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

Scalable Gaussian processes in PyMC3

PyMC3 has support for Gaussian Processes (GPs), but this implementation is too slow for many applications in time series astrophysics. So exoplanet comes with an implementation of scalable GPs powered by celerite. More information about the algorithm can be found in the celerite docs and in the papers (Paper 1 and Paper 2), but this tutorial will give a hands on demo of how to use celerite in PyMC3.


For the best results, we generally recommend the use of the,, and “terms” because the other terms tend to have unphysical behavior at high frequency.

Let’s start with the quickstart demo from the celerite docs. We’ll fit the following simulated dataset using the sum of two objects.

First, generate the simulated data:

import numpy as np
import matplotlib.pyplot as plt


t = np.sort(
    np.append(np.random.uniform(0, 3.8, 57), np.random.uniform(5.5, 10, 68))
)  # The input coordinates must be sorted
yerr = np.random.uniform(0.08, 0.22, len(t))
y = (
    0.2 * (t - 5)
    + np.sin(3 * t + 0.1 * (t - 5) ** 2)
    + yerr * np.random.randn(len(t))

true_t = np.linspace(0, 10, 5000)
true_y = 0.2 * (true_t - 5) + np.sin(3 * true_t + 0.1 * (true_t - 5) ** 2)

plt.errorbar(t, y, yerr=yerr, fmt=".k", capsize=0, label="data")
plt.plot(true_t, true_y, "k", lw=1.5, alpha=0.3, label="truth")
plt.xlim(0, 10)
_ = plt.ylim(-2.5, 2.5)

This plot shows the simulated data as black points with error bars and the true function is shown as a gray line.

Now let’s build the PyMC3 model that we’ll use to fit the data. We can see that there’s some roughly periodic signal in the data as well as a longer term trend. To capture these two features, we will model this as a mixture of two stochastically driven simple harmonic oscillators (SHO) with the power spectrum:

\[S(\omega) = \sqrt{\frac{2}{\pi}}\frac{S_1\,{\omega_1}^4}{(\omega^2 - {\omega_1}^2)^2 + 2\,{\omega_1}^2\,\omega^2} + \sqrt{\frac{2}{\pi}}\frac{S_2\,{\omega_2}^4}{(\omega^2 - {\omega_2}^2)^2 + {\omega_2}^2\,\omega^2/Q^2}\]

The first term is with \(Q=1/\sqrt{2}\) and the second is regular This model has 5 free parameters (\(S_1\), \(\omega_1\), \(S_2\), \(\omega_2\), and \(Q\)) and they must all be positive so we’ll fit for the log of each parameter. Using exoplanet, this is how you would build this model, choosing more or less arbitrary initial values for the parameters.

import pymc3 as pm
import theano.tensor as tt
from import terms, GP

with pm.Model() as model:

    mean = pm.Normal("mean", mu=0.0, sigma=1.0)
    logS1 = pm.Normal("logS1", mu=0.0, sigma=15.0, testval=np.log(np.var(y)))
    logw1 = pm.Normal("logw1", mu=0.0, sigma=15.0, testval=np.log(3.0))
    logS2 = pm.Normal("logS2", mu=0.0, sigma=15.0, testval=np.log(np.var(y)))
    logw2 = pm.Normal("logw2", mu=0.0, sigma=15.0, testval=np.log(3.0))
    logQ = pm.Normal("logQ", mu=0.0, sigma=15.0, testval=0)

    # Set up the kernel an GP
    kernel = terms.SHOTerm(log_S0=logS1, log_w0=logw1, Q=1.0 / np.sqrt(2))
    kernel += terms.SHOTerm(log_S0=logS2, log_w0=logw2, log_Q=logQ)
    gp = GP(kernel, t, yerr ** 2, mean=mean)

    # Condition the GP on the observations and add the marginal likelihood
    # to the model
    gp.marginal("gp", observed=y)

A few comments here:

  1. The term interface in exoplanet only accepts keyword arguments with names given by the parameter_names property of the term. But it will also interpret keyword arguments with the name prefaced by log_ to be the log of the parameter. For example, in this case, we used log_S0 as the parameter for each term, but S0=tt.exp(log_S0) would have been equivalent. This is useful because many of the parameters are required to be positive so fitting the log of those parameters is often best.

  2. The third argument to the constructor should be the variance to add along the diagonal, not the standard deviation as in the original celerite implementation.

  3. Finally, the constructor takes an optional argument J which specifies the width of the problem if it is known at compile time. Just to be confusing, this is actually two times the J from the celerite paper. There are various technical reasons why this is difficult to work out in general and this code will always work if you don’t provide a value for J, but you can get much better performance (especially for small J) if you know what it will be for your problem. In general, most terms cost J=2 with the exception of a (which costs J=1) and a (which costs J=4).

