def covariance_to_internal_jacobian(external_values, constr):
r"""Jacobian of ``covariance_to_internal``.
For reference see docstring of ``jacobian_covariance_from_internal``. In
comparison to that function, however, here we want to differentiate the
reverse graph
external --> cov --> cholesky --> internal
Again use the vectors :math:`c` and :math:`x` to denote the external and
internal values, respectively. To solve for the jacobian we make use of the
identity
.. math::
\frac{\mathrm{d}x}{\mathrm{d}c} = (\frac{\mathrm{d}c}{\mathrm{d}x})^{-1}
Args:
external_values (np.ndarray): Row-wise half-vectorized covariance matrix
Returns:
deriv: The Jacobian matrix.
"""
cov = cov_params_to_matrix(external_values)
chol = robust_cholesky(cov)
internal = chol[np.tril_indices(len(chol))]
deriv = covariance_from_internal_jacobian(internal, constr=None)
deriv = np.linalg.pinv(deriv)
return deriv
def covariance_to_internal(external_values, constr):
"""Do a cholesky reparametrization."""
cov = cov_params_to_matrix(external_values)
chol = robust_cholesky(cov)
return chol[np.tril_indices(len(cov))]
def test_robust_cholesky_with_extreme_cases():
for cov in [np.ones((5, 5)), np.zeros((5, 5))]:
chol = robust_cholesky(cov)
aaae(chol.dot(chol.T), cov)
def sdcorr_to_internal(external_values, constr):
"""Convert sdcorr to cov and do a cholesky reparametrization."""
cov = sdcorr_params_to_matrix(external_values)
chol = robust_cholesky(cov)
return chol[np.tril_indices(len(cov))]
def test_robust_cholesky_with_zero_variance(dim, seed):
cov = random_cov(dim, seed)
chol = robust_cholesky(cov)
aaae(chol.dot(chol.T), cov)
assert (chol[np.triu_indices(len(cov), k=1)] == 0).all()