Note

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

In this tutorial, we will demonstrate how to fit radial velocity observations of an exoplanetary system using exoplanet. We will follow the getting started tutorial from the exellent RadVel package where they fit for the parameters of the two planets in the K2-24 system.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

x = np.array(data.t)
y = np.array(data.vel)
yerr = np.array(data.errvel)

plt.errorbar(x, y, yerr=yerr, fmt=".k")
plt.xlabel("time [days]") Now, we know the periods and transit times for the planets from the K2 light curve, so let’s start by using the exoplanet.estimate_semi_amplitude() function to estimate the expected RV semi-amplitudes for the planets.

import exoplanet as xo

periods = [20.8851, 42.3633]
period_errs = [0.0003, 0.0006]
t0s = [2072.7948, 2082.6251]
t0_errs = [0.0007, 0.0004]
Ks = xo.estimate_semi_amplitude(periods, x, y, yerr, t0s=t0s)
print(Ks, "m/s")

[5.05069163 5.50983542] m/s


## The radial velocity model in PyMC3¶

Now that we have the data and an estimate of the initial values for the parameters, let’s start defining the probabilistic model in PyMC3 (take a look at A quick intro to PyMC3 for exoplaneteers if you’re new to PyMC3). First, we’ll define our priors on the parameters:

import pymc3 as pm
import theano.tensor as tt

with pm.Model() as model:

# Gaussian priors based on transit data (from Petigura et al.)
t0 = pm.Normal("t0", mu=np.array(t0s), sd=np.array(t0_errs), shape=2)
P = pm.Normal("P", mu=np.array(periods), sd=np.array(period_errs), shape=2)

# Wide log-normal prior for semi-amplitude
logK = pm.Normal("logK", mu=np.log(Ks), sd=10.0, shape=2)

# This is a sanity check that restricts the semiamplitude to reasonable
# values because things can get ugly as K -> 0
pm.Potential("logK_bound", tt.switch(logK < -2., -np.inf, 0.0))

# We also want to keep period physical but this probably won't be hit
pm.Potential("P_bound", tt.switch(P <= 0, -np.inf, 0.0))

# Eccentricity & argument of periasteron
ecc = pm.Uniform("ecc", lower=0, upper=0.99, shape=2,
testval=np.array([0.1, 0.1]))
omega = xo.distributions.Angle("omega", shape=2, testval=np.zeros(2))

# Jitter & a quadratic RV trend
logs = pm.Normal("logs", mu=np.log(np.median(yerr)), sd=5.0)
trend = pm.Normal("trend", mu=0, sd=10.0**-np.arange(3)[::-1], shape=3)


Now we’ll define the orbit model:

with model:

# Set up the orbit
orbit = xo.orbits.KeplerianOrbit(
period=P, t0=t0,
ecc=ecc, omega=omega)

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

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

# Sum over planets and add the background to get the full model
rv_model = pm.Deterministic("rv_model", tt.sum(vrad, axis=-1) + bkg)


For plotting purposes, it can be useful to also save the model on a fine grid in time.

t = np.linspace(x.min()-5, x.max()+5, 1000)

with model:
A_pred = np.vander(t - 0.5*(x.min() + x.max()), 3)
bkg_pred = pm.Deterministic("bkg_pred", tt.dot(A_pred, trend))
rv_model_pred = pm.Deterministic("rv_model_pred",


Now, we can plot the initial model:

plt.errorbar(x, y, yerr=yerr, fmt=".k")

with model:
plt.plot(t, xo.eval_in_model(bkg_pred), ":k", alpha=0.5)
plt.plot(t, xo.eval_in_model(rv_model_pred), label="model")

plt.legend(fontsize=10)
plt.xlim(t.min(), t.max())
plt.xlabel("time [days]")
plt.title("initial model"); In this plot, the background is the dotted line, the individual planets are the dashed lines, and the full model is the blue line.

It doesn’t look amazing so let’s add in the likelihood and fit for the maximum a posterior parameters.

with model:

err = tt.sqrt(yerr**2 + tt.exp(2*logs))
pm.Normal("obs", mu=rv_model, sd=err, observed=y)

map_soln = pm.find_MAP(start=model.test_point, vars=[trend])
map_soln = pm.find_MAP(start=map_soln)

logp = -80.668, ||grad|| = 7.2169: 100%|██████████| 21/21 [00:00<00:00, 1197.85it/s]
logp = -59.047, ||grad|| = 11.57: 100%|██████████| 2176/2176 [00:01<00:00, 1466.70it/s]

plt.errorbar(x, y, yerr=yerr, fmt=".k")
plt.plot(t, map_soln["bkg_pred"], ":k", alpha=0.5)
plt.plot(t, map_soln["rv_model_pred"], label="model")

plt.legend(fontsize=10)
plt.xlim(t.min(), t.max())
plt.xlabel("time [days]")
plt.title("MAP model"); That looks better.

