giddy.ergodic.var_fmpt¶

giddy.ergodic.var_fmpt(P)[source]¶

Variances of first mean passage times for an ergodic transition probability matrix.

Parameters

Parray: (k, k), an ergodic Markov transition probability matrix.

Returns

: array: (k, k), elements are the variances for the number of intervals required for a chain starting in state i to first enter state j.

Notes

Uses formulation (and examples on p. 83) in [KS67].

Examples

>>> import numpy as np
>>> from giddy.ergodic import var_fmpt
>>> p=np.array([[.5, .25, .25],[.5,0,.5],[.25,.25,.5]])
>>> vfm=var_fmpt(p)
>>> vfm
array([[ 5.58333333, 12.        ,  6.88888889],
       [ 6.22222222, 12.        ,  6.22222222],
       [ 6.88888889, 12.        ,  5.58333333]])