msmtools.analysis.expected_counts

msmtools.analysis.expected_counts(T, p0, N)

Compute expected transition counts for Markov chain with n steps.

Parameters:

• T ((M, M) ndarray or sparse matrix) – Transition matrix
• p0 ((M,) ndarray) – Initial (probability) vector
• N (int) – Number of steps to take

Returns:

EC – Expected value for transition counts after N steps

Return type:

(M, M) ndarray or sparse matrix

Notes

Expected counts can be computed via the following expression

\[\mathbb{E}[C^{(N)}]=\sum_{k=0}^{N-1} \text{diag}(p^{T} T^{k}) T\]

Examples

>>> import numpy as np
>>> from msmtools.analysis import expected_counts

>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> p0 = np.array([1.0, 0.0, 0.0])
>>> N = 100
>>> EC = expected_counts(T, p0, N)

>>> EC
array([[ 45.44616147,   5.0495735 ,   0.        ],
       [  4.50413223,   0.        ,   4.50413223],
       [  0.        ,   4.04960006,  36.44640052]])