Note

This tutorial was generated from an IPython notebook that can be downloaded here.

# Case study: K2-24, putting it all together¶

theano version: 1.0.4
pymc3 version: 3.7
exoplanet version: 0.2.1.dev0


In this tutorial, we will combine many of the previous tutorials to perform a fit of the K2-24 system using the K2 transit data and the RVs from Petigura et al. (2016). This is the same system that we fit in the Radial velocity fitting tutorial and we’ll combine that model with the transit model from the Transit fitting tutorial and the Gaussian Process noise model from the Gaussian process models for stellar variability tutorial.

## Datasets and initializations¶

To get started, let’s download the relevant datasets. First, the transit light curve from Everest:

import numpy as np
import matplotlib.pyplot as plt

from astropy.io import fits
from scipy.signal import savgol_filter

lc_url = "https://archive.stsci.edu/hlsps/everest/v2/c02/203700000/71098/hlsp_everest_k2_llc_203771098-c02_kepler_v2.0_lc.fits"
with fits.open(lc_url) as hdus:
lc = hdus.data

# Work out the exposure time
texp = lc_hdr["FRAMETIM"] * lc_hdr["NUM_FRM"]
texp /= 60.0 * 60.0 * 24.0

m = (np.arange(len(lc)) > 100) & np.isfinite(lc["FLUX"]) & np.isfinite(lc["TIME"])
bad_bits=[1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17]
qual = lc["QUALITY"]
for b in bad_bits:
m &= qual & 2 ** (b - 1) == 0

# Convert to parts per thousand
x = lc["TIME"][m]
y = lc["FLUX"][m]
mu = np.median(y)
y = (y / mu - 1) * 1e3

# Identify outliers
m = np.ones(len(y), dtype=bool)
for i in range(10):
y_prime = np.interp(x, x[m], y[m])
smooth = savgol_filter(y_prime, 101, polyorder=3)
resid = y - smooth
sigma = np.sqrt(np.mean(resid**2))
m0 = np.abs(resid) < 3*sigma
if m.sum() == m0.sum():
m = m0
break
m = m0

# Only discard positive outliers
m = resid < 3*sigma

# Shift the data so that the K2 data start at t=0. This tends to make the fit
# better behaved since t0 covaries with period.
x_ref = np.min(x[m])
x -= x_ref

# Plot the data
plt.plot(x, y, "k", label="data")
plt.plot(x, smooth)
plt.plot(x[~m], y[~m], "xr", label="outliers")
plt.legend(fontsize=12)
plt.xlim(x.min(), x.max())
plt.xlabel("time")
plt.ylabel("flux")

# Make sure that the data type is consistent
x = np.ascontiguousarray(x[m], dtype=np.float64)
y = np.ascontiguousarray(y[m], dtype=np.float64)
smooth = np.ascontiguousarray(smooth[m], dtype=np.float64) Then the RVs from RadVel:

import pandas as pd

data = pd.read_csv(url, index_col=0)

# Don't forget to remove the time offset from above!
x_rv = np.array(data.t) - x_ref
y_rv = np.array(data.vel)
yerr_rv = np.array(data.errvel)

plt.errorbar(x_rv, y_rv, yerr=yerr_rv, fmt=".k")
plt.xlabel("time [days]") We can initialize the transit parameters using the box least squares periodogram from AstroPy. (Note: you’ll need AstroPy v3.1 or more recent to use this feature.) A full discussion of transit detection and vetting is beyond the scope of this tutorial so let’s assume that we know that there are two periodic transiting planets in this dataset.

from astropy.stats import BoxLeastSquares

m = np.zeros(len(x), dtype=bool)
period_grid = np.exp(np.linspace(np.log(5), np.log(50), 50000))
bls_results = []
periods = []
t0s = []
depths = []

# Compute the periodogram for each planet by iteratively masking out
# transits from the higher signal to noise planets. Here we're assuming
# that we know that there are exactly two planets.
for i in range(2):
bls = BoxLeastSquares(x[~m], y[~m] - smooth[~m])
bls_power = bls.power(period_grid, 0.1, oversample=20)
bls_results.append(bls_power)

# Save the highest peak as the planet candidate
index = np.argmax(bls_power.power)
periods.append(bls_power.period[index])
t0s.append(bls_power.transit_time[index])
depths.append(bls_power.depth[index])

