{data}) \propto P(\text{data}|\theta)P(\theta)$$

The prior $P(\theta)$ is uniform within the given ranges.
The likelihood is calculated as:

$$P(\text{data}|\theta) \propto \exp(Y)$$

where

$$Y=-\frac{1}{2 N}\sum_i^N \left(\frac{(y_{\text{data},i}-y_{\text{model},i})^2}{y_{\text{err},i}^2}\right)$$

The error at each data point is assumed to be 20%. (given)

Each iteration proposes a new set of parameters. If the new parameters give a higher probability, they are accepted; otherwise, they may still be accepted with a probability proportional to their likelihood ratio.
This helps the algorithm explore the entire parameter space instead of getting stuck in local maxima.

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Outputs

The program estimates the most likely values of α, β, and γ, and also provides uncertainty ranges.

We have plotted the following:

• Posterior distributions for each parameter
• A reconstructed signal overlaid on the noisy data
• Histograms or corner plots showing parameter correlations and convergence

Histrograms and Trace Plots

Parameter Covariance Plots

Alpha vs Beta	Beta vs Gamma	Gamma vs Alpha

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Results

The algorithm ran with an acceptance rate of $22.2$%

The predicted parameters are:

Parameter	Alpha $\alpha$	Beta $\beta$	Gamma $\gamma$
Median Value	$1.38$	$3.91 $	$10.00$
95% Interval	$(0.92 - 1.91)$	$(3.64 - 4.19)$	$(9.92 - 10.08)$

The model graph using the predicted parameters is shown in the image above and Results/Original_Unknown/Gravitational_Wave_pred.png

Concepts Used

• Metropolis–Hastings algorithm (MCMC sampling)
• Bayesian inference
• Parameter estimation under noise
• Gravitational wave signal modeling

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Project Structure

Gravity-Wave-Identification/
├── main.py                     # Main orchestrator with mode routing
├── run.sh                      # Bash launcher for interactive execution
├── gw_data.csv                 # Unknown gravitational wave data (problem input)
├── requirements.txt            # Python dependencies (numpy, pandas, matplotlib, etc.)
├── README.md                   # Repository documentation
│
├── Configurations/             # Configuration and model definitions
│   ├── Grav_Wave.yaml          # MCMC parameters for generated/unknown modes
│   ├── variance_test.yaml      # MCMC parameters for variance testing
│   └── gw_functions.py         # Gravitation_Wave class, likelihood functions
│
├── Source/                     # Core algorithm and visualization
│   ├── metropolis_hasting.py   # MetroHaste MCMC implementation
│   └── plotting.py             # Visualization functions (histogram_gw, corner_plot, etc.)
│
├── Data_Gen/                   # Data generation pipeline
│   ├── generator.py            # Generate_Data function
│   └── Data_Grav_Wave.csv      # Synthetic noisy data (generated mode output)
│
├── Results/                    # Output directory for experiments
│   ├── [experiment_id]/        # Custom experiment outputs (generated mode)
│   │   ├── Gravitational_Wave_data.png
│   │   ├── Gravitational_Wave_pred.png
│   │   ├── MH_hist.png
│   │   └── Covariance/         # Alpha_vs_Beta.png, Beta_vs_Gamma.png, Gamma_vs_Alpha.png
│   ├── Original_Unknown/       # Unknown mode outputs
│   │   ├── Gravitational_Wave_pred.png
│   │   ├── Histogram.png
│   │   └── Covariance/         # Alpha_vs_Beta.png, Beta_vs_Gamma.png, Gamma_vs_Alpha.png
│   └── Variance_Test.png       # Variance mode diagnostic plot
│   └── Likelih_Comparison.png  # Comparison between Likelihood Functions
│
├── Typst_Presentation/         # Documentation and presentation
│   └── slides.typ              # Typst source for scientific presentation
│
└── manim_animation/            # Parameter effect animations

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How to Run

1. Clone the Repository

   git clone https://github.com/Anirudh0616/Gravity-Wave-Identification.git
   cd Gravity-Wave-Identification

2. Create and Activate Virtual Environment

# MacOS/Linux
python3 -m venv .venv
source .venv/bin/activate
# Windows ( Command Prompt )
py -3 -m venv .venv
..venv\Scripts\activate.bat

3. Install the dependencies

pip install -r requirements.txt

4. Run the Launcher

# MacOS/Linux
chmod +x run.sh
./run.sh
# Windows
bash run.sh

The Launcher will prompt you to enter values for test experiment and experiment name ( Directory name for plots to be saved in )

5. Deactivate afterwards

deactivate

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References

• Metropolis et al. (1953), Equation of State Calculations by Fast Computing Machines
• Hastings (1970), Monte Carlo Sampling Methods Using Markov Chains
• Gregory, P. C. (2005), Bayesian Logical Data Analysis for the Physical Sciences
• Charles Zaiontz, Effective Sample Size for Metropolis Algorithm
• Gareth O. Roberts, Jeffrey S. Rosenthal "General state space Markov chains and MCMC algorithms," Probability Surveys, Probab. Surveys 1(none), 20-71, (2004)
• Taboga, Marco (2021). "Metropolis-Hastings algorithm", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. StatLect MCMC Notes

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Authors

This repository is our collective project for Dr. Kirit Makwana's Computational Physics EP4210 at IIT Hyderabad (Fall 2025)

Anirudh Bhat -- Samyak Rai -- Shanmukh Machiraju