Matched Filter Burst Search

Figure 10.25

A matched filter search for a burst signal in time series data. A simulated data set generated from a model of the form y(t) = b0 for t < T and y = b0 + A exp[-a(t - T)] for t > T , with homoscedastic Gaussian errors with sigma = 2, is shown in the top-right panel. The posterior pdf for the four model parameters is determined using MCMC and shown in the other panels.

Code output:

@pickle_results: computing results and saving to 'matchedfilt_burst.pkl'

Python source code:

# Author: Jake VanderPlas (adapted to PyMC3 by Brigitta Sipocz)
# License: BSD
#   The figure produced by this code is published in the textbook
#   "Statistics, Data Mining, and Machine Learning in Astronomy" (2013)
#   For more information, see http://astroML.github.com
#   To report a bug or issue, use the following forum:
#    https://groups.google.com/forum/#!forum/astroml-general
import numpy as np
from matplotlib import pyplot as plt

import pymc3 as pm

from astroML.plotting.mcmc import plot_mcmc
from astroML.utils.decorators import pickle_results

# ----------------------------------------------------------------------
# This function adjusts matplotlib settings for a uniform feel in the textbook.
# Note that with usetex=True, fonts are rendered with LaTeX.  This may
# result in an error if LaTeX is not installed on your system.  In that case,
# you can set usetex to False.
if "setup_text_plots" not in globals():
    from astroML.plotting import setup_text_plots
setup_text_plots(fontsize=8, usetex=True)


# ----------------------------------------------------------------------
# Set up toy dataset
def burst(t, b0, A, alpha, T):
    """Burst model"""
    mask = (t >= T)
    y = b0 + mask * A * np.exp(-alpha * (t - T))
    return y


np.random.seed(0)

N = 100
b0_true = 10
A_true = 5
alpha_true = 0.1
T_true = 50
sigma = 1.0

t = 100 * np.random.random(N)

y_true = burst(t, b0_true, A_true, alpha_true, T_true)
y_obs = np.random.normal(y_true, sigma)


# ----------------------------------------------------------------------
# Set up and run the MCMC sampling
# We need to wrap it in a function in order to be able to pickle the result
@pickle_results('matchedfilt_burst.pkl')
def compute_MCMC_results(draws=3000, tune=1000):
    with pm.Model():
        b0 = pm.Uniform('b0', 0, 50)
        A = pm.Uniform('A', 0, 50)
        T = pm.Uniform('T', 0, 100)
        log_alpha = pm.Uniform('log_alpha', -10, 10)

        y = pm.Normal('y', mu=burst(t, b0, A, np.exp(log_alpha), T),
                      sd=sigma, observed=y_obs)

        traces = pm.sample(draws=draws, tune=tune)

        return traces


traces = compute_MCMC_results()
mean_vals = pm.summary(traces)['mean']
mean_vals['alpha'] = np.exp(mean_vals.pop('log_alpha'))

labels = ['$b_0$', '$A$', '$T$', r'$\alpha$']

limits = [(9.2, 11.2), (2, 12), (45, 55), (0.0, 0.25)]
true = [b0_true, A_true, T_true, alpha_true]

# ------------------------------------------------------------
# Plot the results
fig = plt.figure(figsize=(5, 5))
fig.subplots_adjust(bottom=0.1, top=0.95,
                    left=0.1, right=0.95,
                    hspace=0.05, wspace=0.05)

# This function plots multiple panels with the traces
plot_mcmc([traces[i] for i in ['b0', 'A', 'T']] + [np.exp(traces['log_alpha'])],
          labels=labels, limits=limits,
          true_values=true, fig=fig, bins=30, colors='k')

# Plot the model fit
ax = fig.add_axes([0.5, 0.7, 0.45, 0.25])
t_fit = np.linspace(0, 100, 101)
y_fit = burst(t_fit, **mean_vals)

ax.scatter(t, y_obs, s=9, lw=0, c='k')
ax.plot(t_fit, y_fit, '-k')
ax.set_xlim(0, 100)
ax.set_xlabel('$t$')
ax.set_ylabel(r'$h_{\rm obs}$')

plt.show()

[download source: fig_matchedfilt_burst.py]