Note

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

# Gaussian process models for stellar variability¶

When fitting exoplanets, we also need to fit for the stellar variability and Gaussian Processes (GPs) are often a good descriptive model for this variation. PyMC3 has support for all sorts of general GP models, but exoplanet includes support for scalable 1D GPs (see Scalable Gaussian processes in PyMC3 for more info) that can work with large datasets. In this tutorial, we go through the process of modeling the light curve of a rotating star observed by Kepler using exoplanet.

import numpy as np
import lightkurve as lk
import matplotlib.pyplot as plt

)
lc = lcf.PDCSAP_FLUX.normalize().remove_nans().remove_outliers()

x = np.ascontiguousarray(lc.time, dtype=np.float64)
y = np.ascontiguousarray(lc.flux, dtype=np.float64)
yerr = np.ascontiguousarray(lc.flux_err, dtype=np.float64)
mu = np.mean(y)
y = (y / mu - 1) * 1e3
yerr = yerr * 1e3 / mu

plt.plot(x, y, "k")
plt.xlim(x.min(), x.max())
plt.xlabel("time [days]")
plt.ylabel("relative flux [ppt]")
plt.title("KIC 5809890");


## A Gaussian process model for stellar variability¶

This looks like the light curve of a rotating star, and it has been shown that it is possible to model this variability by using a quasiperiodic Gaussian process. To start with, let’s get an estimate of the rotation period using the Lomb-Scargle periodogram:

import exoplanet as xo

results = xo.estimators.lomb_scargle_estimator(
x, y, max_peaks=1, min_period=5.0, max_period=100.0, samples_per_peak=50
)

peak = results["peaks"][0]
freq, power = results["periodogram"]
plt.plot(1 / freq, power, "k")
plt.axvline(peak["period"], color="k", lw=4, alpha=0.3)
plt.xlim((1 / freq).min(), (1 / freq).max())
plt.yticks([])
plt.xlabel("period [days]")
plt.ylabel("power");


Now, using this initialization, we can set up the GP model in exoplanet. We’ll use the exoplanet.gp.terms.RotationTerm kernel that is a mixture of two simple harmonic oscillators with periods separated by a factor of two. As you can see from the periodogram above, this might be a good model for this light curve and I’ve found that it works well in many cases.

import pymc3 as pm
import theano.tensor as tt

with pm.Model() as model:

# The mean flux of the time series
mean = pm.Normal("mean", mu=0.0, sd=10.0)

# A jitter term describing excess white noise
logs2 = pm.Normal("logs2", mu=2 * np.log(np.mean(yerr)), sd=2.0)

# A term to describe the non-periodic variability
logSw4 = pm.Normal("logSw4", mu=np.log(np.var(y)), sd=5.0)
logw0 = pm.Normal("logw0", mu=np.log(2 * np.pi / 10), sd=5.0)

# The parameters of the RotationTerm kernel
logamp = pm.Normal("logamp", mu=np.log(np.var(y)), sd=5.0)
BoundedNormal = pm.Bound(pm.Normal, lower=0.0, upper=np.log(50))
logperiod = BoundedNormal("logperiod", mu=np.log(peak["period"]), sd=5.0)
logQ0 = pm.Normal("logQ0", mu=1.0, sd=10.0)
logdeltaQ = pm.Normal("logdeltaQ", mu=2.0, sd=10.0)
mix = xo.distributions.UnitUniform("mix")

# Track the period as a deterministic
period = pm.Deterministic("period", tt.exp(logperiod))

# Set up the Gaussian Process model
kernel = xo.gp.terms.SHOTerm(log_Sw4=logSw4, log_w0=logw0, Q=1 / np.sqrt(2))
kernel += xo.gp.terms.RotationTerm(
log_amp=logamp, period=period, log_Q0=logQ0, log_deltaQ=logdeltaQ, mix=mix
)
gp = xo.gp.GP(kernel, x, yerr ** 2 + tt.exp(logs2), mean=mean)

# Compute the Gaussian Process likelihood and add it into the
# the PyMC3 model as a "potential"
gp.marginal("gp", observed=y)