# Mask the data points that are in transit for this candidate
m |= bls.transit_mask(x, periods[-1], 0.5, t0s[-1])


Let’s plot the initial transit estimates based on these periodograms:

fig, axes = plt.subplots(len(bls_results), 2, figsize=(15, 10))

for i in range(len(bls_results)):
# Plot the periodogram
ax = axes[i, 0]
ax.axvline(np.log10(periods[i]), color="C1", lw=5, alpha=0.8)
ax.plot(np.log10(bls_results[i].period), bls_results[i].power, "k")
ax.annotate("period = {0:.4f} d".format(periods[i]),
(0, 1), xycoords="axes fraction",
xytext=(5, -5), textcoords="offset points",
va="top", ha="left", fontsize=12)
ax.set_ylabel("bls power")
ax.set_yticks([])
ax.set_xlim(np.log10(period_grid.min()), np.log10(period_grid.max()))
if i < len(bls_results) - 1:
ax.set_xticklabels([])
else:
ax.set_xlabel("log10(period)")

# Plot the folded transit
ax = axes[i, 1]
p = periods[i]
x_fold = (x - t0s[i] + 0.5*p) % p - 0.5*p
m = np.abs(x_fold) < 0.4
ax.plot(x_fold[m], y[m] - smooth[m], ".k")

# Overplot the phase binned light curve
bins = np.linspace(-0.41, 0.41, 32)
denom, _ = np.histogram(x_fold, bins)
num, _ = np.histogram(x_fold, bins, weights=y - smooth)
denom[num == 0] = 1.0
ax.plot(0.5*(bins[1:] + bins[:-1]), num / denom, color="C1")

ax.set_xlim(-0.4, 0.4)
ax.set_ylabel("relative flux [ppt]")
if i < len(bls_results) - 1:
ax.set_xticklabels([])
else:
ax.set_xlabel("time since transit") The discovery paper for K2-24 (Petigura et al. (2016)) includes the following estimates of the stellar mass and radius in Solar units:

M_star_petigura = 1.12, 0.05
R_star_petigura = 1.21, 0.11


Finally, using this stellar mass, we can also estimate the minimum masses of the planets given these transit parameters.

import exoplanet as xo
import astropy.units as u

msini = xo.estimate_minimum_mass(periods, x_rv, y_rv, yerr_rv, t0s=t0s, m_star=M_star_petigura)
msini = msini.to(u.M_earth)
print(msini)

[32.80060146 23.89885976] earthMass


## A joint transit and radial velocity model in PyMC3¶

Now, let’s define our full model in PyMC3. There’s a lot going on here, but I’ve tried to comment it and most of it should be familiar from the previous tutorials (Radial velocity fitting, Transit fitting, Scalable Gaussian processes in PyMC3, and Gaussian process models for stellar variability). In this case, I’ve put the model inside a model “factory” function because we’ll do some sigma clipping below.

import pymc3 as pm
import theano.tensor as tt

t_rv = np.linspace(x_rv.min()-5, x_rv.max()+5, 1000)

if mask is None:
mask = np.ones(len(x), dtype=bool)
with pm.Model() as model:

# Parameters for the stellar properties
mean = pm.Normal("mean", mu=0.0, sd=10.0)
BoundedNormal = pm.Bound(pm.Normal, lower=0, upper=3)
m_star = BoundedNormal("m_star", mu=M_star_petigura, sd=M_star_petigura)
r_star = BoundedNormal("r_star", mu=R_star_petigura, sd=R_star_petigura)

# Orbital parameters for the planets
logm = pm.Normal("logm", mu=np.log(msini.value), sd=1, shape=2)
logP = pm.Normal("logP", mu=np.log(periods), sd=1, shape=2)
t0 = pm.Normal("t0", mu=np.array(t0s), sd=1, shape=2)
logr = pm.Normal("logr", mu=0.5*np.log(1e-3*np.array(depths)) + np.log(R_star_petigura),
sd=1.0, shape=2)
r_pl = pm.Deterministic("r_pl", tt.exp(logr))
ror = pm.Deterministic("ror", r_pl / r_star)
b = xo.distributions.UnitUniform("b", shape=2)

