Enhancement: Replace wrongly specified transition matrices. about willowtree HOT 2 CLOSED

Commit 02d210a solves the issue.

Now, if the last transition matrix in the Markov chain is correctly specified (i.e. it has all the properties of a Markov matrix), function lp automatically replaces all bad matrices with their interpolated versions.

In the case above, since matrix P[-1] (the last one) is well-behaved, P[14] and P[24] are replaced by matrices interpolated from, respectively, P[13] and P[15], and P[23] and P[25].

from willowtree.

Update: Following commit ab317d5 the willow tree is now extremely precise. All transition matrices (including the interpolated ones) have probabilities in rows summing to 1, and the Markov chain is stationary within a very narrow tolerance level (~1e-13).

For n = 20, gamma = 0, and k = 50,

• Probabilities in rows summing to 1:

[P[i].sum(axis=1) for i in range(len(P))]

[array([ 1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,
         1.,  1.,  1.,  1.,  1.,  1.,  1.]),
...

 array([ 1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,
         1.,  1.,  1.,  1.,  1.,  1.,  1.])]

• Stationary distribution:

q @ np.linalg.multi_dot([P[i] for i in range(len(P))]) - q

array([ -2.37962428e-13,  -7.45986606e-13,  -6.41493803e-13,
        -5.86211635e-13,  -5.96439564e-13,  -5.60218538e-13,
        -5.26273469e-13,  -4.96949704e-13,  -4.93674546e-13,
        -4.50209314e-13,  -4.37018477e-13,  -4.27768931e-13,
        -3.93025890e-13,  -4.16118529e-13,  -2.53387589e-13,
        -2.46046239e-13,  -1.70363723e-13,  -1.71175574e-13,
         5.07094366e-14,  -3.53536644e-14])

Since the correctly generated transition matrices are fully well-behaved, the interpolated ones (black patches in the figure below) are as well:

from willowtree.