deeptime.markov.tools.analysis.expectation

deeptime.markov.tools.analysis.expectation(T, a, mu=None)

Equilibrium expectation value of a given observable.

Parameters:

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

• a ((M,) ndarray) – Observable vector

• mu ((M,) ndarray (optional)) – The stationary distribution of T. If given, the stationary distribution will not be recalculated (saving lots of time)

Returns:

val – Equilibrium expectation value fo the given observable

Return type:

float

Notes

The equilibrium expectation value of an observable a is defined as follows

\[\mathbb{E}_{\mu}[a] = \sum_i \mu_i a_i\]

\(\mu=(\mu_i)\) is the stationary vector of the transition matrix \(T\).

Examples

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

>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> a = np.array([1.0, 0.0, 1.0])
>>> m_a = expectation(T, a)
>>> m_a 
0.909090909...