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rrlrae.py
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rrlrae.py
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import importlib.util
import arviz as az
import matplotlib.pyplot as plt
import numpy as np
from matplotlib import pyplot as plt
from matplotlib.text import Text
from astroML.datasets import fetch_rrlyrae_templates
from scipy.signal import fftconvolve
from scipy import fftpack
from matplotlib import mlab
from astroML.datasets import fetch_LIGO_large
from scipy.fftpack import fft
from scipy.stats import norm
from astroML.fourier import PSD_continuous
from astroML.plotting import setup_text_plots
templates = fetch_rrlyrae_templates()
x, y = templates['115r'].T
fig = plt.figure(figsize=(5, 5))
fig.subplots_adjust(hspace=0)
kvals = [1, 3, 8]
subplots = [311, 312, 313]
for (k, subplot) in zip(kvals, subplots):
ax = fig.add_subplot(subplot)
# Use FFT to fit a truncated Fourier series
y_fft = np.fft.fft(y)
y_fft[k + 1:-k] = 0
y_fit = np.fft.ifft(y_fft).real
# plot the true value and the k-term reconstruction
ax.plot(np.concatenate([x, 1 + x]),
np.concatenate([y, y]), '--k', lw=2)
ax.plot(np.concatenate([x, 1 + x]),
np.concatenate([y_fit, y_fit]), color='gray')
label = "%i mode" % k
if k > 1:
label += 's'
ax.text(0.02, 0.1, label, ha='left', va='bottom',
transform=ax.transAxes)
if subplot == subplots[-1]:
ax.set_xlabel('phase')
else:
ax.xaxis.set_major_formatter(plt.NullFormatter())
if subplot == subplots[1]:
ax.set_ylabel('amplitude')
ax.yaxis.set_major_formatter(plt.NullFormatter())
ax.set_xlim(0, 2)
ax.set_ylim(1.1, -0.1)
plt.show()
# Generate data
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 function
w = np.zeros_like(x)
w[(x > 0.12) & (x < 0.28)] = 1
# Set up the plots
fig = plt.figure(figsize=(5, 5))
ax1 = fig.add_subplot(111)
# plot the data and window function
ax1.plot(x, y, '-k', label=r'data $D(x)$')
ax1.fill(x, w, color='gray', alpha=0.5,
label=r'window $W(x)$')
ax1.fill(x, w[::-1], color='gray', alpha=0.5)
ax1.legend()
ax1.xaxis.set_major_formatter(plt.NullFormatter())
ax1.set_xlabel('$x$')
ax1.set_ylabel('$D$')
ax1.set_xlim(0.01, 0.99)
ax1.set_ylim(0.0, 2.0)
(0.0, 2.0)
plt.show()
# Generate data
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 function
w = np.zeros_like(x)
w[(x > 0.12) & (x < 0.28)] = 1
# Set up the plots
fig = plt.figure(figsize=(5, 5))
ax1 = fig.add_subplot(111)
# plot the data and window function
ax1.plot(x, y, '-k', label=r'data $D(x)$')
ax1.fill(x, w, color='gray', alpha=0.5,
label=r'window $W(x)$')
ax1.fill(x, w[::-1], color='gray', alpha=0.5)
ax1.legend()
ax1.xaxis.set_major_formatter(plt.NullFormatter())
ax1.set_xlabel('$x$')
ax1.set_ylabel('$D$')
ax1.set_xlim(0.01, 0.99)
ax1.set_ylim(0.0, 2.0)
(0.0, 2.0)
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))
# plot the Fourier transforms
N = len(x)
k = - 0.5 * N + np.arange(N) * 1. / N / (x[1] - x[0])
ax = fig.add_subplot(211)
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 = fig.add_subplot(212)
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.set_xlabel('$x$')
Text(0.5, 0, '$x$')
fig = plt.figure(figsize=(5, 5))
ax = fig.add_subplot(111)
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'), zorder=2)
