times[i, 1] = benchmark("gp.log_likelihood(y[:{0}])".format(n),
"from __main__ import gp, y")
if n <= 4096:
times[i, 2] = benchmark(
"""
C = gp.get_matrix(t[:{0}])
C[np.diag_indices_from(C)] += yerr[:{0}]**2
cho_factor(C)
""".format(n), """
from __main__ import gp, t, yerr
import numpy as np
from scipy.linalg import cho_factor
""")
print(n, times[i])
fig, ax = plt.subplots(1, 1, figsize=plot_setup.get_figsize(1, 1))
ax.plot(N, N / N[-1] * 2.0, "k", label="$\mathcal{O}(N)$")
ax.plot(N, times[:, 0], ".-", label="compute")
ax.plot(N, times[:, 1], ".--", label="log likelihood")
m = np.isfinite(times[:, 2])
ax.plot(N[m], times[:, 2][m], ".:", label="Cholesky")
ax.set_xscale("log")
ax.set_yscale("log")
ax.set_xlim(N.min(), N.max())
ax.set_ylim(2e-5, 3.0)
ax.set_xlabel("number of data points")
ax.set_ylabel("computational cost [seconds]")
fig.savefig("benchmark.pdf", bbox_inches="tight")
if not np.isfinite(lp):
return -np.inf
ll = gp.log_likelihood(y)
return ll + lp if np.isfinite(ll) else -np.inf
ndim = len(soln.x)
nwalkers = 32
coords = soln.x + 1e-4 * np.random.randn(nwalkers, ndim)
sampler = emcee.EnsembleSampler(nwalkers, ndim, log_probability)
coords, _, _ = sampler.run_mcmc(coords, 500)
sampler.reset()
coords, _, _ = sampler.run_mcmc(coords, 2000)
# Plot the results
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=plot_setup.get_figsize(1, 2))
ax1.errorbar(t, y, yerr=yerr, fmt=".k", capsize=0)
ax1.set_ylim(-3.25, 3.25)
ax1.set_xlim(0, 20)
ax1.set_xlabel("time [day]")
ax1.set_ylabel("relative flux [ppm]")
ax1.annotate("simulated data",
xy=(0, 0),
xycoords="axes fraction",
xytext=(5, 5),
textcoords="offset points",
ha="left",
va="bottom")
n, b, p = ax2.hist(np.exp(-sampler.flatchain[:, -2]) * (2 * np.pi),
fn = os.path.join(args.directory, fn)
data = pd.read_csv(fn, comment="#")
data_matrix = np.empty((data.xi.max() + 1, data.yi.max() + 1))
data_matrix[:] = np.nan
data_matrix[data.xi, data.yi] = data.comp_time + data.ll_time
np_time = np.array(data.numpy_comp_time + data.numpy_ll_time)
np_m = np.isfinite(np_time)
np_time = np_time[np_m]
np_n = np.array(data.n)[np_m]
np_j = np.array(data.j)[np_m]
J = np.sort(np.array(data.j.unique()))
N = np.sort(np.array(data.n.unique()))
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=get_figsize(1, 2), sharey=True)
for i, j in enumerate(J):
x = N
y = data_matrix[i]
m = np.isfinite(y)
ax1.plot(x[m], y[m], ".-", color=COLOR_CYCLE[i], label="{0:.0f}".format(j))
if len(np_time):
ax1.plot(np_n, np_time, ":k", label="direct")
if suffix == "_george":
f = N * np.log(N)**2
ax1.plot(N, 4.0 * f / f[-1], ":k", label=r"$\mathcal{O}(N\,\log^2N)$")
ax1.plot(N, 4e-2 * N / N[-1], "k", label=r"$\mathcal{O}(N)$")
ax1.legend(loc="lower right", bbox_to_anchor=(1.05, 0), fontsize=8)
x.append(x0[m])
mu = np.median(y0[m])
y.append((y0[m] / mu - 1.0) * 1e6)
yerr.append(1e6 * data["PDCSAP_FLUX_ERR"][m] / mu)
x = np.concatenate(x)
y = np.concatenate(y)
yerr = np.concatenate(yerr)
inds = np.argsort(x)
x = np.ascontiguousarray(x[inds], dtype=float)
y = np.ascontiguousarray(y[inds], dtype=float)
yerr = np.ascontiguousarray(yerr[inds], dtype=float)
