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.

First, let’s download and plot the data:

import numpy as np
import matplotlib.pyplot as plt
from import fits

url = ""
with as hdus:
    data = hdus[1].data

x = data["TIME"]
y = data["PDCSAP_FLUX"]
yerr = data["PDCSAP_FLUX_ERR"]
m = (data["SAP_QUALITY"] == 0) & np.isfinite(x) & np.isfinite(y)

x = np.ascontiguousarray(x[m], dtype=np.float64)
y = np.ascontiguousarray(y[m], dtype=np.float64)
yerr = np.ascontiguousarray(yerr[m], 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,

peak = results["peaks"][0]
freq, power = results["periodogram"]
plt.plot(-np.log10(freq), power, "k")
plt.axvline(np.log10(peak["period"]), color="k", lw=4, alpha=0.3)
plt.xlim((-np.log10(freq)).min(), (-np.log10(freq)).max())

Now, using this initialization, we can set up the GP model in exoplanet. We’ll use the 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.min(yerr)), sd=5.0)

    # The parameters of the RotationTerm kernel
    logamp = pm.Normal("logamp", mu=np.log(np.var(y)), sd=5.0)
    logperiod = pm.Normal("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 = pm.Uniform("mix", lower=0, upper=1.0)

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

    # Set up the Gaussian Process model
    kernel =
    gp =, x, yerr**2 + tt.exp(logs2), J=4)

    # Compute the Gaussian Process likelihood and add it into the
    # the PyMC3 model as a "potential"
    pm.Potential("loglike", gp.log_likelihood(y - mean))

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

    # Optimize to find the maximum a posteriori parameters
    map_soln = pm.find_MAP(start=model.test_point)
logp = 694.35, ||grad|| = 0.014649: 100%|██████████| 23/23 [00:00<00:00, 238.27it/s]

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.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 a exoplanet.PyMC3Sampler.

sampler = xo.PyMC3Sampler()
with model:
    sampler.tune(tune=2000, start=map_soln, step_kwargs=dict(target_accept=0.9))
    trace = sampler.sample(draws=2000)
Sampling 4 chains: 100%|██████████| 308/308 [00:24<00:00,  2.61draws/s]
Sampling 4 chains: 100%|██████████| 108/108 [00:10<00:00,  9.86draws/s]
Sampling 4 chains: 100%|██████████| 208/208 [00:04<00:00, 50.18draws/s]
Sampling 4 chains: 100%|██████████| 408/408 [00:06<00:00, 65.17draws/s]
Sampling 4 chains: 100%|██████████| 808/808 [00:12<00:00, 64.76draws/s]
Sampling 4 chains: 100%|██████████| 1608/1608 [00:21<00:00, 76.45draws/s]
Sampling 4 chains: 100%|██████████| 4608/4608 [01:11<00:00, 64.33draws/s]
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [mix, logdeltaQ, logQ0, logperiod, logamp, logs2, mean]
Sampling 4 chains: 100%|██████████| 8200/8200 [02:44<00:00, 49.92draws/s]
There was 1 divergence after tuning. Increase target_accept or reparameterize.
The acceptance probability does not match the target. It is 0.9722822521454982, but should be close to 0.9. Try to increase the number of tuning steps.

Now we can do the usual convergence checks:

pm.summary(trace, varnames=["mix", "logdeltaQ", "logQ0", "logperiod", "logamp", "logs2", "mean"])
mean sd mc_error hpd_2.5 hpd_97.5 n_eff Rhat
mix 0.612192 0.246607 0.004753 0.184111 0.999968 2390.827873 1.001618
logdeltaQ 1.258932 9.498497 0.154109 -16.370452 21.533913 3799.856881 0.999809
logQ0 0.634067 0.577331 0.009861 -0.482119 1.838512 3253.935654 1.000323
logperiod 3.367242 0.121002 0.002092 3.189152 3.651945 3173.838509 1.000147
logamp -0.085608 0.572374 0.010271 -1.113512 1.038443 2806.723951 1.000188
logs2 -4.968817 0.121865 0.001696 -5.212865 -4.737115 5660.208712 0.999860
mean -0.017287 0.216807 0.003465 -0.426309 0.413271 3592.780697 0.999810

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.xlim(bins.min(), bins.max())
plt.xlabel("rotation period [days]")
plt.ylabel("posterior density");