ax.set_xlim(-100, 100)
ax.set_ylim(-100, 3500)
ax.set_xlabel('$k$')
Text(0.5, 0, '$k$')
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
# plot the convolution
fig = plt.figure(figsize=(5, 5))
ax = fig.add_subplot(111)
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'), 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)
ax.set_xlabel('$x$')
ax.set_ylabel('$D_W$')
ax.set_xlim(0.01, 0.99)
ax.set_ylim(0, 1.99)
(0.0, 1.99)
# 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'), 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'), 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()
def gaussian(x, a=1.0):
return np.exp(-0.5 * (x / a) ** 2)
def gaussian_FT(f, a=1.0):
return np.sqrt(2 * np.pi * a ** 2) * np.exp(-2 * (np.pi * a * f) ** 2)
a = 1.0
t = np.linspace(-5, 5, 1000)
h = gaussian(t, a)
f = np.linspace(-2, 2, 1000)
H = gaussian_FT(f, a)
# Two plots: one well-sampled, one over-sampled
N = 12
def plot_aliasing(dt):
"""
Parameters
----------
dt : float
"""
# define time-space sampling
t_sample = dt * (np.arange(N) - N / 2)
h_sample = gaussian(t_sample, a)
# Fourier transform of time-space sampling
df = 1. / dt
f_sample = df * (np.arange(N) - N / 2)
# Plot the results
fig = plt.figure(figsize=(7, 3.5))
fig.subplots_adjust(left=0.07, right=0.95, wspace=0.16,
bottom=0.1, top=0.85, hspace=0.05)
# First plot: sampled time-series
ax = fig.add_subplot(221)
ax.plot(t, h, '-k')
for ts in t_sample:
ax.annotate('', (ts, 0.5), (ts, 0), ha='center', va='center',
arrowprops=dict(arrowstyle='->'))
ax.text(0.03, 0.95,
("Signal and Sampling Window\n" +
r"Sampling Rate $\Delta t$"),
ha='left', va='top', transform=ax.transAxes)
ax.set_ylabel('$h(t)$')
ax.set_xlim(-5, 5)
ax.set_ylim(0, 1.4)
ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.yaxis.set_major_formatter(plt.NullFormatter())
ax.set_title('Time Domain: Multiplication')
# second plot: frequency space
ax = fig.add_subplot(222)
ax.plot(f, H, '-k')
for fs in f_sample:
ax.annotate('', (fs, 1.5), (fs, 0), ha='center', va='center',
arrowprops=dict(arrowstyle='->'))
ax.text(0.03, 0.95,
("FT of Signal and Sampling Window\n" +
r"$\Delta f = 1 / \Delta t$"),
ha='left', va='top', transform=ax.transAxes)
ax.set_ylabel('$H(f)$')
ax.set_xlim(-1.5, 1.5)
ax.set_ylim(0, 3.8)
ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.yaxis.set_major_formatter(plt.NullFormatter())
ax.set_title('Frequency Domain: Convolution')
# third plot: windowed function
ax = fig.add_subplot(223)
for (ts, hs) in zip(t_sample, h_sample):
if hs < 0.1:
continue
ax.annotate('', (ts, hs), (ts, 0), ha='center', va='center',
arrowprops=dict(arrowstyle='->'))
ax.plot(t, h, ':k')
ax.text(0.03, 0.95, "Sampled signal: pointwise\nmultiplication",
ha='left', va='top', transform=ax.transAxes)
ax.set_xlabel('$t$')
ax.set_ylabel('$h(t)$')
ax.set_xlim(-5, 5)
ax.set_ylim(0, 1.4)
ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.yaxis.set_major_formatter(plt.NullFormatter())
# fourth plot: convolved PSD
ax = fig.add_subplot(224)
window = np.array([gaussian_FT(f - fs, a) for fs in f_sample])
ax.plot(f, window.sum(0), '-k')
if dt > 1:
ax.plot(f, window.T, ':k')
ax.text(0.03, 0.95, "Convolution of signal FT\nand window FT",
ha='left', va='top', transform=ax.transAxes)