# This is the eccentricity prior from Kipping (2013):
# https://arxiv.org/abs/1306.4982
BoundedBeta = pm.Bound(pm.Beta, lower=0, upper=1-1e-5)
ecc = BoundedBeta("ecc", alpha=0.867, beta=3.03, shape=2,
testval=np.array([0.1, 0.1]))
omega = xo.distributions.Angle("omega", shape=2)

# RV jitter & a quadratic RV trend
logs_rv = pm.Normal("logs_rv", mu=np.log(np.median(yerr_rv)), sd=5)
trend = pm.Normal("trend", mu=0, sd=10.0**-np.arange(3)[::-1], shape=3)

# Transit jitter & GP parameters
logs2 = pm.Normal("logs2", mu=np.log(np.var(y[mask])), sd=10)
logw0_guess = np.log(2*np.pi/10)
logw0 = pm.Normal("logw0", mu=logw0_guess, sd=10)

# We'll parameterize using the maximum power (S_0 * w_0^4) instead of
# S_0 directly because this removes some of the degeneracies between
# S_0 and omega_0
logpower = pm.Normal("logpower",
sd=10)
logS0 = pm.Deterministic("logS0", logpower - 4 * logw0)

# Tracking planet parameters
period = pm.Deterministic("period", tt.exp(logP))
m_pl = pm.Deterministic("m_pl", tt.exp(logm))

# Orbit model
orbit = xo.orbits.KeplerianOrbit(
r_star=r_star, m_star=m_star,
period=period, t0=t0, b=b, m_planet=m_pl,
ecc=ecc, omega=omega,
m_planet_units=msini.unit)

# Compute the model light curve using starry
light_curves = xo.LimbDarkLightCurve(u_star).get_light_curve(
orbit=orbit, r=r_pl, t=x[mask], texp=texp)*1e3
light_curve = pm.math.sum(light_curves, axis=-1) + mean
pm.Deterministic("light_curves", light_curves)

# GP model for the light curve
kernel = xo.gp.terms.SHOTerm(log_S0=logS0, log_w0=logw0, Q=1/np.sqrt(2))
gp = xo.gp.GP(kernel, x[mask], tt.exp(logs2) + tt.zeros(mask.sum()), J=2)
pm.Potential("transit_obs", gp.log_likelihood(y[mask] - light_curve))
pm.Deterministic("gp_pred", gp.predict())

# Set up the RV model and save it as a deterministic
# for plotting purposes later

# Define the background RV model
A = np.vander(x_rv - 0.5*(x_rv.min() + x_rv.max()), 3)
bkg = pm.Deterministic("bkg", tt.dot(A, trend))

# The likelihood for the RVs
rv_model = pm.Deterministic("rv_model", tt.sum(vrad, axis=-1) + bkg)
err = tt.sqrt(yerr_rv**2 + tt.exp(2*logs_rv))
pm.Normal("obs", mu=rv_model, sd=err, observed=y_rv)

A_pred = np.vander(t_rv - 0.5*(x_rv.min() + x_rv.max()), 3)
bkg_pred = pm.Deterministic("bkg_pred", tt.dot(A_pred, trend))
pm.Deterministic("rv_model_pred", tt.sum(vrad_pred, axis=-1) + bkg_pred)

# Fit for the maximum a posteriori parameters, I've found that I can get
# a better solution by trying different combinations of parameters in turn
if start is None:
start = model.test_point
map_soln = xo.optimize(start=start, vars=[trend])
map_soln = xo.optimize(start=map_soln, vars=[logs2])
map_soln = xo.optimize(start=map_soln, vars=[logr, b])
map_soln = xo.optimize(start=map_soln, vars=[logP, t0])
map_soln = xo.optimize(start=map_soln, vars=[logs2, logpower])
map_soln = xo.optimize(start=map_soln, vars=[logw0])
map_soln = xo.optimize(start=map_soln)

return model, map_soln

model0, map_soln0 = build_model()

