This documentation is for astroML version 0.2

This page

Links

astroML Mailing List

GitHub Issue Tracker

Videos

Scipy 2012 (15 minute talk)

Scipy 2013 (20 minute talk)

Citing

If you use the software, please consider citing astroML.

The effect of SamplingΒΆ

Figure 10.14

An illustration of the impact of measurement errors on the Lomb-Scargle power (cf. figure 10.4). The top-left panel shows a simulated data set with 40 points drawn from the function y(t|P) = sin(t) (i.e., f = 1/(2pi) ~ 0.16) with random sampling. Heteroscedastic Gaussian noise is added to the observations, with a width drawn from a uniform distribution with 0.1 < sigma < 0.2 (this error level is negligible compared to the amplitude of variation). The spectral window function (PSD of sampling times) is shown in the bottom-left panel. The PSD (P_{LS}) computed for the data set from the top-left panel is shown in the top-right panel; it is equal to a convolution of the single peak (shaded in gray) with the window PSD shown in the bottom-left panel (e.g., the peak at f ~ 0.42 in the top-right panel can be traced to a peak at f ~ 0.26 in the bottom-left panel). The bottom-right panel shows the PSD for a data set with errors increased by a factor of 10. Note that the peak f ~ 0.16 is now much shorter, in agreement with eq. 10.47. In addition, errors now exceed the amplitude of variation and the data PSD is no longer a simple convolution of a single peak and the spectral window.

../../_images_1ed/fig_sampling_1.png
# Author: Jake VanderPlas
# 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

from astroML.time_series import lomb_scargle

#----------------------------------------------------------------------
# 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.
from astroML.plotting import setup_text_plots
setup_text_plots(fontsize=8, usetex=True)

#------------------------------------------------------------
# Generate the data
np.random.seed(42)
t_obs = 100 * np.random.random(40)  # 40 observations in 100 days
y_obs1 = np.sin(np.pi * t_obs / 3)
dy1 = 0.1 + 0.1 * np.random.random(y_obs1.shape)
y_obs1 += np.random.normal(0, dy1)

y_obs2 = np.sin(np.pi * t_obs / 3)
dy2 = 10 * dy1
y_obs2 = y_obs2 + np.random.normal(dy2)

y_window = np.ones_like(y_obs1)

t = np.linspace(0, 100, 10000)
y = np.sin(np.pi * t / 3)

#------------------------------------------------------------
# Compute the periodogram
omega = np.linspace(0, 5, 1001)[1:]
P_obs1 = lomb_scargle(t_obs, y_obs1, dy1, omega)
P_obs2 = lomb_scargle(t_obs, y_obs2, dy2, omega)
P_window = lomb_scargle(t_obs, y_window, 1, omega,
                        generalized=False, subtract_mean=False)
P_true = lomb_scargle(t, y, 1, omega)

omega /= 2 * np.pi

#------------------------------------------------------------
# Prepare the figures
fig = plt.figure(figsize=(5, 2.5))
fig.subplots_adjust(bottom=0.15, hspace=0.35, wspace=0.25,
                    left=0.11, right=0.95)

ax = fig.add_subplot(221)
ax.plot(t, y, '-', c='gray')
ax.errorbar(t_obs, y_obs1, dy1, fmt='.k', capsize=1, ecolor='#444444')
ax.text(0.96, 0.92, "Data", ha='right', va='top', transform=ax.transAxes)
ax.set_ylim(-1.5, 1.8)
ax.set_xlabel('$t$')
ax.set_ylabel('$y(t)$')

ax = fig.add_subplot(223)
ax.plot(omega, P_window, '-', c='black')
ax.text(0.96, 0.92, "Window PSD", ha='right', va='top', transform=ax.transAxes)
ax.set_ylim(-0.1, 1.1)
ax.set_xlabel('$f$')
ax.set_ylabel(r'$P_{\rm LS}(f)$')

ax = fig.add_subplot(222)
ax.fill(omega, P_true, fc='gray', ec='gray')
ax.plot(omega, P_obs1, '-', c='black')
ax.text(0.96, 0.92, "Data PSD", ha='right', va='top', transform=ax.transAxes)
ax.set_ylim(-0.1, 1.1)
ax.set_xlabel('$f$')
ax.set_ylabel(r'$P_{\rm LS}(f)$')

ax = fig.add_subplot(224)
ax.fill(omega, P_true, fc='gray', ec='gray')
ax.plot(omega, P_obs2, '-', c='black')
ax.text(0.96, 0.92, "Data PSD\n(10x errors)",
        ha='right', va='top', transform=ax.transAxes)
ax.set_ylim(-0.1, 1.1)
ax.set_xlabel('$f$')
ax.set_ylabel(r'$P_{\rm LS}(f)$')

plt.show()