stationary

Builtin index · Statistics & ML

result = chain.stationary()

Input: no input · Output: result

What it is

Computes the long-run state distribution of a markov_chain chain by power iteration; converges for any irreducible, aperiodic chain.

Overloads

One call shape — this function is not overloaded.

Examples

From the documentation, and run by the build -- the output and the figure below are what the engine actually produced. Simplest first.

P = [0.9, 0.5, 0.1, 0.5] as matrix(2, 2)   # column-major: rows [0.9 0.1], [0.5 0.5]

chain = markov_chain(P, 0)
print("stationary: {round(chain.stationary(), 4)}")
print("3-step:     {round(chain.transition ^ 3, 4)}")

trans = [0.7, 0.4, 0.3, 0.6] as matrix(2, 2)
emit  = [0.5, 0.1, 0.4, 0.3, 0.1, 0.6] as matrix(2, 3)
h = hmm(trans, emit, [0.6, 0.4])
print("P(normal, cold, dizzy) = {round(h.forward([0, 1, 2]), 5)}")
print("most likely path       = {h.viterbi([0, 1, 2])}")

stationary: [0.8333, 0.1667]
3-step:     [0.844, 0.156; 0.78, 0.22]
P(normal, cold, dizzy) = 0.03628
most likely path       = [0, 0, 1]

from statistics-ml

How it works

stationary(chain) — callable as chain.stationary(). The Markov chain's long-run state distribution, found by power iteration (v <- normalize(v * P) from a uniform start) rather than an eigenvector extraction: eig only computes eigenvectors for a SYMMETRIC input (see numeric::linalg::eig's own doc comment), and a general transition matrix has no reason to be symmetric. See markov_stationary's own doc comment.

From the note above this builtin in the interpreter, where the reasoning was written down as it was built.