optimizing logp for variables: ['trend']
message: Optimization terminated successfully.
logp: -8250.572324633431 -> -8233.972131420609
optimizing logp for variables: ['logs2']
message: Optimization terminated successfully.
logp: -8233.972131420609 -> 1985.112196637086
optimizing logp for variables: ['b_logodds__', 'logr']
message: Optimization terminated successfully.
logp: 1985.112196637086 -> 2679.6622840411287
optimizing logp for variables: ['t0', 'logP']
message: Desired error not necessarily achieved due to precision loss.
logp: 2679.6622840411287 -> 3334.1377479694006
optimizing logp for variables: ['logpower', 'logs2']
message: Desired error not necessarily achieved due to precision loss.
logp: 3334.137747969397 -> 4291.60331519934
optimizing logp for variables: ['logw0']
message: Optimization terminated successfully.
logp: 4291.60331519934 -> 4365.555598191493
optimizing logp for variables: ['logpower', 'logw0', 'logs2', 'trend', 'logs_rv', 'omega_angle__', 'ecc_interval__', 'b_logodds__', 'logr', 't0', 'logP', 'logm', 'r_star_interval__', 'm_star_interval__', 'u_star_quadlimbdark__', 'mean']
message: Desired error not necessarily achieved due to precision loss.
logp: 4365.555598191475 -> 4777.10345480088


Now let’s plot the map radial velocity model.

def plot_rv_curve(soln):
fig, axes = plt.subplots(2, 1, figsize=(10, 5), sharex=True)

ax = axes
ax.errorbar(x_rv, y_rv, yerr=yerr_rv, fmt=".k")
ax.plot(t_rv, soln["vrad_pred"], "--k", alpha=0.5)
ax.plot(t_rv, soln["bkg_pred"], ":k", alpha=0.5)
ax.plot(t_rv, soln["rv_model_pred"], label="model")
ax.legend(fontsize=10)

ax = axes
err = np.sqrt(yerr_rv**2+np.exp(2*soln["logs_rv"]))
ax.errorbar(x_rv, y_rv - soln["rv_model"], yerr=err, fmt=".k")
ax.axhline(0, color="k", lw=1)
ax.set_ylabel("residuals [m/s]")
ax.set_xlim(t_rv.min(), t_rv.max())
ax.set_xlabel("time [days]")

plot_rv_curve(map_soln0) That looks pretty similar to what we got in Radial velocity fitting. Now let’s also plot the transit model.

def plot_light_curve(soln, mask=None):
if mask is None:
mask = np.ones(len(x), dtype=bool)

fig, axes = plt.subplots(3, 1, figsize=(10, 7), sharex=True)

ax = axes
gp_mod = soln["gp_pred"] + soln["mean"]
ax.plot(x[mask], gp_mod, color="C2", label="gp model")
ax.legend(fontsize=10)
ax.set_ylabel("relative flux [ppt]")

ax = axes
ax.plot(x[mask], y[mask] - gp_mod, "k", label="de-trended data")
for i, l in enumerate("bc"):
mod = soln["light_curves"][:, i]
ax.plot(x[mask], mod, label="planet {0}".format(l))
ax.legend(fontsize=10, loc=3)
ax.set_ylabel("de-trended flux [ppt]")

ax = axes
mod = gp_mod + np.sum(soln["light_curves"], axis=-1)
ax.axhline(0, color="#aaaaaa", lw=1)
ax.set_ylabel("residuals [ppt]")
ax.set_xlabel("time [days]")

return fig

plot_light_curve(map_soln0); There are still a few outliers in the light curve and it can be useful to remove those before doing the full fit because both the GP and transit parameters can be sensitive to this.

## Sigma clipping¶

To remove the outliers, we’ll look at the empirical RMS of the residuals away from the GP + transit model and remove anything that is more than a 7-sigma outlier.

mod = map_soln0["gp_pred"] + map_soln0["mean"] + np.sum(map_soln0["light_curves"], axis=-1)
resid = y - mod
rms = np.sqrt(np.median(resid**2))
mask = np.abs(resid) < 7 * rms

plt.plot(x, resid, "k", label="data")
plt.axhline(0, color="#aaaaaa", lw=1)
plt.ylabel("residuals [ppt]")
plt.xlabel("time [days]")
plt.legend(fontsize=12, loc=4)
plt.xlim(x.min(), x.max()); That looks better. Let’s re-build our model with this sigma-clipped dataset.

