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
forx
whereK
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 resultx
.
 Raises
RuntimeError – If
GaussianProcess.compute()
is not called first.ValueError – When the inputs are not valid (shape, number, etc.).

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 thatt
is correctly sorted. AValueError
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’sLinAlgError
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()
usingDeterministic
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
fromGausianProcess.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
fromGaussianProcess.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
whereK = L.L^T
andK
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 resultx
.
 Raises
RuntimeError – If
GaussianProcess.compute()
is not called first.ValueError – When the inputs are not valid (shape, number, etc.).

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 byGaussianProcess.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
RuntimeError – If
GaussianProcess.compute()
is not called first.ValueError – When the inputs are not valid (shape, number, etc.).

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 byGaussianProcess.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
fromGaussianProcess.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
RuntimeError – If
GaussianProcess.compute()
is not called first.ValueError – When the inputs are not valid (shape, number, etc.).

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.