function is_reversible

deeptime.markov.tools.analysis.is_reversible(T, mu=None, tol=1e-12)

Check reversibility of the given transition matrix.

Parameters:

• T ((M, M) ndarray or scipy.sparse matrix) – Transition matrix

• mu ((M,) ndarray (optional)) – Test reversibility with respect to this vector

• tol (float (optional)) – Floating point tolerance to check with

Returns:

is_reversible – True, if T is reversible, False otherwise

Return type:

bool

Notes

A transition matrix $T=(t_{ij})$ is reversible with respect to a probability vector $\mu=(\mu_i)$ if the follwing holds,

$$\mu_i \, t_{ij}= \mu_j \, t_{ji}.$$

In this case $\mu$ is the stationary vector for $T$, so that $\mu^T T = \mu^T$.

If the stationary vector is unknown it is computed from $T$ before reversibility is checked.

A reversible transition matrix has purely real eigenvalues. The left eigenvectors $(l_i)$ can be computed from right eigenvectors $(r_i)$ via $l_i=\mu_i r_i$.

Examples

>>> import numpy as np
>>> from deeptime.markov.tools.analysis import is_reversible

>>> P = np.array([[0.8, 0.1, 0.1], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> is_reversible(P)
False

>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> is_reversible(T)
True