# Plot the light curve.
fig, ax = plt.subplots(1, 1, figsize=get_figsize())
ax.plot(x, y, "k", rasterized=True)
ax.set_xlim(x.min(), x.max())
ax.set_ylim(np.std(y) * np.array([-5.0, 5.0]))
ax.set_xlabel("time [KBJD]")
ax.set_ylabel("relative flux [ppm]")
ax.xaxis.set_major_locator(plt.MaxNLocator(4))
fig.savefig(format_filename("time_series"), bbox_inches="tight", dpi=300)
plt.close(fig)
# Define a frequency grid for the periodogram
freq_uHz = np.linspace(1, 300, 100000)
freq = freq_uHz * uHz_conv
# Compute the periodogram on the full dataset
model = LombScargle(x, y)
print(gp.get_parameter_dict())
gp.set_parameter_vector(best[1])
ml_params = np.array(best[1])
# Compute the model predictions
gp.set_parameter_vector(ml_params)
x = np.linspace(t.min(), t.max(), 5000)
mu, var = gp.predict(y, x, return_var=True)
omega = np.exp(np.linspace(np.log(0.1), np.log(10), 5000))
psd = gp.kernel.get_psd(omega)
period = np.exp(gp.get_parameter("kernel:log_period"))
tau = np.linspace(0, 4 * period, 5000)
acf = gp.kernel.get_value(tau)
# Set up the figure
fig, axes = plt.subplots(1, 3, figsize=get_figsize(1, 3))
# Plot the data
ax = axes[0]
color = COLORS["MODEL_1"]
ax.plot(t - 380, y, ".k", rasterized=True, zorder=-1)
ax.plot(x - 380, mu, color=color, zorder=100)
ax.fill_between(x - 380,
mu + np.sqrt(var),
mu - np.sqrt(var),
color=color,
alpha=0.3,
zorder=100)
ax.set_xlim(0, 50)
ax.set_ylim(-1.2, 1.2)
ax.set_xlabel("time [days]")
tau = np.max(tau)
burnin = int(10 * tau)
tau2 = emcee3.autocorr.integrated_time(mean_traces[burnin:], c=3, axis=0)
print("tau: {0}".format(tau2))
print("N_ind: {0}".format(np.prod(samples.shape[:2]) / tau2))
# Plot the traces
names = list(gp.get_parameter_names())
for i in range(len(names)):
name = names[i].split(":")[-1]
if name.startswith("log"):
name = "log("+name[4:]+")"
names[i] = name.replace("_", " ")
fig, axes = plt.subplots(samples.shape[-1] + 1, 1, sharex=True,
figsize=get_figsize(samples.shape[-1]//2, 2))
for i in range(samples.shape[-1]):
axes[i].plot(samples[:, :, i], color="k", alpha=0.3, rasterized=True)
axes[i].set_ylabel(names[i])
axes[i].yaxis.set_major_locator(plt.MaxNLocator(5))
axes[i].axvline(burnin, color=COLORS["MODEL_1"])
axes[-1].plot(backend.get_log_probability(), color="k", alpha=0.3,
rasterized=True)
axes[-1].set_ylabel("log(prob)")
axes[-1].yaxis.set_major_locator(plt.MaxNLocator(5))
axes[-1].axvline(burnin, color=COLORS["MODEL_1"])
fig.savefig(format_filename("trace"))
plt.close(fig)
data_matrix[:] = np.nan
data_matrix[data.xi, data.yi] = data.comp_time + data.ll_time
try:
np_time = np.array(data.numpy_comp_time + data.numpy_ll_time)
except AttributeError:
np_time = np.nan + np.zeros(len(data))
np_m = np.isfinite(np_time)
np_time = np_time[np_m]
np_n = np.array(data.n)[np_m]
np_j = np.array(data.j)[np_m]
J = np.sort(np.array(data.j.unique()))
N = np.sort(np.array(data.n.unique()))
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=get_figsize(1, 2), sharey=True)
for i, j in enumerate(J):
x = N
y = data_matrix[i]
m = np.isfinite(y)
ax1.plot(x[m], y[m], ".-", color=COLOR_CYCLE[i],
label="{0:.0f}".format(j))
if len(np_time):
ax1.plot(np_n, np_time, ":k", label="direct")
if suffix == "_george":
f = N * np.log(N)**2
ax1.plot(N, 4.0 * f / f[-1], ":k", label=r"$\mathcal{O}(N\,\log^2N)$")
ax1.plot(N, 3e-2 * N / N[-1], "k", label=r"$\mathcal{O}(N)$")
if np.allclose(Q, 0.5):
return np.exp(-t) * (1.0 + t)
b = 1.0 / np.sqrt(4*Q**2 - 1)
c = 0.5 / Q
d = 0.5 * np.sqrt(4*Q**2 - 1) / Q
return np.exp(-c * t) * (np.cos(d*t)+b*np.sin(d*t))
def lorentz_psd(Q, x):
return Q**2 * (1.0 / ((x - 1)**2 * (2*Q)**2 + 1) +
1.0 / ((x + 1)**2 * (2*Q)**2 + 1))
def lorentz_acf(Q, tau):
t = np.abs(tau)
return np.exp(-0.5*t/Q) * np.cos(t)
fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=get_figsize(1, 3))
x = 10**np.linspace(-1.1, 1.1, 5000)