To start, let’s fit for the maximum a posteriori (MAP) parameters and look the the predictions that those make.

import exoplanet as xo

with model:
    map_soln = xo.optimize(start=model.test_point)
optimizing logp for variables: [logQ, logw2, logS2, logw1, logS1, mean]
49it [00:00, 73.77it/s, logp=-2.569892e+00]
message: Optimization terminated successfully.
logp: -41.9875698144546 -> -2.569892168573091

We’ll use the exoplanet.eval_in_model() function to evaluate the MAP GP model.

with model:
    mu, var = xo.eval_in_model(
        gp.predict(true_t, return_var=True, predict_mean=True), map_soln
plt.errorbar(t, y, yerr=yerr, fmt=".k", capsize=0, label="data")
plt.plot(true_t, true_y, "k", lw=1.5, alpha=0.3, label="truth")

# Plot the prediction and the 1-sigma uncertainty
sd = np.sqrt(var)
art = plt.fill_between(true_t, mu + sd, mu - sd, color="C1", alpha=0.3)
plt.plot(true_t, mu, color="C1", label="prediction")

plt.xlim(0, 10)
_ = plt.ylim(-2.5, 2.5)

Now we can sample this model using PyMC3. There are strong covariances between the parameters so we’ll use the custom exoplanet.get_dense_nuts_step() to fit for these covariances during burn-in.

with model:
    trace = pm.sample(
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [logQ, logw2, logS2, logw1, logS1, mean]
Sampling 2 chains, 36 divergences: 100%|██████████| 8000/8000 [00:42<00:00, 186.34draws/s]
There were 35 divergences after tuning. Increase target_accept or reparameterize.
There was 1 divergence after tuning. Increase target_accept or reparameterize.
The number of effective samples is smaller than 10% for some parameters.

Now we can compute the standard PyMC3 convergence statistics (using pymc3.summary) and make a trace plot (using pymc3.traceplot).

mean sd hpd_3% hpd_97% mcse_mean mcse_sd ess_mean ess_sd ess_bulk ess_tail r_hat
mean 0.006 0.868 -1.719 1.568 0.017 0.017 2555.0 1306.0 2578.0 2294.0 1.00
logS1 5.189 3.471 -0.544 11.633 0.102 0.072 1156.0 1156.0 629.0 397.0 1.01
logw1 -2.528 1.324 -5.135 -0.076 0.063 0.045 442.0 442.0 377.0 119.0 1.01
logS2 -4.176 0.868 -4.993 -3.111 0.101 0.077 74.0 64.0 422.0 120.0 1.01
logw2 1.129 0.051 1.032 1.224 0.001 0.001 2137.0 2137.0 2351.0 1764.0 1.00
logQ 2.951 1.353 1.102 5.377 0.101 0.087 180.0 120.0 472.0 203.0 1.01

That all looks pretty good, but I like to make two other results plots: (1) a corner plot and (2) a posterior predictive plot.

The corner plot is easy using pymc3.trace_to_dataframe and I find it useful for understanding the covariances between parameters when debugging.

import corner

samples = pm.trace_to_dataframe(trace)
_ = corner.corner(samples)

The “posterior predictive” plot that I like to make isn’t the same as a “posterior predictive check” (which can be a good thing to do too). Instead, I like to look at the predictions of the model in the space of the data. We could have saved these predictions using a pymc3.Deterministic distribution, but that adds some overhead to each evaluation of the model so instead, we can use exoplanet.utils.get_samples_from_trace() to loop over a few random samples from the chain and then the exoplanet.eval_in_model() function to evaluate the prediction just for those samples.

# Generate 50 realizations of the prediction sampling randomly from the chain
N_pred = 50
pred_mu = np.empty((N_pred, len(true_t)))
pred_var = np.empty((N_pred, len(true_t)))
with model:
    pred = gp.predict(true_t, return_var=True, predict_mean=True)
    for i, sample in enumerate(xo.get_samples_from_trace(trace, size=N_pred)):
        pred_mu[i], pred_var[i] = xo.eval_in_model(pred, sample)

# Plot the predictions
for i in range(len(pred_mu)):
    mu = pred_mu[i]
    sd = np.sqrt(pred_var[i])
    label = None if i else "prediction"
    art = plt.fill_between(true_t, mu + sd, mu - sd, color="C1", alpha=0.1)
    plt.plot(true_t, mu, color="C1", label=label, alpha=0.1)

plt.errorbar(t, y, yerr=yerr, fmt=".k", capsize=0, label="data")
plt.plot(true_t, true_y, "k", lw=1.5, alpha=0.3, label="truth")
plt.legend(fontsize=12, loc=2)
plt.xlim(0, 10)
_ = plt.ylim(-2.5, 2.5)


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()
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...")
  author = {Daniel Foreman-Mackey and Ian Czekala and Rodrigo Luger and
            Eric Agol and Geert Barentsen and Tom Barclay},
   title = {exoplanet-dev/exoplanet v0.2.6},
   month = mar,
    year = 2020,
     doi = {10.5281/zenodo.1998447},
     url = {}