# Compute the mean model prediction for plotting purposes
pm.Deterministic("pred", gp.predict())

# Optimize to find the maximum a posteriori parameters
map_soln = xo.optimize(start=model.test_point)

optimizing logp for variables: [mix, logdeltaQ, logQ0, logperiod, logamp, logw0, logSw4, logs2, mean]
81it [00:10,  7.60it/s, logp=7.031724e+02]
message: Optimization terminated successfully.
logp: 267.25686032199354 -> 703.1723809089816


Now that we have the model set up, let’s plot the maximum a posteriori model prediction.

plt.plot(x, y, "k", label="data")
plt.plot(x, map_soln["pred"], color="C1", label="model")
plt.xlim(x.min(), x.max())
plt.legend(fontsize=10)
plt.xlabel("time [days]")
plt.ylabel("relative flux [ppt]")
plt.title("KIC 5809890; map model");


That looks pretty good! Now let’s sample from the posterior using exoplanet.get_dense_nuts_step().

np.random.seed(5809890)
with model:
trace = xo.sample(tune=2500, draws=2000, start=map_soln, target_accept=0.95)

Multiprocess sampling (4 chains in 4 jobs)
NUTS: [mix, logdeltaQ, logQ0, logperiod, logamp, logw0, logSw4, logs2, mean]
Sampling 4 chains, 2 divergences: 100%|██████████| 18000/18000 [08:44<00:00, 34.29draws/s]
There was 1 divergence after tuning. Increase target_accept or reparameterize.
There was 1 divergence after tuning. Increase target_accept or reparameterize.


Now we can do the usual convergence checks:

pm.summary(
trace,
var_names=[
"mix",
"logdeltaQ",
"logQ0",
"logperiod",
"logamp",
"logSw4",
"logw0",
"logs2",
"mean",
],
)

mean sd hpd_3% hpd_97% mcse_mean mcse_sd ess_mean ess_sd ess_bulk ess_tail r_hat
mix 0.650 0.240 0.227 1.000 0.005 0.003 2738.0 2738.0 2353.0 2167.0 1.0
logdeltaQ 0.402 9.413 -17.770 18.972 0.169 0.119 3113.0 3113.0 3092.0 4379.0 1.0
logQ0 0.780 0.540 -0.162 1.809 0.009 0.008 3603.0 2559.0 3997.0 3938.0 1.0
logperiod 3.330 0.090 3.178 3.502 0.002 0.001 2698.0 2635.0 4044.0 2263.0 1.0
logamp 0.364 0.596 -0.619 1.468 0.012 0.010 2665.0 1882.0 3430.0 2614.0 1.0
logSw4 6.600 0.790 5.089 8.004 0.014 0.010 3214.0 3214.0 3535.0 2724.0 1.0
logw0 3.859 0.204 3.479 4.235 0.003 0.002 3814.0 3814.0 4165.0 2690.0 1.0
logs2 -6.242 0.746 -7.647 -5.158 0.019 0.014 1481.0 1385.0 2183.0 1495.0 1.0
mean -0.019 0.216 -0.421 0.394 0.003 0.003 4925.0 2504.0 5324.0 3905.0 1.0

And plot the posterior distribution over rotation period:

period_samples = trace["period"]
bins = np.linspace(20, 45, 40)
plt.hist(period_samples, bins, histtype="step", color="k")
plt.yticks([])
plt.xlim(bins.min(), bins.max())
plt.xlabel("rotation period [days]")
plt.ylabel("posterior density");


## 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:exoplanet, exoplanet:foremanmackey17,
exoplanet:foremanmackey18, exoplanet:pymc3, exoplanet:theano}.

print("\n".join(bib.splitlines()[:10]) + "\n...")

@misc{exoplanet:exoplanet,
author = {Daniel Foreman-Mackey and Rodrigo Luger and Ian Czekala and
Eric Agol and Adrian Price-Whelan and Tom Barclay},
title = {exoplanet-dev/exoplanet v0.3.2},
month = may,
year = 2020,
doi = {10.5281/zenodo.1998447},
url = {https://doi.org/10.5281/zenodo.1998447}
}
...