# Theano interface¶

This celerite2.theano submodule provides access to the celerite2 models within the Theano framework. Of special interest, this adds support for probabilistic model building using PyMC3.

The TBD tutorial demonstrates the use of this interface, while this page provides the details for the theano.GaussianProcess class which provides all this functionality. This page does not include documentation for the term models defined in Theano, but you can refer to the Model building section of the Python interface documentation. All of those models are implemented in Theano and you can access them using something like the following:

import theano.tensor as tt
import celerite2.theano
from celerite2.theano import terms

term = terms.SHOTerm(S0=tt.dscalar(), w0=tt.dscalar(), Q=tt.dscalar())
gp = celerite2.theano.GaussianProcess(term)


The celerite2.theano.GaussianProcess class is detailed below:

class celerite2.theano.GaussianProcess(kernel, t=None, *, mean=0.0, **kwargs)
apply_inverse(y)

Apply the inverse of the covariance matrix to a vector or matrix

Solve K.x = y for x where K is the covariance matrix of the GP.

Note

The mean function is not applied in this method.

Parameters
• y (shape[N] or shape[N, M]) – The vector or matrix y described above.

• inplace (bool, optional) – If True, y will be overwritten with the result x.

Raises
compute(t, *, yerr=None, diag=None, check_sorted=True, quiet=False)

Compute the Cholesky factorization of the GP covariance matrix

Parameters
• t (shape[N]) – The independent coordinates of the observations. This must be sorted in increasing order.

• yerr (shape[N], optional) – If provided, the diagonal standard deviation of the observation model.

• diag (shape[N], optional) – If provided, the diagonal variance of the observation model.

• check_sorted (bool, optional) – If True, a check is performed to make sure that t is correctly sorted. A ValueError will be thrown when this check fails.

• quiet (bool, optional) – If True, when the matrix cannot be factorized (because of numerics or otherwise) the solver’s LinAlgError will be silenced and the determiniant will be set to zero. Otherwise, the exception will be propagated.

Raises
• ValueError – When the inputs are not valid (shape, number, etc.).

• LinAlgError – When the matrix is not numerically positive definite.

conditional(name, y, t=None, include_mean=True, kernel=None, **kwargs)

Add a variable representing the conditional density to a PyMC3 model

Note

The performance of this method will generally be poor since the sampler will numerically sample this parameter. Depending on your use case, you might be better served by tracking the results of GaussianProcess.predict() using Deterministic variables and computing the predictions as a postprocessing step.

Parameters
• name (str) – The name of the random variable

• y (shape[N]) – The observations at coordinates x from GausianProcess.compute().

• t (shape[M], optional) – The independent coordinates where the prediction should be made. If this is omitted the coordinates will be assumed to be x from GaussianProcess.compute() and an efficient method will be used to compute the mean prediction.

• include_mean (bool, optional) – Include the mean function in the prediction.

• kernel (optional) – If provided, compute the conditional distribution using a different kernel. This is generally used to separate the contributions from different model components.

Returns

A pymc3.MvNormal distribution representing the conditional density.

dot_tril(y)

Dot the Cholesky factor of the GP system into a vector or matrix

Compute x = L.y where K = L.L^T and K is the covariance matrix of the GP.

Note

The mean function is not applied in this method.

Parameters
• y (shape[N] or shape[N, M]) – The vector or matrix y described above.

• inplace (bool, optional) – If True, y will be overwritten with the result x.

Raises
log_likelihood(y)

Compute the marginalized likelihood of the GP model

The factorized matrix from the previous call to GaussianProcess.compute() is used so that method must be called first.

Parameters
• y (shape[N]) – The observations at coordinates t as defined by GaussianProcess.compute().

• inplace (bool, optional) – If True, y will be overwritten in the process of the calculation. This will reduce the memory footprint, but should be used with care since this will overwrite the data.

Raises
marginal(name, **kwargs)

Add the marginal likelihood to a PyMC3 model

Parameters
• name (str) – The name of the random variable.

• observed (optional) – The observed data

Returns

A celerite2.theano.CeleriteNormal distribution representing the marginal likelihood.

predict(y, t=None, *, return_cov=False, return_var=False, include_mean=True, kernel=None, _fast_mean=True)

Compute the conditional distribution

The factorized matrix from the previous call to GaussianProcess.compute() is used so that method must be called first.

Parameters
• y (shape[N]) – The observations at coordinates t as defined by GaussianProcess.compute().

• t (shape[M], optional) – The independent coordinates where the prediction should be evaluated. If not provided, this will be evaluated at the observations t from GaussianProcess.compute().

• return_var (bool, optional) – Return the variance of the conditional distribution.

• return_cov (bool, optional) – Return the full covariance matrix of the conditional distribution.

• include_mean (bool, optional) – Include the mean function in the prediction.

• kernel (optional) – If provided, compute the conditional distribution using a different kernel. This is generally used to separate the contributions from different model components. Note that the computational cost and scaling will be worse when using this parameter.

Raises

## PyMC3 support¶

This implementation comes with a custom PyMC3 Distribution that represents a multivariate normal with a celerite covariance matrix. This is used by the celerite2.theano.GaussianProcess.marginal() method documented above which adds a marginal likelihood node to a PyMC3 model.

class celerite2.theano.CeleriteNormal(*args: Any, **kwargs: Any)

A multivariate normal distribution with a celerite covariance matrix

Parameters

gp (celerite2.theano.GaussianProcess) – The Gaussian Process computation engine.