## Sampling¶

Now that we have our model set up and a good estimate of the initial parameters, let’s start sampling. There are substantial covariances between some of the parameters so we’ll use a exoplanet.PyMC3Sampler to tune the sampler (see the PyMC3 extras tutorial for more information).

np.random.seed(42)
sampler = xo.PyMC3Sampler(finish=200)
with model:
burnin = sampler.tune(tune=2500, start=model.test_point)
trace = sampler.sample(draws=3000)

Sampling 4 chains: 100%|██████████| 308/308 [00:04<00:00, 71.96draws/s]
Sampling 4 chains: 100%|██████████| 108/108 [00:01<00:00, 72.27draws/s]
Sampling 4 chains: 100%|██████████| 208/208 [00:02<00:00, 99.87draws/s]
Sampling 4 chains: 100%|██████████| 408/408 [00:01<00:00, 228.50draws/s]
Sampling 4 chains: 100%|██████████| 808/808 [00:02<00:00, 311.28draws/s]
Sampling 4 chains: 100%|██████████| 1608/1608 [00:04<00:00, 340.12draws/s]
The chain contains only diverging samples. The model is probably misspecified.
Sampling 4 chains: 100%|██████████| 6608/6608 [00:16<00:00, 397.62draws/s]
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [trend, logs, omega, ecc, logK, P, t0]
Sampling 4 chains: 100%|██████████| 12800/12800 [00:27<00:00, 460.69draws/s]
There were 13 divergences after tuning. Increase target_accept or reparameterize.
There were 6 divergences after tuning. Increase target_accept or reparameterize.
There were 3 divergences after tuning. Increase target_accept or reparameterize.
There were 35 divergences after tuning. Increase target_accept or reparameterize.
The number of effective samples is smaller than 10% for some parameters.


After sampling, it’s always a good idea to do some convergence checks. First, let’s check the number of effective samples and the Gelman-Rubin statistic for our parameters of interest:

pm.summary(trace, varnames=["trend", "logs", "omega", "ecc", "t0", "logK", "P"])

mean sd mc_error hpd_2.5 hpd_97.5 n_eff Rhat
trend__0 0.000937 0.000769 0.000011 -0.000579 0.002413 4954.818624 1.000000
trend__1 -0.038581 0.023025 0.000305 -0.084751 0.005695 5929.418679 1.000270
trend__2 -1.951777 0.807705 0.010839 -3.533702 -0.374656 4884.021342 0.999880
logs 1.044249 0.224468 0.003847 0.600236 1.489714 3070.836387 1.000529
omega__0 -0.304417 0.786889 0.012632 -1.470657 1.573662 3561.756315 1.000257
omega__1 -0.630168 2.115762 0.029849 -3.133547 2.935825 4958.825223 1.000549
ecc__0 0.233891 0.111863 0.002168 0.000924 0.420804 2392.654832 1.000650
ecc__1 0.199180 0.149677 0.004537 0.000024 0.493935 993.886096 1.003839
t0__0 2072.794804 0.000702 0.000010 2072.793413 2072.796168 6014.819818 1.000091
t0__1 2082.625097 0.000400 0.000005 2082.624311 2082.625855 6931.218536 1.000032
logK__0 1.546506 0.263250 0.006481 1.055484 2.010602 1460.708419 1.002108
logK__1 1.561388 0.295920 0.009996 1.080357 2.068549 823.387263 1.000183
P__0 20.885093 0.000300 0.000004 20.884515 20.885686 6592.192857 1.000022
P__1 42.363301 0.000605 0.000008 42.362124 42.364498 5822.137680 1.000242

It looks like everything is pretty much converged here. Not bad for 14 parameters and about a minute of runtime…

Then we can make a corner plot of any combination of the parameters. For example, let’s look at period, semi-amplitude, and eccentricity:

import corner
samples = pm.trace_to_dataframe(trace, varnames=["P", "logK", "ecc"])
corner.corner(samples); Finally, let’s plot the plosterior constraints on the RV model and compare those to the data:

plt.errorbar(x, y, yerr=yerr, fmt=".k")

# Compute the posterior predictions for the RV model
pred = np.percentile(trace["rv_model_pred"], [16, 50, 84], axis=0)
plt.plot(t, pred, color="C0", label="model")
art = plt.fill_between(t, pred, pred, color="C0", alpha=0.3)
art.set_edgecolor("none")

plt.legend(fontsize=10)
plt.xlim(t.min(), t.max())
plt.xlabel("time [days]")
plt.title("posterior constraints"); ## Phase plots¶

It might be also be interesting to look at the phased plots for this system. Here we’ll fold the dataset on the median of posterior period and then overplot the posterior constraint on the folded model orbits.

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

# Get the posterior median orbital parameters
p = np.median(trace["P"][:, 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 - t0 + 0.5*p) % p - 0.5*p
plt.errorbar(x_fold, y - other, yerr=yerr, fmt=".k")

# Compute the posterior prediction for the folded RV model for this
# planet
t_fold = (t - 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="C0", label="model")
art = plt.fill_between(t_fold[inds], pred, pred, color="C0", alpha=0.3)
art.set_edgecolor("none")

plt.legend(fontsize=10)
plt.xlim(-0.5*p, 0.5*p)
plt.xlabel("phase [days]")  