model, map_soln = build_model(mask, map_soln0)

optimizing logp for variables: ['trend']
message: Optimization terminated successfully.
logp: 5227.11678093738 -> 5227.11678093738
optimizing logp for variables: ['logs2']
message: Optimization terminated successfully.
logp: 5227.11678093738 -> 5309.2238272757595
optimizing logp for variables: ['b_logodds__', 'logr']
message: Optimization terminated successfully.
logp: 5309.2238272757595 -> 5320.261298459935
optimizing logp for variables: ['t0', 'logP']
message: Desired error not necessarily achieved due to precision loss.
logp: 5320.261298459927 -> 5321.717806799228
optimizing logp for variables: ['logpower', 'logs2']
message: Optimization terminated successfully.
logp: 5321.717806799212 -> 5322.451996129972
optimizing logp for variables: ['logw0']
message: Desired error not necessarily achieved due to precision loss.
logp: 5322.451996129972 -> 5322.486246327689
optimizing logp for variables: ['logpower', 'logw0', 'logs2', 'trend', 'logs_rv', 'omega_angle__', 'ecc_interval__', 'b_logodds__', 'logr', 't0', 'logP', 'logm', 'r_star_interval__', 'm_star_interval__', 'u_star_quadlimbdark__', 'mean']
message: Desired error not necessarily achieved due to precision loss.
logp: 5322.486246327689 -> 5324.305565307076 Great! Now we’re ready to sample.

## Sampling¶

The sampling for this model is the same as for all the previous tutorials, but it takes a bit longer (about 2 hours on my laptop). This is partly because the model is more expensive to compute than the previous ones and partly because there are some non-affine degeneracies in the problem (for example between impact parameter and eccentricity). It might be worth thinking about reparameterizations (in terms of duration instead of eccentricity), but that’s beyond the scope of this tutorial. Besides, using more traditional MCMC methods, this would have taken a lot more than 2 hours to get >1000 effective samples!

np.random.seed(123)
with model:
trace = pm.sample(tune=5000, draws=3000, start=map_soln, chains=4,
step=xo.get_dense_nuts_step(target_accept=0.9))

Multiprocess sampling (4 chains in 4 jobs)
NUTS: [logpower, logw0, logs2, trend, logs_rv, omega, ecc, b, logr, t0, logP, logm, r_star, m_star, u_star, mean]
Sampling 4 chains: 100%|██████████| 32000/32000 [1:09:12<00:00,  1.92draws/s]
There were 114 divergences after tuning. Increase target_accept or reparameterize.
There were 166 divergences after tuning. Increase target_accept or reparameterize.
There were 136 divergences after tuning. Increase target_accept or reparameterize.
There were 140 divergences after tuning. Increase target_accept or reparameterize.
The number of effective samples is smaller than 10% for some parameters.


Let’s look at the convergence diagnostics for some of the key parameters:

pm.summary(trace, varnames=["period", "r_pl", "m_pl", "ecc", "omega", "b"])

mean sd mc_error hpd_2.5 hpd_97.5 n_eff Rhat
period__0 42.363458 0.000391 0.000004 42.362673 42.364193 11163.024503 1.000061
period__1 20.885349 0.000204 0.000002 20.884944 20.885740 11103.437023 0.999851
r_pl__0 0.077545 0.004941 0.000110 0.067859 0.087183 1983.684755 1.000486
r_pl__1 0.056425 0.003891 0.000098 0.048451 0.063781 1443.990700 1.001740
m_pl__0 28.141724 5.527235 0.068485 16.910058 38.732736 6917.523690 1.000225
m_pl__1 22.932688 4.118384 0.048495 14.878016 31.150777 7163.706833 0.999940
ecc__0 0.082030 0.067857 0.001349 0.000004 0.219479 2055.104770 1.001326
ecc__1 0.197313 0.083118 0.002285 0.000008 0.324079 1337.096836 1.000551
omega__0 -0.034281 2.253449 0.043221 -3.109510 3.047754 2868.839537 1.000010
omega__1 -0.437822 0.606192 0.018317 -1.551361 0.876164 930.600441 1.001205
b__0 0.589293 0.046759 0.000968 0.499378 0.670839 2346.014626 1.001360
b__1 0.578615 0.123196 0.005263 0.309723 0.743786 342.720670 1.010249

