OperatorPosterior

class OperatorPosterior(means, precisions, *, alreadyinverted=False)

Posterior distribution for operator matrices.

Operator inference models are uniquely determined by operator matrices \(\Ohat\in\RR^{r\times d}\) that concatenate the entries of all operators in the model. For example, the time-continuous model

\[\ddt\qhat(t) = \chat + \Ahat\qhat(t) + \Hhat[\qhat(t)\otimes\qhat(t)]\]

is uniquely determined by the operator matrix

\[\Ohat = [~\chat~~\Ahat~~\Hhat~] \in \RR^{r \times d}.\]

Typical deterministic operator inference learns a single operator matrix \(\Ohat\) from state measurements, while probabilistic or Bayesian operator inference constructs a distribution of operator matrices, \(p(\Ohat)\). This class implements an operator matrix distribution where the rows of \(\Ohat\) are multivariate Normal (Gaussian) random variables, i.e.,

\[\begin{split}p(\ohat_{i}) = \mathcal{N}(\ohat_i\mid\bfmu_i,\bfSigma_i), \\ \bfmu_i \in \RR^{d}, \quad \bfSigma_i \in \RR^{d\times d}, \quad i = 0, \ldots, r-1,\end{split}\]

where \(\ohat_i \in \RR^{d}\) is the \(i\)-th row of \(\Ohat\).

The BayesianROM class has a posterior attribute that is an OperatorPosterior object.

Parameters:

meanslist of r (d,) ndarrays

Mean values for each row of the operator matrix.

precisionslist of r (d, d) ndarrays

INVERSE covariance matrices for each row of the operator matrix.

alreadyinvertedbool

If True, assume precisions is the collection of covariance matrices, not their inverses.

Properties:

covs

Covariance matrices \(\bfSigma_0,\ldots,\bfSigma_{r-1}\in\RR^{d\times d}\) for the rows of the operator matrix.

means

Mean vectors \(\bfmu_0,\ldots,\bfmu_{r-1}\in\RR^{d}\) for the rows of the operator matrix.

nrows

Number of rows \(r\) in the data matrix. This is also the state dimension of the corresponding model.

randomvariables

Multivariate normal random variables for the rows of the operator matrix.

Methods:

load	Load a previously saved posterior operator distribution.
rvs	Draw a random operator matrix from the posterior operator distribution.
save	Save the posterior operator distribution.