emcee for Schechter function

3 min read 20-10-2024

The Schechter function is a crucial mathematical model used in astrophysics to describe the luminosity distribution of galaxies. When analyzing astronomical data, researchers often need a reliable way to fit models to observational data, and that’s where the emcee package, a popular Python tool for Bayesian data analysis, comes into play.

In this article, we'll explore how to effectively use emcee to fit the Schechter function to observational data, guiding you through the process and providing useful insights and resources.

The Problem Scenario

Suppose we have observational data regarding the luminosity of galaxies, and we want to fit this data to the Schechter function, which is represented mathematically as:

[ \phi(L) = \phi^* \left( \frac{L}{L^} \right)^{\alpha} e^{-\frac{L}{L}} ]

Where:

( \phi(L) ) is the number density of galaxies with luminosity ( L ).
( \phi^* ) is the normalization factor.
( L^* ) is the characteristic luminosity.
( \alpha ) is the faint-end slope.

Original Code Example

Here is a simple example of Python code utilizing the emcee package to fit the Schechter function:

import numpy as np
import emcee
import matplotlib.pyplot as plt

def schechter(L, phi_star, L_star, alpha):
    return phi_star * (L / L_star) ** alpha * np.exp(-L / L_star)

# Mock data for demonstration
L_data = np.linspace(0.1, 1.5, 100)
phi_data = schechter(L_data, 0.01, 1.0, -1.5) + 0.001 * np.random.randn(100)

def ln_prior(theta):
    phi_star, L_star, alpha = theta
    if 0 < phi_star < 0.1 and 0 < L_star < 2.0 and -3 < alpha < 0:
        return 0.0
    return -np.inf

def ln_likelihood(theta, L, phi):
    phi_star, L_star, alpha = theta
    return -0.5 * np.sum((phi - schechter(L, phi_star, L_star, alpha)) ** 2)

def ln_posterior(theta, L, phi):
    return ln_prior(theta) + ln_likelihood(theta, L, phi)

# Initial guess for the parameters
nwalkers = 32
ndim = 3
pos = [1e-3, 1.0, -1.5] + 1e-4 * np.random.randn(nwalkers, ndim)

# Set up the sampler
sampler = emcee.EnsembleSampler(nwalkers, ndim, ln_posterior, args=(L_data, phi_data))

# Run the MCMC
sampler.run_mcmc(pos, 5000)

# Plotting the results
samples = sampler.get_chain(discard=100, flat=True)
plt.plot(L_data, phi_data, 'o')
plt.xlabel('Luminosity (L)')
plt.ylabel('Number Density (φ)')
plt.title('Fit of the Schechter Function to Mock Data')
plt.show()

Analysis and Explanation

Key Components of the Code

The Schechter Function: The code defines the schechter function, which computes the number density of galaxies based on the given parameters. This function forms the core of our analysis.
Mock Data Creation: We simulate some mock data for luminosity, incorporating random noise to mimic real observational data.
Likelihood and Prior Functions: The ln_prior function ensures that our parameters fall within reasonable bounds, while ln_likelihood computes the likelihood of our observed data under the model.
Posterior Probability: The ln_posterior function combines the prior and likelihood, which is the essence of Bayesian inference.
MCMC Sampling: Using emcee, we sample from the posterior distribution using the ensemble sampler method.
Result Visualization: Finally, we visualize the fit of our model against the mock data.

Practical Example

Suppose you have observational data from a galaxy survey and you wish to characterize the galaxy luminosity function. By adapting the above code, you can efficiently fit the Schechter function to your data, which can yield critical insights into the population of galaxies in the observed region.

Conclusion

Using emcee to fit the Schechter function provides an effective way to analyze and interpret astronomical data. By leveraging Bayesian methods, researchers can obtain more reliable estimates of the underlying parameters, which can lead to better understanding in astrophysics.

Useful Resources

By following these steps, you can harness the power of the Schechter function and the emcee package to deepen your understanding of galaxy distributions and properties. Happy coding!