Compute the transition probability matrix of a thinned periodically inhomogeneous Markov chain.

If the transition probability matrix of an inhomogeneous Markov chain varies only periodically (with period length \(L\)), it converges to a so-called periodically stationary distribution. This happens, because the thinned Markov chain, which has a full cycle as each time step, has homogeneous transition probability matrix $$\Gamma_t = \Gamma^{(t)} \Gamma^{(t+1)} \dots \Gamma^{(t+L-1)}$$ for all \(t = 1, \dots, L.\) This function calculates the matrix above efficiently as a preliminery step to calculating the periodically stationary distribution.

Usage

tpm_thinned(Gamma, t)

Arguments

Gamma

array of transition probability matrices of dimension c(N,N,L).

t

integer index of the time point in the cycle, for which to calculate the thinned transition probility matrix

Value

thinned transition probabilty matrix of dimension c(N,N)

Examples

# setting parameters for trigonometric link
beta = matrix(c(-1, -2, 2, -1, 2, -4), nrow = 2, byrow = TRUE)
# calculating periodically varying t.p.m. array (of length 24 here)
Gamma = tpm_p(beta = beta)
# calculating t.p.m. of thinned Markov chain
tpm_thinned(Gamma, 4)
#>           [,1]      [,2]
#> [1,] 0.8926642 0.1073358
#> [2,] 0.8926642 0.1073358