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.

Plot a Diagram explaining a ConvolutionΒΆ

Figure 10.2

A schematic of how the convolution of two functions works. The top-left panel shows simulated data (black line); this time series is convolved with a top-hat function (gray boxes); see eq. 10.8. The top-right panels show the Fourier transform of the data and the window function. These can be multiplied together (bottom-right panel) and inverse transformed to find the convolution (bottom-left panel), which amounts to integrating the data over copies of the window at all locations. The result in the bottom-left panel can be viewed as the signal shown in the top-left panel smoothed with the window (top-hat) function.

../../_images/fig_convolution_diagram_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 scipy.signal import fftconvolve

#----------------------------------------------------------------------
# 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 random x, y with a given covariance length
np.random.seed(1)
x = np.linspace(0, 1, 500)
h = 0.01
C = np.exp(-0.5 * (x - x[:, None]) ** 2 / h ** 2)
y = 0.8 + 0.3 * np.random.multivariate_normal(np.zeros(len(x)), C)

#------------------------------------------------------------
# Define a normalized top-hat window function
w = np.zeros_like(x)
w[(x > 0.12) & (x < 0.28)] = 1

#------------------------------------------------------------
# Perform the convolution
y_norm = np.convolve(np.ones_like(y), w, mode='full')
valid_indices = (y_norm != 0)
y_norm = y_norm[valid_indices]

y_w = np.convolve(y, w, mode='full')[valid_indices] / y_norm

# trick: convolve with x-coordinate to find the center of the window at
#        each point.
x_w = np.convolve(x, w, mode='full')[valid_indices] / y_norm

#------------------------------------------------------------
# Compute the Fourier transforms of the signal and window
y_fft = np.fft.fft(y)
w_fft = np.fft.fft(w)

yw_fft = y_fft * w_fft
yw_final = np.fft.ifft(yw_fft)

#------------------------------------------------------------
# Set up the plots
fig = plt.figure(figsize=(5, 5))
fig.subplots_adjust(left=0.09, bottom=0.09, right=0.95, top=0.95,
                    hspace=0.05, wspace=0.05)

#----------------------------------------
# plot the data and window function
ax = fig.add_subplot(221)
ax.plot(x, y, '-k', label=r'data $D(x)$')
ax.fill(x, w, color='gray', alpha=0.5,
        label=r'window $W(x)$')
ax.fill(x, w[::-1], color='gray', alpha=0.5)

ax.legend()
ax.xaxis.set_major_formatter(plt.NullFormatter())

ax.set_ylabel('$D$')

ax.set_xlim(0.01, 0.99)
ax.set_ylim(0, 2.0)

#----------------------------------------
# plot the convolution
ax = fig.add_subplot(223)
ax.plot(x_w, y_w, '-k')

ax.text(0.5, 0.95, "Convolution:\n" + r"$[D \ast W](x)$",
        ha='center', va='top', transform=ax.transAxes,
        bbox=dict(fc='w', ec='k', pad=8), zorder=2)

ax.text(0.5, 0.05,
        (r'$[D \ast W](x)$' +
         r'$= \mathcal{F}^{-1}\{\mathcal{F}[D] \cdot \mathcal{F}[W]\}$'),
        ha='center', va='bottom', transform=ax.transAxes)

for x_loc in (0.2, 0.8):
    y_loc = y_w[x_w <= x_loc][-1]
    ax.annotate('', (x_loc, y_loc), (x_loc, 2.0), zorder=1,
                arrowprops=dict(arrowstyle='->', color='gray', lw=2))

ax.set_xlabel('$x$')
ax.set_ylabel('$D_W$')

ax.set_xlim(0.01, 0.99)
ax.set_ylim(0, 1.99)

#----------------------------------------
# plot the Fourier transforms
N = len(x)
k = - 0.5 * N + np.arange(N) * 1. / N / (x[1] - x[0])

ax = fig.add_subplot(422)
ax.plot(k, abs(np.fft.fftshift(y_fft)), '-k')

ax.text(0.95, 0.95, r'$\mathcal{F}(D)$',
        ha='right', va='top', transform=ax.transAxes)

ax.set_xlim(-100, 100)
ax.set_ylim(-5, 85)

ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.yaxis.set_major_formatter(plt.NullFormatter())

ax = fig.add_subplot(424)
ax.plot(k, abs(np.fft.fftshift(w_fft)), '-k')

ax.text(0.95, 0.95,  r'$\mathcal{F}(W)$', ha='right', va='top',
        transform=ax.transAxes)

ax.set_xlim(-100, 100)
ax.set_ylim(-5, 85)

ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.yaxis.set_major_formatter(plt.NullFormatter())

#----------------------------------------
# plot the product of Fourier transforms
ax = fig.add_subplot(224)
ax.plot(k, abs(np.fft.fftshift(yw_fft)), '-k')

ax.text(0.95, 0.95, ('Pointwise\nproduct:\n' +
                     r'$\mathcal{F}(D) \cdot \mathcal{F}(W)$'),
        ha='right', va='top', transform=ax.transAxes,
        bbox=dict(fc='w', ec='k', pad=8), zorder=2)

ax.set_xlim(-100, 100)
ax.set_ylim(-100, 3500)

ax.set_xlabel('$k$')

ax.yaxis.set_major_formatter(plt.NullFormatter())

#------------------------------------------------------------
# Plot flow arrows
ax = fig.add_axes([0, 0, 1, 1], xticks=[], yticks=[], frameon=False)

arrowprops = dict(arrowstyle="simple",
                  color="gray", alpha=0.5,
                  shrinkA=5, shrinkB=5,
                  patchA=None,
                  patchB=None,
                  connectionstyle="arc3,rad=-0.35")

ax.annotate('', [0.57, 0.57], [0.47, 0.57],
            arrowprops=arrowprops,
            transform=ax.transAxes)
ax.annotate('', [0.57, 0.47], [0.57, 0.57],
            arrowprops=arrowprops,
            transform=ax.transAxes)
ax.annotate('', [0.47, 0.47], [0.57, 0.47],
            arrowprops=arrowprops,
            transform=ax.transAxes)

plt.show()