Note

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.

Note

For the best results, we generally recommend the use of the exoplanet.gp.terms.SHOTerm, exoplanet.gp.terms.Matern32Term, and exoplanet.gp.terms.RotationTerm “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 exoplanet.gp.terms.SHOTerm objects.

First, generate the simulated data:

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(42)

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.legend(fontsize=12)
plt.xlabel("t")
plt.ylabel("y")
plt.xlim(0, 10)
_ = plt.ylim(-2.5, 2.5)
../../_images/gp_6_0.png

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 exoplanet.gp.terms.SHOterm with \(Q=1/\sqrt{2}\) and the second is regular exoplanet.gp.terms.SHOterm. This model has g free parameters: \(S_1\), \(\omega_1\), \(S_2\), \(\omega_2\), \(Q\), and a constant mean value. Most of the parameters will have weakly informative inverse Gamma priors (see this blog post for a discussion of these priors) where the parameters are chosen to have reasonable tail probabilities. Using exoplanet, this is how you would build this model:

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

with pm.Model() as model:

    mean = pm.Normal("mean", mu=0.0, sigma=1.0)
    S1 = pm.InverseGamma(
        "S1", **estimate_inverse_gamma_parameters(0.5 ** 2, 10.0 ** 2)
    )
    S2 = pm.InverseGamma(
        "S2", **estimate_inverse_gamma_parameters(0.25 ** 2, 1.0 ** 2)
    )
    w1 = pm.InverseGamma(
        "w1", **estimate_inverse_gamma_parameters(2 * np.pi / 10.0, np.pi)
    )
    w2 = pm.InverseGamma(
        "w2", **estimate_inverse_gamma_parameters(0.5 * np.pi, 2 * np.pi)
    )
    log_Q = pm.Uniform("log_Q", lower=np.log(2), upper=np.log(10))

    # Set up the kernel an GP
    kernel = terms.SHOTerm(S_tot=S1, w0=w1, Q=1.0 / np.sqrt(2))
    kernel += terms.SHOTerm(S_tot=S2, w0=w2, log_Q=log_Q)
    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_Q as a parameter, but Q=tt.exp(log_Q) 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 exoplanet.gp.GP constructor should be the variance to add along the diagonal, not the standard deviation as in the original celerite implementation.

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: [log_Q, w2, w1, S2, S1, mean]
18it [00:00, 32.45it/s, logp=1.008735e+01]
message: Optimization terminated successfully.
logp: -4.1063696496869095 -> 10.087352559326797

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)
art.set_edgecolor("none")
plt.plot(true_t, mu, color="C1", label="prediction")

plt.legend(fontsize=12)
plt.xlabel("t")
plt.ylabel("y")
plt.xlim(0, 10)
_ = plt.ylim(-2.5, 2.5)
../../_images/gp_13_0.png

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(
        tune=2000,
        draws=2000,
        start=map_soln,
        cores=2,
        chains=2,
        step=xo.get_dense_nuts_step(target_accept=0.9),
    )
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [log_Q, w2, w1, S2, S1, mean]
Sampling 2 chains, 0 divergences: 100%|██████████| 8000/8000 [00:25<00:00, 316.14draws/s]

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

pm.traceplot(trace)
pm.summary(trace)
mean sd hpd_3% hpd_97% mcse_mean mcse_sd ess_mean ess_sd ess_bulk ess_tail r_hat
mean 0.019 0.444 -0.778 0.855 0.007 0.007 4411.0 1981.0 4477.0 2975.0 1.0
S1 0.986 0.626 0.241 1.974 0.011 0.008 3423.0 3038.0 4644.0 2904.0 1.0
S2 0.271 0.118 0.098 0.495 0.002 0.001 4380.0 3626.0 4875.0 3041.0 1.0
w1 0.971 0.239 0.543 1.403 0.004 0.003 4426.0 4039.0 4536.0 3002.0 1.0
w2 3.198 0.207 2.805 3.592 0.003 0.002 4359.0 4359.0 4613.0 2545.0 1.0
log_Q 1.807 0.351 1.182 2.302 0.005 0.004 4667.0 4667.0 4385.0 2399.0 1.0
../../_images/gp_17_1.png

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, varnames=["S1", "S2", "w1", "w2", "log_Q"]
)
_ = corner.corner(samples)
../../_images/gp_19_0.png

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)
    art.set_edgecolor("none")
    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.xlabel("t")
plt.ylabel("y")
plt.xlim(0, 10)
_ = plt.ylim(-2.5, 2.5)
../../_images/gp_21_0.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: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}
}
...