As you see, the effective number of samples for the impact parameters and eccentricites are lower than for the other parameters. This is because of the correlations that I mentioned above:

import corner
varnames = ["b", "ecc"]
samples = pm.trace_to_dataframe(trace, varnames=varnames)
fig = corner.corner(samples); ## Phase plots¶

Finally, as in the Radial velocity fitting and Transit fitting tutorials, we can make folded plots of the transits and the radial velocities and compare to the posterior model predictions. (Note: planets b and c in this tutorial are swapped compared to the labels from Petigura et al. (2016))

for n, letter in enumerate("bc"):
plt.figure()

# Compute the GP prediction
gp_mod = np.median(trace["gp_pred"] + trace["mean"][:, None], axis=0)

# Get the posterior median orbital parameters
p = np.median(trace["period"][:, n])
t0 = np.median(trace["t0"][:, n])

# Compute the median of posterior estimate of the contribution from
# the other planet. Then we can remove this from the data to plot
# just the planet we care about.
other = np.median(trace["light_curves"][:, :, (n + 1) % 2], axis=0)

# Plot the folded data
x_fold = (x[mask] - t0 + 0.5*p) % p - 0.5*p
plt.plot(x_fold, y[mask] - gp_mod - other, ".k", label="data", zorder=-1000)

# Plot the folded model
inds = np.argsort(x_fold)
inds = inds[np.abs(x_fold)[inds] < 0.3]
pred = trace["light_curves"][:, inds, n]
pred = np.percentile(pred, [16, 50, 84], axis=0)
plt.plot(x_fold[inds], pred, color="C1", label="model")
art = plt.fill_between(x_fold[inds], pred, pred, color="C1", alpha=0.5,
zorder=1000)
art.set_edgecolor("none")

# Annotate the plot with the planet's period
txt = "period = {0:.4f} +/- {1:.4f} d".format(
np.mean(trace["period"][:, n]), np.std(trace["period"][:, n]))
plt.annotate(txt, (0, 0), xycoords="axes fraction",
xytext=(5, 5), textcoords="offset points",
ha="left", va="bottom", fontsize=12)

plt.legend(fontsize=10, loc=4)
plt.xlim(-0.5*p, 0.5*p)
plt.xlabel("time since transit [days]")
plt.ylabel("de-trended flux")
plt.title("K2-24{0}".format(letter));
plt.xlim(-0.3, 0.3)  for n, letter in enumerate("bc"):
plt.figure()

# Get the posterior median orbital parameters
p = np.median(trace["period"][:, n])
t0 = np.median(trace["t0"][:, n])

# Compute the median of posterior estimate of the background RV
# and the contribution from the other planet. Then we can remove
# this from the data to plot just the planet we care about.
other = np.median(trace["vrad"][:, :, (n + 1) % 2], axis=0)
other += np.median(trace["bkg"], axis=0)

# Plot the folded data
x_fold = (x_rv - t0 + 0.5*p) % p - 0.5*p
plt.errorbar(x_fold, y_rv - other, yerr=yerr_rv, fmt=".k", label="data")

# Compute the posterior prediction for the folded RV model for this
# planet
t_fold = (t_rv - t0 + 0.5*p) % p - 0.5*p
inds = np.argsort(t_fold)
pred = np.percentile(trace["vrad_pred"][:, inds, n], [16, 50, 84], axis=0)
plt.plot(t_fold[inds], pred, color="C1", label="model")
art = plt.fill_between(t_fold[inds], pred, pred, color="C1", alpha=0.3)
art.set_edgecolor("none")

plt.legend(fontsize=10)
plt.xlim(-0.5*p, 0.5*p)
plt.xlabel("phase [days]")
plt.title("K2-24{0}".format(letter));  We can also compute the posterior constraints on the planet densities.

volume = 4/3*np.pi*trace["r_pl"]**3
density = u.Quantity(trace["m_pl"] / volume, unit=u.M_earth / u.R_sun**3)
density = density.to(u.g / u.cm**3).value

bins = np.linspace(0, 1.1, 45)
for n, letter in enumerate("bc"):
plt.hist(density[:, n], bins, histtype="step", lw=2,
label="K2-24{0}".format(letter), density=True)
plt.yticks([])
plt.legend(fontsize=12)
plt.xlim(bins, bins[-1])
plt.xlabel("density [g/cc]")
plt.ylabel("posterior density"); 