tau = np.linspace(0, 20, 1000)
for i, (Q_name, Q) in enumerate(
[("1/2", 0.5), ("1/\\sqrt{2}", 1./np.sqrt(2)), ("2", 2.0),
("10", 10.0)]):
l, = ax1.plot(x, sho_psd(Q, x), label="$Q = {0}$".format(Q_name), lw=1.5)
c = l.get_color()
ax2.plot(tau, sho_acf(Q, tau), label="$Q = {0}$".format(Q_name), lw=1.5,
color=c)
K = sho_acf(Q, tau[:, None] - tau[None, :])
y = np.random.multivariate_normal(np.zeros(len(tau)), K, size=3)
ax3.axhline(-5*i, color="k", lw=0.75)
ax3.plot(tau, -5*i + (y - np.mean(y, axis=1)[:, None]).T, color=c,
if np.allclose(Q, 0.5):
return np.exp(-t) * (1.0 + t)
b = 1.0 / np.sqrt(4*Q**2 - 1)
c = 0.5 / Q
d = 0.5 * np.sqrt(4*Q**2 - 1) / Q
return np.exp(-c * t) * (np.cos(d*t)+b*np.sin(d*t))
def lorentz_psd(Q, x):
return Q**2 * (1.0 / ((x - 1)**2 * (2*Q)**2 + 1) +
1.0 / ((x + 1)**2 * (2*Q)**2 + 1))
def lorentz_acf(Q, tau):
t = np.abs(tau)
return np.exp(-0.5*t/Q) * np.cos(t)
fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=get_figsize(1, 3))
x = 10**np.linspace(-1.1, 1.1, 5000)
tau = np.linspace(0, 20, 1000)
for i, (Q_name, Q) in enumerate(
[("1/2", 0.5), ("1/\\sqrt{2}", 1./np.sqrt(2)), ("2", 2.0),
("10", 10.0)]):
l, = ax1.plot(x, sho_psd(Q, x), label="$Q = {0}$".format(Q_name), lw=1.5)
c = l.get_color()
ax2.plot(tau, sho_acf(Q, tau), label="$Q = {0}$".format(Q_name), lw=1.5,
color=c)
K = sho_acf(Q, tau[:, None] - tau[None, :])
y = np.random.multivariate_normal(np.zeros(len(tau)), K, size=3)
ax3.axhline(-5*i, color="k", lw=0.75)
ax3.plot(tau, -5*i + (y - np.mean(y, axis=1)[:, None]).T, color=c,
ml_params = np.array(r.x)
print("Maximum log-likelihood: {0}".format(gp.log_likelihood(y)))
# Compute the maximum likelihood predictions
x = np.linspace(t.min(), t.max(), 5000)
trend = gp.predict(y, t, return_cov=False)
trend -= gp.mean.get_value(t) - gp.mean.mean_flux
mu, var = gp.predict(y, x, return_var=True)
std = np.sqrt(var)
mean_mu = gp.mean.get_value(x)
mu -= mean_mu
wn = np.exp(gp.log_white_noise.value)
ml_yerr = np.sqrt(yerr**2 + wn)
# Plot the maximum likelihood predictions
fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True, figsize=get_figsize(1, 2))
ax1.errorbar(t - t.min(), y, yerr=ml_yerr, fmt=".k", capsize=0)
ax1.plot(x - t.min(), mu)
ax1.set_ylim(-0.72, 0.72)
ax1.yaxis.set_major_locator(plt.MaxNLocator(5))
ax1.set_ylabel("raw [ppt]")
ax2.errorbar(t - t.min(), y - trend, yerr=ml_yerr, fmt=".k", capsize=0)
ax2.plot(x - t.min(), mean_mu - gp.mean.mean_flux)
ax2.set_xlim(0, t.max() - t.min())
ax2.set_ylim(-0.41, 0.1)
ax2.yaxis.set_major_locator(plt.MaxNLocator(5))
ax2.set_ylabel("de-trended [ppt]")
ax2.set_xlabel("time [days]")
fig.savefig("transit-ml.pdf")
for i in range(8):
full[-2 * d - 2 * d * i:-d - 2 * d * i, d:2 * d] = 8 - i
# Seaborn set1
c = [(0.89411765336990356, 0.10196078568696976, 0.10980392247438431),
(0.21602460800432691, 0.49487120380588606, 0.71987698697576341),
(0.30426760128900115, 0.68329106055054012, 0.29293349969620797),
(0.60083047361934883, 0.30814303335021526, 0.63169552298153153),
(1.0, 0.50591311045721465, 0.0031372549487095253),
(0.99315647868549117, 0.9870049982678657, 0.19915417450315812),
(0.65845446095747107, 0.34122261685483596, 0.1707958535236471),
(0.95850826852461868, 0.50846600392285513, 0.74492888871361229),
(0.60000002384185791, 0.60000002384185791, 0.60000002384185791),
(0.89411765336990356, 0.10196078568696976, 0.10980392247438431)]
cmap = matplotlib.colors.ListedColormap(["white"] + list(c), name="cmap")
fig, ax = plt.subplots(1, 1, figsize=get_figsize(2.3, 2.3))
ax.pcolor(full, cmap=cmap, vmin=0, vmax=len(c))
# Plot the edges
for i, j in product(range(full_dim), range(full_dim)):
if full[i, j] == 0:
continue
ax.plot((j, j, j + 1, j + 1, j), (i, i + 1, i + 1, i, i), "k", lw=0.5)
ax.set_ylim(full_dim, 0)
ax.set_xlim(0, full_dim)
ax.set_frame_on(False)
ax.set_xticks([])
ax.set_yticks([])
names = [