msmbuilder.msm_analysis.get_reversible_eigenvectors¶

msmbuilder.msm_analysis.get_reversible_eigenvectors(t_matrix, k, populations=None, right=False, dense_cutoff=50, normalized=False, **kwargs)[source]¶

Find the k largest left eigenvalues and eigenvectors of a reversible row-stochastic matrix, sorted by eigenvalue magnitude

Parameters:	t_matrix : sparse or dense matrix The row-stochastic transition probability matrix k : int The number of eigenpairs to calculate. The k eigenpairs with the highest eigenvalues will be returned. populations : np.ndarray, optional The equilibrium, stationary distribution of t_matrix. If not supplied, it can be re-computed from t_matrix. But this substantially increases the runtime of the routine, so if the stationary distribution is known, it’s more efficient to supply it. right : bool, optional The default behavior is to compute the left eigenvectors of the transition matrix. This option may be invoked to instead compute the right eigenvectors. dense_cutoff : int, optional use dense eigensolver if dimensionality is below this normalized : bool, optional normalize the vectors such that \[\phi_i^T \psi_j = \delta_{ij}\] where \(\phi_i\) is the \(i^{th}\) left eigenvector, and \(\psi_j\) is the \(j^{th}\) right eigenvector
Returns:	eigenvalues : ndarray 1D array of eigenvalues eigenvectors : ndarray 2D array of eigenvectors
Other Params:	**kwargs : Additional keyword arguments are passed directly to scipy.sparse.linalg.eigsh. Refer to the scipy documentation for further details.

See also

get_eigenvectors

computes the eigenpairs of a general row-stochastic matrix, without requiring that the matrix be reversible.

scipy.sparse.linalg.eigsh

Notes

• A reversible transition matrix is one that satisifies the detailed balance condition

  \[\pi_i T_{i,j} = \pi_j T_{j,i}\]

• A reversible transition matrix satisifies a number of special conditions. In particular, it is similar to a symmetric matrix \(S_{i, j} = \sqrt{\frac{pi_i}{\pi_j}} T_{i, j} = S_{j, i}\). This property enables a much more robust solution to the eigenvector problem, because of the superior numerical stability of hermetian eigensolvers.