Original Version

The original implementation of this code was created by Ben Moseley. You can view the original repository here: GitHub - Bayesian TDOA Positioning by Ben Moseley.

Overview

Bayesian TDOA positioning is utilized for estimating the position of a signal emitter based on the differences in signal arrival times at multiple sensor locations. This approach applies Bayesian inference methods to provide probabilistic estimations of the emitter’s location.

Installation

This package can be installed via pip. Ensure that you have Python 3.12.9 or higher installed.

pip install bayes-positioner

Usage Example

The following is an example script demonstrating Bayesian TDOA positioning using this package:

"""
Bayesian Time Difference of Arrival (TDOA) Positioning System using PyMC.
Original author bmoseley: https://github.com/benmoseley/bayesian-time-difference-of-arrival-positioning
Updated to work with PyMC 5 and made into an installable package by CBeckwith
"""

from bayes_positioner import BayesianTDOAPositioner
import pymc as pm
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as patches

if __name__ == "__main__":
    # Define problem parameters

    np.random.seed(1)  # Set random seed for reproducibility
    N_STATIONS = 4  # Number of receiver stations
    x_true = np.array([500, 500])  # True source position
    v_true = 346.0  # True wave speed
    t1_true = 0.5 * (np.sqrt(2) * 500 / 346)  # True time offset
    stations = np.random.randint(250, 750, size=(N_STATIONS, 2))  # Receiver positions

    # Generate noisy observations
    d_true = np.linalg.norm(stations - x_true, axis=1)
    t0_true = d_true / v_true
    t_obs = t0_true - t1_true + 0.05 * np.random.randn(*t0_true.shape)

    # Bayesian inference
    B = BayesianTDOAPositioner(stations)
    trace = B.sample(t_obs)

    # Posterior analysis
    mu, sd = B.fit_xy_posterior(trace)
    t0_pred = B.forward(mu)

    # Print results
    print(f"Posterior mean position: {mu}")
    print(f"Posterior std-dev: {sd}")
    print(f"True TOA: {t0_true}")
    print(f"Predicted TOA: {t0_pred}")
    print(f"True Time Offset (t1): {t1_true}")

    # Plot trace (optional)
    pm.plot_trace(trace)
    plt.show()

    # Plot results
    plt.figure(figsize=(6, 6))
    plt.scatter(stations[:, 0], stations[:, 1], marker="^", s=80, label="Receivers", c="blue")
    plt.scatter(x_true[0], x_true[1], s=50, label="True Source", c="red")
    plt.gca().add_patch(
        patches.Ellipse(
            xy=(mu[0], mu[1]), width=4 * sd[0], height=4 * sd[1], color="black",
            alpha=0.5, label="Posterior ($2\\sigma$)"
        )
    )
    plt.legend()
    plt.xlim(0, B.x_lim)
    plt.ylim(0, B.x_lim)
    plt.xlabel("x (m)")
    plt.ylabel("y (m)")
    plt.title("Bayesian TDOA Positioning Results")
    plt.grid()
    plt.show()

References

• PyMC Documentation

• Bayesian Time-Difference-of-Arrival Positioning by Ben Moseley