Compute the stationary distribution of a homogeneous Markov chain

A homogeneous, finite state Markov chain that is irreducible and aperiodic converges to a unique stationary distribution, here called \(\delta\). As it is stationary, this distribution satisfies $$\delta \Gamma = \delta,$$ subject to \(\sum_{j=1}^N \delta_j = 1\), where \(\Gamma\) is the transition probability matrix. This function solves the linear system of equations above.

Usage

stationary(Gamma)

Arguments

Gamma

transition probability matrix of dimension c(N,N) or array of such matrices of dimension c(N,N,nTracks) if the stationary distribution should be computed for several matrices at once

Value

either a single stationary distribution of the Markov chain (vector of length N) or a matrix of stationary distributions of dimension c(nTracks,N) with one stationary distribution in each row

See also

tpm to create a transition probabilty matrix using the multinomial logistic link (softmax)

Other stationary distribution functions: stationary_cont(), stationary_p()

Examples

# single matrix
Gamma = tpm(c(rep(-2,3), rep(-3,3)))
delta = stationary(Gamma)
# multiple matrices
Gamma = array(Gamma, dim = c(3,3,10))
Delta = stationary(Gamma)