Note

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

Radial velocity fitting

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.

First, let’s download the data from RadVel:

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

url = "https://raw.githubusercontent.com/California-Planet-Search/radvel/master/example_data/epic203771098.csv"
data = pd.read_csv(url, index_col=0)

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]")
plt.ylabel("radial velocity [m/s]");
../../_images/rv_4_0.png

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.Bound(pm.Normal, lower=0)(
        "P",
        mu=np.array(periods),
        sd=np.array(period_errs),
        shape=2,
        testval=np.array(periods),
    )

    # Wide log-normal prior for semi-amplitude
    logK = pm.Bound(pm.Normal, lower=0)(
        "logK", mu=np.log(Ks), sd=10.0, shape=2, testval=np.log(Ks)
    )

    # Eccentricity & argument of periasteron
    ecc = xo.distributions.UnitUniform("ecc", shape=2, testval=np.array([0.1, 0.1]))
    omega = xo.distributions.Angle("omega", shape=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
    vrad = orbit.get_radial_velocity(x, K=tt.exp(logK))
    pm.Deterministic("vrad", vrad)

    # 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:
    vrad_pred = orbit.get_radial_velocity(t, K=tt.exp(logK))
    pm.Deterministic("vrad_pred", vrad_pred)
    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", tt.sum(vrad_pred, axis=-1) + bkg_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(vrad_pred), "--k", alpha=0.5)
    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.ylabel("radial velocity [m/s]")
plt.title("initial model");
../../_images/rv_14_0.png

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 = xo.optimize(start=model.test_point, vars=[trend])
    map_soln = xo.optimize(start=map_soln)
optimizing logp for variables: [trend]
13it [00:04,  3.01it/s, logp=-6.484820e+01]
message: Optimization terminated successfully.
logp: -79.73266285618838 -> -64.8482026233154
optimizing logp for variables: [trend, logs, omega, ecc, logK, P, t0]
94it [00:00, 180.82it/s, logp=-1.427676e+01]
message: Optimization terminated successfully.
logp: -64.8482026233154 -> -14.276760262380932
plt.errorbar(x, y, yerr=yerr, fmt=".k")
plt.plot(t, map_soln["vrad_pred"], "--k", alpha=0.5)
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.ylabel("radial velocity [m/s]")
plt.title("MAP model");
../../_images/rv_17_0.png

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.get_dense_nuts_step() to tune the sampler (see the PyMC3 extras tutorial for more information).

np.random.seed(42)
with model:
    trace = pm.sample(
        tune=4000, draws=4000, step=xo.get_dense_nuts_step(target_accept=0.95)
    )
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [trend, logs, omega, ecc, logK, P, t0]
Sampling 4 chains: 100%|██████████| 32000/32000 [01:38<00:00, 324.88draws/s]
There was 1 divergence after tuning. Increase target_accept or reparameterize.
There were 5 divergences after tuning. Increase target_accept or reparameterize.
There were 4 divergences after tuning. Increase target_accept or reparameterize.
There were 6 divergences after tuning. Increase target_accept or reparameterize.
The number of effective samples is smaller than 25% 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 0.000959 0.000767 0.000008 -0.000533 0.002476 8822.213670 0.999924
trend__1 -0.038355 0.022526 0.000198 -0.082882 0.005619 12375.503833 0.999947
trend__2 -1.969807 0.798796 0.007218 -3.486318 -0.396062 10515.726879 1.000064
logs 1.035119 0.223164 0.002460 0.595741 1.471820 7444.652535 1.000065
omega__0 -0.295243 0.770839 0.013249 -1.457638 1.357533 3367.327667 1.001333
omega__1 -0.626204 2.109072 0.028147 -3.138377 2.923905 5452.408556 1.000018
ecc__0 0.238510 0.111980 0.001492 0.010760 0.440057 5346.847668 1.000452
ecc__1 0.199042 0.152331 0.002676 0.000026 0.499889 3250.981569 1.000073
t0__0 2072.794800 0.000704 0.000006 2072.793387 2072.796154 16908.246398 0.999888
t0__1 2082.625099 0.000396 0.000003 2082.624299 2082.625864 19029.275910 0.999969
logK__0 1.555357 0.254439 0.004343 1.022535 1.985683 3303.559263 0.999944
logK__1 1.573699 0.232897 0.002672 1.110036 2.019354 6864.314828 1.000101
P__0 20.885096 0.000299 0.000003 20.884487 20.885659 11727.407960 1.000444
P__1 42.363304 0.000606 0.000004 42.362107 42.364480 17521.191304 0.999982

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", "omega"])
corner.corner(samples);
../../_images/rv_23_0.png

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[1], color="C0", label="model")
art = plt.fill_between(t, pred[0], pred[2], color="C0", alpha=0.3)
art.set_edgecolor("none")

plt.legend(fontsize=10)
plt.xlim(t.min(), t.max())
plt.xlabel("time [days]")
plt.ylabel("radial velocity [m/s]")
plt.title("posterior constraints");
../../_images/rv_25_0.png

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[1], color="C0", label="model")
    art = plt.fill_between(t_fold[inds], pred[0], pred[2], 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]")
    plt.ylabel("radial velocity [m/s]")
    plt.title("K2-24{0}".format(letter));
../../_images/rv_27_0.png ../../_images/rv_27_1.png

Citations

As described in the Citing exoplanet & its dependencies tutorial, we can use exoplanet.citations.get_citations_for_model() to construct an acknowledgement and BibTeX listing that includes the relevant citations for this model.

with model:
    txt, bib = xo.citations.get_citations_for_model()
print(txt)
This research made use of textsf{exoplanet} citep{exoplanet} and its
dependencies citep{exoplanet:astropy13, exoplanet:astropy18,
exoplanet:exoplanet, exoplanet:pymc3, exoplanet:theano}.
print("\n".join(bib.splitlines()[:10]) + "\n...")
@misc{exoplanet:exoplanet,
  author = {Daniel Foreman-Mackey and Ian Czekala and Rodrigo Luger and
            Eric Agol and Geert Barentsen and Tom Barclay},
   title = {dfm/exoplanet: exoplanet v0.2.1},
   month = sep,
    year = 2019,
     doi = {10.5281/zenodo.3462740},
     url = {https://doi.org/10.5281/zenodo.3462740}
}
...