ax.set_xlabel('$f$')
ax.set_ylabel('$H(f)$')
ax.set_xlim(-1.5, 1.5)
ax.set_ylim(0, 3.8)
ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.yaxis.set_major_formatter(plt.NullFormatter())
if dt > 1:
fig.suptitle(r"Undersampled data: $\Delta t > t_c$")
else:
fig.suptitle(r"Well-sampled data: $\Delta t < t_c$")
plot_aliasing(0.9)
plt.show()
plot_aliasing(1.2)
plt.show()
Nbins = 2 ** 15
Nobs = 40
f = lambda t: np.sin(np.pi * t / 3)
t = np.linspace(-100, 200, Nbins)
dt = t[1] - t[0]
y = f(t)
# select observations
np.random.seed(42)
t_obs = 100 * np.random.random(40)
D = abs(t_obs[:, np.newaxis] - t)
i = np.argmin(D, 1)
t_obs = t[i]
y_obs = y[i]
window = np.zeros(Nbins)
window[i] = 1
# Compute PSDs
Nfreq = int(Nbins / 2)
dt = t[1] - t[0]
df = 1. / (Nbins * dt)
f = df * np.arange(Nfreq)
PSD_window = abs(np.fft.fft(window)[:Nfreq]) ** 2
PSD_y = abs(np.fft.fft(y)[:Nfreq]) ** 2
PSD_obs = abs(np.fft.fft(y * window)[:Nfreq]) ** 2
PSD_window /= 500
PSD_y /= PSD_y.max()
PSD_obs /= 500
# Prepare the figures
fig = plt.figure(figsize=(8, 4))
fig.subplots_adjust(bottom=0.15, hspace=0.2, wspace=0.25,
left=0.12, right=0.95)
# First panel: data vs time
ax = fig.add_subplot(221)
ax.plot(t, y, '-', c='gray')
ax.plot(t_obs, y_obs, '.k', ms=4)
ax.text(0.95, 0.93, "Data", ha='right', va='top', transform=ax.transAxes)
ax.set_ylabel('$y(t)$')
ax.set_xlim(0, 100)
ax.set_ylim(-1.5, 1.8)
# Second panel: PSD of data
ax = fig.add_subplot(222)
ax.fill(f, PSD_y, fc='gray', ec='gray')
ax.plot(f, PSD_obs, '-', c='black')
ax.text(0.95, 0.93, "Data PSD", ha='right', va='top', transform=ax.transAxes)
ax.set_ylabel('$P(f)$')
ax.set_xlim(0, 1.0)
ax.set_ylim(-0.1, 1.1)
# Third panel: window vs time
ax = fig.add_subplot(223)
ax.plot(t, window, '-', c='black')
ax.text(0.95, 0.93, "Window", ha='right', va='top', transform=ax.transAxes)
ax.set_xlabel('$t$')
ax.set_ylabel('$y(t)$')
ax.set_xlim(0, 100)
ax.set_ylim(-0.2, 1.5)
# Fourth panel: PSD of window
ax = fig.add_subplot(224)
ax.plot(f, PSD_window, '-', c='black')
ax.text(0.95, 0.93, "Window PSD", ha='right', va='top', transform=ax.transAxes)
ax.set_xlabel('$f$')
ax.set_ylabel('$P(f)$')
ax.set_xlim(0, 1.0)
ax.set_ylim(-0.1, 1.1)
plt.show()
np.random.seed(1)
tj = np.linspace(-25, 25, 512)
hj = np.sin(tj)
hj *= norm(0, 10).pdf(tj)
# plot the results
fig = plt.figure(figsize=(5, 3.75))
fig.subplots_adjust(hspace=0.35)
ax1 = fig.add_subplot(211)
ax2 = fig.add_subplot(212)
offsets = (0, 0.15)
colors = ('black', 'gray')
linewidths = (1, 2)
errors = (0.005, 0.05)
for (offset, color, error, linewidth) in zip(offsets, colors,
errors, linewidths):
# compute the PSD
err = np.random.normal(0, error, size=hj.shape)
hj_N = hj + err + offset
fk, PSD = PSD_continuous(tj, hj_N)
# plot the data and PSD
ax1.scatter(tj, hj_N, s=4, c=color, lw=0)
ax1.plot(tj, 0 * tj + offset, '-', c=color, lw=1)
ax2.plot(fk, PSD, '-', c=color, lw=linewidth)
# vertical line marking the expected peak location
ax2.plot([0.5 / np.pi, 0.5 / np.pi], [-0.1, 1], ':k', lw=1)
ax1.set_xlim(-25, 25)
ax1.set_ylim(-0.1, 0.3001)
ax1.set_xlabel('$t$')
ax1.set_ylabel('$h(t)$')
ax1.yaxis.set_major_locator(plt.MultipleLocator(0.1))
ax2.set_xlim(0, 0.8)
ax2.set_ylim(-0.101, 0.801)
ax2.set_xlabel('$f$')
ax2.set_ylabel('$PSD(f)$')
plt.show()